## Abstract

Resonances of the six-wave mixing susceptibility ${\chi}^{(5)}({\omega}_{1},{\omega}_{1},{\omega}_{1},-{\omega}_{2},-{\omega}_{2})$ at frequency $3{\omega}_{1}-2{\omega}_{2}$ of a system of two-level atoms dissipating through the spontaneous emission and collisions and driven by a weak bichromatic external field of frequencies ${\omega}_{1}$ and ${\omega}_{2}$ 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.

© 2000 Optical Society of America

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### Equations (24)

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(1)
$$\frac{\mathrm{d}\rho}{\mathrm{d}t}=L\rho +{L}_{e}(t)\rho ,$$
(2)
$$L\rho =-i[{H}_{0},\rho ]+{L}_{a}\rho +{L}_{f}\rho +{L}_{c}\rho ,$$
(3)
$${L}_{e}(t)\rho =-i[{H}_{e}(t),\rho ],$$
(4)
$${H}_{0}={\omega}_{0}{S}_{z}+{\omega}_{0}{a}^{\u2020}a+g({S}_{+}a+{a}^{\u2020}{S}_{-}),$$
(5)
$${L}_{f}\rho =\kappa (2a\rho {a}^{\u2020}-{a}^{\u2020}a\rho -\rho {a}^{\u2020}a)$$
(6)
$${L}_{a}\rho =\gamma (2{S}_{-}\rho {S}_{+}-{S}_{+}{S}_{-}\rho -\rho {S}_{+}{S}_{-}),$$
(7)
$${L}_{c}\rho ={\gamma}_{c}(2{S}_{z}\rho {S}_{z}-S_{z}{}^{2}\rho -\rho S_{z}{}^{2})$$
(8)
$${H}_{e}(t)=d[{\u220a}_{1}exp(-i{\omega}_{1}t)+{\u220a}_{2}exp(-i{\omega}_{2}t)]{S}_{+}+\mathrm{h}.\mathrm{c}.,$$
(9)
$${\rho}^{(5)}={\rho}^{(5)}(1)+{\rho}^{(5)}(2)+{\rho}^{(5)}(3)+{\rho}^{(5)}(4),$$
(10)
$${\rho}^{(5)}(1)={d}^{*2}{d}^{3}\frac{1}{L+i\mathrm{\Omega}}\left[{S}_{+},\frac{1}{L+2i({\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\left\{\frac{1}{L+i({\omega}_{1}-2{\omega}_{2})}\left[{S}_{+},\frac{1}{L-2i{\omega}_{2}}\times \left[{S}_{-},\frac{1}{L-i{\omega}_{2}}[{S}_{-},\rho (0)]\right]\right]+\frac{1}{L+i({\omega}_{1}-2{\omega}_{2})}\left[{S}_{-},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\frac{1}{L-i{\omega}_{2}}[{S}_{-},\rho (0)]\right]\right]+\frac{1}{L+i({\omega}_{1}-2{\omega}_{2})}\left[{S}_{-},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\times \left[{S}_{-},\frac{1}{L+i{\omega}_{1}}[{S}_{+},\rho (0)]\right]\right]\right\}\right]\right],$$
(11)
$${\rho}^{(5)}(2)={d}^{*2}{d}^{3}\frac{1}{L+i\mathrm{\Omega}}\left[{S}_{+},\frac{1}{L+2i({\omega}_{1}-{\omega}_{2})}\times \left[{S}_{-},\left\{\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\left[{S}_{-},\frac{1}{L+i{\omega}_{1}}\times [{S}_{+},\rho (0)]\right]\right]+\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\left[{S}_{+},\frac{1}{L-i{\omega}_{2}}\times [{S}_{-},\rho (0)]\right]\right]+\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{-},\frac{1}{L+2i{\omega}_{1}}\times \left[{S}_{+},\frac{1}{L+i{\omega}_{1}}[{S}_{+},\rho (0)]\right]\right]\right\}\right]\right],$$
(12)
$${\rho}^{(5)}(3)={d}^{*2}{d}^{3}\frac{1}{L+i\mathrm{\Omega}}\left[{S}_{-},\frac{1}{L+i(3{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\left\{\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\left[{S}_{-},\frac{1}{L+i{\omega}_{1}}\times [{S}_{+},\rho (0)]\right]\right]+\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{+},\frac{1}{L+i({\omega}_{1}-{\omega}_{2})}\left[{S}_{+},\frac{1}{L-i{\omega}_{2}}\times [{S}_{-},\rho (0)]\right]\right]+\frac{1}{L+i(2{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{-},\frac{1}{L+2i{\omega}_{1}}\times \left[{S}_{+},\frac{1}{L+i{\omega}_{1}}[{S}_{+},\rho (0)]\right]\right]\right\}\right]\right],$$
(13)
$${\rho}^{(5)}(4)={d}^{*2}{d}^{3}\frac{1}{L+i\mathrm{\Omega}}\left[{S}_{-},\frac{1}{L+i(3{\omega}_{1}-{\omega}_{2})}\times \left[{S}_{-},\left\{\frac{1}{L+3i{\omega}_{1}}\left[{S}_{+},\frac{1}{L+2i{\omega}_{1}}\times \left[{S}_{+},\frac{1}{L+i{\omega}_{1}}[{S}_{+},\rho (0)]\right]\right]\right\}\right]\right].$$
(14)
$$\rho (0)=|0,-1/2\u3009\u30080,-1/2|,$$
(15)
$$|{\psi}_{\pm}^{m}\u3009=\frac{1}{\sqrt{2}}[|m,\xbd\u3009\pm |m+1,-\xbd\u3009],$$
(16)
$$m=0,1,2,3,\dots ,$$
(17)
$${H}_{0}|{\psi}_{\pm}^{m}\u3009={\mathrm{\lambda}}_{\pm}^{m}|{\psi}_{\pm}^{m}\u3009,$$
(18)
$${\mathrm{\lambda}}_{\pm}^{m}=(m+\xbd){\omega}_{0}\pm g\sqrt{m+1}.$$
(19)
$$(L+i\alpha )X=A(\alpha )X+\mathit{BY},$$
(20)
$$(L+i\alpha {)}^{-1}X={A}^{-1}X-{A}^{-1}\xb7B\xb7(L+i\alpha {)}^{-1}Y.$$
(21)
$$(L+i\alpha )X=\mathit{AX}+\mathit{BY},\hspace{1em}\hspace{1em}(L+i\alpha )Y=\mathit{CY}+\mathit{DZ},$$
(22)
$$(L+i\alpha )Z=\mathit{FZ},$$
(23)
$$(L+i\alpha {)}^{-1}X=\mathit{EX}-{E}^{\prime}Y+{E}^{\u2033}Z,$$
(24)
$$n({\omega}_{1}-{\omega}_{2})=\pm {\omega}_{\mathrm{hfs}}^{m},\hspace{1em}\mathrm{or}\hspace{1em}({\omega}_{1}-{\omega}_{2})=\pm \frac{{\omega}_{\mathrm{hfs}}^{m}}{n},$$