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Physics Today

Nonclassical Excitation in Spectroscopy with Squeezed Light

Z. Ficek, P. D. Drummond

Published Sep 1, 1997• 17 citations
Publisher

Abstract

The amplitude and phase of all light fields are subject to fluctuations. Much of this irregularity arises from random and uncontrolled changes that occur in any light source and lead to random changes in the light wave's frequency and amplitude. Most of these factors can in principle be eliminated by a careful design of the light source. Even if all of these defects in the source are removed, however, light fields are still subject to fluctuations arising from the laws of quantum mechanics. Even in a highly stabilized laser, the resulting coherent electromagnetic field has an uncertainty in the phase and magnitude of its amplitude. All conventional sources of light fields have at least this noise level, which, in a fully quantum description, arises from vacuum fluctuations. At one time it was believed that this vacuum or coherent-state noise could not be eliminated.

DOI
10.1063/1.881934

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The amplitude and phase of all light fields are subject to fluctuations. Much of this irregularity arises from random and uncontrolled changes that occur in any light source and lead to random changes in the light wave's frequency and amplitude. Most of these factors can in principle be eliminated by a careful design of the light source. Even if all of these defects in the source are removed, however, light fields are still subject to fluctuations arising from the laws of quantum mechanics. Even in a highly stabilized laser, the resulting coherent electromagnetic field has an uncertainty in the phase and magnitude of its amplitude. All conventional sources of light fields have at least this noise level, which, in a fully quantum description, arises from vacuum fluctuations. At one time it was believed that this vacuum or coherent-state noise could not be eliminated.
Well-stabilized lasers emit radiation that is a close approximation to a coherent state: a classical mean field which varies sinusoidally in time at optical frequency ω0, superposed with broad-band noise fields due to the field's quantum uncertainty in its ground state (i.e. vacuum fluctuations). When detected by a standard photodetector, the interference between these vacuum fields and the coherent mean field causes the noise floor commonly referred to as the "shot-noise limit." The vacuum noise also causes fluctuations in the phase of the beam that are independent and uncorrelated with the amplitude fluctuations, and to which photodetectors are insensitive. The amplitude and phase fluctuations of the field can be represented by quadrature amplitudes X ^ A ( δ ) and X ^ ϕ ( δ ) which are the Fourier components of amplitude and phase modulation or fluctuation at a particular modulation frequency δ. These quadrature amplitudes are non-commuting quantum observables which obey the uncertainty principle.
Squeezed states of the electromagnetic field are characterized by a reduction of fluctuations for one of two quadrature phase amplitudes below the level of fluctuations for the vacuum state. This reduction is a manifestly quantum or nonclassical feature which is of great interest in its own right but also potentially important for precision measurement with sensitivity beyond the shot-noise or vacuum-state limit. We describe three different experiments in which squeezed light has been generated in our laboratory involving parametric downconversion, intracavity frequency doubling, and optical bistability with two-state atoms. By employing the squeezed light produced by a subthreshold optical parametric oscillator, we have achieved improvements in sensitivity beyond the vacuum-state limit for the detection of both phase and amplitude changes of the electromagnetic field. An alternate route to sensitivity beyond the vacuum-state limit is through quantum nondemolition measurement, where we have suggested and are implementing a scheme involving parametric conversion and polarization mixing.
Electromagnetic fields, with the noise on one quadrature component reduced to below the quantum mechanical zero-point fluctuation level and the noise on the other quadrature component enhanced to above it, are currently of great interest in quantum optics because of their potential applications to various precision measurements. Such squeezed states of light are usually produced by imposing nonlinear unitary evolution on coherent (or vacuum) states. On the other hand, squeezed states with reduced photon number noise and enhanced phase noise are generated directly by a constant current-driven semiconductor laser. This is the simplest scheme for the generation of nonclassical light, and so far it has yielded the largest quantum noise reduction. The mutual coupling between a lasing junction and an external electrical circuit provides opportunities for exploring the macroscopic and microscopic quantum effects in open systems.
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 showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise . Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states , provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.
As a direct result of the quantum nature of electromagnetic radiation, all forms of light exhibit inherent and unavoidable random fluctuations. Because these fluctuations reveal information about the nature of light and the underlying processes that generate it, they have received considerable attention since the time of Einstein. But they are also a source of noise that limits the accuracy with which information can be transmitted by a beam of light. Although experiments often still struggle with instrumental noise, some are quickly approaching the measurement limits set by the quantum-statistical nature of light. With the help of new techniques to “squeeze” the uncertainty of light, researchers can now conduct experiments with greater precision than possible with laser light. The generation of various forms of squeezed light in recent years has raised hopes for its application to such diverse areas as gravity-wave detection based on optical interferometry and reduced-error lightwave communications.