Time-dependent Variational Principles for Hybrid Non-Unitary Dynamics: Application to Driven-Dissipative Superconductors

TL;DR

The paper develops time-dependent variational principles for non-unitary open quantum dynamics, unifying Lindblad, no-click non-Hermitian, and hybrid Liouvillians through an action-based DMVP framework and a non-Hermitian Dirac-Frenkel extension. It provides practical implementations using Gaussian states and a BCS variational ansatz, deriving effective Hartree-Fock Liouvillians and explicit equations of motion for normalized observables. Applied to driven-dissipative BCS superconductors with two-body losses and pumps, the work reveals that the non-Hermitian limit acts as a singular regime, suppressing single-particle losses via pseudospin-length conservation and producing a non-Hermitian Zeno plateau as well as negative-temperature-like steady states. Overall, the approach offers a flexible, physically transparent tool to study complex open-system dynamics and reveals novel steady-state and dynamical phenomena arising from the interplay of quantum jumps and post-selected evolution.

Abstract

We introduce time-dependent variational principles to study the non-unitary dynamics of open quantum many-body systems, including dynamics described by the full Lindblad master equation, the non-Hermitian dynamics corresponding to the no-click limit of the fully post-selected quantum trajectories, and the dynamics described by a hybrid Lindbladian with a control parameter $α$ which interpolates between the full post-selection and averaging over all quantum trajectories. As an application we study the non-unitary dynamics of a lossy or driven-dissipative BCS superconductors, evolving in presence of two-body losses and two-body pumps. We show that the non-Hermitian limit acts as a singular limit of the hybrid dissipative dynamics, leading to a sharp modification of the universal approach to the driven-dissipative steady-states. By considering the dissipative dynamics with pair losses, we show that, as the non-Hermitian limit is approached, the density dynamics sharply evolves from a universal power-law to exponential decay that converges towards a quasi-steady plateau characterized by the freezing of the particle depletion due to pair losses. The reached quasi-stationary density increases as a function of the dissipation rate highlighting the emergence of a non-Hermitian Zeno effect in the lossy dynamics. For the driven-dissipative case, we show that, in the non-Hermitian limit, the system gets trapped into an effective negative temperature state, thus skipping the infinite temperature steady-state reached in the presence of finite contribution of the quantum jumps. We rationalize these findings in terms of the conservation of the length of the pseudospins which, in the non-Hermitian limit, suppresses the effective single-particle losses and pumps acting on the non-condensed particles.

Figures (8)

• Figure 1: Dynamics of the particles density (top panel) and the superconducting order parameter amplitude (bottom panel) for the different values of the parameter $\alpha$. $|U|/W=1.0$, $\Gamma/|U| = 0.08$ and $P/|U|=0.0$.
• Figure 2: Dynamics of the particles density (top panel) and the superconducting order parameter amplitude (bottom panel) for the different values of the parameter $\alpha$. $|U|/W=0.5$, $\Gamma/|U| = 0.08$ and $P/|U|=0.0$.
• Figure 3: Dynamics of the pseudospin length for the different values of the parameter $\alpha$. $U/W=1.0$, $\Gamma/|U| = 0.08$ and $P/|U|=0.0$.
• Figure 4: Dynamics of the particle density (top panel) and the superconducting order parameter amplitude (bottom panel) in the NH limit for different values of the dissipation rate $\Gamma$. $|U|/W=1.0$, $\alpha=0$ and $P/|U|=0.0$. Inset in the top panel: density in the quasi-steady plateau as a function of the dissipation rate.
• Figure 5: Dynamics of the order parameter with simultaneous pair pumps and losses for different values of the parameter $\alpha$. $|U|/W=1.0$ and $\Gamma/|U| = P/|U|= 0.08$.