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Journal of Optics B Quantum and Semiclassical Optics

Quantum mechanical rotations, information transfer and decoherence in the converter limit of parametric processes

A. Bandilla, G. Drobný, Dragomir M Davidović

Published Apr 9, 2002• 0 citations
Publisher

Abstract

In classical three-wave interaction there is a regime where only the phases move but no energy is exchanged. This is strictly valid for classical trajectories with sharp amplitudes and phases. Quantum mechanically, such a phase motion regime can be realized only approximately if all three waves start in coherent states because the states deform due to their initial uncertainties during rotation. In the limit of a strong signal or idler wave, what amounts to frequency conversion, we show that exact rotations of coherent states become possible. This completes our understanding of three-wave interaction. Different initial quantum states are exchanged periodically if the strong wave is treated classically and undepleted. This limit is investigated by numerical diagonalization of the exact Hamiltonian via nonclassical states in the pump and coherent states in the signal and idler. Classical phase motion turns out to be very helpful to understand the strong demands for the converter limit. In the converter, we consider the impurity of the two modes and show that the coherent state in one mode, which enables the no-energy transfer regime, has no influence on the impurity of the other wave and can be substituted by a vacuum state. Thus, the impurity in such a regime behaves exactly as under simple one-photon losses although the energy can be held constant. The different rotations are illustrated by calculating the Wigner functions via classical trajectories. In addition, we also consider the equivalent emission from a prepared atomic system (squeezed atoms) into one mode.

DOI
10.1088/1464-4266/4/3/305

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In classical three-wave interaction there is a regime where only the phases move but no energy is exchanged. This is strictly valid for classical trajectories with sharp amplitudes and phases. Quantum mechanically, such a phase motion regime can be realized only approximately if all three waves start in coherent states because the states deform due to their initial uncertainties during rotation. In the limit of a strong signal or idler wave, what amounts to frequency conversion, we show that exact rotations of coherent states become possible. This completes our understanding of three-wave interaction. Different initial quantum states are exchanged periodically if the strong wave is treated classically and undepleted. This limit is investigated by numerical diagonalization of the exact Hamiltonian via nonclassical states in the pump and coherent states in the signal and idler. Classical phase motion turns out to be very helpful to understand the strong demands for the converter limit. In the converter, we consider the impurity of the two modes and show that the coherent state in one mode, which enables the no-energy transfer regime, has no influence on the impurity of the other wave and can be substituted by a vacuum state. Thus, the impurity in such a regime behaves exactly as under simple one-photon losses although the energy can be held constant. The different rotations are illustrated by calculating the Wigner functions via classical trajectories. In addition, we also consider the equivalent emission from a prepared atomic system (squeezed atoms) into one mode.
We consider the classical and quantum-mechanical processes of three-wave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the out-of-phase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly sub-Poissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sum-frequency generation. \textcopyright{} 1996 The American Physical Society.
A description of the parametric amplifier and frequency converter is presented without introducing the classical (i.e., parametric) approximation for the pumping field. Constants of the motion are found which reduce the solution of the Schr\"odinger equation to the diagonalization of a matrix. This diagonalization is accomplished numerically, and the eigenvalues and eigen-functions of a system with fixed energy are calculated. The time-dependent behavior of the mean number of photons in the amplified or frequency up-converted field is presented. The time evolution of the probability distributions is illustrated. The technique is extended to the problem of coherent spontaneous emission from a system of NNN two-level atoms interacting with the radiation field where both the atomic system and the radiation field are quantized.
We re-examine the properties of the states produced by nondegenerate coherently stimulated parametric down-conversion wherein the signal and idler modes are seeded with coherent states of light and where the nonlinear crystal is driven by a strong classical field as described by the parametric approximation. The states produced are the two-mode squeezed coherent states defined with a specific ordering of operators, namely, the displacement operators of the two modes acting on the double vacuum state followed by the action of the two-mode squeeze operator representing the down-conversion process. Though mathematically equivalent to the reverse ordering of operators, but with different displacement parameters, the ordering we consider is closely related to what could most easily be implemented in the laboratory. The statistical properties of the state are studied with an emphasis on how they, and its average photon number, are affected by the various controllable phases, namely, those of the classical pump field of the two input coherent states. We then consider the multiphoton interference effects that arise if the two beams are overlapped on a 50:50 beam splitter, investigating the role of the phases in controlling the statistical properties of the output states. Finally, we study the prospects for the application of the states to quantum-optical interferometry to obtain sensitivities in phase-shift measurements beyond the standard quantum limit.
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 study spontaneous parametric processes starting from the quantized field equations in the Heisenberg picture using the common approximation of treating the laser pump as classical. After a (standard) linearization procedure we pass to the Wigner representation and the process becomes formally equivalent to the parametric interaction of two incoming zero-point modes with a pumping laser wave coupled by an optically nonlinear medium. Then the treatment of parametric down conversion and parametric up conversion are described by the Maxwell equations and look like a parametric amplification of the vacuum field. The fourth-order correlation function between the signal and idler fields is obtained in a form that makes for easy comparison with experimental results.
We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}}_{I}=i\ensuremath{\hbar}\ensuremath{\kappa}\left(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{\ifmmode\dagger\else\textdagger\fi{}}c\right)$, wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by nondegenerate coherently stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two-mode state have been shown to depend greatly on the cumulative phase $\mathrm{\ensuremath{\Phi}}={\ensuremath{\theta}}_{s}+{\ensuremath{\theta}}_{i}\ensuremath{-}2\ensuremath{\phi}$ where ${\ensuremath{\theta}}_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2\ensuremath{\phi}$ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the near-thermal-like nature of the steady state valid for long times. Finally, we examine the effect of the cumulative phase $\mathrm{\ensuremath{\Phi}}$ on the correlations between the bipartite system of pump and signal/idler modes, and the effect of the quantized pump on the correlations between signal and idler modes.