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Landau-Zener Transitions in the Pechukas-Yukawa Formalism Under the Influence of Brownian Noise

Mumnuna A. Qureshi, Johnny Q. Zhong, Peter Mason, Joseph J. Betouras, Alexandre M. Zagoskin

Published Mar 13, 2018• 0 citations
OpenAlex

Abstract

Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of 1D classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearisation. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.

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Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of 1D classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearisation. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.
Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions.
We analyze the influence of colored classical Gaussian noise on Landau-Zener transitions during a two-level crossing in a time-dependent regular external field. Transition probabilities and coherence factors become random due to the noise. We calculate their two-time correlation functions, which describe the response of this two-level system to a weak external pulse signal. The spectrum and intensity of the magnetic response are derived. Although the noise enters the equation of motion for the Bloch vector in a multiplicative way, nonperturbative analytic results are obtained by a resummation of diagrams in the limit of a short-noise correlation time. Our results also cover regimes where fluctuations are of the same order of magnitude as averages.
The degenerate Landau–Zener–Majorana–Stückelberg model consists of two degenerate energy levels whose energies vary with time and in the presence of an interaction which couples the states of the two levels. In the adiabatic limit, it allows for the populations transfer from states of one level to the states of the other level. The presence of an interaction with the environment influences the efficiency of the process. Nevertheless, identification of possible decoherence-free subspaces permits to engineer coupling schemes for which the effects of quantum noise can be made negligible.
We investigate the Landau-Zener transition in two- and three-level systems subject to a classical Gaussian noise. Two complementary limits of the noise being fast and slow compared to characteristic Landau-Zener tunnel times are discussed. The analytical solution of a density matrix (Bloch) equation is given for a long-time asymptotic of transition probability. It is demonstrated that the transition probability induced or assisted by the fast noise can be obtained through a procedure of Bloch's equation averaging with further reducing it to a master equation. In contrast to the case of fast noise, the transition probability for LZ transition induced, or assisted, by the slow classical noise can be obtained by averaging the solution of Bloch's equation over the noise realization. As a result, the transition probability is described by the activation Arrhenius law. The approximate solution of the Bloch's equation at finite times is written in terms of Fresnel's integrals and interpreted in terms of interference pattern. We discuss consequences of a local isomorphism between SU(2) and SO(3) groups and connections between Schr\"odinger and Bloch descriptions of spin dynamics. Based on this isomorphism, we establish the relations between S=1/2S=1/2S=1/2 and 1 transition probabilities influenced by the noise. A possibility to use the slow noise as a probe for tunnel time is discussed.
When the drive, which causes the level crossing in a qubit, is slow, the probability PLZ{P}_{LZ}PLZ​ of the Landau-Zener transition is close to 1. In this regime, which is most promising for applications, the noise due to the coupling to the environment reduces the average PLZ{P}_{LZ}PLZ​. At the same time, the survival probability, $1\ensuremath{-}{P}_{LZ}$, which is exponentially small for a slow drive, can be completely dominated by noise-induced correction. Our main message is that the effect of weak classical noise can be captured analytically by treating it as a perturbation in the Schr\"odinger equation. This allows us to study the dependence of the noise-induced correction to PLZ{P}_{LZ}PLZ​ on the correlation time of the noise. As this correlation time exceeds the bare Landau-Zener transition time, the effect of noise becomes negligible. On the physical level, the mechanism of enhancement of the survival probability can be viewed as an absorption of the ``noise quanta'' across the gap. With characteristic energy of the quantum governed by the noise spectrum, the slower the noise is, the lower the number of quanta for which absorption is allowed energetically is. We consider two conventional realizations of noise: Gaussian noise and telegraph noise.
We determine the Landau-Zener transition probability in a dissipative environment including both longitudinal as well as transversal quantum-mechanical noise originating from a single noise source. For this, we use the numerically exact quasiadiabatic path integral, as well as the approximative nonequilibrium Bloch equations. We find that transversal quantum noise in general influences the Landau-Zener probability much more strongly than longitudinal quantum noise does at a given temperature and system-bath coupling strength. In other words, transversal noise contributions become important even when the coupling strength of transversal noise is smaller than that of longitudinal noise. We furthermore reveal that transversal noise renormalizes the tunnel coupling independent of temperature. Finally, we show that the effect of mixed longitudinal and transversal noise originating from a single bath cannot be obtained from an incoherent sum of purely longitudinal and purely transversal noise.