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It is found that displacement and squeezing of arbitrary initial states of the field can be produced in the model of a lossless cavity when a large number of two-level atoms is injected, which are prepared in a superposition state coupled to the single mode via a multi-photon transition under the weak atom-field interaction. The dependence of the field density matrix elements on the number of injected atoms indicates that due to the same initial atomic coherence the emission of individual atoms is a co-operative process.

The phase properties of the cavity field of one- as well as two-photon JCM in a Kerr medium are studied by means of a Pegg - Barnett Hermitian phase operator formalism. The time evolution of the phase probability distribution, the phase fluctuations and the number - phase correlation are obtained. The effect of nonlinear interaction of a Kerr-like medium on the phase properties is analysed by comparing our results with those of usual JCMs.

Phase properties of the micromaser field are studied for different initial states of the cavity field and the atom. It is shown that a multipeak phase structure develops when the cavity field is initially in a coherent state and the atoms enter the cavity in their excited state, which eventually leads to the randomization of the phase. If initially the field is thermal and the atoms entering the cavity are polarized, the steady state with a considerably well defined phase is reached, which asymptotically can be one of the cotangent states or some mixed state depending on the atom-cavity interaction time. An interesting effect of switching between the two- and three-peak phase structure caused by each subsequent atom passing the cavity has been found.

Phase properties of binomial and negative binomial states are studied within the Pegg-Barnett hermitian phase formalism.

We show that macroscopic superposition (Schrodinger cat) states of a quantized single-mode cavity field can be produced via the interaction of this field with a two-level atom which is driven by a classical field even for small initial intensities of the quantized cavity mode. We show that with a properly chosen driving field an almost pure superposition state with arbitrary amplitudes and phases of component states can be produced.

We analyze the phase properties of a strong cavity field interacting with an ensemble of initially excited N two-level atoms. Using the Pegg-Barnett phase formalism [Phys. Rev. A 39, 1665 (1989)], we calculate the phase probability distribution as well as the phase variance. The phase probability density exhibits a (N+1)-peak structure at the initial stages of the evolution. The phase variance is used to illustrate the progressive randomization of the phase on the long-time evolution. The difference in the phase dynamics for the N even and the N odd case is pointed out.

Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P(W)( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P(W)( theta ) in the limit of highly excited states.

We study phase properties of quantum mechanical superposition states decaying into phase-sensitive reservoirs. We show that the Wigner phase quasiprobability distribution reflects the quantum interference effects very well while the Pegg–Barnett and the Husimi phase probability distributions are quite insensitive with respect to the character of the quantum interference between component states. We study in detail phase properties of the even coherent state which decays into the thermal and the squeezed reservoirs. In thermal reservoirs the phase of the superposition state becomes randomized under the influence of thermal fluctuations. The higher the number of thermal photons the faster the randomization process is. On the other hand we show that depending on the relative phase between the superposition state and the phase-sensitive reservoir the process of phase randomization can be either enhanced or completely suppressed.

The phase distribution obtained within the Pegg–Barnett Hermitian phase formalism is compared to the phase distributions obtained from the s-parametrized quasiprobability distributions integrated over the "radial" variable for some real states of the field. Exact analytical formulas for the s-parametrized phase distributions of coherent states, squeezed states, and displaced number states are obtained. A general formula relating the s-parametrized phase distributions to the Pegg–Barnett distribution is derived. Numerical examples illustrating the similarities and differences are presented in a graphical form.

The long-time behaviour of the phase quantum fluctuations in the field produced by the parametric down-conversion with a quantum pump is studied. It is shown that if the pump is initially in a coherent state the phase variances for both the signal and the pump modes show collapses and revivals in their long-time evolution.

A comparison is made of predictions for the phase variances and the phase distribution functions obtained from the Garrison-Wong and Pegg-Barnett formalisms for real field states that include number states, coherent states, and squeezed vacuum states. It is shown that both approaches lead to qualitatively different phase distributions. The Garrison-Wong approach predicts an anisotropy of the phase distribution that is inconsistent with the symmetry of the Wigner and Q functions

The photon number and phase quantum fluctuations in the field produced by the down-conversion process with a quantum pump are studied. The fully quantum approach using the method of numerical diagonalization of the interaction Hamiltonian is applied to find the evolution of the system. The evolution of the photon number fluctuations, the joint number of photons probability distribution, the quadrature variances, the joint phase probability distribution, the marginal number and phase distributions for the signal mode, the number and phase uncertainty products and squeezing parameters are calculated and illustrated graphically. The results for the signal mode are compared to the corresponding results for the ideal squeezed vacuum to show the range of validity of the parametric approximation.

Exact analytical expressions describing the evolution of the expectation values and the variances of the Stokes operators are derived for the elliptically polarized light propagating in a Kerr medium with dissipation. It is shown that quantum fluctuations of the field essentially affect the polarization of the field. The explicit quantum formulas describing the azimuth and the ellipticity of the polarization ellipse as well as the degree of polarization are derived and illustrated graphically. Quantum fluctuations in the Stokes parameters are discussed, and the evolution of the signal-to-noise ratio is shown to be reduced by the quantum field fluctuations. Role of the dissipation is shown explicitly in a fully quantitative way from the exact analytical solutions.

Quantum theory of propagation of elliptically polarized light in a nonlinear Kerr medium with dissipation is applied to describe changes in the polarization state of the field. Exact analytical formulas describing the degree of polarization and the parameters of the polarization ellipse during the evolution are derived. A number of purely quantum effects that arise during the propagation are discussed. It is shown that the Stokes parameters of light propagating in a Kerr medium can be considered as a good measure of the mean value of the phase-difference cosine or sine, and this relation may be treated as an operational way of measuring such phase quantities.

Phase properties of the displaced number states are studied. Exact analytical formulae describing phase distributions based on different phase approaches are obtained and illustrated graphically. It is shown that the Pegg-Barnett phase distribution P( theta ) and the phase distribution Pw( theta ) associated with the Wigner function are very close to each other, while the phase distribution PQ( theta ) associated with the Q function carries less phase information. The results have clear interpretation in terms of the area of overlap in phase space.

Photon number and phase fluctuations and correlations in the second harmonic generation are discussed. The new Pegg-Barnett Hermitian phase formalism is used to deal with the phase properties of the field. The method of numerical diagonalization of the interaction Hamiltonian is applied to find the state evolution and, consequently, the number and phase quantum fluctuations. It is shown that the joint phase probability distribution evolves into a multi-peak structure indicating clearly the transition from the second harmonic to the down-conversion regime, and back. The evolution of all relevant quantities is illustrated graphically, and their dependence on the mean number of photons of the initial field is shown explicitly. The number-phase uncertainty product for the fundamental mode is plotted making evident the abrupt transition from the low to the high level of quantum fluctuations.

Phase properties of elliptically polarized light propagating through a nonlinear Kerr medium are considered within the framework of the Pegg-Barnett Hermitian phase formalism. The joint phase proba...

Phase properties of a damped anharmonic oscillator are studied within the Hermitian phase formalism of Pegg and Barnett. Exact analytical formulas for the phase distribution function, the expectation value and variance of the phase operator, and the expectation value and variance of the cosine of the phase operator are obtained assuming the zero-temperature reservoir and a coherent initial state of the system. It is shown that quantum periodicity in the evolution of phase quantities is destroyed by damping. The effect of damping on the formation of discrete superpositions of coherent states is discussed. A comparison is made between different phase approaches.

The phase properties of the fractional coherent states are discussed from the point of view of the Pegg-Barnett Hermitian phase formalism. Exact analytical formulas for the phase variance are obtained and illustrated graphically. The results can serve as a test for the range of validity of the scaling law for the phase variance.

The phase properties of the self-squeezed states generated during the evolution of the anharmonic oscillator are discussed from the point of view of the new phase formalism of Pegg and Barnett. The phase distribution, the expectation values and the variances of the Hermitian phase operator are obtained and their evolution illustrated graphically. The mean values for the phase cosine and sine functions as well as their variances are also calculated. The results are compared to the Susskind-Glogower formalism results and the results based on the measured phase concept. The relation between squeezing and the phase properties of the field is discussed briefly.

The joint probability distribution P (θ a , θ b ) for the phases θ a and θ b of the fundamental and second-harmonic modes is calculated, according to the Pegg-Barnett hermitian phase formalism, for different initial states of the fundamental mode undergoing second harmonic generation. The evolution of P (θ a , θ b ) is shown for four different initial states of the field (coherent, squeezed vacuum, Fock, and chaotic). Phase properties of the resulting fields are shortly discussed.

Quantum phase fluctuations in the second-harmonic generation are examined from the point of view of the Hermitian phase formalism introduced by Pegg and Barnett. The joint probability distribution as well as the variances for the phases of the fundamental and second-harmonic modes are calculated numerically. Their evolution is illustrated graphically, and the phase properties of the generated light are discussed.

Phase properties of pair coherent states are re-examined from the point of view of the Pegg-Barnett hermitian phase formalism. The joint probability distribution for the phases of the two modes is calculated and illustrated graphically. Strong correlation between the two phases is shown to exist, and the correlation coefficient calculated. The variance of the cosine of the phase-sum- operator is calculated. The results are compared to the earlier results of Agarwal.

The effect of quantum fluctuations of the pump on the quantum phase properties of the signal mode in the parametric down-conversion process is considered. The Pegg-Barnett hermitian phase formalism is used to calculate the joint phase probability distribution and the phase variances of the two modes. The time evolution of the field state is obtained by means of numerical diagonalization of the interaction hamiltonian. The results are illustrated graphically and compared to those for the ideal squeezed state.

The problem of the formation of discrete superpositions of coherent states when elliptically polarized light is propagating through a nonlinear Kerr medium is considered, It is shown that superpositions with any number of components can be obtained if the evolution time is taken as a fraction MIN of the period, where M and N are mutually prime integers. Exact analytical formulae for finding the superposition coeffi- cients are given. It is shown that the coupling between the two circularly polarized components of the ellpitically polarized light caused by the asymmetry of the nonlinear properties of the medium can suppress the number of components in the superposition from N' to N, if the asymmetry parameter takes appropriate values. The phase distri- bution function P(.9,, e.) for the two-mode field is obtained according to the new Pegg-Barnett phase formalism. This function exhibits a well resolved, multi-peak struc- ture, clearly indicating the formation of the discrete superpositions of coherent states. Examples of the phase distribution function for several superposition states are illustrated graphiczlly. showing in a very spectacular way the formation of such superpositions.

The phase properties of the two-mode squeezed vacuum states are re-examined from the point of view of the Hermitian phase formalism introduced by Pegg and Barnett. The joint probability distribution for the phases of the two models is obtained, and the phase properties associated with this distribution are discussed thoroughly.

In one-atom-maser (OAM) experiments [1] the atom enters a high-Q microwave resonator in the upper one of the two Rydberg states of the maser transition, interacts for a certain time with a single resonant mode of the quantized electromagnetic field, and is them probed for its final states. These final states, |A〉 or |B〉, can be either pure Rydberg states themselves or their coherent superpositions depending on the specific experimental setup. Direct measurements of the properties of the photon state are virtually impossible, but the population statistics of the emerging atom — or rather corresponding detector clicks — can be determined experimentally. In this paper we present an analytical method for calculating the mean number of successive detector clicks of the same kind.

Publisher Summary This chapter discusses the quantum phase properties of nonlinear optical phenomena. Optical fields produced in nonlinear optical processes have specific phase properties, which depend on the nonlinear process in which the field is produced, and on the state of the field before it undergoes the nonlinear transformation. The phase properties of a two-mode field are constructed from the single-mode phases. The chapter focuses on the real fields that can be generated in practice in various nonlinear optical processes. The phase properties can be described based on two different formalisms that include the Pegg–Barnett Hermitian phase formalism and the formalism based on s-parametrized phase distributions. Using the examples of real field, the chapter discusses the similarities and differences that are encountered when various phase distributions are applied to describe a particular field state.

We study the phase properties of quantum superpositions of two coherent states (Schr\"odinger-cat states) amplified by phase-sensitive (squeezed) amplifiers. We show that a phase-sensitive amplifier with a properly chosen phase can preserve the phase distribution of the Schr\"odinger-cat-state input.