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arXiv:1205.0296 · Physical Review E

Dephasing by a continuous-time random walk process

Daniel M. Packwood, Yoshitaka Tanimura

Published Jul 26, 2012• 4 citations
arXivHTMLPDF

Abstract

Stochastic treatments of magnetic resonance spectroscopy and optical spectroscopy require evaluations of functions such as (exp(i ∫(0)(t) Q(s)ds)), where t is time, Q(s) is the value of a stochastic process at time s, and the angular brackets denote ensemble averaging. This paper gives an exact evaluation of these functions for the case where Q is a continuous-time random walk process. The continuous-time random walk describes an environment that undergoes slow steplike changes in time. It also has a well-defined Gaussian limit and so allows for non-Gaussian and Gaussian stochastic dynamics to be studied within a single framework. We apply the results to extract qubit-lattice interaction parameters from dephasing data of P-doped Si semiconductors (data collected elsewhere) and to calculate the two-dimensional spectrum of a three-level harmonic oscillator undergoing random frequency modulations.

DOI
10.1103/physreve.86.011130

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A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
We study the dynamics of dephasing in a quantum two-level system by modeling both 1/f1/f1/f and high-frequency noise by random telegraph processes. Our approach is based on a so-called spin-fluctuator model in which a noisy environment is modeled by a large number of fluctuators. In the continuous limit we obtain an effective random process (ERP) that is described by a distribution function of the fluctuators. In a simplified model, we reduce the ERP to the two (slow and fast) ensembles of fluctuators. Using this model, we study decoherence in a superconducting flux qubit and we compare our theoretical results with the available experimental data. We demonstrate good agreement of our theoretical predictions with the experiments. Our approach can be applied to many quantum systems, such as biological complexes, semiconductors, superconducting, and spin qubits, where the effects of interaction with the environment are essential.
A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.
The time evolution of the trace distance between quantum states of a qubit which is placed under the influence of stochastic dephasing is investigated within the framework of the stochastic Liouville equation. When stochastic dephasing is subject to the homogeneous Gauss-Markov process, the trace distance is exactly calculated in the presence of the initial correlation between the qubit and the stochastic process, where the stochastic process is inevitably a nonstationary process. It is found that even the initial correlation with the classical environment can make the trace distance greater than the initial value if stochastic dephasing causes the slow modulation of the qubit.
Two-level systems (TLS) are the major source of dephasing of spin qubits in numerous quantum computing platforms. In spite of much effort, it has been difficult to substantially mitigate the effects of this noise or, in many cases, to fully understand its physical origin. We propose a method to make progress on both of these issues. When an oscillating field is applied to a TLS, stochastic resonance can occur and the noise spectrum is moved to higher frequencies. This shift in the TLS noise spectrum will increase the dephasing times of the qubits that they influence. Furthermore, the details of this effect depend on the physical properties of the noise sources. Thus one can use qubit spectroscopy to investigate their physical properties, specifically the extent to which the TLS themselves possess quantum coherence. We find that it should be possible to determine the dephasing rate and the energy level separation of the TLS themselves in this way.