Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Quantum Electrodynamics of the Stimulated Emission of Radiation. II

R.M. Bevensee

Published Dec 1, 1965• 0 citations
Publisher

Abstract

The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.

DOI
10.1063/1.1704734

Related papers

Quantum Electrodynamics of the Stimulated Emission of Radiation. I

R.M. Bevensee

The quantum electrodynamic formalism of Senitzky is developed so as to yield long-time differential equations for the time-average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two-level molecules, and a dissipation mechanism. The cavity-mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T1 and T2 time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules. Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time-average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off-resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well-behaved differential equations are obtained for the time-average expectation values of field energy and molecular excess population, justified on a long-time basis by the application of an intermittent similarity transformation. A differential equation for the time-average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct-product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum-entropy inference. In conclusion, the equations of motion for the time-average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid-state laser beam during the emission period.

Journal of Mathematical Physics•Mar 1, 1964•1 citation

Dissipation in Quantum Mechanics. The Harmonic Oscillator

I. R. Senitzky

The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.

Physical Review•Jul 15, 1960•466 citations

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.

Journal of Mathematical Physics•Sep 1, 1964•26 citations

Markovian Dynamics of Josephson Parametric Amplification

Waldemar Kaiser, Michael Haider, Johannes A. Russer +2 more

Abstract. In this work, we derive the dynamics of the lossy DC pumped non-degenerate Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the superconducting Josephson junction. The DC bias generates the AC Josephson current varying the nonlinear inductance of the junction. By this way the Josephson junction acts as the pump oscillator as well as the time varying reactance of the parametric amplifier. In quantum-limited amplification, losses and noise have an increased impact on the characteristics of an amplifier. We outline the classical model of the lossy DCPJPA and derive the available noise power spectral densities. A classical treatment is not capable of including properties like spontaneous emission which is mandatory in case of amplification at the quantum limit. Thus, we derive a quantum mechanical model of the lossy DCPJPA. Thermal losses are modeled by the quantum Langevin approach, by coupling the quantized system to a photon heat bath in thermodynamic equilibrium. The mode occupation in the bath follows the Bose-Einstein statistics. Based on the second quantization formalism, we derive the Heisenberg equations of motion of both resonator modes. We assume the dynamics of the system to follow the Markovian approximation, i.e. the system only depends on its actual state and is memory-free. We explicitly compute the time evolution of the contributions to the signal mode energy and give numeric examples based on different damping and coupling constants. Our analytic results show, that this model is capable of including thermal noise into the description of the DC pumped non-degenerate Josephson parametric amplifier.

Advances in radio science•Sep 21, 2017•2 citations

Induced and Spontaneous Emission in a Coherent Field. II

I. R. Senitzky

The interaction between the electromagnetic field and a number of identical atomic systems, individually characterized by an electric dipole moment and two energy levels, is analyzed for the case where the atomic systems are inside a lossy cavity and exposed to a coherent driving field, resonance being assumed between atomic system, cavity, and driving field.The problem of introducing loss into a quantum-mechanical formalism is treated first. Formal operator expressions are obtained for the field variables which include the absorption and the fluctuation (both thermal and quantum-mechanical) effects of the loss mechanism. Expectation values are then obtained for the field strength and the field energy which are valid for times short compared to the lifetime of the excited state. It is shown that the spontaneous-emission energy in the field increases initially as the square of the time and approaches a steady-state value after a transient period which is of the order of the cavity relaxation time. The induced emission contains two parts: incoherent emission induced by the thermal field, and coherent emission induced by the driving field. The incoherent induced-emission energy has the same time dependence as the spontaneous-emission energy, and the ratio of the former to the latter is the number of photons in the thermal (inducing) field. The coherent induced-emission energy does not approach a steady-state value, but, after a transient period, increases linearly with the time. The ratio of the coherent induced-emission energy to the spontaneous-emission energy is equal, initially, to the number of photons in the driving-field energy, but becomes $\ensuremath{\beta}t$ times as large after the transient period, where $\ensuremath{\beta}$ is the reciprocal of the cavity relaxation time.The expectation value of the rate of energy emission by the atomic systems is also obtained. It is shown that the ratio of the downward to the upward transition probabilities has the well-known value of $\frac{(n+1)}{n}$, where $n$ is the field energy in units of the photon energy, only in the absence of a coherent field.

Physical Review•Jul 15, 1959•45 citations

Dissipation in Quantum Mechanics. The Two-Level System

I. R. Senitzky

The study of a two-level system (TLS) coupled to a loss mechanism (LM), the fluctuations of which are taken into account, is motivated by relating the problem to spin-lattice relaxation, radiation damping, and Brownian motion. A TLS of the electric dipole type (coupled to the LM through one variable only) is discussed first. The problem is formulated in terms of the Pauli spin matrices, and the Langevin equations for Brownian motion of a TLS are derived. Their solution---made possible by taking expectation values in LM space---contains the Weisskopf-Wigner exponential decay formula of radiation theory, as well as a second-order shift in frequency of the TLS produced by the (nondispersive) LM. A driving force is added to the problem and expressions are obtained for a driven lossy TLS of the electric dipole type. The same analysis is applied to a TLS of the magnetic dipole type and differential equations for the spin matrices---for which the expectation values in LM space has been taken---are obtained. Taking expectation values in TLS space converts these equations into the Bloch equations. Unlike the electric dipole case, no frequency shift is present.

Physical Review•Sep 15, 1963•38 citations
The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.
The quantum electrodynamic formalism of Senitzky is developed so as to yield long-time differential equations for the time-average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two-level molecules, and a dissipation mechanism. The cavity-mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T1 and T2 time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules. Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time-average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off-resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well-behaved differential equations are obtained for the time-average expectation values of field energy and molecular excess population, justified on a long-time basis by the application of an intermittent similarity transformation. A differential equation for the time-average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct-product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum-entropy inference. In conclusion, the equations of motion for the time-average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid-state laser beam during the emission period.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.
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. In this work, we derive the dynamics of the lossy DC pumped non-degenerate Josephson parametric amplifier (DCPJPA). The main element in a DCPJPA is the superconducting Josephson junction. The DC bias generates the AC Josephson current varying the nonlinear inductance of the junction. By this way the Josephson junction acts as the pump oscillator as well as the time varying reactance of the parametric amplifier. In quantum-limited amplification, losses and noise have an increased impact on the characteristics of an amplifier. We outline the classical model of the lossy DCPJPA and derive the available noise power spectral densities. A classical treatment is not capable of including properties like spontaneous emission which is mandatory in case of amplification at the quantum limit. Thus, we derive a quantum mechanical model of the lossy DCPJPA. Thermal losses are modeled by the quantum Langevin approach, by coupling the quantized system to a photon heat bath in thermodynamic equilibrium. The mode occupation in the bath follows the Bose-Einstein statistics. Based on the second quantization formalism, we derive the Heisenberg equations of motion of both resonator modes. We assume the dynamics of the system to follow the Markovian approximation, i.e. the system only depends on its actual state and is memory-free. We explicitly compute the time evolution of the contributions to the signal mode energy and give numeric examples based on different damping and coupling constants. Our analytic results show, that this model is capable of including thermal noise into the description of the DC pumped non-degenerate Josephson parametric amplifier.
The interaction between the electromagnetic field and a number of identical atomic systems, individually characterized by an electric dipole moment and two energy levels, is analyzed for the case where the atomic systems are inside a lossy cavity and exposed to a coherent driving field, resonance being assumed between atomic system, cavity, and driving field.The problem of introducing loss into a quantum-mechanical formalism is treated first. Formal operator expressions are obtained for the field variables which include the absorption and the fluctuation (both thermal and quantum-mechanical) effects of the loss mechanism. Expectation values are then obtained for the field strength and the field energy which are valid for times short compared to the lifetime of the excited state. It is shown that the spontaneous-emission energy in the field increases initially as the square of the time and approaches a steady-state value after a transient period which is of the order of the cavity relaxation time. The induced emission contains two parts: incoherent emission induced by the thermal field, and coherent emission induced by the driving field. The incoherent induced-emission energy has the same time dependence as the spontaneous-emission energy, and the ratio of the former to the latter is the number of photons in the thermal (inducing) field. The coherent induced-emission energy does not approach a steady-state value, but, after a transient period, increases linearly with the time. The ratio of the coherent induced-emission energy to the spontaneous-emission energy is equal, initially, to the number of photons in the driving-field energy, but becomes $\ensuremath{\beta}t$ times as large after the transient period, where $\ensuremath{\beta}$ is the reciprocal of the cavity relaxation time.The expectation value of the rate of energy emission by the atomic systems is also obtained. It is shown that the ratio of the downward to the upward transition probabilities has the well-known value of (n+1)n\frac{(n+1)}{n}n(n+1)​, where nnn is the field energy in units of the photon energy, only in the absence of a coherent field.
The study of a two-level system (TLS) coupled to a loss mechanism (LM), the fluctuations of which are taken into account, is motivated by relating the problem to spin-lattice relaxation, radiation damping, and Brownian motion. A TLS of the electric dipole type (coupled to the LM through one variable only) is discussed first. The problem is formulated in terms of the Pauli spin matrices, and the Langevin equations for Brownian motion of a TLS are derived. Their solution---made possible by taking expectation values in LM space---contains the Weisskopf-Wigner exponential decay formula of radiation theory, as well as a second-order shift in frequency of the TLS produced by the (nondispersive) LM. A driving force is added to the problem and expressions are obtained for a driven lossy TLS of the electric dipole type. The same analysis is applied to a TLS of the magnetic dipole type and differential equations for the spin matrices---for which the expectation values in LM space has been taken---are obtained. Taking expectation values in TLS space converts these equations into the Bloch equations. Unlike the electric dipole case, no frequency shift is present.