Hermitian Phase Operator Research Articles

Generally deformed oscillator, isospectral oscillator system and Hermitian phase operator

The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q->1.

Phase properties of one- and two-photon Jaynes - Cummings models with a Kerr medium

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.

Even and odd phase coherent states for Hermitian phase operator theory

Even and odd phase coherent states associated with the Hermetian phase operator theory are introduced in terms of the creation operation of the phase quanta defined in a finite-dimensional phase state space. Some mathematical and physical properties of these quantum states are studied in some detail. It is shown that the even phase coherent states together with the odd ones build an overcomplete Hilbert space. Even and odd coherent-state formalism of the Pegg - Barnett phase operator is given in terms of the projection operator in the even and odd phase coherent-state space. The number - phase uncertainty relation is investigated for these quantum states. It is shown that even and odd phase coherent states are not minimum uncertainty and intelligent states for the number and phase operators.

Number-phase uncertainty product for generalized squeezed states arising from the Pegg-Barnett Hermitian phase operator formalism.

Number-phase uncertainty product for generalized squeezed states arising from the Pegg-Barnett Hermitian phase operator formalism.

Q-DEFORMED OSCILLATOR ALGEBRA AND AN INDEX THEOREM FOR THE PHOTON PHASE OPERATOR

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a viewpoint of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q=exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a–dim ker a†=1 which allows only a nonhermitian phase operator with dim ker eiφ–dim ker(eiφ)†=1. For q=exp(2πiθ) with a rational θ, one formally obtains the singular situation dim ker a=∞ and dim ker a†=∞, which allows a hermitian phase operator with dim ker eiΦ–dim ker(eiΦ)†=0 as well as the nonhermitian one with dim ker eiφ– dim ker(eiφ)†=1. Implications of this interpretation of the quantum deformation are discussed. We also show how to overcome the problem of negative norm for q=exp(2πiθ).

Phase operator for the photon field and an index theorem.

An index relation $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 1$ is satisfied by the creation and annihilation operators $a^{\dagger}$ and $a$ of a harmonic oscillator. A hermitian phase operator, which inevitably leads to $dim\ ker\ a^{\dagger}a - dim\ ker\ aa^{\dagger} = 0$, cannot be consistently defined. If one considers an $s+1$ dimensional truncated theory, a hermitian phase operator of Pegg and Barnett which carries a vanishing index can be defined. However, for arbitrarily large $s$, we show that the vanishing index of the hermitian phase operator of Pegg and Barnett causes a substantial deviation from minimum uncertainty in a characteristically quantum domain with small average photon numbers. We also mention an interesting analogy between the present problem and the chiral anomaly in gauge theory which is related to the Atiyah-Singer index theorem. It is suggested that the phase operator problem related to the above analytic index may be regarded as a new class of quantum anomaly. From an anomaly view point ,it is not surprising that the phase operator of Susskind and Glogower, which carries a unit index, leads to an anomalous identity and an anomalous commutator.

Phase Properties of a Strong Field Interacting with Many Atoms with and Without Initial Atomic Coherences

Abstract The phase distributions of an initially strong, coherent, single-mode field interacting with one, two, three and four identical atoms with and without initial atomic coherences using the Pegg-Barnett Hermitian phase operator formalism have been examined. A number of interesting features of the phase distributions are revealed. A link between the coincidences of the peaks in the phase probability distribution and the revivals in the two-atom case is also shown.

Q-PHASE OPERATOR IN A FINITE-DIMENSIONAL HILBERT SPACE

In this letter, the unitary and Hermitian phase operators for the single-mode electromagnetic field are given in the q-deformed case. A new kind of q-coherent states of a q-harmonic oscillator in a finite-dimensional Hilbert space are introduced. Some properties of these operators and q-coherent states are discussed.

Phase Properties of the Displaced Number State in a Kerr Medium

Abstract Influences of the interaction of the displaced-number-state field with a Kerr medium on phase properties of the field are investigated. The interaction brings about the symmetrical splitting of the phase probability distribution (PPD) sometimes. Collapses and revivals have also been found in the long-time behaviours of the Hermitian, cosine and sine phase operators at high intensities. In terms of the PPD, the origin for generating these revivals has been revealed.

Number-phase uncertainty product for displaced number states.

The number-phase uncertainty product for displaced number states of the harmonic oscillator is examined within the framework of three different approaches: (i) the Carruthers-Nieto number-phase uncertainty relation in terms of the Susskind-Glogower sine and cosine phase operators, (ii) a similar relation to this calculated with the Pegg-Barnett unitary phase operator, and (iii) the number-phase uncertainty relation arising from the Pegg-Barnett Hermitian phase operator. The corresponding number-phase uncertainty product is calculated extending from average photon numbers 30 down to m for the first few classes of displaced number states (m=0,1,...,5). It is found that, for a displaced number state with a reasonable average number of excited quanta, all three rival phase formalisms yield similar number-phase uncertainty products, tending, for increasingly large magnitude of the displacement parameter, to the constant value m+1/2. On the other hand, for a small number of excited quanta it is found that, according to the first two formalisms, the number-phase uncertainty product for a given class of displaced number states tends to the maximum value \ensuremath{\surd}2m+1 /(\ensuremath{\surd}m+1 - \ensuremath{\surd}m ), while the third phase formalism predicts an entirely different behavior; with decreasing magnitude of displacement parameter, after passing through a maximum, the number-phase uncertainty product falls off eventually to zero making, in particular, the search for minimum number-phase uncertainty states futile. It is argued in favor of this last result, and the possibility of experimental verification, in the realm of quantum optics, is briefly considered.

Dependence of phase dynamics of light interacting with two-level atoms on the initial state of atoms

Abstract We have studied the phase properties of initially coherent light interacting with two identical two-level atoms using the Hermitian phase operator of Pegg and Barnett. The results show that, in a certain initial state of the atoms, the field would, to a certain extent, preserve its order in phase, even for a long interaction time with the two atoms.

Phase distribution in the Yurke-Stoler states

The phase distribution of the Yurke-Stoler state generated as a result of non-linear interaction by an optical signal, is studied making use of the Hermitian phase operator formalism of Pegg and Barnett. The time evolution of the mean and variance of the phase operator and illustrated graphically.

Coherent state formalism for the Pegg-Barnett Hermitian phase theory

The coherent state associated with the Pegg-Barnett Hermitian phase operator are constructed. Their properties are studied in detail. The coherent state representation of the Pegg-Barnett phase operators is obtained. The number-phase squeezing and the number-phase uncertainty relation are discussed for a two-state system.

Many-port homodyne detection of an optical phase.

We present a quantum analysis of the measurement of a relative optical phase using a many-port method that can be implemented using arrays of photodetectors in the limit of a large number of detector elements. Its application to coherent states and states not describable by the semiclassical theory shows that the many-port method has in principle several advantages compared to an eight-port method previously studied. In some cases, especially for very weak fields, the many-port method provides a more faithful characterization (i.e., a smoother distribution) of the distribution of the relative phase than does the eight-port method. The definition used here for the relative phase between two fields is an operational one, not corresponding to any presently known Hermitian phase operator. We discuss the conditions under which our phase definition corresponds to the best estimate in the maximum-likelihood sense. We find that if the many-port data set is analyzed in a way that retains the contributions from zero photon counts, the many-port phase distribution takes on a character that resembles other phase distributions, such as the Pegg-Barnett or Wigner phase distributions, that show very small modulation depth in the case of weak fields.

Phase and squeezing properties of two copropagating fields in a Kerr medium.

The paper deals with a system of two single-mode radiation fields copropagating through a nonabsorbing Kerr medium modeled as an anharmonic oscillator. The quadrature squeezing of the signal light is investigated and the phase dynamics, including the distribution, shift, and variance of the phase of the probe light, is studied using the Pegg-Barnett Hermitian phase operator. The results show that the squeezing is considerably influenced by the coupling between the two fields and the phase dynamics shows new features compared with that predicted by the Susskind-Glogower phase formalism.

Phase properties of a field in the Jaynes-Cummings model for non-resonant behaviour

The phase properties of a coherent field non-resonant interacting with a two-level atom in a lossless cavity are studied using the new hermitian phase formalism of Pegg and Barnett. The phase probability distribution, the expectation value and the phase fluctuation of the hermitian phase operator are calculated. The results show that phase properties of the field in non-resonance JCM are strongly dependent on the intensity of the coherent field and the detuning parameter. The phase probability distribution splits into four separate asymmetry distributions as time goes on.

Hermitian Phase Operator of Quantized Light Fields. A Fundamental Problem of Quantum Optics.

For a long time, about 60 years, the phase as an observable of the quantized light fields has remained a fundamental problem with considerable difficulties on quantum mechanics and also quantum optics. Recently, however, the hermitian phase operator and its eigenstates in a finite Hilbert space are found by Pegg and Barnett. In this paper, we provide a historical review and the background of the problem on the phase operator, explore the Pegg-Barnett formalism, and show the recent developements of this fields and its application to various light fields. Further problems on this formalism and other approaches are also discussed.

Properties of the Measured Phase Operators in the Squeezed Number States

Abstract The phase properties of the squeezed number states have been studied from the lower order to the higher order via the measured phase operators (MPO) introduced by Barnett and Pegg and compared with those obtained via the Hermitian and Susskind-Glogower phase operators. We have found many new, interesting results. For the lower order, the phase fluctuations of the MPO are exactly independent of the photon number n. For the higher order, the corresponding fluctuations are a decreasing function of n.

Unitary and Hermitian phase operators for the electromagnetic field.

Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)] have proposed a formalism for a Hermitian optical phase operator, constructed from the so-called phase states. In this paper we give an alternative derivation. We begin by defining a unitary phase operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ from the relationship $\stackrel{^}{a}={e}^{i\stackrel{^}{\ensuremath{\varphi}}}R(\stackrel{^}{N})$ ($\stackrel{^}{a}$ is an annihilation operator for a single-mode boson field), but borrow the idea of a finite- (but arbitrarily large) dimensional space from Pegg and Barnett. Finally, we comment on an experiment to determine quantum phase uncertainty. Our analysis uses the operator ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$, which we feel is a more natural choice than $\stackrel{^}{\ensuremath{\varphi}}$. The use of ${e}^{i\stackrel{^}{\ensuremath{\varphi}}}$ also avoids the multivalued problems associated with $\stackrel{^}{\ensuremath{\varphi}}$.

Fluctuation of Phase in the Displaced Number States

Using Pegg-Barnett formalism the author has studied the phase properties of displaced number states. It has been found that the phase probability distribution exhibits the same oscillations as the photon number distribution. Using the Hermitian phase operator the fluctuations of the phase of the displaced number states have been studied. The dependence of the fluctuations on the value of the reference phase have been analysed in detail. It has been found that phase of displaced number states exhibit squeezing in the high-intensity limit of the driving field. These states are approximately equivalent to phase states with precisely defined phase. It has been shown that displaced number states are not minimum uncertainty states with respect to the phase and the number operators.