Nonequilibrium quantum dynamics of second order phase transitions

Abstract

We use the so-called Liouville-von Neumann (LvN) approach to study the nonequilibrium quantum dynamics of time-dependent second order phase transitions. The LvN approach is a canonical method that unifies the functional Schrödinger equation for the quantum evolution of pure states and the LvN equation for the quantum description of mixed states of either equilibrium or nonequilibrium. As nonequilibrium quantum mechanical systems we study a time-dependent harmonic and an anharmonic oscillator and find the exact Fock space and density operator for the harmonic oscillator and the nonperturbative Gaussian Fock space and density operator for the anharmonic oscillator. The density matrix and the coherent, thermal and coherentthermal states are found in terms of their classical solutions, for which the effective Hamiltonians and equations of motion are derived. The LvN approach is further extended to quantum fields undergoing time-dependent second order phase transitions. We study an exactly solvable model with a finite smooth quench and find the two-point correlation functions. Due to the * E-mail: sangkim@ks.kunsan.ac.kr; spkim@phys.ualberta.ca † On leave to Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 ‡ E-mail: chlee@hepth.hanyang.ac.kr 1 spinodal instability of long wavelength modes the two-point correlation functions lead to the t 1/4 -scaling relation during the quench and the Cahn-Allen scaling relation t 1/2 after the completion of quench. Further, after the finite quench the domain formation shows a time-lag behavior at the cubic power of quench period. Finally we study the time-dependent phase transition of a self-interacting scalar field.

Third Quantization and Quantum Universes

“…The quantization of the closed universe is more intriguing than the open or flat universe since the frequency for each oscillator changes the sign after e α 0 = √ p and thus becomes unstable. The inverted quantum oscillators occur in the slow-rolling inflation model [42] and in the second order phase transition [43]. Now the solutions for sub-Planckian size universes are…”

Second quantized scalar QED in homogeneous time-dependent electromagnetic fields

“…In the case of time-dependent electric fields, the Hamiltonian (8) is a system of decoupled, time-dependent oscillators, for which we may use a pair of invariants for each oscillator as the time-dependent annihilation and the creation operators, and construct the Fock space of number states [14][15][16][17][18]. From Eq.…”

“…It has long been known that an oscillator with time-dependent frequency and/or mass has a quantum invariant, known as the Lewis-Riesenfeld invariant, whose eigenstate provides an exact solution of the Schrödinger equation up to a time-dependent phase factor [13]. Hence, in the former case of electric fields, we may employ the time-dependent annihilation and the creation operators, also quantum invariants, and construct not only excited states for each time-dependent oscillator [14][15][16] but also a thermal state [17,18]. In the second case of magnetic fields, however, the Hamiltonian for the functional Schrödinger equation is equivalent to coupled, time-dependent oscillators, and hence, we should employ the invariants for coupled, time-dependent oscillators [19][20][21][22].The quantum invariant method for time-dependent oscillators is very useful in constructing various quantum states from the vacuum state, ranging from excited states to coherent states and even to thermal states (for review and references, see Refs.…”

“…By analogy with time-independent oscillators, the time-dependent annihilation and the creation operators for time-dependent oscillators, quantum invariants linear in momentum and position operators, may be used to find the Fock space of all the number states, which are the exact solutions for the time-dependent Schrödinger equation [14,17]. Furthermore, it can be readily used in constructing thermal states and coherent thermal states, which are also the exact quantum states [17,18]. In scalar quantum electrodynamics (QED) in time-dependent electric fields, the time-dependent vacuum state leads to the pair-production rate [25,26] and also to the in-out scattering matrix between the remote past and the remote future, which yields the renormalized one-loop effective action [5].…”

Interference in Phase Space of Squeezed States for the Time-Dependent Hamiltonian System

“…The system whose Hamiltonian explicitly depends on time has attracted considerable attention (Bandyopadhyay et al, 2001;Choi, 2004a;Dodonov and Man'ko, 1978;Kim and Lee, 2000;Nieto and Truax, 2000;Pedrosa and Guedes, 2002;Samaj, 2002;Song, 2000;Um et al, 2002;Wei et al, 2002) for several decades because of its applications to various branches in physics. The dynamical invariant operator method has been widely employed to find exact quantum states of TDHS after the report of simple relation between the solutions of the Schrödinger equation and the eigenstates of dynamical invariants for the timedependent harmonic oscillator by Lewis and Riesenfeld (1969).…”