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Physical Review A

Quantum-trajectory Monte Carlo method for internal-state evolution of fast ions traversing amorphous solids

Tatsuya Minami, Carlos O Reinhold, Joachim Burgdörfer

Published Feb 28, 2003• 25 citations
Publisher

Abstract

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 ${\mathrm{Kr}}^{35+}$ ion traversing carbon foils. Simulations exhibit clear signatures of partially coherent transitions and are found to be in good agreement with experimental data.

DOI
10.1103/physreva.67.022902

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•Jan 1, 2002•0 citations
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.
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.
A theoretical framework is briefly reviewed that treats the multiple collisions that a quantum system undergoes with particles in its environment. The dynamics is governed by the quantum Lindblad master equation for the density matrix, which can be solved using a Monte Carlo technique. Its classical limit is identified and provides an opportunity to study the role of decoherence in the quantum to classical transition. Applications are discussed related to the dynamics of ions and atoms interacting with solids and gases, and Rydberg wavepackets subject to coloured noise.
We introduce a method to simulate open quantum many-body dynamics by combining time-dependent variational Monte Carlo (tVMC) with quantum trajectory techniques. Our approach unravels the Lindblad master equation into an ensemble of stochastic Schrödinger equations for a variational ansatz, avoiding the exponential cost of density matrix evolution. The method is compatible with generic ansatzë, including expressive neural-network wavefunctions. We derive the nonlinear stochastic equations of motion for the variational parameters and employ suitable Stratonovich numerical solvers. To validate our approach, we simulate quenches in the locally dissipative long-range Ising model in a transverse field, accurately capturing non-equilibrium magnetization and spin squeezing dynamics relevant to trapped-ion and Rydberg atom experiments. The framework is computationally efficient, scalable on high-performance computing platforms, and can be readily integrated into existing tVMC implementations. This work enables large-scale simulations of complex, dissipative quantum systems in higher dimensions, with broad implications for quantum technology and fundamental science.
The theoretical description of materials' properties driven out of equilibrium has important consequences in various fields such as semiconductor spintronics, nonlinear optics, and continuous and discrete quantum information science and technology. The coupling of a quantum many-body system to an external bath can dramatically modify its dynamics compared to that of closed systems; new phenomena like relaxation and decoherence appear as a consequence of the non-unitary evolution of the quantum system. In addition, electron-electron correlations must be properly accounted for in order to go beyond a simple one-electron or mean-field description of the electronic system. Here, we discuss a first-principles methodology based on the evolution of the electronic density matrix capable of treating electron-environment interactions and electron-electron correlations at the same level of description. The effect of the environment is separated into a coherent contribution, like the coupling to applied external electromagnetic fields, and an incoherent contribution, like the interaction with lattice vibrations or the thermal background of radiation. Electron-electron interactions are included using the nonequilibrium Green's function plus the generalized Kadanoff-Baym ansatz. The obtained non-Markovian coupled set of equations reduces to the ordinary Lindblad quantum master equation form in the Markovian limit.
Dissipation, the irreversible loss of energy and coherence, from a microsystem is the result of coupling to a much larger macrosystem (or reservoir) that is so large that one has no chance of keeping track of all of its degrees of freedom. The microsystem evolution is then described by tracing over the reservoir states, which results in an irreversible decay as excitation leaks out of the initially excited microsystems into the outer reservoir environment. Earlier treatments of this dissipation used density matrices to describe an ensemble of microsystems, either in the Schr\"odinger picture with master equations, or in the Heisenberg picture with Langevin equations. The development of experimental techniques to study single quantum systems (for example, single trapped ions, or cavity-radiation-field modes) has stimulated the construction of theoretical methods to describe individual realizations conditioned on a particular observation record of the decay channel. These methods, variously described as quantum-jump, Monte Carlo wave function, and quantum-trajectory methods, are the subject of this review article. We discuss their derivation, apply them to a number of current problems in quantum optics, and relate them to ensemble descriptions.
We provide a unified view for the master equation (ME) approach and the quantum trajectory (QT) approach to the measurement problem of an electron tunnelling between two coherently coupled quantum dots (CQD’s) using a low transparency point contact (PC) or tunnel junction as a detector. We show that the master equation for the reduced or “partially” reduced density matrix can be obtained when an average or “partial” average is taken on the conditional, stochastic density matrix over the possible outcomes of the measurement. Then we simulate ) , ( t N P , the probability of finding N electron that have tunnelled through the PC in time t , using 10000 QT’s and measurement records. The simulation results of ) , ( t N P are in very good agreement with those obtained from the Fourier analysis of the “partially” reduced density matrix. The theory of QT’s (also known as stochastic Schrodinger equation or stochastic ME) has been developed in last ten years mainly in the quantum optics community to describe open quantum system subject to continuous quantum measurements. But it was introduced to the context of solid-state mesoscopics only recently (e.g., Goan et. al. 2001 and references therein). The CQD system, considered here, forms a single charge quantum bit (qubit), and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dots. The QT or stochastic quantum-jump ME of the density matrix operator ) (t c