Entropy exchange and thermal fluctuations in the Jaynes–Cummings model

Abstract

The time dependence of the quantum entropy for a two-level atom interacting with a single-cavity mode is computed using the Jaynes–Cummings model, when the initial state of the radiation field is prepared in a thermal state with temperature fluctuations. In order to describe the out-of-equilibrium situation, the Super-Statistics approximation is implemented so that the gamma and the multi-level distribution functions are used to introduce the inverse temperature fluctuations. In the case of the gamma distribution, paralleling the Tsallis non-additive formalism, the entropy for the system is computed with the q-logarithm prescription, and the impact of the initial state of the atom is also taken into account. The results show that, in the first distribution, the q-parameter (related to the thermal fluctuations) modifies the partial entropies appreciably. In contrast, the way the inverse temperatures are distributed in the second one may lead to changes in the entropy functions.

Appendices

A Super-Statistics and Tsallis thermodynamics

The connection of SS and Tsallis thermodynamics comes from the entropy generalization for non-additive systems [52], namely:

$$\begin{aligned} S=\frac{1}{q-1}\left( 1-\text {Tr}\left[ {\hat{\rho }}^q\right] \right) \forall \;q \in {\mathbb {R}}, \end{aligned}$$

(50)

where q is the non-additive index and ${\hat{\rho }}$ is the density operator of the system. In order to mimic the main results of the Boltzmann statistics, it is argued that the Tsallis framework has to fulfil the so-called Legendre structure of thermodynamics [42], which can be demanded by fundamental arguments related to an increasing entropy and a positive-definite specific heat [53, 54]. In order to implement that idea, the internal energy U is constrained by the following average prescription:

$$\begin{aligned} U=\frac{\text {Tr}\left[ {\hat{\rho }}^q\hat{H}\right] }{\text {Tr}\left[ {\hat{\rho }}^q\right] }, \end{aligned}$$

(51)

where $\hat{H}$ is the Hamiltonian. Then, by maximizing the functional associated with S and U, the density operator is given by:

$$\begin{aligned} {\hat{\rho }}=\frac{1}{Z}\exp _q\left( -\beta \frac{\hat{H}-U}{\text {Tr}\left[ {\hat{\rho }}^q\right] }\right) , \end{aligned}$$

(52)

where $\beta$ is the inverse physical temperature and the partition function Z is given by

$$\begin{aligned} Z=\text {Tr}\left[ \exp _q\left( -\beta \frac{\hat{H}-U}{\text {Tr}\left[ {\hat{\rho }}^q\right] }\right) \right] . \end{aligned}$$

(53)

Equations (51)-(53) are implicit for ${\hat{\rho }}$, and their solution is not trivial. Nevertheless, there is an auxiliary form to avoid that problem. If an auxiliary density matrix ${\hat{\varrho }}$ is defined as

$$\begin{aligned} {\hat{\varrho }}=\frac{1}{{\mathcal {Z}}}\exp _q\left( -\beta ^\star \hat{H}\right) , \end{aligned}$$

(54)

where

$$\begin{aligned} {\mathcal {Z}}=\text {Tr}\left[ \exp _q\left( -\beta ^\star \hat{H}\right) \right] \end{aligned}$$

(55)

and $\beta ^\star$ is a quasi-temperature parameter given by

$$\begin{aligned} \beta =\frac{\beta ^\star \,\text {Tr}\left[ {\hat{\varrho }}^q(\beta ^\star )\right] }{1-(1-q) \beta ^\star {\mathcal {U}}\left( \beta ^\star \right) / \text {Tr}\left[ {\hat{\varrho }}^q(\beta ^\star )\right] }, \end{aligned}$$

(56)

the following relation is found:

$$\begin{aligned} {\hat{\rho }}(\beta )= & {} {\hat{\varrho }}(\beta ^\star ), \end{aligned}$$

(57)

where ${\mathcal {U}}$ is defined in Eq. (30). Therefore, the problem for ${\hat{\rho }}$ is solved by working with ${\hat{\varrho }}$ and with Eqs. (57), together with the physical temperature parametrization of Eq. (56).

B The non-additive average photon number

For the non-extensive case, the average photon number is given by

$$\begin{aligned} {\bar{n}}_q=\frac{\text {Tr}_R\left[ {\hat{\varrho }}^q_R(\beta ^\star )\hat{n}\right] }{\text {Tr}_R\left[ {\hat{\varrho }}^q_R(\beta ^\star )\right] }=\frac{1}{\text {Tr}_R\left[ {\hat{\varrho }}^q_R(\beta ^\star )\right] }\sum _{n=0}^\infty n\left( 1+\alpha n\right) ^{\nu }, \end{aligned}$$

(58)

where $\alpha \equiv -(1-q)\beta ^\star \omega$ and $\nu \equiv q/(1-q)$.

In order to compute the sum above, let me introduce the regulator $\eta$ in the following way:

$$\begin{aligned} {\mathcal {S}}(\alpha ,\nu )= & {} \sum _{n=0}^\infty n\left( 1+\alpha n\right) ^{\nu }\nonumber \\= & {} -\lim _{\eta \rightarrow 0}\frac{\partial }{\partial \eta }\sum _{n=0}^\infty \left( 1+\alpha n\right) ^{\nu }e^{-\eta n}, \end{aligned}$$

(59)

but

$$\begin{aligned} \sum _{n=0}^\infty \left( 1+\alpha n\right) ^{\nu }e^{-\eta n}=\alpha ^\nu \Phi \left( e^{-\eta },-\nu ,\frac{1}{\alpha }\right) , \end{aligned}$$

(60)

where $\Phi (z,s,r)$ is the so-called Hurwitz–Lerch transcendent function, defined as

$$\begin{aligned} \Phi (z,s,r)=\sum _{n=0}^\infty \frac{z^n}{(n+r)^s}. \end{aligned}$$

(61)

Then,

$$\begin{aligned} {\mathcal {S}}(\alpha ,\nu )= & {} -\alpha ^\nu \lim _{\eta \rightarrow 0}\frac{\partial }{\partial \eta }\Phi \left( e^{-\eta },-\nu ,\frac{1}{\alpha }\right) \nonumber \\= & {} \alpha ^\nu \left[ \Phi \left( 1,-1-\nu ,\frac{1}{\alpha }\right) -\frac{1}{\alpha }\Phi \left( 1,-\nu ,\frac{1}{\alpha }\right) \right] . \end{aligned}$$

(62)

Therefore,

$$\begin{aligned} {\bar{n}}_q= & {} \frac{\left[ (q-1)\beta ^\star \omega \right] ^{\frac{q}{1-q}}}{\text {Tr}_R\left[ {\hat{\varrho }}^q_R(\beta ^\star )\right] }\Bigg [\Phi \left( 1,\frac{1}{q-1},\frac{1}{(q-1)\beta ^\star \omega }\right) -\frac{1}{(q-1)\beta ^\star \omega }\Phi \left( 1,\frac{q}{q-1},\frac{1}{(q-1)\beta ^\star \omega }\right) \Bigg ], \end{aligned}$$

(63)

where

$$\begin{aligned} \text {Tr}_R\left[ {\hat{\varrho }}^q_R(\beta ^\star )\right] =\left[ (q-1)\beta ^\star \omega \right] ^{\frac{q}{1-q}}\zeta _\text {H}\left( \frac{q}{q-1},\frac{1}{(q-1)\beta ^\star \omega }\right) .\nonumber \\ \end{aligned}$$

(64)