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arXiv:1812.08582 · Physical Review A

Optimal stochastic unraveling of disordered open quantum systems: Application to driven-dissipative photonic lattices

Filippo Vicentini, Fabrizio Minganti, Alberto Biella, Giuliano Orso, Cristiano Ciuti

Published Mar 14, 2019• 21 citations
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Abstract

We propose an efficient numerical method to compute configuration averages of observables in disordered open quantum systems whose dynamics can be unraveled via stochastic trajectories. We prove that the optimal sampling of trajectories and disorder configurations is simply achieved by considering one random disorder configuration for each individual trajectory. As a first application, we exploit the present method to study the role of disorder on the physics of the driven-dissipative Bose-Hubbard model in two different regimes: (i) for strong interactions, we explore the dissipative physics of fermionized bosons in disordered one-dimensional chains; (ii) for weak interactions, we investigate the role of on-site inhomogeneities on a first-order dissipative phase transition in a two-dimensional square lattice.

DOI
10.1103/physreva.99.032115

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We propose an efficient numerical method to compute configuration averages of observables in disordered open quantum systems whose dynamics can be unraveled via stochastic trajectories. We prove that the optimal sampling of trajectories and disorder configurations is simply achieved by considering one random disorder configuration for each individual trajectory. As a first application, we exploit the present method to study the role of disorder on the physics of the driven-dissipative Bose-Hubbard model in two different regimes: (i) for strong interactions, we explore the dissipative physics of fermionized bosons in disordered one-dimensional chains; (ii) for weak interactions, we investigate the role of on-site inhomogeneities on a first-order dissipative phase transition in a two-dimensional square lattice.
This thesis is devoted to the theoretical study of driven-dissipative many-body quantum systems with a particular focus on advanced computational methods. After reviewing several state-of-the-art numerical methods to determine the steady-state and dynamics of open quantum systems described by a Lindblad master equation for the density matrix, this thesis manuscript presents two original methods: (i) an optimal unraveling of stochastic quantum trajectories for disordered systems; (ii) a variational neural network method. The stochastic methods have been applied to investigate the driven-dissipative Bose-Hubbard lattice model that can for instance describe lattices of coupled nonlinear electromagnetic resonators such those obtained with semiconductor microcavities or superconducting quantum circuits. Our study has shown that a critical slowing down of the dynamics occurs for some parameters in dimensions higher than one. The phenomena is linked to a first-order dissipative phase transition and to lattice-level collective bistability. Furthermore, we reveal the role of disorder on the spatial correlation properties of these phenomena. In the last part of the thesis, we present the strong connection between fundamental concepts of machine learning and variational approaches in quantum physics. We present a novel computational method for open quantum systems based on neural network ans\"{a}tze that is compatible with a stochastic minimisation of a cost function through Monte Carlo sampling. As a first application, we have applied this method to the dissipative transverse quantum Ising model. Exciting perspectives and possible developments are also drawn in the concluding remarks.
We develop a real-time full configuration-interaction quantum Monte Carlo approach to model driven-dissipative open quantum systems with Markovian system-bath coupling. The method enables stochastic sampling of the Liouville--von Neumann time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the nonequilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce the statistical error. Finally, we demonstrate the efficiency of our approach by applying it to the driven-dissipative two-dimensional XYZXYZXYZ spin-12\frac{1}{2}21​ model on a lattice.
We present a new technique for efficiently simulating (in polynomial time) a class of one-dimensional (1D) dissipative spin chains that, when mapped to fermions, have quadratic Hamiltonians, with the only nonlinearity coming from Jordan-Wigner strings appearing in the jump operators, despite the fact that these models cannot be mapped to quadratic fermionic master equations. We show that many such Lindblad master equations admit an exact stochastic unraveling, with individual trajectories evolving as Gaussian fermionic states, even though the full master equation describes a system inequivalent to free fermions. This allows one to calculate arbitrary observables efficiently without sign problems, and with bounded sampling complexity. We utilize this new technique to study three paradigmatic dissipative effects: the melting of antiferromagnetic order in the presence of local loss, many-body subradiant phenomenon in systems with correlated loss, and nonequilibrium steady states of a 1D dissipative transverse-field Ising model. Beyond simply providing a powerful numerical technique, our method can also be used to gain both qualitative and quantitative insights into the role of interactions in these models.
We propose a variational framework for the efficient simulation of stochastic quantum dynamics in interacting bosonic systems. Our method generalizes the time-dependent variational principle to accommodate quantum state diffusion processes and is tailored for describing quantum trajectories unraveled from a master equation. We adopt a wavefunction ansatz composed of a coherent superposition of Gaussian wavepackets, which allows the fully analytical assembly of the variational equations of motion while capturing both the quantum trajectory dynamics and the Lindblad evolution beyond semiclassical approximations. The method is carefully benchmarked on a Bose-Hubbard dimer, and applied to extended Bose-Hubbard lattices in both 1D and 2D. In particular, we show the emergence of a symmetry-breaking phase transition in 2D lattices, which is absent in 1D chains.
The dynamics of open quantum systems can be simulated by unraveling it into an ensemble of pure state trajectories undergoing nonunitary monitored evolution, which has recently been shown to undergo measurement-induced entanglement phase transition. Here, we show that, for an arbitrary decoherence channel, one can optimize the unraveling scheme to lower the threshold for entanglement phase transition, thereby enabling efficient classical simulation of the open dynamics for a broader range of decoherence rates. Taking noisy random unitary circuits as a paradigmatic example, we analytically derive the optimum unraveling basis that on average minimizes the threshold. Moreover, we present a heuristic algorithm that adaptively optimizes the unraveling basis for given noise channels, also significantly extending the simulatable regime. When applied to noisy Hamiltonian dynamics, the heuristic approach indeed extends the regime of efficient classical simulation based on matrix product states beyond conventional quantum trajectory methods. Finally, we assess the possibility of using a quasi-local unraveling, which involves multiple qubits and time steps, to efficiently simulate open systems with an arbitrarily small but finite decoherence rate.
We define quantum chaos and integrability in open quantum many-body systems as a dynamical property of single stochastic realizations, referred to as quantum trajectories. This definition relies on the predictions of random matrix theory applied to the subset of the Liouvillian eigenspectrum involved in each quantum trajectory. Our approach, which we name (SSQT), enables a natural distinction between transient and steady-state quantum chaos as general phenomena in open setups. We test the generality and reliability of the SSQT criterion on several dissipative systems, further showing that an open system with a chaotic structure can evolve towards either a chaotic or integrable steady state. We apply our theoretical framework to two driven-dissipative bosonic systems. First, we study the driven-dissipative Bose-Hubbard model, a paradigmatic example of a quantum simulator, clarifying the interplay of integrability, transient, and steady-state chaos across its phase diagram. Our analysis shows the existence of an emergent dissipative quantum chaotic phase, whereas the classical and semiclassical limits display an integrable behavior. In this regime, chaos arises from the quantum and classical fluctuations associated with the dissipation mechanisms. Second, we investigate dissipative quantum chaos in the dispersive readout of a transmon qubit: a measurement technique ubiquitous in superconducting-based quantum hardware. Through the SSQT, we distinguish several regimes where the performance of the measurement instrument can be connected to the integrable or chaotic nature of the underlying driven-dissipative bosonic system. Our work offers a general understanding of the integrable and chaotic dynamics of open quantum systems and paves the way for the investigation of dissipative quantum chaos and its consequences on state-of-the-art noisy intermediate-scale quantum devices.