Steady-State Coherences under Partial Collective non-Markovian Decoherence

1. Problem Statement

The paper addresses the fundamental challenge of understanding steady-state coherence in open quantum systems, specifically the interplay between collective decoherence (where oscillators share a common environment) and individual decoherence (where oscillators interact with separate environments).

• The Conflict: Theoretical models suggest that collective decoherence can induce or preserve steady-state coherence (via decoherence-free subspaces). However, experimental observations often fail to detect this because unavoidable individual decoherence competes with and suppresses collective effects.
• The Gap: While exactly solvable models exist for single oscillators or specific spectral densities, analytical solutions for two interacting oscillators coupled to both individual and collective reservoirs are scarce. Furthermore, the role of non-Markovianity (memory effects) in this mixed-decoherence regime remains poorly understood.

2. Methodology

The authors employ an exact analytical approach within the Heisenberg picture to model a system of two non-interacting harmonic oscillators.

• System Model:
  • Two harmonic oscillators with frequencies $\omega_1$ and $\omega_2$.
  • Coupled to three reservoirs: two individual bosonic reservoirs (at temperatures $T_1, T_2$) and one shared common bosonic reservoir (at temperature $T_0$).
  • Tunable Parameter ($\theta$): A parameter is introduced to continuously interpolate between pure collective decoherence ($\theta=0$) and pure individual decoherence ($\theta=\pi/2$). The coupling strengths are defined as $g_{0,\alpha} = g_\alpha \cos^2\theta$ (common) and $g_{i,\alpha} = g_\alpha \sin^2\theta$ (individual).
• Spectral Density: The analysis assumes a Lorentzian (Drude-Lorentz) spectral density, $J(\omega)$, characterized by a central frequency $\omega_0$, coupling strength $\Gamma$, and spectral width $\gamma$ (inverse correlation time). This choice allows for exact analytical solutions by converting integro-differential equations into ordinary differential equations.
• Mathematical Approach:
  1. Derivation of exact Heisenberg equations of motion for system operators.
  2. Transformation of coupled integro-differential equations into decoupled second-order differential equations using symmetric and antisymmetric combinations of coefficients.
  3. Analytical solution of these equations to determine time-dependent coefficients.
  4. Calculation of steady-state observables: specifically the coherence measures $\langle J_x \rangle$ and $\langle J_y \rangle$ (related to $\langle \hat{c}_1^\dagger \hat{c}_2 \rangle$) and population difference $\langle J_z \rangle$.
  5. Decomposition of steady-state coherence into Vacuum Coherence (induced by zero-point fluctuations) and Thermal Coherence (induced by thermal fluctuations).

3. Key Contributions

• Exact Analytical Solution: The paper provides the first exact analytical solution for the steady-state coherence of a two-oscillator system under a mix of collective and individual decoherence with a Lorentzian spectral density.
• Benchmarking: The exact solutions serve as a rigorous benchmark for evaluating approximate methods (e.g., Reaction Coordinate mapping, perturbative master equations), particularly in the strong-coupling and non-Markovian regimes where approximations often fail.
• Decoupling Mechanism: The authors demonstrate how the interplay of decoherence channels can be mathematically decoupled into symmetric and antisymmetric modes, revealing distinct decay behaviors for each.

4. Key Results

A. Dependence on Initial State

• Pure Collective Decoherence ($\theta = 0$): The steady-state coherence depends on the initial state. For specific initial states (e.g., the antisymmetric Bell state $|01\rangle - |10\rangle$), coherence is maximized; for others (e.g., symmetric state), it vanishes.
• Partial Collective Decoherence ($\theta \neq 0$): The presence of even a small amount of individual decoherence suppresses the dependence on the initial state. The system relaxes to a unique steady state regardless of the initial preparation.

B. Role of Non-Markovianity ($\gamma$)

• Non-Markovian Regime ($\gamma \ll \Gamma$):
  • Steady-state coherence exhibits rich, non-monotonic behavior.
  • As the proportion of individual decoherence ($\theta$) increases, coherence does not simply vanish; it can initially decrease, reach zero, and then increase again before vanishing at high $\theta$.
  • Strong non-Markovianity allows for the observation of steady-state coherence even when individual decoherence is present, provided parameters are tuned correctly.
• Markovian Regime ($\gamma \gg \Gamma$):
  • Coherence generally decays monotonically as individual decoherence increases.
  • In the limit of strong individual decoherence, coherence vanishes completely.

C. Vacuum vs. Thermal Coherence

• Vacuum Coherence (SSVC): Vanishes in the presence of any partial individual decoherence ($\theta \neq 0$) in the steady state. It is strictly a feature of the collective channel.
• Thermal Coherence (SSTC): Persists under partial collective decoherence. The paper finds that SSTC can be stronger in the non-Markovian regime than in the Markovian regime for certain parameter ranges, contradicting previous assumptions that non-Markovianity always suppresses coherence.

D. Strong Coupling Effects

• The exact solution reveals a multi-peak structure in coherence as a function of coupling strength $\Gamma$ in the non-Markovian regime.
• Standard weak-coupling master equations fail to capture these features, highlighting the necessity of exact models for strong-coupling quantum technologies.

5. Significance and Implications

• Theoretical Insight: The work clarifies that the "survival" of steady-state coherence is not solely a property of collective decoherence but is critically determined by the balance between collective and individual channels and the memory time of the environment.
• Experimental Feasibility: The findings are directly applicable to state-of-the-art platforms like superconducting circuits and optomechanical systems, where dissipation channels can be engineered to control the ratio of collective to individual noise.
• Quantum Technologies: The results offer strategies for maintaining quantum coherence in noisy environments. By tuning environmental parameters (spectral width, coupling strength) and the degree of collective coupling, one can potentially stabilize quantum states for applications in quantum computing, metrology, and thermodynamics.
• Methodological Impact: The paper establishes a gold standard for testing approximate dynamical methods, proving that they often miss critical non-Markovian and strong-coupling phenomena.

In conclusion, this paper demonstrates that while individual decoherence is detrimental to coherence, the non-Markovian nature of the environment can be harnessed to preserve or even enhance steady-state coherence, provided the system is not subjected to complete individual decoherence. This challenges the conventional view that collective decoherence is the only path to steady-state coherence.