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.
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.
A squeezed state of light is usually taken to mean a state in which the fluctuations of one of the quadratures are below the level associated with a coherent or vacuum state. These squeezed fluctuations are phase sensitive in the sense that they vary periodically as the field evolves, but the relationship between squeezing and phase fluctuations is not trivial. While squeezed states can exhibit an arbitrarily low level of phase fluctuations, the phase states, which are eigenstates of the Hermitian optical phase operator are not themselves quadrature squeezed. A phasesqueezed state is one with a lower level of phase fluctuations than a coherent state with the same mean photon number. Phase squeezing is fundamentally different from quadrature squeezing in much the same way that sub-Poissonian photon statistics is different. For this reason optimally squeezed states and states with minimum phase noise will be distinct. We derive the form of the states which minimum phase noise for a given intensity. These have an Airy function behavior for the photon number probability distribution. These phase squeezed states have a far better phase resolution than coherent states of similar intensity, but do not seem to offer a significant reduction of phase fluctuations compared with conventional squeezed states.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
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.
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.
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, \ensuremath{\Lambda}, and ``ladder'' systems).We show that for V and \ensuremath{\Lambda} 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.