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.
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.
Abstract If a magnetic polarization excess is locally injected in a crystal of interacting spins in thermal equilibrium, this ‘excitation’ would spread as consequence of spin–spin interactions. Such an apparently irreversible process is known as spin diffusion and it can lead the system back to ‘equilibrium’. Even so, a unitary quantum dynamics would ensure a precise memory of the non-equilibrium initial condition. Then, if at a certain time, say t / 2 , an experimental protocol reverses the many-body dynamics by changing the sign of the effective Hamiltonian, it would drive the system back to the initial non-equilibrium state at time t . As a matter of fact, the reversal is always perturbed by small experimental imperfections and/or uncontrolled internal or environmental degrees of freedom. This limits the amount of signal M ( t ) recovered locally at time t . The degradation of M ( t ) accounts for these perturbations, which can also be seen as the sources of decoherence. This general idea defines the Loschmidt echo (LE), which embodies the various time-reversal procedures implemented in nuclear magnetic resonance. Here, we present an invitation to the study of the LE following the pathway induced by the experiments. With such a purpose, we provide a historical and conceptual overview that briefly revisits selected phenomena that underlie the LE dynamics including chaos, decoherence, localization and equilibration. This guiding thread ultimately leads us to the discussion of decoherence and irreversibility as an emergent phenomenon. In addition, we introduce the LE formalism by means of spin–spin correlation functions in a manner suitable for presentation in a broad scope physics journal. Last, but not least, we present new results that could trigger new experiments and theoretical ideas. In particular, we propose to transform an initially localized excitation into a more complex initial state, enabling a dynamically prepared LE. This induces a global definition of the LE in terms of the raw overlap between many-body wave functions. Our results show that as the complexity of the prepared state increases, it becomes more fragile towards small perturbations.
If a magnetic polarization excess is locally injected in a crystal of interacting spins in thermal equilibrium, this 'excitation' would spread as consequence of spin-spin interactions. Such an apparently irreversible process is known as spin diffusion and it can lead the system back to 'equilibrium'. Even so, a unitary quantum dynamics would ensure a precise memory of the non-equilibrium initial condition. Then, if at a certain time, say , an experimental protocol reverses the many-body dynamics by changing the sign of the effective Hamiltonian, it would drive the system back to the initial non-equilibrium state at time t. As a matter of fact, the reversal is always perturbed by small experimental imperfections and/or uncontrolled internal or environmental degrees of freedom. This limits the amount of signal M(t) recovered locally at time t. The degradation of M(t) accounts for these perturbations, which can also be seen as the sources of decoherence. This general idea defines the Loschmidt echo (LE), which embodies the various time-reversal procedures implemented in nuclear magnetic resonance. Here, we present an invitation to the study of the LE following the pathway induced by the experiments. With such a purpose, we provide a historical and conceptual overview that briefly revisits selected phenomena that underlie the LE dynamics including chaos, decoherence, localization and equilibration. This guiding thread ultimately leads us to the discussion of decoherence and irreversibility as an emergent phenomenon. In addition, we introduce the LE formalism by means of spin-spin correlation functions in a manner suitable for presentation in a broad scope physics journal. Last, but not least, we present new results that could trigger new experiments and theoretical ideas. In particular, we propose to transform an initially localized excitation into a more complex initial state, enabling a dynamically prepared LE. This induces a global definition of the LE in terms of the raw overlap between many-body wave functions. Our results show that as the complexity of the prepared state increases, it becomes more fragile towards small perturbations.
The problem of reconciling the irreversibility of thermodynamics with the completely reversible mechanics of the ultimate constituents of the thermodynamical system is examined from an operational viewpoint. The informational nature of entropy is demonstrated, and the famous paradoxes of statistical mechanics, due to Loschmidt and Zermelo, are resolved with its aid. Spin echo experiments are shown to realize the conditions of Loschmidt's reflection paradox, and used to illustrate how reversibility occurs only with perfect “memory” or information storage, while “forgetting” or loss of information implies irreversibility.
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.
We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the phenomenon of relaxation to equilibrium. When the number of spins becomes very large, the magnetization satisfies an autonomous equation as a function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical ensemble. This is proved as a law of large numbers with respect to a class of initial data. The corresponding Gibbs–von Neumann entropy is also computed and its monotonicity in time discussed.
An approximate numerical method is proposed and discussed for solving the evolution of the spin density operator when the quantum system has an interaction with an external electromagnetic field.In this method by separating the relaxation and field interaction processes at small steps, instead of solving the conventional Liouville-von Neumann or Bloch differential equations, the time evolution of the density operator is efficiently obtained by a two-stage numerical algorithm.Here we have compared the results of this approach with Bloch equation results for a twolevel quantum system.The proposed approach has potential applications in calculation of the time evolution for different atomic system including nuclear or electron spin resonance systems.