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Physical Review A

Nonunitary quantum trajectory Monte Carlo method for open quantum systems

Marek Seliger, Carlos O Reinhold, Tatsuya Minami, Joachim Burgdörfer

Published Jun 15, 2005• 9 citations
Publisher

Abstract

We have developed a generalized nonunitary Lindblad equation and its quantum trajectory Monte Carlo implementation for the evolution of open quantum systems (OQSs) whose coupling to the environment features not only energy exchange but also probability flux to the environment. This generalization allows the treatment of a class of problems where the state space of the system includes bound and continuum states. We show the equivalence between the solution of the generalized Lindblad equation and the Monte Carlo average over open quantum trajectories. As a first test case we study the multilevel radiative decay of a hydrogenic ion. As a second test case we apply the theory to the time development of the internal state of fast highly charged ${\mathrm{Kr}}^{35+}$ ions traversing carbon foils with varying thickness subject to collisions and to spontaneous radiative decay. We find significantly improved agreement of the nonunitary transport theory with experimental data.

DOI
10.1103/physreva.71.062901

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We have developed a generalized nonunitary Lindblad equation and its quantum trajectory Monte Carlo implementation for the evolution of open quantum systems (OQSs) whose coupling to the environment features not only energy exchange but also probability flux to the environment. This generalization allows the treatment of a class of problems where the state space of the system includes bound and continuum states. We show the equivalence between the solution of the generalized Lindblad equation and the Monte Carlo average over open quantum trajectories. As a first test case we study the multilevel radiative decay of a hydrogenic ion. As a second test case we apply the theory to the time development of the internal state of fast highly charged Kr35+{\mathrm{Kr}}^{35+}Kr35+ ions traversing carbon foils with varying thickness subject to collisions and to spontaneous radiative decay. We find significantly improved agreement of the nonunitary transport theory with experimental data.
We present a theoretical framework for the evolution of the internal state of a fast highly charged one-electron ion traversing an amorphous solid. We employ an open quantum system approach which incorporates the complex array of collisions with electrons and ionic cores in the solid within the framework of system-reservoir interactions. Interactions with the solid environment and the radiation field are treated on the same footing and the quantum master equation for the reduced density matrix of the electronic state of the ion is approximated by a Lindblad equation. The latter allows the solution of this multistate problem in terms of Monte Carlo sampling of quantum trajectories. Similarities and extensions to methods used in quantum optics and previously employed in ion-solid interactions are discussed. Our focus is on the transient buildup and destruction of coherences by stochastic processes. We apply our method to the study of coherence properties of the internal state of a fast Kr35+{\mathrm{Kr}}^{35+}Kr35+ ion traversing carbon foils. Simulations exhibit clear signatures of partially coherent transitions and are found to be in good agreement with experimental data.
The theory of open systems is used in quantum optics in two distinct ways: to model sources of light such as lasers and parametric oscillators, and to describe the response of irradiated cavities and atoms. Sometimes the two applications are used in combination; first, statistical properties of the field radiated by a source are calculated, and these are then used in a separate calculation to determine the response of a system that is irradiated by the source. But the usefulness of this approach is quite hunted. It is essentially limited to coherent sources and certain broadband sources such as broadband squeezed or chaotic light. There exists no general theory for cascaded open systems - a theory that gives the response of system B to radiation emitted by system A when there exists an open-systems treatment for A and B separately. In this paper I develop such a theory using the quantum trajectory formulation of open systems. The source A and irradiated system B are described by a single stochastic wave-function. Generally the wavefunction describes an entangled state of the A and B subsystems. In the quantum trajectory formalism the wavefunction undergoes a coherent evolution governed by a nonunitary Schrödinger equation, interrupted at random times by wavefunction collapses. I derive the nonunitary Schrödinger equation and the form of the collapses. I illustrate the theory with some simple examples and discuss potential applications to problems involving the interaction of atoms with nonclassical light.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm---utilizing a decomposition of nonunitary operators approach---that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.
Quantum simulation of dynamics in open quantum systems is crucial but poses a significant challenge due to the non-Hermitian nature leading to nonunitary evolution and the limited quantum resources on current quantum computers. Here we introduce a variational hybrid quantum-classical algorithm designed for simulating the time evolution governed by the Lindblad master equation. Our approach involves on a stochastic unveiling of the density matrix, transforming the Lindblad equation into a wave function-based quantum state diffusion (QSD) method with the aim of reducing qubit requirements. We then apply variational quantum simulation (VQS) to efficiently capture the nonunitary evolution in QSD. We demonstrate our QSD-VQS algorithm by investigating the quantum dynamics in a two-level system subjected to an amplitude damping channel and a four-level transverse field Ising model within a dissipative environment including time-independent and periodic Hamiltonian cases. The results reveal its promising utility with upcoming hardware in the near future.