The idea behind this work is the supposition that it should be possible to introduce damping into the quantum mechanical discussion of a harmonic oscillator without a detailed description of the damping mechanism. A study is therefore made of an L, C circuit coupled to an artificial (or transmission) line, using a Hamiltonian formalism. The whole system is conservative but, for the transmission line case, the L, C circuit behaves as if it is damped. The most probable form for the Hamiltonian does not exhibit the damping in an obvious way. It is shown that by a suitable change of variables the Hamiltonian can be split into two time-dependent commuting parts, one of which has the form e -2α t ( P 2 /2 C ) + e 2α t ( Q 2 /2 L ) which has been previously used in a study of 2-level masers, and which does display the damping. The new operators Q and P are studied and it is shown that they are related to the charges and currents in the L, C circuit, but with the noise components due to the resistive loss (the transmission line) omitted. They are thus perhaps more natural variables to use than the actual charges and currents which do contain noise components. The problem of coupling a spin system to the current in the inductance is studied and the transition is made to the analogous problem of a spin system interacting with a damped mode of a cavity resonator. The analysis is compared with one given previously by Stevens and Josephson in 1959 and it is shown that the results previously obtained in a heuristic way are valid provided that noise fluctuations are negligible.
The idea behind this work is the supposition that it should be possible to introduce damping into the quantum mechanical discussion of a harmonic oscillator without a detailed description of the damping mechanism. A study is therefore made of an L, C circuit coupled to an artificial (or transmission) line, using a Hamiltonian formalism. The whole system is conservative but, for the transmission line case, the L, C circuit behaves as if it is damped. The most probable form for the Hamiltonian does not exhibit the damping in an obvious way. It is shown that by a suitable change of variables the Hamiltonian can be split into two time-dependent commuting parts, one of which has the form e -2α t ( P 2 /2 C ) + e 2α t ( Q 2 /2 L ) which has been previously used in a study of 2-level masers, and which does display the damping. The new operators Q and P are studied and it is shown that they are related to the charges and currents in the L, C circuit, but with the noise components due to the resistive loss (the transmission line) omitted. They are thus perhaps more natural variables to use than the actual charges and currents which do contain noise components. The problem of coupling a spin system to the current in the inductance is studied and the transition is made to the analogous problem of a spin system interacting with a damped mode of a cavity resonator. The analysis is compared with one given previously by Stevens and Josephson in 1959 and it is shown that the results previously obtained in a heuristic way are valid provided that noise fluctuations are negligible.
The classical problem of a damped harmonic motion is formulated wave mechanically, with particular reference to the charge in a tuned circuit, and the electromagnetic field in a resonant cavity. Results in close agreement with the classical problem are obtained, without using perturbation theory. The classical frequency appears in the time dependence of the eigenfunctions and the classical damping appears as a decay in the eigenvalues. Zero point energy and thermal noise are discussed. The formulation is expected to break down, at very high frequencies, for tuned circuits and it is suggested that the same may be true for cavity oscillations and travelling electromagnetic waves.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.
We study the radiation damping and the spin noise of a magnetic material coupled with a resonant circuit.Radiation damping in magnetic resonance is a dissipation phenomenon, where magnetization prepared after a Rabi pulse decays toward its equilibrium state.The magnetic sample loses its energy by the coupling with resonant circuit, that must be tuned in Larmor frequency of the sample spins.Even though this phenomenon had been studied many years ago, no full quantum description was done.We present a quantum Hamiltonian model, that explains the radiation damping.We use quantum Langevin equation method for this task.Beyond radiation damping, we show the magnetization acquires an unusual intrincate motion, if the circuit initial state is coherent.Using the same Langevin equation, we study the sample inuence on the resonant circuit noise.We calculate the current spectral density in the case of thermal equilibrium of whole system.We can verify the method eectiveness, comparing former papers.Moreover we study modications in the circuit noise, if an external oscillating tension is applied.In this situation, other two peaks emerge in the central peak sidebands of the sample absorption spectrum.It leads to appear three dips in circuit current spectrum.This eect is due to the splitting of the spin energy states.We comment about the analogy between this phenomenon and the resonance uorescence in Quantum Optics.
In the previous chapter, we developed the equation of motion for a system as it evolved under the influence of an unobserved (reservoir) system. We used the density matrix approach and worked in the interaction picture. In this chapter, we consider the same problem of the system-reservoir interaction using a quantum operator approach. We again eliminate the reservoir variables. The resulting equations for the system operators include, in addition to the damping terms, the noise operators which produce fluctuations. These equations have the form of classical Langevin equations, which describe, for example, the Brownian motion of a particle suspended in a liquid. The Heisenberg–Langevin approach discussed in this chapter is particularly suitable for the calculation of two-time correlation functions of the system operator as is, for example, required for the determination of the natural linewidth of a laser. We first consider the damping of the harmonic oscillator by an interaction with a reservoir consisting of many other simple harmonic oscillators. This system describes, for example, the damping of a single-mode field inside a cavity with lossy mirrors. The reservoir, in this case, consists of a large number of phonon-like modes in the mirrors. We also consider the decay of the field due to its interaction with an atomic reservoir. An interesting application of the theory of the system–reservoir interaction is the evolution of an atom inside a damped cavity. It is shown that the spontaneous transition rate of the atom can be substantially enhanced if it is placed in a resonant cavity.
The usual treatment of damping forces in quantum mechanics starts from the introduction of the explicitly time dependent Kanai Hamiltonian, which actually represents a variable mass particle, and the misinterpretation of this Hamiltonian as representing a particle of constant mass leads to certain physical difficulties. However the Hamiltonian can be modified so that it can be reinterpreted as describing a constant mass particle. Here we explicitly introduce the mass as a new dynamical variable, which allows us to write a linear, time independent Hamiltonian for the system, which can be solved by conventional methods. The damped harmonic oscillator and damped free particle are treated in detail, both for the Kanai Hamiltonian and for our case, and the solutions are compared. Our solution can be reduced to the Kanai one in appropriate circumstances, but in general it has a much greater versatility, as a result of which it can be more easily reinterpreted as describing a constant mass particle subject to a damping force, which reinterpretation is of course necessary if the method is to have practical applicability. We also show how such a reinterpretation can be carried out in detail by introducing a ’’dissipation variable’’, in terms of which one may avoid the concept of a variable mass altogether.
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.