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Journal of the Optical Society of America B

Collision-induced quantum effects in four-wave mixing in a high-Q cavity

Ravinder Rupchand Puri, Aditi Ray

Published Jun 1, 1997• 2 citations
Publisher

Abstract

The four-wave mixing susceptibility χ(3)(ω1, ω1, -ω2) of a system of two-level atoms driven by weak external fields of frequencies ω1 and ω2 in a high-Q cavity is evaluated in the presence of atomic collisions. That susceptibility exhibits resonances corresponding to dipole-allowed transitions between the first two excited manifolds of the dressed states and the ground state and also the collision-induced Bloembergen resonances. The resonances corresponding to transitions between the first two manifolds of excited states that characterize the quantum and nonlinear nature of the atom–field interaction are, however, flattened in the absence of collisions. The collisions thus induce the quantum effects in four-wave mixing in a high-Q cavity.

DOI
10.1364/josab.14.001289

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The four-wave mixing susceptibility χ(3)(ω1, ω1, -ω2) of a system of two-level atoms driven by weak external fields of frequencies ω1 and ω2 in a high-Q cavity is evaluated in the presence of atomic collisions. That susceptibility exhibits resonances corresponding to dipole-allowed transitions between the first two excited manifolds of the dressed states and the ground state and also the collision-induced Bloembergen resonances. The resonances corresponding to transitions between the first two manifolds of excited states that characterize the quantum and nonlinear nature of the atom–field interaction are, however, flattened in the absence of collisions. The collisions thus induce the quantum effects in four-wave mixing in a high-Q cavity.
Resonances of the six-wave mixing susceptibility χ(5)(ω1, ω1, ω1,-ω2,-ω2) at frequency 3ω1-2ω2 of a system of two-level atoms dissipating through the spontaneous emission and collisions and driven by a weak bichromatic external field of frequencies ω1 and ω2 in a finite Q cavity are analyzed. It is shown that the atomic collisions not only induce the usual Bloembergen resonances but that they also play a major role in enhancing the transitions among the higher-order dressed states, thereby bringing out some of the quantum resonances that are suppressed in the absence of collisions. The important contribution of collisions in this higher-order nonlinear process is the generation of the subharmonic resonances at ±1/2 times the hyperfine splitting of the first and the second excited manifolds.
The four-wave-mixing signal at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ from a collective system of NNN two-level atoms dissipating through spontaneous emission and dephasing and driven by weak external fields of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ in a cavity of arbitrary QQQ is found by evaluating exactly the susceptibility ${\ensuremath{\chi}}^{(3)}({\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{1},\ensuremath{-}{\ensuremath{\omega}}_{2})$. The exact results show that the customarily employed secular approximation for treating the problem of a strongly coupled atom-cavity system is hopelessly inadequate for describing the process of four-wave mixing.
The entanglement properties of two mode fields generated from four-wave mixing are discussed in the system of an ensemble of VVV-type three-level atoms embedded in a two-mode cavity, in which the atomic transitions from the two excited states to the ground state are driven by one strong and one relative weak laser field, respectively. The nondegenerate four-wave mixing process occurs in the strong laser-driven transition, which is also coupled by the two cavity modes. With the help of the auxiliary atomic transition driven by the weak laser field and adjusting the frequency difference of the two modes, highly squeezed and entangled light with high intensity can be generated.
A theory of four-wave mixing including effects of velocity-changing collisions is presented. Three fields with frequencies \ensuremath{\omega}, \ensuremath{\omega}, and \ensuremath{\omega}+\ensuremath{\delta} are incident on a vapor of ``two-level'' atoms having upper state b and lower state a. Two of the fields are counterpropagating and the third (of frequency \ensuremath{\omega}+\ensuremath{\delta}) makes a small angle with one of the others. The frequency \ensuremath{\omega} is a nearly resonant (inside the Doppler width) with the a-b transition frequency. The phase-conjugate signal emitted at frequency \ensuremath{\omega}-\ensuremath{\delta} is calculated as a function of \ensuremath{\delta}. Using a simple collision model in which collisions are phase interrupting in their effect on atomic coherence and velocity-changing in their effect on level populations, we discuss the conditions under which resonances characterized by the upper or lower radiative and collision rates can be observed. Assuming that the total (a+b) state population is conserved in the absence of collisions, it is shown that velocity-changing collisions can ``open'' the system and lead to a resonance characterized by the lower-state width (convoluted with the residual Doppler width). With increasing pressure, the width of this induced resonance structure decreases monotonically. For sufficiently high pressure, the collisional redistribution of velocity classes is complete---the system ``recloses'' and the narrow resonance disappears. The interplay of the collision-induced opening, line narrowing, and reclosing of the system is discussed, as is the relationship of these narrow resonances to the so-called pressure-induced extra resonances of Bloembergen and co-workers [Indian J. Pure Appl. Phys. 16, 151 (1978); Phys. Rev. Lett. 46, 111 (1981)].
It is shown that the collision-induced four-wave mixing signal in two-photon quasi-resonant excitation has the property of frequency non-mutuality, i.e. when the probe field frequency omega s is replaced by the signal frequency omega out the four-wave mixing coherent signal does not arise. The reason for such frequency non-mutuality lies in the nature of the collision-induced coherent scattering. It is shown that this scattering occurs because of double non-adiabatic transitions (as affected by collisions) between the states of a dressed atom without changing the environment (buffer) state during the four-photon-scattering elementary process; this scattering will be referred to as coherent collisionally double induced scattering (CCDIS).
We study two-photon resonant nondegenerate four-wave mixing (NFWM) in a dressed cascade four-level system. In the presence of a strong coupling field, the two-photon resonant NFWM spectrum exhibits Autler-Townes splitting, accompanied by either suppression or enhancement of the NFWM signal. Such phenomena are demonstrated in Ba through inducing of atomic coherence between the ground state 6s26{s}^{2}6s2 and the doubly excited autoionizing Rydberg state 6pnd6pnd6pnd. This technique provides a spectroscopic tool for measuring not only the resonant frequency and dephasing rate but also the transition dipole moment between two highly excited atomic states.