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Journal of Modern Optics

Phase fluctuations of coherent light coupled to a driven quantum oscillator with time-dependent mass and frequency

S. K. Singh, Swapan Mandal

Published Jun 10, 2008• 3 citations
Publisher

Abstract

We investigate the phase fluctuations of the input coherent light coupled to a quantum driven harmonic oscillator with time-dependent mass and frequency (DHTDMF). Under the Barnett–Pegg formalism, we calculate the phase fluctuation parameters U, Q, and S as functions of interaction time, coupling strength, intensity and phase of the input coherent light. In order to make the analytical studies feasible, we use ω(t) = ω(0) = ω, , and the periodically driven term . For far off-resonance condition (i.e. or ), the driven term hardly affects the phase fluctuation parameters U, Q, and S. The exact analytical expressions for the phase fluctuation parameters for various special cases are exhibited. The effects of vacuum field on the fluctuations of the phase parameters are clearly indicated. In order to have a sense for these analytical results, we give symbolic calculations for a few special situations. Even for the on-resonance condition , the effects of the driven term are negligible for low damping constant γ. Interestingly, for the strong input field, the values of the phase fluctuation parameters are independent of the intensity of the input field. For high intensity of the input field, low damping constant and for far off-resonance condition, the phase fluctuation parameters U, S and Q exhibit an increase and decrease compared to their initial values (i.e. t = 0). All possible cases of physical interest are also investigated. In particular, the effects of input vacuum field on the phase fluctuation parameters are discussed in detail. The phase of the input coherent light plays an important role in the calculations of the phase fluctuation parameters.

DOI
10.1080/09500340701749935

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We investigate the phase fluctuations of the input coherent light coupled to a quantum driven harmonic oscillator with time-dependent mass and frequency (DHTDMF). Under the Barnett–Pegg formalism, we calculate the phase fluctuation parameters U, Q, and S as functions of interaction time, coupling strength, intensity and phase of the input coherent light. In order to make the analytical studies feasible, we use ω(t) = ω(0) = ω, , and the periodically driven term . For far off-resonance condition (i.e. or ), the driven term hardly affects the phase fluctuation parameters U, Q, and S. The exact analytical expressions for the phase fluctuation parameters for various special cases are exhibited. The effects of vacuum field on the fluctuations of the phase parameters are clearly indicated. In order to have a sense for these analytical results, we give symbolic calculations for a few special situations. Even for the on-resonance condition , the effects of the driven term are negligible for low damping constant γ. Interestingly, for the strong input field, the values of the phase fluctuation parameters are independent of the intensity of the input field. For high intensity of the input field, low damping constant and for far off-resonance condition, the phase fluctuation parameters U, S and Q exhibit an increase and decrease compared to their initial values (i.e. t = 0). All possible cases of physical interest are also investigated. In particular, the effects of input vacuum field on the phase fluctuation parameters are discussed in detail. The phase of the input coherent light plays an important role in the calculations of the phase fluctuation parameters.
The higher order moments and hence the amplitude-squared squeezing of input coherent light coupled to a quantum driven harmonic oscillator with time-dependent mass and frequency (DHTDMF) are investigated. Exact analytical expressions for the purpose of investigating the amplitude-squared squeezing of the input coherent light are exhibited. The squeezing in the normal sense of the input coherent light is independent of the driven term. On the other hand, the higher order (amplitude-squared) squeezing is sensitive to the driven term. In order to realize the flavour of the analytical approach, we give an interesting special case for the DHTDMF having ω(t) = ω(0) = ω, and driven term . It is found that the effects of the driven term are negligible if the oscillator is in the off-resonance situation. The condition of strong input coherent light also ensures the suppression of the effects of driven terms. By using a complete analytical method, we give an approximate analytical form of the relevant equations for investigating the amplitude-squared squeezed states. We carefully examine the special situations when the oscillator is underdamped, overdamped and critically damped (from both directions). The possibility of getting amplitude-squared squeezing of the input vacuum field is also discussed. To provide the feelings of the analytical results and hence the deep realizations of the physical system, we give a few results graphically. The realization of the present model in classical and in quantum mechanical pictures are clearly indicated.
In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
The dynamics of the harmonic oscillator, with a fluctuating, time-dependent frequency, is studied in the quantum framework. For periodic fluctuations of period \ensuremath{\tau}, it is found that the mean energy diverges exponentially in time, provided \ensuremath{\tau} falls into certain ``bands,'' centered around ``quasiresonant'' values, for which the rate of increase is maximum. In agreement with the correspondence principle, this rate is shown to coincide exactly with the one obtained for the classic oscillator, in the same conditions [Loris Ferrari, Phys. Rev. B 56, 593 (1997)]. The notion of adiabaticity has been revisited in the case of exponentially diverging mean energy. It is found that the Heisenberg form of the Hamiltonian operator can be expressed as a new effective Hamiltonian, which is still harmonic, but has exponentially diverging coefficients. It is argued that the same results hold in general (that is, without any quasiresonance condition) if the frequency fluctuation in time is disordered. A brief discussion is presented about nonharmonic interactions leading to dissipative effects. Applications to real physical systems are outlined, with special reference to the electrons in a uniform and constant magnetic field, to which a small fluctuating component is added.
We explore the role of quantum fluctuations in the strong-coupling limit of the dissipative Jaynes–Cummings oscillator driven on resonance. For weak excitation, we derive analytical expressions for the spectrum and the intensity correlation function for the photons scattered by the two-state atom coupled to the coherently driven cavity mode. We do so by writing down a birth–death process adding the higher orders in the excitation strength needed to go beyond the pure-state factorization, following the method introduced in Carmichael (2008). Our results for the first and second-order correlation functions are complemented by the numerical investigation of the waiting-time distribution for the photon emissions directed sideways, and the comparison with ordinary resonance fluorescence. To close out our discussion, we increase the driving field amplitude and approach the critical point organizing a second-order dissipative quantum phase transition by depicting the excitation pathways in the intracavity field distribution for a finite system size.
Abstract Post-selected measurement is used as a fundamental axiom in quantum optics. Here, a standard model of post-selected measurement with the input coherent pointer state is considered and the influence of weak value parameters on the quantum phase distribution function is studied on the basis of some analytical calculations. On the other hand, the effects of post-selected measurement on quantum phase fluctuations of the input coherent pointer states have been investigated, where the Barnett-Pegg formalism for the measured phase operators is assumed. It has been observed that the quantum phase fluctuations depend on some parameters such as the coupling strength, average value of input photon number, phase angle of the input coherent pointer state and two fundamental parameters of the complex weak value. Lastly, the presence of nonclassical effect is also checked for different parameters of quantum phase fluctuations.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.