Dynamics of field nonclassicality in the Jaynes–Cummings model

Abstract

Employing an informational quantifier for nonclassicality of bosonic field based on the Wigner–Yanase skew information, we investigate the dynamics of field nonclassicality in the Jaynes–Cummings model evolving from several typical initial atom–field states: coherent states, Fock states, and thermal states for the field, and ground state, excited state, symmetric superposition state, and maximally mixed state for the atom. We reveal some intriguing features of the field dynamics and their sensitivity on the initial states from an informational perspective. The results show that field nonclassicality can be generated in most cases and exhibits complex dynamic patterns involving collapses and revivals. A remarkable observation is that the initial coherent field states, assisted by the atom, can be converted to almost maximally nonclassical field states constrained by the average photon number, even though all coherent states are the most classical and possess the same minimal nonclassicality among pure states. The field dynamics may be exploited to estimate the initial atom as well as the field states, prepare desired evolving states and process quantum information.

Appendix

For the purpose of completeness and convenience of evaluating the field nonclassicality, we first prove Eq. (8) and then derive the standard results concerning the reduced field states of the Jaynes–Cummings model evolving from several typical initial atom–field states.

To establish Eq.(8), noting that $[H_a\otimes \mathbf{1}, \mathbf{1}\otimes H_f]=0,$ and

$$\begin{aligned} {[}\sigma _z\otimes \mathbf{1},\ H_I]= & {} \lambda (\sigma _z \sigma _+ - \sigma _+\sigma _z)\otimes a + \lambda (\sigma _z\sigma _--\sigma _-\sigma _z)\otimes a^\dag \\= & {} 2\lambda (\sigma _+\otimes a-\sigma _-\otimes a^\dag ),\\ {[}{} \mathbf{1}\otimes a^\dag a,\ H_I]= & {} \lambda \sigma _+ \otimes (a^\dag a^2-aa^\dag a) +\lambda \sigma _-\otimes (a^\dag aa^\dag -a^{\dag 2}a)\\= & {} \lambda (\sigma _-\otimes a^\dag -\sigma _+\otimes a). \end{aligned}$$

Therefore, if $\mu =\omega ,$ noting that $H_a=\omega \sigma _z/2$ and $H_f=\mu a^\dagger a,$ we have

$$\begin{aligned} {[}H_a\otimes \mathbf{1}+\mathbf{1}\otimes H_f, \ H_I]=0, \end{aligned}$$

from which we obtain

$$\begin{aligned} e^{-itH}= & {} e^{-it(H_a\otimes \mathbf{1}+\mathbf{1}\otimes H_f)}e^{-itH_I}=(e^{-itH_a}\otimes \mathbf{1})(\mathbf{1}\otimes e^{-itH_f})e^{-itH_I}. \end{aligned}$$

Consequently,

$$\begin{aligned}&\mathrm{tr}_a \big (e^{-itH}(\rho _a\otimes \rho _f) e^{itH}\big )\\&\quad =\mathrm{tr}_a \big ((e^{-itH_a}\otimes \mathbf{1})(\mathbf{1}\otimes e^{-itH_f})e^{-itH_I}(\rho _a\otimes \rho _f) e^{itH_I}(\mathbf{1}\otimes e^{itH_f})(e^{itH_a}\otimes \mathbf{1})\big )\\&\quad =e^{-itH_f} \mathrm{tr}_a\big (e^{-itH_I}(\rho _a\otimes \rho _f) e^{itH_I}\big )e^{itH_f}. \end{aligned}$$

By the phase-space rotation invariance of $N(\cdot )$ in Eq. (6), we obtain

$$\begin{aligned} N\Big (\mathrm{tr}_a \big (e^{-itH}(\rho _a\otimes \rho _f) e^{itH}\big )\Big )= & {} N\Big (\mathrm{tr}_a\big ( e^{-itH_I}(\rho _a\otimes \rho _f) e^{itH_I}\big )\Big ), \end{aligned}$$

which is the desired result.

Next, we derive reduced field states in the Jaynes–Cummings model. If the atom–field is initially in a pure product state $|\psi _a \rangle |\psi _f \rangle$ with the initial atomic state

$$\begin{aligned} |\psi _a\rangle =\sqrt{p}|e\rangle +\sqrt{1-p} |g\rangle , \qquad 0\le p\le 1 \end{aligned}$$

and the initial field

$$|\psi _f \rangle =\sum _{n=0}^\infty q_{n}|n\rangle ,$$

then the atom–field state at time t is

$$\begin{aligned} |\Psi (t)\rangle= & {} e^{-itH_I}(|\psi _a\rangle |\psi _f \rangle )\\= & {} \sum _{n=0}^\infty \big (a_{n}(t)|g\rangle |n\rangle +b_{n}(t)|e\rangle |n\rangle \big ) \end{aligned}$$

with

$$\begin{aligned} a_{n}(t)= & {} \sqrt{1-p}q_{n}\cos (\lambda t \sqrt{n})-i\sqrt{p}q_{n-1}\sin (\lambda t \sqrt{n}),\\ b_{n}(t)= & {} \sqrt{p}q_{n}\cos (\lambda t\sqrt{n+1})-i\sqrt{1-p}q_{n+1}\sin (\lambda t \sqrt{n+1}). \end{aligned}$$

The reduced field state can be expressed as

$$\begin{aligned} \rho _f (t)= & {} \mathrm{tr}_a |\Psi (t)\rangle \langle \Psi (t)|\\= & {} \sum _{m,n=0}^\infty \big (a_{n}(t)a_{m}^{*}(t)+b_{n}(t)b_{m}^{*}(t)\big )|n\rangle \langle m|. \end{aligned}$$

For the mixed initial atomic state

$$\begin{aligned} \rho _a=p_\mathrm{e}|e\rangle \langle e|+(1-p_\mathrm{e})|g\rangle \langle g|,\qquad 0\le p_\mathrm{e}\le 1 \end{aligned}$$

and mixed initial field state

$$\begin{aligned} \rho _f =\sum _{n=0}^\infty p_n|n\rangle \langle n|, \end{aligned}$$

the evolved state can be obtained by linear combinations and the reduced field state can be expressed as

$$\begin{aligned} \rho _f (t)= & {} \mathrm{tr}_\mathrm{a} \left( e^{-itH_I}(\rho _a\otimes \rho _f) e^{itH_I}\right) \\= & {} \sum _{n=0}^\infty c_n(t)|n\rangle \langle n|. \end{aligned}$$

which is a Fock-diagonal state with the coefficients

$$\begin{aligned} c_n(t)= & {} p_\mathrm{e}\left( p_n\mathrm{cos}^2(\lambda t\sqrt{n+1})+p_{n-1}\mathrm{sin}^2(\lambda t\sqrt{n}) \right) \\+ & {} (1-p_\mathrm{e})\left( p_n\mathrm{cos}^2(\lambda t\sqrt{n})+p_{n+1}\mathrm{sin}^2(\lambda t\sqrt{n+1}) \right) . \end{aligned}$$

In the following, we present the explicit expressions of $a_n(t),b_n(t)$ and $c_n(t)$ when the initial field states $\rho _f$ are the coherent states $|\alpha \rangle$, the Fock states $|n\rangle$ and the thermal states

$$\begin{aligned} \rho _{\bar{n}}=\frac{1}{1+\bar{n}} \sum _{n=0}^\infty \Big (\frac{\bar{n}}{1+\bar{n}}\Big )^{n}|n\rangle \langle n| \end{aligned}$$

with $\bar{n}$ being the average photon numbers.

A. Field initially in coherent states

Let the field be initially in the coherent state

$$\begin{aligned} |\psi _f \rangle =|\alpha \rangle =e^{-\frac{|\alpha |^{2}}{2}} \sum _{n=0}^\infty \frac{\alpha ^{n}}{\sqrt{n!}}|n\rangle , \end{aligned}$$

with the average photon number $\bar{n}=|\alpha |^{2}.$ For the initial state of the atom, we consider two cases:

(1) If the atom is initially in the superposition state $|\psi _a\rangle =\sqrt{p}|e\rangle +\sqrt{1-p} |g\rangle ,$ then the initial atom–field state is $|\psi _a \rangle \otimes |\alpha \rangle ,$ and the reduced field state at time t can be obtained as

$$\begin{aligned} \rho _f (t)=\sum _{m,n=0}^\infty \big (a_{n}(t)a_{m}^{*}(t)+b_{n}(t)b_{m}^{*}(t)\big )|n\rangle \langle m| \end{aligned}$$

with

$$\begin{aligned} a_{n}(t)= & {} \sqrt{1-p}e^{-\frac{|\alpha |^{2}}{2}}\frac{\alpha ^{n}}{\sqrt{n!}}\cos (\lambda t\sqrt{n})\\&-i\sqrt{p}e^{-\frac{|\alpha |^{2}}{2}}\frac{\alpha ^{n-1}}{\sqrt{(n-1)!}}\sin (\lambda t \sqrt{n}),\\ b_{n}(t)= & {} \sqrt{p}e^{-\frac{|\alpha |^{2}}{2}}\frac{\alpha ^{n}}{\sqrt{n!}}\cos (\lambda t\sqrt{n+1})\\&-i\sqrt{1-p}e^{-\frac{|\alpha |^{2}}{2}}\frac{\alpha ^{n+1}}{\sqrt{(n+1)!}}\sin (\lambda t\sqrt{n+1}). \end{aligned}$$

(2) If the atom is initially in the mixed state $\rho _a=p_\mathrm{e}|e\rangle \langle e|+(1-p_\mathrm{e})|g\rangle \langle g|,$ then the initial atom–field state is $\rho _a \otimes |\alpha \rangle \langle \alpha |,$ and the reduced field state at time t can be obtained from linear combinations as

$$\begin{aligned} \rho _f (t)= & {} e^{-|\alpha |^2}\sum _{m,n=0}^\infty \left( \frac{p_e\alpha ^{n-1}\alpha ^{*m-1}\sin (\lambda t\sqrt{n})\sin (\lambda t\sqrt{m})}{\sqrt{(n-1)!(m-1)!}}\right. \\+ & {} \frac{(1-p_e)\alpha ^{n+1}\alpha ^{*m+1}\sin (\lambda t\sqrt{n+1})\sin (\lambda t\sqrt{m+1})}{\sqrt{(n+1)!(m+1)!}}\\+ & {} \frac{p_e\alpha ^{n}\alpha ^{*m}\cos (\lambda t\sqrt{n+1})\cos (\lambda t\sqrt{m+1})}{\sqrt{n!m!}}\\+ & {} \left. \frac{(1-p_e)\alpha ^{n}\alpha ^{*m}\cos (\lambda t\sqrt{n})\cos (\lambda t\sqrt{m})}{\sqrt{n!m!}}\right) |n\rangle \langle m|. \end{aligned}$$

B. Field initially in Fock states

Next we consider the case when the field is initially in a Fock state $|\psi _f \rangle =|n\rangle .$ For the initial atomic state, we consider two cases:

(1) If the atom is initially in the superposition state $|\psi _a\rangle =\sqrt{p}|e\rangle +\sqrt{1-p} |g\rangle ,$ then the initial atom–field state is $|\psi _a\rangle \otimes |n\rangle ,$ and the reduced field state at time t is

$$\begin{aligned} \rho _f(t)= & {} \left( (1-p)\cos ^2(\lambda t\sqrt{n})+p\cos ^2(\lambda t\sqrt{n+1})\right) |n\rangle \langle n|\\&+p\sin ^2(\lambda t\sqrt{n+1})|n+1\rangle \langle n+1|\\&+(1-p)\sin ^2(\lambda t\sqrt{n})|n-1\rangle \langle n-1|\\&+i\sqrt{p(1-p)}\sin (\lambda t\sqrt{n+1})\cos (\lambda t\sqrt{n})|n\rangle \langle n+1|\\&-i\sqrt{p(1-p)}\sin (\lambda t\sqrt{n+1})\cos (\lambda t\sqrt{n})|n+1\rangle \langle n|\\&+i\sqrt{p(1-p)}\cos (\lambda t\sqrt{n+1})\sin (\lambda t\sqrt{n})|n\rangle \langle n-1|\\&-i\sqrt{p(1-p)}\cos (\lambda t\sqrt{n+1})\sin (\lambda t\sqrt{n})|n-1\rangle \langle n|. \end{aligned}$$

(2) If the atom is initially in a mixed state $\rho _a=p_\mathrm{e}|e\rangle \langle e|+(1-p_\mathrm{e})|g\rangle \langle g|,$ then the initial atom–field state is $\rho _a \otimes |n\rangle \langle n| ,$ and the reduced field state at time t is

$$\begin{aligned} \rho _f (t)= & {} (1-p_\mathrm{e})\mathrm{sin}^2(\lambda t\sqrt{n})|n-1\rangle \langle n-1|\\+ & {} p_\mathrm{e}\mathrm{sin}^2(\lambda t\sqrt{n+1}) |n+1\rangle \langle n+1|\\+ & {} \left( p_\mathrm{e}\mathrm{cos}^2(\lambda t \sqrt{n+1})+(1-p_\mathrm{e})\mathrm{cos}^2 (\lambda t\sqrt{n})\right) |n\rangle \langle n|. \end{aligned}$$

C. Field initially in thermal states

Finally we consider the case when the field is initially in the thermal state

$$ \rho _f =\rho _{\bar{n}}=\sum _{n=0}^\infty \frac{1}{1+\bar{n}}(\frac{\bar{n}}{1+\bar{n}})^{n}|n\rangle \langle n|.$$

For the initial atomic state, we also consider two cases:

(1) If the atom is initially in the pure state $|\psi _a\rangle =p|e\rangle +\sqrt{1-p^2} |g\rangle ,$ then the initial atom–field state is $|\psi _a\rangle \langle \psi _a|\otimes \rho _{\bar{n}},$ and by linear combination, the reduced field state at time t is

$$\begin{aligned} \rho _f (t)= & {} \sum _{n=0}^\infty \frac{1}{1+\bar{n}}(\frac{\bar{n}}{1+\bar{n}})^{n} \left[ \left( (1-p)\cos ^2(\lambda t\sqrt{n})\right. \right. \\&+\left. p\cos ^2(\lambda t\sqrt{n+1})\right) |n\rangle \langle n|\\&+p\sin ^2(\lambda t\sqrt{n+1})|n+1\rangle \langle n+1|\\&+(1-p)\sin ^2(\lambda t\sqrt{n})|n-1\rangle \langle n-1|\\&+i\sqrt{p(1-p)}\sin (\lambda t\sqrt{n+1})\cos (\lambda t\sqrt{n})|n\rangle \langle n+1|\\&-i\sqrt{p(1-p)}\sin (\lambda t\sqrt{n+1}) \cos (\lambda t\sqrt{n})|n+1\rangle \langle n|\\&+i\sqrt{p(1-p)}\cos (\lambda t\sqrt{n+1}) \sin (\lambda t\sqrt{n})|n\rangle \langle n-1|\\&-\left. ip\sqrt{1-p^2}\cos (\lambda t\sqrt{n+1}) \sin (\lambda t\sqrt{n})|n-1\rangle \langle n|\right] . \end{aligned}$$

(2) If the atom is initially in the mixed state $\rho _a=p_\mathrm{e}|e\rangle \langle e|+(1-p_\mathrm{e})|g\rangle \langle g|,$ then the initial atom–field state is $\rho _a \otimes \rho _{\bar{n}},$ and the reduced field state at time t is

$$\rho _f (t)=\sum _{n=0}^{\infty }c_n(t)|n\rangle \langle n|$$

with

$$\begin{aligned} c_n(t)= & {} \frac{{\bar{n}}^n}{(\bar{n}+1)^{n+1}}(p_\mathrm{e}u_n(t)+(1-p_\mathrm{e})v_n(t)) \end{aligned}$$

where

$$\begin{aligned} u_n(t)= & {} \mathrm{cos}^2(\lambda t\sqrt{n+1})+\frac{\bar{n}+1}{\bar{n}}\mathrm{sin}^2(\lambda t\sqrt{n}), \\ v_n(t)= & {} \mathrm{cos}^2(\lambda t\sqrt{n})+\frac{\bar{n}}{\bar{n}+1}\mathrm{sin}^2(\lambda t\sqrt{n+1}). \end{aligned}$$

D. Field initially in vacuum states

This is actually included in the above cases.