Improving Energy Efficiency of Uplink RSMA via Rydberg Atomic Quantum Receiver

Introduction

The rapid deployment of massive IoT devices in next-generation (6G) wireless networks has dramatically increased the demands for massive connectivity, high spectral efficiency, and sustainable energy consumption. To address these challenges, RSMA has emerged as a flexible multiple-access paradigm. By splitting user messages into common and private parts, RSMA effectively bridges SDMA and NOMA and enables efficient interference management. Meanwhile, maximizing energy efficiency (EE) remains a critical objective for the green evolution of 6G communication networks.

Literature Review

Evolution Towards Flexible and Efficient Multiple Access

To address the spectral-efficiency limitations of orthogonal multiple access (OMA), power-domain NOMA was introduced, allowing multiple users to share the same time-frequency resources through superposition coding and successive interference cancellation (SIC). However, NOMA’s rigid SIC ordering and sensitivity to channel uncertainty have motivated the search for more flexible frameworks. RSMA has emerged as a generalized paradigm that bridges OMA, NOMA, and spatial-division multiple access by splitting user messages into common and private parts, thereby enabling robust interference management even without perfect channel state information at the transmitter [1, 2]. We focus on uplink RSMA because, in massive IoT, the energy bottleneck lies in battery-limited devices. In contrast, downlink RSMA would require such receivers at the user side, which is impractical for cost- and energy-constrained IoT terminals. This motivates our focus on uplink RSMA empowered by a BS-side RAQR. Early investigations of uplink RSMA confirmed its potential for sum-rate improvement and enhanced user fairness [3, 4].

This flexibility has made RSMA effective in a wide range of emerging communication scenarios, including RIS-assisted systems [5], UAV communications [6], satellite networks [7, 8], V2X scenarios [9], and massive IoT applications employing random access protocols [10]. Advanced RSMA variants have also been proposed to further improve capacity and spatial efficiency, including distributed uplink RSMA (DU-RSMA) [11], user cooperation enhanced RSMA [12], and pinching-antenna-based RSMA. These developments demonstrate the versatility of RSMA as a unifying multiple access framework for diverse 6G use cases.

A critical challenge specific to uplink RSMA is the optimization of the decoding order at the receiver, whose complexity grows factorially with the number of users, rendering joint power and decoding-order optimization computationally demanding for large-scale deployments [13]. Hierarchical approaches to splitting and decoding have been explored to manage this complexity [14]. In satellite RSMA systems, channel-dependent power allocation and decoding order selection have been investigated to improve outage performance [7], and data-oriented designs have been analysed under practical constraints [8]. Notably, the work in [13] proposed a low-complexity user-pairing strategy that transforms the joint power-and-decoding-order problem into a tractable power-and-bandwidth allocation problem with closed-form solutions. The concept of user pairing, widely examined in NOMA to reduce complexity [15], offers a promising direction for practical RSMA design. Despite these advances, existing RSMA EE optimizations still rely heavily on computationally burdensome iterative algorithms. Consequently, leveraging user-pairing concepts to derive exact closed-form solutions and reduce iterative inner-loop processing remains necessary for latency-sensitive and resource-constrained IoT deployments.

Rydberg Atomic Quantum Receivers

The detection of ultra-weak uplink signals from energy-constrained devices constitutes a fundamental physical bottleneck in next-generation networks, as the thermal noise floor and limited linearity of conventional RF receivers impose irreducible sensitivity constraints [16]. Quantum-assisted sensing has emerged as a transformative approach to overcome these classical limitations, and Rydberg atomic quantum receivers (RAQRs) have been identified as a promising candidate for future wireless communication systems [17], [18].

The sensing mechanism of RAQRs exploits the exaggerated electric dipole moments of highly excited Rydberg states together with electromagnetically induced transparency (EIT) to transduce incident RF electric fields into optical readouts [19]. This atomic-level field-to-optical conversion offers several advantages over conventional electronic receivers, including quantum-limited sensitivity, immunity to Johnson–Nyquist thermal noise, ultra-broad spectral tunability, and high linearity [20], [21]. These advantages have been highlighted in surveys on quantum sensing and its prospective role in 6G systems [22], [23].

Experimental studies have gradually advanced RAQR-based communication from proof-of-concept demonstrations to practical prototypes. Early works verified the feasibility of recovering amplitude-modulated and frequency-modulated waveforms using room-temperature atomic vapour cells [24]–[26]. A major breakthrough was achieved with the atomic superheterodyne architecture, where a microwave-dressed local oscillator substantially amplifies the atomic response and improves sensitivity by several orders of magnitude over direct EIT readout [27]. Subsequent studies further characterized RAQRs for RF field metrology and sensing [28]–[30], evaluated wideband operation [31], compared noise temperature with conventional receivers [32], and demonstrated RF phase measurement via atom-based mixing [33].

On the theoretical side, equivalent baseband channel models have been developed for both SISO and MIMO RAQR systems, enabling the analysis of achievable performance gains and capacity limits relative to conventional antenna arrays [34]–[36]. A defining characteristic of standard RAQR implementations, however, is that they perform magnitude-only measurements of the incident electric-field envelope and discard phase information. This amplitude-only constraint introduces phase ambiguity and prevents the direct application of conventional coherent detection and interference cancellation techniques [37]. To address this issue, dedicated signal processing methods have been developed, including RAQR-specific MIMO precoding [38], hardware-constrained channel estimation under magnitude observations [39], and multi-user uplink reception schemes for the amplitude-only regime [40]. The integration of RIS with atomic receivers has also been explored to provide controlled phase references for detection [41]. Further extensions include multi-band and multi-carrier operation for spectrum aggregation [42], [43], as well as spatial signal processing functions such as multi-target direction-of-arrival estimation and polarization-aware detection using a single atomic receiver [44], [45]. Nevertheless, mapping this fundamentally non-linear, amplitude-only detection mechanism into the multi-stream interference structure of RSMA remains a challenging open problem.

Motivation and Contributions

Although existing research has made significant progress in RSMA for next-generation networks and has separately explored the physical mechanism of RAQR, there remains a clear research gap in their integrated design for communication systems. Specifically, by leveraging the ultra-weak uplink signal detection capability of RAQR, the EE of resource-constrained IoT devices can be improved at the physical layer. This direction remains largely unexplored in existing RSMA research. Moreover, although EE is a key goal of RSMA, most existing strategies focus on power allocation and interference management at the algorithm level. The phase-coherent RAQR array provides a new physical-layer method, which is crucial for energy-constrained 6G IoT applications [32]. At the same time, by further combining the user pairing strategy, this framework transforms the joint power and decoding-sequence optimization problem into a lower-complexity form and provides a feasible path for deriving exact closed-form solutions [13]. Therefore, it is necessary to study a RAQR-based RSMA framework to systematically address hardware measurement limitations, algorithmic complexity, and system EE in 6G wireless deployment.

To bridge this gap, we conceive a novel uplink RSMA system assisted by an RAQR at the base station and develop a comprehensive optimization framework. To the best of the authors’ knowledge, RAQR-empowered RSMA networks remain largely unexplored. The main contributions of this paper are summarized as follows:

• We establish a novel equivalent system model for RAQR-empowered uplink RSMA. To overcome the fundamental physical limitation and phase ambiguity of atomic measurements, we design a phase alignment mechanism that linearizes the received signal while retaining full channel energy, thereby enabling tractable optimization.

• We formulate a non-convex EE maximization problem and recast it via the Dinkelbach transform. To address heterogeneous scenarios, we devise tailored solutions: closed-form optimal power allocation via KKT conditions for IoT networks with symmetric rate requirements, a generalized block coordinate descent (BCD) framework for asymmetric rate-requirement settings, and a low-complexity paired allocation that exploits strong–weak user pairing to obviate iterative convex solvers.

• The proposed frameworks feature high scalability and computational efficiency, enabling globally optimal bi-section searches with negligible latency and eliminating

reliance on interior-point methods. Simulations confirm that Scheme A achieves the highest EE by fully leveraging multiuser diversity, while Scheme B offers a compelling trade-off, reducing computational cost by orders of magnitude with only moderate EE degradation, making both well suited for RSMA-based 6G deployments.

The remainder of this paper is organized as follows. Section II presents the system model, including the RAQR-empowered uplink RSMA transmission framework and the achievable rate analysis for both Scheme A and Scheme B. Section III details the proposed unified three-layer decomposition framework, covering the inner-layer phase alignment via the extended EM-GS algorithm, the middle-layer resource allocation strategies for symmetric and asymmetric rate-requirement scenarios, and the outer-layer Dinkelbach iteration for EE maximization. Simulation results and performance comparisons are provided in Section IV. Finally, Section V concludes the paper.

Notations

For clarity, the following notation is adopted throughout this paper. Italic letters denote scalars, bold lowercase letters denote vectors, and bold uppercase letters denote matrices. $\mathbb{R}$ and $\mathbb{C}$ represent the real and complex number fields, respectively. The operators $\lvert\cdot\rvert$, $\lVert\cdot\rVert$, and $\lVert\cdot\rVert_F$ denote the element-wise modulus, the Euclidean norm, and the Frobenius norm, respectively. $\mathcal{R}\{\cdot\}$, $\mathcal{I}\{\cdot\}$, and $\angle(\cdot)$ denote the real part, imaginary part, and phase of a complex variable, respectively. The superscripts $(\cdot)^T$, $(\cdot)^H$, and $(\cdot)^*$ stand for the transpose, Hermitian transpose, and the optimal value under current conditions, respectively. The superscript $(\cdot)^{(i)}$ denotes the value at the $i$-th iteration. Moreover, $\hat{x}$ and $\tilde{x}$ denote the estimated and transformed values of variable $x$, while $(\cdot)^{\mathrm{opt}}$ refers to the optimally phase-aligned equivalent real-valued channel. $\mathbb{E}[\cdot]$ denotes the expectation operator. $Q(\cdot)$ represents the pulse amplitude modulation (PAM) demapping operation. Finally, $\mathcal{CN}(\boldsymbol{\mu},\boldsymbol{\Sigma})$ denotes a circularly symmetric complex Gaussian distribution with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. The operator $\circ$ represents the Hadamard product, and $\mathbf{I}_M$ denotes an $M \times M$ identity matrix.

System Model and Problem Statement

System Model

This section considers a single-cell uplink network where a single base station (BS) serves three users, denoted as $U_k$ ($k \in\{1,2,3\}$). While the detailed physical principles of the RAQR can be found in [34], our focus is on the equivalent communication model. As illustrated in Figure 1, the BS utilizes an $M$-element RAQR instead of a conventional RF front-end. This array comprises $M$ vapor cells functioning as atomic antennas, each filled with Rydberg atoms and coupled to a dedicated photodetector (PD). The optical readouts from these PDs are aggregated and digitally processed for signal detection.

Unlike conventional OMA or NOMA, the adopted RSMA framework [13] enables flexible interference management via

message splitting. Specifically, each user $U_k$ splits its information message into a common part and a private part. These parts are encoded into independent Gaussian symbols $x_k^c$ and $x_k^p$, and transmitted with allocated powers $p_k^c$ and $p_k^p$, respectively.

Based on this transmission protocol, the transmitted signal of user $k$ in each symbol interval is expressed as

where $x_k^c$ and $x_k^p$ are independent real-valued PAM symbols with unit power. The complex baseband signal vector $\mathbf{y} \in\mathbb{C}^{M \times1}$ at the output of the RAQR is modeled as

where $\mathbf{h}_k \in\mathbb{C}^{M \times1}$ denotes the equivalent baseband channel vector from user $U_k$ to the $M$-element atomic array, $\mathbf{b} \in\mathbb{C}^{M \times1}$ is the deterministic local-oscillator (LO) reference signal, and $\mathbf{n} \sim\mathcal{CN}(\mathbf{0},\sigma_n^2\mathbf{I}_M)$ is the additive noise. Equation (2) represents the equivalent complex baseband model derived from the Rydberg EIT principle. Under small-signal conditions, the down-converted optical field is a linear superposition of the users’ symbols $\{s_k\}$ weighted by the channel vectors $\{\mathbf{h}_k\}$, plus the deterministic LO term $\mathbf{b}$. Consequently, the PD output follows the linear model in (2). Although the dominant microscopic noise in practical Rydberg receivers is signal-dependent shot noise [37], this paper adopts the circularly symmetric complex Gaussian approximation based on the central limit theorem to ensure analytical tractability for subsequent rate analysis and resource allocation.

Having established the equivalent baseband model in (2), we now clarify the physical origin of the channel vector $\mathbf{h}_k$. It is not an abstract parameter but is explicitly determined by the atomic parameters of the Rydberg vapor cells. For the $m$-th cell, let $\Omega_p^{(m)}$ and $\Omega_c^{(m)}$ denote the probe and coupling Rabi frequencies, $\Delta_p^{(m)}$ and $\Delta_c^{(m)}$ the corresponding detunings, and $\Gamma_{21}, \Gamma_{31}, \Gamma_{32}$ the decay rates of the relevant atomic transitions. Under the small-signal condition, the probe-transition density matrix element $\rho_{21}^{(m)}$ is linearized around its RF-free value $\rho_{21,0}$ as

where $\Omega_{\mathrm{RF},k}^{(m)} = \frac{\mu_{T_g}\varepsilon_{m,k}}{\hbar}\sqrt{P_k\rho_{m,k}}|s_k|$ is the RF Rabi frequency induced by user $k$, with $P_k$ being the total transmit power of user $k$, and $\mu_{eg}$, $\epsilon_{m,k}$, $\rho_{m,k}$ denoting the transition dipole moment, polarization, and path loss, respectively. The atomic susceptibility is

Accordingly, the equivalent channel coefficient from user $k$ to cell $m$ is

with $\eta_{\mathrm{PD}}^{(m)}$ being the optical readout gain of the $m$-th photodetector. Here, the factor $\sqrt{P_k}$ originates from the fact that the RF Rabi frequency is proportional to the incident field amplitude, which scales with $\sqrt{P_k}$. In (2), this factor is absorbed into $h_k$, while $s_k$ carries only the normalized modulation symbol. Finally, stacking $h_{m,k}$ over all $M$ cells yields $\mathbf{h}_k=[h_{1,k},\ldots,h_{M,k}]^{T}$, which shows that $\mathbf{h}_k$ is jointly determined by $\{\Omega_p^{(m)},\Omega_c^{(m)},\Delta_p^{(m)},\Delta_c^{(m)},\Gamma_{21},\Gamma_{31},\Gamma_{32},\rho_{21}^{(m)}\}$ and the wireless propagation coefficients $\{\rho_{m,k},\epsilon_{m,k}\}$. With the atomic origin of $\mathbf{h}_k$ clarified, we now return to the RAQR observation model.

Unlike traditional RF receivers that directly acquire the complex signal $\mathbf{y}$, the considered RAQR only measures its amplitude. While coherent detection using lock-in amplifiers [25], [33] is a potential alternative, it is beyond the scope of this work. The amplitude-only observation is therefore expressed as

where $\mathbf{z}$ is linked to the Autler–Townes (AT) splitting interval of the underlying Rydberg EIT resonance. This non-linear amplitude relationship complicates signal detection. Nevertheless, $\mathbf{z}$ can be linearized by establishing phase coherence between the received signal and the strong LO. Provided that the phase alignment condition $\angle(\sum_{k=1}^{K}\mathbf{h}_k s_k)\approx\angle\mathbf{b}$ holds and the strong-LO assumption $|\mathbf{b}|\gg\|\sum_{k=1}^{K}\mathbf{h}_k s_k\|_2$ is satisfied, the small-signal approximation yields

where $\boldsymbol{\psi}=[\psi_1,\ldots,\psi_M]^T$ is the phase alignment vector applied at the $M$ atomic cells, and $\mathbf{h}_k^{\mathrm{opt}}(\boldsymbol{\psi})$ denotes the equivalent real-valued optical channel for user $k$ under this phase alignment. The equivalent real-valued optical channel is defined as

In the ideal case where the phase alignment exactly compensates the channel phase of user $k$, i.e., $\psi_m=-\angle h_{m,k}$ for all $m$, we have $\Re\{h_{m,k}e^{j\psi_m}\}=|h_{m,k}|$, so that $\mathbf{h}_k^{\mathrm{opt}}(\boldsymbol{\psi})$ reduces to the elementwise magnitude vector $[|h_{1,k}|,\ldots,|h_{M,k}|]^T$. However, since a single common phase vector $\boldsymbol{\psi}$ is shared by all users, this reduction holds simultaneously for all $k$ only when the channel phases $\{\angle h_{m,k}\}$ are aligned across users; otherwise, $\mathbf{h}_k^{\mathrm{opt}}(\boldsymbol{\psi})$ retains the general form in (8), which is adopted in the subsequent optimization. The effective channel gain is therefore $G_k(\boldsymbol{\psi})=\|\mathbf{h}_k^{\mathrm{opt}}(\boldsymbol{\psi})\|_2^2$, consistent with (9). Precise phase alignment thus transforms the non-linear observation in (6) into the linear model in (7), facilitating efficient user signal recovery and interference cancellation; the detailed derivation and phase alignment algorithm are presented in Section III-A.

Effective Channel Gain and User Ordering

Based on the linearized observation model in (7), we define the effective channel gain $G_k(\boldsymbol{\psi})$ as the squared Euclidean norm of the equivalent real-valued channel vector under a specific phase alignment $\boldsymbol{\psi}=[\psi_1,\ldots,\psi_M]^T$:

This metric quantifies the aggregate signal combining gain provided by the $M$-element atomic array under a specific phase alignment $\boldsymbol{\psi}$. When phase alignment is achieved, $G_k(\boldsymbol{\psi})$ reduces to $G_k=\sum_{m=1}^{M}|h_{m,k}|^2=\|\mathbf{h}_k\|_2^2$. Without loss of generality, the users are indexed in non-decreasing order of their effective channel gains:

Two Operational Schemes

To exploit the potential of Rydberg receivers for uplink RSMA, we consider two complementary transmission schemes. These schemes embody a performance-complexity trade-off: Scheme A (Full RSMA) targets maximum EE through global message splitting and full bandwidth sharing, whereas Scheme B (Paired RSMA) reduces decoding complexity via strong-weak user pairing and localized sub-band allocation, at the expense of partial multiplexing gain.

$a)$ Scheme A (Full RSMA): All $K=3$ users simultaneously share the entire bandwidth $B$. Each user $k$ splits its message into a common stream $s_k^p$ and a private stream $s_k^p$, which are transmitted with powers $p_k^c$ and $p_k^p$, respectively. The BS performs SIC to decode all $2K$ streams according to a predetermined decoding order. Scheme A fully exploits the multiplexing gain, which requires a combinatorial decoding-order search.

$b)$ Scheme B (Paired RSMA): Users are grouped based on a strong-weak pairing strategy [13]. Following the user ordering in (10), the strongest user ($\mathrm{U}_3$) is paired with the weakest ($\mathrm{U}_1$) to form Pair 1, while the median user ($\mathrm{U}_2$) operates as a standalone user. The total bandwidth $B$ is orthogonally divided: Pair 1 is allocated a fraction $\alpha_1B$, and $\mathrm{U}_2$ is allocated $\alpha_0B$, subject to $\alpha_0+\alpha_1\leq1$. Within Pair 1, only $\mathrm{U}_1$ splits its message into two sub-streams $x_{11}$ and $x_{12}$ with powers $p_{11}^{\mathrm{B}}$ and $p_{12}^{\mathrm{B}}$, while $\mathrm{U}_3$ transmits a single stream $x_3$ with power $p_3^{\mathrm{B}}$. The SIC decoding order within the pair is fixed as $x_{11}\to x_3\to x_{12}$ [13]. The standalone $\mathrm{U}_2$ transmits over its dedicated orthogonal sub-band without rate splitting, utilizing power $p_2^{\mathrm{B}}$. This orthogonal separation eliminates inter-pair interference, and the deterministic intra-pair decoding order avoids the combinatorial search required in Scheme A.

Scheme A: Achievable Rates with Full RSMA

In Scheme A, the BS applies maximal ratio combining (MRC) to the phase-aligned real-valued observation in (7). Specifically, the combining vector for user $k$ is

where $\mathbf{h}_k^{\mathrm{opt}}(\psi)$ is defined in (8). Applying $\mathbf{w}_k^{T}$ to $\mathbf{z}$ yields the post-combining observation:

Accordingly, the effective channel gain after combining equals $G_k=\left\|\mathbf{h}_k^{\mathrm{opt}}(\psi)\right\|_2^2$, which is consistent with (9). The effective real-valued noise after combining is $\widetilde{n}_k\triangleq\mathbf{w}_k^{T}\widetilde{\mathbf{n}}$, whose variance is $N_0\equiv\mathbb{E}\{\widetilde{n}_k^2\}=\frac{\sigma_n^2}{2}$, where the factor $1/2$ arises because the complex baseband noise $\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma_n^2\mathbf{I}_M)$ has independent real and imaginary parts, each with variance $\sigma_n^2/2$, and the real-valued observation model in (7) retains only the real component.

To accommodate heterogeneous channel conditions, the SIC decoding order is specified as follows: common streams are decoded from the strongest user to the weakest (user $3\rightarrow2\rightarrow1$), since they mutually interfere and are removed sequentially via SIC; private streams are decoded from the weakest user to the strongest (user $1\rightarrow2\rightarrow3$) after all common streams have been cancelled.

Under this ordering, the post-combining achievable common rates are

Similarly, the private rates are

The total achievable rate of user $k$ under Scheme A is given by $R_k^A=r_k^c+r_k^p$.

Scheme B: Achievable Rates with Paired RSMA

In Scheme B, under the fixed intra-pair SIC decoding order $x_{11}\rightarrow x_3\rightarrow x_{12}$, the achievable rates for Pair 1 are:

where $G_k$ is the post-combining effective channel gain, and $N_0$ is the effective real-valued noise variance. The same combining and noise definitions apply to Scheme B. The total rate of U1 is $R_1^B=r_{11}^B+r_{12}^B$, and the rate of U3 is $R_3^B=r_3^B$.

Since U2 occupies an orthogonal sub-band, it experiences no multi-user interference. Its achievable rate is

Compared with Scheme A, Scheme B replaces the combinatorial SIC decoding-order search with a deterministic intra-pair decoding order and continuous bandwidth allocation variables $(\alpha_0,\alpha_1)$.

Problem Formulation

Based on the derived achievable rates, we formulate the EE maximization problem for both schemes. The EE is defined as the ratio of the sum rate to the total power consumption:

where the total power consumption is $P_{\mathrm{total}}=\sum_{k=1}^{K}(p_k^c+p_k^p)+P_c$, and $P_c=M(P_{\mathrm{over}}+P_{\mathrm{PD}})+P_{\mathrm{LO}}$ represents the hardware power overhead of the RAQR. Although both schemes share the same EE objective, their distinct transmission structures result in different optimization formulations.

Problem Formulation for Scheme A

1. Problem Formulation for Scheme A: Using the EE definition in (13), the joint power allocation and phase alignment optimization problem for Scheme A is formulated as:

where $\mathbf{H}^{\mathrm{eq}}(\psi)=[\mathbf{h}_1\circ e^{j\psi},\mathbf{h}_2\circ e^{j\psi},\mathbf{h}_3\circ e^{j\psi}]$ is the composite effective channel matrix, and $\circ$ denotes the Hadamard product. Constraint (14b) ensures that each user’s transmit power is within the hardware budget $P_k^{\max}$. Constraint (14c) guarantees that the achievable rate meets the individual minimum-rate requirement $R_k^{\min}$. Constraint (14d) maintains phase alignment by restricting the energy of the residual imaginary component below a small tolerance $\epsilon>0$, thereby ensuring the validity of the linear model in (7).

Problem Formulation for Scheme B

2. Problem Formulation for Scheme B: For Scheme B, the optimization variables include the transmit powers $\{p_{11}^B,p_{12}^B,p_3^B,p_2^B\}$, the bandwidth allocation factors $(\alpha_0,\alpha_1)$, and the phase alignment vector $\psi$. The problem is formulated as:

Here, constraints (15b)–(15d) impose individual power budgets, (15e) guarantees the rate requirements, (15f) enforces the orthogonal bandwidth partitioning, and (15g) maintains the phase alignment validity identical to constraint (14d).

Non-Convexity Analysis

Both $\mathcal{P}_{\mathrm{A}}$ and $\mathcal{P}_{\mathrm{B}}$ are non-convex due to four factors: (i) the non-convex phase alignment constraint $\left\lVert\{\mathsf{H}_{\mathrm{eq}}(\boldsymbol{\psi})\} \right\rVert_{\mathrm{F}}^{2} \leq\varepsilon$, which involves trigonometric functions of $\boldsymbol{\psi}$ and couples with the power allocation variables; (ii) the fractional structure of the EE objective; (iii) the logarithmic rate expressions involving multi-user interference; and (iv) the coupling between the phase vector $\boldsymbol{\psi}$ and the power variables. Compared to $\mathcal{P}_{\mathrm{A}}$, problem $\mathcal{P}_{\mathrm{B}}$ reduces the number of power variables and eliminates the combinatorial SIC decoding-order search by introducing bandwidth allocation variables. While $\mathcal{P}_{\mathrm{A}}$ yields superior EE by fully exploiting spatial and multiplexing gains, $\mathcal{P}_{\mathrm{B}}$ enables low-complexity resource allocation.

Proposed Unified Three-Layer Decomposition Framework

To address the non-convexity of Problems (14) and (15), we propose a unified hierarchical three-layer optimization framework tailored to the RAQR constraints. The framework systematically decomposes both schemes as follows:

1. Inner Layer (Phase Alignment – Common to Both Schemes): Given the current power allocation, this layer estimates the phase compensation parameters $\{\theta_m\}$ to satisfy the phase alignment constraints (14d) or (15g) using an extended EM-GS algorithm. This process ensures a valid real-valued channel linearization for both schemes.

2. Middle Layer (Resource Allocation – Scheme-Specific):

   • Scheme A (Full RSMA): Based on the aligned phases, the power variables $\{p_k^{\mathrm{c}},p_k^{\mathrm{p}}\}$ are optimized via closed-form solutions in symmetric scenarios or block coordinate descent power allocation (BCD-PA) in asymmetric scenarios, as detailed in Section III-B.

   • Scheme B (Paired RSMA): Given the aligned phases, the power $\{p_{11}^{m},p_{12}^{m},p_{2}^{m}\}$ and bandwidth allocation $\{\alpha_m\}$ are jointly optimized via closed-form expressions and bisection search, as detailed in Section III-C.

3. Outer Layer (EE Maximization – Common to Both Schemes): The Dinkelbach parameter $\eta$ is iteratively updated based on the rate and power allocation obtained from the middle layer. The Dinkelbach iteration structure is identical for both schemes, differing only in the specific rate and power expressions. Upon convergence, minimum mean square error (MMSE) detection is performed for final signal recovery.

This unified framework exploits the underlying structural commonality of the system: both schemes require phase alignment (due to the RAQR hardware constraints) and EE maximization, differing only in the multi-user spectrum access mechanism (full sharing versus orthogonal pairing).

Inner Layer: Phase Alignment via Extended EM-GS Algorithm

The amplitude-only measurement of the RAQR introduces a biased phase retrieval (PR) problem, as the phase information of the received signal is lost in the presence of a known local-oscillator (LO) field b. This mathematical structure is equivalent to the biased PR model established in [34] for atomic MIMO receivers. Consequently, we adopt the expectation-maximization Gerchberg-Saxton (EM-GS) framework and extend it with an explicit gradient-descent step to optimize the phase compensation vector $\boldsymbol{\psi}$ under the linearization constraint of the RAQR.

For each user $k$, the total transmit power is $p_k=p_k^{\mathrm{c}}+p_k^{\mathrm{p}}$. The transmitted composite symbol is given by (1), where $x_k^{\mathrm{c}}$ and $x_k^{\mathrm{p}}$ are independent, unit-power symbols. For phase alignment and channel estimation, the normalized composite symbol is defined as $\bar{x}_k=s_k/\sqrt{p_k}=(\sqrt{p_k^{\mathrm{c}}}x_k^{\mathrm{c}}+\sqrt{p_k^{\mathrm{p}}}x_k^{\mathrm{p}})/\sqrt{p_k}$, yielding $s_k=\sqrt{p_k}x_k$ with $\mathbb{E}[|x_k|^2]=1$. This representation is mathematically consistent with the RSMA signal model, as the RAQR measures the aggregate field $\sum_k h_ks_k$ without separating the individual data streams at the physical layer.

At the $m$-th vapor cell, the amplitude observation follows directly from (6) with the total symbol vector $\mathbf{s}=[\sqrt{p_1}x_1,\ldots,\sqrt{p_K}x_K]^{T}$, i.e., $\mathbf{z}=|\mathbf{A}^{H}\mathbf{s}+\mathbf{b}+\mathbf{n}|$, where $\mathbf{A}$ is the equivalent channel matrix and $\mathbf{b}$ is the known LO field.

It is important to clarify how the circular dependence between the EM-GS phase retrieval and symbol detection is broken in practice. The proposed inner-layer algorithm operates in two stages: a pilot-based initialization stage and a data-aided iterative stage.

During the pilot phase, each user $k$ transmits a known pilot sequence $\{x_k^{\mathrm{p}}\}$ with a predefined power. Stacking the pilot symbols from all users into $\mathbf{s}^{\mathrm{p}}=[\sqrt{p_1}x_1^{\mathrm{p}},\ldots,\sqrt{p_K}x_K^{\mathrm{p}}]^{T}$, the RAQR collects $N_{\mathrm{p}}$ amplitude observations $\{\mathbf{z}_n\}_{n=1}^{N_{\mathrm{p}}}$ according to (6). Since the pilots are known, the initial total-symbol estimate $\hat{\mathbf{s}}^{(0)}$ can be obtained by the pilot-based least-squares solution

Alternatively, $\hat{\mathbf{s}}^{(0)}$ can be obtained by the spectral initialization method in [34], which uses the principal eigenvector of the augmented covariance matrix $\mathbf{M}=\sum_{n=1}^{N_{\mathrm{p}}}\mathbf{z}_{n}^{p}\mathbf{a}_{n}\mathbf{a}_{n}^{H}$ to obtain a coarse estimate without requiring any prior symbol knowledge. This removes the need for data symbols in the initialization of the EM-GS iteration.

Once $\hat{\mathbf{s}}^{(0)}$ is available, the EM-GS iteration alternates between estimating the latent phase $\theta_m$ and refining the total-symbol estimate $\hat{\mathbf{s}}^{(i)}$. At each iteration, $\hat{\mathbf{s}}^{(i)}$ is not a hard decision on the transmitted symbols; it is a soft, linear least-squares estimate obtained from the previous M-step. Therefore, the iteration does not require the final symbol decisions and can be carried out before demodulation.

After the EM-GS iteration converges, the phase alignment vector $\boldsymbol{\psi}^{*}$ is fixed and the equivalent real-valued channel $\bar{h}_{m,k}$ is obtained. The subsequent resource allocation and MMSE detection then recover the data symbols. In other words, the phase retrieval stage uses only pilots (for initialization) and soft symbol estimates (for iteration), while the data symbols are detected only after the phase alignment is completed.

Given the current total-symbol estimate $\hat{\boldsymbol{s}}^{(i)}$, the conditional distribution of the observed amplitude $z_m$ is Rician. The posterior distribution of the latent phase $\theta_m \triangleq\angle(\boldsymbol{a}_m^{H}\boldsymbol{s} + b_m)$ follows a von Mises distribution:

where $\mu_m^{(i)} = \angle\left(\boldsymbol{a}_m^{H}\hat{\boldsymbol{s}}^{(i)} + b_m\right)$ represents the posterior phase mean, and $\kappa_m^{(i)} = \frac{2z_m\left|\boldsymbol{a}_m^{H}\hat{\boldsymbol{s}}^{(i)} + b_m\right|}{\sigma^2}$ is the concentration parameter.

The E-step produces $\tilde{\boldsymbol{z}}^{(i)} = \boldsymbol{z} \circ e^{j\boldsymbol{\mu}^{(i)}} \circ R(\boldsymbol{\kappa}^{(i)})$, where $R(x)=I_1(x)/I_0(x)$ is the ratio of the first-kind modified Bessel functions, and $\circ$ denotes the Hadamard product. As established in [34], $R(x)$ functions as an SNR-dependent high-pass filter, suppressing unreliable phase estimates under low-SNR conditions while converging to 1 at high SNR.

Using $\tilde{\boldsymbol{z}}$ as the effective linear observation, the M-step updates $\hat{\boldsymbol{s}}^{(i+1)} = (\boldsymbol{A}^{H}\boldsymbol{A})^{-1}\boldsymbol{A}^{H}(\tilde{\boldsymbol{z}}^{(i)}-\boldsymbol{b})$. This closed-form update maximizes the expected log-likelihood and is identical to the standard EM-GS algorithm.

To minimize the residual imaginary energy of the equivalent channel for RAQR linearization, we augment the EM-GS iteration with a gradient descent step on the phase compensation vector $\boldsymbol{\psi}$. We define the cost function penalizing the residual imaginary components as

where $\psi_k$ denotes the phase compensation applied to user $k$’s channel across all vapor cells. The gradient with respect to $\psi_k$ is given by

Following each EM-GS iteration, the phase vector is updated via $\boldsymbol{\psi}^{(i+1)}=\boldsymbol{\psi}^{(i)}-\mu\nabla J(\boldsymbol{\psi}^{(i)})$, where $\mu>0$ is the step size. This extension ensures adherence to the RAQR linearization requirements while maintaining the theoretical advantages of the EM framework.

Middle Layer for Scheme A: Power Allocation

Given the phase-aligned real-valued channel matrix $\tilde{\boldsymbol{H}}$ with entries $\bar{h}_{m,k}=\Re\{h_{m,k}e^{j\psi_k}\}$ from the inner layer, the effective channel gain for user $k$ is defined as $G_k=\sum_{m=1}^{M}\bar{h}_{m,k}^{2}$. Without loss of generality, we order the users such that $G_1\le G_2\le\cdots\le G_K$. The middle layer optimizes the power allocation based on these gains, distinguishing between symmetric and asymmetric rate-requirement scenarios.

Symmetric Rate-Requirement Scenario: Closed-Form Solution

For homogeneous deployments where users share similar rate requirements, the Symmetric Rate-Requirement condition is quantified as:

where $\bar{R}=\frac{1}{K}\sum_{k=1}^{K}R_k^{\min}$ and $\delta$ is a predefined threshold (e.g., 0.1). When (29) is met, a common target rate $R$ is enforced for all users with rate-splitting factors satisfying $\alpha+\beta=1$. Note that symmetric rate requirement does not imply symmetric channel gains; the derived closed-form solution is valid for arbitrarily ordered channel gains $G_1\le G_2\le G_3$. It is worth clarifying why the rate-splitting factors $\alpha$ and $\beta$ are treated as fixed parameters rather than optimization variables in Problem (14).

In the considered symmetric-rate scenario, all users share the same target rate $R$, and $\alpha+\beta=1$ only determines how $R$ is split between the common and private streams. Once $R$ is fixed, any feasible pair $(\alpha,\beta)$ yields the same total rate $R$; the split affects neither the sum rate nor the total transmit power required to achieve $R$, and hence does not change the achievable EE. Including $\alpha$ and $\beta$ as additional optimization variables would therefore introduce a redundancy without improving the objective. Moreover, the rate-splitting factors are typically chosen offline according to the RSMA protocol design and the target rate ratio between common and private streams [13]; fixing them reduces the dimensionality of the optimization and enables the closed-form KKT solution in (21)–(22).

Assuming a three-user scenario ($K=3$) with standard SIC ordering (weakest user decoded first for private streams), applying the KKT conditions under the equal-rate constraint $r_k^p=\beta R$ yields the following closed-form optimal private power allocation:

For the common streams decoded in reverse order ($3\rightarrow2\rightarrow1$), we define $\gamma_{\mathrm{th}}=2^{\alpha R/B}-1$ and the interference floor as $I=\sum_{j=1}^{3}G_jp_j^{p*}+N_0$. The optimal common power allocation is derived as:

The structural recursion scales identically for arbitrary $K$. For a fixed rate $R$, the EE is defined as

The unimodality of $\mathrm{EE}(R)$ over the interval $[R_{\min},R_{\max}]$ is established as follows. The total transmit power $P_{\mathrm{tx}}(R)=\sum_k\left(p_k^{c*}(R)+p_k^{p*}(R)\right)$ is a positive linear combination of exponential functions $2^{\beta R/B}$ and $2^{\alpha R/B}$. Consequently, its logarithmic derivative $h(R)=\frac{P_{\mathrm{tx}}'(R)}{P_{\mathrm{tx}}(R)+P_c}$ is strictly monotonically increasing. Since $\frac{d}{dR}\ln\mathrm{EE}(R)=\frac{1}{R}-h(R)$, the derivative $\frac{d}{dR}\ln\mathrm{EE}(R)$ is positive for small $R$ and negative for large $R$, guaranteeing a unique maximizer $R^*$.

Asymmetric-Rate Scenario: BCD-PA Solution

When the symmetric condition (29) is violated, the non-convex EE maximization problem is solved using BCD-PA. The fractional program is $\max_{\boldsymbol{p}\in\mathcal{F}}\mathcal{R}(\boldsymbol{p})/\mathcal{P}(\boldsymbol{p})$, where $\mathcal{R}(\boldsymbol{p})=\sum_k(r_k^c+r_k^p)$ and $P(\mathbf{p})=\sum_k(p_k^c+p_k^p)+P_c$.

Applying Dinkelbach’s transform, the optimal EE $\eta^*$ satisfies $\max_{\mathbf{p}\in\mathcal{F}}[R(\mathbf{p})-\eta^*P(\mathbf{p})]=0$. For a given parameter $\eta$, the subproblem objective is

Due to the coupled power variables inside the logarithmic rate expressions, (24) is non-convex. We apply block coordinate descent, updating one power variable at a time while fixing the others, thereby decomposing the problem into a sequence of convex single-variable subproblems.

For private streams decoded in weakest-first order ($1\to2\to3$), the SINR for user $k$ is $\mathrm{SINR}_k^p=\frac{G_kp_k^p}{I_k^p}$, where the effective interference is $I_3^p=N_0$, $I_2^p=G_3p_3^p+N_0$, and $I_1^p=G_2p_2^p+G_3p_3^p+N_0$.

Setting the partial derivative of the objective with respect to $p_k^p$ to zero yields the unconstrained optimum $p_k^{p,\mathrm{unc}}=\frac{B}{\eta\ln2}-\frac{I_k^p}{G_k}$. Projecting this onto the feasible region yields the update rule:

Similarly, for common streams decoded in strongest-first order ($3\to2\to1$), the SINR is $\mathrm{SINR}_k^c=\frac{G_kp_k^c}{I_k^c}$, where the interference $I_3^c=G_2p_2^c+G_1p_1^c+\sum_{i=1}^{K}G_ip_i^p+N_0$, $I_2^c=G_1p_1^c+\sum_{i=1}^{K}G_ip_i^p+N_0$ and $I_1^c=\sum_{i=1}^{K}G_ip_i^p+N_0$.

The optimal common power update is similarly obtained via projection:

Middle Layer for Scheme B: Joint Power and Bandwidth Allocation

Given the phase-aligned channel gains $\{\bar{G}_1^m,\bar{G}_2^m\}$ and the Dinkelbach parameter $\eta$, the middle layer for Scheme B maximizes the decoupled objective:

For a specific bandwidth fraction $\alpha_m$, the achievable rate region $(r_1^m,r_2^m)$ of pair $m$ is bounded by three line segments:

AB: $p_{12}^m=0$, $p_2^m=P_{\max,2}^m$, $0\leq p_{11}^m\leq P_{\max,1}^m$, (28a)

BC: $p_{11}^m+p_{12}^m=P_{\max,1}^m$, $p_2^m=P_{\max,2}^m$, (28b)

CD: $p_{11}^m=0$, $p_{12}^m=P_{\max,1}^m$, $0\leq p_2^m\leq P_{\max,2}^m$. (28c)

The corresponding maximum rates and sum-rate are

Under Dinkelbach’s transform, optimal rate allocation aligns with the rate target ratio when constraints are active, i.e., $r_1^m/r_2^m=R_{\min,1}^m/R_{\min,2}^m\triangleq\rho_m$. The closed-form power allocation for the corresponding operating segments is given by:

Case 1 (on Segment AB):

Case 2 (on Segment BC):

Case 3 (on Segment CD):

Under Dinkelbach’s transform, optimal rate allocation aligns with the rate target ratio when constraints are active, i.e., $r_1^m/r_2^m=R_{\min,1}^m/R_{\min,2}^m\triangleq\rho_m$. The closed-form power allocation for the corresponding operating segments is given by:

Case 1:

Case 2 (with $\Gamma_m^{BC}=\frac{\bar{G}_1^mP_{\max,1}^m+\bar{G}_2^mP_{\max,2}^m}{N_0}$ and $\bar{R}_m=R_{\min,1}^m+R_{\min,2}^m$):

Case 3:

The optimal case index is selected by maximizing the per-pair EE over the feasible set $\mathcal{I}_{\mathrm{feas}}=\{1,2,3\}$:

Overall bandwidth feasibility $\sum_{m=1}^{M}\alpha_m^*\leq1$ is maintained by adjusting the rate targets dynamically if required.

Algorithm 1. Middle-Layer Resource Allocation for Scheme A and Scheme B

1. Input: Phase-aligned real-valued channel $\tilde{\mathbf{H}}$, channel gains $\{G_k\}$, Dinkelbach parameter $\eta$, scheme selection flag $\mathcal{S}\in\{A,B\}$, rate requirements $\{R_k^{\min}\}$, power bounds $\{P_k^{\max}\}$, tolerance $\epsilon_{\mathrm{mid}}$.

2. Output: Resource allocation: Scheme A: $\{p_k^{c*},p_k^{p*}\}$; Scheme B: $\{p_{11}^{m*},p_{12}^{m*},p_2^{m*},\alpha_m^*\}$.

3. If $S=A$:

a) Scheme A: Full RSMA Power Allocation

i) Compute $G_k=\sum_{m=1}^{M}\bar{h}_{m,k}^{2}$, and sort $G_1\le G_2\le G_3$.

ii) Evaluate rate-requirement symmetry metric: $\delta_{\mathrm{sym}}=\max_k\frac{\lvert R_k^{\min}-\bar{R}\rvert}{\bar{R}}$.

iii) If $\delta_{\mathrm{sym}}<0.1$:

A) Determine $R_{\min}=\max_k R_k^{\min}$.

B) Find $R_{\max}$ via feasibility bisection ensuring $p_k^{c*}+p_k^{p*}\le P_k^{\max},\ \forall k$.

C) Initialize $a\leftarrow R_{\min}$, $b\leftarrow R_{\max}$, $\varphi=(\sqrt{5}-1)/2$.

D) Compute $c=b-\varphi(b-a)$, $d=a+\varphi(b-a)$.

E) While $(b-a)>\epsilon_R$:

• Compute EE$(c)$ and EE$(d)$ using the closed-form private powers in (21), common powers in (22), and EE$(R)$ in (23).

• If EE$(c)>\mathrm{EE}(d)$: $b\leftarrow d$, $d\leftarrow c$, $c\leftarrow b-\varphi(b-a)$.

• Else: $a\leftarrow c$, $c\leftarrow d$, $d\leftarrow a+\varphi(b-a)$.

F) Set $R^*\leftarrow(a+b)/2$.

G) Compute final powers $\{p_k^{c*}\}$, $\{p_k^{p*}\}$ at $R^*$.

iv) Else:

A) Initialize $p_k^c=p_k^p=0.25P_k^{\max}$ for all $k$.

B) Repeat:

• Stage 1: Private Power Update (order $k=K,\ldots,1$).

  • Compute interference: $I_k^p=\sum_{i>k}G_i p_i^p+N_0$.

  • Update $p_k^p$ via (25).

  • If $r_k^p<R_k^{\min}-r_k^c$, increase $p_k^p$ to meet the rate constraint.

• Stage 2: Common Power Update (order $k=K,\ldots,1$).

  • Compute interference: $I_k^c=\sum_{j<k}G_jp_j^c+\sum_{i=1}^{K}G_ip_i^p+N_0$.

  • Update $p_k^c$ via (26).

  • If $r_k^c<R_k^{\min}-r_k^p$, increase $p_k^c$ to meet the rate constraint.

C) Until $\lVert p^{(t+1)}-p^{(t)}\rVert_2<\epsilon_{\mathrm{mid}}$.

D) Output $\{p_k^{c*},p_k^{p*}\}$.

4. Else If $S=B$:

a) Scheme B: Paired RSMA Joint Power and Bandwidth Allocation

i) For each pair $m=1,\ldots,M$:

A) Construct the rate region with segments AB, BC, CD.

B) For each feasible case $i\in\{1,2,3\}$, compute closed-form powers via (29)–(31) and bandwidth via (32)–(34).

C) Select the best case by maximizing per-pair EE: $\mathrm{Case}^*=\arg\max_{i\in\mathcal{I}_{\mathrm{feas}}}\frac{R_m^{(i)}}{P_m^{(i)}+\alpha_m^{(i)}P_{c,m}}$.

D) If needed, refine $\alpha_m$ by bisection search on the KKT condition.

ii) Enforce overall bandwidth feasibility $\sum_{m=1}^{M}\alpha_m^*\le1$ by adjusting rate targets if necessary.

b) Output $\{p_{11}^{m*},p_{12}^{m*},p_2^{m*},\alpha_m^*\}$.

Outer Layer: Unified Dinkelbach Iteration

The outer layer performs the generic Dinkelbach parameter update for both schemes. At iteration $t$, the update rule is: $\eta^{(t+1)}=R_{\mathrm{total}}^{(t)}/P_{\mathrm{total}}^{(t)}$, where the scheme-specific rate and power aggregates are defined as: for Scheme A, $R_{\mathrm{total}}^{(t)}=\sum_k(r_k^c+r_k^p)$ and $P_{\mathrm{total}}^{(t)}=\sum_k(p_k^c+p_k^p)+P_c$; for Scheme B, $R_{\mathrm{total}}^{(t)}=\sum_m(r_1^m+r_2^m)$ and $P_{\mathrm{total}}^{(t)}=\sum_m(p_{11}^m+p_{12}^m+p_2^m)+P_c$.

Overall Algorithm and Performance Analysis

Complete Three-Layer Framework

Integrating the inner, middle, and outer layers constructs the comprehensive optimization framework outlined in Algorithm 2. The process iterates through phase alignment, power allocation, and efficiency evaluation, culminating in final signal recovery.

Algorithm 2. Three-Layer Decomposition Framework for RAQR-RSMA

1. Input: Complex channel $\mathbf{H}$, amplitude observations $\mathbf{z}$, LO field $b$, bandwidth $B$, noise variance $\sigma^2$, power bounds $\{P_k^{\max}\}$, rate requirements $\{R_k^{\min}\}$, scheme selection flag $S\in\{A,B\}$, tolerances $\epsilon_{\mathrm{in}},\epsilon_{\mathrm{mid}},\epsilon_{\mathrm{out}}$.

2. Output: Phase parameters $\psi^*$, resource allocation, maximum EE $\eta^*$, and detected symbols $\hat{\mathbf{s}}$.

3. Stage 1: Initialization

a) Initialize Dinkelbach parameter $\eta^{(0)}=0$, outer iteration index $t=0$.

b) Initialize power allocation $\mathbf{p}^{(0)}$.

4. Stage 2: Iterative Optimization

a) Repeat:

• Layer 1: Inner Layer (Phase Alignment via Extended EM-GS). i) Initialize $\psi^{(0)}=0$. ii) Obtain $\hat{\mathbf{s}}^{(0)}$ from the known pilot observations via spectral initialization or pilot-based least squares in (16). iii) For $i=0,1,\ldots$ until convergence:

  • E-step: Compute posterior phase mean $\mu^{(i)}$ with $\mu_m^{(i)}=\angle(a_m^{(i)}\hat{s}^{(i)}+b_m)$, and concentration $\kappa_m^{(i)}=\frac{2z_m\lvert a_m^{(i)}\hat{s}^{(i)}+b_m\rvert}{\sigma^2}$; form $\tilde{z}=z\circ e^{j\mu^{(i)}}\circ R(\kappa^{(i)})$.

  • M-step: Update $\hat{\mathbf{s}}^{(i+1)}=(A^HA)^{-1}A^H(\tilde{\mathbf{z}}^{(i)}-\mathbf{b})$.

  • Gradient step: Update $\psi^{(i+1)}=\psi^{(i)}-\mu\nabla J(\psi^{(i)})$. iv) Set $\phi^*\leftarrow\psi^{(i+1)}$, $\bar{h}_{m,k}=\Re\{h_{m,k}e^{j\psi_k^*}\}$, and $\hat{\mathbf{s}}\leftarrow\hat{\mathbf{s}}^{(i+1)}$.

v) Obtain the phase-aligned real-valued channel $\widetilde{\mathbf{H}}$ with entries $\bar{h}_{m,k}=\Re\{h_{m,k}e^{j\psi_k^*}\}$.

• Layer 2: Middle Layer (Resource Allocation). i) Compute channel gains: $G_k=\sum_m|\widetilde{H}_{m,k}|^2$, and sort $G_1\le G_2\le\cdots\le G_K$. ii) Execute Algorithm 1 with scheme flag $S$ and Dinkelbach parameter $\eta^{(t)}$ to obtain the resource allocation:

  • If $S=A$: obtain $\{p_k^c,p_k^p\}$.

  • Else If $S=B$: obtain $\{p_{11}^m,p_{12}^m,p_2^m,\alpha^*\}$.

• Layer 3: Outer Layer (EE Update via Dinkelbach Iteration). i) If $S=A$: $R_{\mathrm{total}}^{(t)}=\sum_{k=1}^{K}(r_k^c+r_k^p)$, $P_{\mathrm{total}}^{(t)}=\sum_{k=1}^{K}(p_k^c+p_k^p)+P_c$. ii) Else If $S=B$: $R_{\mathrm{total}}^{(t)}=\sum_{m=1}^{M}(r_1^m+r_2^m)$, $P_{\mathrm{total}}^{(t)}=\sum_{m=1}^{M}(p_{11}^m+p_{12}^m+p_2^m)+P_c$. iii) Compute objective $F(\eta^{(t)})=\left|R_{\mathrm{total}}^{(t)}-\eta^{(t)}P_{\mathrm{total}}^{(t)}\right|$. iv) If $F(\eta^{(t)})<\epsilon_{\mathrm{out}}$: Break. v) Update $\eta^{(t+1)}=\frac{R_{\mathrm{total}}^{(t)}}{P_{\mathrm{total}}^{(t)}}$, and increment $t\leftarrow t+1$.

b) Until convergence criteria are met.

5. Stage 3: Signal Recovery

a) Apply MMSE detection using the phase-aligned real channel: $\hat{\mathbf{s}}=(\widetilde{\mathbf{H}}^{T}\widetilde{\mathbf{H}}+\sigma^2\mathbf{P}^{-1})^{-1}\widetilde{\mathbf{H}}^{T}(\widetilde{\mathbf{z}}-\bar{\mathbf{b}})$, where $\mathbf{P}=\operatorname{diag}(p_1,\ldots,p_K)$ is the power allocation matrix, and $\bar{\mathbf{b}}$ is the equivalent real-valued LO component.

6. Return: $\psi^*$, resource allocation, $\eta^*$, and $\hat{\mathbf{s}}$.

Convergence Analysis

2. Convergence Analysis: The convergence of the proposed framework to a stationary point is analyzed layer by layer.

• Inner Layer Convergence: The extended EM-GS algorithm produces a non-decreasing sequence of the expected complete-data log-likelihood and strictly reduces the residual imaginary energy. Hence, the phase alignment subproblem converges to a stationary point that yields a valid real-valued channel representation.

• Middle Layer Convergence: In symmetric deployments, the strict unimodality of EE$(R)$ ensures that the golden-section search converges globally to the unique maximum $R^*$ at a linear rate. In asymmetric cases, for a given Dinkelbach parameter $\eta$, the BCD-PA algorithm produces a monotonically non-decreasing objective sequence that converges to a coordinate-wise stationary point of the subproblem.

• Outer Layer Convergence: Under Dinkelbach’s fractional programming theory, the auxiliary function $F(\eta)$ is convex and strictly decreasing in $\eta$. The update $\eta^{(t+1)}=R(\mathbf{p}^{(t)})/P(\mathbf{p}^{(t)})$ strictly increases whenever $F(\eta^{(t)})>0$. Since the sum rate is bounded above and the transmit power is strictly positive, $\{\eta^{(t)}\}$ converges monotonically to the optimum $\eta^*$.

Since each layer monotonically improves its objective under bounded system constraints, the overall algorithm converges to a stationary point of the optimization problem.

Complexity Comparison

3. Complexity Comparison: For Scheme A (Full RSMA), the BCD-PA algorithm updates $2K$ power variables per Dinkelbach iteration. Each coordinate update computes local-zed interference terms and applies water-filling projections, giving an inner-iteration complexity of $\mathcal{O}(K\cdot I_{\mathrm{BCD}})$, where $I_{\mathrm{BCD}}$ is the BCD loop count.

Scheme B (Paired RSMA) evaluates closed-form expressions over $M=\lfloor K/2\rfloor$ independent pairs, with a bisection search of precision $\epsilon_{\mathrm{pair}}$. Its complexity is $\mathcal{O}(M\cdot\log_2(1/\epsilon_{\mathrm{pair}}))$, which is considerably lower than that of Scheme A for large $K$, since it avoids coupled iterative optimization within each pair.

For the common components, the inner EM-GS layer requires $\mathcal{O}(N_{\mathrm{atom}}\cdot I_{\mathrm{EM}})$ operations in both schemes, where $N_{\mathrm{atom}}$ is the number of Rydberg vapor cells and $I_{\mathrm{EM}}$ is the number of EM iterations. The outer Dinkelbach layer has the standard convergence complexity of $\mathcal{O}(\log(1/\epsilon_{\mathrm{out}}))$.

Simulation Results and Analysis

In this section, we present numerical results to evaluate the convergence behavior and performance of the proposed energy-efficient uplink RSMA framework enabled by the RAQR.

Unless otherwise specified, the following common simulation parameters are adopted. We consider a single-cell uplink RSMA network, where one base station equipped with a 36-element RAQR array serves three users. The effective noise power $N_0$ is determined by the quantum projection noise of the atomic ensemble. By substituting $N_a\approx5\times10^5$, $\tau_{\mathrm{coh}}\approx10\ \mu\mathrm{s}$, and the typical dipole moment corresponding to the $52D_{5/2}\to53P_{3/2}$ transition, we obtain $N_0\approx-114$ dBm. The circuit power consumption is set to 24.6 W, and the maximum transmit power is 23 dBm. The wireless channel includes both path loss and Rayleigh small-scale fading, and the effective channel gain is obtained by coherently combining the responses of the 36 atomic array elements. The transmit power budget is swept from $-10$ to 30 dBm with a step size of 2 dB.

Figure 2 depicts the mean square error (MSE) of phase estimation versus the iteration index of the expectation-maximization (EM) algorithm under different local oscillator (LO) field intensities, denoted by $|b|^2$. It can be observed that the EM algorithm converges monotonically for all tested LO settings. Moreover, the LO field intensity has a clear impact on both the convergence speed and the final estimation accuracy. When the LO field is strong, e.g., $|b|^2=10$, the MSE decreases rapidly and stabilizes within approximately 5 to 8 iterations. In contrast, when the LO field is weak, e.g., $|b|^2=0.1$, the algorithm requires more than 20 iterations to converge and settles at a higher steady-state MSE. This behavior indicates that a sufficiently strong LO field can effectively reduce the phase ambiguity in amplitude-only RAQR measurements, thereby improving the robustness of the EM-based phase retrieval process.

The convergence of the outer-layer optimization is shown in Figure 3, where the system EE is plotted against the Dinkelbach iteration index for Scheme A (full RSMA in the asymmetric-rate setting solved by BCD-PA) and Scheme B (paired RSMA) under different user scales. In all considered cases, the system EE increases monotonically and converges to a stable value within a small number of iterations, which verifies the effectiveness of the proposed fractional programming framework. For full RSMA, the number of required iterations increases with the total number of users $K$. For example, when $K = 8$, convergence is typically reached after about 8 iterations. At the same time, full RSMA achieves a higher EE because it can more effectively exploit multiuser diversity. By comparison, paired RSMA converges faster, typically within 2 to 3 iterations, but its final EE is lower than that of full RSMA. In addition, for paired RSMA, increasing the number of user pairs $S$ from 2 to 4 further reduces the achievable EE, since orthogonal bandwidth partitioning among more pairs limits the available spectral resources for each pair.

Figure 4 further investigates the trade-off between EE and computational complexity as the total number of users $K$ increases from 2 to 10 under asymmetric rate requirements.

The left vertical axis corresponds to EE, while the right vertical axis indicates the total number of algorithm iterations on a logarithmic scale. As $K$ increases, the EE achieved by Scheme A gradually improves, which can be attributed to the stronger multiuser diversity available in larger systems. However, this gain is accompanied by a rapid increase in computational cost. In particular, the total number of iterations approaches $10^{3}$ when $K = 10$. In contrast, Scheme B decomposes the original problem into multiple lower-dimensional intra-pair optimizations, which substantially reduces the computational burden. Even when $K = 10$, Scheme B requires fewer than 20 iterations while maintaining a relatively stable EE level. These results indicate that Scheme B provides a practical low-complexity alternative, with a moderate loss in EE performance compared with Scheme A in large-scale multi-user settings.

Figure 5 shows the EE as a function of the transmit power budget under symmetric rate requirements. As the power budget increases from $-5$ dBm to 35 dBm, the EE of all considered schemes first increases and then gradually saturates in the high-power region, particularly beyond 25 dBm. This trend indicates that, under the EE maximization criterion, increasing the transmit power beyond the optimal operating region does not provide additional EE improvement. Among all considered schemes, the proposed Scheme A with the closed-form symmetric solution achieves the highest EE over the entire power range, due to its ability to fully exploit the available multiuser diversity. Scheme B, i.e., paired RSMA, achieves the second-best performance and remains close to Scheme A. Both proposed schemes outperform the benchmark methods including traditional RSMA, NOMA, and OMA. In particular, OMA exhibits the lowest EE because orthogonal resource allocation is less efficient in multi-user interference-limited scenarios.

Figure 6 presents the EE performance under asymmetric rate requirements. The overall trend is similar to that observed in the symmetric case. Specifically, the EE increases in the low-power region and then approaches a plateau when the power budget exceeds approximately 20 dBm. Under heterogeneous channel and rate conditions, Scheme A dynamically adjusts the power allocation between common and private streams through the BCD-PA algorithm, and therefore achieves the best EE among all compared schemes. Scheme B maintains a slightly lower but still competitive EE, while avoiding the higher optimization complexity associated with Scheme A. In addition, both RAQR-enabled schemes consistently outperform the classical RF-based RSMA, NOMA, and OMA benchmarks. Classical NOMA shows a more noticeable performance degradation in this asymmetric setting, mainly because its fixed SIC decoding order is less adaptable to heterogeneous user conditions.

To further evaluate the proposed energy-efficient design, Figure 7 compares the EE achieved under EE maximization and spectral efficiency (SE) maximization for six configurations over the transmit power budget range from −10 to 30 dBm. It can be observed that, in all three scenarios, the SE-maximizing strategies tend to use the entire available transmit power budget, and their EE decreases once the power budget exceeds the optimal operating point. In contrast, the EE-maximizing schemes maintain a stable high-power plateau by avoiding unnecessary power expenditure after the optimal EE has been

reached. Figure 7 also provides a direct comparison among the proposed schemes. The EE-maximizing Scheme A achieves the highest EE, reaching approximately 780 to 800 Kbits/J, while Scheme B optimized for EE achieves around 700 Kbits/J. In addition, the symmetric and asymmetric versions of Scheme A perform closely in the high-power regime, indicating that the asymmetric BCD-PA algorithm can effectively approach the performance of the KKT-based closed-form solution for the symmetric case.

Conclusions

This paper investigated a RAQR-empowered uplink RSMA network for energy efficiency (EE) maximization. To address the amplitude-only measurement constraint of RAQRs, a phase alignment mechanism was introduced to obtain a tractable linearized channel model. The resulting non-convex fractional EE problem was handled via Dinkelbach’s method. For symmetric IoT scenarios, exact closed-form power allocation was derived from the KKT conditions, enabling a globally optimal bisection search with low complexity. For asymmetric settings, a generalized BCD-PA framework was developed for heterogeneous rate requirements. In addition, a paired resource allocation scheme (Scheme B) based on Lambert-W-function closed-form solutions was proposed to further reduce complexity. Numerical results showed that Scheme A achieves the highest EE, while Scheme B provides a favorable complexity-performance trade-off, making it suitable for massive IoT deployment in RSMA-based 6G networks. Phase-sensitive RAQR readout remains a topic for future work due to its additional hardware requirements.

References