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arXiv:1911.01482 · Physical Review Research

Generic phase diagram of spin relaxation in solids and the Loschmidt echo

Gábor Csősz, Lénárd Szolnoki, Annamária Kiss, Balázs Dóra, Ferenc Simon

Published Jul 13, 2020• 4 citations
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Abstract

The authors study the generic phase diagram of spin relaxation time, ${\ensuremath{\tau}}_{\text{s}}$, in semiconductors using a stochastic model which considers the temporal evolution of a spin ensemble on the Bloch sphere. The strength of the quasiparticle scattering rate, $\mathrm{\ensuremath{\Gamma}}$, the spin-orbit coupling, $L$, and that of a magnetic term, ${\mathrm{\ensuremath{\Delta}}}_{\text{Z}}$ due to the Zeeman effect was considered.

DOI
10.1103/physrevresearch.2.033058

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The authors study the generic phase diagram of spin relaxation time, \ensuremathτs{\ensuremath{\tau}}_{\text{s}}\ensuremathτs​, in semiconductors using a stochastic model which considers the temporal evolution of a spin ensemble on the Bloch sphere. The strength of the quasiparticle scattering rate, \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ, the spin-orbit coupling, LLL, and that of a magnetic term, \ensuremathΔZ{\mathrm{\ensuremath{\Delta}}}_{\text{Z}}\ensuremathΔZ​ due to the Zeeman effect was considered.
The entropy change during spin relaxation for a realistic model system is studied, whose spin dynamics can be handled with the Boltzmann equation. The time evolution of the von Neumann entropy is monitored during the process and is compared with the recently introduced concept of the Loschmidt echo envelope. The time evolution of the two quantities is remarkably similar which helps to distinguish reversible and irreversible changes to the ensemble spin state. The method is also demonstrated for a toy model of nuclear magnetic resonance, where the usual π spin echo is performed numerically, and the echo envelope also follows the time evolution of the von Neumann entropy. The numerical approach highlights the utility of the entropy concept in analyzing various processes which occur during spin relaxation.
We revisit the problem of decoherence of a qubit centrally coupled to an interacting spin environment, here modeled by a quantum compass chain or an extended XY model in a staggered magnetic field. These two models both support distinct spin-liquid phases, adding a new element to the problem. By analyzing their Loschmidt echoes when perturbed by the qubit we find that a fast decoherence of the qubit is conditioned on the presence of propagating quasiparticles which couple to it. Differently from expectations based on earlier works on central spin models, our result implies that the closeness of an environment to a quantum phase transition is neither a sufficient nor a necessary condition for an accelerated decoherence rate of a qubit.
A detailed investigation is made of the approach to equilibrium of system of spins with I=12I=\frac{1}{2}I=21​. The spin-spin interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with Bjk{B}_{\mathrm{jk}}Bjk​ constant is treated exactly, while the additional interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{L}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with Ljk{L}_{\mathrm{jk}}Ljk​ depending on lattice vibrations is treated by means of the assumption of sufficiently short correlation times for the Ljk{L}_{\mathrm{jk}}Ljk​ operators. All correlations between the Ljk{L}_{\mathrm{jk}}Ljk​ are included. The effect of the correlations usually neglected is expressed in terms of a sum over states somewhat resembling an Ising-model partition function with the time replacing interaction strength. The oscillatory relaxation via the Bjk{B}_{\mathrm{jk}}Bjk​ and the monotonic relaxation via the phonons compete with each other; interference effects between the two relaxation modes also occur; the origin and nature of the irreversibility are very different for the two relaxation modes.
We study the Loschmidt echo (LE) in a central spin model in which a central spin is globally coupled to an environment (EEE) which is subjected to a small and sudden quench at t=0t=0t=0, so that its state at t=0+t={0}^{+}t=0+ remains the same as the ground state of the initial environmental Hamiltonian before the quench; this leads to a nonequilibrium situation. This state now evolves with two Hamiltonians, the final Hamiltonian following the quench and its modified version which incorporates an additional term arising due to the coupling of the central spin to the environment. Using a generic short-time scaling of the decay rate, we establish that in the early-time limit, the rate of decay of the LE close to the quantum critical point (QCP) of EEE is independent of the quenching. We also study the temporal evolution of the LE and establish the presence of a crossover to a situation where the quenching becomes irrelevant. In the limit of large quench amplitude the nonequilibrium initial condition is found to result in a drastic increase in decoherence at large times, even far away from a QCP. These generic results are verified analytically as well as numerically, choosing EEE to be a transverse Ising chain where the transverse field is suddenly quenched.
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.
Abstract We study the non-equilibrium evolution of concurrence of a Bell pair constituted of two qubits, through the measurement of Loschmidt echo (LE) under the scope of generalized central spin model. The qubits are locally coupled to a one dimensional disordered Ising spin chain. We first show that in equilibrium situation the derivative of LE is able to detect the extent of Griffiths phase that appeared in presence of disordered transverse field only. While in the non-equilibrium situation, the spin chain requires a temporal window to realize the effect of disorder. We show that within this timescale, LE falls off exponentially and this decay is maximally controlled by the initial states and coupling strength. Our detail investigation suggests that there actually exist three types of exponential decay, a Gaussian decay in ultra short time scale followed by two exponential decay in the intermediate time with two different decay exponents. The effect of the disorder starts appearing in the late time power law fall of LE where the power law exponent is strongly dependent on disorder strength and the final state but almost independent of initial states and coupling strength. This feature allows us to indicate the presence of Griffiths phase. To be precise, continuously varying critical exponent and wide distribution of relaxation time imprint their effect in LE in the late time limit where the power law fall is absent for quenching to a Griffiths phase. Here, LE vanishes following the fast exponential fall. Interestingly, for off-critical quenching LE attains a higher saturation value for increasing disorder strength, otherwise vanishes for a clean spin chain, referring to the fact that disorder prohibits the rapid decay of entanglement in long time limit. Moreover, we show that disorder is also able to destroy the light cone like nature of post quench quasi-particles as LE does not sense the singular time scales appearing for clean spin chain with qubits coupled at symmetric positions.