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数理解析研究所講究録

A Calculable Model for a Cavity and an Atomic Beam (Applications of Renormalization Group Methods in Mathematical Sciences)

博志 田村

Published Jul 1, 2014• 0 citations
OpenAlex

Abstract

We consider a simple model for the physical system consists of a cavity and a beam of atoms which pass the cavity successively.The Hamiltonian contains time-dependent (piecewise constant) term de- scribing interaction between the cavity and the atom in the beam which is passing the cavity at the prescribed moment.We deal with the model in which the radiation field inside the cavity and each atoms of the beam is modeled by simple harmonic oscillators.We calculate the time evolution of the density matrix of the system and the asymptotic behavior of the cavity, both in the Hamiltonian dynamics and the Markovian dynamics of Kossakowski-Lindblad-Davies type.We also dis-

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A model of a cavity and a beam of harmonic atoms (Mathematical aspects of quantum fields and related topics)

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We consider a simplified mathematical model for the physical system consists of a cavity and a beam of atoms which pass the cavity successively.The Hamiltonian contains time-dependent (piecewise constant) term de- scribing interaction between the cavity and the atom in the beam which is passing the cavity at the prescribed moment.We suppose that the radiation

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A Model of Interacting Radiation and Matter

C. R. Willis

We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.

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Open Quantum Systems, Decoherence, Atom-Field Coupling

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Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity

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Quantum jump dynamics in cavity QED

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We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field ℰ and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state |ψ(t)〉 of the cavity field. For weak atom-field coupling, the evolution of |ψ(t)〉 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field ℰ. For ℰ=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector |ψ(t)〉〈ψ(t)| over all results of the measurements, is independent of ℰ and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large ℰ, an arbitrary initial field state |ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states |α,re2iφ〉, moving in the α-complex plane but with almost constant squeezing parameters r and φ. The values of r and φ are determined analytically. On the other hand, for ℰ=0, the dynamics transforms the initial state into Fock states |n〉 with fluctuating numbers of photons n, as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.

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Existence of the thermodynamic limit and asymptotic behavior of some irreversible quantum dynamical systems

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We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that the dynamics with a such generator on a finite system is a well-defined quantum dynamics in a sense of a norm-continuous cocycle of unit preserving completely positive maps. After proving Lieb-Robinson bounds for a finite lattice systems we also show the existence of the thermodynamic limit of the dynamics. We show that in a strong limit there exits a strongly continuous cocycle of unit preserving completely positive maps. Which means that the dynamics exists in an infinite system, where Lieb-Robinson bounds also holds. In the second part of the paper we consider a system that consists of a beam of two-level atoms that pass one by one through the microwave cavity. The atoms are randomly excited and there is exactly one atom present in the cavity at any given moment. We consider both the ideal and leaky cavity and study the time asymptotic behavior of the state of the cavity. We show that the number of photons increases indefinitely in the case of the ideal cavity. In case of leaky cavity we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the energy flux in both the ideal and leaky and the entropy production for the ideal cavity.

arXiv:1207.5763•Jul 24, 2012•0 citations
We consider a simple model for the physical system consists of a cavity and a beam of atoms which pass the cavity successively.The Hamiltonian contains time-dependent (piecewise constant) term de- scribing interaction between the cavity and the atom in the beam which is passing the cavity at the prescribed moment.We deal with the model in which the radiation field inside the cavity and each atoms of the beam is modeled by simple harmonic oscillators.We calculate the time evolution of the density matrix of the system and the asymptotic behavior of the cavity, both in the Hamiltonian dynamics and the Markovian dynamics of Kossakowski-Lindblad-Davies type.We also dis-
We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field ℰ and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state |ψ(t)〉 of the cavity field. For weak atom-field coupling, the evolution of |ψ(t)〉 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field ℰ. For ℰ=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector |ψ(t)〉〈ψ(t)| over all results of the measurements, is independent of ℰ and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large ℰ, an arbitrary initial field state |ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states |α,re2iφ〉, moving in the α-complex plane but with almost constant squeezing parameters r and φ. The values of r and φ are determined analytically. On the other hand, for ℰ=0, the dynamics transforms the initial state into Fock states |n〉 with fluctuating numbers of photons n, as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.
We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that the dynamics with a such generator on a finite system is a well-defined quantum dynamics in a sense of a norm-continuous cocycle of unit preserving completely positive maps. After proving Lieb-Robinson bounds for a finite lattice systems we also show the existence of the thermodynamic limit of the dynamics. We show that in a strong limit there exits a strongly continuous cocycle of unit preserving completely positive maps. Which means that the dynamics exists in an infinite system, where Lieb-Robinson bounds also holds. In the second part of the paper we consider a system that consists of a beam of two-level atoms that pass one by one through the microwave cavity. The atoms are randomly excited and there is exactly one atom present in the cavity at any given moment. We consider both the ideal and leaky cavity and study the time asymptotic behavior of the state of the cavity. We show that the number of photons increases indefinitely in the case of the ideal cavity. In case of leaky cavity we prove that the mean photon number in the cavity stabilizes in time. The limiting state of the cavity in this case exists and it is independent of the initial state. We calculate the characteristic functional of this non-quasi-free non-equilibrium state. We also calculate the energy flux in both the ideal and leaky and the entropy production for the ideal cavity.