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Journal of Physics A Mathematical and General

Quantum evolution near unstable equilibrium point: an algebraic approach

Zai-Qiao Bai, Wei‐Mou Zheng

Published Mar 6, 2003• 0 citations
Publisher

Abstract

We study the quantum evolution of an unstable system in su(1, 1) algebra: The evolution of any initial state \k, v> is recursively obtained. When t --> infinity, \\(2) decays as e(-4vt) or t(-4v) in the hyperbolic (H = 2K(1)) or parabolic (H = 2K(1) + 2K(3)) unstable cases, respectively. The quantum-classic correspondence independent of the Bargmann index v is established based on the long-time and large-scale behaviour of wavefunctions.

DOI
10.1088/0305-4470/36/11/306

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We study the quantum evolution of an unstable system in su(1, 1) algebra: The evolution of any initial state \k, v> is recursively obtained. When t --> infinity, \(2) decays as e(-4vt) or t(-4v) in the hyperbolic (H = 2K(1)) or parabolic (H = 2K(1) + 2K(3)) unstable cases, respectively. The quantum-classic correspondence independent of the Bargmann index v is established based on the long-time and large-scale behaviour of wavefunctions.
We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is nonunitary. This allows to describe observables that are initially noncommutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples.
The authors contrast the classical and quantum dynamics of SU(1,1) coherent states by the use of a positive-definite quasiprobability distribution, a Q-function, defined over these states. For Hamiltonians that are linear in the SU(1,1) generators, therefore coherence preserving, the quantum and classical Liouville equations are identical. This is illustrated for a degenerate parametric amplifier. For an anharmonic oscillator, the quantum Liouville equation contains additional terms containing second-order derivatives with non-positive-definite coefficients. These terms give rise to the quantum recurrences not seen in the classical evolution of this system. It is pointed out that the quantum equation for the Q-function goes over to the classical equation as the Bargmann index, k, becomes large, in agreement with previous semiclassical considerations.
The time evolution of an unstable quantum system in terms of the dynamical semigroup is described. Using the canonical decomposition theorem, the evolution semigroup is uniquely split into unitary and completely nonunitary parts. This feature provides a natural superselection principle. An association is then made of the evolution of the steady-state density matrix with the unitary part and that of the decaying-state density matrix with the completely nonunitary part. Thus the reversible dynamics goes with the unitary evolution and the irreversible one with the completely nonunitary evolution. A discussion of the von Neumann entropy of this system is also presented. It is shown that for long times, this entropy settles to a constant value different from the initial entropy. A simple example of an exponentially decaying state is discussed.
The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near t=0t=0t=0 are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters T1{T}_{1}T1​ and T2{T}_{2}T2​ separating the three domains. The parameter T1{T}_{1}T1​ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while T2{T}_{2}T2​ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.
The evolution of a three-level quantum system interacting with the environment with the symmetry of the Lie algebra su(3) (qutrit) is considered. The evolution of qutrit is obtained when interacting with the environment. The transverse relaxation of the qutrit (due to the diagonal Lindblad operators) has been studied, leading to zero (with the same speedfor the real and imaginary parts) off-diagonal elements of the density matrix. The dynamics of qutrit is obtained in the presence of both transverse and longitudinal relaxation, i.e. when the Lindblad operators are off-diagonal and there are transitions in the system between all energy levels. It is shown that the addition of the longitudinal relaxation ofthe qutrit in addition to the conversion of the offdiagonallements to zero, also affects the diagonal elements of its density matrix. As a result of dissipation, the quantum system becomes incoherent over time. In both cases, initially non-commuting quantum observables in the process of evolution acquire the properties of commuting classical observables,which algebraically manifests itself in the contraction of the algebra su(3).
We studied the non-equilibrium quench dynamics from free to hard-core 1D bosons in the presence of a hard-wall confining potential. The density profile and the two-point fermionic correlation function in the stationary state as well as their full time evolution was characterised. It was found that for long times the system relaxes to a uniform density profile, but the correlation function memorises the initial state with a stationary algebraic long-distance decay, which is opposite to the exponential behaviour found for the same quench in the periodic setup. We also compute the stationary bosonic two-point correlator which was found to decay exponentially for large distances. A two-step mechanism was shown to govern the time evolution; a quick approach to an almost stationary value was followed by a slow algebraic relaxation to the true stationary state.