Introduction
Matter-wave interferometers (MWIs) are sensitive to dark matter (DM) through observables that are intrinsically quantum. Indeed, a DM background need neither produce a resolvable recoil displacement, nor deposit energy, in order to leave a detectable signature. Rather, DM can modify the reduced density matrix of the probe, by introducing a relative phase between spatially separated wavepackets and suppressing quantum coherence.
Thanks to their exceptional metrological precision, MWIs are poised to become versatile probes of DM across a wide range of masses. For instance, atom gradiometers, where pairs of atom interferometers (AIs) are driven by common laser pulses, have been proposed to probe linearly coupled and quadratically coupled ultralight scalar DM [1 , 2 , 3 , 4 ] , as well as spin-2 DM [5 ] , in the 10 − 18 10^{-18} 1 0 − 18 10 − 12 10^{-12} 1 0 − 12 [6 ] , while atom interferometers in which atoms are prepared in a superposition of spin states could probe axion-like particle DM via ALP-nucleon interactions [7 ] . Co-located interferometers employing different isotopes also offer sensitivity to B − L B-L B − L [8 , 9 , 10 ] . Composite-particle interferometers (CPIs), which are MWIs interfering superpositions of mesoscopic objects, could detect collisional decoherence from elastic, spin-independent scattering with sub-GeV DM [11 , 12 , 13 , 14 , 15 ] . Futuristic space-based gradiometers proposed for mid-frequency gravitational wave detection may also probe a DM subcomponent populated by small sub-asteroid mass DM clumps and constrain small ultralight DM overdensities via purely gravitational interactions [16 ] . Trapped-ion interferometers originally conceived for rotation sensing could detect small magnetic fields sourced by axion-like particles or kinetically mixed dark photons in otherwise unexplored regions of ultralight dark matter parameter space [17 ] .
As a result of these disparate phenomenological regimes, the theoretical description of MWIs interacting with a DM background remains fragmented. A unified treatment should interpolate between particle-like and wave-like DM, while retaining dependence on the quantum state of the DM, including its occupation number, spin statistics, and coherence time. It should also distinguish between two regimes, depending on whether the DM coherence time is shorter or longer than the interferometric time. In the former case, a Markovian (or Lindblad-type) description is adequate, while in the latter a non-Markovian treatment would be appropriate, since interactions between the MWI and the DM environment are non-local in time [18 ] . Furthermore, the commonly used heavy-probe approximation, which neglects DM-induced displacement recoils, must be revisited once momentum transfer produces a resolvable recoil phase or becomes comparable to the wavepacket width. Finally, such a framework should account for the many-body structure of a realistic probe. As shown in the context of elastic spin-independent DM-nucleon scattering [14 , 19 ] , different many-atom states can give rise to different form factors that parametrically change the expected reach of MWIs to sub-GeV DM.
In this work, we construct such a framework. We model the MWI as an open quantum system coupled to the environment, which we take to be the dark sector, and derive its dynamics from first principles using the Schwinger–Keldysh, or in-in, formalism [20 , 21 ] (for a modern review, see Ref. [22 ] ). This formalism preserves causality and unitarity at the level of the full theory and implements real-time evolution along a closed-time path (CTP). The evolution of kets and bras is captured by the forward (“+”) and backward (“−”) branches, respectively, with fields defined on both time branches. Tracing out the environment yields an open EFT for MWIs, as summarized in Figure 1 . The effective action for a MWI then depends on the influence action S I F S_{\mathrm{IF}} S IF
Figure 1. Summary of the logical structure of the open EFT construction presented in this work. We assume that the system and environment are described by fields ψ \psi ψ { Ξ } \{\Xi\} { Ξ } S s S_s S s S e S_e S e S i n t S_{\mathrm{int}} S int ψ + \psi_+ ψ + { Ξ + } \{\Xi_+\} { Ξ + } ψ − \psi_- ψ − { Ξ − } \{\Xi_-\} { Ξ − } + + + − - − S I F S_{\mathrm{IF}} S IF X ± X_\pm X ± x ˉ L , R \bar{x}_{L,R} x ˉ L , R
The Schwinger–Keldysh approach has been used in closely related interferometric settings, including gravitationally induced decoherence in interferometers [23 , 24 , 25 ] , observable effects due to soft graviton [26 , 27 ] and soft photon [28 ] bremsstrahlung, decoherence and phase shifts induced by chameleon fields in atom interferometers [29 , 30 , 31 ] , and more generally to study the decay of the interference term between two Gaussian packets caused by a weakly coupled nonlinear bath of oscillators [32 , 33 ] . For a review on the use of this formalism in the context of open quantum systems more generally, see Ref. [34 ] . More broadly, open EFT methods have recently found applications in cosmology and astrophysics [35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 ] , gravity [44 , 45 ] and gauge theories [46 , 47 , 48 ] . EFTs have been widely employed to study the signatures of DM in other direct detection experiments, e.g., Refs. [49 , 50 , 51 , 52 , 53 , 54 ] .
Our framework cleanly separates the phenomenology of contact and long-range DM-MWI interactions. For contact DM-nucleon interactions, the environment kernel is the DM matter response, which depends on the finite density of the DM background. Because the interferometer observables depend on these kernels, our formalism immediately identifies MWI observables that inherit Bose enhancement or Pauli blocking, and those that remain at most linear in the DM occupation number. For long-range interactions, our construction also shows how the finite DM background density dresses the mediator propagator, which in turn gives rise to screening or anti-screening effects. For attractive interactions, this correction can give rise to a tachyonic mediator mass, which signals the onset of DM condensation and the breakdown of the plane wave approximation underpinning our calculations. These in-medium effects are closely analogous to those discussed in the context of BSM physics in dense astrophysical media [55 , 56 , 57 , 58 , 59 ] and nuclear physics at finite density [60 ] .
A concise account of this construction in the case of contact interactions appeared in Ref. [61 ] . Here, we give the detailed EFT derivation of those results and extend the analysis in several directions. Firstly, we present a unified many-body treatment spanning single atoms, dilute clouds, and bound mesoscopic objects, making clear where coherent enhancements enter. Secondly, we discuss corrections beyond the heavy-probe limit. Thirdly, we consider the phenomenology of contact interactions at and below the DM wave-particle boundary, and show that ambitious CPIs may be able to probe unexplored regions of Higgs-portal DM parameter space. Fourthly, we discuss the case of long-range interactions, where we show how the DM environment dresses the mediator propagator appearing in the observables. This is especially relevant in light of the anticipated sensitivity of AIs
to sub-keV DM through elastic DM-nucleon scattering via a light hadrophilic mediator [13 , 15 ] .
The paper is organized as follows. In section II we derive the effective action starting from a microscopic theory. We then connect the influence action to interferometer observables by projecting the system fields to worldlines. For pedagogical reasons, we first focus in section III on the single-atom case (already presented in Ref. [61 ] ) and in section IV we generalize our results to multi-atom platforms, such as AIs and CPIs. In § 0.6 we show how the tools developed in previous sections can be used to compute the signatures of sub-GeV elastic spin-independent DM-nucleon scattering, distinguishing between contact and long-range interactions. We summarize our results in section VI. Appendices A–D provide additional details.
Conventions and notation
We use natural units and the Minkowski metric η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) \eta_{\mu\nu} = \operatorname{diag}(+1,-1,-1,-1) η μν = diag ( + 1 , − 1 , − 1 , − 1 ) x μ = ( t , x ) x^\mu= (t,\mathbf{x}) x μ = ( t , x ) x μ p μ = x ⋅ p x_\mu p^\mu= \mathbf{x} \cdot\mathbf{p} x μ p μ = x ⋅ p ∫ d n p ( 2 π ) n \int\frac{\mathrm{d}^n p}{(2\pi)^n} ∫ ( 2 π ) n d n p δ ˉ ( n ) ( ⋅ ) = ( 2 π ) n δ ( n ) ( ⋅ ) \bar{\delta}^{(n)}(\cdot) = (2\pi)^n\delta^{(n)}(\cdot) δ ˉ ( n ) ( ⋅ ) = ( 2 π ) n δ ( n ) ( ⋅ ) ∫ t i t f d 4 x ≡ ∫ t i t f d t ∫ − ∞ ∞ d 3 x \int_{t_i}^{t_f} \mathrm{d}^4x \equiv\int_{t_i}^{t_f} \mathrm{d}t \int_{-\infty}^{\infty} \mathrm{d}^3x ∫ t i t f d 4 x ≡ ∫ t i t f d t ∫ − ∞ ∞ d 3 x h ( x ) h(x) h ( x )
h ~ ( p ) = ∫ d 4 x h ( x ) e i p ⋅ x , h ( x ) = ∫ d 4 p h ~ ( p ) e − i p ⋅ x . (1) \widetilde{h}(p) = \int\mathrm{d}^4x\,h(x)e^{ip\cdot x}, \qquad h(x) = \int\mathrm{d}^4p\,\widetilde{h}(p)e^{-ip\cdot x}. \tag*{(1)} h ( p ) = ∫ d 4 x h ( x ) e i p ⋅ x , h ( x ) = ∫ d 4 p h ( p ) e − i p ⋅ x . (1) We will also use the 3-momentum FT and its inverse defined as
h ~ ( p , t ) = ∫ d 3 x h ( x ) e − i p ⋅ x , h ( x ) = ∫ d 3 p h ~ ( p , t ) e i p ⋅ x , (2) \widetilde{h}(\mathbf{p},t) = \int\mathrm{d}^3x\,h(x)e^{-i\mathbf{p}\cdot\mathbf{x}}, \qquad h(x) = \int\mathrm{d}^3p\,\widetilde{h}(\mathbf{p},t)e^{i\mathbf{p}\cdot\mathbf{x}}, \tag*{(2)} h ( p , t ) = ∫ d 3 x h ( x ) e − i p ⋅ x , h ( x ) = ∫ d 3 p h ( p , t ) e i p ⋅ x , (2) respectively.
Unless otherwise stated, we will assume throughout the Standard Halo Model (SHM) in the lab frame with a finite escape velocity, which is described by the velocity distribution
f χ ( v ) = 1 N 0 e − ∣ v + v ⊕ ∣ 2 / v 0 2 Θ ( v e s c − ∣ v + v ⊕ ∣ ) , f_\chi(\mathbf{v}) = \frac{1}{N_0}e^{-|\mathbf{v}+\mathbf{v}_\oplus|^2/v_0^2}\Theta(v_{\mathrm{esc}}-|\mathbf{v}+\mathbf{v}_\oplus|), f χ ( v ) = N 0 1 e − ∣ v + v ⊕ ∣ 2 / v 0 2 Θ ( v esc − ∣ v + v ⊕ ∣ ) , N 0 = π 3 / 2 v 0 2 [ v 0 erf ( v e s c / v 0 ) − 2 v e s c π exp ( − v e s c 2 / v 0 2 ) ] . (3) N_0 = \pi^{3/2}v_0^2\left[v_0\operatorname{erf}(v_{\mathrm{esc}}/v_0)-\frac{2v_{\mathrm{esc}}}{\sqrt{\pi}}\exp(-v_{\mathrm{esc}}^2/v_0^2)\right]. \tag*{(3)} N 0 = π 3/2 v 0 2 [ v 0 erf ( v esc / v 0 ) − π 2 v esc exp ( − v esc 2 / v 0 2 ) ] . (3) with v 0 ≈ 230 k m / s v_0 \approx230\ \mathrm{km/s} v 0 ≈ 230 km/s v e s c ≈ 600 k m / s v_{\mathrm{esc}} \approx600\ \mathrm{km/s} v esc ≈ 600 km/s ∣ v ⊕ ∣ ≈ 240 k m / s |\mathbf{v}_\oplus| \approx240\ \mathrm{km/s} ∣ v ⊕ ∣ ≈ 240 km/s ρ χ \rho_\chi ρ χ m χ m_\chi m χ
n χ ( p ) ≡ ρ χ m χ f χ ( p ) , (4) n_\chi(\mathbf{p}) \equiv\frac{\rho_\chi}{m_\chi}f_\chi(\mathbf{p}), \tag*{(4)} n χ ( p ) ≡ m χ ρ χ f χ ( p ) , (4) where f χ ( p ) = ( 2 π ) 3 f χ ( v ) / m χ 3 f_\chi(\mathbf{p}) = (2\pi)^3 f_\chi(\mathbf{v})/m_\chi^3 f χ ( p ) = ( 2 π ) 3 f χ ( v ) / m χ 3 ∫ d 3 p f χ ( p ) = 1 \int\mathrm{d}^3p\,f_\chi(\mathbf{p}) = 1 ∫ d 3 p f χ ( p ) = 1
The open EFT from the closed theory
Laboratory probes used to search for dark matter are inherently open systems: they interact continuously, albeit weakly, with an external environment constituted by dark matter particles. Even when the total system, i.e., probe plus environment, evolves unitarily, the dynamics of the probe alone is generically non-unitary once the environmental degrees of freedom are traced out. This loss of unitarity manifests physically as decoherence, dissipation, and stochastic fluctuations, and can be naturally captured by the Schwinger–Keldysh (or closed time-path) formalism [20 , 21 ] , which implements real-time evolution while preserving causality and unitarity at the level of the full system. In this section we introduce the key ingredients of this formalism to the unfamiliar reader, and set the conventions and notation used throughout.
Consider a quantum system (the laboratory probe, labeled by “s”) coupled to an environment (the dark matter background, labeled by “e”). Let ψ \psi ψ { Ξ } \{\Xi\} { Ξ } χ \chi χ ϕ \phi ϕ
H ^ [ ψ , { Ξ } ] = H ^ s [ ψ ] + H ^ e [ { Ξ } ] + H ^ i n t [ ψ , { Ξ } ] , (5) \hat{H}[\psi,\{\Xi\}] = \hat{H}_{s}[\psi] + \hat{H}_{e}[\{\Xi\}] + \hat{H}_{\mathrm{int}}[\psi,\{\Xi\}], \tag*{(5)} H ^ [ ψ , { Ξ }] = H ^ s [ ψ ] + H ^ e [{ Ξ }] + H ^ int [ ψ , { Ξ }] , (5) with H ^ s [ ψ ] \hat{H}_{s}[\psi] H ^ s [ ψ ] H ^ e [ { Ξ } ] \hat{H}_{e}[\{\Xi\}] H ^ e [{ Ξ }] H ^ i n t [ ψ , { Ξ } ] \hat{H}_{\mathrm{int}}[\psi,\{\Xi\}] H ^ int [ ψ , { Ξ }] t f t_f t f t i t_i t i t f t_f t f U ^ ( t f , t i ) = T exp [ − i ∫ t i t f d t H ^ ( t ) ] \hat{U}(t_f,t_i) = \mathrm{T}\exp\left[-i\int_{t_i}^{t_f}\mathrm{d}t\,\hat{H}(t)\right] U ^ ( t f , t i ) = T exp [ − i ∫ t i t f d t H ^ ( t ) ] T r e [ ⋅ ] \mathrm{Tr}_{e}[\cdot] Tr e [ ⋅ ] ρ ^ i = ρ ^ s ( t i ) ⊗ ρ ^ e \hat{\rho}_i = \hat{\rho}_s(t_i) \otimes\hat{\rho}_e ρ ^ i = ρ ^ s ( t i ) ⊗ ρ ^ e t i t_i t i ρ ^ e \hat{\rho}_e ρ ^ e [62 ] , the reduced density matrix of the system at time t f t_f t f
ρ ^ s ( t f ) = T r e [ U ^ ( t f , t i ) ρ ^ i U ^ † ( t f , t i ) ] = T r e [ U ^ ( t f , t i ) ρ ^ i U ^ ( t i , t f ) ] , (6) \hat{\rho}_s(t_f) = \mathrm{Tr}_{e}\left[\hat{U}(t_f,t_i)\,\hat{\rho}_i\,\hat{U}^{\dagger}(t_f,t_i)\right] = \mathrm{Tr}_{e}\left[\hat{U}(t_f,t_i)\,\hat{\rho}_i\,\hat{U}(t_i,t_f)\right], \tag*{(6)} ρ ^ s ( t f ) = Tr e [ U ^ ( t f , t i ) ρ ^ i U ^ † ( t f , t i ) ] = Tr e [ U ^ ( t f , t i ) ρ ^ i U ^ ( t i , t f ) ] , (6) In the Schwinger–Keldysh (SK) formulation, also known as the in-in formalism, the RHS of Eq. (6) admits a path-integral representation along a closed time contour extending from time t i t_i t i t f t_f t f ( ψ , { Ξ } ) → ( ψ + , ψ − ; { Ξ + } , { Ξ − } ) (\psi, \{\Xi\}) \to(\psi_+,\psi_-; \{\Xi_+\}, \{\Xi_-\}) ( ψ , { Ξ }) → ( ψ + , ψ − ; { Ξ + } , { Ξ − })
S [ ψ + , ψ − ; { Ξ + } , { Ξ − } ] = S [ ψ + , { Ξ + } ] − S [ ψ − , { Ξ − } ] , S [ ψ ± , { Ξ ± } ] = S s [ ψ ± ] + S e [ { Ξ ± } ] + S i n t [ ψ ± , { Ξ ± } ] . (7) \begin{aligned}
S[\psi_+,\psi_-; \{\Xi_+\}, \{\Xi_-\}] &= S[\psi_+, \{\Xi_+\}] - S[\psi_-, \{\Xi_-\}], \\
S[\psi_\pm, \{\Xi_\pm\}] &= S_s[\psi_\pm] + S_e[\{\Xi_\pm\}] + S_{\mathrm{int}}[\psi_\pm, \{\Xi_\pm\}]. \tag*{(7)}
\end{aligned} S [ ψ + , ψ − ; { Ξ + } , { Ξ − }] S [ ψ ± , { Ξ ± }] = S [ ψ + , { Ξ + }] − S [ ψ − , { Ξ − }] , = S s [ ψ ± ] + S e [{ Ξ ± }] + S int [ ψ ± , { Ξ ± }] . (7) The doubling of degrees of freedom is a direct consequence of enforcing unitarity and causality in real-time evolution, and provides a powerful tool for distinguishing between genuine non-unitary and unitary evolution. For example, in the absence of system-environment interactions, or if the evolution of the system can be expressed in terms of a unitary operator, the evolution along the forward and backward time branches is identical up to complex conjugation. Non-unitary effects emerge whenever the trace over the environment gives rise to terms that mix system degrees of freedom evaluated along the forward and backward time branches. After the environment is traced out, the system evolves with the open effective action
S [ ψ + , ψ − ; { Ξ + } , { Ξ − } ] → S e f f [ ψ + , ψ − ] = S s [ ψ + ] − S s [ ψ − ] + S I F [ ψ + , ψ − ] . (8) S[\psi_+,\psi_-; \{\Xi_+\}, \{\Xi_-\}] \to S_{\mathrm{eff}}[\psi_+,\psi_-] = S_s[\psi_+] - S_s[\psi_-] + S_{\mathrm{IF}}[\psi_+,\psi_-]. \tag*{(8)} S [ ψ + , ψ − ; { Ξ + } , { Ξ − }] → S eff [ ψ + , ψ − ] = S s [ ψ + ] − S s [ ψ − ] + S IF [ ψ + , ψ − ] . (8) The influence action is thus the field-level object that depends on the initial state of the environment, the field content of the environment and the interactions between system and environment. As we shall explicitly see in the next sections, all dependence on the DM model and on the DM state is packaged into retarded and symmetric correlation functions, which preserve path ordering. Correlation functions of “+” fields are time ordered, while correlation functions of “−” fields are anti-time ordered. In correlators involving both “+” and “−” fields, all “+” fields are placed to the right of all “−” fields. For a pedagogical and modern introduction, see Refs. [22 , 63 ] .
It is especially convenient to introduce the Keldysh basis, which facilitates the rewriting of all degrees of freedom in terms of quantum (“q”) and classical (“cl”) fields.1
O c l = 1 2 ( O + + O − ) , O q = O + − O − . (9) O^{\mathrm{cl}} = \frac{1}{2}(O_+ + O_-), \qquad O^{q} = O_+ - O_-. \tag*{(9)} O cl = 2 1 ( O + + O − ) , O q = O + − O − . (9) In this basis, the effective action satisfies the following non-equilibrium constraints [64 , 65 ] :
S e f f [ ψ c l , ψ q = 0 ] = 0 , S e f f [ ψ c l , ψ q ] = − S e f f ∗ [ ψ c l , − ψ q ] , Im { S e f f [ ψ c l , ψ q ] } ≥ 0. (10) S_{\mathrm{eff}}[\psi^{\mathrm{cl}},\psi^{q}=0] = 0, \qquad S_{\mathrm{eff}}[\psi^{\mathrm{cl}},\psi^{q}] = -S_{\mathrm{eff}}^{*}[\psi^{\mathrm{cl}},-\psi^{q}], \qquad\operatorname{Im}\{S_{\mathrm{eff}}[\psi^{\mathrm{cl}},\psi^{q}]\} \ge0. \tag*{(10)} S eff [ ψ cl , ψ q = 0 ] = 0 , S eff [ ψ cl , ψ q ] = − S eff ∗ [ ψ cl , − ψ q ] , Im { S eff [ ψ cl , ψ q ]} ≥ 0. (10) Crucially, this implies that (i) the effective action is at least linear in ψ q \psi^{q} ψ q ψ q \psi^{q} ψ q [66 ] .
From the microscopic theory to the influence action
Having laid out the logic behind the Schwinger–Keldysh construction, we now show how to obtain the influence action for a laboratory probe, whose degrees of freedom consist of atoms.
For simplicity, we focus on monopole-monopole interactions, i.e., we shall consider interaction Lagrangians of the type
L i n t = g Λ D O s [ ψ ] O e [ Ξ ] (11) \mathcal{L}_{\mathrm{int}}=\frac{g}{\Lambda^{D}}O_s[\psi]O_e[\Xi] \tag*{(11)} L int = Λ D g O s [ ψ ] O e [ Ξ ] (11) where O s O_s O s O e O_e O e g ≪ 1 g \ll1 g ≪ 1 Λ \Lambda Λ D D D O e = O χ O_e=O_\chi O e = O χ O e = ϕ O_e=\phi O e = ϕ
At energies relevant for DM direct detection with MWIs, atoms, although built from nucleons and electrons, can be treated as a single heavy non-relativistic (NR) complex field ψ \psi ψ
L s = ψ † ( i ∂ t + ∇ 2 2 m a t ) ψ , (12) \mathcal{L}_s=\psi^\dagger\left(i\partial_t+\frac{\nabla^2}{2m_{\mathrm{at}}}\right)\psi, \tag*{(12)} L s = ψ † ( i ∂ t + 2 m at ∇ 2 ) ψ , (12) where m a t m_{\mathrm{at}} m at U ( 1 ) U(1) U ( 1 ) ψ \psi ψ
O s ( x ) = ∫ d 3 r ϱ ( r ) ψ † ( t , x − r ) ψ ( t , x − r ) , O_s(x)=\int\mathrm{d}^3r\,\varrho(\mathbf{r})\psi^\dagger(t,\mathbf{x}-\mathbf{r})\psi(t,\mathbf{x}-\mathbf{r}), O s ( x ) = ∫ d 3 r ϱ ( r ) ψ † ( t , x − r ) ψ ( t , x − r ) , F A ( q ) = ∫ d 3 r e − i q ⋅ r ϱ ( r ) , (13) F_A(\mathbf{q})=\int\mathrm{d}^3r\,e^{-i\mathbf{q}\cdot\mathbf{r}}\varrho(\mathbf{r}), \tag*{(13)} F A ( q ) = ∫ d 3 r e − i q ⋅ r ϱ ( r ) , (13) where ϱ ( r ) \varrho(\mathbf{r}) ϱ ( r ) F A ( q ) F_A(\mathbf{q}) F A ( q ) [67 ] .2
Using Eqs. (9) and (11) , and introducing the notation ⟨ ⟨ ⋅ ⟩ ⟩ ≡ Tr [ ⋅ ρ ^ e ] \langle\!\langle\cdot\rangle\!\rangle\equiv\operatorname{Tr}[\cdot\hat{\rho}_e] ⟨ ⟨ ⋅ ⟩ ⟩ ≡ Tr [ ⋅ ρ ^ e ] O ( g 2 ) \mathcal{O}(g^2) O ( g 2 ) [68 ] :
exp { i S I F } = ⟨ ⟨ exp { i g Λ D ∫ t i t f d 4 x [ O s c l ( x ) O e q ( x ) + O s q ( x ) O e c l ( x ) ] } ⟩ ⟩ = exp { i g Λ D ∫ t i t f d 4 x O s q ( x ) ⟨ ⟨ O e c l ( x ) ⟩ ⟩ − i 2 ( g Λ D ) 2 ∫ t i t f d 4 x ∫ t i t f d 4 x ′ O ⃗ s T ( x ) K ( x , x ′ ) O ⃗ s ( x ′ ) + O ( g 3 ) } . (14) \begin{aligned}
\exp\{iS_{\mathrm{IF}}\}=\left\langle\!\left\langle\exp\left\{i\frac{g}{\Lambda^D}\int_{t_i}^{t_f}\mathrm{d}^4x\left[O_s^{\mathrm{cl}}(x)O_e^{\mathrm{q}}(x)+O_s^{\mathrm{q}}(x)O_e^{\mathrm{cl}}(x)\right]\right\}\right\rangle\!\right\rangle\\
&=\exp\left\{i\frac{g}{\Lambda^D}\int_{t_i}^{t_f}\mathrm{d}^4x\,O_s^{\mathrm{q}}(x)\langle\!\langle O_e^{\mathrm{cl}}(x)\rangle\!\rangle-\frac{i}{2}\left(\frac{g}{\Lambda^D}\right)^2\int_{t_i}^{t_f}\mathrm{d}^4x\int_{t_i}^{t_f}\mathrm{d}^4x'\,\vec{O}_s^{\,T}(x)K(x,x')\vec{O}_s(x')+\mathcal{O}(g^3)\right\}. \tag*{(14)}
\end{aligned} exp { i S IF } = ⟨ ⟨ exp { i Λ D g ∫ t i t f d 4 x [ O s cl ( x ) O e q ( x ) + O s q ( x ) O e cl ( x ) ] } ⟩ ⟩ = exp { i Λ D g ∫ t i t f d 4 x O s q ( x ) ⟨ ⟨ O e cl ( x )⟩ ⟩ − 2 i ( Λ D g ) 2 ∫ t i t f d 4 x ∫ t i t f d 4 x ′ O s T ( x ) K ( x , x ′ ) O s ( x ′ ) + O ( g 3 ) } . (14) where we introduced the system current vector
O ⃗ s T ( x ) = [ O s c l ( x ) O s q ( x ) ] , (15) \vec{O}_s^{\mathrm{T}}(x)=\begin{bmatrix} O_s^{\mathrm{cl}}(x) & O_s^{\mathrm{q}}(x) \end{bmatrix}, \tag*{(15)} O s T ( x ) = [ O s cl ( x ) O s q ( x ) ] , (15) and the environment matrix kernel
K ( x , x ′ ) = [ 0 K a ( x , x ′ ) K r ( x , x ′ ) K k ( x , x ′ ) ] , (16) K(x,x')=\begin{bmatrix} 0 & K_{\mathrm{a}}(x,x') \\ K_{\mathrm{r}}(x,x') & K_{\mathrm{k}}(x,x') \end{bmatrix}, \tag*{(16)} K ( x , x ′ ) = [ 0 K r ( x , x ′ ) K a ( x , x ′ ) K k ( x , x ′ ) ] , (16) whose entries depend explicitly on the connected correlation functions of environment currents:
K r ( x , x ′ ) = − i ⟨ ⟨ O e c l ( x ) O e q ( x ′ ) ⟩ ⟩ c = − i θ ( t − t ′ ) ⟨ ⟨ [ O ^ e ( x ) , O ^ e ( x ′ ) ] ⟩ ⟩ c , (17) K_{\mathrm{r}}(x,x')=-i\langle\langle O_e^{\mathrm{cl}}(x)O_e^{\mathrm{q}}(x')\rangle\rangle_{\mathrm{c}}=-i\theta(t-t')\langle\langle[\hat{O}_e(x),\hat{O}_e(x')]\rangle\rangle_{\mathrm{c}}, \tag*{(17)} K r ( x , x ′ ) = − i ⟨⟨ O e cl ( x ) O e q ( x ′ )⟩ ⟩ c = − i θ ( t − t ′ ) ⟨⟨[ O ^ e ( x ) , O ^ e ( x ′ )]⟩ ⟩ c , (17) K a ( x , x ′ ) = − i ⟨ ⟨ O e q ( x ) O e c l ( x ′ ) ⟩ ⟩ c = + i θ ( t ′ − t ) ⟨ ⟨ [ O ^ e ( x ) , O ^ e ( x ′ ) ] ⟩ ⟩ c , (18) K_{\mathrm{a}}(x,x')=-i\langle\langle O_e^{\mathrm{q}}(x)O_e^{\mathrm{cl}}(x')\rangle\rangle_{\mathrm{c}}=+i\theta(t'-t)\langle\langle[\hat{O}_e(x),\hat{O}_e(x')]\rangle\rangle_{\mathrm{c}}, \tag*{(18)} K a ( x , x ′ ) = − i ⟨⟨ O e q ( x ) O e cl ( x ′ )⟩ ⟩ c = + i θ ( t ′ − t ) ⟨⟨[ O ^ e ( x ) , O ^ e ( x ′ )]⟩ ⟩ c , (18) K k ( x , x ′ ) = − i ⟨ ⟨ O e c l ( x ) O e c l ( x ′ ) ⟩ ⟩ c = − 1 2 ⟨ ⟨ { O ^ e ( x ) , O ^ e ( x ′ ) } ⟩ ⟩ c . (19) K_{\mathrm{k}}(x,x')=-i\langle\langle O_e^{\mathrm{cl}}(x)O_e^{\mathrm{cl}}(x')\rangle\rangle_{\mathrm{c}}=-\frac{1}{2}\langle\langle\{\hat{O}_e(x),\hat{O}_e(x')\}\rangle\rangle_{\mathrm{c}}. \tag*{(19)} K k ( x , x ′ ) = − i ⟨⟨ O e cl ( x ) O e cl ( x ′ )⟩ ⟩ c = − 2 1 ⟨⟨{ O ^ e ( x ) , O ^ e ( x ′ )}⟩ ⟩ c . (19) Note that the correlation function ⟨ ⟨ O e q ( x ) O e q ( x ′ ) ⟩ ⟩ c \langle\langle O_e^{\mathrm{q}}(x)O_e^{\mathrm{q}}(x')\rangle\rangle_{\mathrm{c}} ⟨⟨ O e q ( x ) O e q ( x ′ )⟩ ⟩ c O e ( x ) O_e(x) O e ( x ) K k ( x , x ′ ) K_{\mathrm{k}}(x,x') K k ( x , x ′ ) K a ( x , x ′ ) K_{\mathrm{a}}(x,x') K a ( x , x ′ ) K r ( x , x ′ ) K_{\mathrm{r}}(x,x') K r ( x , x ′ )
K r ( x , x ′ ) = K a ∗ ( x ′ , x ) . (20) K_{\mathrm{r}}(x,x')=K_{\mathrm{a}}^*(x',x). \tag*{(20)} K r ( x , x ′ ) = K a ∗ ( x ′ , x ) . (20) Crucially, the explicit form of the environment kernels depends on the properties of the environment. In the remainder of this work we specialize to a stationary and homogeneous DM background, whose three-momentum distribution is given by a boosted Maxwell–Boltzmann distribution f χ ( p ) f_\chi(\mathbf{p}) f χ ( p ) ∫ d 3 p f χ ( p ) = 1 \int\mathrm{d}^3p\,f_\chi(\mathbf{p})=1 ∫ d 3 p f χ ( p ) = 1 [69 ] . This assumption is valid whenever the DM phase-space distribution varies negligibly over the interferometric time and length scales, which requires that:
DM-SM interactions are sufficiently weak that screening [70 ] and direct deflection/anti-screening [71 ] of DM by the local environment are negligible; for quadratic couplings in particular, the detector environment can suppress or distort the local DM field amplitude, potentially violating both the homogeneity and stationarity assumptions [72 , 73 ] ;
any daily or annual modulations of the local DM density, arising for instance from gravitational focusing by the Earth or Sun, occur on timescales much longer than the interferometric time [74 , 75 ] ; and
the DM distribution is smooth on experimental scales, as supported by N N N [76 , 77 ] .
Under these assumptions, K ( x , x ′ ) → K ( x − x ′ ) K(x,x')\to K(x-x') K ( x , x ′ ) → K ( x − x ′ )
Furthermore, we assume that the initial DM state is Gaussian, as argued in Ref. [78 ] . In the case of long-range DM-probe interactions, we take the initial state of the mediator to be the vacuum, which is Gaussian by definition. As a result, Wick’s theorem applies and the environmental kernels K ( x − x ′ ) K(x-x') K ( x − x ′ ) [79 ] : for real bosonic Ξ \Xi Ξ
G r Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ c l ( x ) Ξ q ( x ′ ) ⟩ ⟩ = − i θ ( t − t ′ ) ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ( x ′ ) ] ⟩ ⟩ G^{\Xi}_{r}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{cl}}(x)\Xi^{\mathrm{q}}(x')\right\rangle\!\right\rangle= -i\theta(t-t')\left\langle\!\left\langle[\hat{\Xi}(x),\hat{\Xi}(x')]\right\rangle\!\right\rangle G r Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ cl ( x ) Ξ q ( x ′ ) ⟩ ⟩ = − i θ ( t − t ′ ) ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ( x ′ )] ⟩ ⟩ G a Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ q ( x ) Ξ c l ( x ′ ) ⟩ ⟩ = + i θ ( t ′ − t ) ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ( x ′ ) ] ⟩ ⟩ G^{\Xi}_{a}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{q}}(x)\Xi^{\mathrm{cl}}(x')\right\rangle\!\right\rangle= +i\theta(t'-t)\left\langle\!\left\langle[\hat{\Xi}(x),\hat{\Xi}(x')]\right\rangle\!\right\rangle G a Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ q ( x ) Ξ cl ( x ′ ) ⟩ ⟩ = + i θ ( t ′ − t ) ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ( x ′ )] ⟩ ⟩ G k Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ c l ( x ) Ξ c l ( x ′ ) ⟩ ⟩ = − i 2 ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ( x ′ ) } ⟩ ⟩ (21) G^{\Xi}_{k}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{cl}}(x)\Xi^{\mathrm{cl}}(x')\right\rangle\!\right\rangle= -\frac{i}{2}\left\langle\!\left\langle\{\hat{\Xi}(x),\hat{\Xi}(x')\}\right\rangle\!\right\rangle \tag*{(21)} G k Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ cl ( x ) Ξ cl ( x ′ ) ⟩ ⟩ = − 2 i ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ( x ′ )} ⟩ ⟩ (21) while for fermionic environmental fields the free-theory propagators read as
G r Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ c l ( x ) Ξ ‾ c l ( x ′ ) ⟩ ⟩ = − i θ ( t − t ′ ) ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ‾ ( x ′ ) } ⟩ ⟩ G^{\Xi}_{r}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{cl}}(x)\overline{\Xi}^{\mathrm{cl}}(x')\right\rangle\!\right\rangle= -i\theta(t-t')\left\langle\!\left\langle\{\hat{\Xi}(x),\overline{\hat{\Xi}}(x')\}\right\rangle\!\right\rangle G r Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ cl ( x ) Ξ cl ( x ′ ) ⟩ ⟩ = − i θ ( t − t ′ ) ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ( x ′ )} ⟩ ⟩ G a Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ q ( x ) Ξ ‾ q ( x ′ ) ⟩ ⟩ = + i θ ( t ′ − t ) ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ‾ ( x ′ ) } ⟩ ⟩ G^{\Xi}_{a}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{q}}(x)\overline{\Xi}^{\mathrm{q}}(x')\right\rangle\!\right\rangle= +i\theta(t'-t)\left\langle\!\left\langle\{\hat{\Xi}(x),\overline{\hat{\Xi}}(x')\}\right\rangle\!\right\rangle G a Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ q ( x ) Ξ q ( x ′ ) ⟩ ⟩ = + i θ ( t ′ − t ) ⟨ ⟨ { Ξ ^ ( x ) , Ξ ^ ( x ′ )} ⟩ ⟩ G k Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ c l ( x ) Ξ ‾ q ( x ′ ) ⟩ ⟩ = − i ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ‾ ( x ′ ) ] ⟩ ⟩ (22) G^{\Xi}_{k}(x-x') \equiv-i\left\langle\!\left\langle\Xi^{\mathrm{cl}}(x)\overline{\Xi}^{\mathrm{q}}(x')\right\rangle\!\right\rangle= -i\left\langle\!\left\langle[\hat{\Xi}(x),\overline{\hat{\Xi}}(x')]\right\rangle\!\right\rangle \tag*{(22)} G k Ξ ( x − x ′ ) ≡ − i ⟨ ⟨ Ξ cl ( x ) Ξ q ( x ′ ) ⟩ ⟩ = − i ⟨ ⟨ [ Ξ ^ ( x ) , Ξ ^ ( x ′ )] ⟩ ⟩ (22) where in the latter case we used the Larkin-Ovchinnikov convention [80 ]
Ξ c l = 1 2 ( Ξ + + Ξ − ) , Ξ q = 1 2 ( Ξ + − Ξ − ) , Ξ ‾ c l = 1 2 ( Ξ ‾ + − Ξ ‾ − ) , Ξ ‾ q = 1 2 ( Ξ ‾ + + Ξ ‾ − ) , (23) \Xi^{\mathrm{cl}} = \frac{1}{\sqrt{2}}(\Xi_{+}+\Xi_{-}),\quad\Xi^{\mathrm{q}} = \frac{1}{\sqrt{2}}(\Xi_{+}-\Xi_{-}),\quad\overline{\Xi}^{\mathrm{cl}} = \frac{1}{\sqrt{2}}(\overline{\Xi}_{+}-\overline{\Xi}_{-}),\quad\overline{\Xi}^{\mathrm{q}} = \frac{1}{\sqrt{2}}(\overline{\Xi}_{+}+\overline{\Xi}_{-}), \tag*{(23)} Ξ cl = 2 1 ( Ξ + + Ξ − ) , Ξ q = 2 1 ( Ξ + − Ξ − ) , Ξ cl = 2 1 ( Ξ + − Ξ − ) , Ξ q = 2 1 ( Ξ + + Ξ − ) , (23) with Ξ ‾ = Ξ † γ 0 \overline{\Xi}=\Xi^{\dagger}\gamma^{0} Ξ = Ξ † γ 0 § 0.6 , the entries of the matrix K ( x − x ′ ) K(x-x') K ( x − x ′ )
Taking the limit t f = − t i → ∞ t_f=-t_i\to\infty t f = − t i → ∞ (1) , and noting that O ~ s ∗ ( q ) = O ~ s ( − q ) \widetilde{O}^{*}_{s}(q)=\widetilde{O}_{s}(-q) O s ∗ ( q ) = O s ( − q ) K ~ a ( q ) = K ~ r ∗ ( q ) \widetilde{K}_{a}(q)=\widetilde{K}^{*}_{r}(q) K a ( q ) = K r ∗ ( q ) (20) ), the influence action may be expressed in four-momentum representation as
S I F [ ψ + , ψ − ] = g Λ D ∫ d 4 q O ~ s q ( q ) ⟨ ⟨ O ~ e c l ( − q ) ⟩ ⟩ − ( g Λ D ) 2 ∫ d 4 q { Re [ O ~ s c l ( q ) O ~ s q ( − q ) ] Re [ K ~ r ( q ) ] − Im [ O ~ s c l ( q ) O ~ s q ( − q ) ] Im [ K ~ r ( q ) ] } − i 2 ( g Λ D ) 2 ∫ d 4 q ∣ O ~ s q ( q ) ∣ 2 Im [ K ~ k ( q ) ] + O ( g 3 ) , (24) \begin{aligned}
S_{\mathrm{IF}}[\psi_{+},\psi_{-}] ={}& \frac{g}{\Lambda^{D}}\int d^{4}q\,\widetilde{O}^{q}_{s}(q)\left\langle\!\left\langle\widetilde{O}^{\mathrm{cl}}_{e}(-q)\right\rangle\!\right\rangle\\
&-\left(\frac{g}{\Lambda^{D}}\right)^{2}\int d^{4}q\left\{\operatorname{Re}\left[\widetilde{O}^{\mathrm{cl}}_{s}(q)\widetilde{O}^{q}_{s}(-q)\right]\operatorname{Re}\left[\widetilde{K}_{r}(q)\right]-\operatorname{Im}\left[\widetilde{O}^{\mathrm{cl}}_{s}(q)\widetilde{O}^{q}_{s}(-q)\right]\operatorname{Im}\left[\widetilde{K}_{r}(q)\right]\right\} \\
&-\frac{i}{2}\left(\frac{g}{\Lambda^{D}}\right)^{2}\int d^{4}q\,\left|\widetilde{O}^{q}_{s}(q)\right|^{2}\operatorname{Im}\left[\widetilde{K}_{k}(q)\right] \\
&+O(g^{3}), \tag*{(24)}
\end{aligned} S IF [ ψ + , ψ − ] = Λ D g ∫ d 4 q O s q ( q ) ⟨ ⟨ O e cl ( − q ) ⟩ ⟩ − ( Λ D g ) 2 ∫ d 4 q { Re [ O s cl ( q ) O s q ( − q ) ] Re [ K r ( q ) ] − Im [ O s cl ( q ) O s q ( − q ) ] Im [ K r ( q ) ] } − 2 i ( Λ D g ) 2 ∫ d 4 q O s q ( q ) 2 Im [ K k ( q ) ] + O ( g 3 ) , (24) where q q q
We conclude by noting that, because the environment is described by quantum fields, the kernels K r , a , k K_{r,a,k} K r , a , k K k K_{k} K k
From the influence action to interferometer observables
Having established the field-level open EFT for an arbitrary system, we now focus on MWIs. In particular, we will show explicitly how the reduced density matrix of the MWI can be mapped directly to the worldline influence action.
For pedagogical reasons, we start with the single-atom case, which was already discussed in Ref. [61 ] and contains all the essential structure. The generalization to an ensemble of N a t N_{\mathrm{at}} N at
A toy example: a single-atom interferometer
In a single-atom MWI, the atom as a whole is prepared in a superposition of states that are sent along different spacetime trajectories. In practice, this is achieved by entangling the c.o.m. degree of freedom with an internal degree of freedom, e.g., the atomic spin or energy level, which labels the trajectory. To make this structure manifest in the field theory, we introduce a discrete path label α ∈ { L , R } \alpha\in\{L,R\} α ∈ { L , R } Ψ = ( ψ L , ψ R ) T \Psi= (\psi_L,\psi_R)^T Ψ = ( ψ L , ψ R ) T (11) –(13) under the replacement ψ ( t , x − r ) → Ψ ( t , x − r ) \psi(t,\mathbf{x}-\mathbf{r}) \to\Psi(t,\mathbf{x}-\mathbf{r}) ψ ( t , x − r ) → Ψ ( t , x − r )
Since atoms are neither created nor destroyed at sub-GeV scales, the system dynamics and measurements are restricted to the one-particle sector of the Fock space of Ψ ^ \hat{\Psi} Ψ ^ ∣ α , y ⟩ ≡ ψ ^ α † ( y ) ∣ 0 ⟩ s \lvert\alpha,y\rangle\equiv\hat{\psi}^{\dagger}_{\alpha}(y)\lvert0\rangle_s ∣ α , y ⟩ ≡ ψ ^ α † ( y ) ∣ 0 ⟩ s ∣ 0 ⟩ s \lvert0\rangle_s ∣ 0 ⟩ s
O ^ s ( x ) ψ ^ α † ( y ) ∣ 0 ⟩ s = ϱ ( x − y ) ψ ^ α † ( y ) ∣ 0 ⟩ s . (25) \hat{O}_s(x)\hat{\psi}^{\dagger}_{\alpha}(y)\lvert0\rangle_s = \varrho(x-y)\hat{\psi}^{\dagger}_{\alpha}(y)\lvert0\rangle_s . \tag*{(25)} O ^ s ( x ) ψ ^ α † ( y ) ∣ 0 ⟩ s = ϱ ( x − y ) ψ ^ α † ( y ) ∣ 0 ⟩ s . (25) This projection then implies that O s → ϱ O_s \to\varrho O s → ϱ
We define the initial state of the system at time t i t_i t i L L L R R R
ρ ^ s ( t i ) = ∣ Φ ⟩ ⟨ Φ ∣ , ∣ Φ ⟩ = 1 2 ∫ d 3 y Φ ( y ) ( ψ ^ L † ( y ) + ψ ^ R † ( y ) ) ∣ 0 ⟩ s , (26) \hat{\rho}_s(t_i) = \lvert\Phi\rangle\langle\Phi\rvert\,, \quad\lvert\Phi\rangle= \frac{1}{\sqrt{2}}\int d^3y\,\Phi(y)\left(\hat{\psi}^{\dagger}_L(y)+\hat{\psi}^{\dagger}_R(y)\right)\lvert0\rangle_s \,, \tag*{(26)} ρ ^ s ( t i ) = ∣ Φ ⟩ ⟨ Φ ∣ , ∣ Φ ⟩ = 2 1 ∫ d 3 y Φ ( y ) ( ψ ^ L † ( y ) + ψ ^ R † ( y ) ) ∣ 0 ⟩ s , (26) where Φ ( y ) \Phi(y) Φ ( y ) σ \sigma σ J ^ \hat{J} J ^ { L , R } \{L,R\} { L , R } [81 ] . As a result, components of the time-evolved reduced density matrix in the α , β ∈ { L , R } \alpha,\beta\in\{L,R\} α , β ∈ { L , R } { x f , + , x f , − } \{\mathbf{x}_{f,+},\mathbf{x}_{f,-}\} { x f , + , x f , − }
⟨ x f , + ∣ ρ ^ s α β ( t f ) ∣ x f , − ⟩ ≡ ⟨ α , x f , + ∣ ρ ^ s ( t f ) ∣ β , x f , − ⟩ = ⟨ α , x f , + ∣ Tr e [ U ^ ( t f , t i ; J ^ ) ρ ^ i U ^ ( t i , t f ; J ^ ) ] ∣ β , x f , − ⟩ . (27) \langle\mathbf{x}_{f,+}|\hat{\rho}^{\alpha\beta}_{s}(t_f)|\mathbf{x}_{f,-}\rangle\equiv\langle\alpha,\mathbf{x}_{f,+}|\hat{\rho}_{s}(t_f)|\beta,\mathbf{x}_{f,-}\rangle
= \langle\alpha,\mathbf{x}_{f,+}|\operatorname{Tr}_{e}\left[\hat{U}(t_f,t_i;\hat{J})\hat{\rho}_{i}\hat{U}(t_i,t_f;\hat{J})\right]|\beta,\mathbf{x}_{f,-}\rangle. \tag*{(27)} ⟨ x f , + ∣ ρ ^ s α β ( t f ) ∣ x f , − ⟩ ≡ ⟨ α , x f , + ∣ ρ ^ s ( t f ) ∣ β , x f , − ⟩ = ⟨ α , x f , + ∣ Tr e [ U ^ ( t f , t i ; J ^ ) ρ ^ i U ^ ( t i , t f ; J ^ ) ] ∣ β , x f , − ⟩ . (27) In what follows, we will focus on interferometric readouts that depend on the path label. This is standard in MWI protocols with internal-state labeling [82 ] , which entangle the probe’s internal degree of freedom with the interferometer arms. As a result, the arm-dependent history is encoded in the coherence between the L L L R R R
ρ s α β ( t f ) ≡ ∫ d 3 x f , + ∫ d 3 x f , − δ ( 3 ) ( x f , + − x f , − ) ⟨ x f , + ∣ ρ ^ s α β ( t f ) ∣ x f , − ⟩ . (28) \rho^{\alpha\beta}_{s}(t_f) \equiv\int\mathrm{d}^{3}x_{f,+}\int\mathrm{d}^{3}x_{f,-}\,\delta^{(3)}(\mathbf{x}_{f,+}-\mathbf{x}_{f,-})\langle\mathbf{x}_{f,+}|\hat{\rho}^{\alpha\beta}_{s}(t_f)|\mathbf{x}_{f,-}\rangle. \tag*{(28)} ρ s α β ( t f ) ≡ ∫ d 3 x f , + ∫ d 3 x f , − δ ( 3 ) ( x f , + − x f , − ) ⟨ x f , + ∣ ρ ^ s α β ( t f ) ∣ x f , − ⟩ . (28) Because the dynamics is confined to the one-particle sector and the initial state is given by Eq. (26) , Eq. (28) admits a path-integral representation over worldlines X ± \mathbf{X}_{\pm} X ± [32 , 33 ] driven by external impulsive, path-dependent currents J α ∝ m a t \mathbf{J}_{\alpha} \propto m_{\mathrm{at}} J α ∝ m at J ^ \hat{J} J ^ x f , + = x f \mathbf{x}_{f,+}=\mathbf{x}_{f} x f , + = x f (28) reads as
ρ s α β ( t f ) = ∫ d 3 x i , + ∫ d 3 x i , − ⟨ α , x i , + ∣ ρ ^ s ( t i ) ∣ β , x i , − ⟩ × ∫ d 3 x f ∫ x i , + x f D X + ∫ x i , − x f D X − e i S s [ X + , J α , + ] − i S s [ X − , J β , − ] e i S I F [ X + , X − ] . (29) \begin{aligned}
\rho^{\alpha\beta}_{s}(t_f)={}&\int\mathrm{d}^{3}x_{i,+}\int\mathrm{d}^{3}x_{i,-}\,\langle\alpha,\mathbf{x}_{i,+}|\hat{\rho}_{s}(t_i)|\beta,\mathbf{x}_{i,-}\rangle\\
&\times\int\mathrm{d}^{3}x_f\int_{\mathbf{x}_{i,+}}^{\mathbf{x}_f}\mathcal{D}\mathbf{X}_{+}\int_{\mathbf{x}_{i,-}}^{\mathbf{x}_f}\mathcal{D}\mathbf{X}_{-}\,e^{iS_s[\mathbf{X}_{+},\mathbf{J}_{\alpha,+}]-iS_s[\mathbf{X}_{-},\mathbf{J}_{\beta,-}]}e^{iS_{\mathrm{IF}}[\mathbf{X}_{+},\mathbf{X}_{-}]} . \tag*{(29)}
\end{aligned} ρ s α β ( t f ) = ∫ d 3 x i , + ∫ d 3 x i , − ⟨ α , x i , + ∣ ρ ^ s ( t i ) ∣ β , x i , − ⟩ × ∫ d 3 x f ∫ x i , + x f D X + ∫ x i , − x f D X − e i S s [ X + , J α , + ] − i S s [ X − , J β , − ] e i S IF [ X + , X − ] . (29) with boundary conditions X ± ( t i ) = x i , ± \mathbf{X}_{\pm}(t_i)=\mathbf{x}_{i,\pm} X ± ( t i ) = x i , ± X ± ( t f ) = x f \mathbf{X}_{\pm}(t_f)=\mathbf{x}_f X ± ( t f ) = x f
S s [ X ± , J α , ± ] = ∫ t i t f d t [ m a t 2 ∣ X ˙ ± ∣ 2 + J α , ± ⋅ X ± ] . (30) S_s[\mathbf{X}_{\pm},\mathbf{J}_{\alpha,\pm}]=\int_{t_i}^{t_f}\mathrm{d}t\left[\frac{m_{\mathrm{at}}}{2}|\dot{\mathbf{X}}_{\pm}|^2+\mathbf{J}_{\alpha,\pm}\cdot\mathbf{X}_{\pm}\right] . \tag*{(30)} S s [ X ± , J α , ± ] = ∫ t i t f d t [ 2 m at ∣ X ˙ ± ∣ 2 + J α , ± ⋅ X ± ] . (30) We assume that both arms couple to the environment with the same charge. We also assume that the interferometer closes, as shown in Figure 2 . As a result, the O ( g ) \mathcal{O}(g) O ( g ) (14) ). Introducing the Keldysh representation of the atom’s charge density profile,
Figure 2. Spacetime diagram for the interferometric geometry considered in this work. The total interferometric time is T s T_s T s Δ x \Delta x Δ x x ˉ L , R ( t ) \bar{x}_{L,R}(t) x ˉ L , R ( t ) (35) , defined in Eq. (36) , are shown in solid orange and dashed green, together with the initial wavepacket of size σ \sigma σ T ′ ≪ T T' \ll T T ′ ≪ T ∣ Δ x ∣ |\Delta\mathbf{x}| ∣Δ x ∣
ρ c l ( x ; X ± ) = 1 2 [ ρ ( x − X + ( t ) ) + ρ ( x − X − ( t ) ) ] , (31) \rho^{\mathrm{cl}}(x;\mathbf{X}_{\pm})=\frac{1}{2}\left[\rho(x-\mathbf{X}_{+}(t))+\rho(x-\mathbf{X}_{-}(t))\right] , \tag*{(31)} ρ cl ( x ; X ± ) = 2 1 [ ρ ( x − X + ( t )) + ρ ( x − X − ( t )) ] , (31) ρ q ( x ; X ± ) = ρ ( x − X + ( t ) ) − ρ ( x − X − ( t ) ) , (32) \rho^{\mathrm{q}}(x;\mathbf{X}_{\pm})=\rho(x-\mathbf{X}_{+}(t))-\rho(x-\mathbf{X}_{-}(t)) , \tag*{(32)} ρ q ( x ; X ± ) = ρ ( x − X + ( t )) − ρ ( x − X − ( t )) , (32) and the system current vector
ρ ⃗ ( x ) T = ( ρ c l ( x ; X ± ) ρ q ( x ; X ± ) ) , (33) \vec{\rho}(x)^{\mathsf{T}}=\left(\rho^{\mathrm{cl}}(x;\mathbf{X}_{\pm})\ \ \rho^{\mathrm{q}}(x;\mathbf{X}_{\pm})\right) , \tag*{(33)} ρ ( x ) T = ( ρ cl ( x ; X ± ) ρ q ( x ; X ± ) ) , (33) the leading-order contribution to the influence action takes the compact form
S I F [ X + , X − ] = − 1 2 ( g Λ 2 ) 2 ∫ t i t f d 4 x ∫ t i t f d 4 x ′ ρ ⃗ ( x ; X ± ) T K ( x − x ′ ) ρ ⃗ ( x ′ ; X ± ) + O ( g 3 ) , (34) S_{\mathrm{IF}}[X_+,X_-] = -\frac{1}{2}\left(\frac{g}{\Lambda^2}\right)^2 \int_{t_i}^{t_f} \mathrm{d}^4x \int_{t_i}^{t_f} \mathrm{d}^4x'\,\vec{\rho}(x;X_\pm)^{\mathsf{T}} K(x-x')\vec{\rho}(x';X_\pm) + \mathcal{O}(g^3), \tag*{(34)} S IF [ X + , X − ] = − 2 1 ( Λ 2 g ) 2 ∫ t i t f d 4 x ∫ t i t f d 4 x ′ ρ ( x ; X ± ) T K ( x − x ′ ) ρ ( x ′ ; X ± ) + O ( g 3 ) , (34) with K ( x − x ′ ) K(x-x') K ( x − x ′ ) (16) ).
To evaluate the path integral over worldlines, we decompose the worldlines X ± X_\pm X ± classical (driven) trajectory X ‾ α , ± \overline{X}_{\alpha,\pm} X α , ± J α , ± J_{\alpha,\pm} J α , ±
m a t X ‾ ¨ α , ± = J α , ± , X ‾ α , ± ( t i ) = x i , ± , X ‾ α , ± ( t f ) = x f , (35) m_{\mathrm{at}}\ddot{\overline{X}}_{\alpha,\pm}=J_{\alpha,\pm},\qquad\overline{X}_{\alpha,\pm}(t_i)=x_{i,\pm},\qquad\overline{X}_{\alpha,\pm}(t_f)=x_f, \tag*{(35)} m at X ¨ α , ± = J α , ± , X α , ± ( t i ) = x i , ± , X α , ± ( t f ) = x f , (35) which admits the solution
X ‾ α , ± ( t ) = x i , ± t f − t t f − t i + x f t − t i t f − t i + ∫ t i t f d t ′ G d ( t , t ′ ) J α , ± ( t ′ ) , (36) \overline{X}_{\alpha,\pm}(t)=x_{i,\pm}\frac{t_f-t}{t_f-t_i}+x_f\frac{t-t_i}{t_f-t_i}+\int_{t_i}^{t_f}\mathrm{d}t'\,G_d(t,t')J_{\alpha,\pm}(t'), \tag*{(36)} X α , ± ( t ) = x i , ± t f − t i t f − t + x f t f − t i t − t i + ∫ t i t f d t ′ G d ( t , t ′ ) J α , ± ( t ′ ) , (36) in terms of the Dirichlet Green’s function [83 ]
G d ( t , t ′ ) = ( t < − t i ) ( t > − t f ) m a t ( t f − t i ) , t < = min ( t , t ′ ) , t > = max ( t , t ′ ) , (37) G_d(t,t')=\frac{(t_<-t_i)(t_>-t_f)}{m_{\mathrm{at}}(t_f-t_i)},\qquad t_< = \min(t,t'),\quad t_> = \max(t,t'), \tag*{(37)} G d ( t , t ′ ) = m at ( t f − t i ) ( t < − t i ) ( t > − t f ) , t < = min ( t , t ′ ) , t > = max ( t , t ′ ) , (37) and the (interferometer geometry-dependent but boundary condition-independent) particular solution x ‾ α , ± ( t ) \overline{x}_{\alpha,\pm}(t) x α , ± ( t ) worldline fluctuations δ X ± \delta X_\pm δ X ±
δ X ± ( t i ) = δ X ± ( t f ) = 0. (38) \delta X_\pm(t_i)=\delta X_\pm(t_f)=0. \tag*{(38)} δ X ± ( t i ) = δ X ± ( t f ) = 0. (38) As a result, recalling from Eq. (26) that ⟨ α , x i , + ∣ ρ ^ s ( t i ) ∣ β , x i , − ⟩ = 1 2 Φ ( x i , + ) Φ ∗ ( x i , − ) \langle\alpha,x_{i,+}|\hat{\rho}_s(t_i)|\beta,x_{i,-}\rangle=\frac{1}{2}\Phi(x_{i,+})\Phi^*(x_{i,-}) ⟨ α , x i , + ∣ ρ ^ s ( t i ) ∣ β , x i , − ⟩ = 2 1 Φ ( x i , + ) Φ ∗ ( x i , − ) (29) can be rewritten as a path integral over worldline fluctuations:
ρ s α β ( t f ) = 1 2 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) ∫ d 3 x f e i S s [ X ‾ α , + , J α , + ] − i S s [ X ‾ β , − , J β , − ] ∫ 0 0 D δ X + ∫ 0 0 D δ X − e i S s [ δ X + ] − i S s [ δ X − ] e i S I F [ X ‾ α , + + δ X + , X ‾ β , − − δ X − ] . (39) \rho_s^{\alpha\beta}(t_f)=\frac{1}{2}\int\mathrm{d}^3x_{i,+}\int\mathrm{d}^3x_{i,-}\,\Phi(x_{i,+})\Phi^*(x_{i,-})\int\mathrm{d}^3x_f\,e^{iS_s[\overline{X}_{\alpha,+},J_{\alpha,+}]-iS_s[\overline{X}_{\beta,-},J_{\beta,-}]}
\int_0^0\mathcal{D}\delta X_+\int_0^0\mathcal{D}\delta X_-e^{iS_s[\delta X_+]-iS_s[\delta X_-]}e^{iS_{\mathrm{IF}}[\overline{X}_{\alpha,+}+\delta X_+,\overline{X}_{\beta,-}-\delta X_-]}. \tag*{(39)} ρ s α β ( t f ) = 2 1 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) ∫ d 3 x f e i S s [ X α , + , J α , + ] − i S s [ X β , − , J β , − ] ∫ 0 0 D δ X + ∫ 0 0 D δ X − e i S s [ δ X + ] − i S s [ δ X − ] e i S IF [ X α , + + δ X + , X β , − − δ X − ] . (39) Following Ref. [61 ] , one can solve the path integral perturbatively by: (i) introducing fictitious currents j ± j_\pm j ± S s [ δ X ± ] → S s [ δ X ± , j ± ] S_s[\delta X_\pm]\to S_s[\delta X_\pm,j_\pm] S s [ δ X ± ] → S s [ δ X ± , j ± ] (30) ; (ii) rewriting δ X ± → ∓ i δ / δ j ± \delta X_\pm\to\mp i\delta/\delta j_\pm δ X ± → ∓ i δ / δ j ± S I F S_{\mathrm{IF}} S IF
We further focus on symmetric interferometric sequences, such that J α = − J β J_\alpha=-J_\beta J α = − J β α ≠ β \alpha\ne\beta α = β t f = − t i = T t_f=-t_i=T t f = − t i = T
Integrating Eq. (30) by parts and substituting the classical trajectories of Eq. (36) , the classical contribution to the system phase reduces to
e i S s [ X ‾ α , + , J α , + ] − i S s [ X ‾ β , − , J β , − ] = exp [ i m a t 4 T ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) ] . (40) e^{iS_s[\overline{X}_{\alpha,+},J_{\alpha,+}]-iS_s[\overline{X}_{\beta,-},J_{\beta,-}]}=\exp\left[\frac{i m_{\mathrm{at}}}{4T}\left(\lvert x_f-x_{i,+}\rvert^2-\lvert x_f-x_{i,-}\rvert^2\right)\right]. \tag*{(40)} e i S s [ X α , + , J α , + ] − i S s [ X β , − , J β , − ] = exp [ 4 T i m at ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) ] . (40) The contribution from the worldline fluctuations is captured by the generating functional [83 ]
Z 0 [ j + , j − ] = ( m a t 4 π T ) − 3 ∫ 0 0 D δ X + ∫ 0 0 D δ X − e i S s [ δ X + , j + ] − i S s [ δ X − , j − ] = exp { i 2 ∫ − T T d t ∫ − T T d t ′ G d ( t , t ′ ) [ j ˙ + ( t ) ⋅ j ˙ + ( t ′ ) − j ˙ − ( t ) ⋅ j ˙ − ( t ′ ) ] } , (41) \begin{aligned}
Z_0[j_+,j_-] = \left(\frac{m_{\mathrm{at}}}{4\pi\mathcal{T}}\right)^{-3}\int_0^0 \mathcal{D}\delta X_+ \int_0^0 \mathcal{D}\delta X_-\,e^{iS_s[\delta X_+,j_+]-iS_s[\delta X_-,j_-]} \\
&= \exp\left\{\frac{i}{2}\int_{-\mathcal{T}}^{\mathcal{T}}dt\int_{-\mathcal{T}}^{\mathcal{T}}dt'\,G_d(t,t')\left[\dot{j}_+(t)\cdot\dot{j}_+(t')-\dot{j}_-(t)\cdot\dot{j}_-(t')\right]\right\}, \tag*{(41)}
\end{aligned} Z 0 [ j + , j − ] = ( 4 π T m at ) − 3 ∫ 0 0 D δ X + ∫ 0 0 D δ X − e i S s [ δ X + , j + ] − i S s [ δ X − , j − ] = exp { 2 i ∫ − T T d t ∫ − T T d t ′ G d ( t , t ′ ) [ j ˙ + ( t ) ⋅ j ˙ + ( t ′ ) − j ˙ − ( t ) ⋅ j ˙ − ( t ′ ) ] } , (41) with Z 0 [ 0 , 0 ] = 1 Z_0[0,0]=1 Z 0 [ 0 , 0 ] = 1 (40) and (41) , Eq. (39) reads as
ρ s α β ( T ) = 1 2 ( m a t 4 π T ) 3 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) × ∫ d 3 x f e i m a t 4 T ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) e i S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] Z 0 [ j + , j − ] ∣ j ± = 0 = 1 2 + i 2 ⟨ S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] ⟩ Z 0 [ j + , j − ] ∣ j ± = 0 + O ( a 3 ) , (42) \begin{aligned}
\rho_s^{\alpha\beta}(\mathcal{T})=\frac{1}{2}\left(\frac{m_{\mathrm{at}}}{4\pi\mathcal{T}}\right)^3\int d^3x_{i,+}\int d^3x_{i,-}\,\Phi(x_{i,+})\Phi^*(x_{i,-}) \\
&\times\int d^3x_f\,e^{\frac{im_{\mathrm{at}}}{4\mathcal{T}}\left(\lvert x_f-x_{i,+}\rvert^2-\lvert x_f-x_{i,-}\rvert^2\right)}e^{iS_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-i\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+i\frac{\delta}{\delta j_-}\right]}Z_0[j_+,j_-]\bigg|_{j_\pm=0} \\
&=\frac{1}{2}+\frac{i}{2}\left\langle S_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-i\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+i\frac{\delta}{\delta j_-}\right]\right\rangle Z_0[j_+,j_-]\bigg|_{j_\pm=0}+\mathcal{O}(a^3), \tag*{(42)}
\end{aligned} ρ s α β ( T ) = 2 1 ( 4 π T m at ) 3 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) × ∫ d 3 x f e 4 T i m at ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) e i S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] Z 0 [ j + , j − ] j ± = 0 = 2 1 + 2 i ⟨ S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] ⟩ Z 0 [ j + , j − ] j ± = 0 + O ( a 3 ) , (42) where in going to the last line we linearized the influence functional and defined ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩
Henceforth, we will work in the heavy-probe limit, corresponding to sending m a t → ∞ m_{\mathrm{at}}\to\infty m at → ∞ J ± / m a t J_\pm/m_{\mathrm{at}} J ± / m at S I F S_{\mathrm{IF}} S IF 3
Since G d ∝ m a t − 1 G_d\propto m_{\mathrm{at}}^{-1} G d ∝ m at − 1 (29) reduces to the influence functional evaluated on the driven classical paths. Furthermore, as we carefully derive in Appendix A, in this limit the external currents J ± J_\pm J ± x f x_f x f x i , + = x i , − x_{i,+}=x_{i,-} x i , + = x i , − particular solution x ‾ α , ± ( t ) \overline{x}_{\alpha,\pm}(t) x α , ± ( t ) (36) ), i.e.,
lim m a t → ∞ ⟨ S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] ⟩ Z 0 [ j + , j − ] ∣ j ± = 0 = S I F [ x ‾ α , + , x ‾ β , − ] . (43) \lim_{m_{\mathrm{at}}\to\infty}\left\langle S_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-i\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+i\frac{\delta}{\delta j_-}\right]\right\rangle Z_0[j_+,j_-]\bigg|_{j_\pm=0}=S_{\mathrm{IF}}[\overline{x}_{\alpha,+},\overline{x}_{\beta,-}] . \tag*{(43)} m at → ∞ lim ⟨ S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] ⟩ Z 0 [ j + , j − ] j ± = 0 = S IF [ x α , + , x β , − ] . (43) Introducing the 3-momentum FTs of the system currents
ρ ~ c l , q ( q , t ; x ‾ ± ) = ∫ d 3 x ρ c l , q ( x ; x ‾ ± ( t ) ) e − i q ⋅ x = F A ( q ) W ~ c l , q ( q , t ; x ‾ ± ) , (44) \widetilde{\rho}^{\mathrm{cl,q}}(\mathbf{q},t;\overline{x}_\pm)=\int d^3x\,\rho^{\mathrm{cl,q}}(x;\overline{x}_\pm(t))e^{-i\mathbf{q}\cdot\mathbf{x}}=F_A(\mathbf{q})\widetilde{\mathcal{W}}^{\mathrm{cl,q}}(\mathbf{q},t;\overline{x}_\pm), \tag*{(44)} ρ cl , q ( q , t ; x ± ) = ∫ d 3 x ρ cl , q ( x ; x ± ( t )) e − i q ⋅ x = F A ( q ) W cl , q ( q , t ; x ± ) , (44) where F A ( q ) F_A(\mathbf{q}) F A ( q ) (13) ) and
W ~ c l ( q , t ; x ‾ ± ) = 1 2 [ e − i q ⋅ x ‾ α , + ( t ) + e − i q ⋅ x ‾ β , − ( t ) ] , \widetilde{\mathcal{W}}^{\mathrm{cl}}(\mathbf{q},t;\overline{x}_\pm)=\frac{1}{2}\left[e^{-i\mathbf{q}\cdot\overline{x}_{\alpha,+}(t)}+e^{-i\mathbf{q}\cdot\overline{x}_{\beta,-}(t)}\right], W cl ( q , t ; x ± ) = 2 1 [ e − i q ⋅ x α , + ( t ) + e − i q ⋅ x β , − ( t ) ] , W ~ q ( q , t ; x ‾ ± ) = e − i q ⋅ x ‾ α , + ( t ) − e − i q ⋅ x ‾ β , − ( t ) , (45) \widetilde{\mathcal{W}}^{\mathrm{q}}(\mathbf{q},t;\overline{x}_\pm)=e^{-i\mathbf{q}\cdot\overline{x}_{\alpha,+}(t)}-e^{-i\mathbf{q}\cdot\overline{x}_{\beta,-}(t)}, \tag*{(45)} W q ( q , t ; x ± ) = e − i q ⋅ x α , + ( t ) − e − i q ⋅ x β , − ( t ) , (45) capture the worldline histories, the influence action evaluated along the particular solution of the classical worldlines reads as
S I F [ x ˉ α , + , x ˉ β , − ] = − ( g Λ D ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 Re [ W ~ q ( − q , t ; x ˉ ± ) W ~ c l ( q , t ′ ; x ˉ ± ) ] Re [ K ~ r ( q , t − t ′ ) ] + ( g Λ D ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 Im [ W ~ q ( − q , t ; x ˉ ± ) W ~ c l ( q , t ′ ; x ˉ ± ) ] Im [ K ~ r ( q , t − t ′ ) ] − i 2 ( g Λ D ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 W ~ q ( − q , t ; x ˉ ± ) W ~ q ( q , t ′ ; x ˉ ± ) Im [ K ~ k ( q , t − t ′ ) ] + O ( g 3 ) . (46) \begin{aligned}
S_{\mathrm{IF}}[\bar{x}_{\alpha,+},\bar{x}_{\beta,-}]={}&-\left(\frac{g}{\Lambda^{D}}\right)^2\int dt\int dt'\int d^3q\,\lvert F_A(\mathbf{q})\rvert^2\operatorname{Re}\left[\widetilde{W}^{q}(-\mathbf{q},t;\bar{x}_{\pm})\widetilde{W}^{\mathrm{cl}}(\mathbf{q},t';\bar{x}_{\pm})\right]\operatorname{Re}\left[\widetilde{K}_{r}(\mathbf{q},t-t')\right]\\
&+\left(\frac{g}{\Lambda^{D}}\right)^2\int dt\int dt'\int d^3q\,\lvert F_A(\mathbf{q})\rvert^2\operatorname{Im}\left[\widetilde{W}^{q}(-\mathbf{q},t;\bar{x}_{\pm})\widetilde{W}^{\mathrm{cl}}(\mathbf{q},t';\bar{x}_{\pm})\right]\operatorname{Im}\left[\widetilde{K}_{r}(\mathbf{q},t-t')\right]\\
&-\frac{i}{2}\left(\frac{g}{\Lambda^{D}}\right)^2\int dt\int dt'\int d^3q\,\lvert F_A(\mathbf{q})\rvert^2\widetilde{W}^{q}(-\mathbf{q},t;\bar{x}_{\pm})\widetilde{W}^{q}(\mathbf{q},t';\bar{x}_{\pm})\operatorname{Im}\left[\widetilde{K}_{k}(\mathbf{q},t-t')\right]\\
&+O(g^3). \tag*{(46)}
\end{aligned} S IF [ x ˉ α , + , x ˉ β , − ] = − ( Λ D g ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 Re [ W q ( − q , t ; x ˉ ± ) W cl ( q , t ′ ; x ˉ ± ) ] Re [ K r ( q , t − t ′ ) ] + ( Λ D g ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 Im [ W q ( − q , t ; x ˉ ± ) W cl ( q , t ′ ; x ˉ ± ) ] Im [ K r ( q , t − t ′ ) ] − 2 i ( Λ D g ) 2 ∫ d t ∫ d t ′ ∫ d 3 q ∣ F A ( q ) ∣ 2 W q ( − q , t ; x ˉ ± ) W q ( q , t ′ ; x ˉ ± ) Im [ K k ( q , t − t ′ ) ] + O ( g 3 ) . (46) Hence, substituting Eq. (43) into Eq. (42) and re-exponentiating the influence functional, the heavy-probe and long-time limit of the system’s reduced density matrix takes the form
lim T → ∞ lim m a t → ∞ ρ ^ s ( T ) ≃ 1 2 ( e i S I F [ x ˉ L , x ˉ L ] e i S I F [ x ˉ L , x ˉ R ] e i S I F [ x ˉ R , x ˉ L ] e i S I F [ x ˉ R , x ˉ R ] ) ≡ 1 2 ( 1 e − D − i P e − D + i P 1 ) , (47) \lim_{T\to\infty}\lim_{m_{\mathrm{at}}\to\infty}\hat{\rho}_{s}(T)\simeq\frac{1}{2}
\begin{pmatrix}
e^{iS_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{L}]} & e^{iS_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{R}]}\\
e^{iS_{\mathrm{IF}}[\bar{x}_{R},\bar{x}_{L}]} & e^{iS_{\mathrm{IF}}[\bar{x}_{R},\bar{x}_{R}]}
\end{pmatrix}
\equiv\frac{1}{2}
\begin{pmatrix}
1 & e^{-\mathbb{D}-i\mathbb{P}}\\
e^{-\mathbb{D}+i\mathbb{P}} & 1
\end{pmatrix}, \tag*{(47)} T → ∞ lim m at → ∞ lim ρ ^ s ( T ) ≃ 2 1 ( e i S IF [ x ˉ L , x ˉ L ] e i S IF [ x ˉ R , x ˉ L ] e i S IF [ x ˉ L , x ˉ R ] e i S IF [ x ˉ R , x ˉ R ] ) ≡ 2 1 ( 1 e − D + i P e − D − i P 1 ) , (47) with
D = Im { S I F [ x ˉ L , x ˉ R ] } , P = − Re { S I F [ x ˉ L , x ˉ R ] } , (48) \mathbb{D}=\operatorname{Im}\{S_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{R}]\},\qquad\mathbb{P}=-\operatorname{Re}\{S_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{R}]\}, \tag*{(48)} D = Im { S IF [ x ˉ L , x ˉ R ]} , P = − Re { S IF [ x ˉ L , x ˉ R ]} , (48) decoherence and phase shift, respectively.
These quantities can then be related to the expectation value of the projective measurements P ^ ± ; θ = ∣ ± ; θ ⟩ ⟨ ± ; θ ∣ \hat{P}_{\pm;\theta}=\lvert\pm;\theta\rangle\langle\pm;\theta\rvert P ^ ± ; θ = ∣ ± ; θ ⟩ ⟨ ± ; θ ∣ ∣ ± ; θ ⟩ = 1 2 ( ∣ L ⟩ ± e i θ ∣ R ⟩ ) \lvert\pm;\theta\rangle=\frac{1}{\sqrt{2}}(\lvert L\rangle\pm e^{i\theta}\lvert R\rangle) ∣ ± ; θ ⟩ = 2 1 (∣ L ⟩ ± e i θ ∣ R ⟩)
lim T → ∞ lim m a t → ∞ Tr [ ρ ^ s ( T ) P ^ ± ; θ ] = 1 2 [ 1 ± e − D cos ( θ − P ) ] . (49) \lim_{T\to\infty}\lim_{m_{\mathrm{at}}\to\infty}\operatorname{Tr}\left[\hat{\rho}_{s}(T)\hat{P}_{\pm;\theta}\right]=\frac{1}{2}\left[1\pm e^{-\mathbb{D}}\cos(\theta-\mathbb{P})\right]. \tag*{(49)} T → ∞ lim m at → ∞ lim Tr [ ρ ^ s ( T ) P ^ ± ; θ ] = 2 1 [ 1 ± e − D cos ( θ − P ) ] . (49) At leading order, these quantities can then be extracted from the total number of clicks in the plus or minus ports, e.g.,
Expected number of counts in the ‘−’ port = { 1 2 D + O ( g 3 ) for θ = 0 , 1 2 ( 1 − P ) + O ( g 3 ) for θ = π / 2 , (50) \text{Expected number of counts in the ‘−’ port}=
\begin{cases}
\frac{1}{2}\mathbb{D}+O(g^3) & \text{for }\theta=0,\\
\frac{1}{2}(1-\mathbb{P})+O(g^3) & \text{for }\theta=\pi/2,
\end{cases} \tag*{(50)} Expected number of counts in the ‘−’ port = { 2 1 D + O ( g 3 ) 2 1 ( 1 − P ) + O ( g 3 ) for θ = 0 , for θ = π /2 , (50) As a result, the shot-noise error on the count is Poissonian and, by repeating the experiment N e x p N_{\mathrm{exp}} N exp 1 / N e x p 1/\sqrt{N_{\mathrm{exp}}} 1/ N exp
Several features are worth emphasizing before specializing to a particular interferometer geometry. Firstly, in the off-diagonal components of the system’s reduced density matrix, the two Schwinger–Keldysh branches are identified with the two arms of the interferometer. Secondly, D \mathbb{D} D P \mathbb{P} P D \mathbb{D} D ϱ q \varrho^{q} ϱ q P \mathbb{P} P ϱ c l \varrho^{\mathrm{cl}} ϱ cl ϱ q \varrho^{q} ϱ q
This asymmetry between phase and contrast on the system side explains why, for a system of N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at
The derivation above also clarifies how a MWI could detect interactions with the environment that would be invisible to classical detectors, which rely on recoil displacements or internal-state transitions. In the heavy-probe limit, the atom’s c.o.m. motion is frozen onto its classical trajectory. As a result, wavepacket spreading and recoil are both neglected at this order. Nevertheless, the reduced density matrix is modified via D \mathbb{D} D P \mathbb{P} P L L L R R R
To find explicit expressions for Eq. (48) , it is necessary to specify a concrete interferometric geometry. Here, we limit our attention to the interferometric geometry depicted in Figure 2 . In this case, J R = − J L \mathbf{J}_R=-\mathbf{J}_L J R = − J L
J L ( t ) = m a t v 2 [ − δ ( t + T + T ′ 2 ) + δ ( t + T 2 ) + δ ( t − T 2 ) − δ ( t − T + T ′ 2 ) ] . (51) \mathbf{J}_L(t)=\frac{m_{\mathrm{at}}\mathbf{v}}{2}\left[-\delta\left(t+\frac{T+T'}{2}\right)+\delta\left(t+\frac{T}{2}\right)+\delta\left(t-\frac{T}{2}\right)-\delta\left(t-\frac{T+T'}{2}\right)\right]. \tag*{(51)} J L ( t ) = 2 m at v [ − δ ( t + 2 T + T ′ ) + δ ( t + 2 T ) + δ ( t − 2 T ) − δ ( t − 2 T + T ′ ) ] . (51) where v \mathbf{v} v T s = T + T ′ T_s=T+T' T s = T + T ′ ∣ Δ x ∣ = ∣ v ∣ T ′ / 2 |\Delta\mathbf{x}|=|\mathbf{v}|T'/2 ∣Δ x ∣ = ∣ v ∣ T ′ /2 T ≫ T ′ T\gg T' T ≫ T ′ [84 , 85 ] , Bordé interferometers [86 ] and other atom interferometer geometries [87 , 88 , 89 ] .
In the asymptotic time limit, corresponding to T → ∞ T \to\infty T → ∞ (46) ) as an integral over the four momentum transfer q q q (24) ). In this limit, the left/right particular solutions (see Eq. (36) ), defined by the external currents of Eq. (51) in the limit T ≫ T ′ T \gg T' T ≫ T ′
x ˉ α ( t ) = ∫ d t ′ ∣ t − t ′ ∣ 2 m a t J α ( t ′ ) ≈ { − Δ x / 2 for α = L and ∣ t ∣ ≤ T / 2 , + Δ x / 2 for α = R and ∣ t ∣ ≤ T / 2 , 0 otherwise. (52) \bar{x}_{\alpha}(t)=\int dt'\frac{\lvert t-t'\rvert}{2m_{\mathrm{at}}}J_{\alpha}(t')\approx
\begin{cases}
-\Delta x/2 & \text{for }\alpha=L\text{ and }\lvert t\rvert\le T/2,\\
+\Delta x/2 & \text{for }\alpha=R\text{ and }\lvert t\rvert\le T/2,\\
0 & \text{otherwise.}
\end{cases} \tag*{(52)} x ˉ α ( t ) = ∫ d t ′ 2 m at ∣ t − t ′ ∣ J α ( t ′ ) ≈ ⎩ ⎨ ⎧ − Δ x /2 + Δ x /2 0 for α = L and ∣ t ∣ ≤ T /2 , for α = R and ∣ t ∣ ≤ T /2 , otherwise. (52) The 4-momentum FT of the system charge density for this geometry evaluates to
ϱ ~ ( q ; x ˉ α , ± ) = ∫ d 4 x ϱ ( x − x ˉ α , ± ( t ) ) e i q ⋅ x ≈ F A ( q ) W ~ ( q ; x ˉ α , ± ) , (53) \widetilde{\varrho}(q;\bar{x}_{\alpha,\pm})=\int d^{4}x\,\varrho(x-\bar{x}_{\alpha,\pm}(t))e^{iq\cdot x}\approx F_{A}(q)\widetilde{W}(q;\bar{x}_{\alpha,\pm}), \tag*{(53)} ϱ ( q ; x ˉ α , ± ) = ∫ d 4 x ϱ ( x − x ˉ α , ± ( t )) e i q ⋅ x ≈ F A ( q ) W ( q ; x ˉ α , ± ) , (53) with
W ~ ( q ; x ˉ α , ± ) = { δ ˉ ( q 0 ) + T sinc ( q 0 T / 2 ) ( e i q ⋅ Δ x / 2 − 1 ) for α = L , δ ˉ ( q 0 ) + T sinc ( q 0 T / 2 ) ( e − i q ⋅ Δ x / 2 − 1 ) for α = R , (54) \widetilde{W}(q;\bar{x}_{\alpha,\pm})=
\begin{cases}
\bar{\delta}(q_{0})+T\,\operatorname{sinc}(q_{0}T/2)(e^{iq\cdot\Delta x/2}-1) & \text{for }\alpha=L,\\
\bar{\delta}(q_{0})+T\,\operatorname{sinc}(q_{0}T/2)(e^{-iq\cdot\Delta x/2}-1) & \text{for }\alpha=R,
\end{cases} \tag*{(54)} W ( q ; x ˉ α , ± ) = { δ ˉ ( q 0 ) + T sinc ( q 0 T /2 ) ( e i q ⋅ Δ x /2 − 1 ) δ ˉ ( q 0 ) + T sinc ( q 0 T /2 ) ( e − i q ⋅ Δ x /2 − 1 ) for α = L , for α = R , (54) such that the decoherence and phase shift observables may be expressed as
D = − ( g Λ D ) 2 ∫ d 4 q ∣ F A ( q ) ∣ 2 T 2 sinc 2 ( q 0 T 2 ) [ 1 − cos ( q ⋅ Δ x ) ] Im [ K ~ k ( q ) ] + O ( g 3 ) , (55) \mathbb{D}=-\left(\frac{g}{\Lambda_{D}}\right)^{2}\int d^{4}q\,\lvert F_{A}(q)\rvert^{2}T^{2}\operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right)\left[1-\cos(q\cdot\Delta x)\right]\operatorname{Im}\left[\widetilde{K}_{k}(q)\right]+\mathcal{O}(g^{3}), \tag*{(55)} D = − ( Λ D g ) 2 ∫ d 4 q ∣ F A ( q ) ∣ 2 T 2 sinc 2 ( 2 q 0 T ) [ 1 − cos ( q ⋅ Δ x ) ] Im [ K k ( q ) ] + O ( g 3 ) , (55) and
P = ( g Λ D ) 2 ∫ d 4 q ∣ F A ( q ) ∣ 2 sin ( q ⋅ Δ x 2 ) [ 2 T δ ˉ ( q 0 ) − 4 T 2 sinc 2 ( q 0 T 2 ) sin 2 ( q ⋅ Δ x 4 ) ] Im [ K ~ r ( q ) ] + O ( g 3 ) , (56) \mathbb{P}=\left(\frac{g}{\Lambda_{D}}\right)^{2}\int d^{4}q\,\lvert F_{A}(q)\rvert^{2}\sin\left(\frac{q\cdot\Delta x}{2}\right)\left[2T\bar{\delta}(q_{0})-4T^{2}\operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right)\sin^{2}\left(\frac{q\cdot\Delta x}{4}\right)\right]\operatorname{Im}\left[\widetilde{K}_{r}(q)\right]+\mathcal{O}(g^{3}), \tag*{(56)} P = ( Λ D g ) 2 ∫ d 4 q ∣ F A ( q ) ∣ 2 sin ( 2 q ⋅ Δ x ) [ 2 T δ ˉ ( q 0 ) − 4 T 2 sinc 2 ( 2 q 0 T ) sin 2 ( 4 q ⋅ Δ x ) ] Im [ K r ( q ) ] + O ( g 3 ) , (56) respectively (see Appendix B for a derivation and for a discussion on other interferometric geometries).
Eqs. (55) and (56) are the key result of this subsection. Decoherence is governed entirely by fluctuations of the environment, encoded in the noise kernel K ~ k \widetilde{K}_{k} K k (19) ), while P \mathbb{P} P K ~ r \widetilde{K}_{r} K r (17) ). The remaining factors, which depend upon the interferometer geometry via the external currents J L , R J_{L,R} J L , R 1 − cos ( q ⋅ Δ x ) 1-\cos(q\cdot\Delta x) 1 − cos ( q ⋅ Δ x ) D \mathbb{D} D [90 , 91 ] . For momentum transfers too small to resolve the arm separation, this factor goes to zero, since the environment cannot extract which-path information. In P \mathbb{P} P sin ( q ⋅ Δ x / 2 ) \sin(q\cdot\Delta x/2) sin ( q ⋅ Δ x /2 ) sinc 2 ( q 0 T 2 ) \operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right) sinc 2 ( 2 q 0 T ) 1 / T 1/T 1/ T T T T
Beyond the heavy-probe limit
Before studying multi-atom platforms, we conclude by commenting on corrections to the heavy-probe limit, which can be organized as a series expansion in 1 / m a t 1/m_{\mathrm{at}} 1/ m at q \mathbf{q} q P \mathbf{P} P σ p 2 \sigma_p^2 σ p 2 σ p \sigma_p σ p
Δ E a t = ∣ P + q ∣ 2 2 m a t − ∣ P ∣ 2 2 m a t = ∣ q ∣ 2 2 m a t + P ⋅ q m a t , (57) \Delta E_{\mathrm{at}} = \frac{|\mathbf{P}+\mathbf{q}|^2}{2m_{\mathrm{at}}} - \frac{|\mathbf{P}|^2}{2m_{\mathrm{at}}} = \frac{|\mathbf{q}|^2}{2m_{\mathrm{at}}} + \frac{\mathbf{P}\cdot\mathbf{q}}{m_{\mathrm{at}}}, \tag*{(57)} Δ E at = 2 m at ∣ P + q ∣ 2 − 2 m at ∣ P ∣ 2 = 2 m at ∣ q ∣ 2 + m at P ⋅ q , (57) and is equivalent to the energy deposited by DM. At the level of the influence action (see Eq. (46) ), and therefore of the interferometric observables, the kinetic-energy shift ∣ q ∣ 2 / 2 m a t |\mathbf{q}|^2/2m_{\mathrm{at}} ∣ q ∣ 2 /2 m at ∣ q ∣ 2 ∣ t − t ′ ∣ / 2 m a t |\mathbf{q}|^2|t-t'|/2m_{\mathrm{at}} ∣ q ∣ 2 ∣ t − t ′ ∣/2 m at ∣ q ∣ σ p ∣ t − t ′ ∣ / m a t |\mathbf{q}|\sigma_p|t-t'|/m_{\mathrm{at}} ∣ q ∣ σ p ∣ t − t ′ ∣/ m at ∣ t − t ′ ∣ ≲ T |t-t'|\lesssim T ∣ t − t ′ ∣ ≲ T ∣ t − t ′ ∣ ≳ τ e |t-t'|\gtrsim\tau_e ∣ t − t ′ ∣ ≳ τ e τ e \tau_e τ e τ e f f ≡ min ( T , τ e ) \tau_{\mathrm{eff}}\equiv\min(T,\tau_e) τ eff ≡ min ( T , τ e )
ϵ φ ∼ ∣ q ∣ 2 τ e f f m a t , ϵ σ ∼ ( ∣ q ∣ σ p τ e f f m a t ) 2 . (58) \epsilon_\varphi\sim\frac{|\mathbf{q}|^2\tau_{\mathrm{eff}}}{m_{\mathrm{at}}}, \qquad\epsilon_\sigma\sim\left(\frac{|\mathbf{q}|\sigma_p\tau_{\mathrm{eff}}}{m_{\mathrm{at}}}\right)^2. \tag*{(58)} ϵ φ ∼ m at ∣ q ∣ 2 τ eff , ϵ σ ∼ ( m at ∣ q ∣ σ p τ eff ) 2 . (58) Hence, the form of the influence action (see Eq. (43) ) entering the system’s reduced density matrix at time T T T (42) ) reads as
⟨ S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] Z 0 [ j + , j − ] ⟩ j ± = 0 = S I F [ x ‾ α , + , x ‾ β , − ] + O ( ϵ φ , ϵ σ ) . (59) \left\langle S_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-i\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+i\frac{\delta}{\delta j_-}\right]Z_0[j_+,j_-]\right\rangle_{j_\pm=0}=S_{\mathrm{IF}}[\overline{x}_{\alpha,+},\overline{x}_{\beta,-}]+\mathcal{O}(\epsilon_\varphi,\epsilon_\sigma). \tag*{(59)} ⟨ S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] Z 0 [ j + , j − ] ⟩ j ± = 0 = S IF [ x α , + , x β , − ] + O ( ϵ φ , ϵ σ ) . (59) where S I F [ x ‾ α , + , x ‾ β , − ] S_{\mathrm{IF}}[\overline{x}_{\alpha,+},\overline{x}_{\beta,-}] S IF [ x α , + , x β , − ] (43) and (46) ).
For virialized dark matter, v 0 ∼ 10 − 3 v_0\sim10^{-3} v 0 ∼ 1 0 − 3 ∣ q ∣ ≲ m χ v 0 |\mathbf{q}|\lesssim m_\chi v_0 ∣ q ∣ ≲ m χ v 0 τ e ≃ τ c = ( m χ v 0 2 ) − 1 \tau_e\simeq\tau_c=(m_\chi v_0^2)^{-1} τ e ≃ τ c = ( m χ v 0 2 ) − 1
ϵ φ ≲ m χ m a t min ( 1 , T τ c ) , ϵ σ ≲ ( σ p m a t v 0 ) 2 min ( 1 , T 2 τ c 2 ) . (60) \epsilon_\varphi\lesssim\frac{m_\chi}{m_{\mathrm{at}}}\min\left(1,\frac{T}{\tau_c}\right), \qquad\epsilon_\sigma\lesssim\left(\frac{\sigma_p}{m_{\mathrm{at}}v_0}\right)^2\min\left(1,\frac{T^2}{\tau_c^2}\right). \tag*{(60)} ϵ φ ≲ m at m χ min ( 1 , τ c T ) , ϵ σ ≲ ( m at v 0 σ p ) 2 min ( 1 , τ c 2 T 2 ) . (60) Thus, the heavy-probe approximation is controlled by m χ ≪ m a t m_\chi\ll m_{\mathrm{at}} m χ ≪ m at σ p / m a t ≪ v 0 \sigma_p/m_{\mathrm{at}}\ll v_0 σ p / m at ≪ v 0 m χ ≪ 1 G e V m_\chi\ll1\ \mathrm{GeV} m χ ≪ 1 GeV σ p ≲ ( 100 μ m ) − 1 \sigma_p\lesssim(100\ \mu\mathrm{m})^{-1} σ p ≲ ( 100 μ m ) − 1
Interferometer observables in many-atom systems
Having established the map between MWI observables and the influence action for a single-atom system, we now construct the open EFT for a system of N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at
Composite-particle interferometers (CPIs) , which employ a mesoscopic bound object (e.g. a silica or gold nanosphere) of total mass M = N a t m a t M = N_{\mathrm{at}}m_{\mathrm{at}} M = N at m at N a t N_{\mathrm{at}} N at { r n } \{\mathbf{r}_n\} { r n } [92 , 93 , 94 , 95 ] , magnetically levitated superconducting microspheres [96 ] , and molecules [97 , 98 , 99 , 100 ] .
Atom interferometers (AIs) , which employ a diffuse cloud of N a t N_{\mathrm{at}} N at 4 and initially uncorrelated atoms of mass m a t m_{\mathrm{at}} m at [86 , 103 , 104 ] and optically trapped atom interferometers [85 , 105 ] .
Before we explore each class of experiments, a few general comments are in order. The interaction Lagrangian is identical to the single-atom case (see Eqs. (11) and (13) ). The multi-atom generalization enters exclusively through the action of the system operators in second-quantization on the many-body state. Recalling that the quantized system operator appearing in the interaction Lagrangian is O ^ s ( y ) = ∫ d 3 r ϱ ( r ) ∑ α ψ ^ α † ( y − r ) ψ ^ α ( y − r ) \hat{O}_{s}(y) = \int d^{3}r\,\varrho(\mathbf{r}) \sum_{\alpha} \hat{\psi}_{\alpha}^{\dagger}(y-\mathbf{r})\hat{\psi}_{\alpha}(y-\mathbf{r}) O ^ s ( y ) = ∫ d 3 r ϱ ( r ) ∑ α ψ ^ α † ( y − r ) ψ ^ α ( y − r ) 4
O ^ s ( y ) ∏ n = 1 N a t ψ ^ α † ( x + r n ) ∣ 0 ⟩ s = [ ∑ n = 1 N a t ϱ ( y − x − r n ) ] ∏ n = 1 N a t ψ ^ α † ( x + r n ) ∣ 0 ⟩ s (61) \hat{O}_{s}(y)\prod_{n=1}^{N_{\mathrm{at}}}\hat{\psi}_{\alpha}^{\dagger}(\mathbf{x}+\mathbf{r}_{n})|0\rangle_{s} = \left[\sum_{n=1}^{N_{\mathrm{at}}}\varrho(y-\mathbf{x}-\mathbf{r}_{n})\right]\prod_{n=1}^{N_{\mathrm{at}}}\hat{\psi}_{\alpha}^{\dagger}(\mathbf{x}+\mathbf{r}_{n})|0\rangle_{s} \tag*{(61)} O ^ s ( y ) n = 1 ∏ N at ψ ^ α † ( x + r n ) ∣0 ⟩ s = [ n = 1 ∑ N at ϱ ( y − x − r n ) ] n = 1 ∏ N at ψ ^ α † ( x + r n ) ∣0 ⟩ s (61) such that O s → ∑ n = 1 N a t ϱ ( y − x − r n ) O_s \to\sum_{n=1}^{N_{\mathrm{at}}}\varrho(y-\mathbf{x}-\mathbf{r}_{n}) O s → ∑ n = 1 N at ϱ ( y − x − r n )
As shown in the following subsections, many-body effects enter entirely through the decoherence and phase form factors F D ( q ) F_{\mathrm{D}}(\mathbf{q}) F D ( q ) F P ( q ) F_{\mathrm{P}}(\mathbf{q}) F P ( q )
D = − ( g Λ D ) 2 ∫ d 4 q ∣ F D ( q ) ∣ 2 T 2 sinc 2 ( q 0 T 2 ) [ 1 − cos ( q ⋅ Δ x ) ] Im [ K ~ κ ( q ) ] + O ( g 3 ) , P = ( g Λ D ) 2 ∫ d 4 q ∣ F P ( q ) ∣ 2 sin 2 ( q ⋅ Δ x 2 ) [ 2 T δ ~ ( q 0 ) − 4 T 2 sinc 2 ( q 0 T 2 ) sin 2 ( q ⋅ Δ x 4 ) ] Im [ K ~ r ( q ) ] + O ( g 3 ) . (62) \begin{aligned}
\mathbb{D} &= -\left(\frac{g}{\Lambda_{\mathrm{D}}}\right)^{2}\int d^{4}q\,|F_{\mathrm{D}}(\mathbf{q})|^{2}T^{2}\operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right)\left[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})\right]\operatorname{Im}\left[\widetilde{K}_{\kappa}(\mathbf{q})\right]+\mathcal{O}(g^{3}),\\
\mathbb{P} &= \left(\frac{g}{\Lambda_{\mathrm{D}}}\right)^{2}\int d^{4}q\,|F_{\mathrm{P}}(\mathbf{q})|^{2}\sin^{2}\left(\frac{\mathbf{q}\cdot\Delta\mathbf{x}}{2}\right)\left[2T\widetilde{\delta}(q_{0})-4T^{2}\operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right)\sin^{2}\left(\frac{\mathbf{q}\cdot\Delta\mathbf{x}}{4}\right)\right]\operatorname{Im}\left[\widetilde{K}_{r}(\mathbf{q})\right]+\mathcal{O}(g^{3}). \tag*{(62)}
\end{aligned} D P = − ( Λ D g ) 2 ∫ d 4 q ∣ F D ( q ) ∣ 2 T 2 sinc 2 ( 2 q 0 T ) [ 1 − cos ( q ⋅ Δ x ) ] Im [ K κ ( q ) ] + O ( g 3 ) , = ( Λ D g ) 2 ∫ d 4 q ∣ F P ( q ) ∣ 2 sin 2 ( 2 q ⋅ Δ x ) [ 2 T δ ( q 0 ) − 4 T 2 sinc 2 ( 2 q 0 T ) sin 2 ( 4 q ⋅ Δ x ) ] Im [ K r ( q ) ] + O ( g 3 ) . (62) Notably, for composite-particle interferometers (CPIs), the decoherence and phase shift form factors are identical. They both depend on the Debye-Waller form factor e − 2 W ( q ) e^{-2W(\mathbf{q})} e − 2 W ( q ) ∣ F s p h e r e ( q ) ∣ 2 ≤ 1 |F_{\mathrm{sphere}}(\mathbf{q})|^{2}\leq1 ∣ F sphere ( q ) ∣ 2 ≤ 1 N a t ≫ 1 N_{\mathrm{at}}\gg1 N at ≫ 1 N a t 2 N_{\mathrm{at}}^{2} N at 2 5 F A ( q ) F_A(\mathbf{q}) F A ( q ) ∣ F c l o u d ( q ) ∣ 2 ≤ 1 |F_{\rm cloud}(\mathbf{q})|^2 \le1 ∣ F cloud ( q ) ∣ 2 ≤ 1 N a t N_{\rm at} N at
Table 1. Many-body factors relevant to the results of this section, where N a t N_{\rm at} N at e − 2 W ( q ) e^{-2W(\mathbf{q})} e − 2 W ( q ) (71) ) and ∣ F s p h e r e ( q ) ∣ 2 |F_{\rm sphere}(\mathbf{q})|^2 ∣ F sphere ( q ) ∣ 2 (74) ), where the latter depends on the (assumed to be spherical) geometry of the mesoscopic object. For atom interferometers (AIs), instead, the decoherence and phase shift form factors differ: they both depend on the atomic form factor F A ( q ) F_A(\mathbf{q}) F A ( q ) (13) ), but only the phase shift form factor is sensitive to the cloud form factor ∣ F c l o u d ( q ) ∣ 2 |F_{\rm cloud}(\mathbf{q})|^2 ∣ F cloud ( q ) ∣ 2 (89) ). For both classes of MWI, phase and decoherence can be extracted from the expected number of counts in two output ports, corresponding to projective measurements in the qubit basis. Assuming shot-noise limited sensitivity, the error on D A I \mathcal{D}_{\rm AI} D AI P A I \mathcal{P}_{\rm AI} P AI 1 / N a t 1/\sqrt{N_{\rm at}} 1/ N at D C P I \mathcal{D}_{\rm CPI} D CPI P C P I \mathcal{P}_{\rm CPI} P CPI N a t N_{\rm at} N at 1 / N e x p 1/\sqrt{N_{\rm exp}} 1/ N exp N e x p N_{\rm exp} N exp
Composite-particle interferometers
In a composite-particle interferometer (CPI), the c.o.m. of the mesoscopic object is directly entangled with some measured degree of freedom, while the relative coordinates of the probe’s constituents remain unobserved. It is therefore natural to write the worldline of the n n n
X n , ± ( t ) = X ± ( t ) + r n , ± ( t ) , X ± ( t ) = 1 N a t ∑ n = 1 N a t X n , ± ( t ) , ∑ n = 1 N a t m a t r n , ± ( t ) = 0 , (63) \mathbf{X}_{n,\pm}(t)=\mathbf{X}_{\pm}(t)+\mathbf{r}_{n,\pm}(t),\qquad\mathbf{X}_{\pm}(t)=\frac{1}{N_{\rm at}}\sum_{n=1}^{N_{\rm at}}\mathbf{X}_{n,\pm}(t),\qquad\sum_{n=1}^{N_{\rm at}}m_{\rm at}\mathbf{r}_{n,\pm}(t)=0, \tag*{(63)} X n , ± ( t ) = X ± ( t ) + r n , ± ( t ) , X ± ( t ) = N at 1 n = 1 ∑ N at X n , ± ( t ) , n = 1 ∑ N at m at r n , ± ( t ) = 0 , (63) where X ± \mathcal{X}_{\pm} X ± r n , ± \mathbf{r}_{n,\pm} r n , ± n n n 1 ≤ n ≤ N a t 1 \le n \le N_{\rm at} 1 ≤ n ≤ N at
ϱ ~ C P I ( q , t ; X ± + { r n , ± } ) = ϱ ~ ( q , t ; X ± ) ∑ n = 1 N a t e − i q ⋅ r n , ± ( t ) . (64) \widetilde{\varrho}_{\rm CPI}\left(\mathbf{q},t;\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\}\right)=\widetilde{\varrho}\left(\mathbf{q},t;\mathcal{X}_{\pm}\right)\sum_{n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot\mathbf{r}_{n,\pm}(t)}. \tag*{(64)} ϱ CPI ( q , t ; X ± + { r n , ± } ) = ϱ ( q , t ; X ± ) n = 1 ∑ N at e − i q ⋅ r n , ± ( t ) . (64) Here, ϱ ~ ( q , t ; X ± ) \widetilde{\varrho}(\mathbf{q},t;\mathcal{X}_{\pm}) ϱ ( q , t ; X ± ) X ± → X ± \mathbf{X}_{\pm}\to\mathcal{X}_{\pm} X ± → X ± (44) ).
For translationally invariant binding interactions, which in a crystalline solid correspond to the lattice potential whose harmonic expansion yields the solid’s phonon modes [106 ] , this change of variables separates the free action into a c.o.m. sector and N a t − 1 N_{\rm at}-1 N at − 1 [107 ] . As a result, the c.o.m. and relative coordinates sectors decouple in the absence of system-environment interactions, and the external currents defining the left and right arms of the interferometer couple only to the c.o.m. coordinate. Therefore, after tracing over the relative coordinate sector, the 3-momentum system current bilinears entering the influence action read as
⟨ ϱ ~ C P I ( q , t ; X ± + { r n , ± } ) ϱ ~ C P I ( − q , t ′ ; X ± + { r n , ± } ) ⟩ r e l = ϱ ~ ( q , t ; X ± ) ϱ ~ ( − q , t ′ ; X ± ) S r e l ± ± ( q , t − t ′ ) , (65) \left\langle\widetilde{\varrho}_{\rm CPI}\left(\mathbf{q},t;\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\}\right)\widetilde{\varrho}_{\rm CPI}\left(-\mathbf{q},t';\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\}\right)\right\rangle_{\rm rel}
=\widetilde{\varrho}\left(\mathbf{q},t;\mathcal{X}_{\pm}\right)\widetilde{\varrho}\left(-\mathbf{q},t';\mathcal{X}_{\pm}\right)S_{\rm rel}^{\pm\pm}(\mathbf{q},t-t'), \tag*{(65)} ⟨ ϱ CPI ( q , t ; X ± + { r n , ± } ) ϱ CPI ( − q , t ′ ; X ± + { r n , ± } ) ⟩ rel = ϱ ( q , t ; X ± ) ϱ ( − q , t ′ ; X ± ) S rel ±± ( q , t − t ′ ) , (65) ⟨ ϱ ~ C P I ( q , t ; X ± + { r n , ± } ) ϱ ~ C P I ( − q , t ′ ; X ∓ + { r n , ∓ } ) ⟩ r e l = ϱ ~ ( q , t ; X ± ) ϱ ~ ( − q , t ′ ; X ∓ ) S r e l ± ∓ ( q , t − t ′ ) , (66) \left\langle\widetilde{\varrho}_{\rm CPI}\left(\mathbf{q},t;\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\}\right)\widetilde{\varrho}_{\rm CPI}\left(-\mathbf{q},t';\mathcal{X}_{\mp}+\{\mathbf{r}_{n,\mp}\}\right)\right\rangle_{\rm rel}
=\widetilde{\varrho}\left(\mathbf{q},t;\mathcal{X}_{\pm}\right)\widetilde{\varrho}\left(-\mathbf{q},t';\mathcal{X}_{\mp}\right)S_{\rm rel}^{\pm\mp}(\mathbf{q},t-t'), \tag*{(66)} ⟨ ϱ CPI ( q , t ; X ± + { r n , ± } ) ϱ CPI ( − q , t ′ ; X ∓ + { r n , ∓ } ) ⟩ rel = ϱ ( q , t ; X ± ) ϱ ( − q , t ′ ; X ∓ ) S rel ±∓ ( q , t − t ′ ) , (66) where the common relative-sector dependent quantity is the object’s structure factor:
S r e l ± ± ( q , t − t ′ ) = ⟨ ∑ m , n = 1 N a t e − i q ⋅ [ r n , ± ( t ) − r m , ± ( t ′ ) ] ⟩ r e l , S r e l ± ∓ ( q , t − t ′ ) = ⟨ ∑ m , n = 1 N a t e − i q ⋅ [ r n , ± ( t ) − r m , ∓ ( t ′ ) ] ⟩ r e l . (67) S_{\rm rel}^{\pm\pm}(\mathbf{q},t-t')=\left\langle\sum_{m,n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot[\mathbf{r}_{n,\pm}(t)-\mathbf{r}_{m,\pm}(t')]}\right\rangle_{\rm rel},\qquad
S_{\rm rel}^{\pm\mp}(\mathbf{q},t-t')=\left\langle\sum_{m,n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot[\mathbf{r}_{n,\pm}(t)-\mathbf{r}_{m,\mp}(t')]}\right\rangle_{\rm rel}. \tag*{(67)} S rel ±± ( q , t − t ′ ) = ⟨ m , n = 1 ∑ N at e − i q ⋅ [ r n , ± ( t ) − r m , ± ( t ′ )] ⟩ rel , S rel ±∓ ( q , t − t ′ ) = ⟨ m , n = 1 ∑ N at e − i q ⋅ [ r n , ± ( t ) − r m , ∓ ( t ′ )] ⟩ rel . (67) To find a closed-form expression for the structure factor, we rewrite the relative coordinates as
r n , ± ( t ) = r ‾ n + u n , ± ( t ) , (68) \mathbf{r}_{n,\pm}(t)=\overline{\mathbf{r}}_{n}+\mathbf{u}_{n,\pm}(t), \tag*{(68)} r n , ± ( t ) = r n + u n , ± ( t ) , (68) where r ‾ n \overline{\mathbf{r}}_{n} r n u n , ± ( t ) \mathbf{u}_{n,\pm}(t) u n , ± ( t ) ⟨ u n , ± ( t ) ⟩ r e l = 0 \langle\mathbf{u}_{n,\pm}(t)\rangle_{\rm rel}=0 ⟨ u n , ± ( t ) ⟩ rel = 0 [108 ] .
Hence, using Eq. (68) , Eq. (67) may be expressed as
S r e l ± ± ( q , t − t ′ ) = ∑ m , n = 1 N a t e − i q ⋅ ( r ‾ n − r ‾ m ) ⟨ e − i q ⋅ ( u n , ± ( t ) − u m , ± ( t ′ ) ) ⟩ r e l = ∑ m , n = 1 N a t e − i q ⋅ ( r ‾ n − r ‾ m ) [ ⟨ e − i q ⋅ u n ( t ) ⟩ r e l ⟨ e i q ⋅ u m ( t ′ ) ⟩ r e l + O ( phonon excitations ) ] = e − 2 W ( q ) ∑ m , n = 1 N a t e − i q ⋅ ( r ‾ n − r ‾ m ) ⏟ S r e l ( q ) + O ( phonon excitations ) . (69) \begin{aligned}
S_{\rm rel}^{\pm\pm}(\mathbf{q},t-t')&=\sum_{m,n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot(\overline{\mathbf{r}}_{n}-\overline{\mathbf{r}}_{m})}\left\langle e^{-i\mathbf{q}\cdot(\mathbf{u}_{n,\pm}(t)-\mathbf{u}_{m,\pm}(t'))}\right\rangle_{\rm rel}\\
&=\sum_{m,n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot(\overline{\mathbf{r}}_{n}-\overline{\mathbf{r}}_{m})}\left[\left\langle e^{-i\mathbf{q}\cdot\mathbf{u}_{n}(t)}\right\rangle_{\rm rel}\left\langle e^{i\mathbf{q}\cdot\mathbf{u}_{m}(t')}\right\rangle_{\rm rel}+\mathcal{O}(\text{phonon excitations})\right]\\
&=e^{-2W(\mathbf{q})}\underbrace{\sum_{m,n=1}^{N_{\rm at}}e^{-i\mathbf{q}\cdot(\overline{\mathbf{r}}_{n}-\overline{\mathbf{r}}_{m})}}_{S_{\rm rel}(\mathbf{q})}+\mathcal{O}(\text{phonon excitations}) . \tag*{(69)}
\end{aligned} S rel ±± ( q , t − t ′ ) = m , n = 1 ∑ N at e − i q ⋅ ( r n − r m ) ⟨ e − i q ⋅ ( u n , ± ( t ) − u m , ± ( t ′ )) ⟩ rel = m , n = 1 ∑ N at e − i q ⋅ ( r n − r m ) [ ⟨ e − i q ⋅ u n ( t ) ⟩ rel ⟨ e i q ⋅ u m ( t ′ ) ⟩ rel + O ( phonon excitations ) ] = e − 2 W ( q ) S rel ( q ) m , n = 1 ∑ N at e − i q ⋅ ( r n − r m ) + O ( phonon excitations ) . (69) and similarly
S r e l ± ∓ ( q , t − t ′ ) = S r e l ( q ) + O ( phonon excitations ) , (70) S_{\mathrm{rel}}^{\pm\mp}(\mathbf{q},t-t')=S_{\mathrm{rel}}(\mathbf{q})+\mathcal{O}(\text{phonon excitations}) , \tag*{(70)} S rel ±∓ ( q , t − t ′ ) = S rel ( q ) + O ( phonon excitations ) , (70) where the first term is the time-independent elastic contribution. The neglected terms describe inelastic processes, in which energy deposited by the environment excites phonons in the object [109 ] .
Here, e − 2 W ( q ) e^{-2W(\mathbf{q})} e − 2 W ( q ) [110 ] , which quantifies the scattering rate suppression due to the mean relative spatial uncertainty of the atoms, whether from thermal or quantum fluctuations. For an isotropic and homogeneous solid at inverse temperature β \beta β ω p h \omega_{\mathrm{ph}} ω ph
2 W ( q ) = ⟨ ( q ⋅ u n ( t ) ) 2 ⟩ r e l = ∣ q ∣ 2 σ u 2 , with σ u 2 ≃ 1 ω p h m a t max ( 1 , 1 β ω p h ) , (71) 2W(\mathbf{q})=\left\langle\left(\mathbf{q}\cdot\mathbf{u}_{n}(t)\right)^{2}\right\rangle_{\mathrm{rel}}=|\mathbf{q}|^{2}\sigma_{u}^{2},\quad\text{with}\quad\sigma_{u}^{2}\simeq\frac{1}{\omega_{\mathrm{ph}}m_{\mathrm{at}}}\max\left(1,\frac{1}{\beta\omega_{\mathrm{ph}}}\right), \tag*{(71)} 2 W ( q ) = ⟨ ( q ⋅ u n ( t ) ) 2 ⟩ rel = ∣ q ∣ 2 σ u 2 , with σ u 2 ≃ ω ph m at 1 max ( 1 , β ω ph 1 ) , (71) where σ u 2 \sigma_{u}^{2} σ u 2 ∣ q ∣ σ u ≫ 1 |\mathbf{q}|\sigma_{u}\gg1 ∣ q ∣ σ u ≫ 1
The sum over the rigid positions of the atoms in the target, instead, depends on the geometry of the target, and can be expressed as
∑ m , n = 1 N a t e − i q ⋅ ( r n − r m ) = N a t + ∑ n ≠ m e − i q ⋅ ( r n − r m ) , (72) \sum_{m,n=1}^{N_{\mathrm{at}}}e^{-i\mathbf{q}\cdot(\mathbf{r}_{n}-\mathbf{r}_{m})}=N_{\mathrm{at}}+\sum_{n\ne m}e^{-i\mathbf{q}\cdot(\mathbf{r}_{n}-\mathbf{r}_{m})}, \tag*{(72)} m , n = 1 ∑ N at e − i q ⋅ ( r n − r m ) = N at + n = m ∑ e − i q ⋅ ( r n − r m ) , (72) where the first and second terms account for incoherent and coherent effects, respectively. In the large N a t N_{\mathrm{at}} N at r c r_{c} r c n s p h e r e ( r ) = 3 4 π r c 3 Θ ( r c − ∣ r ∣ ) n_{\mathrm{sphere}}(\mathbf{r})=\frac{3}{4\pi r_{c}^{3}}\Theta(r_{c}-|\mathbf{r}|) n sphere ( r ) = 4 π r c 3 3 Θ ( r c − ∣ r ∣ )
∑ n ≠ m e − i q ⋅ ( r n − r m ) ≈ N a t ( N a t − 1 ) ∣ ∫ d 3 r n s p h e r e ( r ) e − i q ⋅ r ∣ 2 = N a t ( N a t − 1 ) ∣ F s p h e r e ( q ) ∣ 2 , (73) \sum_{n\ne m}e^{-i\mathbf{q}\cdot(\mathbf{r}_{n}-\mathbf{r}_{m})}\approx N_{\mathrm{at}}(N_{\mathrm{at}}-1)\left|\int d^{3}r\,n_{\mathrm{sphere}}(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\right|^{2}=N_{\mathrm{at}}(N_{\mathrm{at}}-1)|F_{\mathrm{sphere}}(\mathbf{q})|^{2}, \tag*{(73)} n = m ∑ e − i q ⋅ ( r n − r m ) ≈ N at ( N at − 1 ) ∫ d 3 r n sphere ( r ) e − i q ⋅ r 2 = N at ( N at − 1 ) ∣ F sphere ( q ) ∣ 2 , (73) where the geometry is captured by the form factor
∣ F s p h e r e ( q ) ∣ 2 = ( 3 j 1 ( ∣ q ∣ r c ) ∣ q ∣ r c ) 2 , (74) |F_{\mathrm{sphere}}(\mathbf{q})|^{2}=\left(\frac{3j_{1}(|\mathbf{q}|r_{c})}{|\mathbf{q}|r_{c}}\right)^{2}, \tag*{(74)} ∣ F sphere ( q ) ∣ 2 = ( ∣ q ∣ r c 3 j 1 ( ∣ q ∣ r c ) ) 2 , (74) with j 1 j_{1} j 1
Neglecting phonon excitations, which require non-zero energy deposition, the classical-quantum bilinears appearing in the influence action hence take the simplified form
⟨ ρ ^ C P I q ( q , t ; X ± + { r n , ± } ) ρ ^ C P I c l ( − q , t ′ ; X ± + { r n , ± } ) ⟩ r e l = ρ ^ q ( q , t ; X ± ) ρ ^ c l ( − q , t ′ ; X ± ) S r e l ( q ) , (75) \left\langle\hat{\rho}_{\mathrm{CPI}}^{q}(\mathbf{q},t;\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\})\hat{\rho}_{\mathrm{CPI}}^{\mathrm{cl}}(-\mathbf{q},t';\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\})\right\rangle_{\mathrm{rel}}=\hat{\rho}^{q}(\mathbf{q},t;\mathcal{X}_{\pm})\hat{\rho}^{\mathrm{cl}}(-\mathbf{q},t';\mathcal{X}_{\pm})S_{\mathrm{rel}}(\mathbf{q}), \tag*{(75)} ⟨ ρ ^ CPI q ( q , t ; X ± + { r n , ± }) ρ ^ CPI cl ( − q , t ′ ; X ± + { r n , ± }) ⟩ rel = ρ ^ q ( q , t ; X ± ) ρ ^ cl ( − q , t ′ ; X ± ) S rel ( q ) , (75) ⟨ ρ ^ C P I q ( q , t ; X ± + { r n , ± } ) ρ ^ C P I q ( − q , t ′ ; X ± + { r n , ± } ) ⟩ r e l = ρ ^ q ( q , t ; X ± ) ρ ^ q ( − q , t ′ ; X ± ) S r e l ( q ) . (76) \left\langle\hat{\rho}_{\mathrm{CPI}}^{q}(\mathbf{q},t;\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\})\hat{\rho}_{\mathrm{CPI}}^{q}(-\mathbf{q},t';\mathcal{X}_{\pm}+\{\mathbf{r}_{n,\pm}\})\right\rangle_{\mathrm{rel}}=\hat{\rho}^{q}(\mathbf{q},t;\mathcal{X}_{\pm})\hat{\rho}^{q}(-\mathbf{q},t';\mathcal{X}_{\pm})S_{\mathrm{rel}}(\mathbf{q}). \tag*{(76)} ⟨ ρ ^ CPI q ( q , t ; X ± + { r n , ± }) ρ ^ CPI q ( − q , t ′ ; X ± + { r n , ± }) ⟩ rel = ρ ^ q ( q , t ; X ± ) ρ ^ q ( − q , t ′ ; X ± ) S rel ( q ) . (76) From this point, the heavy-probe derivation of the observables proceeds exactly as for the single-atom MWI of section III A, by replacing the atom with the c.o.m. of the CPI, such that m a t → M m_{\mathrm{at}} \to M m at → M (47) , with the influence functional evaluated on the classical c.o.m. trajectories X ± \mathcal{X}_{\pm} X ± M = N a t m a t M = N_{\mathrm{at}}m_{\mathrm{at}} M = N at m at F D ( q ) = F P ( q ) = F C P I ( q ) F_{\mathbb{D}}(\mathbf{q}) = F_{\mathbb{P}}(\mathbf{q}) = F_{\mathrm{CPI}}(\mathbf{q}) F D ( q ) = F P ( q ) = F CPI ( q )
∣ F C P I ( q ) ∣ 2 = e − 2 W ( q ) N a t [ 1 + ( N a t − 1 ) ∣ F s p h e r e ( q ) ∣ 2 ] . (77) \left|F_{\mathrm{CPI}}(\mathbf{q})\right|^2 = e^{-2W(\mathbf{q})}N_{\mathrm{at}}\left[1 + (N_{\mathrm{at}} - 1)\left|F_{\mathrm{sphere}}(\mathbf{q})\right|^2\right]. \tag*{(77)} ∣ F CPI ( q ) ∣ 2 = e − 2 W ( q ) N at [ 1 + ( N at − 1 ) ∣ F sphere ( q ) ∣ 2 ] . (77) Each factor has a simple interpretation: F s p h e r e F_{\mathrm{sphere}} F sphere e − 2 W e^{-2W} e − 2 W 7
The mapping between the observables P \mathbb{P} P D \mathbb{D} D (49) ), e.g.,
Expected number of counts in the ‘−’ port = { 1 2 D C P I + O ( g 3 ) for θ = 0 , 1 2 ( 1 − P C P I ) + O ( g 3 ) for θ = π / 2. (78) \text{Expected number of counts in the ‘−’ port} =
\begin{cases}
\frac{1}{2}\mathbb{D}_{\mathrm{CPI}} + \mathcal{O}(g^3) & \text{for }\theta= 0, \\
\frac{1}{2}(1 - \mathbb{P}_{\mathrm{CPI}}) + \mathcal{O}(g^3) & \text{for }\theta= \pi/2.
\end{cases} \tag*{(78)} Expected number of counts in the ‘−’ port = { 2 1 D CPI + O ( g 3 ) 2 1 ( 1 − P CPI ) + O ( g 3 ) for θ = 0 , for θ = π /2. (78) Since the entire target either clicks in the left or right port, the shot-noise error on the count is Poissonian. Hence, by repeating the experiment N e x p N_{\mathrm{exp}} N exp D C P I \mathbb{D}_{\mathrm{CPI}} D CPI P C P I \mathbb{P}_{\mathrm{CPI}} P CPI 1 / N e x p 1/\sqrt{N_{\mathrm{exp}}} 1/ N exp N a t = 1 N_{\mathrm{at}} = 1 N at = 1 F s p h e r e = 1 F_{\mathrm{sphere}} = 1 F sphere = 1 [11 , 12 , 13 , 14 ] . Notably, these form factors apply for both Markovian and non-Markovian dynamics, and are independent of the specific form of the DM operator and the DM occupation number.
Finally, note that our treatment neglected environment-induced phonon excitations which, if included, modify the form of the structure factor (see Eq. (69) ). We leave a study in this direction to future work.
Atom interferometers
In an AI, N a t N_{\mathrm{at}} N at
Typically, the atoms are initially prepared in some trap and cooled to some temperature, such that the center of mass of each wavepacket is neither at rest nor located at the center of the trap. As a result, the atom cloud, which we assume to be dilute for simplicity, is expected to grow considerably during the interferometric sequence [111 ] . However, it is possible to suppress atom cloud expansion during the interferometric sequence via matter-wave lensing techniques, such as delta-kick collimation, which effectively reduces the momentum spread of the cloud at the expense of a larger initial size [112 , 113 ] . For simplicity, we therefore specialize to atomic fountains employing matter-wave lensing, which allow the atoms to freely propagate in spacetime during the interferometric sequence with negligible cloud expansion. Assuming that the c.o.m. of the n n n r ˉ n \bar{r}_n r ˉ n (68) ), with P ( { r ˉ n } ) P(\{\bar{r}_n\}) P ({ r ˉ n }) { r ˉ n } \{\bar{r}_n\} { r ˉ n }
ρ ^ s ( t i ) = ( ∏ n = 1 N a t ∫ d 3 r ˉ n P ( { r ˉ n } ) ) ⨂ n = 1 N a t ∣ Φ n ⟩ ⟨ Φ n ∣ (79) \hat{\rho}_s(t_i) = \left(\prod_{n=1}^{N_{\rm at}} \int\mathrm{d}^3\bar{r}_n P(\{\bar{r}_n\})\right) \bigotimes_{n=1}^{N_{\rm at}} \lvert\Phi_n\rangle\langle\Phi_n\rvert \tag*{(79)} ρ ^ s ( t i ) = ( n = 1 ∏ N at ∫ d 3 r ˉ n P ({ r ˉ n }) ) n = 1 ⨂ N at ∣ Φ n ⟩ ⟨ Φ n ∣ (79) where
∣ Φ n ⟩ = 1 2 ∫ d 3 y Φ ( y ) ( ψ L † ( y + r ˉ n ) + ψ R † ( y + r ˉ n ) ) ∣ 0 ⟩ s (80) \lvert\Phi_n\rangle= \frac{1}{\sqrt{2}}\int\mathrm{d}^3 y\,\Phi(\mathbf{y})\left(\psi_L^\dagger(\mathbf{y}+\bar{\mathbf{r}}_n)+\psi_R^\dagger(\mathbf{y}+\bar{\mathbf{r}}_n)\right)\lvert0\rangle_s \tag*{(80)} ∣ Φ n ⟩ = 2 1 ∫ d 3 y Φ ( y ) ( ψ L † ( y + r ˉ n ) + ψ R † ( y + r ˉ n ) ) ∣ 0 ⟩ s (80) is the single-atom state centered at r ˉ n \bar{\mathbf{r}}_n r ˉ n 8 r c r_c r c
P ( { r ˉ n } ) = ∏ n = 1 N a t P ( r ˉ n ) , (81) P(\{\bar{\mathbf{r}}_n\}) = \prod_{n=1}^{N_{\rm at}} P(\bar{\mathbf{r}}_n), \tag*{(81)} P ({ r ˉ n }) = n = 1 ∏ N at P ( r ˉ n ) , (81) where P ( r ˉ n ) P(\bar{\mathbf{r}}_n) P ( r ˉ n ) n n n 9
As a result, the influence action depends on N a t N_{\rm at} N at { X n , ± ( t ) } \{X_{n,\pm}(t)\} { X n , ± ( t )}
ρ c l ( x ; { X n , ± + r ˉ n } ) = 1 2 ∑ n = 1 N a t [ ϱ ( x − X n , + ( t ) − r ˉ n ) + ϱ ( x − X n , − ( t ) − r ˉ n ) ] , ρ q ( x ; { X n , ± + r ˉ n } ) = ∑ n = 1 N a t [ ϱ ( x − X n , + ( t ) − r ˉ n ) − ϱ ( x − X n , − ( t ) − r ˉ n ) ] , (82) \begin{aligned}
\rho^{\rm cl}(x;\{X_{n,\pm}+\bar{r}_n\}) &= \frac{1}{2}\sum_{n=1}^{N_{\rm at}}\left[\varrho(x-X_{n,+}(t)-\bar{r}_n)+\varrho(x-X_{n,-}(t)-\bar{r}_n)\right],\\
\rho^{\rm q}(x;\{X_{n,\pm}+\bar{r}_n\}) &= \sum_{n=1}^{N_{\rm at}}\left[\varrho(x-X_{n,+}(t)-\bar{r}_n)-\varrho(x-X_{n,-}(t)-\bar{r}_n)\right], \tag*{(82)}
\end{aligned} ρ cl ( x ; { X n , ± + r ˉ n }) ρ q ( x ; { X n , ± + r ˉ n }) = 2 1 n = 1 ∑ N at [ ϱ ( x − X n , + ( t ) − r ˉ n ) + ϱ ( x − X n , − ( t ) − r ˉ n ) ] , = n = 1 ∑ N at [ ϱ ( x − X n , + ( t ) − r ˉ n ) − ϱ ( x − X n , − ( t ) − r ˉ n ) ] , (82) and the ( { α } , { β } ) (\{\alpha\},\{\beta\}) ({ α } , { β }) N a t N_{\rm at} N at t f t_f t f { α 1 , α 2 , … , α N a t } \{\alpha_1,\alpha_2,\ldots,\alpha_{N_{\rm at}}\} { α 1 , α 2 , … , α N at }
⟨ { α } ∣ ρ ^ s ( t f ) ∣ { β } ⟩ = ρ s { α } , { β } ( t f ) ≡ 1 2 N a t ∏ n = 1 N a t ( ∫ d 3 x i , n , + ∫ d 3 x i , n , − Φ ( x i , n , + ) Φ ∗ ( x i , n , − ) ) ∏ n = 1 N a t ( ∫ d 3 x f , n ∫ x i , n , + x f , n D X n , + ∫ x i , n , − x f , n D X n , − ) exp { i ∑ n = 1 N a t S s [ X n , + , J α n , + ] − i ∑ n = 1 N a t S s [ X n , − , J β n , − ] } exp { i S I F [ { X n , + + r ˉ n } , { X n , − + r ˉ n } ] } , (83) \langle\{\alpha\}|\hat{\rho}_s(t_f)|\{\beta\}\rangle=\rho_s^{\{\alpha\},\{\beta\}}(t_f)\equiv\frac{1}{2^{N_{\rm at}}}\prod_{n=1}^{N_{\rm at}}\left(\int\mathrm{d}^3x_{i,n,+}\int\mathrm{d}^3x_{i,n,-}\,\Phi(x_{i,n,+})\Phi^*(x_{i,n,-})\right)
\prod_{n=1}^{N_{\rm at}}\left(\int\mathrm{d}^3x_{f,n}\int_{x_{i,n,+}}^{x_{f,n}}\mathcal{D}X_{n,+}\int_{x_{i,n,-}}^{x_{f,n}}\mathcal{D}X_{n,-}\right)
\exp\left\{i\sum_{n=1}^{N_{\rm at}}S_s[X_{n,+},J_{\alpha_n,+}]-i\sum_{n=1}^{N_{\rm at}}S_s[X_{n,-},J_{\beta_n,-}]\right\}
\exp\left\{iS_{\rm IF}[\{X_{n,+}+\bar{r}_n\},\{X_{n,-}+\bar{r}_n\}]\right\}, \tag*{(83)} ⟨{ α } ∣ ρ ^ s ( t f ) ∣ { β }⟩ = ρ s { α } , { β } ( t f ) ≡ 2 N at 1 n = 1 ∏ N at ( ∫ d 3 x i , n , + ∫ d 3 x i , n , − Φ ( x i , n , + ) Φ ∗ ( x i , n , − ) ) n = 1 ∏ N at ( ∫ d 3 x f , n ∫ x i , n , + x f , n D X n , + ∫ x i , n , − x f , n D X n , − ) exp { i n = 1 ∑ N at S s [ X n , + , J α n , + ] − i n = 1 ∑ N at S s [ X n , − , J β n , − ] } exp { i S IF [{ X n , + + r ˉ n } , { X n , − + r ˉ n }] } , (83) where, in direct analogy to Eq. (29) , x i , n , ± x_{i,n,\pm} x i , n , ± x f , n x_{f,n} x f , n n n n J α n , ± \mathbf{J}_{\alpha_n,\pm} J α n , ± n n n (83) has the same structure as the single-atom path integral. The atom’s initial position r ˉ n \bar{r}_n r ˉ n t f = − t i = T → ∞ t_f=-t_i=T\to\infty t f = − t i = T → ∞ X n , ± ( t ′ ) → x ˉ n , ± ( t ′ ) X_{n,\pm}(t')\to\bar{x}_{n,\pm}(t') X n , ± ( t ′ ) → x ˉ n , ± ( t ′ ) x ˉ n , ± ( t ′ ) \bar{x}_{n,\pm}(t') x ˉ n , ± ( t ′ ) J α n , ± ( t ′ ) \mathbf{J}_{\alpha_n,\pm}(t') J α n , ± ( t ′ )
Differently from CPIs, typical AI measurements rely on one-body observables, i.e., counting the number of atoms clicking in one of two ports in the experiment [114 ] . The measurement is thus given by ∑ m = 1 N a t P ^ ± ; θ ( m ) \sum_{m=1}^{N_{\rm at}}\hat{P}^{(m)}_{\pm;\theta} ∑ m = 1 N at P ^ ± ; θ ( m ) P ^ ± ; θ ( m ) \hat{P}^{(m)}_{\pm;\theta} P ^ ± ; θ ( m ) m m m
Tr [ ρ ^ s ( T ) ∑ m = 1 N a t P ^ ± ; θ ( m ) ] = N a t Tr [ ρ ^ s ( m ) ( T ) P ^ ± ; θ ( m ) ] . (84) \operatorname{Tr}\left[\hat{\rho}_s(T)\sum_{m=1}^{N_{\rm at}}\hat{P}^{(m)}_{\pm;\theta}\right]=N_{\rm at}\operatorname{Tr}\left[\hat{\rho}_s^{(m)}(T)\hat{P}^{(m)}_{\pm;\theta}\right]. \tag*{(84)} Tr [ ρ ^ s ( T ) m = 1 ∑ N at P ^ ± ; θ ( m ) ] = N at Tr [ ρ ^ s ( m ) ( T ) P ^ ± ; θ ( m ) ] . (84) where ρ ^ s ( m ) \hat{\rho}_s^{(m)} ρ ^ s ( m ) m m m N a t − 1 N_{\rm at}-1 N at − 1
Crucially, the trace over N a t − 1 N_{\rm at}-1 N at − 1 α m ′ = β m ′ \alpha_{m'}=\beta_{m'} α m ′ = β m ′ m ′ ≠ m m'\ne m m ′ = m 2 N a t − 1 2^{N_{\rm at}-1} 2 N at − 1 [14 , 19 ] . Equivalently, taking the trace corresponds to setting J α m ′ , + = J β m ′ , − \mathbf{J}_{\alpha_{m'},+}=\mathbf{J}_{\beta_{m'},-} J α m ′ , + = J β m ′ , − m ′ ≠ m m'\ne m m ′ = m α m ′ ∈ { L , R } \alpha_{m'}\in\{L,R\} α m ′ ∈ { L , R } (82) reads as
ρ q ( x ′ ; { x ˉ n , ± + r ˉ n } ) → ρ q ( x ′ ; x ˉ m , ± + r ˉ m ) , \rho^{\rm q}(x';\{\bar{x}_{n,\pm}+\bar{r}_n\})\to\rho^{\rm q}(x';\bar{x}_{m,\pm}+\bar{r}_m), ρ q ( x ′ ; { x ˉ n , ± + r ˉ n }) → ρ q ( x ′ ; x ˉ m , ± + r ˉ m ) , ρ c l ( x ′ ; { x ˉ n , + + r ˉ n } ) → ρ c l ( x ′ ; x ˉ m , + + r ˉ m ) + ∑ n ≠ m ρ ( x ′ ; x ˉ n + r ˉ n ) . (85) \rho^{\rm cl}(x';\{\bar{x}_{n,+}+\bar{r}_n\})\to\rho^{\rm cl}(x';\bar{x}_{m,+}+\bar{r}_m)+\sum_{n\ne m}\rho(x';\bar{x}_n+\bar{r}_n). \tag*{(85)} ρ cl ( x ′ ; { x ˉ n , + + r ˉ n }) → ρ cl ( x ′ ; x ˉ m , + + r ˉ m ) + n = m ∑ ρ ( x ′ ; x ˉ n + r ˉ n ) . (85) Substituting the FT of Eq. (85) into the influence functional, three bilinear structures appear. Firstly, the K ~ k \widetilde{K}_{k} K k m m m K ~ r , a \widetilde{K}_{r,a} K r , a m m m N a t − 1 N_{\mathrm{at}}-1 N at − 1 m m m
lim T → ∞ lim m a t → ∞ ρ ^ ( m ) ( T ) = 1 2 ( e i ⟨ S I F [ x ˉ L , x ˉ L ] ⟩ 0 e − i ⟨ S I F [ x ˉ L , x ˉ R ] ⟩ 0 e i ⟨ S I F [ x ˉ R , x ˉ L ] ⟩ 0 e i ⟨ S I F [ x ˉ R , x ˉ R ] ⟩ 0 ) ≡ 1 2 ( 1 e − D A I − i P A I e − D A I + i P A I 1 ) + O ( g 3 ) , (86) \begin{aligned}
\lim_{\mathcal{T}\to\infty}\lim_{m_{\mathrm{at}}\to\infty}\hat{\rho}^{(m)}(\mathcal{T})=\frac{1}{2}\begin{pmatrix} e^{i\langle S_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{L}]\rangle_{0}} & e^{-i\langle S_{\mathrm{IF}}[\bar{x}_{L},\bar{x}_{R}]\rangle_{0}} \\ e^{i\langle S_{\mathrm{IF}}[\bar{x}_{R},\bar{x}_{L}]\rangle_{0}} & e^{i\langle S_{\mathrm{IF}}[\bar{x}_{R},\bar{x}_{R}]\rangle_{0}} \end{pmatrix} \\
&\equiv\frac{1}{2}\begin{pmatrix} 1 & e^{-\mathbb{D}_{\mathrm{AI}}-i\mathbb{P}_{\mathrm{AI}}} \\ e^{-\mathbb{D}_{\mathrm{AI}}+i\mathbb{P}_{\mathrm{AI}}} & 1 \end{pmatrix}+\mathcal{O}(g^{3}) \,, \tag*{(86)}
\end{aligned} T → ∞ lim m at → ∞ lim ρ ^ ( m ) ( T ) = 2 1 ( e i ⟨ S IF [ x ˉ L , x ˉ L ] ⟩ 0 e i ⟨ S IF [ x ˉ R , x ˉ L ] ⟩ 0 e − i ⟨ S IF [ x ˉ L , x ˉ R ] ⟩ 0 e i ⟨ S IF [ x ˉ R , x ˉ R ] ⟩ 0 ) ≡ 2 1 ( 1 e − D AI + i P AI e − D AI − i P AI 1 ) + O ( g 3 ) , (86) where ⟨ ⋅ ⟩ 0 \langle\cdot\rangle_{0} ⟨ ⋅ ⟩ 0
D A I = − T 2 ( g Λ D ) 2 ∫ d 4 q sinc 2 ( q 0 T 2 ) ∣ F A ( q ) ∣ 2 ( 1 − cos ( q ⋅ Δ x ) ) Im [ K ~ k ( q ) ] , (87) \mathbb{D}_{\mathrm{AI}}=-\mathcal{T}^{2}\left(\frac{g}{\Lambda_{D}}\right)^{2}\int\mathrm{d}^{4}q\ \operatorname{sinc}^{2}\left(\frac{q_{0}\mathcal{T}}{2}\right)\lvert F_{A}(\mathbf{q})\rvert^{2}\left(1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})\right)\operatorname{Im}\left[\widetilde{K}_{k}(q)\right] \,, \tag*{(87)} D AI = − T 2 ( Λ D g ) 2 ∫ d 4 q sinc 2 ( 2 q 0 T ) ∣ F A ( q ) ∣ 2 ( 1 − cos ( q ⋅ Δ x ) ) Im [ K k ( q ) ] , (87) and
P A I = T ( g Λ D ) 2 ∫ d 4 q sin ( q ⋅ Δ x 2 ) ∣ F A I ( q ) ∣ 2 [ 2 δ ˉ ( q 0 ) − 4 T sinc 2 ( q 0 T 2 ) sin 2 ( q ⋅ Δ x 4 ) ] Im [ K ~ r ( q ) ] , (88) \mathbb{P}_{\mathrm{AI}}=\mathcal{T}\left(\frac{g}{\Lambda_{D}}\right)^{2}\int\mathrm{d}^{4}q\ \sin\left(\frac{\mathbf{q}\cdot\Delta\mathbf{x}}{2}\right)\lvert F_{\mathrm{AI} }(\mathbf{q})\rvert^{2}\left[2\bar{\delta}(q_{0})-4\mathcal{T}\operatorname{sinc}^{2}\left(\frac{q_{0}\mathcal{T}}{2}\right)\sin^{2}\left(\frac{\mathbf{q}\cdot\Delta\mathbf{x}}{4}\right)\right]\operatorname{Im}\left[\widetilde{K}_{r}(q)\right] \,, \tag*{(88)} P AI = T ( Λ D g ) 2 ∫ d 4 q sin ( 2 q ⋅ Δ x ) ∣ F AI ( q ) ∣ 2 [ 2 δ ˉ ( q 0 ) − 4 T sinc 2 ( 2 q 0 T ) sin 2 ( 4 q ⋅ Δ x ) ] Im [ K r ( q ) ] , (88) respectively. Notably, while the decoherence observable only depends on the atomic form factor, the phase shift observable depends on the AI form factor
∣ F A I ( q ) ∣ 2 ≡ ∣ F A ( q ) ∣ 2 [ 1 + ( N a t − 1 ) ∣ F c l o u d ( q ) ∣ 2 ] , (89) \lvert F_{\mathrm{AI}}(\mathbf{q})\rvert^{2}\equiv\lvert F_{A}(\mathbf{q})\rvert^{2}\left[1+(N_{\mathrm{at}}-1)\lvert F_{\mathrm{cloud}}(\mathbf{q})\rvert^{2}\right] \,, \tag*{(89)} ∣ F AI ( q ) ∣ 2 ≡ ∣ F A ( q ) ∣ 2 [ 1 + ( N at − 1 ) ∣ F cloud ( q ) ∣ 2 ] , (89) where F c l o u d ( q ) = P ~ ( q ) F_{\mathrm{cloud}}(\mathbf{q})=\widetilde{P}(\mathbf{q}) F cloud ( q ) = P ( q ) r c r_{c} r c
∣ F c l o u d ( q ) ∣ 2 = e − ∣ q ∣ 2 r c 2 . (90) \lvert F_{\mathrm{cloud}}(\mathbf{q})\rvert^{2}=e^{-\lvert\mathbf{q}\rvert^{2}r_{c}^{2}} \,. \tag*{(90)} ∣ F cloud ( q ) ∣ 2 = e − ∣ q ∣ 2 r c 2 . (90) By construction, ∣ F c l o u d ( 0 ) ∣ 2 = 1 \lvert F_{\mathrm{cloud}}(0)\rvert^{2}=1 ∣ F cloud ( 0 ) ∣ 2 = 1 ∣ F A I ( q ) ∣ 2 = ∣ F A ( q ) ∣ 2 \lvert F_{\mathrm{AI}}(\mathbf{q})\rvert^{2}=\lvert F_{A}(\mathbf{q})\rvert^{2} ∣ F AI ( q ) ∣ 2 = ∣ F A ( q ) ∣ 2 N a t = 1 N_{\mathrm{at}}=1 N at = 1 N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at
As a result of the measurement protocol (see Eq. (84) ), which gives rise to the single-atom reduced density matrix (see Eq. (86) ), the observable number of counts in either detection port is directly related to Eqs. (87) –(88) , e.g.,
Expected number of counts in the ‘−’ port = N a t × { 1 2 D A I + O ( g 3 ) for θ = 0 , 1 2 ( 1 − P A I ) + O ( g 3 ) for θ = π / 2. (91) \text{Expected number of counts in the ‘−’ port}=N_{\mathrm{at}}\times\begin{cases}\frac{1}{2}\mathbb{D}_{\mathrm{AI}}+\mathcal{O}(g^{3}) & \text{for }\theta=0, \\ \frac{1}{2}\left(1-\mathbb{P}_{\mathrm{AI}}\right)+\mathcal{O}(g^{3}) & \text{for }\theta=\pi/2.\end{cases} \tag*{(91)} Expected number of counts in the ‘−’ port = N at × { 2 1 D AI + O ( g 3 ) 2 1 ( 1 − P AI ) + O ( g 3 ) for θ = 0 , for θ = π /2. (91) Since each atom can now independently click in either port, in contrast with the CPI case, the shot-noise error on the count is Poissonian and scales with 1 / N a t 1/\sqrt{N_{\mathrm{at}}} 1/ N at N e x p N_{\mathrm{exp}} N exp D A I \mathcal{D}_{\mathrm{AI}} D AI P A I \mathcal{P}_{\mathrm{AI}} P AI 1 / N a t N e x p 1/\sqrt{N_{\mathrm{at}}N_{\mathrm{exp}}} 1/ N at N exp
The structure of the form factors, which agrees with Ref. [14 ] in the limit of a point-like cloud and with Ref. [19 ] in the context of a static cloud, reveals a clear asymmetry between unitary and non-unitary effects. D A I \mathcal{D}_{\mathrm{AI}} D AI N a t N_{\mathrm{at}} N at P A I \mathcal{P}_{\mathrm{AI}} P AI N a t N_{\mathrm{at}} N at [14 ] in the context of coherent collisional decoherence.10 N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at O ( g 2 ) \mathcal{O}(g^2) O ( g 2 ) N a t N_{\mathrm{at}} N at (84) ). In the coherent regime, by contrast, the phase acquires an N a t N_{\mathrm{at}} N at
We conclude by highlighting that these expressions remain independent of the precise coupling between the environment and the system. They are also consistent with the general bounds of Ref. [117 ] .
Spin-independent sub-GeV dark matter–nucleon scattering
We now turn our attention to computing the signatures of concrete DM models in MWIs. In particular, we will focus on elastic spin-independent DM-nucleon interactions, where DM and nucleon fields are represented by χ \chi χ N N N contact interactions, i.e.,
L ⊃ − g Λ 2 N ‾ N O χ (92) \mathcal{L} \supset-\frac{g}{\Lambda^2}\overline{N}N O_\chi \tag*{(92)} L ⊃ − Λ 2 g N N O χ (92) and (ii) long-range interactions mediated by a light hadrophilic real scalar ϕ \phi ϕ m ϕ m_\phi m ϕ
L ⊃ − y n ϕ N ‾ N − y χ ϕ O χ (93) \mathcal{L} \supset-y_n\phi\overline{N}N-y_\chi\phi O_\chi \tag*{(93)} L ⊃ − y n ϕ N N − y χ ϕ O χ (93) where y n y_n y n y χ y_\chi y χ
O χ = { m χ χ ‾ χ for real scalar DM , χ ‾ χ for Dirac fermionic DM . (94) O_\chi=
\begin{cases}
m_\chi\overline{\chi}\chi& \text{for real scalar DM},\\
\overline{\chi}\chi& \text{for Dirac fermionic DM}.
\end{cases} \tag*{(94)} O χ = { m χ χ χ χ χ for real scalar DM , for Dirac fermionic DM . (94) Clearly, O s ≡ N ‾ N O_s \equiv\overline{N}N O s ≡ N N { χ , ϕ } \{\chi,\phi\} { χ , ϕ }
Equipped with these models, we can explicitly compute the entries of the environment matrix kernel K ~ \widetilde{K} K (17) –(19) ), which directly enter the interferometer observables (see Eq. (62) ). Notably, for contact interactions the environment kernels can be understood in terms of in-in bubble diagrams involving only χ \chi χ m χ ≲ 10 e V m_\chi\lesssim10\ \mathrm{eV} m χ ≲ 10 eV dressed in-in ϕ \phi ϕ y χ y_\chi y χ m χ m_\chi m χ
We first consider the case of contact interactions, distinguishing between fermionic and bosonic DM. In particular, we will show a simple, diagrammatic way of computing the environment-kernels using in-in bubble diagrams.
Recall from section II A that, before tracing out environmental degrees of freedom, the total action contains the interaction term
S i n t [ O s , + , O χ , + ] − S i n t [ O s , − , O χ , − ] ⊃ − g Λ 2 ( O s , + O χ , + − O s , − O χ , − ) = − g Λ 2 ( O s c l O χ q + O s q O χ c l ) , (95) S_{\mathrm{int}}[O_{s,+},O_{\chi,+}]-S_{\mathrm{int}}[O_{s,-},O_{\chi,-}] \supset-\frac{g}{\Lambda^2}\left(O_{s,+}O_{\chi,+}-O_{s,-}O_{\chi,-}\right)
= -\frac{g}{\Lambda^2}\left(O_s^{\mathrm{cl}}O_\chi^{\mathrm{q}}+O_s^{\mathrm{q}}O_\chi^{\mathrm{cl}}\right), \tag*{(95)} S int [ O s , + , O χ , + ] − S int [ O s , − , O χ , − ] ⊃ − Λ 2 g ( O s , + O χ , + − O s , − O χ , − ) = − Λ 2 g ( O s cl O χ q + O s q O χ cl ) , (95) where in the last line we used the Keldysh basis (see Eq. (9) ). Writing out the environment currents explicitly in terms of χ \chi χ O ( g 2 ) \mathcal{O}(g^2) O ( g 2 ) P \mathcal{P} P D \mathcal{D} D ( g Λ 2 ) 2 O s c l O s q K ~ r , a \left(\frac{g}{\Lambda^2}\right)^2 O_s^{\mathrm{cl}}O_s^{\mathrm{q}}\widetilde{K}_{r,a} ( Λ 2 g ) 2 O s cl O s q K r , a ( g Λ 2 ) 2 O s q O s q K ~ k \left(\frac{g}{\Lambda^2}\right)^2 O_s^{\mathrm{q}}O_s^{\mathrm{q}}\widetilde{K}_k ( Λ 2 g ) 2 O s q O s q K k (62) ), the kernels, which were defined in terms of operators in Eqs. (17) –(19) , are diagrammatically obtained by joining appropriate combinations of two of the vertices above with χ \chi χ (21) and (22) ). In detail, K ~ r , a \widetilde{K}_{r,a} K r , a O ~ s c l \widetilde{O}_s^{\mathrm{cl}} O s cl O ~ s q \widetilde{O}_s^{\mathrm{q}} O s q K ~ k \widetilde{K}_k K k O ~ s q \widetilde{O}_s^{\mathrm{q}} O s q (24) ), where in both cases the system currents are stripped off. The assignment of the internal lines is fixed by the cl/q labels at each end, with each internal line corresponding to the appropriate in-in propagator. By sewing together χ \chi χ
Real scalar dark matter
Using Eq. (9) and the real scalar model in Eq. (94) , O χ c l , q \mathcal{O}_{\chi}^{\mathrm{cl,q}} O χ cl , q χ c l , q \chi^{\mathrm{cl,q}} χ cl , q (95) reads as
S [ O s c l , O s q , χ c l , χ q ] ⊃ − m χ g Λ 2 ( 2 O s c l χ c l χ q + O s q χ c l χ c l + 1 4 O s q χ q χ q ) = O s c l ( in-in Feynman diagram ) + O s q ( in-in Feynman diagram ) + O s q ( in-in Feynman diagram ) , − 2 m χ g Λ 2 , − 2 m χ g Λ 2 , − 1 2 m χ g Λ 2 (96) \begin{aligned}
S[O_s^{\mathrm{cl}},O_s^{q},\chi^{\mathrm{cl}},\chi^{q}] \supset-m_{\chi}\frac{g}{\Lambda^2}\left(2O_s^{\mathrm{cl}}\chi^{\mathrm{cl}}\chi^{q}+O_s^{q}\chi^{\mathrm{cl}}\chi^{\mathrm{cl}}+\frac{1}{4}O_s^{q}\chi^{q}\chi^{q}\right) \\
&=O_s^{\mathrm{cl}}\;(\text{in-in Feynman diagram})+O_s^{q}\;(\text{in-in Feynman diagram})+O_s^{q}\;(\text{in-in Feynman diagram}),
\qquad-2m_{\chi}\frac{g}{\Lambda^2},\quad-2m_{\chi}\frac{g}{\Lambda^2},\quad-\frac{1}{2}m_{\chi}\frac{g}{\Lambda^2} \tag*{(96)}
\end{aligned} S [ O s cl , O s q , χ cl , χ q ] ⊃ − m χ Λ 2 g ( 2 O s cl χ cl χ q + O s q χ cl χ cl + 4 1 O s q χ q χ q ) = O s cl ( in-in Feynman diagram ) + O s q ( in-in Feynman diagram ) + O s q ( in-in Feynman diagram ) , − 2 m χ Λ 2 g , − 2 m χ Λ 2 g , − 2 1 m χ Λ 2 g (96) where in the second line we wrote down the in-in Feynman diagrams for each vertex and represented q- and cl-fields with dotted and solid lines, respectively. Single and double lines depict χ \chi χ O s O_s O s χ \chi χ χ \chi χ (21) ):
G ~ a χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ q ( x ) χ c l ( 0 ) ⟩ ⟩ e i p ⋅ x = 1 ( p 0 − i ϵ ) 2 − E p 2 = P [ 1 p 0 2 − E p 2 ] − i 4 E p [ δ ˉ ( p 0 + E p ) − δ ˉ ( p 0 − E p ) ] ≡ ⋯ , \begin{aligned}
\widetilde{G}_{a}^{\chi}(p)=-i\int d^4x\,\left\langle\!\left\langle\chi^{q}(x)\chi^{\mathrm{cl}}(0)\right\rangle\!\right\rangle e^{ip\cdot x} \\
&=\frac{1}{(p_0-i\epsilon)^2-E_p^2}
=\mathrm{P}\left[\frac{1}{p_0^2-E_p^2}\right]-\frac{i}{4E_p}\left[\bar{\delta}(p_0+E_p)-\bar{\delta}(p_0-E_p)\right] \\
&\equiv\cdots,
\end{aligned} G a χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ q ( x ) χ cl ( 0 ) ⟩ ⟩ e i p ⋅ x = ( p 0 − i ϵ ) 2 − E p 2 1 = P [ p 0 2 − E p 2 1 ] − 4 E p i [ δ ˉ ( p 0 + E p ) − δ ˉ ( p 0 − E p ) ] ≡ ⋯ , G ~ r χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ c l ( x ) χ q ( 0 ) ⟩ ⟩ e i p ⋅ x = 1 ( p 0 + i ϵ ) 2 − E p 2 = P [ 1 p 0 2 − E p 2 ] + i 4 E p [ δ ˉ ( p 0 + E p ) − δ ˉ ( p 0 − E p ) ] ≡ ⋯ , (97) \begin{aligned}
\widetilde{G}_{r}^{\chi}(p)=-i\int d^4x\,\left\langle\!\left\langle\chi^{\mathrm{cl}}(x)\chi^{q}(0)\right\rangle\!\right\rangle e^{ip\cdot x} \\
&=\frac{1}{(p_0+i\epsilon)^2-E_p^2}
=\mathrm{P}\left[\frac{1}{p_0^2-E_p^2}\right]+\frac{i}{4E_p}\left[\bar{\delta}(p_0+E_p)-\bar{\delta}(p_0-E_p)\right] \\
&\equiv\cdots, \tag*{(97)}
\end{aligned} G r χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ cl ( x ) χ q ( 0 ) ⟩ ⟩ e i p ⋅ x = ( p 0 + i ϵ ) 2 − E p 2 1 = P [ p 0 2 − E p 2 1 ] + 4 E p i [ δ ˉ ( p 0 + E p ) − δ ˉ ( p 0 − E p ) ] ≡ ⋯ , (97) G ~ k χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ c l ( x ) χ c l ( 0 ) ⟩ ⟩ e i p ⋅ x = − i 4 E p [ ( 1 + 2 n χ ( p ) ) δ ˉ ( p 0 − E p ) + ( 1 + 2 n χ ( − p ) ) δ ˉ ( p 0 + E p ) ] ≡ solid line , \begin{aligned}
\widetilde{G}_{k}^{\chi}(p)=-i\int d^4x\,\left\langle\!\left\langle\chi^{\mathrm{cl}}(x)\chi^{\mathrm{cl}}(0)\right\rangle\!\right\rangle e^{ip\cdot x} \\
&=-\frac{i}{4E_p}\left[(1+2n_{\chi}(p))\bar{\delta}(p_0-E_p)+(1+2n_{\chi}(-p))\bar{\delta}(p_0+E_p)\right] \\
&\equiv\text{solid line},
\end{aligned} G k χ ( p ) = − i ∫ d 4 x ⟨ ⟨ χ cl ( x ) χ cl ( 0 ) ⟩ ⟩ e i p ⋅ x = − 4 E p i [ ( 1 + 2 n χ ( p )) δ ˉ ( p 0 − E p ) + ( 1 + 2 n χ ( − p )) δ ˉ ( p 0 + E p ) ] ≡ solid line , where P [ ⋅ ] \mathrm{P}[\,\cdot\,] P [ ⋅ ] E p = m χ 2 + ∣ p ∣ 2 E_p=\sqrt{m_{\chi}^2+|\mathbf{p}|^2} E p = m χ 2 + ∣ p ∣ 2 n χ ( p ) ≃ ρ χ f χ ( p ) / m χ n_{\chi}(p)\simeq\rho_{\chi}f_{\chi}(p)/m_{\chi} n χ ( p ) ≃ ρ χ f χ ( p ) / m χ ⟨ ⟨ χ q χ q ⟩ ⟩ = 0 \langle\!\langle\chi^{q}\chi^{q}\rangle\!\rangle=0 ⟨ ⟨ χ q χ q ⟩ ⟩ = 0 S I F ⊃ ( g Λ 2 ) 2 O s c l O s q K ~ r S_{\mathrm{IF}}\supset\left(\frac{g}{\Lambda^2}\right)^2O_s^{\mathrm{cl}}O_s^{q}\widetilde{K}_r S IF ⊃ ( Λ 2 g ) 2 O s cl O s q K r (24) ), K ~ r \widetilde{K}_r K r χ \chi χ (96) . Similarly, because S I F ⊃ ( g Λ 2 ) 2 O s q O s q K ~ k S_{\mathrm{IF}}\supset\left(\frac{g}{\Lambda^2}\right)^2O_s^{q}O_s^{q}\widetilde{K}_k S IF ⊃ ( Λ 2 g ) 2 O s q O s q K k K ~ k \widetilde{K}_k K k χ \chi χ (96) . Hence, given the interaction term in Eq. (96) and the in-in free theory $\chi propagators, the only non-vanishing one-loop and system-source-stripped diagrams contributing to the observables (see Eq. (62) ) are
\left(\frac{g}{\Lambda^2}\right)^2 \widetilde{K}_{r}(q)=i\;\includegraphics[width=0.12\textwidth]{loop} = i\left(2m_{\chi}\frac{g}{\Lambda^2}\right)^2\int d^4p\,\widetilde{G}^{\chi}_{k}(p)\widetilde{G}^{\chi}_{k}(q-p)\equiv\widetilde{\Pi}^{\chi}_{r}(q),
\begin{aligned}
\left(\frac{g}{\Lambda^2}\right)^2\widetilde{K}_{k}(q)=i\;\includegraphics[width=0.12\textwidth]{loop1}+i\;\includegraphics[width=0.12\textwidth]{loop2}+i\;\includegraphics[width=0.12\textwidth]{loop3} \\
&=\frac{i}{2}\left(2m_{\chi}\frac{g}{\Lambda^2}\right)^2\int d^4p\left[\widetilde{G}^{\chi}_{k}(p)\widetilde{G}^{\chi}_{k}(q-p)+\frac{1}{4}\widetilde{G}^{\chi}_{r}(p)\widetilde{G}^{\chi}_{r}(q-p)+\frac{1}{4}\widetilde{G}^{\chi}_{a}(p)\widetilde{G}^{\chi}_{a}(q-p)\right] \\
&\equiv\widetilde{\Pi}^{\chi}_{k}(q), \tag*{(98)}
\end{aligned}
where in the last diagrams we applied the symmetry factor of 1 / 2 1/2 1/2 (17) –(19) ).
Dirac fermion dark matter
In the case of fermionic dark matter, we rewrite the fermions in the Larkin-Ovchinnikov field basis (see Eq. (23) ) such that, from Eqs. (92) and (94) the closed theory action will contain the terms
S [ O s c l , O s q , χ c l , χ q , χ ‾ c l , χ ‾ q ] ⊃ − g Λ 2 ( O s c l χ ‾ c l χ c l + O s c l χ ‾ q χ q + 1 2 O s q χ ‾ c l χ q + 1 2 O s q χ ‾ q χ c l ) = O s c l ( − g Λ 2 ) + O s c l ( − g Λ 2 ) + O s q ( − 1 2 g Λ 2 ) + O s q ( − 1 2 g Λ 2 ) , (99) \begin{aligned}
S[O_s^{\mathrm{cl}},O_s^{q},\chi^{\mathrm{cl}},\chi^{q},\overline{\chi}^{\mathrm{cl}},\overline{\chi}^{q}]\supset-\frac{g}{\Lambda^2}\left(O_s^{\mathrm{cl}}\overline{\chi}^{\mathrm{cl}}\chi^{\mathrm{cl}}+O_s^{\mathrm{cl}}\overline{\chi}^{q}\chi^{q}+\frac{1}{2}O_s^{q}\overline{\chi}^{\mathrm{cl}}\chi^{q}+\frac{1}{2}O_s^{q}\overline{\chi}^{q}\chi^{\mathrm{cl}}\right) \\
&=O_s^{\mathrm{cl}}\left(-\frac{g}{\Lambda^2}\right)+O_s^{\mathrm{cl}}\left(-\frac{g}{\Lambda^2}\right)+O_s^{q}\left(-\frac{1}{2}\frac{g}{\Lambda^2}\right)+O_s^{q}\left(-\frac{1}{2}\frac{g}{\Lambda^2}\right), \tag*{(99)}
\end{aligned} S [ O s cl , O s q , χ cl , χ q , χ cl , χ q ] ⊃ − Λ 2 g ( O s cl χ cl χ cl + O s cl χ q χ q + 2 1 O s q χ cl χ q + 2 1 O s q χ q χ cl ) = O s cl ( − Λ 2 g ) + O s cl ( − Λ 2 g ) + O s q ( − 2 1 Λ 2 g ) + O s q ( − 2 1 Λ 2 g ) , (99) where single lines denote χ \chi χ A , B A,B A , B (22) ) are given by
[ G ~ a χ ( p ) ] A B = − i ∫ d 4 x ⟨ ⟨ χ q ( x ) χ ‾ q ( 0 ) ⟩ ⟩ A B e i p ⋅ x = ( p̸ + m χ ) A B ( p 0 − i ϵ ) 2 − E p 2 = ( p̸ + m χ ) A B { P [ 1 p 0 2 − E p 2 ] − i 4 E p [ δ ( p 0 + E p ) − δ ( p 0 − E p ) ] } = ⋯ , (100) \begin{aligned}
[\widetilde{G}^{\chi}_{a}(p)]_{AB}=-i\int d^4x\left\langle\!\left\langle\chi^{q}(x)\overline{\chi}^{q}(0)\right\rangle\!\right\rangle_{AB}e^{ip\cdot x} \\
&=\frac{(\not{p}+m_{\chi})_{AB}}{(p_0-i\epsilon)^2-E_p^2}
=(\not{p}+m_{\chi})_{AB}\left\{\mathcal{P}\left[\frac{1}{p_0^2-E_p^2}\right]-\frac{i}{4E_p}\left[\delta(p_0+E_p)-\delta(p_0-E_p)\right]\right\} \\
&=\cdots, \tag*{(100)}
\end{aligned} [ G a χ ( p ) ] A B = − i ∫ d 4 x ⟨ ⟨ χ q ( x ) χ q ( 0 ) ⟩ ⟩ A B e i p ⋅ x = ( p 0 − i ϵ ) 2 − E p 2 ( p + m χ ) A B = ( p + m χ ) A B { P [ p 0 2 − E p 2 1 ] − 4 E p i [ δ ( p 0 + E p ) − δ ( p 0 − E p ) ] } = ⋯ , (100) [ G ~ r χ ( p ) ] A B = − i ∫ d 4 x ⟨ ⟨ χ c l ( x ) χ ‾ c l ( 0 ) ⟩ ⟩ A B e i p ⋅ x = ( p̸ + m χ ) A B ( p 0 + i ϵ ) 2 − E p 2 = ( p̸ + m χ ) A B { P [ 1 p 0 2 − E p 2 ] + i 4 E p [ δ ( p 0 + E p ) − δ ( p 0 − E p ) ] } = ⟶ , (101) \begin{aligned}
[\widetilde{G}^{\chi}_{r}(p)]_{AB}=-i\int d^4x\left\langle\!\left\langle\chi^{\mathrm{cl}}(x)\overline{\chi}^{\mathrm{cl}}(0)\right\rangle\!\right\rangle_{AB}e^{ip\cdot x} \\
&=\frac{(\not{p}+m_{\chi})_{AB}}{(p_0+i\epsilon)^2-E_p^2}
=(\not{p}+m_{\chi})_{AB}\left\{\mathcal{P}\left[\frac{1}{p_0^2-E_p^2}\right]+\frac{i}{4E_p}\left[\delta(p_0+E_p)-\delta(p_0-E_p)\right]\right\} \\
&=\longrightarrow, \tag*{(101)}
\end{aligned} [ G r χ ( p ) ] A B = − i ∫ d 4 x ⟨ ⟨ χ cl ( x ) χ cl ( 0 ) ⟩ ⟩ A B e i p ⋅ x = ( p 0 + i ϵ ) 2 − E p 2 ( p + m χ ) A B = ( p + m χ ) A B { P [ p 0 2 − E p 2 1 ] + 4 E p i [ δ ( p 0 + E p ) − δ ( p 0 − E p ) ] } =⟶ , (101) \begin{aligned}
[\widetilde{G}^{\chi}_{k}(p)]_{AB}=-i\int d^{4}x\,\langle\!\langle\chi^{\mathrm{cl}}(x)\bar{\chi}^{\mathrm{cl}}(0)\rangle\!\rangle_{AB}e^{ip\cdot x} \\
&=-\frac{i}{2E_{\mathbf p}}(\slashed{p}+m_{\chi})_{AB}\left[\bar{\delta}(p_{0}-E_{\mathbf p})(1-n_{\chi}(\mathbf p))+\bar{\delta}(p_{0}+E_{\mathbf p})\right] \\
&=\longrightarrow\cdots, \tag*{(102)}
\end{aligned}
where n χ ( p ) n_{\chi}(\mathbf p) n χ ( p ) spin-summed occupation number per three-momentum p \mathbf p p ⟨ ⟨ χ q χ ˉ c l ⟩ ⟩ \langle\!\langle\chi^{q}\bar{\chi}^{\mathrm{cl}}\rangle\!\rangle ⟨ ⟨ χ q χ ˉ cl ⟩ ⟩ S I F ⊃ ( g Λ 2 ) 2 O ~ s q O ~ s c l K ~ r S_{\mathrm{IF}}\supset\left(\frac{g}{\Lambda^{2}}\right)^{2}\widetilde{O}^{q}_{s}\widetilde{O}^{\mathrm{cl}}_{s}\widetilde{K}_{r} S IF ⊃ ( Λ 2 g ) 2 O s q O s cl K r S I F ⊃ ( g Λ 2 ) 2 O ~ s q O ~ s q K ~ k S_{\mathrm{IF}}\supset\left(\frac{g}{\Lambda^{2}}\right)^{2}\widetilde{O}^{q}_{s}\widetilde{O}^{q}_{s}\widetilde{K}_{k} S IF ⊃ ( Λ 2 g ) 2 O s q O s q K k (24) ), for the interactions in Eq. (99) , the non-vanishing diagrams contributing to the observables (see Eq. (62) ) read as
( g Λ 2 ) 2 K ~ r ( q ) = − i 2 ( g Λ 2 ) 2 ∫ d 4 p tr [ G ~ k χ ( p ) G ~ r χ ( p + q ) + G ~ a χ ( p ) G ~ k χ ( p + q ) ] ≡ Π ~ r χ ( q ) , \left(\frac{g}{\Lambda^{2}}\right)^{2}\widetilde{K}_{r}(q)
=-\frac{i}{2}\left(\frac{g}{\Lambda^{2}}\right)^{2}\int d^{4}p\,\operatorname{tr}\left[\widetilde{G}^{\chi}_{k}(p)\widetilde{G}^{\chi}_{r}(p+q)+\widetilde{G}^{\chi}_{a}(p)\widetilde{G}^{\chi}_{k}(p+q)\right]\equiv\widetilde{\Pi}^{\chi}_{r}(q), ( Λ 2 g ) 2 K r ( q ) = − 2 i ( Λ 2 g ) 2 ∫ d 4 p tr [ G k χ ( p ) G r χ ( p + q ) + G a χ ( p ) G k χ ( p + q ) ] ≡ Π r χ ( q ) , ( g Λ 2 ) 2 K ~ k ( q ) = − i 4 ( g Λ 2 ) 2 ∫ d 4 p tr [ G ~ k χ ( p ) G ~ k χ ( p + q ) + G ~ r χ ( p ) G ~ a χ ( p + q ) + G ~ a χ ( p ) G ~ r χ ( p + q ) ] ≡ Π ~ k χ ( q ) , (103) \left(\frac{g}{\Lambda^{2}}\right)^{2}\widetilde{K}_{k}(q)
=-\frac{i}{4}\left(\frac{g}{\Lambda^{2}}\right)^{2}\int d^{4}p\,\operatorname{tr}\left[\widetilde{G}^{\chi}_{k}(p)\widetilde{G}^{\chi}_{k}(p+q)+\widetilde{G}^{\chi}_{r}(p)\widetilde{G}^{\chi}_{a}(p+q)+\widetilde{G}^{\chi}_{a}(p)\widetilde{G}^{\chi}_{r}(p+q)\right]
\equiv\widetilde{\Pi}^{\chi}_{k}(q), \tag*{(103)} ( Λ 2 g ) 2 K k ( q ) = − 4 i ( Λ 2 g ) 2 ∫ d 4 p tr [ G k χ ( p ) G k χ ( p + q ) + G r χ ( p ) G a χ ( p + q ) + G a χ ( p ) G r χ ( p + q ) ] ≡ Π k χ ( q ) , (103) where tr [ ⋅ ] \operatorname{tr}[\cdot] tr [ ⋅ ]
Equipped with the form of the environment kernels, we now invoke the Markovian limit, corresponding to the regime in which the time scale of the environment is much shorter than the system’s time scale. In our application, the DM coherence time τ c \tau_{c} τ c T T T m χ ≳ 10 − 7 e V m_{\chi}\gtrsim10^{-7}\,\mathrm{eV} m χ ≳ 1 0 − 7 eV τ c ≪ T ∼ O ( 1 s ) \tau_{c}\ll T\sim\mathcal{O}(1\,\mathrm{s}) τ c ≪ T ∼ O ( 1 s ) K ( x − x ′ ) ∝ δ ( t − t ′ ) K(x-x')\propto\delta(t-t') K ( x − x ′ ) ∝ δ ( t − t ′ ) T → ∞ T\to\infty T → ∞
T 2 sinc 2 ( q 0 T 2 ) ⟶ T δ ˉ ( q 0 ) . (104) T^{2}\operatorname{sinc}^{2}\left(\frac{q_{0}T}{2}\right)\longrightarrow T\bar{\delta}(q_{0}). \tag*{(104)} T 2 sinc 2 ( 2 q 0 T ) ⟶ T δ ˉ ( q 0 ) . (104) Under this identification, and after performing the q 0 q_{0} q 0 (62) reads as
D = − T ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] Im [ Π ~ k χ ( 0 , q ) ] + O ( g 3 ) , \mathbb{D}=-T\int d^{3}q\,|F_{\mathrm D}(\mathbf q)|^{2}\left[1-\cos(\mathbf q\cdot\Delta\mathbf x)\right]\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{k}(0,\mathbf q)\right]+\mathcal{O}(g^{3}), D = − T ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] Im [ Π k χ ( 0 , q ) ] + O ( g 3 ) , P = T ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) Im [ Π ~ r χ ( 0 , q ) ] + O ( g 3 ) . (105) \mathbb{P}=T\int d^{3}q\,|F_{\mathrm P}(\mathbf q)|^{2}\sin(\mathbf q\cdot\Delta\mathbf x)\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf q)\right]+\mathcal{O}(g^{3}). \tag*{(105)} P = T ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) Im [ Π r χ ( 0 , q ) ] + O ( g 3 ) . (105) Notably, the emergence of the energy-conserving delta function dictates that open-system dynamics requires no energy deposition from the environment.
Since DM in the laboratory is non-relativistic, the imaginary part of the environment kernels (see Eqs. (98) and (103) for scalar and Dirac fermion DM, respectively) can be approximated by
Im [ Π ~ R χ ( 0 , q ) ] ≈ 1 2 ( g Λ 2 ) 2 ∫ d 3 p [ n χ ( p ) − n χ ( p − q ) ] δ ˉ ( E p − E p − q ) , Im [ Π ~ K χ ( 0 , q ) ] ≈ − 1 2 ( g Λ 2 ) 2 ∫ d 3 p [ n χ ( p ) S χ ( p − q ) + n χ ( p − q ) S χ ( p ) ] δ ˉ ( E p − E p − q ) , (106) \begin{aligned}
\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{\mathrm{R}}(0,\mathbf{q})\right] &\approx\frac{1}{2}\left(\frac{g}{\Lambda^{2}}\right)^{2}\int\mathrm{d}^{3}p\left[n_{\chi}(\mathbf{p})-n_{\chi}(\mathbf{p}-\mathbf{q})\right]\bar{\delta}(E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}),\\
\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{\mathrm{K}}(0,\mathbf{q})\right] &\approx-\frac{1}{2}\left(\frac{g}{\Lambda^{2}}\right)^{2}\int\mathrm{d}^{3}p\left[n_{\chi}(\mathbf{p})S_{\chi}(\mathbf{p}-\mathbf{q})+n_{\chi}(\mathbf{p}-\mathbf{q})S_{\chi}(\mathbf{p})\right]\bar{\delta}(E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}), \tag*{(106)}
\end{aligned} Im [ Π R χ ( 0 , q ) ] Im [ Π K χ ( 0 , q ) ] ≈ 2 1 ( Λ 2 g ) 2 ∫ d 3 p [ n χ ( p ) − n χ ( p − q ) ] δ ˉ ( E p − E p − q ) , ≈ − 2 1 ( Λ 2 g ) 2 ∫ d 3 p [ n χ ( p ) S χ ( p − q ) + n χ ( p − q ) S χ ( p ) ] δ ˉ ( E p − E p − q ) , (106) where we introduced the (spin-statistics dependent) DM statistical factor
S χ ( p − q ) = { 1 + n χ ( p − q ) for bosonic DM , 1 − 1 2 n χ ( p − q ) for fermionic DM . (107) S_{\chi}(\mathbf{p}-\mathbf{q}) =
\begin{cases}
1+n_{\chi}(\mathbf{p}-\mathbf{q}) & \text{for bosonic DM},\\
1-\frac{1}{2}n_{\chi}(\mathbf{p}-\mathbf{q}) & \text{for fermionic DM}.
\end{cases} \tag*{(107)} S χ ( p − q ) = { 1 + n χ ( p − q ) 1 − 2 1 n χ ( p − q ) for bosonic DM , for fermionic DM . (107) Hence, the decoherence and phase shift observables may be expressed as
D = T ( g Λ 2 ) 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] g D ( q ) , (108) \mathbb{D}=T\left(\frac{g}{\Lambda^{2}}\right)^{2}\int\mathrm{d}^{3}q\,\lvert F_{\mathbb{D}}(\mathbf{q})\rvert^{2}\left[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})\right]g_{\mathbb{D}}(\mathbf{q}), \tag*{(108)} D = T ( Λ 2 g ) 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] g D ( q ) , (108) P = T ( g Λ 2 ) 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) g P ( q ) , (109) \mathbb{P}=T\left(\frac{g}{\Lambda^{2}}\right)^{2}\int\mathrm{d}^{3}q\,\lvert F_{\mathbb{P}}(\mathbf{q})\rvert^{2}\sin(\mathbf{q}\cdot\Delta\mathbf{x})g_{\mathbb{P}}(\mathbf{q}), \tag*{(109)} P = T ( Λ 2 g ) 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) g P ( q ) , (109) with F D , P F_{\mathbb{D},\mathbb{P}} F D , P F A ( q ) ∝ A F_{A}(\mathbf{q})\propto A F A ( q ) ∝ A [118 ] , where A A A
g D ( q ) = ∫ d 3 p n χ ( p ) S χ ( p − q ) δ ˉ ( E p − E p − q ) , (110) g_{\mathbb{D}}(\mathbf{q})=\int\mathrm{d}^{3}p\,n_{\chi}(\mathbf{p})S_{\chi}(\mathbf{p}-\mathbf{q})\bar{\delta}(E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}), \tag*{(110)} g D ( q ) = ∫ d 3 p n χ ( p ) S χ ( p − q ) δ ˉ ( E p − E p − q ) , (110) g P ( q ) = ∫ d 3 p n χ ( p ) δ ˉ ( E p − E p − q ) . (111) g_{\mathbb{P}}(\mathbf{q})=\int\mathrm{d}^{3}p\,n_{\chi}(\mathbf{p})\bar{\delta}(E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}). \tag*{(111)} g P ( q ) = ∫ d 3 p n χ ( p ) δ ˉ ( E p − E p − q ) . (111) Introducing the vector q ^ = q / ∣ q ∣ \hat{\mathbf{q}}=\mathbf{q}/\lvert\mathbf{q}\rvert q ^ = q / ∣ q ∣ (3) ) as [67 ]
g P ( q ) = ρ χ m χ 1 ∣ q ∣ ∫ d 3 v f χ ( v ) δ ˉ ( v ⋅ q ^ − ∣ q ∣ 2 m χ ) = ρ χ m χ 2 π 2 v 0 2 N 0 ∣ q ∣ [ exp ( − v min ( q ) 2 v 0 2 ) − exp ( − v e s c 2 v 0 2 ) ] , (112) \begin{aligned}
g_{\mathbb{P}}(\mathbf{q}) &= \frac{\rho_{\chi}}{m_{\chi}}\frac{1}{\lvert\mathbf{q}\rvert}\int\mathrm{d}^{3}v\,f_{\chi}(\mathbf{v})\bar{\delta}\left(\mathbf{v}\cdot\hat{\mathbf{q}}-\frac{\lvert\mathbf{q}\rvert}{2m_{\chi}}\right)\\
&= \frac{\rho_{\chi}}{m_{\chi}}\frac{2\pi^{2}v_{0}^{2}}{N_{0}\lvert\mathbf{q}\rvert}\left[\exp\left(-\frac{v_{\min}(\mathbf{q})^{2}}{v_{0}^{2}}\right)-\exp\left(-\frac{v_{\mathrm{esc}}^{2}}{v_{0}^{2}}\right)\right], \tag*{(112)}
\end{aligned} g P ( q ) = m χ ρ χ ∣ q ∣ 1 ∫ d 3 v f χ ( v ) δ ˉ ( v ⋅ q ^ − 2 m χ ∣ q ∣ ) = m χ ρ χ N 0 ∣ q ∣ 2 π 2 v 0 2 [ exp ( − v 0 2 v m i n ( q ) 2 ) − exp ( − v 0 2 v esc 2 ) ] , (112) with
v min ( q ) = min ( v ⊕ ⋅ q ^ + ∣ q ∣ 2 m χ , v e s c ) . (113) v_{\min}(\mathbf{q})=\min\left(\mathbf{v}_{\oplus}\cdot\hat{\mathbf{q}}+\frac{\lvert\mathbf{q}\rvert}{2m_{\chi}},v_{\mathrm{esc}}\right). \tag*{(113)} v m i n ( q ) = min ( v ⊕ ⋅ q ^ + 2 m χ ∣ q ∣ , v esc ) . (113) The three boxed expressions capture some of the central results of this section. For fixed DM occupation number, decoherence saturates for momentum transfers ∣ q ∣ ≳ ∣ Δ x ∣ − 1 \lvert\mathbf{q}\rvert\gtrsim\lvert\Delta\mathbf{x}\rvert^{-1} ∣ q ∣ ≳ ∣ Δ x ∣ − 1 ∣ q ∣ ≫ ∣ Δ x ∣ − 1 \lvert\mathbf{q}\rvert\gg\lvert\Delta\mathbf{x}\rvert^{-1} ∣ q ∣ ≫ ∣ Δ x ∣ − 1 [90 , 91 ] . We can observe this behavior in Figure 3 , where we plot the experiment-dependent “normalized DM effect”11 [13 ] :
Figure 3. Detector-dependent normalized DM effect (see Eq. (114) ) as a function of the (magnitude of the) three-momentum transfer ∣ q ∣ \lvert\mathbf{q}\rvert ∣ q ∣
p D ( ∣ q ∣ ) = 1 2 ∫ d cos θ ∣ F D ( q ) ∣ 2 [ 1 − cos ( ∣ q ∣ ∣ Δ x ∣ cos θ ) ] = ∣ F D ( q ) ∣ 2 ( 1 − sinc ( ∣ q ∣ ∣ Δ x ∣ ) ) , \begin{aligned}
p_{\mathrm{D}}(\lvert\mathbf{q}\rvert) &= \frac{1}{2}\int\mathrm{d}\cos\theta\lvert F_{\mathrm{D}}(\mathbf{q})\rvert^2 [1-\cos(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert\cos\theta)] \\
&= \lvert F_{\mathrm{D}}(\mathbf{q})\rvert^2 (1-\operatorname{sinc}(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)),
\end{aligned} p D (∣ q ∣) = 2 1 ∫ d cos θ ∣ F D ( q ) ∣ 2 [ 1 − cos (∣ q ∣ ∣ Δ x ∣ cos θ )] = ∣ F D ( q ) ∣ 2 ( 1 − sinc (∣ q ∣ ∣ Δ x ∣)) , p P ( ∣ q ∣ ) = 1 2 ∫ d cos θ ∣ F P ( q ) ∣ 2 sin ( ∣ q ∣ ∣ Δ x ∣ cos θ ) e − ∣ v ⊕ ∣ ∣ q ∣ cos θ / m χ v 0 2 = ∣ F P ( q ) ∣ 2 cos ( ∣ q ∣ ∣ Δ x ∣ ) sinh ( ν ⊕ ∣ q ∣ ∣ Δ x ∣ ) − ν ⊕ sin ( ∣ q ∣ ∣ Δ x ∣ ) cosh ( ν ⊕ ∣ q ∣ ∣ Δ x ∣ ) ∣ q ∣ ∣ Δ x ∣ ( 1 + ν ⊕ 2 ) (114) \begin{aligned}
p_{\mathrm{P}}(\lvert\mathbf{q}\rvert) &= \frac{1}{2}\int\mathrm{d}\cos\theta\lvert F_{\mathrm{P}}(\mathbf{q})\rvert^2 \sin(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert\cos\theta)e^{-\lvert\mathbf{v}_{\oplus}\rvert\lvert\mathbf{q}\rvert\cos\theta/m_{\chi}v_0^2} \\
&= \lvert F_{\mathrm{P}}(\mathbf{q})\rvert^2 \frac{\cos(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)\sinh(\nu_{\oplus}\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)-\nu_{\oplus}\sin(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)\cosh(\nu_{\oplus}\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)}{\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert(1+\nu_{\oplus}^2)} \tag*{(114)}
\end{aligned} p P (∣ q ∣) = 2 1 ∫ d cos θ ∣ F P ( q ) ∣ 2 sin (∣ q ∣ ∣ Δ x ∣ cos θ ) e − ∣ v ⊕ ∣ ∣ q ∣ c o s θ / m χ v 0 2 = ∣ F P ( q ) ∣ 2 ∣ q ∣ ∣ Δ x ∣ ( 1 + ν ⊕ 2 ) cos (∣ q ∣ ∣ Δ x ∣) sinh ( ν ⊕ ∣ q ∣ ∣ Δ x ∣) − ν ⊕ sin (∣ q ∣ ∣ Δ x ∣) cosh ( ν ⊕ ∣ q ∣ ∣ Δ x ∣) (114) where sinc ( x ) = sin ( x ) / x \operatorname{sinc}(x)=\sin(x)/x sinc ( x ) = sin ( x ) / x ν ⊕ = ∣ v ⊕ ∣ / ( m χ v 0 2 ∣ Δ x ∣ ) \nu_{\oplus}=\lvert\mathbf{v}_{\oplus}\rvert/(m_{\chi}v_0^2\lvert\Delta\mathbf{x}\rvert) ν ⊕ = ∣ v ⊕ ∣ / ( m χ v 0 2 ∣ Δ x ∣) ∣ q ∣ \lvert\mathbf{q}\rvert ∣ q ∣ (114) , p D ∝ ( ∣ q ∣ ∣ Δ x ∣ ) 2 p_{\mathrm{D}}\propto(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)^2 p D ∝ (∣ q ∣ ∣ Δ x ∣ ) 2 p P ∝ ( ∣ q ∣ ∣ Δ x ∣ ) 2 ν ⊕ ≃ ∣ q ∣ 2 ∣ Δ x ∣ / m χ v 0 p_{\mathrm{P}}\propto(\lvert\mathbf{q}\rvert\lvert\Delta\mathbf{x}\rvert)^{2\nu_{\oplus}}\simeq\lvert\mathbf{q}\rvert^2\lvert\Delta\mathbf{x}\rvert/m_{\chi}v_0 p P ∝ (∣ q ∣ ∣ Δ x ∣ ) 2 ν ⊕ ≃ ∣ q ∣ 2 ∣ Δ x ∣ / m χ v 0 ∣ q ∣ ≪ ∣ Δ x ∣ − 1 \lvert\mathbf{q}\rvert\ll\lvert\Delta\mathbf{x}\rvert^{-1} ∣ q ∣ ≪ ∣ Δ x ∣ − 1 p P p_{\mathrm{P}} p P ∣ v ⊕ ∣ = 0 \lvert\mathbf{v}_{\oplus}\rvert=0 ∣ v ⊕ ∣ = 0
In light of the many-body and interferometer-specific form factors, and recalling that σ N < r c < ∣ Δ x ∣ \sigma_N<r_c<\lvert\Delta\mathbf{x}\rvert σ N < r c < ∣ Δ x ∣ σ N \sigma_N σ N Figure 3 also shows how the initial state of the interferometer affects the observables as a function of ∣ q ∣ |\mathbf{q}| ∣ q ∣ ∣ q ∣ ≲ 1 / r c |\mathbf{q}|\lesssim1/r_c ∣ q ∣ ≲ 1/ r c ∣ F C P I ( q ) ∣ 2 ∝ A 2 N a t 2 |F_{\mathrm{CPI}}(\mathbf{q})|^2\propto A^2N_{\mathrm{at}}^2 ∣ F CPI ( q ) ∣ 2 ∝ A 2 N at 2 A 2 N a t 2 A^2N_{\mathrm{at}}^2 A 2 N at 2 ∣ F A I ( q ) ∣ 2 ∝ A 2 N a t |F_{\mathrm{AI}}(\mathbf{q})|^2\propto A^2N_{\mathrm{at}} ∣ F AI ( q ) ∣ 2 ∝ A 2 N at ∣ q ∣ ≳ 1 / r c |\mathbf{q}|\gtrsim1/r_c ∣ q ∣ ≳ 1/ r c ∣ q ∣ ≲ 1 / σ N |\mathbf{q}|\lesssim1/\sigma_N ∣ q ∣ ≲ 1/ σ N D A I \mathcal{D}_{\mathrm{AI}} D AI P A I \mathcal{P}_{\mathrm{AI}} P AI N a t N_{\mathrm{at}} N at ∣ q ∣ ≳ 1 / σ N |\mathbf{q}|\gtrsim1/\sigma_N ∣ q ∣ ≳ 1/ σ N ∣ q ∣ ≲ 1 / σ u |\mathbf{q}|\lesssim1/\sigma_u ∣ q ∣ ≲ 1/ σ u ∣ F C P I ( q ) ∣ 2 ∝ N a t 2 A 2 / ( ∣ q ∣ r c ) 4 |F_{\mathrm{CPI}}(\mathbf{q})|^2\propto N_{\mathrm{at}}^2A^2/(|\mathbf{q}|r_c)^4 ∣ F CPI ( q ) ∣ 2 ∝ N at 2 A 2 / ( ∣ q ∣ r c ) 4 (74) ). The DM resolves the position of the individual atoms for ∣ q ∣ ≳ 1 / σ u |\mathbf{q}|\gtrsim1/\sigma_u ∣ q ∣ ≳ 1/ σ u e − 2 W e^{-2W} e − 2 W ω p h ≳ 10 m e V \omega_{\mathrm{ph}}\gtrsim10\ \mathrm{meV} ω ph ≳ 10 meV [67 ] , such that, well below room temperature, the signal is parametrically suppressed for ∣ q ∣ ≳ ω p h m a t ≃ 30 k e V m a t / 100 G e V |\mathbf{q}|\gtrsim\sqrt{\omega_{\mathrm{ph}}m_{\mathrm{at}}}\simeq30\ \mathrm{keV}\sqrt{m_{\mathrm{at}}/100\ \mathrm{GeV}} ∣ q ∣ ≳ ω ph m at ≃ 30 keV m at /100 GeV (71) ).
Through the DM statistical factor S χ ( p − q ) S_\chi(\mathbf{p}-\mathbf{q}) S χ ( p − q ) D \mathcal{D} D n χ ( p ) [ 1 + n χ ( p − q ) ] n_\chi(\mathbf{p})[1+n_\chi(\mathbf{p}-\mathbf{q})] n χ ( p ) [ 1 + n χ ( p − q )] n χ ( p ) [ 1 − 1 2 n χ ( p − q ) ] n_\chi(\mathbf{p})[1-\frac{1}{2}n_\chi(\mathbf{p}-\mathbf{q})] n χ ( p ) [ 1 − 2 1 n χ ( p − q )] n χ ( p ) ≤ 2 n_\chi(\mathbf{p})\leq2 n χ ( p ) ≤ 2 P \mathcal{P} P Figure 4 , where we plot the ratio g D ( q ) / g P ( q ) g_{\mathrm{D}}(q)/g_{\mathrm{P}}(q) g D ( q ) / g P ( q ) m χ m_\chi m χ ∣ q ∣ = m χ v 0 |\mathbf{q}|=m_\chi v_0 ∣ q ∣ = m χ v 0
Figure 4. Ratio g D ( q ) / g P ( q ) g_{\mathrm{D}}(q)/g_{\mathrm{P}}(q) g D ( q ) / g P ( q ) m χ m_\chi m χ ∣ q ∣ = m χ v 0 |\mathbf{q}|=m_\chi v_0 ∣ q ∣ = m χ v 0 q ∥ v ⊕ \mathbf{q}\parallel\mathbf{v}_{\oplus} q ∥ v ⊕ n χ ( q ) ≳ 1 n_\chi(q)\gtrsim1 n χ ( q ) ≳ 1
caption: tbl:2 . Benchmark experimental parameters assumed for the two interferometer configurations considered in this work.
For both examples, we assume shot-noise limited sensitivity and a total measurement campaign T i n t = 10 8 T_{\mathrm{int}} = 10^8 T int = 1 0 8 N e x p = T i n t / T N_{\mathrm{exp}} = T_{\mathrm{int}}/T N exp = T int / T [96 ] .
at n χ ( q ) = 1 n_\chi(\mathbf{q}) = 1 n χ ( q ) = 1 g D ( q ) / g P ( q ) ≈ 1 g_{\mathrm{D}}(\mathbf{q})/g_{\mathrm{P}}(\mathbf{q}) \approx1 g D ( q ) / g P ( q ) ≈ 1 [11 , 12 , 13 ] . For masses below the boundary, the occupation number per mode exceeds unity and g D ( q ) / g P ( q ) ∼ ρ χ / m χ 4 v 0 3 ≃ λ c 3 ρ χ / m χ ≫ 1 g_{\mathrm{D}}(\mathbf{q})/g_{\mathrm{P}}(\mathbf{q}) \sim\rho_\chi/m_\chi^4 v_0^3 \simeq\lambda_c^3\rho_\chi/m_\chi\gg1 g D ( q ) / g P ( q ) ∼ ρ χ / m χ 4 v 0 3 ≃ λ c 3 ρ χ / m χ ≫ 1 λ c \lambda_c λ c
Projected sensitivity
To understand the phenomenological consequences of these previously overlooked finite density effects, we study the reach of the AI and CPI benchmarks (see Table II) to single-component sub-MeV DM, i.e., for ρ χ = 0.4 \rho_\chi= 0.4 ρ χ = 0.4 3 ^3 3 O ( 1 ) \mathcal{O}(1) O ( 1 ) [119 , 120 ] , we assume a non-thermal production mechanism. The simplest model providing the necessary ingredients is Z 2 \mathbb{Z}_2 Z 2
L ⊃ − g H χ 2 H † H → 2 g H m N 9 m h 2 m χ ⏟ − g / Λ 2 m χ χ 2 N ‾ N , (115) \mathcal{L} \supset-g_H \chi^2 H^\dagger H \to\underbrace{\frac{2g_Hm_N}{9m_h^2m_\chi}}_{-g/\Lambda^2}m_\chi\chi^2\overline{N}N, \tag*{(115)} L ⊃ − g H χ 2 H † H → − g / Λ 2 9 m h 2 m χ 2 g H m N m χ χ 2 N N , (115) where g H < 0 g_H < 0 g H < 0 H H H m h m_h m h m N m_N m N [8 ] .12
The hierarchy between the phase-shift and decoherence channels is visible directly in Figure 5 , which shows the projected reach of the AI and CPI benchmarks in Table II to single-component bosonic DM. The main qualitative features are: (i) the oscillation close to the wave-particle boundary, which appears only in the decoherence observable; (ii) the different low- and high-mass scaling of the two observables; and (iii) the different roles played by coherent atom-number enhancement in CPIs and AIs. These features follow from the interplay between the experimental length scales, the DM coherence properties, and the CPI/AI form factors, as summarized below.
Figure 5. Projected constraints on the dimensionful coupling strength g / Λ 2 g/\Lambda^{2} g / Λ 2 m χ m_{\chi} m χ (115) ) for the CPI (green) and AI (purple) benchmarks (see Table II). The projected reach from the phase shift (P \mathbb{P} P D \mathbb{D} D [115 , 121 ] ; in orange we show the regions of parameter space excluded by structure formation [122 ] , the Bullet Cluster [115 , 123 ] , and SN1987A and stellar cooling [124 , 125 ] . In red we show bounds from invisible Higgs decay at the LHC [126 ] . Note that for repulsive interactions the DM is shielded by the laboratory environment [115 , 116 ] , so the projected interferometer constraints are only valid below the dash-dotted black lines for a terrestrial experiment.
a. Phase shifts. The sensitivity from the O ( g 2 ) \mathcal{O}(g^2) O ( g 2 )
m χ ≪ ( ∣ Δ x ∣ v 0 ) − 1 m_\chi\ll(|\Delta x|v_0)^{-1} m χ ≪ ( ∣Δ x ∣ v 0 ) − 1 ∣ q ∣ ≪ ∣ Δ x ∣ − 1 |\mathbf{q}| \ll|\Delta x|^{-1} ∣ q ∣ ≪ ∣Δ x ∣ − 1 p P ∝ ( ∣ q ∣ ∣ Δ x ∣ ) 2 / m χ v 0 Δ x p_{\mathrm{P}} \propto(|\mathbf{q}||\Delta x|)^2/m_\chi v_0\Delta x p P ∝ ( ∣ q ∣∣Δ x ∣ ) 2 / m χ v 0 Δ x (114) ) and DM cannot resolve the superposition. Since r c ≪ ∣ Δ x ∣ r_c \ll|\Delta x| r c ≪ ∣Δ x ∣ ∣ F C P I ∣ 2 ∼ A 2 N a t 2 |F_{\mathrm{CPI}}|^2 \sim A^2N_{\mathrm{at}}^2 ∣ F CPI ∣ 2 ∼ A 2 N at 2 ∣ F A I ∣ 2 ∼ A 2 N a t |F_{\mathrm{AI}}|^2 \sim A^2N_{\mathrm{at}} ∣ F AI ∣ 2 ∼ A 2 N at ∣ q ∣ ∼ m χ v 0 |\mathbf{q}| \sim m_\chi v_0 ∣ q ∣ ∼ m χ v 0 g P ∼ ρ D M / m χ ∣ q ∣ v 0 g_{\mathrm{P}} \sim\rho_{\mathrm{DM}}/m_\chi|\mathbf{q}|v_0 g P ∼ ρ DM / m χ ∣ q ∣ v 0 (112) and (113) ), and P C P I ∼ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M ( m χ v 0 ) 2 ∣ Δ x ∣ \mathbb{P}_{\mathrm{CPI}} \sim T\left(\frac{g}{\Lambda^2}\right)^2N_{\mathrm{at}}^2A^2\rho_{\mathrm{DM}}(m_\chi v_0)^2|\Delta x| P CPI ∼ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM ( m χ v 0 ) 2 ∣Δ x ∣ P A I ∼ T ( g Λ 2 ) 2 N a t A 2 ρ D M ( m χ v 0 ) 2 ∣ Δ x ∣ \mathbb{P}_{\mathrm{AI}} \sim T\left(\frac{g}{\Lambda^2}\right)^2N_{\mathrm{at}}A^2\rho_{\mathrm{DM}}(m_\chi v_0)^2|\Delta x| P AI ∼ T ( Λ 2 g ) 2 N at A 2 ρ DM ( m χ v 0 ) 2 ∣Δ x ∣ 1 / N a t 1/\sqrt{N_{\mathrm{at}}} 1/ N at N e x p N_{\mathrm{exp}} N exp
g Λ 2 ≳ ( A m χ v 0 ) − 1 ( ρ D M ∣ Δ x ∣ T ) − 1 / 2 N e x p − 1 / 4 { N a t − 1 for CPIs , N a t − 3 / 4 for AIs . (116) \frac{g}{\Lambda^2} \gtrsim(Am_\chi v_0)^{-1}(\rho_{\mathrm{DM}}|\Delta x|T)^{-1/2}N_{\mathrm{exp}}^{-1/4}
\begin{cases}
N_{\mathrm{at}}^{-1} & \text{for CPIs},\\
N_{\mathrm{at}}^{-3/4} & \text{for AIs}.
\end{cases} \tag*{(116)} Λ 2 g ≳ ( A m χ v 0 ) − 1 ( ρ DM ∣Δ x ∣ T ) − 1/2 N exp − 1/4 { N at − 1 N at − 3/4 for CPIs , for AIs . (116) m χ ≫ ( ∣ Δ x ∣ v 0 ) − 1 m_\chi\gg(|\Delta x|v_0)^{-1} m χ ≫ ( ∣Δ x ∣ v 0 ) − 1 p P p_{\mathrm{P}} p P ∣ q ∣ ≫ ∣ Δ x ∣ − 1 |\mathbf{q}| \gg|\Delta x|^{-1} ∣ q ∣ ≫ ∣Δ x ∣ − 1 Figure 3 ), the rate is dominated by ∣ q ∣ ∼ ∣ Δ x ∣ − 1 ≪ m χ v 0 |\mathbf{q}| \sim|\Delta x|^{-1} \ll m_\chi v_0 ∣ q ∣ ∼ ∣Δ x ∣ − 1 ≪ m χ v 0 p P ∼ ∣ F P ∣ 2 p_{\mathrm{P}} \sim|F_{\mathrm{P}}|^2 p P ∼ ∣ F P ∣ 2 ∣ F C P I ∣ 2 ∼ N a t 2 A 2 |F_{\mathrm{CPI}}|^2 \sim N_{\mathrm{at}}^2A^2 ∣ F CPI ∣ 2 ∼ N at 2 A 2 ∣ F A I ∣ 2 ∝ N a t A 2 |F_{\mathrm{AI}}|^2 \propto N_{\mathrm{at}}A^2 ∣ F AI ∣ 2 ∝ N at A 2 g P ∼ ρ D M / m χ 2 v 0 2 g_{\mathrm{P}} \sim\rho_{\mathrm{DM}}/m_\chi^2v_0^2 g P ∼ ρ DM / m χ 2 v 0 2 P C P I ∼ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M ( m χ v 0 ) − 2 ∣ Δ x ∣ − 3 \mathbb{P}_{\mathrm{CPI}} \sim T\left(\frac{g}{\Lambda^2}\right)^2N_{\mathrm{at}}^2A^2\rho_{\mathrm{DM}}(m_\chi v_0)^{-2}|\Delta x|^{-3} P CPI ∼ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM ( m χ v 0 ) − 2 ∣Δ x ∣ − 3 P A I ∼ T ( g Λ 2 ) 2 N a t A 2 ρ D M ( m χ v 0 ) − 2 ∣ Δ x ∣ − 3 \mathbb{P}_{\mathrm{AI}} \sim T\left(\frac{g}{\Lambda^2}\right)^2N_{\mathrm{at}}A^2\rho_{\mathrm{DM}}(m_\chi v_0)^{-2}|\Delta x|^{-3} P AI ∼ T ( Λ 2 g ) 2 N at A 2 ρ DM ( m χ v 0 ) − 2 ∣Δ x ∣ − 3
g Λ 2 ≳ ∣ Δ x ∣ 3 / 2 m χ v 0 A − 1 ( ρ D M T ) − 1 / 2 N e x p − 1 / 4 { N a t − 1 for CPIs , N a t − 3 / 4 for AIs . (117) \frac{g}{\Lambda^2} \gtrsim|\Delta x|^{3/2}m_\chi v_0 A^{-1}(\rho_{\mathrm{DM}}T)^{-1/2}N_{\mathrm{exp}}^{-1/4}
\begin{cases}
N_{\mathrm{at}}^{-1} & \text{for CPIs},\\
N_{\mathrm{at}}^{-3/4} & \text{for AIs}.
\end{cases} \tag*{(117)} Λ 2 g ≳ ∣Δ x ∣ 3/2 m χ v 0 A − 1 ( ρ DM T ) − 1/2 N exp − 1/4 { N at − 1 N at − 3/4 for CPIs , for AIs . (117) The scalings above explain why the phase-shift reach is optimized near the transition between the two regimes, m χ ∼ 10 3 / ∣ Δ x ∣ m_\chi\sim10^3/|\Delta x| m χ ∼ 1 0 3 /∣Δ x ∣
g Λ 2 ∣ o p t i m a l ∼ ∣ Δ x ∣ 1 / 2 A − 1 ( ρ D M T ) − 1 / 2 N e x p − 1 / 4 { N a t − 1 for CPIs , N a t − 3 / 4 for AIs . (118) \left.\frac{g}{\Lambda^2}\right|_{\mathrm{optimal}} \sim|\Delta x|^{1/2}A^{-1}(\rho_{\mathrm{DM}}T)^{-1/2}N_{\mathrm{exp}}^{-1/4}
\begin{cases}
N_{\mathrm{at}}^{-1} & \text{for CPIs},\\
N_{\mathrm{at}}^{-3/4} & \text{for AIs}.
\end{cases} \tag*{(118)} Λ 2 g optimal ∼ ∣Δ x ∣ 1/2 A − 1 ( ρ DM T ) − 1/2 N exp − 1/4 { N at − 1 N at − 3/4 for CPIs , for AIs . (118) For the parameters displayed in Table II, this gives g / Λ 2 ∣ o p t i m a l ∼ 10 7 G e V − 2 \left.g/\Lambda^2\right|_{\mathrm{optimal}} \sim10^7\,\mathrm{GeV}^{-2} g / Λ 2 optimal ∼ 1 0 7 GeV − 2 m χ ∼ 4 × 10 − 5 e V m_\chi\sim4 \times10^{-5}\,\mathrm{eV} m χ ∼ 4 × 1 0 − 5 eV g / Λ 2 ∣ o p t i m a l ∼ 10 − 1 G e V − 2 \left.g/\Lambda^2\right|_{\mathrm{optimal}} \sim10^{-1}\,\mathrm{GeV}^{-2} g / Λ 2 optimal ∼ 1 0 − 1 GeV − 2 m χ ∼ 1 e V m_\chi\sim1\,\mathrm{eV} m χ ∼ 1 eV Figure 5 . Thus, even with the coherent atom-number enhancement and shot-noise-limited readout, the phase-shift channel in the AI benchmark does not reach unexplored parameter space once existing astrophysical and laboratory constraints are included (see Appendix § 0.13 for a discussion). By contrast, the CPI benchmark, aided by its much larger effective coherent target and maximal separation ∣ Δ x ∣ ≲ 250 μ m |\Delta x| \lesssim250\,\mu\mathrm{m} ∣Δ x ∣ ≲ 250 μ m
b. Decoherence. The decoherence observable displays much richer phenomenology. This is because the observable is sensitive both to the wave-particle transition and to whether the DM resolves the superposition or the target constituents. The Bose enhancement in the wavelike regime changes the parametric scaling of the two observables with the DM mass and is therefore responsible for the pronounced low-mass improvement of the decoherence curves in Figure 5 . In light of the choice of ∣ Δ x ∣ |\Delta x| ∣Δ x ∣ r c r_c r c
CPI and AI
m χ ≪ ( ∣ Δ x ∣ v 0 ) − 1 m_{\chi} \ll(|\Delta x|v_0)^{-1} m χ ≪ ( ∣Δ x ∣ v 0 ) − 1 ∣ Δ x ∣ |\Delta x| ∣Δ x ∣ m χ ≪ 1 e V m_{\chi} \ll1\,\mathrm{eV} m χ ≪ 1 eV
deep in the wave-like regime for both the CPI and AI benchmarks. As a result, g D ≈ g P ρ D M / m χ 4 v 0 3 g_{\mathrm{D}} \approx g_{\mathrm{P}} \rho_{\mathrm{DM}}/m_{\chi}^{4}v_{0}^{3} g D ≈ g P ρ DM / m χ 4 v 0 3 p D ∝ ∣ q ∣ 2 ∣ Δ x ∣ 2 p_{\mathrm{D}} \propto|\mathbf{q}|^{2}|\Delta\mathbf{x}|^{2} p D ∝ ∣ q ∣ 2 ∣Δ x ∣ 2 ∣ F C P I ∣ 2 ∝ N a t 2 A 2 |F_{\mathrm{CPI}}|^{2} \propto N_{\mathrm{at}}^{2}A^{2} ∣ F CPI ∣ 2 ∝ N at 2 A 2 A 2 A^{2} A 2 ∣ q ∣ ∼ m χ v 0 |\mathbf{q}| \sim m_{\chi}v_{0} ∣ q ∣ ∼ m χ v 0 g D ∼ ρ D M 2 / m χ 5 v 0 4 ∣ q ∣ g_{\mathrm{D}} \sim\rho_{\mathrm{DM}}^{2}/m_{\chi}^{5}v_{0}^{4}|\mathbf{q}| g D ∼ ρ DM 2 / m χ 5 v 0 4 ∣ q ∣ D C P I ∝ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M m χ − 1 ∣ Δ x ∣ 2 \mathbb{D}_{\mathrm{CPI}} \propto T\left(\frac{g}{\Lambda^{2}}\right)^{2}N_{\mathrm{at}}^{2}A^{2}\rho_{\mathrm{DM}}m_{\chi}^{-1}|\Delta\mathbf{x}|^{2} D CPI ∝ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM m χ − 1 ∣Δ x ∣ 2 D A I ∼ T ( g Λ 2 ) 2 A 2 ρ D M m χ − 1 ∣ Δ x ∣ 2 \mathbb{D}_{\mathrm{AI}} \sim T\left(\frac{g}{\Lambda^{2}}\right)^{2}A^{2}\rho_{\mathrm{DM}}m_{\chi}^{-1}|\Delta\mathbf{x}|^{2} D AI ∼ T ( Λ 2 g ) 2 A 2 ρ DM m χ − 1 ∣Δ x ∣ 2
g Λ 2 ≳ m χ 1 / 2 T − 1 / 2 ( A ρ D M ∣ Δ x ∣ ) − 1 N e x p − 1 / 4 { N a t − 1 for CPIs , N a t − 1 / 4 for AIs . (119) \frac{g}{\Lambda^{2}} \gtrsim m_{\chi}^{1/2}T^{-1/2}(A\rho_{\mathrm{DM}}|\Delta\mathbf{x}|)^{-1}N_{\mathrm{exp}}^{-1/4}
\begin{cases}
N_{\mathrm{at}}^{-1} & \text{for CPIs},\\
N_{\mathrm{at}}^{-1/4} & \text{for AIs}.
\end{cases} \tag*{(119)} Λ 2 g ≳ m χ 1/2 T − 1/2 ( A ρ DM ∣Δ x ∣ ) − 1 N exp − 1/4 { N at − 1 N at − 1/4 for CPIs , for AIs . (119) – ( ∣ Δ x ∣ v 0 ) − 1 ≲ m χ ≲ 10 e V (|\Delta\mathbf{x}|v_{0})^{-1} \lesssim m_{\chi} \lesssim10\ \mathrm{eV} ( ∣Δ x ∣ v 0 ) − 1 ≲ m χ ≲ 10 eV g D ≈ g P ρ D M / m χ 4 v 0 3 g_{\mathrm{D}} \approx g_{\mathrm{P}}\rho_{\mathrm{DM}}/m_{\chi}^{4}v_{0}^{3} g D ≈ g P ρ DM / m χ 4 v 0 3 ∣ q ∣ ∼ m χ v 0 |\mathbf{q}| \sim m_{\chi}v_{0} ∣ q ∣ ∼ m χ v 0 ∣ F C P I ∣ 2 ∝ N a t 2 A 2 |F_{\mathrm{CPI}}|^{2} \propto N_{\mathrm{at}}^{2}A^{2} ∣ F CPI ∣ 2 ∝ N at 2 A 2 p D p_{\mathrm{D}} p D q \mathbf{q} q Δ x \Delta\mathbf{x} Δ x g D ∼ ρ D M 2 / m χ 5 v 0 4 ∣ q ∣ g_{\mathrm{D}} \sim\rho_{\mathrm{DM}}^{2}/m_{\chi}^{5}v_{0}^{4}|\mathbf{q}| g D ∼ ρ DM 2 / m χ 5 v 0 4 ∣ q ∣ D C P I ∝ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M m χ − 3 v 0 − 2 \mathbb{D}_{\mathrm{CPI}} \propto T\left(\frac{g}{\Lambda^{2}}\right)^{2}N_{\mathrm{at}}^{2}A^{2}\rho_{\mathrm{DM}}m_{\chi}^{-3}v_{0}^{-2} D CPI ∝ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM m χ − 3 v 0 − 2 D A I ∝ T ( g Λ 2 ) 2 A 2 ρ D M m χ − 3 v 0 − 2 \mathbb{D}_{\mathrm{AI}} \propto T\left(\frac{g}{\Lambda^{2}}\right)^{2}A^{2}\rho_{\mathrm{DM}}m_{\chi}^{-3}v_{0}^{-2} D AI ∝ T ( Λ 2 g ) 2 A 2 ρ DM m χ − 3 v 0 − 2
g Λ 2 ≳ m χ 3 / 2 v 0 T − 1 / 2 ( A ρ D M ) − 1 N e x p − 1 / 4 { N a t − 1 for CPIs , N a t − 1 / 4 for AIs . (120) \frac{g}{\Lambda^{2}} \gtrsim m_{\chi}^{3/2}v_{0}T^{-1/2}(A\rho_{\mathrm{DM}})^{-1}N_{\mathrm{exp}}^{-1/4}
\begin{cases}
N_{\mathrm{at}}^{-1} & \text{for CPIs},\\
N_{\mathrm{at}}^{-1/4} & \text{for AIs}.
\end{cases} \tag*{(120)} Λ 2 g ≳ m χ 3/2 v 0 T − 1/2 ( A ρ DM ) − 1 N exp − 1/4 { N at − 1 N at − 1/4 for CPIs , for AIs . (120) – 10 e V ≲ m χ ≲ ( r c v 0 ) − 1 10\ \mathrm{eV} \lesssim m_{\chi} \lesssim(r_{c}v_{0})^{-1} 10 eV ≲ m χ ≲ ( r c v 0 ) − 1 g D ∼ ρ D M / m χ ∣ q ∣ v 0 g_{\mathrm{D}} \sim\rho_{\mathrm{DM}}/m_{\chi}|\mathbf{q}|v_{0} g D ∼ ρ DM / m χ ∣ q ∣ v 0 ∣ q ∣ = m χ v 0 |\mathbf{q}| = m_{\chi}v_{0} ∣ q ∣ = m χ v 0 ∣ F C P I ∣ 2 ∝ N a t 2 A 2 |F_{\mathrm{CPI}}|^{2} \propto N_{\mathrm{at}}^{2}A^{2} ∣ F CPI ∣ 2 ∝ N at 2 A 2 p D p_{\mathrm{D}} p D q \mathbf{q} q Δ x \Delta\mathbf{x} Δ x D C P I ∝ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M m χ v 0 \mathbb{D}_{\mathrm{CPI}} \propto T\left(\frac{g}{\Lambda^{2}}\right)^{2}N_{\mathrm{at}}^{2}A^{2}\rho_{\mathrm{DM}}m_{\chi}v_{0} D CPI ∝ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM m χ v 0
g Λ 2 ≳ ( ρ D M m χ v 0 T ) − 1 / 2 ( A N a t ) − 1 N e x p − 1 / 4 . (121) \frac{g}{\Lambda^{2}} \gtrsim(\rho_{\mathrm{DM}}m_{\chi}v_{0}T)^{-1/2}(AN_{\mathrm{at}})^{-1}N_{\mathrm{exp}}^{-1/4}. \tag*{(121)} Λ 2 g ≳ ( ρ DM m χ v 0 T ) − 1/2 ( A N at ) − 1 N exp − 1/4 . (121) – m χ ≫ ( r c v 0 ) − 1 m_{\chi} \gg(r_{c}v_{0})^{-1} m χ ≫ ( r c v 0 ) − 1 ∣ q ∣ ≲ r c − 1 |\mathbf{q}| \lesssim r_{c}^{-1} ∣ q ∣ ≲ r c − 1 p D p_{\mathrm{D}} p D q \mathbf{q} q Δ x \Delta\mathbf{x} Δ x g D ∼ ρ D M / m χ ∣ q ∣ v 0 g_{\mathrm{D}} \sim\rho_{\mathrm{DM}}/m_{\chi}|\mathbf{q}|v_{0} g D ∼ ρ DM / m χ ∣ q ∣ v 0 D C P I ∝ T ( g Λ 2 ) 2 N a t 2 A 2 ρ D M ( m χ v 0 ) − 1 r c − 2 \mathbb{D}_{\mathrm{CPI}} \propto T\left(\frac{g}{\Lambda^{2}}\right)^{2}N_{\mathrm{at}}^{2}A^{2}\rho_{\mathrm{DM}}(m_{\chi}v_{0})^{-1}r_{c}^{-2} D CPI ∝ T ( Λ 2 g ) 2 N at 2 A 2 ρ DM ( m χ v 0 ) − 1 r c − 2
g Λ 2 ≳ r c ( m χ v 0 ) 1 / 2 ( ρ D M T ) − 1 / 2 ( A N a t ) − 1 N e x p − 1 / 4 . (122) \frac{g}{\Lambda^{2}} \gtrsim r_{c}(m_{\chi}v_{0})^{1/2}(\rho_{\mathrm{DM}}T)^{-1/2}(AN_{\mathrm{at}})^{-1}N_{\mathrm{exp}}^{-1/4}. \tag*{(122)} Λ 2 g ≳ r c ( m χ v 0 ) 1/2 ( ρ DM T ) − 1/2 ( A N at ) − 1 N exp − 1/4 . (122) − m χ ≫ 10 e V . -\ m_\chi\gg10\ \mathrm{eV}. − m χ ≫ 10 eV . g D ∼ ρ D M / m χ ∣ q ∣ v 0 g_{\mathrm{D}} \sim\rho_{\mathrm{DM}}/m_\chi|\mathbf{q}|v_0 g D ∼ ρ DM / m χ ∣ q ∣ v 0 ∣ q ∣ = m χ v 0 |\mathbf{q}| = m_\chi v_0 ∣ q ∣ = m χ v 0 p D p_{\mathrm{D}} p D q \mathbf{q} q Δ x \Delta x Δ x D A I ∝ T ( g Λ 2 ) 2 A 2 ρ D M m χ v 0 \mathbb{D}_{\mathrm{AI}} \propto T\left(\frac{g}{\Lambda^2}\right)^2 A^2\rho_{\mathrm{DM}}m_\chi v_0 D AI ∝ T ( Λ 2 g ) 2 A 2 ρ DM m χ v 0
g Λ 2 ≳ ( ρ D M m χ v 0 T ) − 1 / 2 A − 1 ( N e x p N a t ) − 1 / 4 . (123) \frac{g}{\Lambda^2} \gtrsim\left(\rho_{\mathrm{DM}}m_\chi v_0T\right)^{-1/2}A^{-1}\left(N_{\mathrm{exp}}N_{\mathrm{at}}\right)^{-1/4}. \tag*{(123)} Λ 2 g ≳ ( ρ DM m χ v 0 T ) − 1/2 A − 1 ( N exp N at ) − 1/4 . (123) The decoherence scalings derived above reproduce the structure of the curves in Figure 5 and clarify the nontrivial hierarchy between the two observables. Importantly, below the wave–particle boundary, Bose enhancement makes decoherence parametrically more sensitive than phase shifts to DM scattering signatures. This also applies to the AI benchmark, despite the large N a t N_{\mathrm{at}} N at
Finally, Figure 5 shows that the Bose-enhanced decoherence reach is not equally useful for the two detector classes once existing constraints and shielding effects are included. For the repulsive interactions assumed here, only the CPI benchmark probes previously unexplored regions of parameter space in which the Earth and atmosphere do not screen the DM as seen by a terrestrial experiment [115 , 116 ] (see the discussion in Appendix § 0.14 ).13
Finally, note that the operator considered here can produce a warm DM subcomponent via freeze-in and thermalize with the SM plasma in the early universe. Therefore, for sufficiently large reheat temperatures, large regions of parameter space are excluded. To evade such bounds, setting a much smaller reheat temperature is consistent with the plot. We therefore leave a detailed study that depends on the specific cosmology to future work.
Long-range interactions
We now consider DM-nucleon interactions mediated by a light hadrophilic scalar ϕ \phi ϕ m ϕ m_\phi m ϕ (93) ). In particular, we shall consider the case where the mediator mass m ϕ m_\phi m ϕ O s = N ‾ N O_s = \overline{N}N O s = N N
S [ O s c l , O s q , ϕ c l , ϕ q , O χ c l , O χ q ] ⊃ − y n ϕ c l O s q − y n ϕ q O s c l − y χ ϕ c l O χ q − y χ ϕ q O χ c l , (124) S[O_s^{\mathrm{cl}}, O_s^{\mathrm{q}}, \phi^{\mathrm{cl}}, \phi^{\mathrm{q}}, O_\chi^{\mathrm{cl}}, O_\chi^{\mathrm{q}}] \supset-y_n\phi^{\mathrm{cl}}O_s^{\mathrm{q}} - y_n\phi^{\mathrm{q}}O_s^{\mathrm{cl}} - y_\chi\phi^{\mathrm{cl}}O_\chi^{\mathrm{q}} - y_\chi\phi^{\mathrm{q}}O_\chi^{\mathrm{cl}}, \tag*{(124)} S [ O s cl , O s q , ϕ cl , ϕ q , O χ cl , O χ q ] ⊃ − y n ϕ cl O s q − y n ϕ q O s cl − y χ ϕ cl O χ q − y χ ϕ q O χ cl , (124) where y χ y_\chi y χ y n y_n y n O χ O_\chi O χ (94) ).
As before, we can obtain the effective action by tracing out environmental degrees of freedom. In this case, however, the environment contains both χ \chi χ ϕ \phi ϕ ρ ^ e = ρ ^ ϕ ⊗ ρ ^ χ \hat{\rho}_e=\hat{\rho}_\phi\otimes\hat{\rho}_\chi ρ ^ e = ρ ^ ϕ ⊗ ρ ^ χ ρ ^ ϕ = ∣ 0 ⟩ ⟨ 0 ∣ \hat{\rho}_\phi=|0\rangle\langle0| ρ ^ ϕ = ∣0 ⟩ ⟨ 0∣ ρ ^ χ \hat{\rho}_\chi ρ ^ χ (124) , it immediately follows that the influence action will depend on the retarded, advanced and Keldysh propagators of both χ \chi χ ϕ \phi ϕ y n y_n y n y χ y_\chi y χ
Because of fifth force constraints from tests of the inverse square law [127 ] and from stellar cooling [55 , 128 , 129 , 130 ] , we will be considering a light mediator with very small couplings to nucleons, i.e., y n ≪ 1 y_n\ll1 y n ≪ 1 m ϕ ≪ 1 G e V m_\phi\ll1\ \mathrm{GeV} m ϕ ≪ 1 GeV χ \chi χ χ \chi χ N e f f N_{\mathrm{eff}} N eff [131 ] . We also fix y χ = 10 − 3 y_\chi=10^{-3} y χ = 1 0 − 3 m ϕ > y χ 2 m χ / ( 4 π ) m_\phi>y_\chi^2m_\chi/(4\pi) m ϕ > y χ 2 m χ / ( 4 π ) [132 , 133 , 134 ] . Furthermore, assuming that this DM subcomponent decoupled from the thermal bath above the QCD phase transition and ensuring that its typical velocity be less than the escape velocity of the Milky Way, we will focus on DM masses m χ ≳ 10 e V m_\chi\gtrsim10\ \mathrm{eV} m χ ≳ 10 eV [13 ] . In the latter case, to be consistent with self-interacting DM (SIDM) constraints [132 ] , we choose y χ 2 / ( 4 π ) = 10 − 10 ( m χ / M e V ) 3 / 2 y_\chi^2/(4\pi)=10^{-10}(m_\chi/\mathrm{MeV})^{3/2} y χ 2 / ( 4 π ) = 1 0 − 10 ( m χ / MeV ) 3/2 [131 ] . Furthermore, to be consistent with warm DM constraints (assuming a thermal production mechanism), we focus on DM masses larger than 10 keV.
As a direct consequence of these assumptions and constraints, we compute the effective action through O ( y n 2 ) \mathcal{O}(y_n^2) O ( y n 2 ) χ \chi χ y χ y_\chi y χ K ^ e \hat{K}_e K ^ e dressed ϕ \phi ϕ D ~ r , a , k ϕ \widetilde{D}^{\phi}_{\mathrm{r,a,k}} D r , a , k ϕ ϕ \phi ϕ ϕ \phi ϕ χ \chi χ
Given the free theory ϕ \phi ϕ
G ~ a ϕ ( p ) = ⋯ = 1 ( p 0 − i ϵ ) 2 − E p 2 , G ~ r ϕ ( p ) = ⋯ = 1 ( p 0 + i ϵ ) 2 − E p 2 , G ~ k ϕ ( p ) = ⋯ = − i 4 E p [ δ ( p 0 − E p ) + δ ( p 0 + E p ) ] . (125) \widetilde{G}^{\phi}_{a}(p)=\cdots=\frac{1}{(p_0-i\epsilon)^2-E_p^2},
\qquad
\widetilde{G}^{\phi}_{r}(p)=\cdots=\frac{1}{(p_0+i\epsilon)^2-E_p^2},
\qquad
\widetilde{G}^{\phi}_{k}(p)=\cdots=-\frac{i}{4E_p}\left[\delta(p_0-E_p)+\delta(p_0+E_p)\right]. \tag*{(125)} G a ϕ ( p ) = ⋯ = ( p 0 − i ϵ ) 2 − E p 2 1 , G r ϕ ( p ) = ⋯ = ( p 0 + i ϵ ) 2 − E p 2 1 , G k ϕ ( p ) = ⋯ = − 4 E p i [ δ ( p 0 − E p ) + δ ( p 0 + E p ) ] . (125) with E p = ∣ p ∣ 2 + m ϕ 2 E_p=\sqrt{|\mathbf{p}|^2+m_\phi^2} E p = ∣ p ∣ 2 + m ϕ 2 ϕ \phi ϕ g Λ 2 O s c l , q O χ c l , q \frac{g}{\Lambda^2}O_s^{\mathrm{cl,q}}O_\chi^{\mathrm{cl,q}} Λ 2 g O s cl , q O χ cl , q (95) to y χ ϕ c l , q χ c l , q y_\chi\phi^{\mathrm{cl,q}}\chi^{\mathrm{cl,q}} y χ ϕ cl , q χ cl , q (124) , it immediately follows that the dressing can be computed using Eq. (98) for real scalar DM and Eq. (103) for fermionic DM. Employing the relation Π ~ a χ ( q ) = [ Π ~ r χ ( q ) ] † \widetilde{\Pi}^{\chi}_{a}(q)=\left[\widetilde{\Pi}^{\chi}_{r}(q)\right]^{\dagger} Π a χ ( q ) = [ Π r χ ( q ) ] † ϕ \phi ϕ (125) ), we find
D ~ r ϕ ( q ) = ⋯ + ⋯ Π ~ r χ ⋯ + ⋯ Π ~ r χ ⋯ Π ~ r χ ⋯ + ⋯ = G ~ r ϕ ( q ) + G ~ r ϕ ( q ) Π ~ r χ ( q ) D ~ r ϕ ( q ) = 1 ( q 0 + i ϵ ) 2 − E q 2 − Π ~ r χ ( q ) , (126) \begin{aligned}
\widetilde{D}^{\phi}_{r}(q)=\cdots+\cdots\widetilde{\Pi}^{\chi}_{r}\cdots+\cdots\widetilde{\Pi}^{\chi}_{r}\cdots\widetilde{\Pi}^{\chi}_{r}\cdots+\cdots\\
&=\widetilde{G}^{\phi}_{r}(q)+\widetilde{G}^{\phi}_{r}(q)\widetilde{\Pi}^{\chi}_{r}(q)\widetilde{D}^{\phi}_{r}(q) \\
&=\frac{1}{(q_{0}+i\epsilon)^{2}-E_{\mathbf{q}}^{2}-\widetilde{\Pi}^{\chi}_{r}(q)}, \tag*{(126)}
\end{aligned} D r ϕ ( q ) = ⋯ + ⋯ Π r χ ⋯ + ⋯ Π r χ ⋯ Π r χ ⋯ + ⋯ = G r ϕ ( q ) + G r ϕ ( q ) Π r χ ( q ) D r ϕ ( q ) = ( q 0 + i ϵ ) 2 − E q 2 − Π r χ ( q ) 1 , (126) D ~ k ϕ ( q ) = ⋯ + ⋯ Π ~ r χ ⋯ + ⋯ Π ~ a χ ⋯ + ⋯ Π ~ k χ ⋯ + ⋯ = D ~ r ϕ ( q ) [ G ~ r ϕ ( q ) − 1 G ~ k ϕ ( q ) G ~ a ϕ ( q ) − 1 + Π ~ k χ ( q ) ] D ~ a ϕ ( q ) , (127) \begin{aligned}
\widetilde{D}^{\phi}_{k}(q)=\cdots+\cdots\widetilde{\Pi}^{\chi}_{r}\cdots+\cdots\widetilde{\Pi}^{\chi}_{a}\cdots+\cdots\widetilde{\Pi}^{\chi}_{k}\cdots+\cdots\\
&=\widetilde{D}^{\phi}_{r}(q)\left[\widetilde{G}^{\phi}_{r}(q)^{-1}\widetilde{G}^{\phi}_{k}(q)\widetilde{G}^{\phi}_{a}(q)^{-1}+\widetilde{\Pi}^{\chi}_{k}(q)\right]\widetilde{D}^{\phi}_{a}(q), \tag*{(127)}
\end{aligned} D k ϕ ( q ) = ⋯ + ⋯ Π r χ ⋯ + ⋯ Π a χ ⋯ + ⋯ Π k χ ⋯ + ⋯ = D r ϕ ( q ) [ G r ϕ ( q ) − 1 G k ϕ ( q ) G a ϕ ( q ) − 1 + Π k χ ( q ) ] D a ϕ ( q ) , (127) where D ~ a ϕ ( q ) = [ D ~ r ϕ ( q ) ] ∗ \widetilde{D}^{\phi}_{a}(q)=\left[\widetilde{D}^{\phi}_{r}(q)\right]^{*} D a ϕ ( q ) = [ D r ϕ ( q ) ] ∗ [135 , 136 ] .
Because of the DM mass range of interest, τ c ≪ T \tau_{c}\ll T τ c ≪ T (104) , such that the MWI observables in Eq. (62) read as
D = − T y n 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] Im [ D ~ k ϕ ( 0 , q ) ] + O ( y n 3 ) , (128) \mathbb{D}=-Ty_{n}^{2}\int\mathrm{d}^{3}q\,\lvert F_{D}(\mathbf{q})\rvert^{2}[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]\operatorname{Im}\left[\widetilde{D}^{\phi}_{k}(0,\mathbf{q})\right]+\mathcal{O}(y_{n}^{3}), \tag*{(128)} D = − T y n 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x )] Im [ D k ϕ ( 0 , q ) ] + O ( y n 3 ) , (128) P = T y n 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) Im [ D ~ r ϕ ( 0 , q ) ] + O ( y n 3 ) . (129) \mathbb{P}=Ty_{n}^{2}\int\mathrm{d}^{3}q\,\lvert F_{P}(\mathbf{q})\rvert^{2}\sin(\mathbf{q}\cdot\Delta\mathbf{x})\operatorname{Im}\left[\widetilde{D}^{\phi}_{r}(0,\mathbf{q})\right]+\mathcal{O}(y_{n}^{3}). \tag*{(129)} P = T y n 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) Im [ D r ϕ ( 0 , q ) ] + O ( y n 3 ) . (129) Because G ~ k ϕ ( 0 , q ) = 0 \widetilde{G}^{\phi}_{k}(0,\mathbf{q})=0 G k ϕ ( 0 , q ) = 0 (125) ), the imaginary parts of the dressed ϕ \phi ϕ
Im [ D ~ r ϕ ( 0 , q ) ] = Im [ Π ~ r χ ( 0 , q ) ] ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 , \operatorname{Im}\left[\widetilde{D}^{\phi}_{r}(0,\mathbf{q})\right]=\frac{\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]}{\left(\lvert\mathbf{q}\rvert^{2}+m_{\phi}^{2}+\operatorname{Re}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]\right)^{2}+\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]^{2}}, Im [ D r ϕ ( 0 , q ) ] = ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q ) ] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 Im [ Π r χ ( 0 , q ) ] , Im [ D ~ k ϕ ( 0 , q ) ] = Im [ Π ~ k χ ( 0 , q ) ] ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 , (130) \operatorname{Im}\left[\widetilde{D}^{\phi}_{k}(0,\mathbf{q})\right]=\frac{\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{k}(0,\mathbf{q})\right]}{\left(\lvert\mathbf{q}\rvert^{2}+m_{\phi}^{2}+\operatorname{Re}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]\right)^{2}+\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]^{2}}, \tag*{(130)} Im [ D k ϕ ( 0 , q ) ] = ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q ) ] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 Im [ Π k χ ( 0 , q ) ] , (130) with Im [ Π ~ r , k χ ( 0 , q ) ] \operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r,k}(0,\mathbf{q})\right] Im [ Π r , k χ ( 0 , q ) ] (106) under the replacement g / Λ 2 → y χ g/\Lambda^{2}\to y_{\chi} g / Λ 2 → y χ
Re [ Π ~ r χ ( 0 , q ) ] ≃ y χ 2 ∫ d 3 p P [ n χ ( p ) − n χ ( p − q ) E p − E p − q ] , (131) \operatorname{Re}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right]\simeq y_{\chi}^{2}\int\mathrm{d}^{3}p\,\mathrm{P}\left[\frac{n_{\chi}(\mathbf{p})-n_{\chi}(\mathbf{p}-\mathbf{q})}{E_{\mathbf{p}}-E_{\mathbf{p}-\mathbf{q}}}\right], \tag*{(131)} Re [ Π r χ ( 0 , q ) ] ≃ y χ 2 ∫ d 3 p P [ E p − E p − q n χ ( p ) − n χ ( p − q ) ] , (131) where P \mathrm{P} P 14
In the heavy-mediator regime, the in-medium ϕ \phi ϕ
y n 2 Im [ D ~ r , k ϕ ( 0 , q ) ] ≃ y n 2 m ϕ 4 Im [ Π ~ r , k χ ( 0 , q ) ] , for m ϕ ≫ ∣ q ∣ . (132) y_{n}^{2}\operatorname{Im}\left[\widetilde{D}^{\phi}_{r,k}(0,\mathbf{q})\right]\simeq\frac{y_{n}^{2}}{m_{\phi}^{4}}\operatorname{Im}\left[\widetilde{\Pi}^{\chi}_{r,k}(0,\mathbf{q})\right],\qquad\text{for }m_{\phi}\gg\lvert\mathbf{q}\rvert. \tag*{(132)} y n 2 Im [ D r , k ϕ ( 0 , q ) ] ≃ m ϕ 4 y n 2 Im [ Π r , k χ ( 0 , q ) ] , for m ϕ ≫ ∣ q ∣ . (132) This reproduces the contact-interaction result under the mapping g → − y χ y n g \to-y_{\chi}y_n g → − y χ y n Λ 2 → m ϕ 2 \Lambda^2 \to m_{\phi}^2 Λ 2 → m ϕ 2 (106) ), as expected from standard EFT arguments. Indeed, in the NR limit and for m ϕ ≫ ∣ q ∣ m_{\phi} \gg\lvert\mathbf{q}\rvert m ϕ ≫ ∣ q ∣ 15
L ⊃ y χ y n m ϕ 2 N ‾ N O χ , [ O χ ] = 3. (133) \mathcal{L} \supset\frac{y_{\chi}y_n}{m_{\phi}^2}\overline{N}N O_{\chi}, \qquad[O_{\chi}] = 3. \tag*{(133)} L ⊃ m ϕ 2 y χ y n N N O χ , [ O χ ] = 3. (133) In the light-mediator regime, the dressed ϕ \phi ϕ Re [ Π ~ r χ ( 0 , q ) ] \operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})] Re [ Π r χ ( 0 , q )] lim ∣ q ∣ → 0 Re [ Π ~ r χ ( 0 , q ) ] ≤ 0 \lim_{\lvert\mathbf{q}\rvert\to0}\operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})] \le0 lim ∣ q ∣ → 0 Re [ Π r χ ( 0 , q )] ≤ 0 ϕ \phi ϕ ϕ \phi ϕ
m ϕ , e f f 2 = m ϕ 2 + lim ∣ q ∣ → 0 Re [ Π ~ r χ ( 0 , q ) ] ≃ m ϕ 2 − 2 y χ 2 ρ χ m χ 2 v 0 2 , (134) m_{\phi,\mathrm{eff}}^2 = m_{\phi}^2 + \lim_{\lvert\mathbf{q}\rvert\to0}\operatorname{Re}\left[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})\right] \simeq m_{\phi}^2 - 2y_{\chi}^2\frac{\rho_{\chi}}{m_{\chi}^2v_0^2}, \tag*{(134)} m ϕ , eff 2 = m ϕ 2 + ∣ q ∣ → 0 lim Re [ Π r χ ( 0 , q ) ] ≃ m ϕ 2 − 2 y χ 2 m χ 2 v 0 2 ρ χ , (134) tachyonic instabilities occur for
m ϕ , e f f 2 < 0 ⟹ m ϕ ≲ y χ 2 ρ χ m χ v 0 ≃ 5 × 10 − 4 e V ( y χ 1 ) ( 1 k e V m χ ) ( 10 − 3 v 0 ) ( ρ χ / ρ D M 0.05 ) 1 / 2 . (135) m_{\phi,\mathrm{eff}}^2 < 0 \Longrightarrow m_{\phi} \lesssim y_{\chi}\frac{\sqrt{2\rho_{\chi}}}{m_{\chi}v_0} \simeq5\times10^{-4}\,\mathrm{eV}\left(\frac{y_{\chi}}{1}\right)\left(\frac{1\,\mathrm{keV}}{m_{\chi}}\right)\left(\frac{10^{-3}}{v_0}\right)\left(\frac{\rho_{\chi}/\rho_{\mathrm{DM}}}{0.05}\right)^{1/2}. \tag*{(135)} m ϕ , eff 2 < 0 ⟹ m ϕ ≲ y χ m χ v 0 2 ρ χ ≃ 5 × 1 0 − 4 eV ( 1 y χ ) ( m χ 1 keV ) ( v 0 1 0 − 3 ) ( 0.05 ρ χ / ρ DM ) 1/2 . (135) Hence, even a small DM subcomponent with large y χ y_{\chi} y χ m χ m_{\chi} m χ m ϕ , e f f 2 m_{\phi,\mathrm{eff}}^2 m ϕ , eff 2 m ϕ , e f f 2 < 0 m_{\phi,\mathrm{eff}}^2 < 0 m ϕ , eff 2 < 0
Using the property Im [ Π ~ r χ ( 0 , − q ) ] = − Im [ Π ~ r χ ( 0 , q ) ] \operatorname{Im}[\widetilde{\Pi}^{\chi}_{r}(0,-\mathbf{q})] = -\operatorname{Im}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})] Im [ Π r χ ( 0 , − q )] = − Im [ Π r χ ( 0 , q )] Re [ Π ~ r χ ( 0 , − q ) ] = Re [ Π ~ r χ ( 0 , q ) ] \operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,-\mathbf{q})] = \operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})] Re [ Π r χ ( 0 , − q )] = Re [ Π r χ ( 0 , q )] (128) and (129) can be written analogously to the contact interactions case:
D = T y n 2 y χ 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] 1 ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 g D ( q ) , (136) \mathbb{D} = T y_n^2 y_{\chi}^2 \int\mathrm{d}^3q \, \lvert F_D(\mathbf{q})\rvert^2 [1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]\frac{1}{\left(\lvert\mathbf{q}\rvert^2+m_{\phi}^2+\operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})]\right)^2+\operatorname{Im}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})]^2}g_D(\mathbf{q}), \tag*{(136)} D = T y n 2 y χ 2 ∫ d 3 q ∣ F D ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x )] ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q )] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 1 g D ( q ) , (136) P = T y n 2 y χ 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) 1 ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 g P ( q ) . (137) \mathbb{P} = T y_n^2 y_{\chi}^2 \int\mathrm{d}^3q \, \lvert F_P(\mathbf{q})\rvert^2 \sin(\mathbf{q}\cdot\Delta\mathbf{x})\frac{1}{\left(\lvert\mathbf{q}\rvert^2+m_{\phi}^2+\operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})]\right)^2+\operatorname{Im}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})]^2}g_P(\mathbf{q}). \tag*{(137)} P = T y n 2 y χ 2 ∫ d 3 q ∣ F P ( q ) ∣ 2 sin ( q ⋅ Δ x ) ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q )] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 1 g P ( q ) . (137) with g D g_D g D g P g_P g P (110) and (111) , respectively. In the dilute DM limit, which is valid in the mass range considered in this section, g P ≈ g D g_P \approx g_D g P ≈ g D ∣ G ~ r ϕ ( 0 , q ) ∣ 2 = ( ∣ q ∣ 2 + m ϕ 2 ) − 2 \lvert\widetilde{G}^{\phi}_{r}(0,\mathbf{q})\rvert^2 = (\lvert\mathbf{q}\rvert^2+m_{\phi}^2)^{-2} ∣ G r ϕ ( 0 , q ) ∣ 2 = (∣ q ∣ 2 + m ϕ 2 ) − 2 [12 , 13 , 15 , 19 ] , the rate depends on ∣ D ~ r ϕ ( 0 , q ) ∣ 2 = [ ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 ] − 1 \lvert\widetilde{D}^{\phi}_{r}(0,\mathbf{q})\rvert^2 = [(\lvert\mathbf{q}\rvert^2+m_{\phi}^2+\operatorname{Re}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})])^2+\operatorname{Im}[\widetilde{\Pi}^{\chi}_{r}(0,\mathbf{q})]^2]^{-1} ∣ D r ϕ ( 0 , q ) ∣ 2 = [(∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q )] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 ] − 1 m ϕ , e f f 2 m_{\phi,\mathrm{eff}}^2 m ϕ , eff 2 ϕ \phi ϕ ∣ q ∣ \lvert\mathbf{q}\rvert ∣ q ∣ ∣ q ∣ − 4 \lvert\mathbf{q}\rvert^{-4} ∣ q ∣ − 4 ∣ q ∣ \lvert\mathbf{q}\rvert ∣ q ∣ ∣ G ~ r ϕ ( 0 , q ) ∣ 2 \lvert\widetilde{G}^{\phi}_{r}(0,\mathbf{q})\rvert^2 ∣ G r ϕ ( 0 , q ) ∣ 2 ∣ D ~ r ϕ ( 0 , q ) ∣ 2 \lvert\widetilde{D}^{\phi}_{r}(0,\mathbf{q})\rvert^2 ∣ D r ϕ ( 0 , q ) ∣ 2 m ϕ − 4 m_{\phi}^{-4} m ϕ − 4 ∼ m ϕ , e f f − 4 \sim m_{\phi,\mathrm{eff}}^{-4} ∼ m ϕ , eff − 4
Projected sensitivity
Equipped with Eqs. (136) and (137) , in Figure 6 and Figure 7 we show the projected reach of the benchmarks to elastic spin-independent DM-nucleon scattering via the exchange of a light hadrophilic scalar. We plot the projected sensitivity to the reference cross section σ ˉ \bar{\sigma} σ ˉ
Figure 6. Projected sensitivity of the CPI (green) benchmark to spin-independent elastic DM-nucleon scattering via the exchange of a light hadrophilic mediator (see Eqs. (93) and (94) ), assuming a 5% DM subcomponent. We plot the reference cross section σ ‾ \overline{\sigma} σ (138) ) as a function of the DM mass m χ m_{\chi} m χ y χ = 10 − 3 y_{\chi}=10^{-3} y χ = 1 0 − 3 R ϕ , χ = m ϕ / m χ R_{\phi,\chi}=m_{\phi}/m_{\chi} R ϕ , χ = m ϕ / m χ 10 − 7 10^{-7} 1 0 − 7 10 − 5 10^{-5} 1 0 − 5 10 − 3 10^{-3} 1 0 − 3 P C P I \mathbb{P}_{\mathrm{CPI}} P CPI D C P I \mathbb{D}_{\mathrm{CPI}} D CPI m ϕ , e f f 2 ≤ 0 m_{\phi,\mathrm{eff}}^{2}\leq0 m ϕ , eff 2 ≤ 0 [127 ] , while in orange we show regions of parameter space excluded by neutron star and stellar cooling [55 , 128 , 129 , 130 ] . Above the black dot-dashed curves, refraction by the Earth or atmosphere can induce order-one fractional changes in the DM speed and density, i.e., focusing.
Figure 7. Projected sensitivity of the CPI (green) benchmark to spin-independent elastic DM-nucleon scattering via the exchange of a light hadrophilic mediator (see Eqs. (93) and (94) ), assuming that χ \chi χ σ ˉ \bar{\sigma} σ ˉ (138) ) as a function of the DM mass m χ m_\chi m χ y χ 2 / ( 4 π ) = 10 − 10 ( m χ / M e V ) 3 / 2 y_\chi^2/(4\pi)=10^{-10}(m_\chi/\mathrm{MeV})^{3/2} y χ 2 / ( 4 π ) = 1 0 − 10 ( m χ / MeV ) 3/2 [131 ] . We fix the ratio R ϕ , χ = m ϕ / m χ R_{\phi,\chi}=m_\phi/m_\chi R ϕ , χ = m ϕ / m χ 10 − 7 10^{-7} 1 0 − 7 10 − 6 10^{-6} 1 0 − 6 10 − 5 10^{-5} 1 0 − 5 Figure 6 .
σ ˉ = y χ 2 y n 2 4 π m χ 2 ( m χ 2 v 0 2 + m ϕ 2 ) 2 , (138) \bar{\sigma} = \frac{y_{\chi}^{2}y_{n}^{2}}{4\pi}\frac{m_{\chi}^{2}}{(m_{\chi}^{2}v_{0}^{2}+m_{\phi}^{2})^{2}}, \tag*{(138)} σ ˉ = 4 π y χ 2 y n 2 ( m χ 2 v 0 2 + m ϕ 2 ) 2 m χ 2 , (138) for fixed ratios R ϕ , χ ≡ m ϕ / m χ R_{\phi,\chi}\equiv m_{\phi}/m_{\chi} R ϕ , χ ≡ m ϕ / m χ m ϕ − 1 m_{\phi}^{-1} m ϕ − 1 [12 ] . Using Eq. (138) and in the dilute DM limit, Eqs. (136) and (137) can be rewritten as
D = 4 π T σ ˉ ρ χ m χ 3 ∫ d 3 q ∣ q ∣ ∣ F D ( q ) ∣ 2 ∣ F ϕ , e f f ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x ) ] g ( q ) + O ( y n 3 ) , P = 4 π T σ ˉ ρ χ m χ 3 ∫ d 3 q ∣ q ∣ ∣ F P ( q ) ∣ 2 ∣ F ϕ , e f f ( q ) ∣ 2 sin ( q ⋅ Δ x ) g ( q ) + O ( y n 3 ) , (139) \begin{aligned}
\mathbb{D} &= 4\pi T\bar{\sigma}\frac{\rho_{\chi}}{m_{\chi}^{3}}\int\frac{\mathrm{d}^{3}q}{|\mathbf{q}|}|F_{\mathbb{D}}(\mathbf{q})|^{2}|\mathcal{F}_{\phi,\mathrm{eff}}(\mathbf{q})|^{2}[1-\cos(\mathbf{q}\cdot\Delta\mathbf{x})]g(\mathbf{q})+\mathcal{O}(y_{n}^{3}),\\
\mathbb{P} &= 4\pi T\bar{\sigma}\frac{\rho_{\chi}}{m_{\chi}^{3}}\int\frac{\mathrm{d}^{3}q}{|\mathbf{q}|}|F_{\mathbb{P}}(\mathbf{q})|^{2}|\mathcal{F}_{\phi,\mathrm{eff}}(\mathbf{q})|^{2}\sin(\mathbf{q}\cdot\Delta\mathbf{x})g(\mathbf{q})+\mathcal{O}(y_{n}^{3}), \tag*{(139)}
\end{aligned} D P = 4 π T σ ˉ m χ 3 ρ χ ∫ ∣ q ∣ d 3 q ∣ F D ( q ) ∣ 2 ∣ F ϕ , eff ( q ) ∣ 2 [ 1 − cos ( q ⋅ Δ x )] g ( q ) + O ( y n 3 ) , = 4 π T σ ˉ m χ 3 ρ χ ∫ ∣ q ∣ d 3 q ∣ F P ( q ) ∣ 2 ∣ F ϕ , eff ( q ) ∣ 2 sin ( q ⋅ Δ x ) g ( q ) + O ( y n 3 ) , (139) with F ϕ , e f f ( q ) \mathcal{F}_{\phi,\mathrm{eff}}(\mathbf{q}) F ϕ , eff ( q ) dressed DM mediator form factor:
∣ F ϕ , e f f ( q ) ∣ 2 = ( m χ 2 v 0 2 + m ϕ 2 ) 2 ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π ~ r χ ( 0 , q ) ] ) 2 + Im [ Π ~ r χ ( 0 , q ) ] 2 . (140) |\mathcal{F}_{\phi,\mathrm{eff}}(\mathbf{q})|^{2}=\frac{(m_{\chi}^{2}v_{0}^{2}+m_{\phi}^{2})^{2}}{(|\mathbf{q}|^{2}+m_{\phi}^{2}+\operatorname{Re}[\widetilde{\Pi}_{r}^{\chi}(0,\mathbf{q})])^{2}+\operatorname{Im}[\widetilde{\Pi}_{r}^{\chi}(0,\mathbf{q})]^{2}}. \tag*{(140)} ∣ F ϕ , eff ( q ) ∣ 2 = ( ∣ q ∣ 2 + m ϕ 2 + Re [ Π r χ ( 0 , q )] ) 2 + Im [ Π r χ ( 0 , q ) ] 2 ( m χ 2 v 0 2 + m ϕ 2 ) 2 . (140) The scaling of the CPI and AI curves can be understood by studying different kinematic regimes. To facilitate this analysis, we define R e f f , ϕ ≡ m ϕ , e f f / m ϕ R_{\mathrm{eff},\phi}\equiv m_{\phi,\mathrm{eff}}/m_{\phi} R eff , ϕ ≡ m ϕ , eff / m ϕ R ϕ , χ ≲ v 0 R_{\phi,\chi}\lesssim v_{0} R ϕ , χ ≲ v 0 m ϕ < m χ v 0 m_{\phi}<m_{\chi}v_{0} m ϕ < m χ v 0 ∣ q ∣ |\mathbf{q}| ∣ q ∣ m ϕ m_{\phi} m ϕ
Phase shift
We begin by studying the sensitivity arising from the phase shift observable.
m χ ≪ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_{\chi}\ll(R_{\phi,\chi}|\Delta\mathbf{x}|)^{-1} m χ ≪ ( R ϕ , χ ∣Δ x ∣ ) − 1 m ϕ ≪ ∣ Δ x ∣ − 1 m_{\phi}\ll|\Delta\mathbf{x}|^{-1} m ϕ ≪ ∣Δ x ∣ − 1 ∣ q ∣ ∼ m ϕ , e f f |\mathbf{q}|\sim m_{\phi,\mathrm{eff}} ∣ q ∣ ∼ m ϕ , eff p P ∝ ∣ q ∣ 2 ∣ Δ x ∣ / m χ v 0 p_{\mathbb{P}}\propto|\mathbf{q}|^{2}|\Delta\mathbf{x}|/m_{\chi}v_{0} p P ∝ ∣ q ∣ 2 ∣Δ x ∣/ m χ v 0 ∣ F ϕ , e f f ∣ 2 ∼ ( m χ v 0 / m ϕ , e f f ) 4 |\mathcal{F}_{\phi,\mathrm{eff}}|^{2}\sim(m_{\chi}v_{0}/m_{\phi,\mathrm{eff}})^{4} ∣ F ϕ , eff ∣ 2 ∼ ( m χ v 0 / m ϕ , eff ) 4 r c ≪ ∣ Δ x ∣ r_{c}\ll|\Delta\mathbf{x}| r c ≪ ∣Δ x ∣ ∣ F C P I ∣ 2 ∝ N a t 2 A 2 |F_{\mathrm{CPI}}|^{2}\propto N_{\mathrm{at}}^{2}A^{2} ∣ F CPI ∣ 2 ∝ N at 2 A 2 ∣ F A I ∣ 2 ∝ N a t A 2 |F_{\mathrm{AI}}|^{2}\propto N_{\mathrm{at}}A^{2} ∣ F AI ∣ 2 ∝ N at A 2 P C P I ∼ T σ ˉ ρ χ A 2 N a t 2 ∣ Δ x ∣ v 0 2 \mathbb{P}_{\mathrm{CPI}}\sim T\bar{\sigma}\rho_{\chi}A^{2}N_{\mathrm{at}}^{2}|\Delta\mathbf{x}|v_{0}^{2} P CPI ∼ T σ ˉ ρ χ A 2 N at 2 ∣Δ x ∣ v 0 2 P A I ∼ T σ ˉ ρ χ A 2 N a t ∣ Δ x ∣ v 0 2 \mathbb{P}_{\mathrm{AI}}\sim T\bar{\sigma}\rho_{\chi}A^{2}N_{\mathrm{at}}|\Delta\mathbf{x}|v_{0}^{2} P AI ∼ T σ ˉ ρ χ A 2 N at ∣Δ x ∣ v 0 2 N e x p N_{\mathrm{exp}} N exp
σ ˉ ≳ N e x p − 1 / 2 ( ρ χ T ∣ Δ x ∣ ) − 1 ( v 0 A ) − 2 { N a t − 2 for CPIs , N a t − 3 / 2 for AIs . (141) \bar{\sigma}\gtrsim N_{\mathrm{exp}}^{-1/2}(\rho_{\chi}T|\Delta\mathbf{x}|)^{-1}(v_{0}A)^{-2}
\begin{cases}
N_{\mathrm{at}}^{-2} & \text{for CPIs},\\
N_{\mathrm{at}}^{-3/2} & \text{for AIs}.
\end{cases} \tag*{(141)} σ ˉ ≳ N exp − 1/2 ( ρ χ T ∣Δ x ∣ ) − 1 ( v 0 A ) − 2 { N at − 2 N at − 3/2 for CPIs , for AIs . (141) Note that the scaling here is independent of m χ m_{\chi} m χ m ϕ , e f f 2 > 0 m_{\phi,\mathrm{eff}}^{2}>0 m ϕ , eff 2 > 0
m χ ≫ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_\chi\gg(R_{\phi,\chi}|\Delta x|)^{-1} m χ ≫ ( R ϕ , χ ∣Δ x ∣ ) − 1 m ϕ ≫ ∣ Δ x ∣ − 1 m_\phi\gg|\Delta x|^{-1} m ϕ ≫ ∣Δ x ∣ − 1 m ϕ , e f f ≫ ∣ Δ x ∣ − 1 m_{\phi,\mathrm{eff}} \gg|\Delta x|^{-1} m ϕ , eff ≫ ∣Δ x ∣ − 1 ∣ q ∣ ∼ ∣ Δ x ∣ − 1 |\mathbf{q}| \sim|\Delta x|^{-1} ∣ q ∣ ∼ ∣Δ x ∣ − 1 ∣ F ϕ , e f f ∣ 2 ∼ ( v 0 / R ϕ , χ R e f f , ϕ ) 4 |F_{\phi,\mathrm{eff}}|^2 \sim(v_0/R_{\phi,\chi}R_{\mathrm{eff},\phi})^4 ∣ F ϕ , eff ∣ 2 ∼ ( v 0 / R ϕ , χ R eff , ϕ ) 4 p P ∝ ( m χ v 0 ∣ Δ x ∣ ) − 1 p_{\mathrm{P}} \propto(m_\chi v_0|\Delta x|)^{-1} p P ∝ ( m χ v 0 ∣Δ x ∣ ) − 1 r c ≪ ∣ Δ x ∣ r_c \ll|\Delta x| r c ≪ ∣Δ x ∣ ∣ F C P I ∣ 2 ∝ N a t 2 A 2 |F_{\mathrm{CPI}}|^2 \propto N_{\mathrm{at}}^2 A^2 ∣ F CPI ∣ 2 ∝ N at 2 A 2 ∣ F A I ∣ 2 ∝ N a t A 2 |F_{\mathrm{AI}}|^2 \propto N_{\mathrm{at}}A^2 ∣ F AI ∣ 2 ∝ N at A 2 P C P I ∼ T σ ˉ ρ χ A 2 N a t 2 v 0 3 ( R ϕ , χ R e f f , ϕ m χ ) − 4 ∣ Δ x ∣ − 3 \mathbb{P}_{\mathrm{CPI}} \sim T\bar{\sigma}\rho_\chi A^2N_{\mathrm{at}}^2v_0^3(R_{\phi,\chi}R_{\mathrm{eff},\phi}m_\chi)^{-4}|\Delta x|^{-3} P CPI ∼ T σ ˉ ρ χ A 2 N at 2 v 0 3 ( R ϕ , χ R eff , ϕ m χ ) − 4 ∣Δ x ∣ − 3 P A I ∼ T σ ˉ ρ χ A 2 N a t v 0 3 ( R ϕ , χ R e f f , ϕ m χ ) − 4 ∣ Δ x ∣ − 3 \mathbb{P}_{\mathrm{AI}} \sim T\bar{\sigma}\rho_\chi A^2N_{\mathrm{at}}v_0^3(R_{\phi,\chi}R_{\mathrm{eff},\phi}m_\chi)^{-4}|\Delta x|^{-3} P AI ∼ T σ ˉ ρ χ A 2 N at v 0 3 ( R ϕ , χ R eff , ϕ m χ ) − 4 ∣Δ x ∣ − 3
σ ˉ ≳ ( R ϕ , χ R e f f , ϕ m χ ) 4 ∣ Δ x ∣ 3 N e x p − 1 / 2 ( ρ χ T ) − 1 ( v 0 A ) − 2 { N a t − 2 for CPIs , N a t − 3 / 2 for AIs . (142) \bar{\sigma} \gtrsim(R_{\phi,\chi}R_{\mathrm{eff},\phi}m_\chi)^4|\Delta x|^3N_{\mathrm{exp}}^{-1/2}(\rho_\chi T)^{-1}(v_0A)^{-2}
\begin{cases}
N_{\mathrm{at}}^{-2} & \text{for CPIs},\\
N_{\mathrm{at}}^{-3/2} & \text{for AIs}.
\end{cases} \tag*{(142)} σ ˉ ≳ ( R ϕ , χ R eff , ϕ m χ ) 4 ∣Δ x ∣ 3 N exp − 1/2 ( ρ χ T ) − 1 ( v 0 A ) − 2 { N at − 2 N at − 3/2 for CPIs , for AIs . (142) In this regime, the reach deteriorates as m χ 4 m_\chi^4 m χ 4 R ϕ , χ 4 R_{\phi,\chi}^4 R ϕ , χ 4 R e f f , ϕ ≪ 1 R_{\mathrm{eff},\phi} \ll1 R eff , ϕ ≪ 1
In light of these scalings, and in the limit where finite density effects are negligible, i.e., R e f f , ϕ ≈ 1 R_{\mathrm{eff},\phi} \approx1 R eff , ϕ ≈ 1 m χ ≲ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_\chi\lesssim(R_{\phi,\chi}|\Delta x|)^{-1} m χ ≲ ( R ϕ , χ ∣Δ x ∣ ) − 1 m χ ≲ ( 4 × 10 − 8 / R ϕ , χ ) e V m_\chi\lesssim(4 \times10^{-8}/R_{\phi,\chi})\,\mathrm{eV} m χ ≲ ( 4 × 1 0 − 8 / R ϕ , χ ) eV m χ ≲ ( 10 − 3 / R ϕ , χ ) e V m_\chi\lesssim(10^{-3}/R_{\phi,\chi})\,\mathrm{eV} m χ ≲ ( 1 0 − 3 / R ϕ , χ ) eV 10 − 35 c m 2 10^{-35}\,\mathrm{cm}^2 1 0 − 35 cm 2 Figure 6 and Figure 7 .
b. Decoherence. We now study the sensitivity from the decoherence observable. Because decoherence in AIs is insensitive to the atom cloud structure, we study the AI and CPI cases separately where appropriate.
CPI and AI
m χ ≪ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_\chi\ll(R_{\phi,\chi}|\Delta x|)^{-1} m χ ≪ ( R ϕ , χ ∣Δ x ∣ ) − 1 m ϕ ≪ ∣ Δ x ∣ − 1 m_\phi\ll|\Delta x|^{-1} m ϕ ≪ ∣Δ x ∣ − 1 ∣ q ∣ ∼ m ϕ , e f f |\mathbf{q}| \sim m_{\phi,\mathrm{eff}} ∣ q ∣ ∼ m ϕ , eff p D ∝ ( ∣ q ∣ ∣ Δ x ∣ ) 2 p_{\mathrm{D}} \propto(|\mathbf{q}||\Delta x|)^2 p D ∝ ( ∣ q ∣∣Δ x ∣ ) 2 N a t N_{\mathrm{at}} N at D C P I ∼ T σ ˉ ρ χ A 2 N a t 2 m χ ∣ Δ x ∣ 2 v 0 3 \mathbb{D}_{\mathrm{CPI}} \sim T\bar{\sigma}\rho_\chi A^2N_{\mathrm{at}}^2m_\chi|\Delta x|^2v_0^3 D CPI ∼ T σ ˉ ρ χ A 2 N at 2 m χ ∣Δ x ∣ 2 v 0 3 D A I ∼ T σ ˉ ρ χ A 2 m χ ∣ Δ x ∣ 2 v 0 3 \mathbb{D}_{\mathrm{AI}} \sim T\bar{\sigma}\rho_\chi A^2m_\chi|\Delta x|^2v_0^3 D AI ∼ T σ ˉ ρ χ A 2 m χ ∣Δ x ∣ 2 v 0 3
σ ˉ ≳ N e x p − 1 / 2 ( ρ χ T m χ ) − 1 ( A ∣ Δ x ∣ ) − 2 v 0 − 3 { N a t − 2 for CPIs , N a t − 1 / 2 for AIs . (143) \bar{\sigma} \gtrsim N_{\mathrm{exp}}^{-1/2}(\rho_\chi Tm_\chi)^{-1}(A|\Delta x|)^{-2}v_0^{-3}
\begin{cases}
N_{\mathrm{at}}^{-2} & \text{for CPIs},\\
N_{\mathrm{at}}^{-1/2} & \text{for AIs}.
\end{cases} \tag*{(143)} σ ˉ ≳ N exp − 1/2 ( ρ χ T m χ ) − 1 ( A ∣Δ x ∣ ) − 2 v 0 − 3 { N at − 2 N at − 1/2 for CPIs , for AIs . (143) As for the phase shift, in this regime finite density effects are not relevant.
AI only
m χ ≫ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_\chi\gg(R_{\phi,\chi}|\Delta x|)^{-1} m χ ≫ ( R ϕ , χ ∣Δ x ∣ ) − 1 m ϕ ≫ ∣ Δ x ∣ − 1 m_\phi\gg|\Delta x|^{-1} m ϕ ≫ ∣Δ x ∣ − 1 p D ∼ ∣ F A ∣ 2 p_{\mathrm{D}} \sim|F_{\mathrm{A}}|^2 p D ∼ ∣ F A ∣ 2 ∣ q ∣ ≳ ∣ Δ x ∣ − 1 |\mathbf{q}| \gtrsim|\Delta x|^{-1} ∣ q ∣ ≳ ∣Δ x ∣ − 1 ∣ q ∣ ∼ m ϕ , e f f |\mathbf{q}| \sim m_{\phi,\mathrm{eff}} ∣ q ∣ ∼ m ϕ , eff m χ = 10 M e V m_\chi= 10\,\mathrm{MeV} m χ = 10 MeV
not suppressed by the nuclear form factor. Hence, D A I ∼ T σ ‾ ρ χ A 2 v 0 3 m χ − 1 ( R ϕ , χ R e f f , ϕ ) − 2 \mathbb{D}_{\mathrm{AI}}\sim T\overline{\sigma}\rho_{\chi}A^{2}v_{0}^{3}m_{\chi}^{-1}(R_{\phi,\chi}R_{\mathrm{eff},\phi})^{-2} D AI ∼ T σ ρ χ A 2 v 0 3 m χ − 1 ( R ϕ , χ R eff , ϕ ) − 2
σ ‾ ≳ ( R ϕ , χ R e f f , ϕ ) 2 m χ ( N e x p N a t ) − 1 / 2 ( ρ χ T ) − 1 A − 2 v 0 − 3 (144) \overline{\sigma}\gtrsim(R_{\phi,\chi}R_{\mathrm{eff},\phi})^{2}m_{\chi}(N_{\mathrm{exp}}N_{\mathrm{at}})^{-1/2}(\rho_{\chi}T)^{-1}A^{-2}v_{0}^{-3} \tag*{(144)} σ ≳ ( R ϕ , χ R eff , ϕ ) 2 m χ ( N exp N at ) − 1/2 ( ρ χ T ) − 1 A − 2 v 0 − 3 (144) Hence, the sensitivity deteriorates as m χ m_{\chi} m χ m ϕ , e f f 2 → 0 m_{\phi,\mathrm{eff}}^{2}\to0 m ϕ , eff 2 → 0
CPI only
– ( R ϕ , χ ∣ Δ x ∣ ) − 1 ≲ m χ ≲ ( R ϕ , χ r c ) − 1 (R_{\phi,\chi}|\Delta x|)^{-1}\lesssim m_{\chi}\lesssim(R_{\phi,\chi}r_{c})^{-1} ( R ϕ , χ ∣Δ x ∣ ) − 1 ≲ m χ ≲ ( R ϕ , χ r c ) − 1 ∣ Δ x ∣ − 1 ≲ m ϕ ≲ r c − 1 |\Delta x|^{-1}\lesssim m_{\phi}\lesssim r_{c}^{-1} ∣Δ x ∣ − 1 ≲ m ϕ ≲ r c − 1 p D ∼ ∣ F D ∣ 2 p_{\mathbb{D}}\sim|F_{\mathbb{D}}|^{2} p D ∼ ∣ F D ∣ 2 ∣ q ∣ ≳ ∣ Δ x ∣ − 1 |\mathbf{q}|\gtrsim|\Delta x|^{-1} ∣ q ∣ ≳ ∣Δ x ∣ − 1 ∣ Δ x ∣ − 1 ≲ m ϕ , e f f ≲ r c − 1 |\Delta x|^{-1}\lesssim m_{\phi,\mathrm{eff}}\lesssim r_{c}^{-1} ∣Δ x ∣ − 1 ≲ m ϕ , eff ≲ r c − 1 ∣ q ∣ ∼ m ϕ , e f f |\mathbf{q}|\sim m_{\phi,\mathrm{eff}} ∣ q ∣ ∼ m ϕ , eff D C P I ∼ T σ ‾ ρ χ A 2 N a t 2 v 0 3 m χ − 1 ( R ϕ , χ R e f f , ϕ ) − 2 \mathbb{D}_{\mathrm{CPI}}\sim T\overline{\sigma}\rho_{\chi}A^{2}N_{\mathrm{at}}^{2}v_{0}^{3}m_{\chi}^{-1}(R_{\phi,\chi}R_{\mathrm{eff},\phi})^{-2} D CPI ∼ T σ ρ χ A 2 N at 2 v 0 3 m χ − 1 ( R ϕ , χ R eff , ϕ ) − 2
σ ‾ ≳ ( R ϕ , χ R e f f , ϕ ) 2 m χ N e x p − 1 / 2 ( ρ χ T ) − 1 ( A N a t ) − 2 v 0 − 3 . (145) \overline{\sigma}\gtrsim(R_{\phi,\chi}R_{\mathrm{eff},\phi})^{2}m_{\chi}N_{\mathrm{exp}}^{-1/2}(\rho_{\chi}T)^{-1}(AN_{\mathrm{at}})^{-2}v_{0}^{-3}. \tag*{(145)} σ ≳ ( R ϕ , χ R eff , ϕ ) 2 m χ N exp − 1/2 ( ρ χ T ) − 1 ( A N at ) − 2 v 0 − 3 . (145) In this regime, the sensitivity also deteriorates with m χ m_{\chi} m χ
− m χ ≳ ( R ϕ , χ r c ) − 1 -m_\chi\gtrsim(R_{\phi,\chi}r_c)^{-1} − m χ ≳ ( R ϕ , χ r c ) − 1 Figure 6 . Here, m ϕ ≫ r c − 1 m_\phi\gg r_c^{-1} m ϕ ≫ r c − 1 ∣ q ∣ ≲ r c − 1 |\mathbf{q}| \lesssim r_c^{-1} ∣ q ∣ ≲ r c − 1 ∣ q ∣ ∼ r c − 1 |\mathbf{q}| \sim r_c^{-1} ∣ q ∣ ∼ r c − 1 m ϕ , e f f ≫ r c − 1 m_{\phi,\mathrm{eff}} \gg r_c^{-1} m ϕ , eff ≫ r c − 1 D C P I ∼ T σ ˉ ρ χ A 2 N a t 2 v 0 3 m χ − 3 ( R ϕ , χ R e f f , ϕ ) − 4 r c − 2 \mathcal{D}_{\mathrm{CPI}} \sim T\bar{\sigma}\rho_\chi A^2N_{\mathrm{at}}^2v_0^3m_\chi^{-3}(R_{\phi,\chi}R_{\mathrm{eff},\phi})^{-4}r_c^{-2} D CPI ∼ T σ ˉ ρ χ A 2 N at 2 v 0 3 m χ − 3 ( R ϕ , χ R eff , ϕ ) − 4 r c − 2
σ ˉ ≳ ( R ϕ , χ R e f f , ϕ ) 4 m χ 3 r c 2 N e x p − 1 / 2 ( ρ χ T ) − 1 ( N a t A ) − 2 v 0 − 3 . (146) \bar{\sigma} \gtrsim(R_{\phi,\chi}R_{\mathrm{eff},\phi})^4m_\chi^3r_c^2N_{\mathrm{exp}}^{-1/2}(\rho_\chi T)^{-1}(N_{\mathrm{at}}A)^{-2}v_0^{-3}. \tag*{(146)} σ ˉ ≳ ( R ϕ , χ R eff , ϕ ) 4 m χ 3 r c 2 N exp − 1/2 ( ρ χ T ) − 1 ( N at A ) − 2 v 0 − 3 . (146) In the limit where finite density effects can be neglected, our results agree exactly with Ref. [13 ] up to differences in the choice of experimental parameters. Because ∣ Δ x ∣ > r c |\Delta\mathbf{x}|>r_c ∣Δ x ∣ > r c m χ ∼ ( R ϕ , χ ∣ Δ x ∣ ) − 1 m_\chi\sim(R_{\phi,\chi}|\Delta\mathbf{x}|)^{-1} m χ ∼ ( R ϕ , χ ∣Δ x ∣ ) − 1 m χ ∼ ( 4 × 10 − 8 / R ϕ , χ ) e V m_\chi\sim(4\times10^{-8}/R_{\phi,\chi})\,\mathrm{eV} m χ ∼ ( 4 × 1 0 − 8 / R ϕ , χ ) eV m χ ∼ ( 10 − 3 / R ϕ , χ ) e V m_\chi\sim(10^{-3}/R_{\phi,\chi})\,\mathrm{eV} m χ ∼ ( 1 0 − 3 / R ϕ , χ ) eV 10 − 35 c m 2 10^{-35}\,\mathrm{cm}^2 1 0 − 35 cm 2 Figure 6 and Figure 7 .
Notably, in light of constraints from stellar cooling and tests of the inverse square law, we can see from Figure 6 that a 5% DM subcomponent with mass between ∼ 10 e V \sim10\,\mathrm{eV} ∼ 10 eV ∼ 10 M e V \sim10\,\mathrm{MeV} ∼ 10 MeV N a t 2 N_{\mathrm{at}}^2 N at 2 not expected to probe any unexplored region of parameter space that is not subject to DM condensation. Furthermore, tachyonic instabilities occur over large regions of parameter space. Finally, note that in large regions of parameter space, DM refraction caused by the Earth and atmosphere can induce order-one changes in the DM speed, and therefore in the DM density, at terrestrial experiments (see Appendix D for a derivation). This is the non-gravitational analogue of gravitational focusing of DM [74 , 75 ] , and has previously been explored in the context of very light scalar mediators [137 ] .16 χ \chi χ Figure 7 . Indeed, the D C P I \mathcal{D}_{\mathrm{CPI}} D CPI
Summary & conclusions
In this work we presented an open EFT for computing DM-induced effects in MWIs. By making use of the SK formalism, we extended the single-atom and contact interactions case considered in Ref. [61 ] to also encompass long-range interactions and realistic MWI experiments employing N a t ≫ 1 N_{\mathrm{at}} \gg1 N at ≫ 1
Firstly, by mapping the open EFT from the field level to system worldlines, we rigorously showed the precise sense in which MWIs are sensitive to DM signatures in the absence of DM-induced corrections to the system worldlines. This is equivalent to the observation that DM can induce changes in the MWI reduced density matrix in the absence of energy deposition. Notably, we showed that this approximation is valid across the DM mass range considered in this work.
Secondly, by extending our analysis to many-atom systems, we recovered multi-atom form factors previously identified in the literature in the context of elastic spin-independent DM-nucleon scattering [11 , 12 , 13 , 14 , 15 ] . Notably, these form factors we derived are independent of the DM-MWI coupling and are valid also in the non-Markovian regime. For CPI experiments, we find that all observables can receive at leading order an N a t 2 N_{\mathrm{at}}^2 N at 2 N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at N a t N_{\mathrm{at}} N at
Most importantly, our results also provide a new phenomenological perspective on DM searches with MWIs. For contact interactions, Bose enhancements parametrically increase the sensitivity of CPIs to scalar DM candidates with masses well below O ( 10 e V ) \mathcal{O}(10\ \mathrm{eV}) O ( 10 eV ) negative correction to the mediator mass squared. For sufficiently small DM masses and/or large mediator-DM coupling strengths, this correction can give rise to a tachyonic instability in relevant regions of DM parameter space. Provided that m ϕ , e f f 2 ≥ 0 m_{\phi,\mathrm{eff}}^2 \ge0 m ϕ , eff 2 ≥ 0
This framework also opens up new avenues for exploring the DM landscape. By neglecting finite DM energy depositions, we automatically excluded contributions to the CPI observable from phonon excitations. Because phonon excitations in solids and other materials are expected to provide promising reach to DM with masses m χ ≳ M e V m_\chi\gtrsim\mathrm{MeV} m χ ≳ MeV [141 ] for a review), we will explore these signatures in future work. Our formalism also clarifies the necessary steps required to compute the reach to heavier DM candidates, where the heavy-probe approximation breaks down. Furthermore, a phenomenological survey of the model-dependent and model-independent constraints at the under-explored wave-particle boundary is warranted.
Finally, our formalism can also be used to go beyond the leading order contribution to the decoherence and phase shift observables, and to study ordinary environmental backgrounds in MWI experiments, such as gas collisions, blackbody radiation, vibration noise, and laser phase noise. Indeed, having identified the relationship between the effective action and MWI observables, and having established how to construct environment kernels using in-in Feynman rules, one can systematically compute corrections to the phase and decoherence for any MWI to arbitrary order in perturbation theory. We leave this more technical direction to future work.
Acknowledgments
We thank Itay Bloch, Federico Cima, Sebastian A. R. Ellis, Angelo Esposito, Roni Harnik, Junwu Huang, Christopher McCabe, Michele Papucci, Pablo Quilez, Jesse Shelfhout, Allic Sivaramakrishnan, and Jordan Wilson-Gerow for discussions. L.B. and K.Z. are supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics under Award Number DE-SC0011632, and by the Walter Burke Institute for Theoretical Physics. K.Z. is also supported by a Simons Investigator award, and by the Heising-Simons Foundation “Observational Signatures of Quantum Gravity” collaboration grant. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. L.B. also thanks the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 390831469), and the CERN Theory group for their hospitality and support during the completion of this work. L.B. would also like to thank the participants of the TH-Institute “Quantum sensing for fundamental physics” for many stimulating discussions. GPT-6 Astra and Claude Fable 5.1 were used under human supervision to produce all figures and to proof-read the paper. We take full responsibility for the accuracy of all aspects of this work.
Single-atom reduced density matrix at finite mass
In this appendix, we show how to derive the expressions for the phase shift and decoherence observable at finite atomic mass m a t m_{\mathrm{at}} m at
In section III A, we showed that, after expanding around the classical trajectories, setting t f = − t i = T t_f=-t_i=\mathcal{T} t f = − t i = T
ρ s α β ( T ) = 1 2 ( m a t 4 π T ) 3 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) ∫ d 3 x f e i m a t 4 T ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) e i S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] Z 0 [ j + , j − ] ∣ j ± = 0 ≡ 1 2 + i 2 ⟨ S I F [ X ‾ α , + − i δ δ j + , X ‾ β , − + i δ δ j − ] Z 0 [ j + , j − ] ⟩ j ± = 0 + O ( g 3 ) , (147) \begin{aligned}
\rho_{\mathrm{s}}^{\alpha\beta}(\mathcal{T}) = \frac{1}{2}\left(\frac{m_{\mathrm{at}}}{4\pi\mathcal{T}}\right)^3 \int\mathrm{d}^3x_{i,+}\int\mathrm{d}^3x_{i,-}\Phi(x_{i,+})\Phi^*(x_{i,-})
\int\mathrm{d}^3x_f\,e^{\frac{\mathrm{i}m_{\mathrm{at}}}{4\mathcal{T}}\left(\lvert x_f-x_{i,+}\rvert^2-\lvert x_f-x_{i,-}\rvert^2\right)}
\left.e^{\mathrm{i}S_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-\mathrm{i}\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+\mathrm{i}\frac{\delta}{\delta j_-}\right]}Z_0[j_+,j_-]\right|_{j_\pm=0} \\
&\equiv\frac{1}{2}+\frac{\mathrm{i}}{2}\left\langle S_{\mathrm{IF}}\left[\overline{X}_{\alpha,+}-\mathrm{i}\frac{\delta}{\delta j_+},\overline{X}_{\beta,-}+\mathrm{i}\frac{\delta}{\delta j_-}\right]Z_0[j_+,j_-]\right\rangle_{j_\pm=0}+\mathcal{O}(g^3), \tag*{(147)}
\end{aligned} ρ s α β ( T ) = 2 1 ( 4 π T m at ) 3 ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) ∫ d 3 x f e 4 T i m at ( ∣ x f − x i , + ∣ 2 − ∣ x f − x i , − ∣ 2 ) e i S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] Z 0 [ j + , j − ] j ± = 0 ≡ 2 1 + 2 i ⟨ S IF [ X α , + − i δ j + δ , X β , − + i δ j − δ ] Z 0 [ j + , j − ] ⟩ j ± = 0 + O ( g 3 ) , (147) (see Eq. (42) and the surrounding discussion). Introducing the vector
W ⃗ ~ ( q , t ; X ± ) T = ( W ~ c l ( q , t ; X ± ) W ~ q ( q , t ; X ± ) ) , (148) \widetilde{\vec{W}}(\mathbf{q},t;X_\pm)^{\mathrm{T}}=\left(\widetilde{\mathcal{W}}^{\mathrm{cl}}(\mathbf{q},t;X_\pm)\ \ \widetilde{\mathcal{W}}^{q}(\mathbf{q},t;X_\pm)\right), \tag*{(148)} W ( q , t ; X ± ) T = ( W cl ( q , t ; X ± ) W q ( q , t ; X ± ) ) , (148) which depends on the worldline histories via Eq. (45) }, and using Eq. (3.10), the second term in last line of {xref}Eq. (A1)` can be rewritten as
\text{2\textsuperscript{nd} term of Eq. (A1)}=-\frac{\mathrm{i}}{4}\left(\frac{g}{\Lambda_\phi}\right)^2\int_{-\mathcal{T}}^{\mathcal{T}}\mathrm{d}t\int_{-\mathcal{T}}^{\mathcal{T}}\mathrm{d}t'\int\mathrm{d}^3q\,\lvert F_A(\mathbf{q})\rvert^2
\left\langle\widetilde{\vec{W}}\left(-\mathbf{q},t;\overline{X}_\pm+\mathrm{i}\frac{\delta}{\delta j_\pm}\right)\overline{K}(\mathbf{q},t-t')\widetilde{\vec{W}}\left(\mathbf{q},t';\overline{X}_\pm+\mathrm{i}\frac{\delta}{\delta j_\pm}\right)\right\rangle Z_0[j_+,j_-]\bigg|_{j_\pm=0}. \tag*{(149)}
In light of Eqs. (148) and (45) , we can factorise the basic worldline-dependent objects entering Eq. (149) into classical path dependent and worldline fluctuation dependent pieces: for m , n ∈ { + , − } m,n\in\{+,-\} m , n ∈ { + , − }
\text{2\textsuperscript{nd} line of Eq. (A3)}\supset\left\langle e^{-\mathrm{i}\mathbf{q}\cdot\overline{X}_m(t)}e^{+\mathrm{i}\mathbf{q}\cdot\overline{X}_n(t')}\right\rangle e^{-m\mathbf{q}\cdot\frac{\delta}{\delta j_m(t)}}e^{n\mathbf{q}\cdot\frac{\delta}{\delta j_n(t')}}Z_0[j_+,j_-]\bigg|_{j_\pm=0}, \tag*{(150)}
where θ m n ( t , t ′ ) \theta_{mn}(t,t') θ mn ( t , t ′ ) (41) ), the phases θ m n ( t , t ′ ) \theta_{mn}(t,t') θ mn ( t , t ′ ) (37) ) as
θ + + ( t , t ′ ) = − θ − − ( t , t ′ ) = ∣ q ∣ 2 2 [ G d ( t , t ) + G d ( t ′ , t ′ ) − 2 G d ( t , t ′ ) ] , \theta_{++}(t,t')=-\theta_{--}(t,t')=\frac{\lvert\mathbf{q}\rvert^2}{2}\left[G_d(t,t)+G_d(t',t')-2G_d(t,t')\right], θ ++ ( t , t ′ ) = − θ −− ( t , t ′ ) = 2 ∣ q ∣ 2 [ G d ( t , t ) + G d ( t ′ , t ′ ) − 2 G d ( t , t ′ ) ] , θ + − ( t , t ′ ) = − θ − + ( t , t ′ ) = ∣ q ∣ 2 2 [ G d ( t , t ) − G d ( t ′ , t ′ ) ] . (151) \theta_{+-}(t,t')=-\theta_{-+}(t,t')=\frac{\lvert\mathbf{q}\rvert^2}{2}\left[G_d(t,t)-G_d(t',t')\right]. \tag*{(151)} θ +− ( t , t ′ ) = − θ −+ ( t , t ′ ) = 2 ∣ q ∣ 2 [ G d ( t , t ) − G d ( t ′ , t ′ ) ] . (151) Using Eq. (36) , the classical (unperturbed) worldline-dependent contribution in Eq. (150) reads as
⟨ e − i q ⋅ X ‾ m ( t ) e + i q ⋅ X ‾ n ( t ′ ) ⟩ = ⟨ e − i q ⋅ [ x i , m ( T − t 2 T ) − x i , n ( T − t ′ 2 T ) + x f ( t − t ′ 2 T ) + x ‾ m ( t ) − x ‾ n ( t ′ ) ] ⟩ = e − i q ⋅ ( x ‾ m ( t ) − x ‾ n ( t ′ ) ) × ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) δ ( 3 ) ( x i , − − x i , + + t ′ − t m a t q ) × e − i q ⋅ [ x i , m ( T − t 2 T ) − x i , n ( T − t ′ 2 T ) ] e i m a t 4 T ( ∣ x i , + ∣ 2 − ∣ x i , − ∣ 2 ) = e − i q ⋅ ( x ‾ m ( t ) − x ‾ n ( t ′ ) ) e i ϕ m n ( t , t ′ ) ∫ d 3 x i , + Φ ( x i , + ) Φ ∗ ( x i , + + t − t ′ m a t q ) , (152) \begin{aligned}
\left\langle e^{-i\mathbf{q}\cdot\overline{\mathbf{X}}_m(t)}e^{+i\mathbf{q}\cdot\overline{\mathbf{X}}_n(t')}\right\rangle
&=\left\langle e^{-i\mathbf{q}\cdot\left[\mathbf{x}_{i,m}\left(\frac{\mathcal{T}-t}{2\mathcal{T}}\right)-\mathbf{x}_{i,n}\left(\frac{\mathcal{T}-t'}{2\mathcal{T}}\right)+\mathbf{x}_f\left(\frac{t-t'}{2\mathcal{T}}\right)+\overline{\mathbf{x}}_m(t)-\overline{\mathbf{x}}_n(t')\right]}\right\rangle\\
&=e^{-i\mathbf{q}\cdot\left(\overline{\mathbf{x}}_m(t)-\overline{\mathbf{x}}_n(t')\right)}\\
&\quad\times\int\mathrm{d}^3\mathbf{x}_{i,+}\int\mathrm{d}^3\mathbf{x}_{i,-}\,\Phi(\mathbf{x}_{i,+})\Phi^*(\mathbf{x}_{i,-})\delta^{(3)}\left(\mathbf{x}_{i,-}-\mathbf{x}_{i,+}+\frac{t'-t}{m_{\mathrm{at}}}\mathbf{q}\right)\\
&\quad\times e^{-i\mathbf{q}\cdot\left[\mathbf{x}_{i,m}\left(\frac{\mathcal{T}-t}{2\mathcal{T}}\right)-\mathbf{x}_{i,n}\left(\frac{\mathcal{T}-t'}{2\mathcal{T}}\right)\right]}e^{\frac{i m_{\mathrm{at}}}{4\mathcal{T}}\left(|\mathbf{x}_{i,+}|^2-|\mathbf{x}_{i,-}|^2\right)}\\
&=e^{-i\mathbf{q}\cdot\left(\overline{\mathbf{x}}_m(t)-\overline{\mathbf{x}}_n(t')\right)}e^{i\phi_{mn}(t,t')}\int\mathrm{d}^3\mathbf{x}_{i,+}\,\Phi(\mathbf{x}_{i,+})\Phi^*\left(\mathbf{x}_{i,+}+\frac{t-t'}{m_{\mathrm{at}}}\mathbf{q}\right), \tag*{(152)}
\end{aligned} ⟨ e − i q ⋅ X m ( t ) e + i q ⋅ X n ( t ′ ) ⟩ = ⟨ e − i q ⋅ [ x i , m ( 2 T T − t ) − x i , n ( 2 T T − t ′ ) + x f ( 2 T t − t ′ ) + x m ( t ) − x n ( t ′ ) ] ⟩ = e − i q ⋅ ( x m ( t ) − x n ( t ′ ) ) × ∫ d 3 x i , + ∫ d 3 x i , − Φ ( x i , + ) Φ ∗ ( x i , − ) δ ( 3 ) ( x i , − − x i , + + m at t ′ − t q ) × e − i q ⋅ [ x i , m ( 2 T T − t ) − x i , n ( 2 T T − t ′ ) ] e 4 T i m at ( ∣ x i , + ∣ 2 − ∣ x i , − ∣ 2 ) = e − i q ⋅ ( x m ( t ) − x n ( t ′ ) ) e i ϕ mn ( t , t ′ ) ∫ d 3 x i , + Φ ( x i , + ) Φ ∗ ( x i , + + m at t − t ′ q ) , (152) where, in going to the second line, the integral over x f \mathbf{x}_f x f
ϕ m n ( t , t ′ ) = ∣ q ∣ 2 ( t − t ′ ) 2 m a t T [ ν n ( T − t ′ ) − ν m ( T − t ) ] = ∣ q ∣ 2 ( t − t ′ ) 2 4 m a t T , with ν + = 0 , ν − = 1 , (153) \phi_{mn}(t,t')=\frac{|\mathbf{q}|^2(t-t')}{2m_{\mathrm{at}}\mathcal{T}}\left[\nu_n(\mathcal{T}-t')-\nu_m(\mathcal{T}-t)\right]=\frac{|\mathbf{q}|^2(t-t')^2}{4m_{\mathrm{at}}\mathcal{T}},
\qquad\text{with }\nu_+=0,\ \nu_-=1, \tag*{(153)} ϕ mn ( t , t ′ ) = 2 m at T ∣ q ∣ 2 ( t − t ′ ) [ ν n ( T − t ′ ) − ν m ( T − t ) ] = 4 m at T ∣ q ∣ 2 ( t − t ′ ) 2 , with ν + = 0 , ν − = 1 , (153) and the time-dependent wavepacket form factor F W ( q , t − t ′ ) F_{\mathrm{W}}(\mathbf{q},t-t') F W ( q , t − t ′ ) (151) –(153) , and defining
φ m n ( t − t ′ ) ≡ ϕ m n ( t , t ′ ) + θ m n ( t , t ′ ) , (154) \varphi_{mn}(t-t')\equiv\phi_{mn}(t,t')+\theta_{mn}(t,t'), \tag*{(154)} φ mn ( t − t ′ ) ≡ ϕ mn ( t , t ′ ) + θ mn ( t , t ′ ) , (154) which parametrizes the eikonal phase due to environment-induced recoils, Eq. (150) reads as
⟨ e − i q ⋅ X ‾ m ( t ) e + i q ⋅ X ‾ n ( t ′ ) ⟩ e i θ m n ( t , t ′ ) = e − i q ⋅ ( x ‾ m ( t ) − x ‾ n ( t ′ ) ) e i φ m n ( t − t ′ ) F W ( q , t − t ′ ) , (155) \left\langle e^{-i\mathbf{q}\cdot\overline{\mathbf{X}}_m(t)}e^{+i\mathbf{q}\cdot\overline{\mathbf{X}}_n(t')}\right\rangle e^{i\theta_{mn}(t,t')}=e^{-i\mathbf{q}\cdot\left(\overline{\mathbf{x}}_m(t)-\overline{\mathbf{x}}_n(t')\right)}e^{i\varphi_{mn}(t-t')}F_{\mathrm{W}}(\mathbf{q},t-t'), \tag*{(155)} ⟨ e − i q ⋅ X m ( t ) e + i q ⋅ X n ( t ′ ) ⟩ e i θ mn ( t , t ′ ) = e − i q ⋅ ( x m ( t ) − x n ( t ′ ) ) e i φ mn ( t − t ′ ) F W ( q , t − t ′ ) , (155) where
φ + + ( t − t ′ ) = − φ − − ( t − t ′ ) = − ∣ q ∣ 2 2 m a t ∣ t − t ′ ∣ , \varphi_{++}(t-t')=-\varphi_{--}(t-t')=-\frac{|\mathbf{q}|^2}{2m_{\mathrm{at}}}|t-t'|, φ ++ ( t − t ′ ) = − φ −− ( t − t ′ ) = − 2 m at ∣ q ∣ 2 ∣ t − t ′ ∣ , φ + − ( t − t ′ ) = − φ − + ( t − t ′ ) = ∣ q ∣ 2 2 m a t ( t − t ′ ) , (156) \varphi_{+-}(t-t')=-\varphi_{-+}(t-t')=\frac{|\mathbf{q}|^2}{2m_{\mathrm{at}}}(t-t'), \tag*{(156)} φ +− ( t − t ′ ) = − φ −+ ( t − t ′ ) = 2 m at ∣ q ∣ 2 ( t − t ′ ) , (156) and
F W ( q , t − t ′ ) = e − ∣ q ∣ 2 ( t − t ′ ) 2 / 8 m a t 2 σ 2 , (157) F_{\mathrm{W}}(\mathbf{q},t-t')=e^{-|\mathbf{q}|^2(t-t')^2/8m_{\mathrm{at}}^2\sigma^2}, \tag*{(157)} F W ( q , t − t ′ ) = e − ∣ q ∣ 2 ( t − t ′ ) 2 /8 m at 2 σ 2 , (157) for Gaussian wavepackets with width σ \sigma σ
As is evident from Eqs. (155) –(157) , finite mass corrections enter the observable via φ m n ( t − t ′ ) \varphi_{mn}(t-t') φ mn ( t − t ′ ) F W ( q , t − t ′ ) F_{\mathrm{W}}(\mathbf{q},t-t') F W ( q , t − t ′ ) t , t ′ ∈ [ − T s / 2 , T s / 2 ] t,t'\in[-T_s/2,T_s/2] t , t ′ ∈ [ − T s /2 , T s /2 ] T s T_s T s K ~ ( q ; t − t ′ ) \widetilde{K}(\mathbf{q};t-t') K ( q ; t − t ′ ) (149) has support for ∣ t − t ′ ∣ ≲ τ e |t-t'|\lesssim\tau_e ∣ t − t ′ ∣ ≲ τ e τ e \tau_e τ e ∣ q ∣ 2 τ e f f / m a t |\mathbf{q}|^2\tau_{\mathrm{eff}}/m_{\mathrm{at}} ∣ q ∣ 2 τ eff / m at ∣ q ∣ 2 τ e f f 2 / m a t 2 σ 2 ∼ ∣ q ∣ 2 τ e f f 2 σ p 2 / m a t 2 |\mathbf{q}|^2\tau_{\mathrm{eff}}^2/m_{\mathrm{at}}^2\sigma^2\sim|\mathbf{q}|^2\tau_{\mathrm{eff}}^2\sigma_p^2/m_{\mathrm{at}}^2 ∣ q ∣ 2 τ eff 2 / m at 2 σ 2 ∼ ∣ q ∣ 2 τ eff 2 σ p 2 / m at 2 τ e f f = min ( T s , τ e ) \tau_{\mathrm{eff}}=\min(T_s,\tau_e) τ eff = min ( T s , τ e ) σ p ∼ 1 / σ \sigma_p\sim1/\sigma σ p ∼ 1/ σ
The heavy-probe limit can then be easily taken at fixed T \mathcal{T} T σ \sigma σ φ m n \varphi_{mn} φ mn F W ( q , t − t ′ ) → 1 F_{\mathrm{W}}(q,t-t')\to1 F W ( q , t − t ′ ) → 1
lim m a t → ∞ ⟨ e − i q ⋅ X ‾ m ( t ) e + i q ⋅ X ‾ n ( t ′ ) ⟩ e i θ m n ( t , t ′ ) = e − i q ⋅ ( x m ( t ) − x n ( t ′ ) ) (158) \lim_{m_{\mathrm{at}}\to\infty}\left\langle e^{-i\mathbf{q}\cdot\overline{\mathbf{X}}_m(t)}e^{+i\mathbf{q}\cdot\overline{\mathbf{X}}_n(t')} \right\rangle e^{i\theta_{mn}(t,t')}=e^{-i\mathbf{q}\cdot(\mathbf{x}_m(t)-\mathbf{x}_n(t'))} \tag*{(158)} m at → ∞ lim ⟨ e − i q ⋅ X m ( t ) e + i q ⋅ X n ( t ′ ) ⟩ e i θ mn ( t , t ′ ) = e − i q ⋅ ( x m ( t ) − x n ( t ′ )) (158) and the driven classical histories remain encoded in the influence action via the particular solution x ˉ ± \bar{x}_{\pm} x ˉ ± (36) ). Taking the asymptotic limit T → ∞ \mathcal{T}\to\infty T → ∞ (149) simplifies to Eq. (46) .
Derivation of the system currents and geometry dependence
In this appendix, we explicitly compute the Fourier transform of the system charge current
∫ d 4 x ρ ( x − x ˉ α ( t ) ) e i q ⋅ x = F A ( q ) W ~ ( q ; x ˉ α ) (159) \int\mathrm{d}^{4}x\,\rho(x-\bar{x}_{\alpha}(t))e^{i\mathbf{q}\cdot x}=F_A(\mathbf{q})\widetilde{\mathcal{W}}(q;\bar{x}_{\alpha}) \tag*{(159)} ∫ d 4 x ρ ( x − x ˉ α ( t )) e i q ⋅ x = F A ( q ) W ( q ; x ˉ α ) (159) where F A ( q ) F_A(\mathbf{q}) F A ( q ) x ˉ α ( t ) \bar{x}_{\alpha}(t) x ˉ α ( t ) α ∈ { L , R } \alpha\in\{L,R\} α ∈ { L , R } W ~ ( q ; x ˉ α ) \widetilde{\mathcal{W}}(q;\bar{x}_{\alpha}) W ( q ; x ˉ α ) (159) depends on the details of the interferometric sequence.
We begin by considering the split and hold interferometric configuration, corresponding to the geometry where the two wavepackets are held at fixed relative separation for the majority of the interferometric sequence (see Figure 2 ). In this case,
x ˉ L ( t ) ≈ { − Δ x / 2 for ∣ t ∣ ≲ T / 2 0 otherwise , x ˉ R ( t ) ≈ { + Δ x / 2 for ∣ t ∣ ≲ T / 2 0 otherwise (160) \bar{x}_L(t)\approx
\begin{cases}
-\Delta x/2 & \text{for } |t|\lesssim\mathcal{T}/2 \\
0 & \text{otherwise}
\end{cases},
\qquad
\bar{x}_R(t)\approx
\begin{cases}
+\Delta x/2 & \text{for } |t|\lesssim\mathcal{T}/2 \\
0 & \text{otherwise}
\end{cases} \tag*{(160)} x ˉ L ( t ) ≈ { − Δ x /2 0 for ∣ t ∣ ≲ T /2 otherwise , x ˉ R ( t ) ≈ { + Δ x /2 0 for ∣ t ∣ ≲ T /2 otherwise (160) where ∣ Δ x ∣ |\Delta x| ∣Δ x ∣ W ~ ( q ; x ˉ L ) \widetilde{\mathcal{W}}(q;\bar{x}_L) W ( q ; x ˉ L ) ∣ t ∣ ≲ T / 2 |t|\lesssim\mathcal{T}/2 ∣ t ∣ ≲ T /2 ∣ t ∣ ≳ T / 2 |t|\gtrsim\mathcal{T}/2 ∣ t ∣ ≳ T /2
W ~ ( q ; x ˉ L ) = ∫ − ∞ ∞ d t e i q 0 t e − i q ⋅ x ˉ L ( t ) ≈ ∫ − ∞ − T / 2 d t e i q 0 t + ∫ − T / 2 T / 2 d t e i q 0 t e i q ⋅ Δ x / 2 + ∫ T / 2 ∞ d t e i q 0 t = δ ˉ ( q 0 ) + T sinc ( q 0 T 2 ) ( e i q ⋅ Δ x / 2 − 1 ) (161) \begin{aligned}
\widetilde{\mathcal{W}}(q;\bar{x}_L)&=\int_{-\infty}^{\infty}\mathrm{d}t\,e^{iq_0t}e^{-i\mathbf{q}\cdot\bar{x}_L(t)}\\
&\approx\int_{-\infty}^{-\mathcal{T}/2}\mathrm{d}t\,e^{iq_0t}+\int_{-\mathcal{T}/2}^{\mathcal{T}/2}\mathrm{d}t\,e^{iq_0t}e^{i\mathbf{q}\cdot\Delta x/2}+\int_{\mathcal{T}/2}^{\infty}\mathrm{d}t\,e^{iq_0t}\\
&=\bar{\delta}(q_0)+\mathcal{T}\operatorname{sinc}\left(\frac{q_0\mathcal{T}}{2}\right)\left(e^{i\mathbf{q}\cdot\Delta x/2}-1\right) \tag*{(161)}
\end{aligned} W ( q ; x ˉ L ) = ∫ − ∞ ∞ d t e i q 0 t e − i q ⋅ x ˉ L ( t ) ≈ ∫ − ∞ − T /2 d t e i q 0 t + ∫ − T /2 T /2 d t e i q 0 t e i q ⋅ Δ x /2 + ∫ T /2 ∞ d t e i q 0 t = δ ˉ ( q 0 ) + T sinc ( 2 q 0 T ) ( e i q ⋅ Δ x /2 − 1 ) (161) where we made use of ∫ − T / 2 T / 2 d t e i q 0 t = T sinc ( q 0 T / 2 ) \int_{-\mathcal{T}/2}^{\mathcal{T}/2}\mathrm{d}t\,e^{iq_0t}=\mathcal{T}\operatorname{sinc}(q_0\mathcal{T}/2) ∫ − T /2 T /2 d t e i q 0 t = T sinc ( q 0 T /2 )
∫ d 4 x ρ ( x − x ˉ L ( t ) ) e i q ⋅ x ≈ F A ( q ) [ δ ˉ ( q 0 ) + T sinc ( q 0 T 2 ) ( e i q ⋅ Δ x / 2 − 1 ) ] (162) \int\mathrm{d}^{4}x\,\rho(x-\bar{x}_L(t))e^{i\mathbf{q}\cdot x}\approx F_A(\mathbf{q})\left[\bar{\delta}(q_0)+\mathcal{T}\operatorname{sinc}\left(\frac{q_0\mathcal{T}}{2}\right)\left(e^{i\mathbf{q}\cdot\Delta x/2}-1\right)\right] \tag*{(162)} ∫ d 4 x ρ ( x − x ˉ L ( t )) e i q ⋅ x ≈ F A ( q ) [ δ ˉ ( q 0 ) + T sinc ( 2 q 0 T ) ( e i q ⋅ Δ x /2 − 1 ) ] (162) where W ~ ( q ; x ˉ R ) = W ~ ( q ; − x ˉ L ) \widetilde{\mathcal{W}}(q;\bar{x}_R)=\widetilde{\mathcal{W}}(q;-\bar{x}_L) W ( q ; x ˉ R ) = W ( q ; − x ˉ L )
Mach–Zehnder sequences
Mach–Zehnder sequences [103 ] as viewed in the mid-point trajectory frame [142 ] are depicted in Figure 8 . These can be derived from Eq. (51) by setting T = 0 T=0 T = 0 T ′ → T T' \to T T ′ → T
Figure 8. Spacetime diagram for the Mach–Zehnder configuration in the mid-point trajectory frame, which is a limiting case of the split and hold geometry for T = 0 T=0 T = 0 T ′ → T T' \to T T ′ → T T s T_s T s Δ x \Delta x Δ x x ˉ L , R ( t ) \bar{x}_{L,R}(t) x ˉ L , R ( t ) (35) and are defined in terms of external currents in Eq. (36) , are shown in solid orange and dashed green.
x ˉ L ( t ) = { − v ( t + T / 2 ) / 2 for − T / 2 ≤ t ≤ 0 , − v ( T / 2 − t ) / 2 for 0 ≤ t ≤ T / 2 , 0 otherwise , x ˉ R ( t ) = { v ( t + T / 2 ) / 2 for − T / 2 ≤ t ≤ 0 , v ( T / 2 − t ) / 2 for 0 ≤ t ≤ T / 2 , 0 otherwise . (163) \begin{aligned}
\bar{x}_L(t) &=
\begin{cases}
-v(t+T/2)/2 & \text{for } -T/2 \le t \le0,\\
-v(T/2-t)/2 & \text{for } 0 \le t \le T/2,\\
0 & \text{otherwise},
\end{cases}\\
\bar{x}_R(t) &=
\begin{cases}
v(t+T/2)/2 & \text{for } -T/2 \le t \le0,\\
v(T/2-t)/2 & \text{for } 0 \le t \le T/2,\\
0 & \text{otherwise}.
\end{cases} \tag*{(163)}
\end{aligned} x ˉ L ( t ) x ˉ R ( t ) = ⎩ ⎨ ⎧ − v ( t + T /2 ) /2 − v ( T /2 − t ) /2 0 for − T /2 ≤ t ≤ 0 , for 0 ≤ t ≤ T /2 , otherwise , = ⎩ ⎨ ⎧ v ( t + T /2 ) /2 v ( T /2 − t ) /2 0 for − T /2 ≤ t ≤ 0 , for 0 ≤ t ≤ T /2 , otherwise . (163) where ∣ Δ x ∣ = ∣ v ∣ T / 2 |\Delta x|=|v|T/2 ∣Δ x ∣ = ∣ v ∣ T /2
With these paths, the worldline-dependent part of Eq. (159) can be evaluated using the same steps as above.
Introducing the variable
α ± = q 0 ± q ⋅ v 2 , (164) \alpha_{\pm}=q_0\pm\frac{\mathbf{q}\cdot\mathbf{v}}{2}, \tag*{(164)} α ± = q 0 ± 2 q ⋅ v , (164) the left path contribution may be expressed as
W ~ ( q ; x L ) = ∫ − ∞ ∞ d t e i q 0 t e − i q ⋅ x L ( t ) = δ ˉ ( q 0 ) − ∫ − T / 2 T / 2 d t e i q 0 t + e i q ⋅ v T / 4 [ ∫ − T / 2 0 d t e i α + t + ∫ 0 T / 2 d t e i α − t ] = δ ˉ ( q 0 ) − T sinc ( q 0 T 2 ) + T 2 e i q ⋅ Δ x / 4 [ e − i q 0 T / 4 sinc ( α + T 4 ) + e i q 0 T / 4 sinc ( α − T 4 ) ] . (165) \begin{aligned}
\widetilde{W}(q;\mathbf{x}_L)=\int_{-\infty}^{\infty}dt\,e^{iq_0t}e^{-i\mathbf{q}\cdot\mathbf{x}_L(t)} \\
&=\bar{\delta}(q_0)-\int_{-T/2}^{T/2}dt e^{iq_0t}+e^{i\mathbf{q}\cdot\mathbf{v}T/4}\left[\int_{-T/2}^{0}dt e^{i\alpha_+t}+\int_{0}^{T/2}dt e^{i\alpha_-t}\right] \\
&=\bar{\delta}(q_0)-T\operatorname{sinc}\left(\frac{q_0T}{2}\right)+\frac{T}{2}e^{i\mathbf{q}\cdot\Delta\mathbf{x}/4}\left[e^{-iq_0T/4}\operatorname{sinc}\left(\frac{\alpha_+T}{4}\right)+e^{iq_0T/4}\operatorname{sinc}\left(\frac{\alpha_-T}{4}\right)\right]. \tag*{(165)}
\end{aligned} W ( q ; x L ) = ∫ − ∞ ∞ d t e i q 0 t e − i q ⋅ x L ( t ) = δ ˉ ( q 0 ) − ∫ − T /2 T /2 d t e i q 0 t + e i q ⋅ v T /4 [ ∫ − T /2 0 d t e i α + t + ∫ 0 T /2 d t e i α − t ] = δ ˉ ( q 0 ) − T sinc ( 2 q 0 T ) + 2 T e i q ⋅ Δ x /4 [ e − i q 0 T /4 sinc ( 4 α + T ) + e i q 0 T /4 sinc ( 4 α − T ) ] . (165) Hence, in the Markovian regime, Eq. (165) implies the appearance of energy conserving delta functions δ ( q 0 ± q ⋅ v 2 ) \delta\left(q_0\pm\frac{\mathbf{q}\cdot\mathbf{v}}{2}\right) δ ( q 0 ± 2 q ⋅ v )
Higgs-portal dark matter constraints
In this appendix, we briefly summarize the constraints shown in Figure 5 , which we collated from Refs. [115 , 116 ] .
SN1987A and stellar cooling. —The effective DM-nucleon coupling provides an additional cooling mechanism in stars and supernovae via N N → N N χ χ NN\to NN\chi\chi N N → N N χχ [125 ] , for which
g Λ 2 ≲ 4.5 G e V − 2 × ( e V m χ ) . (166) \frac{g}{\Lambda^2}\lesssim4.5\ \mathrm{GeV}^{-2}\times\left(\frac{\mathrm{eV}}{m_\chi}\right). \tag*{(166)} Λ 2 g ≲ 4.5 GeV − 2 × ( m χ eV ) . (166) Invisible Higgs decay. —In the broken phase, L ⊃ − g H χ 2 H † H → − g H v h χ 2 \mathcal{L}\supset-g_H\chi^2H^\dagger H\to-g_Hv h\chi^2 L ⊃ − g H χ 2 H † H → − g H v h χ 2 v v v Γ h → χ χ / ( Γ h → S M + Γ h → χ χ ) < 0.11 \Gamma_{h\to\chi\chi}/(\Gamma_{h\to\mathrm{SM}}+\Gamma_{h\to\chi\chi})<0.11 Γ h → χχ / ( Γ h → SM + Γ h → χχ ) < 0.11 [126 ] , where Γ S M = 4.07 M e V \Gamma_{\mathrm{SM}}=4.07\ \mathrm{MeV} Γ SM = 4.07 MeV [143 ] and Γ h → χ χ = g H 2 v 2 / 8 π m h \Gamma_{h\to\chi\chi}=g_H^2v^2/8\pi m_h Γ h → χχ = g H 2 v 2 /8 π m h m χ ≪ m h m_\chi\ll m_h m χ ≪ m h [144 ] . This implies ∣ g H ∣ < 5.1 × 10 − 3 |g_H|<5.1\times10^{-3} ∣ g H ∣ < 5.1 × 1 0 − 3
g Λ 2 ≲ 70 G e V − 2 × ( e V m χ ) . (167) \frac{g}{\Lambda^2}\lesssim70\ \mathrm{GeV}^{-2}\times\left(\frac{\mathrm{eV}}{m_\chi}\right). \tag*{(167)} Λ 2 g ≲ 70 GeV − 2 × ( m χ eV ) . (167) Bullet-cluster. —In the broken phase, L ⊃ − g H χ 2 H † H → − g H v h χ 2 − g H h 2 χ 2 / 2 \mathcal{L}\supset-g_H\chi^2H^\dagger H\to-g_Hv h\chi^2-g_Hh^2\chi^2/2 L ⊃ − g H χ 2 H † H → − g H v h χ 2 − g H h 2 χ 2 /2 m χ ≪ m h m_\chi\ll m_h m χ ≪ m h − λ e f f χ 4 / 4 ! -\lambda_{\mathrm{eff}}\chi^4/4! − λ eff χ 4 /4 ! λ e f f ∼ g H 2 / 4 π 2 \lambda_{\mathrm{eff}}\sim g_H^2/4\pi^2 λ eff ∼ g H 2 /4 π 2 ( 1 + n χ , B C ) σ χ χ → χ χ / m χ ≲ 1 c m 2 / g (1+n_{\chi,\mathrm{BC}})\sigma_{\chi\chi\to\chi\chi}/m_\chi\lesssim1\ \mathrm{cm}^2/\mathrm{g} ( 1 + n χ , BC ) σ χχ → χχ / m χ ≲ 1 cm 2 / g [145 ] , where σ χ χ → χ χ = λ e f f 2 / 128 π m χ 2 \sigma_{\chi\chi\to\chi\chi}=\lambda_{\mathrm{eff}}^2/128\pi m_\chi^2 σ χχ → χχ = λ eff 2 /128 π m χ 2 [146 ] .
Here, ( 1 + n χ , B C ) (1+n_{\chi,\mathrm{BC}}) ( 1 + n χ , BC ) n χ , B C ≃ 23 × ( 1 e V / m χ ) 4 n_{\chi,\mathrm{BC}}\simeq23\times(1\,\mathrm{eV}/m_\chi)^4 n χ , BC ≃ 23 × ( 1 eV / m χ ) 4 [115 ] . Hence,
g Λ 2 ≲ { 0.25 G e V − 2 ( m χ e V ) 3 / 4 for m χ ≲ 2 e V , 0.55 G e V − 2 ( e V m χ ) 1 / 4 for m χ ≳ 2 e V . (168) \frac{g}{\Lambda^2}\lesssim\begin{cases}0.25\,\mathrm{GeV}^{-2}\left(\frac{m_\chi}{\mathrm{eV}}\right)^{3/4} & \text{for }m_\chi\lesssim2\,\mathrm{eV},\\ 0.55\,\mathrm{GeV}^{-2}\left(\frac{\mathrm{eV}}{m_\chi}\right)^{1/4} & \text{for }m_\chi\gtrsim2\,\mathrm{eV}.\end{cases} \tag*{(168)} Λ 2 g ≲ ⎩ ⎨ ⎧ 0.25 GeV − 2 ( eV m χ ) 3/4 0.55 GeV − 2 ( m χ eV ) 1/4 for m χ ≲ 2 eV , for m χ ≳ 2 eV . (168) Large-scale structure formation. —In the early universe, the quartic self-interaction renders the oscillating χ \chi χ L J ≃ ( π ∣ λ e f f ∣ M P l 2 / 16 m χ 4 ) 1 / 2 L_J\simeq(\pi|\lambda_{\mathrm{eff}}|M_{\mathrm{Pl}}^2/16m_\chi^4)^{1/2} L J ≃ ( π ∣ λ eff ∣ M Pl 2 /16 m χ 4 ) 1/2 [122 , 147 ] . Requiring that self-interactions not substantially modify the growth of perturbations with present-day comoving wavelengths ≳ 100 k p c \gtrsim100\,\mathrm{kpc} ≳ 100 kpc ∣ λ e f f ∣ ≲ 1.8 × 10 − 6 ( m χ / e V ) 4 |\lambda_{\mathrm{eff}}|\lesssim1.8\times10^{-6}(m_\chi/\mathrm{eV})^4 ∣ λ eff ∣ ≲ 1.8 × 1 0 − 6 ( m χ / eV ) 4 [122 ] , or equivalently
g Λ 2 ≲ 10 2 G e V − 2 ( m χ e V ) . (169) \frac{g}{\Lambda^2}\lesssim10^2\,\mathrm{GeV}^{-2}\left(\frac{m_\chi}{\mathrm{eV}}\right). \tag*{(169)} Λ 2 g ≲ 1 0 2 GeV − 2 ( eV m χ ) . (169) Modifications to the DM distribution due to the Earth and the atmosphere
In this appendix, we compute the impact of the Earth and the atmosphere on the DM phase space distribution. Regions of parameter space affected lie above the black dot-dashed lines in Figure 5 –Figure 7 . We will specifically distinguish between: (i) contact and repulsive interactions, already considered in Refs. [115 , 116 ] , and (ii) long-range attractive interactions. Here, we also restrict our attention to the regime where the DM de Broglie wavelength is much shorter than the size and thickness of the Earth and atmosphere.
For the interaction L ⊃ − ( g / Λ 2 ) m χ χ 2 N ‾ N \mathcal{L}\supset-(g/\Lambda^2)m_\chi\chi^2\overline{N}N L ⊃ − ( g / Λ 2 ) m χ χ 2 N N χ \chi χ N ‾ N \overline{N}N N N ρ N = ⟨ N ‾ N ⟩ \rho_N=\langle\overline{N}N\rangle ρ N = ⟨ N N ⟩
[ ∂ t 2 − ∇ 2 + m χ 2 + 2 g Λ 2 m χ ρ N ] χ = 0. (170) \left[\partial_t^2-\nabla^2+m_\chi^2+2\frac{g}{\Lambda^2}m_\chi\rho_N\right]\chi=0. \tag*{(170)} [ ∂ t 2 − ∇ 2 + m χ 2 + 2 Λ 2 g m χ ρ N ] χ = 0. (170) As a result, DM acquires an effective mass in a medium of nucleons given by
m χ , e f f 2 = m χ 2 + 2 g Λ 2 m χ ρ N . (171) m_{\chi,\mathrm{eff}}^2=m_\chi^2+\frac{2g}{\Lambda^2}m_\chi\rho_N. \tag*{(171)} m χ , eff 2 = m χ 2 + Λ 2 2 g m χ ρ N . (171) Imposing DM energy conservation inside and outside the medium then implies that, for repulsive interactions (i.e., g > 0 g>0 g > 0 ∣ p ∣ = m χ v 0 |\mathbf{p}|=m_\chi v_0 ∣ p ∣ = m χ v 0 [115 ]
g Λ 2 ≳ m χ v 0 2 2 ρ N . (172) \frac{g}{\Lambda^2} \gtrsim\frac{m_\chi v_0^2}{2\rho_N}. \tag*{(172)} Λ 2 g ≳ 2 ρ N m χ v 0 2 . (172) In the limit that the DM de Broglie wavelength is much shorter than the thickness and size of the shield, strong attenuation simply requires satisfying Eq. (172) [116 ] . Hence, using the typical number density of nucleons in the atmosphere, we find that atmospheric screening for an experiment on Earth becomes significant for
g Λ 2 ≳ 5 × 10 4 G e V − 2 ( m χ 1 e V ) , (173) \frac{g}{\Lambda^2} \gtrsim5 \times10^4\ \mathrm{GeV}^{-2}\left(\frac{m_\chi}{1\ \mathrm{eV}}\right), \tag*{(173)} Λ 2 g ≳ 5 × 1 0 4 GeV − 2 ( 1 eV m χ ) , (173) throughout the mass range shown in Figure 5 .
Similarly, the Earth can screen DM. Approximating the Earth as a homogeneous solid sphere with mean nucleon number density ρ N ≃ 3.3 × 10 24 c m − 3 \rho_N \simeq3.3 \times10^{24}\ \mathrm{cm}^{-3} ρ N ≃ 3.3 × 1 0 24 cm − 3
g Λ 2 ≳ 12 G e V − 2 ( m χ 1 e V ) . (174) \frac{g}{\Lambda^2} \gtrsim12\ \mathrm{GeV}^{-2}\left(\frac{m_\chi}{1\ \mathrm{eV}}\right). \tag*{(174)} Λ 2 g ≳ 12 GeV − 2 ( 1 eV m χ ) . (174) Space-based experiments can avoid atmospheric screening, but can also be sensitive to the Earth’s DM “shadow” during their orbit [116 ] . Furthermore, surrounding spacecraft material may still attenuate the signal. This additional shielding depends on the material density, thickness, and geometry, and is left to future work.
Long-range attractive interactions
We now consider the case of DM-nucleon interactions via a hadrophilic light scalar mediator. In this case, interactions are attractive and the atmosphere and Earth can give rise to DM refraction, similar to the effect of gravitational focusing due to the Sun [74 , 75 ] . Neglecting the DM back-reaction and assuming a static and homogeneous distribution of nucleons, the equations of motion for the matter-sourced static field ϕ ˉ \bar{\phi} ϕ ˉ χ \chi χ L ⊃ − y χ m χ ϕ χ 2 − y n ϕ N ˉ N \mathcal{L} \supset-y_\chi m_\chi\phi\chi^2-y_n\phi\bar{N}N L ⊃ − y χ m χ ϕ χ 2 − y n ϕ N ˉ N
( − ∇ 2 + m ϕ 2 ) ϕ ˉ = − y n ρ N , (175) \left(-\nabla^2+m_\phi^2\right)\bar{\phi}=-y_n\rho_N, \tag*{(175)} ( − ∇ 2 + m ϕ 2 ) ϕ ˉ = − y n ρ N , (175) ( ∂ t 2 − ∇ 2 + m χ 2 + 2 y χ m χ ϕ ˉ ) χ = 0. (176) \left(\partial_t^2-\nabla^2+m_\chi^2+2y_\chi m_\chi\bar{\phi}\right)\chi=0. \tag*{(176)} ( ∂ t 2 − ∇ 2 + m χ 2 + 2 y χ m χ ϕ ˉ ) χ = 0. (176) Imposing the boundary condition ϕ ˉ = 0 \bar{\phi}=0 ϕ ˉ = 0 (175) is given by
ϕ ˉ ( x ) = − y n ∫ d 3 x ′ G ϕ r ( x − x ′ ) ρ N ( x ′ ) , (177) \bar{\phi}(\mathbf{x})=-y_n\int d^3x'\,G_\phi^r(\mathbf{x}-\mathbf{x}')\rho_N(\mathbf{x}'), \tag*{(177)} ϕ ˉ ( x ) = − y n ∫ d 3 x ′ G ϕ r ( x − x ′ ) ρ N ( x ′ ) , (177) where G r ϕ ( x − x ′ ) G_{\mathrm{r}}^{\phi}(\mathbf{x}-\mathbf{x}') G r ϕ ( x − x ′ ) ϕ \phi ϕ χ = e − i E t χ ′ ( x ) + c . c . \chi=e^{-iEt}\chi'(\mathbf{x})+\mathrm{c.c.} χ = e − i E t χ ′ ( x ) + c.c. (176) can be expressed as
[ − ∇ 2 2 m χ + V ( x ) ] χ ′ ( x ) = E 2 − m χ 2 2 m χ χ ′ ( x ) , V ( x ) = y χ ϕ ˉ ( x ) . (178) \left[-\frac{\nabla^2}{2m_{\chi}}+V(\mathbf{x})\right]\chi'(\mathbf{x})=\frac{E^2-m_{\chi}^2}{2m_{\chi}}\chi'(\mathbf{x}),\qquad V(\mathbf{x})=y_{\chi}\bar{\phi}(\mathbf{x}). \tag*{(178)} [ − 2 m χ ∇ 2 + V ( x ) ] χ ′ ( x ) = 2 m χ E 2 − m χ 2 χ ′ ( x ) , V ( x ) = y χ ϕ ˉ ( x ) . (178) where V ( x ) V(\mathbf{x}) V ( x ) m ϕ − 1 m_{\phi}^{-1} m ϕ − 1 V ( x ) V(\mathbf{x}) V ( x )
V ( R ⊕ ) ≃ − y χ y n ρ N 2 m ϕ 2 . (179) V(R_{\oplus})\simeq-\frac{y_{\chi}y_{n}\rho_{N}}{2m_{\phi}^{2}}. \tag*{(179)} V ( R ⊕ ) ≃ − 2 m ϕ 2 y χ y n ρ N . (179) Imposing energy conservation, we find that an O ( 1 ) \mathcal{O}(1) O ( 1 )
∣ V ( R ⊕ ) ∣ ∼ 1 2 m χ v 0 2 ⟹ y χ y n ∼ m ϕ 2 m χ v 0 2 ρ N . (180) \lvert V(R_{\oplus})\rvert\sim\frac{1}{2}m_{\chi}v_{0}^{2}\Longrightarrow y_{\chi}y_{n}\sim\frac{m_{\phi}^{2}m_{\chi}v_{0}^{2}}{\rho_{N}}. \tag*{(180)} ∣ V ( R ⊕ )∣ ∼ 2 1 m χ v 0 2 ⟹ y χ y n ∼ ρ N m ϕ 2 m χ v 0 2 . (180) Hence, in the limit m χ ≪ 1 G e V m_{\chi}\ll1\ \mathrm{GeV} m χ ≪ 1 GeV (138) ), we find that appreciable corrections to the DM phase space properties on the surface of the Earth occur for
σ ˉ ≳ 5 × 10 − 40 c m 2 ( m χ 1 k e V ) 4 ( R ϕ , χ 10 − 5 ) 4 , (181) \bar{\sigma}\gtrsim5\times10^{-40}\ \mathrm{cm}^{2}\left(\frac{m_{\chi}}{1\ \mathrm{keV}}\right)^{4}\left(\frac{R_{\phi,\chi}}{10^{-5}}\right)^{4}, \tag*{(181)} σ ˉ ≳ 5 × 1 0 − 40 cm 2 ( 1 keV m χ ) 4 ( 1 0 − 5 R ϕ , χ ) 4 , (181) σ ˉ ≳ 10 − 32 c m 2 ( m χ 1 k e V ) 4 ( R ϕ , χ 10 − 5 ) 4 , (182) \bar{\sigma}\gtrsim10^{-32}\ \mathrm{cm}^{2}\left(\frac{m_{\chi}}{1\ \mathrm{keV}}\right)^{4}\left(\frac{R_{\phi,\chi}}{10^{-5}}\right)^{4}, \tag*{(182)} σ ˉ ≳ 1 0 − 32 cm 2 ( 1 keV m χ ) 4 ( 1 0 − 5 R ϕ , χ ) 4 , (182) due to the Earth and the atmosphere, respectively.
The effect of laboratory shield, which could be relevant for certain space-based proposals and could be smaller in size than m ϕ − 1 m_{\phi}^{-1} m ϕ − 1
References 
