Effect of the Ensemble of Cold Atoms Position in the Brillouin Zone on the Optical Bistability and Entanglement Dynamics

Abstract

We consider the dynamics of an ensemble of cold atoms coupled to light through radiation pressure in an optical lattice. We show that the steady-state displacement of the atomic ensemble shows bistable behavior which can be tuned by the position of the cold atoms in the Brillouin zone. Further, we describe a scheme to transfer the quantum state of the light field to the atomic field of two ensembles of cold atoms inside two independent optical cavities. This method leads to quantum entanglement which can be used in implementing quantum communication and computing. An Einstein-Podolsky-Rosen (EPR) state may be achieved with a judicious choice of parameters for our system.

Appendices

Appendix: 1

The following correlations are obeyed by the quantum noise operators for the input cavity field [52]:

$$ \begin{array}{@{}rcl@{}} &&\left\langle a_{in}(t)=0 \right\rangle \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} &&<a_{in}(t)a_{in}(t^{\prime})>=<a_{in}^{\dagger}(t)a_{in}(t^{\prime})>=0, \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} &&<a_{in}(t)a_{in}^{\dagger}(t^{\prime})>=\delta(t-t^{\prime}). \end{array} $$

(44)

Also, the quadrature noise operators for the input cavity field in the Fourier space obey the following correlations [52]:

$$ \begin{array}{@{}rcl@{}} &&<x_{1in}(\omega)x_{1in}(\omega^{\prime})>=<y_{1in}(\omega)y_{1in}(\omega^{\prime})>=2 \pi \delta(\omega + \omega^{\prime}), \end{array} $$

(45)

$$ \begin{array}{@{}rcl@{}} &&<x_{1in}(\omega)y_{1in}(\omega^{\prime})>=2 i \pi \delta (\omega +\omega^{\prime}), <y_{1in}(\omega)x_{1in}(\omega^{\prime})>=-2 i \pi \delta(\omega +\omega^{\prime}). \end{array} $$

(46)

Appendix: 2

The values of coefficients A_i,i = 1, 2, 3 in Eq. 41 are defined as follows:

$$ \begin{array}{@{}rcl@{}} A_{1}&=&\frac{2\pi}{({B_{1}^{2}}(0)+{C_{1}^{2}}(0))^{2}}({L_{1}^{2}}(0)+{F_{1}^{2}}(0)), \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} A_{2}&=&\frac{4\pi}{\left[ ({B_{1}^{2}}(0)+{C_{1}^{2}}(0))({B_{2}^{2}}(0)+{C_{2}^{2}}(0))({\kappa_{1}^{2}}+{\Delta}_{d1}^{2})\right] }(\delta_{1}+\delta_{2}+\delta_{3}), \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} A_{3}&=&\frac{2\pi}{\left[({B_{2}^{2}}(0)+{C_{2}^{2}}(0))^{2}({\kappa_{1}^{2}}+{\Delta}_{d1}^{2})^{2} \right] }(\delta_{4}+\delta_{5}+\delta_{6}+\delta_{7}+\delta_{8}+\delta_{9}+\delta_{10}+\delta_{11}), \end{array} $$

(49)

where,

$$ \begin{array}{@{}rcl@{}} \delta_{1}&=&\frac{g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}L_{2}(0)-\kappa_{1}F_{2}(0)\right)\left( {L_{1}^{2}}(0)+{F_{1}^{2}}(0)\right)}{ {B_{1}^{2}}(0)+{C_{1}^{2}}(0)}, \end{array} $$

(50)

$$ \begin{array}{@{}rcl@{}} \delta_{2}&=&\left[({\kappa_{1}^{2}}-{\Delta}_{d1}^{2})L_{2}(0)+2{\Delta}_{d1}\kappa_{1}F_{2}(0)\right]L_{1}(0), \end{array} $$

(51)

$$ \begin{array}{@{}rcl@{}} \delta_{3}&=&\left[({\kappa_{1}^{2}}-{\Delta}_{d1}^{2})F_{2}(0)-2{\Delta}_{d1}\kappa_{1}L_{2}(0)\right]F_{1}(0), \end{array} $$

(52)

$$ \begin{array}{@{}rcl@{}} \delta_{4}&=&\frac{\left( g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}L_{2}(0)-\kappa_{1}F_{2}(0)\right) \right)^{2}\left( {L_{1}^{2}}(0)+{F_{1}^{2}}(0)\right)}{\left( {B_{1}^{2}}(0)+{C_{1}^{2}}(0)\right)^{2}}, \end{array} $$

(53)

$$ \begin{array}{@{}rcl@{}} \delta_{5}&=&\frac{\left( g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}F_{2}(0)+\kappa_{1}L_{2}(0)\right) \right)^{2}\left( {L_{1}^{2}}(0)+{F_{1}^{2}}(0)\right)}{\left( {B_{1}^{2}}(0)+{C_{1}^{2}}(0)\right)^{2}}, \end{array} $$

(54)

$$ \begin{array}{@{}rcl@{}} \delta_{6}&=&\left[({\kappa_{1}^{2}}-{\Delta}_{d1}^{2})L_{2}(0)+2{\Delta}_{d1}\kappa_{1}F_{2}(0)\right]^{2}, \end{array} $$

(55)

$$ \begin{array}{@{}rcl@{}} \delta_{7}&=&\left[({\kappa_{1}^{2}}-{\Delta}_{d1}^{2})F_{2}(0)-2{\Delta}_{d1}\kappa_{1}L_{2}(0)\right]^{2}, \end{array} $$

(56)

$$ \begin{array}{@{}rcl@{}} \delta_{8}&=&2\frac{g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}L_{2}(0)-\kappa_{1}F_{2}(0)\right)\sqrt{\delta_{6}}L_{1}(0)}{ {B_{1}^{2}}(0)+{C_{1}^{2}}(0)}, \end{array} $$

(57)

$$ \begin{array}{@{}rcl@{}} \delta_{9}&=&2\frac{g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}L_{2}(0)-\kappa_{1}F_{2}(0)\right)\sqrt{\delta_{7}}F_{1}(0)}{ {B_{1}^{2}}(0)+{C_{1}^{2}}(0)}, \end{array} $$

(58)

$$ \begin{array}{@{}rcl@{}} \delta_{10}&=&-2\frac{g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}F_{2}(0)+\kappa_{1}L_{2}(0)\right)\sqrt{\delta_{6}}F_{1}(0)}{ {B_{1}^{2}}(0)+{C_{1}^{2}}(0)}, \end{array} $$

(59)

$$ \begin{array}{@{}rcl@{}} \delta_{11}&=&2\frac{g_{c1}\sqrt{2\kappa_{1}}\left( {\Delta}_{d1}F_{2}(0)+\kappa_{1}L_{2}(0)\right)\sqrt{\delta_{7}}L_{1}(0)}{ {B_{1}^{2}}(0)+{C_{1}^{2}}(0)}. \end{array} $$

(60)

In the above equations, we have used the values of B_j(ω), C_j(ω), L_j(ω) and F_j(ω) (j is 1 or 2) from the Eqs. 35–38 at ($\omega \rightarrow 0$) in the main text.