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Europhysics Letters (EPL)

Non-Markovian generalization of a stochastic Schrödinger equation for two interacting qubits

Vitalii Semin

Published Apr 30, 2020• 1 citation
Publisher

Abstract

The non-Markovian generalization of a stochastic Schrödinger equation of diffusion type is considered. Generalization is performed by adding to the system Hamiltonian new stochastic terms some of which are correlated with the environment. The advantage of such an approach is ensuring the complete positivity of the density operator at any time. We illustrate the approach by the example of two interacting qubits in noisy laser fields.

DOI
10.1209/0295-5075/130/10003

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The non-Markovian generalization of a stochastic Schrödinger equation of diffusion type is considered. Generalization is performed by adding to the system Hamiltonian new stochastic terms some of which are correlated with the environment. The advantage of such an approach is ensuring the complete positivity of the density operator at any time. We illustrate the approach by the example of two interacting qubits in noisy laser fields.
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Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.
We present the non-Markovian generalization of the widely used stochastic Schrodinger equation. Our result allows to describe open quantum systems in terms of stochastic state vectors rather than density operators, without approximation. Moreover, it unifies two recent independent attempts towards a stochastic description of non-Markovian open systems, based on path integrals on the one hand and coherent states on the other. The latter approach utilizes the analytical properties of coherent states and enables a microscopic interpretation of the stochastic states. The alternative first approach is based on the general description of open systems using path integrals as originated by Feynman and Vernon.
A nonlinear stochastic Schr\"odinger equation for pure states describing non-Markovian diffusion of quantum trajectories and compatible with non-Markovian master equations is presented. This provides an unraveling of the evolution of any quantum system coupled to a finite or infinite number of harmonic oscillators without any approximation. Its power is illustrated by several examples, including measurementlike situations, dissipation, and quantum Brownian motion. Some examples treat this environment phenomenologically as an infinite reservoir with fluctuations of arbitrary correlation. In other examples the environment consists of a finite number of oscillators. In such a quasiperiodic case we see the reversible decay of a macroscopic quantum-superposition (``Schr\"odinger cat''). Finally, our description of open systems is compatible with different positions of the ``Heisenberg cut'' between system and environment.