Contractivity of time-dependent driven-dissipative systems

In a number of physically relevant contexts, a quantum system interacting with a decohering environment is simultaneously subjected to time-dependent controls and its dynamics is thus described by a time-dependent Lindblad master equation. Of particular interest in such systems is to understand the circumstances in which, despite the ability to apply time-dependent controls, they lose information about their initial state exponentially with time i.e., their dynamics are exponentially contractive. While there exists an extensive framework to study contractivity for time-independent Lindbladians, their time-dependent counterparts are far less well understood. In this paper, we study the contractivity of Lindbladians, which have a fixed dissipator (describing the interaction with an environment), but with a time-dependent driving Hamiltonian. We establish exponential contractivity in the limit of sufficiently small or sufficiently slow drives together with explicit examples showing that, even when the fixed dissipator is exponentially contractive by itself, a sufficiently large or a sufficiently fast Hamiltonian can result in non-contractive dynamics. Furthermore, we provide a number of sufficient conditions on the fixed dissipator that imply exponential contractivity independently of the Hamiltonian. These sufficient conditions allow us to completely characterize Hamiltonian-independent contractivity for unital dissipators and for two-level systems.

💡 Research Summary

The paper investigates the contractivity properties of driven‑dissipative quantum systems whose dynamics are governed by a time‑dependent Lindblad master equation with a fixed set of jump operators (the dissipator) and a Hamiltonian that can be varied arbitrarily in time. The authors first formalize exponential contractivity as a uniform bound on the trace‑norm distance between any two states that decays as K e^{‑γ|t‑s|}. For time‑independent Lindbladians this property is equivalent to the existence of a unique full‑rank stationary state, a fact that follows from linear dynamical‑system theory and the Perron‑Frobenius theorem.

The core contributions are threefold. First, the authors prove that adding a sufficiently small or sufficiently slow time‑dependent Hamiltonian to a contractive Lindbladian does not destroy exponential contractivity. Using Dyson series expansions and a variational bound on the propagator, they show that the contraction rate γ is reduced only by terms proportional to the norm of the Hamiltonian or its time‑derivative, establishing rigorous conditions under which the driven dynamics remain contractive.

Second, they demonstrate that the intuition “if the dissipator alone is contractive, then any Hamiltonian preserves contractivity” is false in the quantum setting. A concrete counter‑example is constructed on a two‑qubit system with an irreducible dissipator and a simple σ_y⊗I Hamiltonian. Although the dissipator alone has a unique full‑rank fixed point, the combined generator possesses a rank‑one stationary state, breaking exponential contractivity. This highlights a stark difference from classical Markov processes, where adding any rate matrix to an irreducible generator preserves contractivity.

Third, the paper provides several Hamiltonian‑independent sufficient conditions for exponential contractivity. One condition is spectral: the superoperator 𝔖 = ∑i 𝔇{L_i} must have a strictly positive smallest non‑zero eigenvalue. Another is algebraic: the linear span of the jump operators together with their products L_i†L_i must generate the full matrix algebra on the Hilbert space. For unital dissipators (those that leave the maximally mixed state invariant) the authors prove a complete equivalence: a unital dissipator is contractive for all Hamiltonians if and only if it is contractive on its own. They also show that for two‑level systems every trace‑preserving dissipator satisfies these conditions, and that for three‑level systems the result holds almost universally.

A detailed case study focuses on “ladder” dissipators, such as the damping of a harmonic oscillator. When the system is truncated to a finite number of energy levels, the ladder dissipator cannot support a non‑full‑rank stationary state, guaranteeing exponential contractivity irrespective of the driving Hamiltonian. This result is particularly relevant for experimental platforms where only a few low‑lying levels are populated.

The technical development is supported by extensive appendices: Appendix A proves the equivalence between uniqueness of the stationary state and exponential contractivity for time‑independent generators; Appendix B derives the small‑drive bound; Appendix C contains the explicit calculations for the counter‑example; Appendix D establishes the unital equivalence theorem. Throughout, the authors employ tools from quantum information theory (trace norm, χ² divergence), operator algebras, and spectral analysis.

In conclusion, the work clarifies when time‑dependent control can be safely applied to open quantum systems without compromising their natural tendency to forget initial conditions. It provides both negative results (showing that strong or fast drives can destroy contractivity) and positive design principles (spectral and algebraic criteria on the dissipator) that can guide the engineering of robust quantum memories, sensors, and controllable open‑system dynamics. Future directions suggested include extending the analysis to infinite‑dimensional systems, non‑Markovian environments, and optimal control strategies that respect the derived contractivity bounds.