Spin-orbit coupling induced geometric squeezing in rotating Bose-Einstein condensates

Abstract: Squeezed states play a key role in diverse frontiers of quantum physics. Geometrically squeezed states, a squeezed state in the orbital phase space of rotating Bose-Einstein condensates (BEC), have been conventionally generated by anisotropic trapping potentials. In this work, we propose a different route to generate geometric squeezing via spin-orbit coupling (SOC) in a pseudospin-1/2 BEC. We show that the SOC enables effective two-phonon transitions within the lowest Landau level via virtual spin-flip processes, leading to exponential squeezing dynamics in both spin components. Furthermore, by applying a $π/2$ spin rotation, the two spin channels can be coherently coupled to produce two-mode geometric squeezing. We also investigate the influence of interatomic interactions on squeezing performance and identify parameters where robust squeezing can be achieved. Our work provides a viable pathway to realize and manipulate geometric squeezing in spinor quantum gases.

• The paper demonstrates that weak Raman-induced spin-orbit coupling produces an effective quadratic squeezing Hamiltonian through second-order virtual spin flips, with strength ζ = α²mω/(4J) and exponential single-mode squeezing under optimized rotation.
• A π/2 spin rotation converts the orthogonal single-mode channels into transient two-mode squeezing, reducing the Duan–Simon noise factor to a minimum of 1/2 at ζt = 1/2 rather than producing indefinite exponential growth.
• Interactions generally suppress squeezing, but tuning the spin-orbit coupling and rotation can restore exponential-like behavior in one channel, while spin-resolved density moments provide a practical measurement method for Gaussian lowest-Landau-level states.

Overview

This paper proposes and analyzes a mechanism for generating geometric squeezing in rotating Bose-Einstein condensates (BECs) that does not rely on anisotropic trapping potentials. Instead, a weak Raman-induced spin-orbit coupling (SOC) of the form $\alpha p_x \sigma_x$ in a pseudospin-1/2 BEC produces, through second-order virtual spin-flip processes within the lowest Landau level (LLL), an effective quadratic Hamiltonian that drives two-phonon transitions in the guiding-center mode. The authors show analytically and numerically that this yields exponential single-mode squeezing in each spin component, that a $\pi/2$ spin rotation converts the two channels into two-mode geometric squeezing, and they characterize how interatomic interactions degrade—and can partially restore—the squeezing. An experimentally practical observation protocol based on spin-resolved density moments is also provided.

Background: anisotropic-potential geometric squeezing

The paper first reviews the established route to geometric squeezing (2606.30187). In a rotating frame with Hamiltonian $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$, near critical rotation $\Omega \approx \omega$ the spectrum separates into a high-frequency cyclotron sector ($\omega_+ = \omega + \Omega$) and a nearly degenerate guiding-center sector ($\omega_- = \omega - \Omega$). At $\Omega = \omega$, the guiding-center manifold becomes exactly degenerate, forming Landau levels with magnetic length $l_B = 1/\sqrt{2m\omega}$. A weak anisotropic potential $\propto \varepsilon(x^2-y^2)$ then induces an effective term $-\frac{\zeta}{2}(b^{\dagger 2}+b^2)$ with $\pi/2$0, producing quadrature variances

$\pi/2$1

i.e., exponential squeezing at rate set by $\pi/2$2. This is the benchmark against which the SOC mechanism is compared.

Effective Hamiltonian from virtual spin flips

The central theoretical result is that although the SOC term $\pi/2$3 is only linear in momentum—unlike the intrinsically quadratic anisotropy—it generates the required quadratic coupling at second order in perturbation theory. With Raman coupling $\pi/2$4 splitting the spin Landau levels by $\pi/2$5, and treating $\pi/2$6 as a perturbation valid for $\pi/2$7, degenerate perturbation theory on each spin's LLL projector yields

$\pi/2$8

with the effective squeezing strength $\pi/2$9. The diagonal terms have opposite signs for the two spins: choosing $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$0 cancels one component's diagonal term, giving ideal single-mode squeezed states with purely exponential variance dynamics; at unshifted $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$1, both components exhibit "sheared" squeezing,

$$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$2

which reproduces the exponential scaling only at short times $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$3. The sign structure of $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$4 also implies that the two spin components squeeze along orthogonal phase-space directions.

Numerical integration of the time-dependent Schrödinger equation confirms these predictions with excellent agreement, using representative parameters $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$5 and $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$6 (so $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$7), consistent with existing Raman-SOC experiments (2606.30187). The real-space densities evolve into anisotropic Gaussians stretched along $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$8, matching closed-form analytical expressions.

Two-mode squeezing via spin rotation

Because the two single-mode squeezing channels act along orthogonal axes, a passive $$h_0 = \mathbf{p}^2/2m + V_0(\mathbf{r}) - \Omega L_z$$9 spin rotation—analogous to a 50:50 beam splitter under the Bloch–Messiah reduction—maps them onto modes $\Omega \approx \omega$0 whose effective Hamiltonian contains a standard two-mode squeezing term $\Omega \approx \omega$1 plus a same-order number-conserving mode-exchange term $\Omega \approx \omega$2. The latter qualitatively alters the finite-time dynamics: rather than monotonic exponential reduction of the joint-noise factor $\Omega \approx \omega$3 (Duan–Simon criterion), the system exhibits transient two-mode squeezing with

$\Omega \approx \omega$4

which dips below the vacuum level only for $\Omega \approx \omega$5 and reaches its minimum value $\Omega \approx \omega$6 at $\Omega \approx \omega$7. This is a notable quantitative distinction from ideal two-mode squeezing Hamiltonians, and it is fully confirmed by numerics. The connection between the single- and two-mode descriptions is additionally framed through equivalent representations of the $\Omega \approx \omega$8 algebra.

Effect of interatomic interactions

Gross-Pitaevskii simulations with equal intra- and inter-spin couplings $\Omega \approx \omega$9 (justified for $\omega_+ = \omega + \Omega$0Na and $\omega_+ = \omega + \Omega$1Rb, where the difference is below a few percent) show that interactions suppress squeezing: at $\omega_+ = \omega + \Omega$2 short-time exponential behavior persists but long-time dynamics drifts upward; at $\omega_+ = \omega + \Omega$3 with the original parameters, squeezing vanishes entirely, replaced by weak periodic oscillations attributed to collective excitations near equilibrium. Importantly, the authors identify a compensating regime: increasing $\omega_+ = \omega + \Omega$4 to $\omega_+ = \omega + \Omega$5 and lowering $\omega_+ = \omega + \Omega$6 to $\omega_+ = \omega + \Omega$7 restores exponential-like squeezing in the spin-up channel even at $\omega_+ = \omega + \Omega$8, and two-mode squeezing emerges for $\omega_+ = \omega + \Omega$9. However, they state plainly that within their numerical exploration no parameter regime was found where both spin components simultaneously exhibit exponential-like squeezing in the interacting case—an explicit open limitation of the proposal.

Experimental realization and observation scheme

Implementing the rotating-frame Hamiltonian requires the Raman recoil direction to co-rotate with the trap, achievable via acousto-optic deflectors or rotating optics following the scheme of Radić et al.; the laboratory-frame Hamiltonian reduces exactly to the rotating-frame form under $\omega_- = \omega - \Omega$0. For measurement, the paper derives a reconstruction protocol exploiting the statistical independence of cyclotron and guiding-center sectors: for Gaussian LLL states the cyclotron mode contributes a fixed zero-point floor $\omega_- = \omega - \Omega$1, so guiding-center variances follow from spin-resolved in-situ density covariance matrices via $\omega_- = \omega - \Omega$2, and the joint-noise factor reduces to $\omega_- = \omega - \Omega$3. Numerical benchmarks confirm that this density-based inference reproduces the calculated squeezing trajectories. The protocol is explicitly restricted to Gaussian states; for non-Gaussian states with Landau-level mixing or strong interactions, the decomposition fails, and suitable measurement schemes remain undeveloped.

Limitations and open questions

Several assumptions bound the results. The perturbative treatment requires $\omega_- = \omega - \Omega$4, restricting accessible squeezing rates $\omega_- = \omega - \Omega$5; the interaction analysis is mean-field (GP) and assumes equal scattering lengths; the observation protocol presumes Gaussian LLL states; and simultaneous squeezing of both spin components under interactions was not achieved. Additionally, the co-rotating Raman geometry demands engineering beyond static beam configurations. The authors also note that extending the mechanism to laser-induced synthetic magnetic fields—which generate Landau levels without mechanical rotation—is left as future work.

Conclusion

This work establishes SOC as an alternative, perturbatively generated route to geometric squeezing in rotating spinor BECs, distinct from anisotropic-potential schemes in that a linear-in-momentum coupling produces an effective quadratic squeezing Hamiltonian via virtual spin flips. It further provides a concrete pathway to transient two-mode geometric squeezing through spin rotations, quantifies the destructive role of interactions along with partial mitigation strategies, and supplies a density-based observable reconstruction directly applicable to current experimental capabilities.