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arXiv:0811.2386 · Physical Review A

Dynamics of the inhomogeneous Dicke model for a single-boson mode coupled to a bath of nonidentical spin-1/2 systems

Oleksandr Tsyplyatyev, Daniel Loss

Published Aug 5, 2009• 28 citations
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Abstract

We study the time dynamics of a single boson coupled to a bath of two-level systems (spins 1/2) with different excitation energies, described by an inhomogeneous Dicke model. Analyzing the time-dependent Schr\"odinger equation exactly, we find that at resonance the boson decays in time to an oscillatory state with a finite amplitude characterized by a single Rabi frequency if the inhomogeneity is below a certain threshold. In the limit of small inhomogeneity, the decay is suppressed and exhibits a complex (mainly Gaussian-like) behavior, whereas the decay is complete and of exponential form in the opposite limit. For intermediate inhomogeneity, the boson decay is partial and governed by a combination of exponential and power laws.

DOI
10.1103/physreva.80.023803

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Using the exact eigenstates of the inhomogeneous Dicke model obtained by numerically solving the Bethe equations, we study the decay of bosonic excitations due to the coupling of the mode to an ensemble of two-level (spin 1/21/21/2) systems. We compare the quantum time evolution of the bosonic mode population with the mean-field description which, for a few bosons, agree up to a relatively long Ehrenfest time. We demonstrate that additional excitations lead to a dramatic shortening of the period of validity of the mean-field analysis. However, even in the limit where the number of bosons equal the number of spins, the initial instability remains adequately described by the mean-field approach leading to a finite, albeit short, Ehrenfest time. Through finite size analysis we also present indications that the mean-field approach could still provide an adequate description for thermodynamically large systems even at long times. However, for mesoscopic systems one cannot expect it to capture the behavior beyond the initial decay stage in the limit of an extremely large number of excitations.
We show that in the few-excitation regime, the classical and quantum time evolution of the inhomogeneous Dicke model for NNN two-level systems coupled to a single boson mode agree for N⪢1N⪢1N⪢1. In the presence of a single excitation only, the leading term in an 1/N1/N1/N expansion of the classical equations of motion reproduces the result of the Schr\"odinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for NNN sufficiently large. By solving the Schr\"odinger equation exactly for two excitations and a particular inhomogeneity, we obtain 1/N1/N1/N corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by ${\ensuremath{\tau}}_{E}=\sqrt{N/⟨{g}^{2}⟩}$, where ⟨g2⟩\sqrt{⟨{g}^{2}⟩}⟨g2⟩​ is an effective coupling strength between the two-level systems and the boson.
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An exactly solvable model for the decoherence of one and two-qubit states interacting with a spin bath, in the presence of a time-dependent magnetic field is studied. The magnetic field is static along $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction and oscillatory in the transverse plane. The transition probability and Rabi oscillations between the spin states of a single qubit is shown to depend on the size of bath, the distribution of qubit-bath couplings, and the initial bath polarization. In contrast to the fast Gaussian decay for short times, the polarization of the qubit shows an oscillatory power-law decay for long times. The loss of entanglement for the maximally entangled two-qubit states can be controlled by tuning the frequency of the rotating field. The decay rates of entanglement and purity for all the Bell states are the same when the qubits are noninteracting, and different when they are interacting.
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We study the time evolution of a single spin coupled by exchange interaction to an environment of interacting spin bath modeled by the XY Hamiltonian. By evaluating the spin correlator of the single spin, we observed that the decay rate of the spin oscillations strongly depends on the relative magnitude of the exchange coupling between the single spin and its nearest neighbor J(') and coupling among the spins in the environment J. The decoherence time varies significantly based on the relative coupling magnitudes of J and J('). The decay rate law has a Gaussian profile when the two exchange couplings are of the same order J(') approximately J but converts to exponential and then a power law as we move to the regimes of J(')>J and J(')<J. We also show that the spin oscillations propagate from the single spin to the environmental spins with a certain speed. The effect of varying the anisotropic parameter, external magnetic field, and temperature on the decaying rate of the spin state is also discussed.