Pump quantization changes optimal measurements for quantum sensing

Introduction

Quantum mechanics not only sets the ultimate limits on the sensitivity of any metrological task, but also offers a path to improve upon classical metrology [1, 2]. A hallmark example is the use of squeezed states [3], which have been shown to improve measurement of gravitational waves [4], sensing beam displacements [5], biomedical imaging [6, 7], providing resources for continuous variable quantum computing [8], and even searching for physics beyond the standard model [9].

Nonclassicality is expensive, however, and its generation requires highly efficient and strong nonlinear interactions [10]. Any fair assessment of a claimed quantum advantage needs to factor in these ‘hidden’ costs, and it is debatable whether quantum advantage survives when this overhead is accounted for [11, 12, 13, 14, 15].

The standard approach to squeezed-state generation [3, 10] treats the pump as a “free” resource, considering it as a classical drive with no fluctuations. As modern platforms, particularly in nanophotonics [16], achieve stronger nonlinearities and highly efficient squeezing [17, 18], significant pump depletion is becoming an experimental reality. In these high-flux regimes, the back-action on the pump is non-negligible, the signal and pump states become entangled [19, 20] and evolve into non-Gaussian states, which fundamentally limits the available squeezing [21], necessitating a fully quantized treatment. In this work, we consider the following question: does the quantum advantage of squeezing persist in the presence of pump quantization?

Intuitively, pump quantization is expected to introduce fluctuations [22], limiting phase coherence. Several works used phase space heuristics to show that pump quantization suppresses the quantum advantage with squeezing, offering no improvement over using the coherent pump for sensing [11, 22, 23, 24, 25, 26, 27]. Nevertheless, to the best of our knowledge, a unified, information-theoretic framework connecting pump quantization, state generation, and metrological performance remains lacking.

We address this gap by considering the effect of pump quantization and depletion on the sensing capacity of squeezed-state phase interferometry. We find that pump depletion reshapes both metrological sensitivity and optimal measurement strategies. Using an analysis based on the quantum Fisher information (QFI), we show the following:

• (a) When the phase is encoded in the squeezed signal mode, the QFI for phase sensing cannot beat the classical shot noise limit (SNL) scaling of $N$, where $N$ is the pump photon number.

• (b) Mixing the squeezed state with a coherent state and performing Mach-Zehnder interferometry (MZI) yields an $N^{3/2}$ QFI scaling, beating the SNL.

• (c) Despite mixing with a coherent state, pump quantization restricts squeezed states from attaining the Heisenberg-limited (HL) scaling of $N^2$.

• (d) Operating near the point of maximum pump depletion, where the signal flux is maximized, recovers HL scaling of QFI.

• (e) Photon counting (or homodyne detection) at the point of maximum depletion is sub-optimal and does not achieve HL scaling. Instead, a nonlinear, time-reversal measurement is shown to saturate the HL [28, 29, 30, 31, 32, 33].

(a) is an information-theoretic reformulation of past results based on phase space heuristics [22, 23, 34]. (b-e) discuss two schemes where the quantum advantage is dynamically reclaimed despite pump quantization. (b) is fundamentally an interference effect between the squeezed signal and the coherent probe [35]. (c) follows directly from (b), showing the limits of squeezed state interferometry. (d) shows that HL scaling can be achieved by tuning dynamics of the nonlinear source generation. Crucially, (e) shows that this transition in source dynamics and information capacity also necessitates a shift in measurement schemes. We start with a description of our model, followed by an exposition of our numerical results on the information capacity in the presence of pump depletion, as captured by the QFI associated with the phase-encoded state.

Squeezed-state Interferometry – Quantum metrology leverages non-classical states to estimate an unknown parameter, $\theta$, with precision surpassing the SNL. In conventional phase sensing, sensitivity is capped by vacuum fluctuations that mix with the probe field at the input of an interferometer [3], as illustrated in Figure 1. Mathematically, we consider a joint input state $|\psi_{\mathrm{in}}\rangle_{s,p}=|0\rangle_s|\alpha\rangle_p$ where the signal starts in vacuum, and the pump is prepared in a coherent state $|\alpha\rangle$ with mean photon number $N=|\alpha|^2$. The measured phase is inserted in one arm of the MZI, with the optical path length $\theta$ being imprinted via the interaction Hamiltonian $e^{i\theta\hat{n}}$, $\hat{n}$ being the number operator in the corresponding mode of the MZI arm. The encoded state $|\psi(\theta)\rangle_{s,p,\mathrm{pr}}$, is the state used for phase estimation, and the measurement protocol typically involves photocount detection at the output ports of the MZI [Figure 1c].

Squeezed states are commonly generated through nonlinear optical processes, for example by parametric down-conversion driven by the pump field. The dynamics are modelled by the fully quantized interaction Hamiltonian of a degenerate parametric oscillator [37, 38, 39, 40, 41] (DPO):

where $\kappa$ denotes the effective single-photon nonlinear coupling rate, while $a_p,a_s$ are the annihilation operators for the pump and signal modes, respectively. To isolate the fundamental physics of pump depletion, we restrict our analysis to a single longitudinal mode for each field and assume perfect phase matching [42].

In the undepleted, parametric approximation, $a_p\rightarrow\alpha_p$ and the resulting Hamiltonian generates an idealized minimum-uncertainty squeezed state which can beat the SNL [43], and even achieve HL scaling, by mixing with a coherent probe in an MZI [28, 36]. While the parametric approximation predicts better phase estimation with harder pumping, it completely neglects any pump depletion [17, 18], decoherence [16], entanglement [19, 20], and non-Gaussianity that restricts available squeezing [21].

Depleted down-conversion – To address these effects, we consider the full quantum Hamiltonian in Eq. (1). Since $[n_p+n_s,H_I]=0$, the dynamics conserve the total excitations in the system, leading to pump depletion and nonlinear Rabi oscillations between the modes [24], as shown in (Figure 2a). For an initial state consisting of a coherent pump ($|\alpha|^2=100$) and a vacuum signal, the signal photon number $\langle n\rangle_s$ in the state $|\psi\rangle_{s,p}=\exp(-iH_I t/\hbar)|\psi_{\mathrm{in}}\rangle_{s,p}$ grows until it reaches a peak corresponding to an incomplete transfer of the pump energy at the full depletion time $t_{\mathrm{fl}}$. While the parametric approximation predicts a monotonic, exponential decrease in the squeezed quadrature variance $\langle\Delta x_s^2\rangle$, full quantum dynamics introduce severe phase diffusion as the pump depletes [34, 44], and the quadrature variance is minimized at $t_{\mathrm{sq}}$, the point of maximum squeezing. Physically, the evolution $e^{-iH_I t/\hbar}$ generates photon-added coherent states in the pump and signal [19, 20] and fundamentally caps the achievable squeezing [21]. Concurrently, the photon number variances $\langle\Delta n^2\rangle$ of both the pump and signal grow substantially [Figure 2c], maximizing slightly before $t_{\mathrm{fl}}$, and not at $t_{\mathrm{sq}}$. Appendix A details the numerical methods used in this work. We now address how the DPO dynamics affect phase estimation.

Sensitivity Analysis – To quantify the phase estimation precision, we utilize the Cramér–Rao bound (CRB), which limits the variance of any unbiased estimator $\hat{\theta}$ as $\langle\Delta\tilde{\theta}^2\rangle\geq1/\mathcal{F}(\theta)$. For the pure state $|\psi(\theta)\rangle_{s,p,\mathrm{pr}}$, the quantum Fisher information (QFI) is defined as $\mathcal{F}(\theta)=2\operatorname{Tr}[(\partial_\theta\rho)^2]$, representing the maximum extractable information about $\theta$ independent of the specific measurement scheme. For the MZI in Figure 1b, the QFI is determined by the photon number fluctuations in the lower arm, $\mathcal{F}=4\langle(\Delta\hat{n}_1)^2\rangle$ [36, 45] [46]. Assuming a signal-probe product state at the MZI input and noting that odd-order signal moments vanish when $|\psi_{\mathrm{in}}\rangle_{s,p}$ evolves under $H_I$, the QFI evaluates to:

where $|\alpha|^2$ and $\phi$ represent the probe intensity and phase, respectively, and $\Delta a = a - \langle a\rangle$.

Equation (2) provides an intuitive framework for quantifying the quantum advantage for any arbitrary signal state. We analyze the scaling of $\mathcal{F}$ versus the total photon resource $N$. When the DPO is biased at $t_{sq}$, we find that $\langle\Delta n_s^2\rangle$ scales as $\sim N$ and $\langle n_s\rangle$ scales as $\sim\sqrt{N}$; results (a-c) in the introduction follow immediately. (a) When the probe is in vacuum ($\alpha= 0$), $\mathcal{F} = \langle\Delta n_s^2\rangle+\langle n_s\rangle$ and we find a scaling of $N^{1.01\pm0.017}$ which is just the semiclassical SNL dictated by the coherent probe. Without the probe, squeezing itself cannot provide any quantum advantage, as also found by phase space arguments in [22, 23] and explained in Appendix B. (b) However, when mixed with a probe with mean photon number $N \approx|\alpha|^2$, the QFI scales as $N^{1.48\pm0.008}$, as shown in Figure 3. The advantage stems from the interference term $2|\alpha|^2\langle\Delta a_s^\dagger\Delta a_s\rangle$ in Eq. (2) which is expected to scale as $\sim N^{3/2}$. Interference between the probe and squeezed state at the beamsplitter [3, 35] preserves the quantum advantage despite pump quantization. (c) Finally, we see that no term scales as $N^2$ in Eq. (2); squeezing itself does not achieve Heisenberg scaling.

Error propagation analysis shows that photon counting at the MZI output achieves this near-$N^{3/2}$ scaling. For an MZI phase-biased near $\theta= \pi/2$ and $\phi= \pm\pi/2$, linearized error propagation yields the SNR:

where $x = i(a_s^\dagger-a_s)$ is the squeezed quadrature and $n_D$ is the photon number difference in the two output ports of the MZI. Our numerically computed $S_{PC}$ scales as $N^{1.52\pm0.012}$ (Figure 3), successfully beating the SNL, as also expected from the $\sqrt{N}$ scaling of $\langle n_s\rangle$ at $t_{sq}$ [47].

Optimal Operation—A natural question is whether HL can be achieved with pump quantization, and if so, what sets the optimal operating point of the squeezer. We therefore consider dynamics beyond $t_{sq}$, where the signal evolves beyond the coherent-state manifold. A key feature is a maximum in the signal flux at $t = t_{fl}$, when most of the pump energy has been transferred and $\langle n_s\rangle\propto N$. Furthermore, the photon number variance is maximized slightly before $t_{fl}$, not at $t_{sq}$, and operating at the optimal squeezing value actually limits the achievable sensitivity. This indicates that it is optimal to bias the DPO for maximal flux, interacting for time $t_{fl}$. By properly accounting for the quantum state of the pump in the measurement state, this highly non-Gaussian, depleted operating point can be leveraged to achieve higher sensitivity.

We find that $\mathcal{F}$ exhibits HL scaling of $N^{2\pm0.00024}$ if the squeezer is biased at $t = t_{fl}$. Note that two terms in Eq. (2) exhibit the quadratic scaling at $t_{fl}$, namely, the signal variance $\langle\Delta n_s^2\rangle$ and the cross term between the signal and probe mean $2\alpha^2\langle\Delta a_s^\dagger\Delta a_s\rangle$. That the signal variance scales as $N^2$ indicates that the signal photon number distribution spans the full support of the Hilbert space associated with each pump Fock state, as explored in Appendix § 8.

Maximizing the flux is an intuitive, first-order, and phase-insensitive design principle that engineers can use to make efficient nonlinear devices in highly non-Gaussian regimes. This could amount to optimizing the length of a nonlinear crystal or an (adiabatic) pump pulse duration. As Figure 2 shows, optimizing flux usually conflicts with, and is often computationally simpler than, optimizing squeezing. Our results demonstrate an optimum of flux and squeezing that results in a higher QFI for phase estimation and pave the way for future experiments to incorporate such design principles.

While the QFI exhibits HL scaling by a simple change in the operating point of the squeezer, it comes at the expense of a more sophisticated measurement requirement to saturate the QFI. As we show in Appendix C, photon counting only achieves shot noise scaling if the squeezer is flux-optimized. Intuitively, this means that photon counting is not utilizing the complete information available in the photon statistics of the flux-optimized state, since Eq. (3) is only sensitive to the signal mean and its quadrature fluctuations. To this end, one would expect an improved SNR scaling with a measurement scheme that is sensitive to higher order photon number moments. Different schemes have been proposed in the squeezing literature, such as parity detection [28, 29], a generalized homodyne scheme measuring $\langle x^2\rangle$ instead of $\langle x\rangle$ [30, 31], or SU (1,1) interferometers using time-reversal schemes [7, 32, 33, 36, 48]. While a complete review of such schemes is beyond the scope of this paper, a recurring theme in these works is the use of nonlinear (in the photon number) schemes to exhaust the information available in the non-Gaussian, nonclassical states of the kind generated in the squeezer when it is flux-optimized.

schemes is beyond the scope of this paper, a recurring theme in these works is the use of nonlinear (in the photon number) schemes to exhaust the information available in the non-Gaussian, nonclassical states of the kind generated in the squeezer when it is flux-optimized. A typical implementation of such a nonlinear measurement is shown in Figure 4, where both the encoding and measurement schemes are now different from that in Figure 1 [49]. The phase is encoded directly in the signal beam, while the measurement involves another evolution under $H_I$ with the same strength and interaction time used in state preparation. Note that this is a joint measurement with the signal and pump beams, and we finally measure the photon number in the signal beam. The QFI, $F_{NL}=4\langle\Delta n_s^2\rangle$ and the SNR, $S_{NL}$, of this nonlinear scheme is defined as $(\partial_\theta\langle n_s\rangle)^2/\langle\Delta n_s^2\rangle$. When biased at the dark fringe at $\theta=\pi/2$, $S_{NL}$ achieves HL scaling, as shown in Figure 3. The nonlinear scheme is not always optimal; when operated at $t_{sq}$ (squeezing-optimized) instead of $t_{fl}$ (flux-optimized), both $F_{NL},S_{NL}$ exhibit shot noise scaling (see Appendix C for details). Thus, neither photon counting nor nonlinear detection provides a quantum advantage for all interaction strengths. The optimality of a detection scheme is highly sensitive to the encoded quantum state, a well-established feature in quantum-enhanced metrology that is further amplified by the effects of pump quantization and depletion.

Discussion and Conclusions

In summary, we have established a framework for quantum sensing that accounts for the finite resources and dynamical back-action on the generating pump. While previous analyses suggest that squeezing cannot beat the SNL and is useful only when high pump power is not feasible such as in the case of photocell? detection [6], radiation pressure [3] or thermal lensing [50], we demonstrate that the genuine quantum enhancement in sensitivity remains robust despite pump back-action. Specifically, squeezing can outperform SNL even with pump quantization, achieving a QFI scaling of $\sim N^{3/2}$. Nevertheless, pump quantization precludes squeezing from achieving HL scaling, which is only approached deep into the depletion regime where the mode is manifestly unsqueezed, and traditional linear detection is suboptimal. Instead, a nonlinear scheme jointly measuring the pump and signal, such as an SU (1,1) interferometer [51, 52], is optimal in the depleted regime.

Our single-mode model can be readily extended to capture the multimodal, open system physics of realistic systems. A critical avenue is to extend our fully quantized 3-body framework to encompass the 4-body $\chi^{(3)}$ dynamics which generate squeezed states in nanophotonic waveguides, optical fibers, and atomic ensembles [42]. An open question is whether the robust $\sim N^{3/2}$ sensitivity scaling observed here persists or if higher-order nonlinear processes or losses enforce stricter practical bounds on the quantum advantage. Furthermore, our analysis is qualified by a local, as opposed to a global, sensing protocol [53], and saturating multi-parameter Fisher information bounds (vs. single parameter in this work) requires judicious estimator designs, priors determined by experimental details, and taming finite data sizes. Such details are germane to any practical application of quantum metrology [54] and offer productive routes to extend our analysis.

By treating the pump as a quantized degree of freedom, we have shown that the fundamental limits of phase estimation are dictated not merely by the properties of the generated squeezed state, but by the joint coherence and entanglement of the entire pump-signal system. The framework, based on the 3-body DPO Hamiltonian of Eq. (1), informs not only quantum optical sensing but also a diverse physical landscape including the $N$-atom Tavis-Cummings model [55, 56], Brillouin scattering [57], phonon lasing [58], circuit QED [59, 60], nonlinear gravitational waves [61], and quantum thermodynamics [62]. These results reconcile the idealized notions of a “probe state” generated by quantum dynamics across these different systems with the constraints of the probe’s physical generation. To achieve any nonclassical advantage, the design of the sensor must encompass the full cycle of the quantum resource—from its creation by a quantized pump to its final readout in a correlated, nonlinear measurement.

Beyond sensing, our work informs on the deeply related question of the impact of quantization of a classical control signal on quantum dynamics [63, 64, 65]. More broadly, we demonstrate that quantizing traditionally semiclassical resources is not merely a bookkeeping exercise for accurate performance bounds, but a fundamental prerequisite for designing optimal quantum protocols.

Acknowledgements

S. A. W. and L. B. acknowledge useful discussions with L. P. McGuinness that triggered this work. S. A. W. acknowledges OpenAI’s LLM for pointing to related work in [66, 67].

L. B. and T. I. contributed equally to this work.

Numerical Methods

In this section, we describe the numerical simulation methods used to obtain the results presented in the main text, closely following approaches employed in previous works [20, 24, 33].

To simulate the dynamics, we expand the input state $\lvert\psi_{\mathrm{in}}(t=0)\rangle_{s,p}=\lvert0\rangle_s\lvert\alpha\rangle_p$ in a joint Fock basis as $\lvert0\rangle_s\lvert\alpha\rangle_p=\sum_n c_n\lvert0\rangle_s\lvert n\rangle_p$, where $\lvert c_n\rvert^2$ is Poisson distributed with mean $N=\lvert\alpha\rvert^2$. Within the rotating-wave approximation, the DPO Hamiltonian in Eq. (1) is time-independent, and thus direct integration of the Schrödinger equation yields the time-evolved state $\lvert\psi_{\mathrm{in}}(t)\rangle_{s,p}=\exp[-iH_I t/\hbar]\lvert\psi_{\mathrm{in}}(t=0)\rangle_{s,p}$. To compute the matrix exponential directly, we can use the matrix elements of the Hamiltonian in the joint Fock basis $\lvert n_s\rangle_s\lvert n_p\rangle_p$. A more efficient approach to evolve the input state $\lvert\psi_{\mathrm{in}}\rangle_{s,p}$ is to use the fact that the Hamiltonian commutes with the total photon number operator $\hat{M}=n_p+n_s/2$, i.e., $[\hat{M},H_I]=0$, and thus a shared set of eigenvectors exists between $H_I$ and $\hat{M}$. Consequently, the evolution of a state with a given eigenvalue $M$ does not couple to states with different total photon number $M'$. Alternatively, the evolution of states with different pump photon numbers in the initial state $\lvert\psi_{\mathrm{in}}\rangle_{s,p}$ is decoupled, and we can evolve the subspace corresponding to each $M$ independently. Denoting $\lvert n\rangle_M\equiv\lvert2(M-n)\rangle_s\lvert n\rangle_p$ as a ket in the subspace containing $M$ photons shared between the signal and pump mode, we obtain the tridiagonal matrix

where $a_{nn'}=(\hbar\kappa/2)\sqrt{n(2M-2n+1)(2M-2n+2)}$ and $0\le n,n'\le M$.

For the simulation, $M$ is varied from $M_{\min}=N-5\sqrt{N}$ to $M_{\max}=N+5\sqrt{N}$, which covers the tails of the Poisson distribution that are at least 5 orders of magnitude smaller than the peak value. For each $M$, the $t=0$ ket state $\lvert n\rangle_M$ is represented by a column vector whose $n^{\mathrm{th}}$ element is equal to unity and the rest are zeros. At $t=0$, $\lvert n=M\rangle_M$, since all the photons are in the pump mode, and the ket is given by $[0,\ldots,0,1]^T$, with $M-1$ entries of zero. We directly exponentiate the $M$-dimensional matrix obtained through Eq. (4), and use the MATLAB function expmv_tspan [68] developed in [69] to compute the time-evolved ket. For the simulation, $\kappa=1$, and the temporal axis is set to $t=[0.001:0.0001:1]$ to efficiently sample the dynamics over different pump powers. We simulated the interaction for $N=[50,100:100:1000]$.

The time-evolved state can then be used to compute the physical observables such as $\langle n_s\rangle$, $\langle n_p\rangle$, $\langle x^2\rangle_s$, $\langle p^2\rangle_s$, $\langle\Delta n_s^2\rangle$ which are plotted in Figure 2 of the main text and are used in computing the QFI and SNR plotted in Figure 3 of the main text.

Previous Work on Phase Estimation with Quantized Pump

In this section we compare our results with past work on limits to squeezing-based phase estimation in the presence of pump depletion [23, 34], pump phase fluctuations in the parametric approximation [22, 27], and energy balance arguments [70, 71].

Phase space SNR – From a phase space perspective, a physically motivated SNR for squeezed state phase estimation can be defined as the ratio of i) mean photon number $\langle n_s\rangle$ in the squeezed state (signal) and ii) the variance in the squeezed quadrature $\langle(\Delta x)^2\rangle$ (noise). For a coherent state $\lvert\alpha\rangle$, for example, $\langle n_s\rangle=\lvert\alpha\rvert^2$, and the variance $\langle\Delta x^2\rangle=1$ of the quadrature $x=i(a^\dagger-a)$. The SNR is then given by $\lvert\alpha\rvert^2/\langle\Delta x^2\rangle=\lvert\alpha\rvert^2=N$, which is the SNL. For a squeezed state, it was shown in [22, 27, 34] that $\langle(\Delta x)^2\rangle$, $\langle n_s\rangle$ respectively scale as $N^{-0.5}$, $N^{+0.5}$ when the DPO is biased at maximum squeezing (or minimum quadrature variance). Thus, the phase space SNR for a squeezed state $\langle n_s\rangle/\langle(\Delta x)^2\rangle$ scales as $N$, equivalent to the SNL. Note that the phase space SNR provides a lower bound for the variance $\langle\Delta\theta^2\rangle$; to compare with the previous literature using the standard deviation $\Delta\theta=\sqrt{\langle\Delta\theta^2\rangle}$, one should use the square root of the SNR defined above. This can be interpreted as a direct consequence of pump quantization: the dynamical phase fluctuations of the pump limit the available squeezing, and the finite pump energy limits the generated flux. This result was interpreted to suggest that the phase resolution of any interferometric scheme employing squeezed vacuum cannot surpass the SNL, and that the advantage of squeezing lies primarily in reduced power requirements rather than in improved scaling.

Such phase space metrics are based on the intrinsic parameters of the generated state and do not consider either the phase encoding schemes or measurement statistics. These results can be recovered in our formalism by setting $\alpha=0$ in Eq. (2) to obtain $\mathcal{F}=\langle n_s\rangle+\langle\Delta n_s^2\rangle$. At the point of maximum squeezing, $\langle n_s\rangle,\langle\Delta n_s^2\rangle$ scale as $N^{0.5},N$ respectively, and $\mathcal{F}$ scales according to the SNL.

Mixing the squeezed state with a coherent probe modifies the QFI and allows beyond-SNL scaling. As shown in Eq. (3) of the main text, photon counting after a Mach-Zehnder interferometer achieves an SNR scaling of $N^{3/2}$; note that this result is true despite $\langle(\Delta x)^2\rangle$, $\langle n_s\rangle$ scaling as $N^{-0.5},N^{+0.5}$ respectively. The conceptual shift from a phase space-based perspective to a measurement-aware analysis of Fisher information reveals the metrological advantage of squeezed states.

Stochastic pump – The possibility of achieving beyond-SNL performance in Mach-Zehnder interferometry, despite the phase space effects associated with pump quantization, can be motivated semiclassically. Rather than treating the pump as a fully quantized mode, let us first consider a classical pump field with stochastic phase fluctuations. In the parametric approximation, the pump mode is replaced by $a_p \to|\alpha|e^{i\beta}$, where $N=|\alpha|^2$ and $\beta$ denotes the pump phase. Solving the Heisenberg equations of motion for $a_s$ generated by the Hamiltonian in Eq. (1) of the main text gives

We now regard the pump phase $\beta$ as a classical random variable with prescribed variance $\overline{\Delta\beta^2}\sim N^{-1}$, as appropriate for the phase uncertainty of a coherent state, where the overline denotes classical ensemble averaging. In phase space, these pump-phase fluctuations lead to a jitter of the squeezing ellipse orientation. As a result, even though each realization produces a squeezed state, averaging over the fluctuating pump phase degrades the observable squeezing. The minimum achievable quadrature variance is then limited to the scaling $\langle\Delta x^2\rangle_{\min}\sim N^{-1/2}$ [22, 23].

While the phase space jitter seems to be an intrinsic limitation due to pump phase fluctuations, its effect can be suppressed by measuring in the frame that corotates with the squeezing ellipse. Operationally, this is precisely what homodyne detection achieves by mixing the squeezed state with a phase-locked coherent probe. To see this mathematically, consider the following. Assuming the signal at $t=0$ is a vacuum mode, and using the expression in Eq. (5), we find $\langle\Delta a_s^2\rangle$ proportional to $e^{i\beta}$. One would expect degraded squeezing due to a fluctuating $\beta$. However, the QFI for phase estimation as given by Eq. (2) has a $\beta$-dependence only in the last term containing the factor $e^{-i2\phi}\langle\Delta a_s^2\rangle$. For a stochastic pump, this term is proportional to $e^{-i(2\phi-\beta)}$. Crucially, because the pump field is generated by second harmonic generation of the probe, as is necessary to maintain a phase-locked probe for homodyne detection, we have the (deterministic) relation $(2\phi=\beta)$ [35, 72] and $e^{-i2\phi}$ is invariant under pump phase fluctuations [73]. Thus, the QFI does not degrade in the presence of a stochastic pump. Similar arguments apply to the SNR of photon counting as given by Eq. (3) of the main text, showing that any claim of a metrological advantage, or lack thereof, is meaningful only in relation to phase encoding and measurement protocols.

Energy Balance – Further modifications to the parametric approximation have also been considered [70, 71]. In particular, Paris [70] approached squeezed-state interferometry from an energy balance perspective. They showed that under a fixed total photon budget within the interferometer, sub-shot-noise scaling of $\langle\Delta^2\phi\rangle\sim N^{-3/2}$ can be achieved, although this enhancement rapidly degrades with detector inefficiency. These works, however, treat the squeezed state as an external resource and do not account for the cost or dynamics of its generation from a quantized pump.

Observable	Interaction time	Scaling
QFI	$t_{\mathrm{fl}}$ (max. flux)	$5.53N^{2\pm0.00024}$
$S_{\mathrm{NL}}$	$t_{\mathrm{fl}}$	$1.03N^{2\pm0.004}$
QFI	$t_{\mathrm{sq}}$ (max. squeezing)	$4.68N^{1.48\pm0.008}$
$S_{\mathrm{PC}}$	$t_{\mathrm{sq}}$	$0.56N^{1.52\pm0.012}$

QFI Scaling for Different Interaction Lengths

In this section, we list the prefactors of the scaling curves shown in Figure 3. We also show the scaling for QFI, photon counting SNR ($S_{\mathrm{PC}}$), and the SNR for nonlinear (NL) measurement $S_{\mathrm{NL}}$ at points other than those corresponding to maximum squeezing and maximum flux.

Different interaction lengths and associated choice of measurements give a huge design space to explore for the DPO operating point. Table I shows the prefactors for the scaling curves shown in Figure 3 of the main text. Table II summarizes the other scalings explored in this Appendix. We find that the nonlinear (NL) scheme can give below-HL performance when operated at points other than $t_{fl}$. As shown in Table II and Figure 5, the NL scheme achieves SNL when operated at $t_{sq}$. Note that the QFI for the NL scheme at $t_{sq}$ shows shot noise scaling (See Figure 3 in main text). In that sense, the NL scheme is optimal, but does not provide any quantum advantage. Conversely, the photon counting SNR $S_{PC}$ cannot go beyond the SNL when the system is flux-optimized at $t_{fl}$. Although both schemes operate at the SNL and are suboptimal, note that the prefactor for $S_{PC}$ is an order of magnitude smaller than the corresponding prefactor for $S_{NL}$ in Table II. Of course, we are not restricted to $t_{sq},t_{fl}$. Figure 5 shows the QFI at the point of maximum signal photon number variance $t_{var}$, where the Heisenberg scaling persists. Notably, $t_{var}$ does not coincide with $t_{fl}$, as is also shown in Figure 2c for $N=100$. Nevertheless, as discussed in Appendix E, we expect the signal photon number variance at $t_{var}$ to scale as $N^2$, which matches the simulated scaling shown in Figure 5. Similarly, instead of targeting maximum squeezing (minimum $\langle\Delta x^2\rangle$), one can bias at the second minimum of $\langle\Delta x^2\rangle$ (corresponding to second maximum of squeezing), an instance of which is shown around $\kappa t=0.6$ in Figure 2b for $N=100$. The QFI at this point scales as $N^{1.69\pm0.004}$, which is better than the $N^{3/2}$ scaling achieved by biasing at $t_{sq}$. This is expected since the signal photon number variance is higher at the second minimum of $\langle\Delta x^2\rangle$ than at the first occurring at $t_{sq}$. At the second minimum of $\langle\Delta x^2\rangle$, $S_{PC}$ (not shown in Figure 5), shows a scaling of $0.01N^{1.64\pm0.06}$. Although this still beats both the SNL and $N^{3/2}$ scaling, it has a very small prefactor of 0.01 and the actual value of the SNR, at least for the pump powers considered here, is smaller than $S_{PC}$ operating at $t_{sq}$. Note that we have restricted our analysis to the first maximum of signal photon mean and variance, and leave the analysis of long-term dynamics to future work.

Observable	Interaction time	Scaling
QFI	$t_{var}$ (max. variance)	$4.82N^{2\pm0.003}$
$S_{\mathrm{NL}}$	$t_{sq}$ (max. squeezing)	$7.36N^{1.01\pm0.017}$
QFI	$t_{sq2}$ (2nd max. squeezing)	$13.66N^{1.69\pm0.004}$
$S_{\mathrm{PC}}$	$t_{fl}$ (max. flux)	$0.05N^{1.01\pm0.002}$

QFI for Phase Encoding with Photon Number Difference

In this section, we compare the QFI scaling for a phase encoding in the Mach-Zehnder photon number difference operator, which is a different phase encoding scheme used in earlier works on quantum-enhanced phase metrology [36, 45]. In the main text, we assumed that the measured phase is inserted in one arm of the MZI, with the optical path length $\theta$ being imprinted via the interaction Hamiltonian $e^{i\hat{n}\theta}$, with $\hat{n}$ being the number operator in the corresponding mode of the MZI arm. Let $n_{1,2}$ denote the photon number in the two arms of the interferometer. Strictly speaking, both arms impart a phase difference $\theta_{1,2}$, making phase measurement a two-parameter estimation problem. It is instructive to reparametrize the phase encoding interaction as $e^{i(n_1\theta_1+n_2\theta_2)}=e^{in_+(\theta_1+\theta_2)/2}e^{in_-(\theta_1-\theta_2)/2}$, where $n_{\pm}=n_1\pm n_2$. For the cases where the optical path length in both arms needs to be estimated, such as in LIGO, the $n_+$ term cannot be ignored, and the metrological performance is limited by the multiparameter QFI. In this work, we consider the case corresponding to placing the phase object in one arm of the interferometer, while the global path lengths of the interferometer are known. The phase encoding can then simply be written as $e^{in_-\theta/2}$, where the phase difference $\theta=(\theta_1-\theta_2)$. The QFI for phase estimation is found as

where the subscript $D$ denotes using the phase difference for the phase encoding. Because we are using $n_-$ to encode the phase, Eq. (6) does not have the signal and probe variance terms that are present in Eq. (2), and the actual values predicted by the two equations will differ, as was also elaborated in [36]. Nevertheless, the second term in Eq. (6) is equivalent to $N\langle n_s\rangle$, which scales as $N^2$ when the signal is flux optimized and $\langle n_s\rangle\propto N$. We find both $\mathcal{F}_D$ and $\mathcal{F}$, the QFI considered in the main text, to have similar scalings and comparable prefactors when biased at $t_{sq},t_{fl}$, and $t_{var}$. Therefore, insofar as scaling is concerned, both encodings have similar performance for the DPO.

Evolution of the Photon Number Distribution

In this section, we show the evolution of the photon number distribution for different pump photon numbers in a Fock state. In particular, we show that when the signal flux is maximized, the photon number distribution spans the full support of the signal Fock space. Qualitatively, this can be interpreted as one of the mechanisms enabling Heisenberg-limited performance of the joint signal-pump state.

Consider the input state $|\psi(t=0)\rangle=|0\rangle_s|N\rangle_p$, where the pump is in a Fock state $|N\rangle_p$. Evolution under the

Hamiltonian $H_{\mathrm{I}}$ given by Eq. (1) preserves the total input photon number $N$, and the time-evolved state is given by $|\psi(t)\rangle= \sum_{n=0}^{N} c_n(t)|2(N-n)\rangle|n\rangle$ with $c_n(0) = \delta_{n,N}$, and $|c_n(t)|^2$ describing the (discrete) probability distribution to observe $n$ pump and $2(N-n)$ signal photons. Figure 6 shows the evolution of $c_n(t)$ for Fock pumps with $N = [50, 200, 400, 600, 800, 1000]$. Note that there are no odd signal photons in the evolved state and the odd indices are not indicated in the figure. To facilitate comparison between different $N$ values, we plot using the normalized coordinate $x = n/N$, where $x = 0(1)$ denotes the amplitude for all $N$ photons to be present in the signal (pump) mode. The evolution is qualitatively similar across the different $N$ values. The initial energy depletes in the pump mode ($x = 1$), and the photon population spreads across the Fock space. Once the probability amplitude reaches the $x = 0$ boundary, it undergoes a reflection and we see oscillations in the probability distribution [74]. These oscillations of probability distribution are qualitatively similar to quantum caustics observed in many-body photonic, spin, and condensed matter systems governed by nonlinear Hamiltonians [66, 67].

Strictly speaking, the probability distribution is discrete, but for large $N$ the sampling in the domain $x = [0, 1]$ increases, and the finer scale of oscillations might appear aliased in the figure. The range of $\kappa t$ in each plot in Figure 6 is restricted to $1.2t_{fl}$ (for $\kappa= 1$), where $t_{fl}$ is the time at which the signal flux is maximized, shown by the vertical dashed cyan line. As expected, $t_{sq}$ decreases with increasing photon number.

The Quantum Fisher Information (QFI) for phase estimation as given by Eq. (2) has one term proportional to the signal photon variance $\langle\Delta n^2\rangle$, and one proportional to $N\langle n_s\rangle$. Figure 6 shows that at $t_{fl}$, the wavefunction spans the full support of the Fock space from $n = 0$ to $n = N$. Therefore, the signal mean and variance are expected to scale as $N, N^2$ respectively. Consequently, the QFI will scale as $N^2$. Note that Figure 6 assumes a Fock state pump. To compute the variance and mean for a coherent state pump considered in the main text, one has to average over the pump Poisson distribution. However, each pump Fock state will evolve into a broad signal distribution spanning the full support of the Hilbert space. The QFI for a coherent state pump is then also expected to scale as $N^2$, as shown in Figure 3 of the main text.