Nonclassical dynamics with time- and intensity-dependent coupling

Abstract

Nonclassical light and collapse-revival dynamics are consequences of dynamical quantum interference in transient photon-atom interaction. We study the time evolution of atom and photons in a high quality cavity for time-dependent atom-field coupling, with different initial field states and initial atomic states. The inversion for initial superposed atomic state seems to be independent of initial classical fields but can be stimulated by the Schrodinger cat field. Interesting effects of the transient coupling are found through analysis of the collapse-revival in population inversion and the features in the Wigner function. Oscillatory coupling coefficient can prolong the occurrence of collapse, in analogy to the Zeno effect. The intensity atom-field coupling duration is an important parameter for controlling atomic inversion and producing frozen nonclassical light in the cavity after the atom-field coupling ceases.

Appendices

Appendix A: density matrix elements

In general, the density matrix elements of the field $\rho _{nm}$ are obtained by tracing out the atomic system (subscript ”s”), $\rho _{nm}=\langle n|\hat{\rho }_{f}|m\rangle =\sum \nolimits _{x}\langle n|\hat{\rho }_{xx}|m\rangle$ where $\hat{\rho }_{f}=Tr_{s}\{\hat{\rho } (t)\}=\sum \nolimits _{x=a,b}\hat{\rho }_{xx}$ is the reduced density matrix of the field (subscript ”f”), $\hat{\rho }_{xx}=\langle x|\hat{\rho }|x\rangle \,(x=a,b)$ and $\langle n|\hat{\rho }_{xx}|m\rangle =C_{x,n}(t)C_{x,m}^{*}(t)\,(x=a,b)$. The transient state of the field is governed by the matrix element of the field $\rho _{nm}(t)$. For the two-level atom,

$$\begin{aligned} \rho _{nm}(t) =\langle n|\{\hat{\rho }_{aa}(t)+\hat{\rho }_{bb} (t)\}|m\rangle =C_{a,n}(t)C_{a,m}^{*}(t)+C_{b,n}(t)C_{b,m}^{*}(t)\quad \end{aligned}$$

(26)

where $C_{x,n}(t)$ is obtained from the solutions of the standard coupled Eqs. (3) and (4) for a two-level atom.

Initially, the atomic system $\hat{\rho }_{s}(0)$ is uncorrelated to the field state $\hat{\rho }_{f}(0)$, i.e. $\hat{\rho }(0)=\, \hat{\rho }_{f}(0)\otimes \hat{\rho }_{s}(0)$ since both are not coupled. Thus, the initial coefficients can be decomposed into products of the atomic (subscript ’$x=a,b$’) and photonic coefficients (subscript ’$n$’),

$$\begin{aligned} C_{x,n}(0)=C_{x}(0)C_{n}(0) \end{aligned}$$

(27)

satisfying $|C_{a}(0)|^{2}+|C_{b}(0)|^{2}= \rho _{aa}(0)+\rho _{bb}(0)=1$ with $\rho _{xx}(0)=\langle x|\hat{\rho }_{s}(0)|x\rangle =C_{x}(0)C_{x}^{*}(0)$. Thus $C_{x,n}(0)C_{y,m}^{*}(0)=C_{x}(0)C_{y}^{*}(0)\rho _{nm}(0)$. The initial coherence between photon numbers is

$$\begin{aligned} {\rho _{nm}}(0)=\langle n|{\hat{\rho }}_{f}(0)|m\rangle =C_{n}(0)C_{m}^{*}(0) \end{aligned}$$

(28)

where we use Eqs. (26) and (27). As expected, the initial matrix element of the field $\rho _{nm}(0)$ does not depend on the atomic initial conditions.

Appendix B: Wigner function versus density matrix elements

Various physical parameters have been defined to determine the nonclassical states of light as well as the degree of nonclassicality [27]. The typical ones are the squeezing parameter $S=\langle \hat{p}^{2}\rangle -\langle \hat{p}\rangle ^{2}$ with $\hat{p}=\hat{a}e^{i\theta }+\hat{a} ^{\dagger }e^{-i\theta }$, antibunching in $G^{(2)}$, entanglement criteria, Mandel’s $Q=(\langle \hat{n}^{2}\rangle -\langle \hat{n}\rangle ^{2})/\langle \hat{n}\rangle ,\hat{n}=\hat{a}^{\dagger }\hat{a}$ and the Wigner function W. Each quantity displays nonclassicality from certain aspects. There are states which do not meet all the nonclassical criteria. For example, Schrödinger cat is not squeezed and has $Q>0$ (super-Poissonian statistics) for certain values of phase $\theta$, but it shows nonclassicality through negativity in W and Agarwal’s parameter [28]. The W is a reliable quantity for studying PhA and PhS processes for producing nonclassical states. The relationship between W and the matrix elements for the field $\rho _{nm}$ can be obtained using [9, 29],

$$\begin{aligned} \frac{W(\alpha ,t)}{K(\alpha )}=\sum _{m,n=0}^{\infty }\rho _{nm}(t) \int \langle -\beta |n\rangle \langle m|\beta \rangle e^{2(\beta ^{*}\alpha -\beta \alpha ^{*})}d^{2}\beta \end{aligned}$$

(29)

where $K(\alpha )=\frac{2e^{-2\left| \alpha \right|^{2}}}{\pi }$. We use $\hat{\rho }_{f}(t)=\sum _{m,n}^{\infty }|n\rangle \rho _{nm}(t)\langle m|$ the field state in photon number basis $\{|n\rangle \}$ with the field matrix element $\rho _{nm}(t)$ given by Eq. (26). The complex coherent state variable $\alpha =re^{i\theta }$ can be mapped into polar coordinates, r and $\theta$. The $|r|^{2}$ corresponds to the number of photons while $\theta$ is the coherent state phase. Similarly for $\beta$. Inserting $\langle n|\alpha \rangle =e^{-|\alpha |^{2}/2}\frac{\alpha ^{n}}{\sqrt{n!}}$ and $\int \int \beta ^{m}\beta ^{*n}e^{-|\beta |^{2}}e^{(\beta ^{*}2\alpha -\beta 2\alpha ^{*})}d^{2}\beta =\pi L_{n}^{m-n}(|2\alpha |^{2})n!e^{-|2\alpha |^{2}}(2\alpha )^{m-n}$ we obtain the required Eq. (14)

In view of $C_{b,0}(t)=0$, straightforward calculation from Eq. (14) gives

$$\begin{aligned} \frac{W(\alpha , t)}{K(\alpha )}=\sum _{m=0}^{\infty }(-1)^{m} \{L_{m}^{0}(x)|C_{a,m}(t)|^{2}-L_{m+1}^{0}(x)|C_{b,m+1}(t)|^{2}\}+ \end{aligned}$$

(30)

$$\begin{aligned}&\sum _{m=1}^{\infty }\sum _{n=0}^{m-1}(-1)^{n}\left[ \sqrt{\frac{n!}{m!}} L_{n}^{k}(x)2Re\{z^{k}C_{a,n}(t) C_{a,m}^{*}(t)\}\right.\nonumber \\&\quad \quad \left.-\sqrt{\frac{n+1!}{m+1!}} L_{n+1}^{k}(x)2Re\{z^{k}C_{b,n+1}(t)C_{b,m+1}^{*}(t)\}\right].\nonumber \end{aligned}$$

(31)