Stabilizing correlated pair tunneling of spin-orbit-coupled bosons in a non-Hermitian driven double well

1. Problem Statement

The paper addresses a fundamental gap in non-Hermitian quantum physics: the stability of correlated many-body processes (specifically, second-order pair tunneling) in the presence of engineered dissipation (gain and loss) and spin-orbit coupling (SOC).

• Context: While periodic driving (Floquet engineering) and non-Hermitian physics (PT symmetry) have been studied individually, their convergence in few-body systems with strong interactions remains largely unexplored.
• Challenge: In open quantum systems, gain and loss typically lead to exponential growth or decay of probability, destabilizing quantum states. The authors investigate whether bound pairs (doublons) of spin-orbit-coupled bosons can maintain stable tunneling dynamics under these conditions.
• Specific Question: Can correlated pair tunneling be stabilized in a periodically driven, non-Hermitian double-well potential, and what are the mechanisms governing this stability for different tunneling channels?

2. Methodology

The authors employ a rigorous analytical framework combining Floquet theory with multiple-scale asymptotic analysis.

• Physical Model:
  • Two ultracold bosons with synthetic SOC confined in a driven non-Hermitian double-well potential.
  • Described by an extended two-site Bose-Hubbard Hamiltonian including:
    • Spin-independent tunneling ($\nu$).
    • SOC strength ($\alpha$) and Raman coupling ($\delta$).
    • Time-dependent Zeeman field ($\Omega(t)$) and AC driving force ($f(t)$).
    • Non-Hermitian gain/loss terms ($i\beta_1, -i\beta_2$) representing particle sources and sinks.
    • Strong on-site interactions ($U$).
• Analytical Approach:
  • Regime: Strong interaction limit ($U \gg \nu$) and high-frequency driving ($\omega \gg \nu$).
  • Perturbation Theory: The authors introduce small parameters $\epsilon = \nu/\omega$ and $\theta = \delta/\omega$. They perform a multiple-scale expansion on the time variable ($\tau = \omega t$) to separate fast oscillations from slow dynamics.
  • Derivation: By eliminating secular terms (terms that grow unbounded over time) order-by-order in $\epsilon$, they derive effective second-order equations of motion for the probability amplitudes of the relevant Fock states.
  • Stability Analysis: They calculate the Floquet quasienergy spectra. Stability is determined by the imaginary parts of these quasienergies:
    • Balanced Gain/Loss: All quasienergies must be real (Im($E$) = 0).
    • Unbalanced Gain/Loss: Some quasienergies can be complex, provided their imaginary parts are non-positive (Im($E$) $\le$ 0).

3. Key Contributions and Results

The study analyzes three distinct tunneling channels and reveals unique stability mechanisms for each:

A. Interwell Spin-Conserving Tunneling

• Setup: $\alpha = 1, \delta = 0$. Dynamics confined to states with two particles in one well or one in each (e.g., $|0020\rangle \leftrightarrow |1010\rangle \leftrightarrow |2000\rangle$).
• Balanced Gain/Loss ($\beta_1 = \beta_2$):
  • Stability occurs in discrete, well-defined parameter regions (islands) in the $(2U_1/\omega, 2f/\omega)$ plane.
  • These regions arise from the resonant structure of Bessel functions in the effective coupling parameters ($\rho_1, \rho_2$).
  • Stability requires the coherent pair tunneling amplitude to dominate over the total dissipation rate ($|2\nu^2\rho_2/\omega| \ge 2\beta$).
• Unbalanced Gain/Loss ($\beta_1 < \beta_2$):
  • Stability is achieved only when parameters satisfy a specific balance condition: $\beta_1\beta_2 = \nu^4\rho_2^2/\omega^2$.
  • Under this condition, the system exhibits stable dynamics where the total probability converges to a constant.

B. Interwell Spin-Flipping Tunneling

• Setup: $\alpha = 0.5, \delta = 0$. Dynamics involve spin-flip transitions between wells.
• Novel Symmetry: Unlike the spin-conserving case, the spin-flipping channel exhibits a characteristic symmetry in its stability landscape.
  • The stability region is symmetric about the axis $2f/\omega = 2\Omega/\omega$.
  • This symmetry is a direct consequence of the even/odd properties of Bessel functions under argument sign changes induced by the interplay of driving and SOC.
• Stability: Similar to the spin-conserving case, stability islands exist for balanced gain/loss, and specific parametric relations enable stability for unbalanced gain/loss.

C. Intrawell Spin-Flipping Tunneling

• Setup: Dynamics confined to transitions within a single well (e.g., $|0020\rangle \leftrightarrow |0011\rangle$), driven by Raman coupling $\delta$.
• Critical Finding: For an initial Fock state (particles entirely in one well), the system is inherently unstable due to the exponential growth/decay caused by gain/loss.
• Coherence as a Control Parameter: The authors demonstrate that stable intrawell spin-flipping becomes accessible if the system is prepared in a coherent superposition state (e.g., $|\psi(0)\rangle = \frac{1}{\sqrt{2}}(|0020\rangle + |2000\rangle)$).
  • This superposition allows the system to populate a subspace that is effectively immune to dissipation (a "dark state" relative to the gain/loss mechanism).
  • This reveals that initial-state coherence can serve as a control knob to stabilize dynamics that are forbidden for simple Fock states.

4. Significance and Implications

• Theoretical Framework: The paper provides a robust analytical method to predict and control correlated tunneling in non-Hermitian, driven many-body systems, bridging the gap between single-particle non-Hermitian physics and interacting many-body dynamics.
• Engineering Dissipation: It demonstrates that dissipation is not merely a source of decoherence but can be engineered (via gain/loss balance and initial state preparation) to stabilize specific quantum correlations.
• New Control Knobs:
  • Floquet Engineering: Using driving frequency and amplitude to create discrete stability islands.
  • Symmetry Control: Exploiting the unique symmetry in spin-flipping channels for robust control.
  • Quantum Coherence: Utilizing initial state superposition to shield subsystems from environmental decay.
• Experimental Relevance: The results are directly applicable to current cold-atom experiments involving synthetic SOC, optical lattices, and Feshbach resonances, offering a pathway to observe stable correlated pair dynamics in open quantum systems.

In conclusion, the work establishes that correlated pair tunneling in non-Hermitian systems is not only possible but can be precisely controlled through the interplay of driving, SOC, dissipation engineering, and quantum coherence.