Quantum and classical correlations in open quantum-spin lattices via truncated-cumulant trajectories

The study of quantum many-body physics in Liouvillian open quantum systems becomes increasingly important with the recent progress in experimental control on dissipative systems and their technological exploitation . A central question in open quantum systems concerns the fate of quantum correlations, and the possibility of controlling them by engineering the competition between the Hamiltonian dynamics and the coupling to a bath. Such a question is challenging from a theoretical point of view, as numerical methods faithfully accounting for quantum correlations are either relying on exact diagonalization, limiting drastically the sizes that can be treated; or on approximations on the range or strength of quantum correlations, associated to the choice of a specific Ansatz for the density matrix. In this work we propose a new method to treat open quantum-spin lattices, based on stochastic quantum trajectories for the solution of the open-system dynamics. Along each trajectory, the hierarchy of equations of motion for many-point spin-spin correlators is truncated to a given finite order, assuming that multivariate $k$-th order cumulants vanish for $k$ exceeding a cutoff $k_c$. This allows tracking the evolution of quantum spin-spin correlations up to order $k_c$ for all length scales. We validate this approach in the paradigmatic case of the phase transitions of the dissipative 2D XYZ lattice, subject to spontaneous decay. We convincingly assess the existence of steady-state phase transitions from paramagnetic to ferromagnetic, and back to paramagnetic, upon increasing one of the Hamiltonian couplings; as well as their classical Ising nature. Moreover, the approach allows us to show the presence of significant quantum correlations in the vicinity of the dissipative critical point, and to unveil the presence of spin squeezing, a tight lower bound to the quantum Fisher information

Dynamical hysteresis properties of the driven-dissipative Bose-Hubbard model with a Gutzwiller Monte Carlo approach

We study the dynamical properties of a driven-dissipative Bose-Hubbard model in the strongly interacting regime through a quantum trajectory approach with a cluster-Gutzwiller Ansatz for the wave function. This allows us to take classical and quantum correlations into account. By studying the dynamical hysteresis surface that arises by sweeping through the coherent driving strength we show that the phase diagram for this system is in qualitative correspondence with the Gutzwiller mean-field result. However, quantitative differences are present and the inclusion of classical and quantum correlations causes a significant shift of the critical parameters. Additionally, we show that approximation techniques relying on a unimodal distribution such as the mean field and $1/z$ expansion drastically underestimate the particle number fluctuations. Finally, we show that a proposed mapping of the driven-dissipative many-body Bose-Hubbard model onto a single driven-dissipative Kerr model is not accurate for parameters in the hysteresis regime

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

International audienceThe characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is $\textit{general and model-independent}$, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization

Canonical pair condensation in a flat-band BCS superconductor

The standard approach of the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity is to introduce a self-consistent mean-field approximation, and a variational ansatz for the many-body ground state. The resulting mean-field Hamiltonian no longer commutes with the total number operator, and the variational search takes place in Fock space rather than in a Hilbert space of states with fixed number of particles. This is a disadvantage when studying small systems where the canonical ensemble predictions differ from the corresponding grand-canonical results. To remedy this, alternative approaches such as Richardson’s method have been put forward. Here, we derive the exact many-body ground state of a model Hamiltonian corresponding to the deep-BCS or flat-band regime, without having to resort to Richardson’s set of coupled nonlinear equations. This allows to write the exact many-body ground state in a way that makes the difference with the BCS variational wave function particularly clear. We show that the exact wave function consists of a superposition of many-pair states in such a way that the mean-field averaging corresponds to a summation over these many-pair states. This explains why many expectation values calculated with the BCS variational wave function give the same result as when calculated with the exact wave function, even though these wave functions are different. In the canonical (fixed-number) approach, pairing is investigated using the second-order reduced density matrix and calculating its largest eigenvalue. When interpreted as the order parameter of the superconducting state, this can be compared directly to the behavior of the mean-field gap. Finally, we show that a clear difference between the canonical approach and the BCS grand canonical estimates appears when evaluating pair condensate fluctuations as well as the pair entanglement entropy

Quantum and classical correlations in open quantum-spin lattices via truncated-cumulant trajectories

The study of quantum many-body physics in Liouvillian open quantum systems becomes increasingly important with the recent progress in experimental control on dissipative systems and their technological exploitation . A central question in open quantum systems concerns the fate of quantum correlations, and the possibility of controlling them by engineering the competition between the Hamiltonian dynamics and the coupling to a bath. Such a question is challenging from a theoretical point of view, as numerical methods faithfully accounting for quantum correlations are either relying on exact diagonalization, limiting drastically the sizes that can be treated; or on approximations on the range or strength of quantum correlations, associated to the choice of a specific Ansatz for the density matrix. In this work we propose a new method to treat open quantum-spin lattices, based on stochastic quantum trajectories for the solution of the open-system dynamics. Along each trajectory, the hierarchy of equations of motion for many-point spin-spin correlators is truncated to a given finite order, assuming that multivariate $k$-th order cumulants vanish for $k$ exceeding a cutoff $k_c$. This allows tracking the evolution of quantum spin-spin correlations up to order $k_c$ for all length scales. We validate this approach in the paradigmatic case of the phase transitions of the dissipative 2D XYZ lattice, subject to spontaneous decay. We convincingly assess the existence of steady-state phase transitions from paramagnetic to ferromagnetic, and back to paramagnetic, upon increasing one of the Hamiltonian couplings; as well as their classical Ising nature. Moreover, the approach allows us to show the presence of significant quantum correlations in the vicinity of the dissipative critical point, and to unveil the presence of spin squeezing, a tight lower bound to the quantum Fisher information