Showing papers on "Quantum state published in 1992"

Quantum cryptography using any two nonorthogonal states

Abstract: Quantum techniques for key distribution---the classically impossible task of distributing secret information over an insecure channel whose transmissions are subject to inspection by an eavesdropper, between parties who share no secret initially---have been proposed using (a) four nonorthogonally polarized single-photon states or low-intensity light pulses, and (b) polarization-entangled or spacetime-entangled two-photon states. Here we show that in principle any two nonorthogonal quantum states suffice, and describe a practical interferometric realization using low-intensity coherent light pulses.

The quantum-state diffusion model applied to open systems

Abstract: A model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation, derived from a density operator equation such as the Bloch equation, and consistent with it. The advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described. The method is applied to the nonlinear absorber, cascades of quantum transitions, second-harmonic generation and a measurement reduction process. The model provides graphic illustrations of these processes, with statistical fluctuations that mimic those of experiments. The stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics. The model is compared with the quantum-jump model of Dalibard (1992), Carmichael and others, which originated among experimenters looking for intuitive pictures and rules of computation.

Weaving a classical metric with quantum threads

Abstract: Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that (i) while local operators such as the metric at a point may not be well defined, there do exist nonlocal operators, such as the area of a given two-surface, which can be regulated diffeomorphism invariantly and which are finite without renormalization; (ii) there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.

Signature of Néel order in exact spectra of quantum antiferromagnets on finite lattices.

Abstract: We show how the broken symmetries of the N\'eel state are embodied in the exact spectrum of the triangular Heisenberg antiferromagnet on finite lattices as small as N=21 (spectra up to N=36 have been computer). We present the first numerical evidence of an extensive set of low-lying levels that are below the softest magnons and collapse to the ground state in the thermodynamic limit. This set of quantum states represents the quantum counterpart of the classical N\'eel ground state. We develop an approach relying on the symmetry analysis and finite-size scaling and we provide new arguments in favor of an ordered ground state for the S=1/2 triangular Heisenberg model.

Observation of quantum frequency conversion

Abstract: Quantum frequency conversion, a process with which an input beam of light can be converted into an output beam of a different frequency while preserving the quantum state, is experimentally demonstrated for the first time. Nonclassical intensity correlation (\ensuremath{\simeq}3 dB) between two beams at 1064 nm is used as the input quantum property. When the frequency of one of the beams is converted from 1064 to 532 nm, nonclassical intensity correlations (\ensuremath{\simeq}1.5 dB) appear between the up-converted beam and the remaining beam. Our measurements are in excellent agreement with the quantum theory of frequency conversion. The development of tunable sources of novel quantum light states seems possible.

Representations of the holonomy algebras of gravity and non-Abelian gauge theories

Abstract: Holonomy algebras arise naturally in the classical description of Yang-Mills fields and gravity, and it has been suggested, at a heuristic level, that they may also play an important role in a non-perturbative treatment of the quantum theory. The aim of this paper is to provide a mathematical basis for this proposal. The quantum holonomy algebra is constructed, and, in the case of real connections, given the structure of a certain C-star algebra. A proper representation theory is then provided using the Gel'fand spectral theory. A corollory of these general results is a precise formulation of the ``loop transform'' proposed by Rovelli and Smolin. Several explicit representations of the holonomy algebra are constructed. The general theory developed here implies that the domain space of quantum states can always be taken to be the space of maximal ideals of the C-star algebra. The structure of this space is investigated and it is shown how observables labelled by ``strips'' arise naturally.

Operational approach to the phase of a quantum field

Abstract: We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.

Poisson-Lie groups. The quantum duality principle and the twisted quantum double

Abstract: The quantum duality principle relates the quantum groups that arise on the quantization of Poisson-Lie dual groups and generalizes Fourier duality. Also considered are the theory of the Heisenberg double, which replaces the cotangent bundle for quantum groups, and its deformations (the twisted double).

Coherent states in a finite-dimensional basis: Their phase properties and relationship to coherent states of light.

Abstract: The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.

Moments of P functions and nonclassical depths of quantum states

Abstract: The practical aspect of the moment problem of the P function is developed; i.e., a simple explicit expression for the P function in terms of its moments and the derivatives of the Dirac δ function in the complex domain is derived. Its connection with the concept of the nonclassical depth of a quantum state introduced recently is explored. The basic formulas derived are used in the calculations of several specific examples that are popular in the quantum-optics community, such as photon-number state, binomial state, two-photon thermal state, squeezed-vacuum state, etc

Quantum mechanics without the projection postulate

Abstract: I show that the quantum state ω can be interpreted as defining a probability measure on a subalgebra of the algebra of projection operators that is not fixed (as in classical statistical mechanics) but changes with ω and appropriate boundary conditions, hence with the dynamics of the theory. This subalgebra, while not embeddable into a Boolean algebra, will always admit two-valued homomorphisms, which correspond to the different possible ways in which a set of “determinate” quantities (selected by ω and the boundary conditions) can have values. The probabilities defined by ω (via the Born rule) are probabilities over these two-valued homomorphisms or value assignments. So any universe of interacting systems, including those functioning as measuring instruments, can be modelled quantum mechanically without the projection postulate.

Rapid Approximate Calculation of Numbers of Quantum States W(E,J) in the Phase Space Theory of Unimolecular Bond Fission Reactions

Abstract: A facile and rapid approximation for counting numbers of quantum states W(E,J) is described for application in the phase space theory of unimolecular bond fission reactions. The method accounts for J-conservation. It is based on an empirical interpolation between analytical expressions of W(E,J) in the low J- and high J-limits, and it is checked by extensive accurate state counting for sets of representative model molecules.

The Physical State Space of Quantum Electrodynamics

Abstract: One consequence of the possible appearance of infrared clouds is that in quantum electrodynamics there are innumerably many sectors satisfying the Borchers selection criterion even for states with the same charge It is then convenient to make a coarser distinction and combine all sectors which coincide in their information about the content of outgoing (or incoming) massive particles in their respective folia of states into one class Since electrically charged particles are massive such a class fixes in particular the electric charge and for this reason it was called a charge class in [Buch 82b] We shall keep this terminology here though these classes still give a much finer distinction than that provided by the charge quantum numbers

Phase from Q function via linear amplification.

Abstract: We relate the phase distribution of a quantum state obtained by integrating its corresponding Q function over radius (1) to a specific measurement scheme using a linear amplifier and (2) to a particular phase operator. A simple relation between the P distribution of the amplified macroscopic state and the Q function of the initial unamplified microscopic state allows us to prove the identity of the phase distributions obtained by integrating the corresponding distributions.

Inhomogeneous quantum groups

Abstract: Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsUq(N), SOq(N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.

Accurate specific molecular state densities by phase space integration. II. Comparison with quantum calculations on H+3 and HD+2

Abstract: The semiclassical determination of N(E;J) and ρ(E;J), the specific number and density of quantum states at energy E, and fixed total angular momentum J, by Monte Carlo integration of phase space is compared to recent exact quantum calculations on H+3 and HD+2, which yielded lists of up to 900 quantum states for single values of J. This allows for the first time tests of such a procedure to be made without assuming anything about separability or harmonicity of the potentials. The excellent agreement between semiclassical and quantum state counts shows that the semiclassical numerical computation is a viable and simple method for the determination of state numbers and densities in small molecules with a precision of the order of 1%. For J=0, the procedure has been extended to state numbers for the different symmetry species occuring in H+3 and HD+2.

Accurate specific molecular state densities by phase space integration. I. Computational method

Abstract: The semiclassical determination of the specific density of quantum states, ρ(E;J), at energy E with fixed total angular momentum J is discussed for small molecules. Monte Carlo integration allows the accurate numerical determination of the phase space volume of systems with J>0 and arbitrary anharmonicity. The corresponding semiclassical number of states can be corrected for the effects of zero point motion in analogy to the well‐known Whitten–Rabinovitch procedure. In this paper, the procedures are tested by comparison with rigid rotor harmonic oscillator models, while a comparison with recent exact quantum calculations on H+3 and HD+2 is described in the following paper. We conclude that, if the intramolecular potential is known or assumed, this numerical semiclassical procedure is a viable and simple way to get state densities of a much improved accuracy.

Recent Developments in Non-Perturbative Quantum Gravity

Abstract: New results from the new variables/loop representation program of nonperturbative quantum gravity are presented, with a focus on results of Ashtekar, Rovelli and the author which greatly clarify the physical interpretation of the quantum states in the loop representation. These include: 1) The construction of a class of states which approximate smooth metrics for length measurements on scales, $L$, to order $l_{Planck}/L$. 2) The discovery that any such state must have discrete structure at the Planck length. 3) The construction of operators for the area of arbitrary surfaces and volumes of arbitrary regions and the discovery that these operators are finite. 4) The diagonalization of these operators and the demonstration that the spectra are discrete, so that in quantum gravity areas and volumes are quantized in Planck units. 5) The construction of finite diffeomorphism invariant operators that measure geometrical quantities such as the volume of the universe and the areas of minimal surfaces. These results are made possible by the use of new techniques for the regularization of operator products that respect diffeomorphism invariance. Several new results in the classical theory are also reviewed including the solution of the hamiltonian and diffeomorphism constraints in closed form of Capovilla, Dell and Jacobson and a new form of the action that induces Chern-Simon theory on the boundaries of spacetime. A new classical discretization of the Einstein equations is also presented.

Quantum emission of a medium with a time-dependent refractive index

Abstract: Quantum emission of photons caused by sudden change of the refractive index of a medium is considered. This emission appears due to time dependence of the zeroth quantum state of the field and is a non-stationary analogue of the Casimir effect. A model-exact solution of the problem is presented. If sudden changes of the refractive index are caused by excitation of a semiconductor near to a band-to-band transition by a synchroneously pumped subpicosecond dye laser the intensity of the quantum emission is approximately 10-9 W in the infrared. The analogy of the effect with and its difference from, the Hawking and the Unruh radiations are discussed.

Quantum-state space metric and correlations.

Abstract: We study the Fubini-Study metric induced on the quantum evolution submanifold of the projective Hilbert space associated with the evolutions generated by a set of independent operators. It is shown that if the operators are Hermitian, then the off-diagonal components of the metric describe the correlations between them. Thus, not only the uncertainty but also the correlation acquires its geometrical interpretation. We also demonstrate this by using a squeezed state

Classical-quantum correspondence for barrier crossing in a driven bistable potential

Abstract: Both quantum and classical analyses are performed to study the barrier crossing dynamics in a driven quartic oscillator. The regions of phase space with regular classical motion are found to be smaller than the size of a quantum state. However, coherent tunnelling is still possible due to the existence of Floquet states localized on two small regular islands. The quantum evolution of localized wavepackets and the classical evolution of the corresponding distributions show similar coherence properties although the degree of coherence is quantally enhanced.

A priori probability and localized observers

Abstract: A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous “wave-packet collapse” across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining thea priori probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement.