Minimum Uncertainty States Research Articles

Minimum-uncertainty angular wave packets and quantized mean values.

Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum. \textcopyright{} 1996 The American Physical Society.

SU(1,1) coherent states defined via a minimum-uncertainty product and an equality of quadrature variances.

The coherent states of a Hamiltonian linear in SU(1,1) operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables. The proposed approach is thus based on a physical characteristic of the harmonic-oscillator coherent states which is in contrast with the existing ones which rely on the generalization of the mathematical methods used for constructing the harmonic-oscillator coherent states. The set of states obtained by following the proposed method contains not only the known SU(1,1) coherent states but also a different class of states. \textcopyright{} 1996 The American Physical Society.

SU(2) coherent states in parametric down-conversion.

The states generated from the vacuum in two parametric down-conversion crystals with aligned idler beams are studied. It is shown that after a measurement of the photon number in some of the modes, the emerging states can be related to SU(2) and SU(1,1) coherent and minimum-uncertainty states. \textcopyright{} 1996 The American Physical Society.

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.

Quantum theory for mesoscopic electric circuits.

A quantum theory for mesoscopic electric circuits in accord with the discreteness of electric charges is proposed. On the basis of the theory, Schr\"{o}dinger equation for the quantum LC-design and L-design is solved exactly. The uncertainty relation for electric charge and current is obtained and a minimum uncertainty state is solved. By introducing a gauge field, a formula for persistent current arising from magnetic flux is obtained from a new point of view.

Considerations on localization of macroscopic bodies

Position holds a very special role in understanding the classical behaviour of macroscopic bodies on the basis of quantum principles. This lead us to examine the localised states of a large condensed object in the context of a realistic model. Following the argument that an isolated macroscopic body is usually described by a linear superposition of low-lying energy eigenstates, it has been found that localised states of this type correspond to a nearly minimum-uncertainty state for the center of mass. An indication is also given of the dependence of the center of mass position spread on the number of constituent particles. This paper is not offered as an answer to the intriguing question of the preferred role played by the position basis, but will hopefully provide some contribution to the quantum modelling of multi-particle systems.

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 space formalism: The generalized harmonic-oscillator functions

We discuss the mathematical properties of phase space harmonic-oscillator orthogonal functions. We introduce the relevant creation-annihilation operators and point out the existence of generalized uncertainty, relations of the Heisenberg type. We define coherent phase space states and show that they are generalized minimum uncertainty states. We apply the methods developed to the analysis of classical and quantum harmonic oscillators and, within such a context propose an operator formalism that permits one to exploit Heisenberg-like equations in both classical and quantum problems.

Uncertainty relations for realistic joint measurements of position and momentum in quantum optics

We derive simple uncertainty relations and find the minimum-uncertainty states for realistic joint measurements of canonically conjugate quadratures in quantum optics. We take into account photodetection inefficiencies as well as a finite intensity of the local oscillator.

Stochastic variational approach to minimum uncertainty states

We introduce a new variational characterization of Gaussian diffusion processes as minimum uncertainty states. We then define a variational method constrained by kinematics of diffusions and Schr\"{o}dinger dynamics to seek states of local minimum uncertainty for general non-harmonic potentials.

Radiation pressure and coherent states of two-level atoms.

Two sets of coherent states of a two-level atom are constructed by the displacement-operator technique from the choices of the ground and excited states as the extremal state. Particular forms of both of these sets of coherent states are minimum-uncertainty states. Radiation-induced forces are determined for a two-level atom prepared in such coherent states. In geneal, these forces are affected by the presence of a second atom. For a minimum-uncertainty state, the force on a two-level atom is inedpendent of the state of the field. The significance of the calculations is discussed in relation to the pure atomic state occurring in the midst of the ``collapse region'' of the atomic inversion in the Jaynes-Cummings model. The existence of such a pure state may provide a possible means of experimentally generating two-level atomic minimum-uncertainty states.

Minimum Uncertainty States Using n-Dependent Annihilation Operator

In the last ten years there has been tremendous interest in the study of quantum mechanical states which give minimum value to the uncertainty product [1]. Several methods [2] have been proposed to obtain such states. The simplest being the annihilation operator method, in which one first constructs the step-down operator which gives the state |n — 1) operating on the state In) of the system. The minimum uncertainty state is then obtained by solving the eigenstate of the annihilation operator. In several problems of physical interest it happens that the annihilation operator depends on n. The purpose of the present work is to develop a formalism to construct minimum uncertainty states using such annihilation operators. We shall use the three-dimensional harmonic oscillator to bring out the essential features of the present formalism, which is given in the next section. Concluding remarks are presented in Sec. 4.

Laguerre polynomial states in single-mode Fock space

Based on the nonlinear realization of the SU (1, 1) algebra whose generators are R †  a † N+1 , R, and N + 1 2 we introduce the concept of field intensity-dependent squeezing and find that the state L M(−yR †) |0〉 , where L M ( x) is a Laguerre polynomial, is a minimum uncertainty state for field intensity-dependent squeezing.

State Reduction for Correlated Photons and Eigenstates of Unitarily and Non-unitarily Displaced Number Operator

Abstract The number-quadrature minimum-uncertainty states after rotation and displacement emerge in the parametric down-conversion followed by photon number measurement in the idler mode when the coherent state inputs. Again, the state generated in the signal mode by the reduction is described by the principally measurable quantum-statistical tools.

Optimum field squeezing from atomic sources: Three-level atoms.

Light emitted from laser-driven atoms often has squeezed quadrature fluctuations reflecting the phase dependence of the atomic source excited into a coherent superposition by the driving field. In this paper, we study the maximum squeezing in the emitted light fields from three-level atoms, paying particular attention to the role of atomic coherences in governing the optimum squeezing which is possible. Some phase dependent nonlinear optical processes have the potential for substantial quantum noise reduction at useful intensities: for example, four-wave mixing involving atomic systems has already been used to generate quadrature-squeezed light. The threoretical description of such a source is generally complicated with damping processes, coherent and incoherent pumping, and so on, all needing to be taken into account. Fully quantum treatments are available, but the basic physics of the process is often difficult to grasp in the necessarily complicated analysis. Although less important from the point of view of efficient practical sources of squeezed light, studying the generation of squeezed light in simpler systems such as two- or three-level atoms interacting with one or two modes (as in Jaynes-Cummings models) or in spontaneous emission or resonance fluorescence gives greater insight into the fundamentals of the squeezing process. In particular, the relationship between the atomic coherences and the degree of squeezing in the generated fields can be explored. We study the maximum (optimized) squeezing that can be obtained in the emitted fields from the irreversible decay from three-level atoms for any choice of initial conditions, excitations and decay process, for all quadrature components and for all possible configurations (V, , and ladder systems). We show that for V and systems, optimum squeezing is actually associated with a two-level state (in a suitable basis) involving a single one-photon coherence, but for ladder systems the state for maximum noise reduction is not equivalent to a two-level system and involves a single two-photon coherence and no intermediate state population. We pay particular attention to whether the total field is in a minimum uncertainty state, and the nature of the atomic state associated with maximum squeezing, especially whether it is a mixed or pure state. In the case where the initial free field has a zero amplitude at the detector and the source atoms are confined to a region small compared to the wavelength (Dicke source) we show that the squeezing in the total field is given in terms of the squeezing in the source field, and hence related to atomic populations and atomic one-photon and two-photon coherences for the case of three level atom. Previously unconsidered source-free field interferences terms are shown to be zero. The choice of quadrature phase and frequency is optimized to minimize the source field normally ordered variance. We then minimize further with respect to the choice of atomic density matrix elements subject to the constraints that the density matrix is Hermitean, positive, and has a trace equal to unity. In all cases we find that the optimum squeezing is produced when the source field is in a minimum uncertainty state.

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.

DIFFUSION PROCESSES AND COHERENT STATES

It is shown that uncertanity relations, as well as coherent and squeezed states, are structural properties of stochastic processes with Fokker–Planck dynamics. The quantum mechanical coherent and squeezed states are explicitly constructed via Nelson stochastic quantization. The method is applied to derive new minimum uncertainty states in time-dependent oscillator potentials.

Generation of nonclassical light by dissipative two-photon processes.

Nonclassical states of light may be generated by processes involving the creation or annihilation of photons in pairs. A quadratic coupling, characteristic of a parametric amplifier, generates a squeezed vacuum from a normal vacuum, and a two-photon absorber can also generate a squeezed state (though not a minimum-uncertainty state) even though it is a purely dissipative process. We consider here the simultaneous action of a quadratic pump on a two-photon absorber and demonstrate how superpositions of distinct coherent states may be generated by their combined effects. We use standard master equations to describe the time development, employing split operators and direct numerical integrations to determine the field density-matrix elements and quasiprobabilities. The purities of the nonclassical states are determined by evaluating the field entropy. Provided one-photon dissipative processes may be ignored, a pure superposition state is formed in the steady state. This superposition is destroyed if one-photon loss processes are important.

Minimum-uncertainty states for noncanonical operators.

The usual uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}〈C${\mathrm{〉}}^{2}$/4 between two Hermitian operators A and B satisfying the noncanonical commutation relation [A,B]=iC, where C is not a constant multiple of the unit operator, fails to give a nontrivial lower bound on the product of the variances \ensuremath{\Delta}${\mathit{A}}^{2}$ and \ensuremath{\Delta}${\mathit{B}}^{2}$ when 〈C〉=0. For those operators, therefore, the general uncertainty relation \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$\ensuremath{\ge}[〈C${\mathrm{〉}}^{2}$+〈F${\mathrm{〉}}^{2}$]/4 where 〈F〉=〈AB+BA〉-2〈A〉〈B〉 is better suited to determine the lower bound on \ensuremath{\Delta}${\mathit{A}}^{2}$\ensuremath{\Delta}${\mathit{B}}^{2}$. The implications of the general uncertainty relation and the properties of the minimum-uncertainty states, i.e., the states for which the general uncertainty relation is satisfied with equality, are discussed. The minimum-uncertainty states are found to fall in two different classes. One is the usually studied class for which 〈F〉=0, that is, the case when the usual uncertainty relation holds. The other is the hitherto unnoticed class for 〈C〉=0, that is, when the usual uncertainty relation is redundant. The squeezing properties of the present class of minimum-uncertainty states is discussed by defining squeezing in the light of the general uncertainty relation.

Coherent Phase State and Displaced Phase State in a Finite-dimensional Basis and Their Light Field Limits

Abstract Using the Pegg-Barnett phase operator formalism we have introduced the coherent phase state and the displaced phase state in a finite dimension. Using the definition of phase creation and annihilation operators we have constructed a displacement operator which yields coherent phase states and displaced phase states within a finite-dimensional basis and numerically evaluated observables as a function of the size of the basis set. We have shown that, when the dimension of the state space is much larger than the average excitation of phase quanta, then these states are the coherent phase state and displaced phase state of a harmonic oscillator which has an infinite number of eigenstates and exhibits phase squeezing up to 100%. The coherent phase state is a minimum-uncertainty state with respect to the phase quadrature operators but not the minimum-uncertainty state with respect to number and phase operators. The properties of coherent phase states for a two-state system are also investigated. It is interesting to realize that these coherent phase states and displaced phase states are exactly the same as the phase vacuum state and different phase states respectively for the single-mode light field case.