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arXiv:quant-ph/0106015 · Physical Review A

Relaxation of a two-level system strongly coupled to a reservoir: Anomalously slow decoherence

Abraham G. Kofman

Published Aug 14, 2001• 18 citations
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Abstract

Relaxation of a two-level system (TLS) into a resonant infinite-temperature reservoir with a Lorentzian spectrum is studied. The reservoir is described by a complex Gaussian-Markovian field coupled to the nondiagonal elements of the TLS Hamiltonian. The theory can be relevant for electromagnetic interactions in microwave high-Q cavities and muon spin depolarization. Analytical results are obtained for the strong-coupling regime, ${\ensuremath{\Omega}}_{0}\ensuremath{\gg}\ensuremath{\nu},$ where ${\ensuremath{\Omega}}_{0}$ is the rms coupling amplitude (Rabi frequency) and $\ensuremath{\nu}$ is the width of the reservoir spectrum. In this regime, the population difference and half of the initial coherence decay with two characteristic rates: most of the decay occurs at $t\ensuremath{\sim}{\ensuremath{\Omega}}_{0}^{\ensuremath{-}1},$ the relaxation being reversible for $t\ensuremath{\ll}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3},$ whereas for $t\ensuremath{\gg}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3}$ the relaxation becomes irreversible and is practically over. The other half of the coherence decays with a rate on the order of $\ensuremath{\nu},$ which may be slower by orders of magnitude than the time scale of the population relaxation. The above features are explained by the fact that at $t\ensuremath{\ll}{\ensuremath{\nu}}^{\ensuremath{-}1},$ the reservoir temporal fluctuations are effectively one-dimensional (adiabatic). Moreover, we identify the pointer basis, in which the reduction of the state vector occurs. The pointer states depend on the initial phase of the reservoir.

DOI
10.1103/physreva.64.033809

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Relaxation of a two-level system strongly coupled to a resevoir: anomalously slow decoherence

Abraham G. Kofman

Relaxation of a two-level system (TLS) into a resonant infinite-temperature reservoir with a Lorentzian spectrum is studied. The reservoir is described by a complex Gaussian-Markovian field coupled to the nondiagonal elements of the TLS Hamiltonian. The theory can be relevant for electromagnetic interactions in microwave high-Q cavities and muon spin depolarization. Analytical results are obtained for the strong-coupling regime, 0>ν, where Ω0 is the rms coupling amplitude (Rabi frequency) and ν is the width of the reservoir spectrum. In this regime, the population difference and half of the initial coherence decay with two characteristic rates: the most part of the decay occurs at t < (Ω0ν)-1/3 the relaxation becomes reversible for t ~ Ω0-1/3, whereas for t > (Ω02ν)-1/3 the relaxation becomes irreversible and is practically over. The other half of the coherence decays with the rate on the order of ν, which may be slower by orders of magnitude than the time scale of the population relaxation. The above features are explained by the fact that at t < ν-1 the reservoir temporal fluctuations are effectively one-dimensional (adiabatic). Moreover, we identify the pointer basis, in which the reduction of the state vector occurs. The pointer states are correlated with the reservoir, being dependent on the reservoir phase.

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Relaxation of a two-level system (TLS) into a resonant infinite-temperature reservoir with a Lorentzian spectrum is studied. The reservoir is described by a complex Gaussian-Markovian field coupled to the nondiagonal elements of the TLS Hamiltonian. The theory can be relevant for electromagnetic interactions in microwave high-Q cavities and muon spin depolarization. Analytical results are obtained for the strong-coupling regime, \ensuremathΩ0\ensuremath≫\ensuremathν,{\ensuremath{\Omega}}_{0}\ensuremath{\gg}\ensuremath{\nu},\ensuremathΩ0​\ensuremath≫\ensuremathν, where \ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​ is the rms coupling amplitude (Rabi frequency) and \ensuremathν\ensuremath{\nu}\ensuremathν is the width of the reservoir spectrum. In this regime, the population difference and half of the initial coherence decay with two characteristic rates: most of the decay occurs at t\ensuremath∼\ensuremathΩ0\ensuremath−1,t\ensuremath{\sim}{\ensuremath{\Omega}}_{0}^{\ensuremath{-}1},t\ensuremath∼\ensuremathΩ0\ensuremath−1​, the relaxation being reversible for t\ensuremath≪(\ensuremathΩ02\ensuremathν)\ensuremath−1/3,t\ensuremath{\ll}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3},t\ensuremath≪(\ensuremathΩ02​\ensuremathν)\ensuremath−1/3, whereas for t\ensuremath≫(\ensuremathΩ02\ensuremathν)\ensuremath−1/3t\ensuremath{\gg}({\ensuremath{\Omega}}_{0}^{2}\ensuremath{\nu}{)}^{\ensuremath{-}1/3}t\ensuremath≫(\ensuremathΩ02​\ensuremathν)\ensuremath−1/3 the relaxation becomes irreversible and is practically over. The other half of the coherence decays with a rate on the order of \ensuremathν,\ensuremath{\nu},\ensuremathν, which may be slower by orders of magnitude than the time scale of the population relaxation. The above features are explained by the fact that at t\ensuremath≪\ensuremathν\ensuremath−1,t\ensuremath{\ll}{\ensuremath{\nu}}^{\ensuremath{-}1},t\ensuremath≪\ensuremathν\ensuremath−1, the reservoir temporal fluctuations are effectively one-dimensional (adiabatic). Moreover, we identify the pointer basis, in which the reduction of the state vector occurs. The pointer states depend on the initial phase of the reservoir.
Relaxation of a two-level system (TLS) into a resonant infinite-temperature reservoir with a Lorentzian spectrum is studied. The reservoir is described by a complex Gaussian-Markovian field coupled to the nondiagonal elements of the TLS Hamiltonian. The theory can be relevant for electromagnetic interactions in microwave high-Q cavities and muon spin depolarization. Analytical results are obtained for the strong-coupling regime, 0>ν, where Ω0 is the rms coupling amplitude (Rabi frequency) and ν is the width of the reservoir spectrum. In this regime, the population difference and half of the initial coherence decay with two characteristic rates: the most part of the decay occurs at t < (Ω0ν)-1/3 the relaxation becomes reversible for t ~ Ω0-1/3, whereas for t > (Ω02ν)-1/3 the relaxation becomes irreversible and is practically over. The other half of the coherence decays with the rate on the order of ν, which may be slower by orders of magnitude than the time scale of the population relaxation. The above features are explained by the fact that at t < ν-1 the reservoir temporal fluctuations are effectively one-dimensional (adiabatic). Moreover, we identify the pointer basis, in which the reduction of the state vector occurs. The pointer states are correlated with the reservoir, being dependent on the reservoir phase.
We study the absorption spectrum of a weak probe field near resonant to an atomic transition, the upper level of which is strongly coupled to a third level by the interaction with a Lorentzian bosonic reservoir, such as, e.g., a mode of a high-QQQ cavity or a local vibration in a solid. The reservoir coupling is approximated by the interaction with a classical complex Gaussian-Markovian random process (control field), which is justified when the reservoir temperature exceeds significantly the mode frequency or when the high-QQQ cavity is pumped by broadband incoherent radiation. The present theory is applicable also when the control field is chaotic laser light. We assume that the rms control-field Rabi frequency V0{V}_{0}V0​ is much greater than the field detuning ${\ensuremath{\Delta}}_{c}$, which, in turn, is much greater than the material relaxation constants. We reveal and describe analytically all qualitatively different regimes of the spectrum modification and obtain their validity conditions. The analytical results are verified by numerical calculations using the exact continued-fraction solution. The analytical formulas obtained allow one to perform fast computer calculations for arbitrarily small values of the reservoir (control-field) bandwidth $\ensuremath{\nu}$, in contrast to the known numerical methods, which require sharply increasing computational resources with a decrease of $\ensuremath{\nu}$. In the most interesting case $\ensuremath{\nu}⪡{V}_{0}$, the spectrum consists of two peaks, the nonvanishing bandwidth and material relaxation affecting mainly the dip between the peaks. The results obtained in the static limit (i.e., a very narrow reservoir) are independent of the reservoir band shape. We reveal reservoir-induced transparency (RIT)---i.e., absorption reduction due to the reservoir coupling. Moreover, two unexpected, remarkable features are uncovered in a range of intermediate values of $\ensuremath{\nu}$ and V0{V}_{0}V0​, ${\ensuremath{\Gamma}}^{2}\ensuremath{\mid}{\ensuremath{\Delta}}_{c}\ensuremath{\mid}⪡{V}_{0}^{2}\ensuremath{\nu}⪡{\ensuremath{\mid}{\ensuremath{\Delta}}_{c}\ensuremath{\mid}}^{3}$ ($\ensuremath{\Gamma}$ is the spectral width in the absence of the control field): an extra peak in the dip and a resonance-absorption decrease by a half (i.e., an increase of RIT) relative to the static limit. The existence of the above spectral features depends on the statistical details of the control-field temporal behavior.
The present paper treats the electromagnetically induced transparency (EIT), i.e., the absorption spectrum of a weak probe field, near-resonant to a material transition, in the presence of a strong amplitude-phase fluctuating field which couples the upper level of the transition to another excited level. The coupling field is assumed to be chaotic (i.e., a complex Gaussian process with a Lorentzian spectrum) and so strong that the material relaxation and the detuning can be neglected. In addition, we consider the Fourier transform of the spectrum Uaa(t),{U}_{\mathrm{aa}}(t),Uaa​(t), which determines absorption of a probe field with a time-dependent intensity. The quantity Uaa(t){U}_{\mathrm{aa}}(t)Uaa​(t) equals a diagonal element of the average reduced evolution operator of the two-level system (TLS), coupled by the strong field. As a starting point, we use the stationary, partially averaged stochastic Schr\"odinger equation, which has the form of a set of two second-order differential equations. The theory is developed by combining analytical and numerical approaches. For the most important regime, when the rms Rabi frequency V0{V}_{0}V0​ is much greater than the coupling-field bandwidth $\ensuremath{\nu},$ we obtain scaling relations for the EIT line shape and the TLS evolution. In particular, the absorption at the minimum (i.e., at the resonance frequency) scales as ${\ensuremath{\nu}}^{1/3}{V}_{0}^{\ensuremath{-}4/3},$ whereas the evolution function Uaa(t){U}_{\mathrm{aa}}(t)Uaa​(t) performs damped oscillations with the decay rate on the order of ${(V}_{0}^{2}\ensuremath{\nu}{)}^{1/3}.$ These results differ significantly from those following from the uncorrelated-jump model. The results of this paper are shown to be applicable to more general models of amplitude-phase fluctuations than the Gaussian-Markovian chaotic field.
The dynamics of a three-level Λ-atom coupled to a structured reservoir has been investigated. Comparing the Lorentzian time-dependent decay rate and population dynamics for the weak coupling regime with those for the strong coupling regime, we note that non-Markovian features exist in both regimes, but population dynamics shows fairly different behaviors. Moreover, the different positions of the two transition frequencies with respect to the structured reservoir reduces the influence of non-Markovian effects on population dynamics. Based on the non-Markovianity, we give an intuitive interpretation of the result that the non-Markovian effects have little influence on population dynamics in the weak coupling regime, but have an obvious influence in the strong coupling regime.
We discuss the long-time relaxation of a qubit linearly coupled to a finite bath of [Formula: see text] spins (two-level systems, TLSs), with the interaction Hamiltonian in rotating wave approximation. We focus on the regime [Formula: see text], assuming that the qubit–bath coupling is weak, that the range of spin frequencies is sufficiently broad, and that all the spins are initialized in the ground state. Despite the model being perfectly integrable, we make two interesting observations about the effective system relaxation. First, as one would expect, the qubit relaxes exponentially towards its zero-temperature state at a well characterized rate. Second, the bath spins, even when mutually coupled, do not relax towards a thermal distribution, but rather form a Lorentzian distribution peaked at the frequency of the initially excited qubit. This behaviour is captured by an analytical approximation that makes use of the property [Formula: see text] to treat the TLS frequencies as a continuum and is confirmed by our numerical simulations.
Driving an open spin system by two strong, nearly degenerate fields enables addressing populations of individual spin states, characterisation of their interaction with thermal bath, and measurements of their relaxation/decoherence rates. With such addressing we observe nested magnetic resonances having nontrivial dependence on microwave field intensity: while the width of one of the resonances undergoes a strong power broadening, the other one exhibits a peculiar field-induced stabilization. We also observe light-induced narrowing of such composite resonances. The observations are explained by the dynamics of bright and dark superposition states and their interaction with reservoir.