Quantum theory of light propagation in a fluctuating laser-active medium

The basic equations are derived which describe the propagation of an electromagnetic field in a fluctuating laser-active medium. The well-known methods of Langevinequations and master-equation for a few discrete modes are generalized to meet also the case of a radiation field with continuous spectrum. The medium is described by two-level atoms which are embedded in a merely passive solid matrix and homogeneously distributed over space. They have an inversion which is kept constant by an externally applied pump. The atomic line may be homogeneously or inhomogeneously broadened. We obtain a complete set of partial differential equations for the field operators with damping terms and fluctuating forces homogeneously distributed over the material. The telegraph equation with a fluctuating force occurs as a special case. After the exact elimination of the atomic variables we obtain a nonlinear field equation for the radiation field alone. By means of a pseudo-Hamiltonian and by a simple one-dimensional example we show that in a certain sense there exists a close formal analogy between the present theory and the theory of an interacting Bose gas. The characteristic differences between the two theories are also discussed. We find, that there occurs a phase transition of the radiation field because above a certain threshold of the pump the photons condense into a single mode and establish an “offdiagonal-long-range order”. The amplitude fluctuations and the phase fluctuations, which restore the broken phase symmetry, are calculated in detail. A new condition for the occurrence of undamped spiking (pulse formation) for a continuum of modes is derived.     

Phase space theory of Bose–Einstein condensates and time-dependent modes

A phase space theory approach for treating dynamical behaviour of Bose–Einstein condensates applicable to situations such as interferometry with BEC in time-dependent double well potentials is presented. Time-dependent mode functions are used, chosen so that one, two,…highly occupied modes describe well the physics of interacting condensate bosons in time dependent potentials at well below the transition temperature. Time dependent mode annihilation, creation operators are represented by time dependent phase variables, but time independent total field annihilation, creation operators are represented by time independent field functions. Two situations are treated, one (mode theory) is where specific mode annihilation, creation operators and their related phase variables and distribution functions are dealt with, the other (field theory) is where only field creation, annihilation operators and their related field functions and distribution functionals are involved. The field theory treatment is more suitable when large boson numbers are involved. The paper focuses on the hybrid approach, where the modes are divided up between condensate (highly occupied) modes and non-condensate (sparsely occupied) modes. It is found that there are extra terms in the Ito stochastic equations both for the stochastic phases and stochastic fields, involving coupling coefficients defined via overlap integrals between mode functions and their time derivatives. For the hybrid approach both the Fokker–Planck and functional Fokker–Planck equations differ from those derived via the correspondence rules, the drift vectors are unchanged but the diffusion matrices contain additional terms involving the coupling coefficients.Results are also presented for the combined approach where all the modes are treated as one set. Here both the Fokker–Planck and functional Fokker–Planck equations are exactly the same as those derived via the correspondence rules. However, although the Ito stochastic field equations are also unchanged, the Ito equations for the stochastic phases contain an extra classical term involving the coupling coefficients.     

Inverse of Radiation Field Operators and the Generalized Jaynes-Cummings Model

We generalize the standard Jaynes-Cummings model (JCM) to a model Hamiltonian with the radiation field operators being the inverse of a harmonic oscillator's creation and annihilation operators. Some new commutative relations about the inverse operators are derived and the generalized JCM Hamiltonian's eigenstates are derived.     

Nuclear Creation and Annihilation Operators and Electromagnetic Multipole Fields

The frequency spectrum of electromagnetic radiation can be written as the Fourier transform of the first-order correlation function of the vector potential. If nuclei are coupled to the radiation field, the Heisenberg equations of motion of the field operators contain nuclear operators and vice versa. Under plausible assumptions the equations of motion for the nuclear operators can be integrated and hence, the equations of the field operators can be solved. The vector potential of the radiated field can then be expressed as a function of solely nuclear quantities. The first-order correlation function deduced from it contains only two-times and one-time averages of simple nuclear creation and annihilation operators. The theory can be used to explain homogeneous line broadening for long-lived nuclei submitted to small fluctuating interactions. This revised version was published online in July 2006 with corrections to the Cover Date.     

四模场中单原子系统的新算符求解方法

提出处理腔场与原子、腔场与腔场等系统的较为一般算符方法。基于此方法,通过构造四对时间依赖的产生和湮灭算符,简捷地求解四模腔场或四腔场与二能级原子非共振相互作用系统,得到其本征态、本征值和一般态矢。特别地,在四模场或四腔场和原子的初态分别为真空态和一般叠加态时,给出四场模平均光子数和原子布居数反转的时间演化。该新方法可应用于其它一些量子系统。     

Exact Solution for a Multiphoton Generalized Jaynes-Cummings Model with Inverse Operators

A new exactly solvable multiphoton generalized Jaynes-Cummings model is presented, whose Hamiltonian is related to the inverse of field mode creation and annihilation operators. Then we use supersymmetric unitary operators to diagonalize the Hamiltonian above and obtain their energy spectra and eigenstates. In addition, its pseudo-invariant eigen-operator is found as well, directly leading to the corresponding energy-level gap.     

Excitation of coherent correlated states in the Jaynes-Cummings model by external deterministic forces: II

In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t ₀, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t ₀.     

Holevo-Ordering and the Continuous-Time Limit for Open Floquet Dynamics

We consider an atomic beam reservoir as a source of quantum noise. The atoms are modelled as two-state systems and interact one-at-a-time with the system. The Floquet operators are described in terms of the Fermionic creation, annihilation and number operators associated with the two-state atom. In the limit where the time between interactions goes to zero and the interaction is suitably scaled, we show that we may obtain a causal (that is, adapted) quantum stochastic differential equation of Hudson—Parthasarathy type, driven by creation, annihilation and conservation processes. The effect of the Floquet operators in the continuous limit is exactly captured by the Holevo ordered form for the stochastic evolution     

High-order realisations of the para-Fermi algebra with parafield operators

Using second-order realisations of Lie algebras by means of creation and annihilation parafield operators the generators of the para-Fermi algebras are expressed as high-order polynomials of para-Bose or para-Fermi creation and annihilation operators.     

Quantum treatment of the time-dependent coupled oscillators under the action of a random force

In this communication we introduce the problem of time-dependent frequency converter under the action of external random force. We have assumed that the coupling parameter and the phase pump are explicitly time dependent. Using the equations of motion in the Heisenberg picture the dynamical operators are obtained, however, under a certain integrability condition. When the system is initially prepared in the even coherent states the squeezing phenomenon is discussed. The correlation function is also considered and it has been shown that the nonclassical properties are apparent and sensitive to any variation in the integrability parameter. Furthermore, the wave function in Schrödinger picture is calculated and used it to derive the wave function in the coherent states. The accurate definition of the creation and annihilation operators are also introduced and employed to diagonalize the Hamiltonian system.     

Energy transfer to an anharmonic diatomic system

We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.     

Normal forms for classical and boson systems

The group of Bogoliubov transformations of annihilation and creation operators is a subgroup of U(n,n) where n is the number of distinct pairs of annihilation and creation operators. Here, it is established that this subgroup of U(n,n) is isomorphic to Sp(2n,$\text{R}$), which appears in classical dynamics as the group of linear canonical transformations on a 2n-dimensional phase space. Well-known results in classical dynamics are then to used to deduce the full set of normal forms for Boson Hamiltonians. These are classified using a para-eigenvalue notation applicable to both classical and Bose field systems. A simple sufficient condition is given for the non-removability of pairs of creation operators. Explicit normal forms have not previously been given for Hamiltonians with this pathology, which may occur even when the corresponding classical Hamiltonian can be diagonalized.     

Theory of intensity and phase fluctuations of a homogeneously broadened laser

We extend our previous quantum mechanical nonlinear treatment of laser noise to the following problem: We consider a set of atoms each with three levels, which support laser action of one or several modes. The laser action can take place either between the upper or the lower two levels. The atomic line is assumed to be homogeneously broadened. The broadening can be caused by the decay into the nonlasing modes, by the pumping process, lattice vibrations and other, non specified sources. The fluctuations of the atomic variables (or operators) are taken into account in a quantum mechanically consistent way using results of previous papers byHaken andWeidlich as well asSchmid andRisken. The laser modes are coupled to the thermal resonator noise usingSenitzky's method. In the first part of the present paper, we treat quite generally multimode laser action. It is shown, that each light mode chooses a specificcollective atomic “mode” to interact with. We introduce a set of suitable collective atomic “modes”, which leads to a simplification of the equations of motion for theHeisenberg operators of the light field and the atomic operators. From the new equations we can eliminate all atomic operators. We are then left with a set of coupled nonlinear, integro-differential equations for the light field operators alone. These equations, which are completely exact and valid both for running and standing waves, represent a considerable simplification of the original problem. In the second part of this paper, these equations are specialized to single mode operation, which is studied above laser threshold. In the vicinity of the threshold the laser equation can be simplified to an operator-equation, whose classical analogue is vander-Pol's equation with a noisy driving force. With increasing inversion, the full equation must be treated, however. Using the method of our previous paper, we decompose the light amplitude into a phase-factor and a real amplitude, which is expanded around its stable value. We determine the Fourier-transform of the intensity correlation function and the total intensity of the fluctuating part of the amplitude. Somewhat above threshold this intensity drops down with the inverse of the photon output power,P, while the inherent relaxation frequency increases withP. The noise intensity stems in this region from the off-diagonal elements of the noise operators and not from the diagonal elements, which are responsible for the shot noise. This result is insofar remarkable, as a rate equation treatment would include only the latter ones. Under certain conditions the intensity fluctuations can show resonances with increasing output power,P. At high inversion the vacuum fluctuations of the light field are dominant, while the other noise sources give rise to contributions which vanish with the inverse of the output power. As a by-product our treatment yields the following formula for the linewidth (half width at half power) which is caused by phase fluctuations:

$$\Delta \nu = \frac{{\gamma _{3 2}^2 \kappa ^2 }}{{(\kappa + \gamma _{3 2} )^2 }}\frac{{\hbar \omega }}{P}\left( {\frac{1}{2}\frac{{(N_3 + N_2 )}}{{N_3 - N_2 }} + n_{Th} + \frac{1}{2}} \right)$$     

Second quantized scalar QED in homogeneous time-dependent electromagnetic fields

We formulate the second quantization of a charged scalar field in homogeneous, time-dependent electromagnetic fields, in which the Hamiltonian is an infinite system of decoupled, time-dependent oscillators for electric fields, but it is another infinite system of coupled, time-dependent oscillators for magnetic fields. We then employ the quantum invariant method to find various quantum states for the charged field. For time-dependent electric fields, a pair of quantum invariant operators for each oscillator with the given momentum plays the role of the time-dependent annihilation and the creation operators, constructs the exact quantum states, and gives the vacuum persistence amplitude as well as the pair-production rate. We also find the quantum invariants for the coupled oscillators for the charged field in time-dependent magnetic fields and advance a perturbation method when the magnetic fields change adiabatically. Finally, the quantum state and the pair production are discussed when a time-dependent electric field is present in parallel to the magnetic field.     

Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Many-electron exchange Hamiltonian in the theory of ferromagnetism

A Hamiltonian describing effects of many-electron interatomic exchange in magnetic crystals is derived by application of an appropriate unitary transformation to the many-electron, many-nucleus Hamiltonian. It is expressed in terms of atomic annihilation and creation operators and j-fold exchange matrix elements for 1 ? j ? N, where N is the number of electrons localized on each crystal site. Explicit formulas are derived for the case of not more than half-filled shells with Hund's rule factorization. An appropriate spin algebra is introduced and the coefficient of the bilinear (Heisenberg) and biquadratic interactions evaluated in terms of single and double-exchange matrix elements. The relative magnitudes of such matrix elements are roughly estimated using determinantal wave functions.     

Neutral and charged quommutators with and without symmetries

We present two sets of qualgebras involving operators which generalize creation and annihilation operators. These two groups of operators satisfy separately quommutation relations rather than commutation or anticommutation relations. The quommutators of the creation and annihilation operators generate new “neutral operators” which themselves are subjected to quommutation relations. Two solutions are presented. In the second one, some new symmetry relations are added to the system. In a certain sense these extra relations, rather than imposing new constraints on the parameters, increase their freedom.     

Mode transformations and entanglement relativity in bipartite Gaussian states

A proper choice of subsystems for a system of identical particles e.g., bosons, is provided by second-quantized modes, i.e., creation/annihilation operators. Here we investigate how the entanglement properties of bipartite Gaussian states of bosons change when modes are changed by means of unitary, number conserving, Bogoliubov transformations. This set of “virtual” bipartitions is then finite-dimensionally parametrized and one can quantitatively address relevant questions such as the determination of the minimal and maximal available entanglement. In particular, we show that in the class of bipartite Gaussian states there are states which remain separable for every possible modes redefinition, while do not exist states which remain entangled for every possible modes redefinition.