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Physical Review A

Quantum state exchange between indirectly coupled modes

Antonio S. M. de Castro, J. F. Peruzzo, V. V. Dodonov

Published Mar 17, 2005• 8 citations
Publisher

Abstract

The problem of quantum state exchange between three coupled quantum harmonic oscillators (which can represent interacting modes of the electromagnetic field in nonlinear crystal or interaction between laser beams and vibrational modes of ions in traps) is considered. We review different definitions of state exchange and find conditions of its possibility for arbitrary initial states in a chain of three oscillators with bilinear resonance coupling (corresponding to the rotating wave approximation), analyzing the structure of integrals of motion and propagators of the Schr\"odinger equation, as well as the Wigner functions. The influence of different parameters (such as detunings between coupling constants and temperatures of the intermediate oscillator and ``receiver'') on the fidelity of exchange is analyzed.

DOI
10.1103/physreva.71.032319

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The problem of quantum state exchange between three coupled quantum harmonic oscillators (which can represent interacting modes of the electromagnetic field in nonlinear crystal or interaction between laser beams and vibrational modes of ions in traps) is considered. We review different definitions of state exchange and find conditions of its possibility for arbitrary initial states in a chain of three oscillators with bilinear resonance coupling (corresponding to the rotating wave approximation), analyzing the structure of integrals of motion and propagators of the Schr"odinger equation, as well as the Wigner functions. The influence of different parameters (such as detunings between coupling constants and temperatures of the intermediate oscillator and ``receiver'') on the fidelity of exchange is analyzed.
The nonlinear interaction of three ions in a linear trap with laser light is studied by taking into account the three vibrational modes within a rotating-wave approximation. It is demonstrated that, for localization conditions far from the Lamb-Dicke limit, all three modes have a significant impact on the dynamics of the system. Preparations of vibrational Fock states can be used to realize two different regimes with an exactly solvable three-atom dynamics. The first one corresponds to a nonlinear version of the Tavis-Cummings coupling, whereas the Tavis-Cummings type interactions are suppressed in the second one.
We discuss the problem of information transfer (exchange of states configuration) between two interacting quantum systems along their evolution in time. We consider the specific case of two modes of an electromagnetic field with a rotating wave coupling interaction (up-conversion in a nonlinear crystal). We verify that for certain initial states of the fields the swapping of state configuration occurs with conservation of the mean energy of each mode (without energy transfer), thus characterizing pure information flow.
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.
Efficient Three-Dimensional Sub-Doppler Cooling of 40^{40}40Ca+^++ in a Penning Trap
We demonstrate efficient sub-Doppler laser cooling of the three eigenmodes of a 40^{40}40Ca+^++ ion confined in a compact Penning trap operating with a magnetic field of 0.91 T. Using the same set of laser beams as required for the initial Doppler laser cooling operation, we detune the laser frequencies to produce a narrow two-photon dark resonance. The process achieves a 1/e cooling time constant of 108(8) μμμs, ultimately reducing the mean thermal axial mode occupation from 72(23) to 1.5(3) in 800 μμμs as measured by resonantly probing an electric quadrupole transition near 729 nm. A parametric drive is applied to the trap electrodes which coherently exchanges the axial mode occupation with that of each radial mode, allowing for three-dimensional sub-Doppler cooling using only the axially-propagating laser beams. This sub-Doppler cooling is achieved for an axial oscillation frequency of $ω_z = 2π~\times~$221 kHz, which places the motion well outside of the Lamb Dicke confinement regime at the Doppler laser cooling limit. Our measured cooling rate and final mode occupation are in good agreement with a semiclassical model which combines a Lindblad master equation solution for ion-photon interactions with classical harmonic oscillator motion of the trapped ion.
Although the problem of electromagnetic radiation by a quantum harmonic oscillator is considered in textbooks on quantum mechanics, some of its aspects have remained unclear until now. By this, we mean that usually the initial quantum states of both the oscillator and the field are assumed to be characterized by a definite energy level of the oscillator and definite occupation numbers of the field modes. In connection with growing interest in squeezed states, it would be interesting to analyze the general case when the initial states of both subsystems are arbitrary superpositions of energy eigenstates. This problem was considered in other work, where the power of the spontaneous emission was calculated in the case of an arbitrary oscillator's initial state, but the field was initially in a vacuum state. In the present article, we calculate the rate of the oscillator average energy, squeezing, and correlation parameter change under the influence of an arbitrary external radiation field. Some other problems relating to the interaction between quantum particles (atoms) or oscillators where the electromagnetic radiation is an arbitrary (in particular squeezed) state were investigated.