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Physical Review A

Dissipation effects on wave packets in level crossings: A comparison between two numerical approaches

W. K. Lai, Kalle‐Antti Suominen, B. M. Garraway, Stig Stenholm

Published Jun 1, 1993• 42 citations
Publisher

Abstract

We study the time evolution of wave packets on coupled energy levels including the irreversible effect of spontaneous emission. As a test case we choose a simple two-state level-crossing problem of the Landau-Zener type. The numerical treatment of this situation is demanding, and we explore two approaches based on different views of the quantum-mechanical dynamics. First, we solve the time evolution of the wave packet using the traditional master equation giving directly the ensemble averages. Second, we use the recently developed Monte Carlo simulation method, which gives the individual members of an ensemble of quantum states undergoing spontaneous quantum jumps. The results are compared and found to agree when large enough ensembles are used. We discuss the computational requirements for the two methods and we find both quite demanding but qualitatively different.

DOI
10.1103/physreva.47.4779

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arXiv:2410.03582•Oct 4, 2024•0 citations
We study the time evolution of wave packets on coupled energy levels including the irreversible effect of spontaneous emission. As a test case we choose a simple two-state level-crossing problem of the Landau-Zener type. The numerical treatment of this situation is demanding, and we explore two approaches based on different views of the quantum-mechanical dynamics. First, we solve the time evolution of the wave packet using the traditional master equation giving directly the ensemble averages. Second, we use the recently developed Monte Carlo simulation method, which gives the individual members of an ensemble of quantum states undergoing spontaneous quantum jumps. The results are compared and found to agree when large enough ensembles are used. We discuss the computational requirements for the two methods and we find both quite demanding but qualitatively different.
We study the changes appearing in Gaussian momentum distributions of quantum-mechanical wave packets after a traversal of a linear level crossing. The numerical results are discussed in terms of the semiclassical Landau-Zener theory. We observe a momentum filtering effect in nondissipative crossings. The results obtained from studies of dissipative level crossings suggest that one can obtain substantial changes in momentum distributions with these systems. The Landau-Zener theory provides qualitative understanding of the processes taking place in the dissipative level crossing.
We study numerically the effects of spontaneous emission from the upper state in a single two-level atom (qubit) driven by a field of constant amplitude and frequency varying linearly in time, crossing the atomic resonance, the Landau-Zener model, using a discontinuous jump quantum trajectory formalism. A single trajectory describes the pure state atomic evolution during the sweep of the field frequency across the atomic transition. Each jump returns the atom to its ground state, but the behavior of reexcitation depends on the time the jump occurred: before, near, or after the resonance, as a result of the nonstationary nature of the Landau-Zener model. The evolution of the Bloch vector during a single trajectory is unitary (a pure state preserves the trace), but shows the stochastic nature of the particular qubit history. The ensemble average, which agrees with the Bloch equations, shows that spontaneous emission causes both the shrinking of the Bloch vector shortly after crossing the resonance and its recovery for longer times. The quantum jump approach allows a simple calculation of the distribution of emissions per sweep. Its mean agrees with the integrated emission rate, the variance increases with the field strength and decay rate, and the zero-jump value of the distribution gives the fraction of trajectories without a jump.
In this paper I study the dynamics of a two-level atom interacting with a standing-wave field. When the atom is subjected to a weak linear force, the problem can be turned into a time-dependent one, and the evolution is understood from the band structure of the spectrum. The presence of level crossings in the spectrum gives rise to Bloch oscillations of the atomic motion. Here I investigate the effects of the atom-field detuning parameter. A variety of different level crossings are obtained by changing the magnitude of the detuning, and the behavior of the atomic motion is strongly affected due to this. I also consider the situation in which the detuning is oscillating in time, and its impact on the atomic motion. Wave-packet simulations of the full problem are treated numerically and the results are compared with analytical solutions given by the standard Landau-Zener and the three-level Landau-Zener models.
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.
A Monte Carlo simulation of the atomic master equation for spontaneous emission in terms of atomic wave functions is developed. Realizations of the time evolution of atomic wave functions are constructed that correspond to an ensemble of atoms driven by laser light undergoing a sequence of spontaneous emissions. The atomic decay times are drawn according to the photon count distribution of the driven atom. Each quantum jump of the atomic electron projects the atomic wave function to the ground state of the atom. Our theory is based on a stochastic interpretation and generalization of Mollow's pure-state analysis of resonant light scattering, and the Srinivas-Davies theory of continuous measurements in photodetection. An extension of the theory to include mechanical light effects and a generalization to atomic systems with Zeeman substructure are given. We illustrate the method by simulating the solutions of the optical Bloch equations for two-level systems, and laser cooling of a two-level atom in an ion trap where the center-of-mass motion of the atom is described quantum mechanically.
We present statistics of quantum jumps in the two-level system with landau-Zener Hamiltonian that undergoes a Markovian process. For the Landau-Zener model, which is successful in simulating adiabatic/non-adiabatic evolution and quantum annealing, we consider two types of dissipation. In the first one, the jump operators project states to the initial ground state and excited state of the Hamiltonian at t→−∞t\to -\inftyt→−∞. In the second type, the jump operators project to the instantaneous eigenstates of the Hamiltonian. By the quantum trajectories approach, we present the probability of the number of jumps in adiabatic and non-adiabatic regimes for both models. Furthermore, we demonstrate the statistics of jumps in time intervals of the evolutions. Also, we show the role of bath temperature, coupling strength to the environment, and spin-coupling directions on the statistics of quantum jumps.