Hermitian Phase Operator Research Articles

Квантовое измерение амплитуды и фазы электромагнитного поля методом оптического гомодинирования

The possibility of measuring the amplitude and phase of the electromagnetic field by optical light homodyning is shown. A new method for determining the relative phase of the field using the interference scheme of balanced homodyne detection is proposed. The method is based on direct measurement of the average complex amplitude of a quantum field mixed with a classical field using a passive symmetric beam splitter under conditions of a balanced homodyne measurement scheme. A "direct" measurement of the average field amplitude is a direct measurement of the real and imaginary parts of the average quantum mechanical value of the complex field amplitude by homodyning. By comparison with the quantum theory of the Hermitian phase operator of the electromagnetic field, the accuracy of this measurement is estimated for various quantum states of the microscopic electromagnetic field.

Fluctuations of the hermitian phase operator of electromagnetic field for quantum phase superpositions of coherent states

ABSTRACTQuantum-statistical properties of the phase of electromagnetic field in the case of phase superpositions of coherent states of the type have been studied. Formulas for the average values and fluctuations (variances) of Hermitian phase operator of the electromagnetic field (so-called Pegg–Barnett optical phase operator) in the case of general superposition of coherent states are obtained and analysed. The states of phase superpositions of the ‘Schrödinger cat’ type , and superpositions of coherent states of the type , are considered separately.

Quantum Encoding and Entanglement in Terms of Phase Operators Associated with Harmonic Oscillator

Realization of qudit quantum computation has been presented in terms of number operator and phase operators associated with one-dimensional harmonic oscillator and it has been demonstrated that the representations of generalized Pauli group, viewed in harmonic oscillator operators, allow the qudits to be explicitly encoded in such systems. The non-Hermitian quantum phase operators contained in decomposition of the annihilation and creation operators associated with harmonic oscillator have been analysed in terms of semi unitary transformations (SUT) and it has been shown that the non-vanishing analytic index for harmonic oscillator leads to an alternative class of quantum anomalies. Choosing unitary transformation and the Hermitian phase operator free from quantum anomalies, the truncated annihilation and creation operators have been obtained for harmonic oscillator and it has been demonstrated that any attempt of removal of quantum anomalies leads to absence of minimum uncertainty.

Complementarity and phases in SU(3)

Phase operators and phase states are introduced for irreducible representations of the Lie algebra using a polar decomposition of ladder operators. In contradistinction with , it is found that the polar decomposition does not uniquely determine a Hermitian phase operator. We describe two possible ways of proceeding: one based on imposing SU(2) invariance and the other based on the idea of complementarity. The generalization of these results to SU(n) is sketched.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Operator formalism for the Wigner phase distribution

We construct a Hermitian phase operator for the radially integrated Wigner distribution, which is known to be sensitive to phase. We show that this operator is complete and also elucidate a set of complete but non-orthogonal states that seems to be naturally associated with such an operator. Further, we show that our operator satisfies a weak equivalence relation with the Pegg–Barnett operator, thus showing that the essential phase information furnished by both formalisms are the same. It is also shown that this operator gives results which are in agreement with the expected uniform phase distribution of a Fock state.

Hermitian Operators Conjugate to Two-Mode Number-Difference Operator Studied in Entangled State Representation

In similar to the derivation of phase angle operator conjugate to the number operator by Arroyo Carrasco–Moya Cessay we deduce the Hermitian phase operators that are conjugate to the two-mode number-difference operator and the three-mode number combination operator. It is shown that these operators are on the same footing in the entangled state representation as the one of Turski in the coherent state representation.

Hermitian-phase-operator treatment of the superfluid states in a quantum two-well problem

We apply the Hermitian phase operator method (HPOM), as discussed by Pegg and Barnett [D. T. Pegg and S. M. Barnett, Phys. Rev. A 39, 1665 (1989)] to a two-well Bose-Hubbard model. Application of the HPOM formalism yields an approximate quantum phase model, or a Schr\"odinger-like differential equation for the phase variable $\ensuremath{\theta}$, which is a conjugate variable to the half of the number difference between the wells, $m=({N}_{1}\ensuremath{-}{N}_{2})∕2$. In the construction of the model we take care so that the Hermiticity of the original Bose-Hubbard model in number representation is inherited. We demonstrate that the quantum phase model supersedes two theoretical models suggested for the group of so-called ``normal'' states, for which $⟨\ensuremath{\theta}⟩=⟨m⟩=0$: a non-Hermitian ``exact quantum phase model'' by Anglin, Drummond, and Smerzi [J. R. Anglin, P. Drummond, and A. Smerzi, Phys. Rev. A 64, 063605 (2001)], and the ``phonon model'' by Javanainen and Ivanov [J. Javanainen and M. Yu. Ivanov, Phys. Rev. A 60, 2351 (1999)]. We apply the HPOM to the so called ``superfluid'' states, for which $⟨m⟩=0$ and $⟨\ensuremath{\theta}⟩=\ensuremath{\pi}$, and find their superfluid phonon model. We find that the superfluid phonon model has a threshold point $\mathcal{M}=\ensuremath{\delta}∕(4gN)=1$, at which the model diverges. Here, $\ensuremath{\delta}$ is the tunneling rate between the wells and $g$ is the self-interaction, while $N$ is the total number of particles. The above threshold phonon model is quite accurate in the description of the number and phase statistics of the superfluid states. We show that at the threshold point HPOM itself can be exactly solved. However, the agreement between the exact solution and the HPOM solution is poor: $⟨{m}^{2}⟩$ close to the threshold assumes values $\ensuremath{\sim}{N}^{2}$, thus voiding the second order expansion the HPOM is based on, $⟨{m}^{2}⟩⪡{N}^{2}$.

Operational Phase Operators and Hermitian Phase Operators

It is shown that there exist two kinds of operational (or measured) phase operators, those of the Hradil type (Phys. Rev. A 47 (1993), 4532), defined on the entire Hilbert space, and those of the Noh-Fougeres-Mandel type (Phys. Rev. A 45 (1992), 424), defined only on a subspace. As a consequence, strictly speaking, the Noh-Fougeres-Mandel measured phase cannot be defined in the case of, for example, coherent states, squeezed states and the vacuum. The difference between these operators is made clear by using the concept of the Bargmann distance between a quantum state and its reduced state. Since Noh et al. did not use the correct variances of the Susskind-Glogower and Pegg-Barnett phase operators when they compared their data with the results predicted through use of these operators, the correct expressions in that case are presented. Comparing the measured operators with the Hermitian phase operator in the extended Susskind-Glogower formalism, we show that the quantum phase given by the Hermitian phase operator is more appropriate than the measured phases. The variance of the Hermitian operator in the vacuum is consistent with the complementarity of the photon number and phase, which implies that the vacuum has a uniform phase distribution. Hradil's measured operator does not have such a property. Although the Noh-Fougeres-Mandel operator has such a property in a particular limit, this operator does not represent, in principle, a meaningful measure of the phase measurement in a typical quantum region, because the Bargmann distance becomes large in such a region.

Generalized phase states and dynamics of generalized coherent states

In terms of Lie theory the evolution of a time-dependent quantum system can be described by the dynamics of its generalized coherent state and its phase properties can be investigated by an Hermitian phase operator constructed from a generalized phase state we introduced. Two illustrative examples of SU(1,1) and SU(2) groups are presented.

Hermitian Phase Operator and Approximate Quantum Measurement in Extended Number-State Space

Considering an approximate quantum phase measurement, we show that the extended Pegg-Barnett formalism with dimension (2s+1) is not plagued by Vorontsov and Rembovsky problem whereby unphysical states can appear after the phase measurement. The approximate phase measurement in this formalism gives, in the large s limit, the same result as that in the Susskind-Glogower formalism. The difficulty of defining a Hermitian phase operator, caused by the periodicity of the phase angle, is overcome by introducing an extended number-state space and by making use of periodic generalized functions (distributions) and their Fourier transforms. This space is the large s limiting case of the extended Pegg-Barnett formalism. Similarly, with the help of periodic generalized functions, Hermitian phase operators can be defined in other infinite-dimensional extended theories. It is also shown that approximate phase measurements give the same results in all these extended formalisms.

Simulations of continuum coherent states and its use in quantum cryptographic systems

Abstract The continuum states formalism is suitable for field quantization in optical fibre; however, they are harder to use than discrete states. On the other hand, a Hermitian phase operator can be defined only in a finite dimensional space. We approximated a coherent continuum state by a finite tensor product of coherent states, each one defined in a finite dimensional space. Using this, in the correct limit, we were able to obtain some statistical properties of the photon number and phase of the continuum coherent states from the probability density functions of the individual, finite dimensional, coherent states. Then, we performed a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography. We observed the fluctuations of the mean photon number in the pulses that arrive at Bob, which occurs in the practical system, introduced by the statistical property of the simulation.

Simulations of continuum coherent states and its use in quantum cryptographic systems

The continuum states formalism is suitable for field quantization in optical fibre; however, they are harder to use than discrete states. On the other hand, a Hermitian phase operator can be defined only in a finite dimensional space. We approximated a coherent continuum state by a finite tensor product of coherent states, each one defined in a finite dimensional space. Using this, in the correct limit, we were able to obtain some statistical properties of the photon number and phase of the continuum coherent states from the probability density functions of the individual, finite dimensional, coherent states. Then, we performed a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography. We observed the fluctuations of the mean photon number in the pulses that arrive at Bob, which occurs in the practical system, introduced by the statistical property of the simulation.

Phase Properties of a Field Mode Interacting with a Three-Level Atom with Kerr Nonlinearity

Phase properties of a field interacting with a three-level atom with a Kerr-like medium are investigated through the phase formalism of Pegg and Barnett. The phase probability distribution, the expectation value and the phase fluctuations of the Hermitian phase operator are calculated. The phase properties of the field reflect the collapses and revivals. A number of interesting features are found and discussed.

Quantum Phase and Phase Difference for Fermion**The project party supported by National Natural Science Foundation of China and partly by the Natural Science Foundation of Shadong Province

In this paper, from the index theorem, we present the Hermitian phase operator for fermion, give its eigenvalues and orthogonal, normalized and complete eigenstates. Furthermore, we also get the phase difference operator for two fermions and discuss its quantum properties.

Visibility limit in multiparticle eigenstates of definite relative phase

We show that if two multiply occupied boson modes are in eigenstates of the Hermitian relative phase operator, then the visibility of fringes formed by the interference between the modes is necessarily less than unity. For large total occupation numbers the visibility becomes $Vl~\ensuremath{\pi}/4.$ States with definite relative phase and unit visibility do exist. They are related to coherent states and are not orthogonal (not eigenstates of a Hermitian phase operator). This visibility limitation may make it possible to investigate experimentally the physical role of the relative phase eigenstates in interference measurements such as those on Bose-Einstein condensates.

A constructive index theorem, phase operators, and phase-difference operators

A constructive reversed index theorem concerning the polar decomposition of an operator into a product of a unitary exponential phase operator and a Hermitian amplitude operator is investigated. Its applications to the construction of Hermitian phase operators of fermions and phase-difference operators between bosons or fermions are presented. Specifically, a Hermitian operator is constructed for the phase difference between a single-mode fermion and boson, which is justified by the fact that it is exactly the interaction part of the Hamiltonian of the Jaynes-Cummings Model. Furthermore, a Hermitian phase operator of a single-mode boson can also be defined referring to a single-mode fermion. All those quantized phases and phase-differences are found to obey a quantum addition rule instead of the ordinary commutative addition rule.

The quantum phase problem and the linear phase insensitive quantum amplifier

We establish a general condition which must be obeyed by every operator referred to the quantum phase. This condition is derived on the basis of the model of the linear phase insensitive amplifier, proposed by Glauber. We demonstrate that the phase operators, most frequently discussed in literature, do not satisfy this condition, and that no Hermitian phase operator can satisfy this condition. We also show by explicit construction that this condition singles out one particular probability operator measure as the only candidate for the correct definition of the phase distribution of the quantum states.

Admissible cyclic representations and an algebraic approach to quantum phase

Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawa's index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity. The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.

Remarks on the construction of a Hermitian phase operator

We consider an oscillator system (whose states are realized in the space of Kravchuk polynomials) in a finite-dimensional Hilbert space. A Hermitian phase operator and the phase quanta raising and lowering operator are constructed using this framework. Also, we have constructed phase coherent states and examined their squeezing property.

Phase Operator Problem and Macroscopic Extension of Quantum Mechanics

To find the Hermitian phase operator of a single-mode electromagnetic field in quantum mechanics, the Schrödinger representation is extended to a larger Hilbert space augmented by states with infinite excitation by nonstandard analysis. The Hermitian phase operator is shown to exist on the extended Hilbert space. This operator is naturally considered as the controversial limit of the approximate phase operators on finite dimensional spaces proposed by Pegg and Barnett. The spectral measure of this operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter found by Helstrom. Eventually, the two promising approaches to the statistics of the phase in quantum mechanics are synthesized by means of the Hermitian phase operator in the macroscopic extension of the Schrödinger representation.