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Optics and Spectroscopy

Non-Markovian dynamics and quantum jumps

Jyrki Piilo, Sabrina Maniscalco, Kari Härkönen, Kalle‐Antti Suominen

Published Mar 1, 2010• 1 citation
Publisher

Abstract

Experiments with trapped particles have demonstrated the existence of quantum jumps and the discrete nature of single-system dynamics in quantum mechanics. The concept of jumps is also a powerful tool for simulating and understanding open quantum systems. In non-Markovian systems jump probabilities can become negative due to memory effects between the system and its environment. We discuss a recently presented method that can handle both positive and negative probabilities and provides powerful insight into the dynamics of open systems with memory. The key element is a reversed quantum jump to a system state that was, in principle, already destroyed by an earlier normal jump. Instead of using artificial extensions of the system or exploiting hidden variables we take advantage of the information stored in the quantum ensemble itself.

DOI
10.1134/s0030400x10030148

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Experiments with trapped particles have demonstrated the existence of quantum jumps and the discrete nature of single-system dynamics in quantum mechanics. The concept of jumps is also a powerful tool for simulating and understanding open quantum systems. In non-Markovian systems jump probabilities can become negative due to memory effects between the system and its environment. We discuss a recently presented method that can handle both positive and negative probabilities and provides powerful insight into the dynamics of open systems with memory. The key element is a reversed quantum jump to a system state that was, in principle, already destroyed by an earlier normal jump. Instead of using artificial extensions of the system or exploiting hidden variables we take advantage of the information stored in the quantum ensemble itself.
We discuss in detail how non-Markovian open system dynamics can be described in terms of quantum jumps [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. Our results demonstrate that it is possible to have a jump description contained in the physical Hilbert space of the reduced system. The developed non-Markovian quantum jump approach is a generalization of the Markovian Monte Carlo wave function (MCWF) method into the non-Markovian regime. The method conserves both the probabilities in the density matrix and the norms of the state vectors exactly and sheds new light on non-Markovian dynamics. The dynamics of the pure state ensemble illustrates how local-in-time master equation can describe memory effects and how the current state of the system carries information on its earlier state. Our approach solves the problem of negative jump probabilities of the Markovian MCWF method in the non-Markovian regime by defining the corresponding jump process with positive probability. The results demonstrate that in the theoretical description of non-Markovian open systems, there occurs quantum jumps which recreate seemingly lost superpositions due to the memory.
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Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics.
Stochastic methods with quantum jumps are often used to solve open quantum system dynamics. Moreover, they provide insight into fundamental topics, such as the role of measurements in quantum mechanics and the description of non-Markovian memory effects. However, there is no unified framework to use quantum jumps to describe open-system dynamics in any regime. We solve this issue by developing the rate operator quantum jump (ROQJ) approach. The method not only applies to both Markovian and non-Markovian evolutions, but also allows us to unravel master equations for which previous methods do not work. In addition, ROQJ yields a rigorous measurement-scheme interpretation for a wide class of dynamics, including a set of master equations with negative decay rates, and sheds light on different types of memory effects which arise when using stochastic quantum jump methods.
Recent developments in practical quantum engineering and control techniques have allowed significant developments for experimental studies of open quantum systems and decoherence engineering. Indeed, it has become possible to test experimentally various theoretical, mathematical, and physical concepts related to non-Markovian quantum dynamics. This includes experimental characterization and quantification of non-Markovian memory effects and proof-of-principle demonstrations how to use them for certain quantum communication and information tasks. We describe here recent experimental advances for open system studies, focussing in particular to non-Markovian dynamics including the applications of memory effects, and discuss the possibilities for ultimate control of decoherence and open system dynamics.
Various Monte Carlo methods allow the simulation of the dynamics of open quantum systems. We highlight the central features a commonly used method for Markovian systems, i.e., Monte Carlo Wave Function method and another one used for non-Markovian systems, i.e., the doubled Hilbert space method. One of the open problems in the field has been the lack of the existence of Monte Carlo method with quantum jumps for non-Markovian systems. We characterize the key problems that should be solved in this context and give indications for the way towards the development of quantum jump method without auxiliary states for non-Markovian systems.