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arXiv:cond-mat/0509395 · Physical Review B

Spin relaxation and decoherence of two-level systems

X. R. Wang, Yanzhe Zheng, Yin Sun

Published Sep 15, 2005• 19 citations
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Abstract

We revisit the concepts of spin relaxation and spin decoherence of two-level (spin-$1∕2$) systems. From two toy models, we clarify two issues related to the spin relaxation and decoherence: (1) For an ensemble of two-level particles each subjected to a different environmental field, there exists an ensemble relaxation time ${T}_{1}^{*}$, which is fundamentally different from ${T}_{1}$. When the off-diagonal coupling of each particle is in a single mode with the same frequency but a random coupling strength, we show that ${T}_{1}^{*}$ is finite while the spin-relaxation time of a single-spin ${T}_{1}$ and the usual ensemble decoherence time ${T}_{2}^{*}$ are infinite. (2) For a two-level particle under only a random diagonal coupling, its relaxation time ${T}_{1}$ shall be infinite, but its decoherence time ${T}_{2}$ is finite.

DOI
10.1103/physrevb.72.121303

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We revisit the concepts of spin relaxation and spin decoherence of two-level (spin-1∕21∕21∕2) systems. From two toy models, we clarify two issues related to the spin relaxation and decoherence: (1) For an ensemble of two-level particles each subjected to a different environmental field, there exists an ensemble relaxation time T1∗{T}_{1}^{*}T1∗​, which is fundamentally different from T1{T}_{1}T1​. When the off-diagonal coupling of each particle is in a single mode with the same frequency but a random coupling strength, we show that T1∗{T}_{1}^{*}T1∗​ is finite while the spin-relaxation time of a single-spin T1{T}_{1}T1​ and the usual ensemble decoherence time T2∗{T}_{2}^{*}T2∗​ are infinite. (2) For a two-level particle under only a random diagonal coupling, its relaxation time T1{T}_{1}T1​ shall be infinite, but its decoherence time T2{T}_{2}T2​ is finite.
A quantum statistical model for the relaxation of two strongly coupled spin 1/2 particles toward thermal equilibrium is formulated microscopically using a master equation. Raman-type dissipation mechanisms are introduced in addition to the usual bilinear coupling between the coupled system and the reservoir. The effects of two different rotating wave approximations to derive the master equation are investigated. Decoherence is considered as a process toward thermal equilibrium for the dissipative system and is studied by observing the phase and energy relaxation.
We present an approach to spin dynamics by extending the optical Bloch equations for the driven two-level system to derive microscopic expressions for the transverse and longitudinal spin-relaxation times. This is done for the six-level system of electron and hole subband states in a semiconductor or a semiconductor quantum structure to account for the degrees of freedom of the carrier spin and the polarization of the exciting light and includes the scattering between carriers and lattice vibrations on a microscopic level. For the subsystem of the spin-split electron subbands we treat the electron-phonon interaction in second order and derive a set of equations of motion for the $2\ifmmode\times\else\texttimes\fi{}2$ spin-density matrix, which describes the electron spin dynamics and contains microscopic expressions for the longitudinal (T1)({T}_{1})(T1​) and transverse (T2)({T}_{2})(T2​) spin-relaxation times. Their meaning will be discussed in relation to experimental investigations of these quantities.
In quantum computation, it is of paramount importance to locate the parameter space where maximal coherence can be preserved in the qubit system. In recent years environment-induced decoherence based the quantum Brownian motion (QBM) models have been applied to two level systems (2LS) interacting with an electromagnetic field, leading to the general belief that 2LS are easily decohered. In a recent paper C. Anastopoulos and B. L. Hu [Phys. Rev. A62, (2000) 033821] derived a new exact non-Markovian master equation at zero temperature, from which they showed that this belief is actually misplaced. For a two-level atom (2LA)- electromagnetic field (EMF) system the decoherence time is rather long, comparable to the relaxation time. Theoretically this is because the dominant interaction is the hatsigmapm\\hat \\sigma_{\\pm}hatsigmapm​ type of coupling between the two levels (what constitutes the qubit) and the field, not the hatsigmaz\\hat \\sigma_zhatsigmaz​ type, which shows the QBM behavior. Depending on the coupling the field can act as a resonator (in an atom cavity) or as a bath (in QBM) and produce very different decoherent behavior in the system. This is not new to Cavity QED experimentalists: the 2LA-EMF system maintaining its coherence in sufficiently long duration is the reason why they can manipulate them so well to show interesting quantum coherence effects.
The dynamics of a two-level system which is coupled to an environmental sea of infinitely many spin-1/2 particles is investigated by use of the resolvent operator approach. Only at zero temperature does this spin-spin-bath model exhibit identical behavior as the more familiar spin-boson model. It is found that increasing temperature favors coherent dynamics. At high temperatures, the spin-spin-bath model for an Ohmic spectral density sustains a coherent dynamics if the dissipation coefficient $\ensuremath{\alpha}$ is sufficiently small, i.e., $\ensuremath{\alpha}<1/2$; while the decoherence exhibits pure exponential decay if $\ensuremath{\alpha}>1/2$.
In Born-Markov approximation,according to Bloch-Redfield equation,the method to calculate the relaxation time T1 and decoherence time T2 of superconducting qubit is given.Based on this,the decoherence phenomenon of superconducting flux qubit with mutual inductance coupling is investigated.The energy relaxation time T1 and decoherence time T2 of this superconducting qubit is also obtained.The results show that for the flux qubit under investigation,the dephasing time T,the energy relaxation time T1 from high level |j〉 to low level |i〉,and the decoherence time T2 of the quantum system are all proportional to dissipation when other conditions are invariant and the environment thermal bath is equivalent to a pure resistance.