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arXiv:cond-mat/0603844 · Physical Review B

Model of qubits as devices to detect the third moment of current fluctuations

Valentina Brosco, Rosario Fazio, F. W. J. Hekking, Jukka P Pekola

Published Jul 27, 2006• 17 citations
arXivHTMLPDF

Abstract

Under appropriate conditions, controllable two-level systems can be used to detect the third moment of current fluctuations. We derive a master equation for a quantum system coupled to a bath valid to the third order in the coupling between the system and the environment. In this approximation the reduced dynamics of the quantum system depends on the frequency-dependent third moment. Specializing to the case of a controllable two-level system (a qubit) and in the limit in which the splitting between the levels is much smaller than the characteristic frequency of the third moment, it is possible to show that the decay of the qubit has additional oscillations whose amplitude is directly proportional to the value of the third moment. We discuss an experimental setup where this effect can be seen.

DOI
10.1103/physrevb.74.024524

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臺灣大學物理研究所學位論文•Jan 1, 2010•0 citations
Under appropriate conditions, controllable two-level systems can be used to detect the third moment of current fluctuations. We derive a master equation for a quantum system coupled to a bath valid to the third order in the coupling between the system and the environment. In this approximation the reduced dynamics of the quantum system depends on the frequency-dependent third moment. Specializing to the case of a controllable two-level system (a qubit) and in the limit in which the splitting between the levels is much smaller than the characteristic frequency of the third moment, it is possible to show that the decay of the qubit has additional oscillations whose amplitude is directly proportional to the value of the third moment. We discuss an experimental setup where this effect can be seen.
A unified theory is given of dynamically modified decay and decoherence in driven two-level and multilevel quantum systems that are weakly coupled to arbitrary finite-temperature reservoirs and undergo random phase fluctuations. Criteria for the optimization of decoherence suppression and the limitations of this approach are obtained. For a driven qubit that is strongly coupled to the continuum edge of reservoir's spectrum, we demonstrate that only an appropriately ordered sequence of abrupt changes of the resonance frequency, near the continuum edge, can effectively protect the qubit state from decoherence.
The problem of Rabi oscillations in a qubit coupled to a fluctuator and in contact with a heath bath is considered. A scheme is developed for taking into account both phase and energy relaxation in a phenomenological way, while taking full account of the quantum dynamics of the four-level system subject to a driving AC field. Significant suppression of the Rabi oscillations is found when the qubit and fluctuator are close to resonance. The effect of the fluctuator state on the read-out signal is discussed. This effect is shown to modify the observed signal significantly. This may be relevant to recent experiments by Simmonds et al. ( Phys. Rev. Lett. 93 (2004) 077003).
Abstract Recent electron spin resonance experiments on CaWO 4_44​ 4 :Gd 3+^{3+}3+ 3 + and other magnetic impurities have demonstrated that sustained Rabi oscillations can be created by driving a magnetic moment with a microwave field frequency slightly larger than the Larmor frequency and tuned to the Floquet resonance, together with another microwave field (image drive) with a frequency smaller than the Larmor frequency. These observations are confirmed by the new experimental results reported in this paper. We use numerical and analytical techniques to study the interplay between the microwave drives and three different mechanisms of relaxation. The first model describes a magnetic moment subject to microwave fields, interacting with a bath of two-level systems which acts as a source of decoherence and dissipation. The second model describes identical, interacting magnetic moments, subject to the same microwave fields. The decay of the Rabi oscillations is now due to the interactions. Third, we study Rabi oscillation decay due to the inhomogeneity of the microwave radiation. We show that under appropriate conditions, and in particular at the Floquet resonance, the magnetization exhibits sustained Rabi oscillations, in some cases with additional beatings. Although these two microscopic models separately describe the experimental data well, a simulation study that simultaneously accounts for both types of interactions is currently prohibitively costly. To gain further insight into the microscopic dynamics of these two different models, we study the time dependence of the bath and system energy and of the correlations of the spins, data that are not readily accessible experimentally. Graphic abstract
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.
We investigate the influence of nearby two-level systems on the dynamics of a qubit. The intrinsic decoherence is given by a coupling of both the qubit and the two-level systems to a heat bath. Assuming weak interactions between the qubit and the two-level systems we show that the dephasing of the qubit follows a multi-exponential decay. For a flat distribution of energy splittings of the two-level systems the multi-exponential behavior simplifies to a good approximation to a single exponential decay. The rate is given by the sum of the dephasing rate of the isolated qubit plus all relaxation rates for the two-level systems out of their present state. This latter contribution sums up to a rate approximately given by the mean coupling between the qubit and the two-level systems times the number of thermally active neighbors.
The thesis is devoted to the study of the non-Markovian dynamical process in several different physical systems. Specifically, we investigate the non-Markovian entangle- ment dynamics of two quantum bits (qubits) in a thermal squeezed bath, the non-Markovian dynamics of a nanomechanical resonator (NMR) measured by a quantum point contact (QPC) detector and, the non-Markovian evolution of two-time correlation functions (CF’s) of a two-level atom coupled to a thermal bosonic bath. First, in the investigation of the non-Markovian entanglement dynamics of two qubits in a common squeezed bath, we see a remarkable difference between the non-Markovian entanglement dynamics and its Markovian counterpart. We show that a non-Markovian decoherence-free state is also decoherence free in the Markovian regime, but all the Markovian decoherence-free states are not necessarily decoherence free in the non-Markovian domain. We extend our calculation from a squeezed vacuum bath to a squeezed thermal bath, where we see the effect of finite bath temperatures on the entanglement dynamics. Second, we also investigate the dynamics of a NMR subject to a measurement by a low-transparency QPC or tunnel junction in the non-Markovian domain. We derive the non-Markovian number-resolved (conditional) and unconditional master equations valid to second order in the tunneling Hamiltonian without making the rotating-wave approximation and the Markovian approximation, generally made for systems in quantum optics. We find considerable difference in dynamics between the non-Markovian cases and its Markovian counterparts. We also calculate the time-dependent transport current through the QPC which contains information about the measured NMR system. We find an extra transient current term proportional to the expectation value of the symmetrized product of the position andmomentum operators of the NMR. This extra term, with a coefficient coming from the combination of the imaginary parts of the QPC reservoir correlation functions, was generally ignored in the studies of the same problem in the literature. But we find that it has a substantial contribution to the total transient current in the Non- Markovian case and differs qualitatively and quantitatively from its Markovian counterpart. Thus it may serve as a witness or signature of non-Markovian features for the coupled NMR-QPC system. Finally, we use the quantum master equation approach to derive, valid to second order in the system-environment interaction Hamiltonian, non-Markovian evolution equations of two-time CF’s of system operators at finite environment temperatures with any initial separable system-environment states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equations are valid for both Her- mitian and non-Hermitian system coupling operator cases. We then give conditions on which the derived evolution equations reduced to the case of quantum regression theorem (QRT)in the weak system-environment coupling case, and apply the derived evolution equations to a problem of a two-level system (atom) coupled to a bosonic environment (electromagnetic fields) with L=L(＋)L = L^(＋)L=L(＋).