Number-phase Minimum Uncertainty States Research Articles

Entangled photon–electron states and the number-phase minimum uncertainty states of the photon field

The exact analytic solutions of the energy eigenvalue equation of the system consisting of a free electron and one mode of the quantized radiation field are used for studying the physical meaning of a class of number-phase minimum uncertainty states. The states of the mode which minimize the uncertainty product of the photon number and the Suskind and Glogower (1964 Physics 1 49–61) cosine operator have been obtained by Jackiw (1968 J. Math. Phys. 9 339–46). However, these states have so far remained mere mathematical constructions without any physical significance. It is proved that the most fundamental interaction in quantum electrodynamics—namely the interaction of a free electron with a mode of the quantized radiation field—leads quite naturally to the generation of the mentioned minimum uncertainty states. It is shown that from the entangled photon–electron states developing from a highly excited number state, due to the interaction with a Gaussian electronic wave packet, the minimum-uncertainty states of Jackiw's type can be constructed. In the electron's coordinate representation, the physical meaning of the expansion coefficients of these states is the joint probability amplitudes of simultaneous detection of an electron and of a definite number of photons. The photon occupation probabilities in these states preserve their functional form as time elapses, but they depend on the location in space-time of the detected electron. An analysis of the entanglement entropies derived from the photon number distribution and from the electron's density operator is given.

Q-phase-coherent states and their squeezing properties

Some kinds of q-phase-coherent states of a q-harmonic osscillator in a finite-dimensional Hilbert space are constructed. Some properties of these states are discussed. Second-order squeezing properties of these states with respect to the phase quadrature operators are studied. The number-phase squeezing and number-phase uncertainty relations are also studied in detail for a two-state system. Some new number-phase minimum uncertainty states are found.

Q-PHASE-COHERENT STATES AND THEIR SQUEEZING PROPERTIES

Squeezing properties of q-phase-coherent states with respect to the phase quadrature operators are studied.The relations between number-phase squeezing and number-phase uncertainty are also studied in detail for a two-state system.Some new number-phase minimum uncertainty states are found.

Number-phase uncertainty relations

The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.

Phase in quantum optics and number-phase minimum uncertainty states

The quantum phase connected with the simultaneous measurement of quadrature operators is investigated. The photon number-phase uncertainty relation is found. The appropriate minimum uncertainty states are the noise minimum states. The explicit expression for the ultimate phase resolution is obtained for the strong mean field approximation. The Shapiro-Wagner phase measurement is included.

Amplitude squeezing through photon fractioning.

A prototype mechanism for producing amplitude-squeezed states, based on the so-called ``photon fractioning'' procedure, is investigated. The coherent state is connected with a quasinumber eigenstate through an intermediate density matrix involving multiphoton processes. The enhancement in the phase fluctuations and the simultaneous reduction of the noise in the photon number result from two competing limits; the ``bright limit,'' which is semiclassical and requires highly excited states, and the photon fractioning limit, which is pure quantum. Asymptotical evaluations of the number probability distribution and of the Q function show that the mixed states are very close to the exact number-phase minimum-uncertainty states.

Noise minimum states and the squeezing and antibunching of light.

The purpose of this paper is to investigate intrinsic restrictions of quantum mechanics put on the correlation function ${g}^{(2)}$ by values of lower-order moments, which characterize noise in interferometric measurements. Some general consequences regarding the relation between antibunching and squeezing are obtained. The class of fields with the minimum value of ${g}^{(2)}$ for given noise is specified and the relation of such states to the Kerr medium is emphasized. The concept of noise minimum states can replace number-phase minimum uncertainty states.

Generation of number-phase minimum-uncertainty states and number states

The difference between the two nonclassical photon states, i.e., the squeezed state and the number-phase minimum-uncertainty state, is discussed. Four different generation principles for number-phase minimum-uncertainty states (and number states) are described. They are the following: (1) unitary evolution, using self-phase modulation and interference; (2) nonunitary state reduction, using either quantum nondemolition photon-number measurement or parametrically amplified idler-wave photon counting; (3) negative-feedback oscillators, incorporating a correlated photon-pair generator; and (4) pump-amplitude noise-suppressed lasers. A number-phase minimum-uncertainty state with photon-number noise reduced to 1 dB (20%) below the standard quantum limit in a frequency range over 100 MHz was produced by the fourth-generation principle.

Realization of number-phase minimum-uncertainty states and number states by quantum nondemolition measurement.

Realization of number-phase minimum-uncertainty states and number states by quantum nondemolition measurement.

Squeezing of light. Manipulation of quantum fluctuations.

The origin of quantum noise is addressed and it can be eliminated when the quantum state is optimally prepared for the single measured observable. Minimum uncertainty states are reviewed using Q (α) -representation, which include squeezed states and number-phase minimum uncertainty states. Squeezing is a concept of trading quantum uncertainties between two conjugate observables to optimize the state for the measurement. Need for optical nonlinearity to achieve squeezing is induced. Principles and experimental results for squeezing quadrature amplitude and photon number are delineated. Applications to optical communications and optical precision measurement are briefly stated.