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The Journal of Chemical Physics

Squeezed spin states

James D. Macomber, Robert Lynch

Published Dec 15, 1985• 8 citations
Publisher

Abstract

The squeezed states of a spin-1/2 particle are described, and the response of such a particle to sudden illumination by a resonant, circularly polarized, coherent electromagnetic wave in the absence of relaxation is given. It is shown that the resultant undamped Torrey nutations (Rabi flops) lead to periodic squeezing of the spin states. The error ellipsoids associated with the minimum-uncertainty states for this system are presented. The implications of these results for the squeezing of the field vectors of the electromagnetic radiation emitted by spins and quasi-two-level atomic systems (in states which have been prepared by coherent transient driving fields) are pointed out.

DOI
10.1063/1.449838

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•Jan 1, 1993•0 citations
The squeezed states of a spin-1/2 particle are described, and the response of such a particle to sudden illumination by a resonant, circularly polarized, coherent electromagnetic wave in the absence of relaxation is given. It is shown that the resultant undamped Torrey nutations (Rabi flops) lead to periodic squeezing of the spin states. The error ellipsoids associated with the minimum-uncertainty states for this system are presented. The implications of these results for the squeezing of the field vectors of the electromagnetic radiation emitted by spins and quasi-two-level atomic systems (in states which have been prepared by coherent transient driving fields) are pointed out.
The nonclassical nature of the so-called squeezed states of electromagnetic fields has recently been discussed in connection with the phenomenon of resonance fluorescence from a coherently driven two-level atom.1 It has been shown that either the in-phase component E1 or the out-of-phase component E2 of the fluorescent field can become squeezed if the Rabi frequency Ω of the ex- citing field is sufficiently low. Squeezing is defined as a negative value of the normally ordered variance 〈: (ΔE1)2:〉 (or 〈:(ΔE2)2:〈). To satisfy this condition in the steady state, if the exciting field is tuned perfectly to the atomic transition, the Rabi frequency Ω has to be less than √2y, with 2y = A—the Einstein coefficient for spontaneous emission. Off-resonance excitation further lowers this critical value of 0. Walls and Zoller1 have indicated, without giving the details, that, in a transient regime, the squeezing effect can become twice as large as in the steady state.
A theory is developed for the interaction of coherent electromagnetic (EM) radiation with a free electron gas and for the influence of the plasma on the photon statistics of the radiation field. Two important possibilities for generating squeezed states are shown: Squeezed states can be generated by the interaction of a one-mode laser with a plasma in a cavity; squeezed states can be generated by transmitting coherent radiation through the plasma. The maximal squeezing of the radiation obtained in the latter case is equal to the reciprocal of the dielectric constant of the plasma. Inside the plasma the EM field is represented by quasiphotons. Their dispersion relations are derived by using Bogoliubov transformations together with a self-consistency requirement for the motion of the electrons interacting with these quasiphotons. The squeezing processes are found to be strong when the frequency of the radiation is a little above the plasma frequency. By adding a constant magnetic field in the direction of propagation of the EM wave different amounts of squeezing are obtained for right- and left-circularly polarized EM waves.
We investigate the properties of angular-momentum states which yield high sensitivity to rotation. We discuss the application of these ``squeezed-spin'' or correlated-particle states to spectroscopy. Transitions in an ensemble of N two-level (or, equivalently, spin-1/2) particles are assumed to be detected by observing changes in the state populations of the particles (population spectroscopy). When the particles' states are detected with 100% efficiency, the fundamental limiting noise is projection noise, the noise associated with the quantum fluctuations in the measured populations. If the particles are first prepared in particular quantum-mechanically correlated states, we find that the signal-to-noise ratio can be improved over the case of initially uncorrelated particles. We have investigated spectroscopy for a particular case of Ramsey's separated oscillatory method where the radiation pulse lengths are short compared to the time between pulses. We introduce a squeezing parameter ${\ensuremath{\xi}}_{\mathit{R}}$ which is the ratio of the statistical uncertainty in the determination of the resonance frequency when using correlated states vs that when using uncorrelated states. More generally, this squeezing parameter quantifies the sensitivity of an angular-momentum state to rotation. Other squeezing parameters which are relevant for use in other contexts can be defined. We discuss certain states which exhibit squeezing parameters ${\ensuremath{\xi}}_{\mathit{R}}$\ensuremath{\simeq}${\mathit{N}}^{\mathrm{\ensuremath{-}}1/2}$. We investigate possible experimental schemes for generation of squeezed-spin states which might be applied to the spectroscopy of trapped atomic ions. We find that applying a Jaynes-Cummings--type coupling between the ensemble of two-level systems and a suitably prepared harmonic oscillator results in correlated states with ${\ensuremath{\xi}}_{\mathit{R}}$1.
The nature of phase noise in quantum optics is analyzed. In an experiment involving the measurement of the electromagnetic field the two quantities of interest are the energy and phase of the field. However, measurements of the quantities produce quantum fluctuations. The quantum fluctuations are regarded as noise in the treatment presented here. The quantum system is represented by a probability distribution, the Wigner function, and the quantum fluctuations are treated as stochastic noise associated with the quantity being measured. The difficulties of associating a quantum operator with the phase of the system are reviewed and the related energy-phase uncertainty relation is discussed. The alternate interpretation of the phase noise of a quantum system as being the classical phase noise of the Wigner function is presented. In particular the energy and phase noise of the vacuum state, the coherent state, the squeezed state and the squeezed vacuum are discussed in this way. The squeezed states of light are minimum uncertainty states with respect to the quadrature operators and exhibit noise of one quadrature below the noise level associated with the vacuum. The reduced noise level in one quadrature of the field underlies the importance of squeezed states in many practical applications where there is a need to reduce the quantum noise of one quadrature of coherent light. The periodic phase operator eliminates the difficulties associated with the multivalued nature of phase. The analysis of the vacuum and intense coherent state of Carruthers and Nieto by employing periodic phase operators is reviewed, particularly with respect to the energy-phase uncertainty relations and we generalize the approach to develop a phase operator analysis of the squeezed state in the intense field and vacuum limits. We demonstrate here for the first time that the phase operator is simply related to the phase of the squeezed state in the intense field limit and that the squeezed state is approximately an energy-phase minimum uncertainty state in the low-squeezing limit. Also we enlarge on previous work to demonstrate that the phase operator corresponds simply and unambiguously to the phase of the squeeze parameter for the strongly squeezed vacuum and the intensely squeezed vacuum is an energy-phase minimum uncertainty state for some values of phase. The occurrence of squeezing for the case of two coupled quantum oscillators is presented. The system consisting of one mode of the electromagnetic field coupled to a spinless nonrelativistic electron subjected to an harmonic potential is represented by two coupled harmonic oscillators. The dynamics are compared for the case that the rotating wave approximation is employed and for the case that the counter-rotating terms are included. These calculations have not been performed before. The parametric amplifier Hamiltonian with a nonresonant coupling is also studied in order to provide insight into the effects of the counter-rotating terms. Squeezing of the field produced by the electron is a consequence of the inclusion of the counter-rotating terms. The case of a spinless nonrelativistic electron subject to an harmonic potential and coupled to a continuum of electromagnetic field modes is also considered. The case of two coupled oscillators discussed above is generalized by replacing the oscillator which represents the single-mode field by a bath of oscillators. The effects of including counter-rotating terms and of ignoring the counter - rotating terms in the Hamiltonian are compared. The interaction is assumed to produce a frequency shift and an exponential damping term for the oscillating electron. The frequency shift is assumed to be small in either case and so the Wigner-Weisskopff approximation is employed to solve the equations of motion. We demonstrate the new results that dissipation-induced phase-dependent noise is a consequence of including the counter-rotating terms and that the noise is phase-independent for the case that the counterrotating terms are excluded. The relation between these results and recent work on quantum tunnelling in superconducting quantum interference devices is discussed. We conclude by suggesting further research related to the work in this thesis.
We examine the transient behaviour of a plane-spherical resonator driven by a continuous wave and filled by an optically thin gas which is suddenly switched on exact resonance with the driving field. Contrary to previous treatments, the axial and transversal field distributions and the molecular motions through the standing wave are taken into account. This is achieved by projecting the induced gas polarization on the fundamental gaussian mode of the resonator and the results are given in a quasi analytic form. An oscillatory response is only obtained when the Rabi frequency is large with respect to the Doppler line width and for a bad quality resonator. Some of the theoretical predictions have been supported by an experiment at a 6 mm-wavelength.