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Laser Physics

Nonstationary dissipative dynamics of a single qubit

H. M. Castro-Beltrán, E. R. Marquina-Cruz, Gabriel Arroyo-Correa, A. Denisov

Published Feb 1, 2008• 1 citation
Publisher

Abstract

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.

DOI
10.1134/s1054660x08020096

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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.
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.
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.
The effect of noise on the populations of the levels of a qubit in individual experimental implementations has been studied by the quantum trajectory method. A transition to the average dynamics obtained by means of multiple measurements of the state of the qubit is analyzed. The developed method is applied to investigate the effect of noise on the interference pattern appearing in the amplitude spectroscopy of the qubit in a strong variable field owing to Landau-Zener transitions. The effect of the number of repeated measurements and the fluctuation of the phase of a pump pulse on the formation of the response of the qubit to the external field has been analyzed. This makes it possible to interpret recent experiments in terms of individual implementations and averaged dynamics.
An analytical study of the dissipative Landau-Zener model is presented. The model where two energy levels at constant speed are brought to cross is a standard model used to describe a large variety of phenomena. In many cases of interest the presence of coupling of the two-state system to an environment is of importance, the accounting for which shall be done here from first principles. Analytical results for the excitation transition from the ground state of the two-state system at large negative times to the excited state at large positive times (as well as the opposite, decay, transition) are obtained in terms of the speed by which the two levels approach each other, the energy gap between the adiabatic energies, the coupling strength to the environment, and its temperature. For the excitation transition we find the following results: In the slow-passage limit of small sweeping speed it is shown that adiabaticity is limited to low temperatures and the quantitative adiabatic criterion is established. Particularly, at zero temperature there is no influence of the environment on the transition probability as a consequence of a compensation property shown to be peculiar to the linear-sweep model. The transition is at low temperatures due to quantum tunneling and, with an increase in temperature, an intermediate region appears where the transition is dominated by thermally assisted transitions across the energy gap before finally at high temperatures a saturated regime is reached with equal population of the levels. In contrast to the dependence on the temperature the dependence of the transition probability as a function of coupling strength is nonmonotonic with maximum influence at intermediate strength. In the fast-passage limit with rapid sweep speed there is no influence of the environment on the transition probability. For the decay transition the adiabatic limit does not exist for the linear-sweep model for physically relevant spectra of the environment and the decay transition is dominated by spontaneous emission, except in the fast-passage limit and the high-temperature limit where the decay transition probability equals the excitation probability.
We consider the dissipative single-qubit circuit QED architecture in which the atomic transition frequency undergoes a weak external time modulation. For sinusoidal modulation with linearly varying frequency we derive effective Hamiltonians that resemble the Landau-Zener problem of finite duration associated with a two- or multilevel systems. The corresponding off-diagonal coupling coefficients originate either from the rotating or the counter-rotating terms in the Rabi Hamiltonian, depending on the values of the modulation frequency. It is demonstrated that in the dissipationless case one can accomplish almost complete transitions between the eigenstates of the bare Rabi Hamiltonian even for relatively short durations of the frequency sweep. To assess the experimental feasibility of our scheme we solved numerically the phenomenological and the microscopic quantum master equations in the Markovian regime at zero temperature. Both models exhibit qualitatively similar behavior and indicate that photon generation from vacuum via effective Landau-Zener transitions could be implemented with the current technology on the time scales of a few microseconds. Moreover, unlike the harmonic dynamical Casimir effect implementations, our proposal does not require precise knowledge of the resonant modulation frequency to accomplish meaningful photon generation.