Phase-Rotated Symbol Spreading in Atomic MIMO

• Phase-Rotated Symbol Spreading (PRSS) is a transmitter coding and modulation technique that overcomes the nonlinear detection challenges of envelope-only Rydberg atomic receivers in MIMO systems.
• It encodes each complex symbol across two consecutive time slots with a fixed phase rotation, enabling the separation of real and imaginary components for efficient linear demultiplexing.
• PRSS yields multi-dB gains in bit error rate performance and scalability, allowing conventional RF-MIMO and OFDM algorithms to be applied in quantum-based wireless communications.

Phase-Rotated Symbol Spreading (PRSS) is a transmitter-side coding and modulation technique introduced to address the scalability and linear detection challenges inherent in quantum-based wireless systems employing Rydberg atomic receivers for Multiple-Input Multiple-Output (MIMO) communication architectures. Rydberg atomic receivers are fundamentally limited to envelope detection, erasing phase information and resulting in a nonlinear detection model that severely impedes conventional MIMO signal processing. PRSS overcomes this barrier by spreading each complex symbol over two consecutive time slots using a fixed, known phase rotation, and by exploiting a strong local oscillator (reference) signal at the receiver to facilitate efficient linear demultiplexing. This method linearizes the originally nonlinear atomic front-end, enabling the full suite of established RF-MIMO and OFDM digital processing algorithms to be applied, with demonstrated multi-dB gains in bit error rate performance and scalable computational complexity (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

1. Detection Problem in Atomic Wireless MIMO Systems

The archetypical Rydberg atomic receiver measures only the envelope (magnitude) of the superposed electromagnetic field, not the phase, yielding a nonlinear measurement model. In the synchronous uplink case with $N$ transmit antennas and $M$ atomic receivers, the baseband received signal vector per slot is

$$\mathbf{z}^{(1)} = |\mathbf{H}\mathbf{x} + \mathbf{v}^{(1)} + \mathbf{r}|,$$

where $\mathbf{x} \in \mathbb{C}^N$ is the transmit vector, $\mathbf{H} \in \mathbb{C}^{M \times N}$ is the fading channel, $\mathbf{r}$ is a strong, known real-valued reference, and $\mathbf{v}^{(1)}$ is additive Gaussian noise. The absolute-value operation is element-wise and destroys all phase information. This prohibits the use of linear MIMO detectors (e.g., ML, ZF, MMSE), and prior solutions such as expectation-maximization Gibbs sampling (EM-GS) or nonlinear ML search are computationally intensive and scale poorly unless $M \gg N$ (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

2. PRSS Encoding: Spreading and Phase Rotation

PRSS counters the nonlinearity by encoding each complex constellation symbol across two consecutive time slots with a fixed phase offset. Let $x_n$ denote the symbol for antenna $n$. The spreading code is

$\mathbf{c} = [1;\ e^{j \theta}],$

with $\theta \neq 0$ the rotation parameter. The transmit vector over the two slots is, per antenna,

$$\tilde{x}_n = x_n \cdot \mathbf{c} = [x_n;\ x_n e^{j \theta}].$$

Stacking across $N$ antennas yields a $2N \times 1$ vector. The stacking operation preserves the original symbol vector structure but spreads each symbol over two instants in a manner analogous to code-division, with a critical phase component (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026). In practice, a phase choice of $\theta = \frac{3\pi}{2}$ or $\theta = \frac{\pi}{2}$ is optimal.

The key to PRSS is that the phase rotation is fixed and known, enabling the real and imaginary components of the original complex symbol to be teased apart during receiver processing.

3. Receiver Linearization and Model Recovery

Upon reception, the atomic receiver provides, per reference branch $k$ and for slots 1 and 2: $$y_{k,1} = |h_k^T x + v_{k,1} + r_k|,\quad y_{k,2} = |h_k^T (x e^{j \theta}) + v_{k,2} + r_k|.$$ Under the strong reference regime $r_k \gg |h_k^T x|$, a first-order Taylor expansion reveals that

$$\begin{aligned} y_{k,1} &\approx r_k + \operatorname{Re}\{z_{k,1}\}, \ y_{k,2} &\approx r_k + \operatorname{Im}\{z_{k,2}\}, \end{aligned}$$

with $z_{k,1} = h_k^T x + v_{k,1}$ and $z_{k,2} = h_k^T x + v_{k,2}$. By subtracting the known bias $r_k$ from both outputs and forming the complex variable

$$\tilde{y}_k = (y_{k,1} + j y_{k,2}) - (r_k + j r_k) = h_k^T x + v_k,$$

where $v_k = v_{k,1} + j v_{k,2}$, the receiver recovers a linear-complex observation of the channel-output plus noise. Stacking over all $K$ receivers yields the standard baseband linear MIMO model

$\tilde{\mathbf{y}} = Hx + v.$

This enables direct application of linear MIMO detectors such as Maximum-Likelihood (ML), Zero-Forcing (ZF), Minimum Mean Squared Error (MMSE), matched filter, or iterative message-passing methods, with complexities matching their RF analogs (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

4. Spectral Efficiency and Optimal Phase Choice

Each PRSS-encoded symbol requires two time slots for transmission, resulting in a fundamental spectral efficiency of $C_{\text{PRSS}} \approx \frac{1}{2} C_{\text{RF}}$, where $C_{\text{RF}}$ is the capacity of the full-rate complex RF-MIMO link. This matches the effective spectral efficiency of single-shot (envelope-only) atomic detection, which also transmits real-valued amplitude information per slot carrying only half the degrees of freedom of complex modulation.

The optimal phase rotation $\theta$ is derived from minimizing the mean squared error of the least-squares estimator that recovers the complex channel projection from two real measurements. The optimality condition $|\sin \theta| = 1$ yields $\theta^\star = \pm \frac{\pi}{2}$, empirically adopted as $\theta = \frac{\pi}{2}$ or $\theta = \frac{3\pi}{2}$ in experimental demonstrations. This allows for noise-preserving separation of in-phase and quadrature components and construction of the complex-valued MIMO system output (Liu et al., 25 Jan 2026).

5. Signal Processing Blocks and Detector Integration

Transmitter:

• Forward each symbol through optional precoding (for MU-MIMO or beamforming).
• Apply PRSS spreading: $x_n \mapsto [x_n;\ x_n e^{j\theta}]$.
• RF upconversion and emission.

Receiver (per atomic channel):

• Measure two consecutive envelope outputs per reference branch.
• Subtract known reference: $\tilde{y}_{k,1} = y_{k,1} - r_k$, $\tilde{y}_{k,2} = y_{k,2} - r_k$.
• Form complex de-spread value: $\tilde{y}_k = \tilde{y}_{k,1} + j \tilde{y}_{k,2}$.
• Stack into $\tilde{\mathbf{y}}$ and apply desired linear detector:
  • Maximum-Likelihood (ML): $$\hat{x}_{ML} = \arg\min_{x \in \Sigma^N} \|\tilde{\mathbf{y}} - Hx\|^2$$, complexity $\mathcal{O}(|\Sigma|^N)$.
  • Zero-Forcing (ZF): $$\hat{x}_{ZF} = \mathcal{Q}_{\Sigma}((H^H H)^{-1} H^H \tilde{\mathbf{y}})$$.
  • LMMSE and Iterative Methods: Include matched filter, p-Jacobi, and other iterative detection algorithms.

Non-PRSS (single-shot) methods include Envelope Matched Filtering (EMF), Envelope-Squared Least Squares (ES-LS) using gradient descent, and EM-Gibbs Sampling (EM-GS), all of which encounter scalability limitations due to their intrinsic nonlinearity (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

6. Performance and Scalability Benchmarks

Empirical evaluations compare PRSS against envelope-only single-transmission atomic MIMO under various detection strategies and antenna configurations:

MIMO Size	Envelope-Only BER @10 dB	PRSS-BER @10 dB
$2 \times 8$	$10^{-3.8}$	$10^{-2.5}$
$4 \times 8$	$10^{-2.7}$	$10^{-1.2}$

For small-scale MIMO (e.g., $N=4, M=8$), PRSS under optimal ML detection achieves $\sim 2.5-3$ dB BER gain at $10^{-3}$ BER. With suboptimal iterative detectors (e.g., p-Jacobi for PRSS vs. EM-GS for envelope-only), PRSS can provide up to $3$ dB gain at high SNR, and for large-scale MIMO ($N=48,64; M=128$), the gain exceeds $10$ dB, with envelope-only methods exhibiting error floors (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

In atomic OFDM scenarios, PRSS allows the IDFT/DFT chains to remain unmodified; with 64 subcarriers and high QAM constellations, atomic-OFDM BER slopes match those of RF-OFDM but are shifted by the atomic SNR advantage ($>20$ dB in some regimes).

PRSS de-spreading complexity is $\mathcal{O}(K)$, negligible relative to the $\mathcal{O}(N^3)$ complexity of typical ML or ZF detectors. This enables scalability to very large MIMO arrays without algorithmic or error-floor bottlenecks (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

7. Interference Resilience, OFDM Compatibility, and Architectural Implications

By linearizing the detection model, PRSS permits mature spatial and frequency-domain linear detection strategies, including spatial nulling and MMSE equalization, to be directly applied to atomic receiver architectures. The two-slot spreading (with corresponding minor spectral penalty) is remediated by using higher-order constellations, thus restoring bit-rate parity with conventional MIMO systems. In frequency-selective channels, PRSS combined with OFDM preserves the full compatibility of the DFT-based demultiplexer, and the atomic SNR advantage is retained throughout the subcarrier ensemble (Liu et al., 27 Apr 2025, Liu et al., 25 Jan 2026).

PRSS also improves resilience to multi-user interference and phase ambiguity, since the strong reference suppresses cross-talk and ambiguous mappings in the envelope domain.

PRSS is thus not only a solution to the fundamental nonlinearity of atomic wireless MIMO, but also an enabling architecture for scalable, interference-robust quantum-native wireless communication, with seamless integration into both time-division (MIMO) and frequency-division (OFDM) multiplexed systems.

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References:

• [SA-MIMO: Scalable Quantum-Based Wireless Communications, (Liu et al., 27 Apr 2025)]
• [Phase-Rotated Symbol Spreading for Scalable Rydberg Atomic-MIMO Detection, (Liu et al., 25 Jan 2026)]