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Quantum theory of heating of a single trapped ion

F. Intravaia, Maniscalco, S, Piilo, J, Messina, Andrea

Published Jun 21, 2002• 9 citations
OpenAlex

Abstract

The heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born-Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve it analytically is proposed. Our master equation is in the Lindblad form with time dependent coefficients, thus allowing the simulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method. For such a system, this is the first demonstration of the applicability of the quantum jump MCWF method in the non-Markovian regime.

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A theory of quantum jumps in a system of few trapped ions is formulated. The V configuration of the energy levels and incoherent pumping are assumed. The master equation is derived in terms of the SU(3) collective operators for the V system. Numerical results for the intensity-intensity correlations for up to five ions are presented. Analytical results for a system of two ions are derived separating out dynamics at two difference time scales and by eliminating adiabatically the fast variables. From the analytical results, the underlying three-state Markov process that can be used to characterize the quantum jumps in the two-ion system is identified. Probability distributions for various off-on time intervals are presented.>

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The heating of trapped ions due to the interaction with a quantized environment is studied without performing the Born-Markov approximation. A generalized master equation local in time is derived and a novel theoretical approach to solve it analytically is proposed. Our master equation is in the Lindblad form with time dependent coefficients, thus allowing the simulation of the dynamics by means of the Monte Carlo Wave Function (MCWF) method. For such a system, this is the first demonstration of the applicability of the quantum jump MCWF method in the non-Markovian regime.
The Monte Carlo wave function method or the quantum-trajectory-jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. We extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Markovian jump approach and the computational efficiency of this method is also discussed.
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.
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.
Open quantum systems are ubiquitous in nature and central to quantum technologies. A common description of their dynamics is given by the celebrated Lindblad master equation, which can be generalized to the non-Markovian scenario. In this work, we introduce the non-Markovian ensemble propagation (NMEP) method, which extends the Monte Carlo wave-function (MCWF) method to the non-Markovian case in a simple and general manner. We demonstrate its accuracy and effectiveness in a selection of examples, and compare the results with either analytic expressions or direct numerical integration of the master equation.
Open quantum systems are ubiquitous in nature and central to quantum technologies. A common description of their dynamics is given by the celebrated Lindblad master equation, which can be generalized to the non-Markovian scenario. In this work, we introduce the non-Markovian ensemble propagation (NMEP) method, which extends the Monte Carlo wave-function (MCWF) method to the non-Markovian case in a simple and general manner. We demonstrate its accuracy and effectiveness in a selection of examples, and compare the results with either analytic expressions or direct numerical integration of the master equation.
A theory of quantum jumps in a system of few trapped ions is formulated. The V configuration of the energy levels and incoherent pumping are assumed. The master equation is derived in terms of the SU(3) collective operators for the V system. Numerical results for the intensity-intensity correlations for up to five ions are presented. Analytical results for a system of two ions are derived separating out dynamics at two difference time scales and by eliminating adiabatically the fast variables. From the analytical results, the underlying three-state Markov process that can be used to characterize the quantum jumps in the two-ion system is identified. Probability distributions for various off-on time intervals are presented.>