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Nonlinear Guided Waves and Their Applications

Squeezed States, Parametric Interactions and Fundamental Noise Sources in Optical Fibers

Robert M. Shelby, M. D. Levenson

Published Jan 1, 1989• 0 citations
Publisher

Abstract

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.

DOI
10.1364/nlgwp.1989.sa1

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•Nov 1, 1987•0 citations
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.
A stabilized laser emits a light beam in a coherent state: a light field with properties very similar to a classical field, but with quantum zero-point fluctuations (vacuum noise) of the field superposed. The nonlinear refractive index of optical fibers can be used to alter the quantum statistics of the light to produce a beam with non-classical properties: a squeezed state. Upon detection, squeezed states exhibit a noise level which is phase-dependent and which can be well below the coherent state level ("shot-noise limit"). The nonlinear refractive index also provides a means for experimental demonstration of the related concept of "quantum non-demolition detection": The measurement of a quantum observable without adding uncertainty to that variable.
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.
Parametric interactions such as four wave mixing have been predicted to produce states of the radiation field where the quantum fluctuations in one phase quadrature are less than those of a coherent state.1 These “squeezed states” have proved remarkably difficult to generate and detect, not least because of the presence of additional classical noise in many laser sources and parametric devices. We have now demonstrated the reduction of classical fluctuations by means of four wave mixing in an optical fiber under conditions very similar to those needed to demonstrate squeezed state generation. This success reinforces the plausibility of related quantum noise experiments, and has implications for fiber optic communications.
Optimizing the sensitivity for interferometric measurement of optical phase involves maximizing the intensity of the illuminating light source and minimizing any noise sources. Limits to the light source intensity may arise from the availability of a stable high-power laser or from damage to the system being measured. For example, we describe here a scanning confocal polarization microscope designed to image the dynamics of nerve networks. In this case the light intensity limit is set by several factors, including the scanning rate, the required response time and damage limits of the biological system. Assuming a stable experimental configuration for the interferometer and light source, the limiting noise source for visible light interferometry is shot-noise associated with the quantum state of the light inputs to the interferometer. Shot-noise can be thought of as arising from vacuum fluctuations entering the dark input port of the interferometer.(1) These vacuum field fluctuations combine with the light entering the other port to cause shot-noise in the detected photocurrent at the output ports of the interferometer. This shot-noise current normally limits the measured phase variance Δϕ to a value of 1/N where N is the average number of photons detected in a measurement interval. We describe an experiment which demonstrates that this shot-noise limit can be improved by injecting squeezed vacuum into the normally dark input port.(2) In this case the phase variance is reduced below the 1/N limit by the degree of squeezing. In contrast to early experiments,(2) we are now operating the interferometer in a dark fringe mode(3–5) in order to avoid setting a limit to the light level caused by detector saturation (near 1 mW).
Improvements in precision beyond the limit set by the vacuum state or zero-point fluctuations of the field are reported for two optical experiments. Both experiments employ the squeezed light from an optical parametric oscillator to reduce the level of fluctuations in one of two quadrature phase amplitudes below the shot-noise level (SNL). The first experiment follows the work of Caves1 on precision interferometry with squeezed states. An increase in signal-to-noise of 2.5 dB relative to the SNL is demonstrated for the detection of phase modulation in a Mach-Zehnder interferometer. In the second experiment amplitude modulation is encoded on a weak signal beam with an acoustooptic modulator. For the case of a squeezed signal beam compared with a beam in a coherent state, an improvement in sensitivity of 2.5 dB is achieved for the detection of amplitude changes of the field. The observed increases in sensitivity in both experiments are currently limited by simple linear losses in propagation and detection and not by the degree of available squeezing.
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.