Effects of Gravitational Waves on Two-Level Atom Moving in a Quantized Traveling Light Field: Exact Solution via Path Integral

Abstract

We adopt a coherent states path integral formalism to study the system of a two-level atom moving in a quantized traveling light field and a gravitational field. By using the phase space and some rotations in the space of coherent states, have enabled greatly simplify the calculations. The propagator is first written in a standard form, $\int \mathcal{D}\left(\mathit{\text{path}}\right)exp\left(i/\mathrm{\hslash }\right)S\left(\mathit{\text{path}}\right)$, by replacing the spin with a unit vector aligned along the polar and azimuthal directions. Then, it is determined exactly due to the auxiliary equation which has a spacial function as a solution. The corresponding wave functions have been deduced by applying the principles of quantum mechanics. The results obtained are perfectly identical with those found by other standard methods.

1. Introduction

Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. Gravitational waves transport energy as gravitational radiation, a form of radiant energy similar to electromagnetic radiation [1].

The effect of gravitational waves on the movement of atoms is important and cannot be neglected. In atomic optics experiments have made it possible to create atomic clouds and beams with very small velocity [2]. For atoms moving with a velocity of a few millimeters or centimeters per second for a time period of several milliseconds or more, the influence of Earth’s acceleration becomes important and cannot be neglected [3].

Among the simplest scheme to investigate the interaction between a two-level atom and a single-mode quantized electromagnetic field is the Jaynes-Cummings model (JCM) [4]. The JCM has received a great deal of experimental as well as theoretical attention [5, 6, 7, 8]. Over the years, the JCM has been extended and generalized in many directions, for example, the effects of finite cavity damping [9, 10], intensity dependent coupling [11, 12] and the introduction of a Kerr-like medium [13]. A very significant and noteworthy generalization of JCM is to include the quantization of atomic momentum and position [14, 15, 16] so that the internal and external dynamics of the atom could be treated into this model.

For this reason, we are devoted to this model of interaction, we use the path integral formalism in the bosonic and fermionic coherent states representation to solve the generalized JCM in the presence of a gravitational field, governed by the Hamiltonian ($\mathrm{\hslash }=1$)

$$\begin{array}{l}H=\frac{p^2}{2m}-\mathit{mgr}+\omega \left(a^{†}a+\frac{1}{2}\right)\\ +\frac{{\omega }_a}{2}{\mathit{\sigma }}_z+\lambda \left(e^{+\mathit{ikr}}a{\mathit{\sigma }}_{+}+e^{-\mathit{ikr}}{a}^{†}{\mathit{\sigma }}_{-}\right).\end{array}$$E1

Here, $a$ and $a^{†}$ are the atomic flipping operators, $\omega$ is the field frequency, $\lambda$ is the atom-field coupling constant, ${\omega }_a$ is the transition frequency between the levels, ${\mathit{\sigma }}_z,{\mathit{\sigma }}_{+},{\mathit{\sigma }}_{-}$ are the usual Pauli matrices, $k$ is the wave vector of the running wave, $p$ and $r$ denote, respectively, the momentum and position operators of the atomic center of mass motion, and $g$ is Earth’s gravitational acceleration.

In this paper, we propose to present an alternative solution to the given problem by the coherent states path integral representation via the Schwinger’s model of spin. It should be noted that, in the semi-classical description of two-level atom interacting with electromagnetic wave, the effect of the gravitational field has been recently studied using fermionic coherent state path integral [17].

This paper is organized as follows. In Section 2, we give a brief review of the bosonic and fermionic coherent states path integral representation for our further computations. In Section 3, after setting up a path integral formalism for the propagator, we perform the direct calculations over the angular variables. Accordingly, the integration over the bosonic variables is easy to carry out and the result is given as a perturbation series, which is summed up exactly. The explicit result of the propagator is directly computed and the wave function is then deduced in Section 4. Finally, we present our conclusions in the last section, we give a brief review of the bosonic and fermionic coherent states path integral representation.

2. Coherent state propagator

At this stage, we briefly give the definitions and some properties related to bosonic and fermionic coherent states in the path integral formalism.

For the coherent states $\left|Z\right.⟩$ relative to bosons, the properties are known. These are

• the eigenstate of the annihilation operator $a\left(\left[a,a^{†}\right]=1\right)$

  $$a\left|Z\right.⟩=Z\left|Z\right.⟩,$$E2

• they can also be created from the vacuum state $\left|0\right.⟩$ by applying a unitary operator called the displacement operator

  $$\left|Z\right.⟩=e^{{\mathit{Za}}^{†}-Z^{\ast }a}\left|0\right.⟩,$$E3

In this case, the scalar product and the projector operator are respectively

$$⟨Z|Z^{\prime }⟩=exp\left(Z^{\ast }{Z}^{\prime }-\frac{1}{2}\left({\left|Z\right|}^2+{\left|Z^{\prime }\right|}^2\right)\right),$$E4

$$\int \frac{d^{2}Z}{\pi }\left|Z\right.⟩\left.⟨Z\right|=1.$$E5

As for spin interaction, we use the approach whose recipe includes replacing the Pauli matrices ${\sigma }_i$ by a unit vector $\mathbf{n}$ directed along $\left(\theta ,\phi \right)$

$$\left|\theta ,\phi \right.⟩=e^{-\mathit{i\phi }{S}_z}{e}^{-\mathit{i\theta }{S}_y}\left|\uparrow \right.⟩$$E6

which is obtained by applying two rotations with angles $\theta$ and $\phi$ around the $z$ and $y$ axes on the state $\left|\uparrow \right.⟩$, and whose scalar product and projector are respectively

$$\left.⟨\theta ,\phi \right|{\theta }^{\prime },{\phi }^{\prime }⟩=cos\frac{\theta }{2}cos\frac{{\theta }^{\prime }}{2}{e}^{\frac{i}{2}\left(\phi -{\phi }^{\prime }\right)}+sin\frac{\theta }{2}sin\frac{{\theta }^{\prime }}{2}{e}^{-\frac{i}{2}\left(\phi -{\phi }^{\prime }\right)}$$E7

$$\frac{1}{2\pi }\int \mathit{d\phi d}cos\left(\theta \right)\left|\theta ,\phi \right.⟩\left.⟨\theta ,\phi \right|=\mathbf{I}.$$E8

Taking into account that

$$\left.⟨\theta ,\phi \right|{\sigma }_z\left|{\theta }_n,{\phi }^{\prime }\right.⟩=cos\frac{\theta }{2}cos\frac{{\theta }^{\prime }}{2}{e}^{+\frac{i}{2}\left(\phi -{\phi }^{\prime }\right)}-sin\frac{\theta }{2}sin\frac{{\theta }^{\prime }}{2}{e}^{-\frac{i}{2}\left(\phi -{\phi }^{\prime }\right)},$$E9

$$\left.⟨\theta ,\phi \right|{\sigma }_{+}\left|{\theta }_n,{\phi }^{\prime }\right.⟩=cos\frac{\theta }{2}sin\frac{{\theta }^{\prime }}{2}{e}^{+\frac{i}{2}\left(\phi +{\phi }^{\prime }\right)},$$E10

$$\left.⟨\theta ,\phi \right|{\sigma }_{-}\left|\theta ,{\phi }^{\prime }\right.⟩=sin\frac{\theta }{2}cos\frac{{\theta }^{\prime }}{2}{e}^{-\frac{i}{2}\left(\phi +{\phi }^{\prime }\right)}.$$E11

According to the habitual construction procedure of the path integral. We consider the quantum state $\left|r,Z,\theta ,\phi \right.⟩,$ where $Z$ is a complex variable generating the dynamics of the field, $\left(\theta ,\phi \right)$ are the polar angles variables generating the dynamics of the spin and $r$ the real variable describing the atom position, with the corresponding projector

$$\int \left|r\right.⟩\left.⟨r\right|{\mathit{dr}}^3=1.$$E12

The transition amplitude from the initial state $\left|r_i,Z_i,{\theta }_i,{\phi }_i\right.⟩$ and the final state $\left|r_f,Z_f,{\theta }_f,{\phi }_f\right.⟩$ at $t_i=0$ to the final state at $t_f=T$ is defined with the matrix elements of the evolution operator

$$K\left(f,i;T\right)=⟨r_f,Z_f,{\theta }_f,{\phi }_f|U\left(T\right)|r_i,Z_i,{\theta }_i,{\phi }_i⟩,$$E13

where

$$U\left(T\right)={\mathbf{T}}_{D}exp\left(-\frac{i}{\mathrm{\hslash }}{\int }_0^{T}H\left(t\right)\mathit{dt}\right),$$E14

with $T_D$ the Dyson chronological operator.

For moving to the path-integral representation, we first subdivide the time interval $\left[0,T\right]$ into $N+1$ intermediate moments of length $\epsilon$. Using the Trotter’s formula, we then introduce in (13) the projectors according to these $N$ intermediate instants, which are regularly distributed between $0$ and $T$. We obtain the discrete path-integral form of the propagator

$$\begin{array}{l}K\left(f,i;T\right)=\underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i\epsilon }}\right)}^{3N/2}\int \prod _{n=1}^N{d}^3{r}_n\int \prod _{n=1}^N\frac{d^2{Z}_n}{\pi }{e}^{-Z_n^{\ast }{Z}_n}\\ \times \prod _{n=1}^{N+1}⟨r_n,Z_n|e^{-\mathit{i\epsilon }{H}_0}|r_{n-1},Z_{n-1}⟩\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{\mathit{d\phi d}cos\left(\theta \right)}{2\pi }\prod _{n=1}^{N+1}⟨Z_n,{\theta }_n,{\phi }_n|e^{-\mathit{i\epsilon }{H}_{\mathit{int}}}|Z_{n-1},{\theta }_{n-1},{\phi }_{n-1}⟩,\end{array}$$E15

where

$$\left\{\begin{array}{l}{r}_{N+1}=r_f,Z_{N+1}=Z_f,{\theta }_{N+1}={\theta }_f,{\phi }_{N+1}={\phi }_f\\ \text{and}\\ r_0=r_i,Z_0=Z_i,{\theta }_0={\theta }_i,{\phi }_0={\phi }_i\end{array}\right.,$$E16

and the propagator related to our problem takes the form of Feynman path integral

$$K=\int {\mathcal{D}}_{\mathit{\text{path}}}{e}^{i\left(\mathit{\text{Action}}\right)}.$$E17

After having obtained the conventional form, it remains to integrate it, in order to extract the interesting physical properties. We thus proceed to the calculation of $K\left(f,i;T\right)$ in the next section.

3. The propagator calculation

We note that (15) is written like the following discrete-time form

$$\begin{array}{l}K\left(f,i;T\right)=\underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i\epsilon }}\right)}^{3N/2}\int \prod _{n=1}^N{d}^3{r}_n\prod _{n=1}^{N+1}exp\left\{i\left[\frac{m}{2\epsilon }{\left(r_n-r_{n-1}\right)}^2+\epsilon {\mathit{mgr}}_n\right]\right\}\\ \\ \times \underset{N\to \infty }{lim}\frac{1}{2\mathit{\pi i}}\int \prod _{n=1}^N{\mathit{dZ}}_n^{\ast }{\mathit{dZ}}_n{e}^{-Z_n^{\ast }{Z}_n}\prod _{n=1}^{N+1}exp\left\{\frac{\mathit{i\epsilon }}{\mathrm{\hslash }}\omega \left(Z_n^{\ast }{Z}_{n-1}+\frac{1}{2}\right)\right\}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{\mathit{d\phi d}cos\left(\theta \right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n}{2}{e}^{+\frac{i}{2}{\phi }_n},cos\frac{{\theta }_n}{2}{e}^{-\frac{i}{2}{\phi }_n}\right)R\left(r_n,t_n\right)\left(\begin{array}{c}cos\frac{{\theta }_{n-1}}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}}\\ sin\frac{{\theta }_{n-1}}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}}\end{array}\right),\end{array}$$E18

with

$$R\left(r_n,t_n\right)=\left[e^{-\mathit{i\epsilon }\frac{{\omega }_a}{2}{\sigma }_z}+\mathit{i\epsilon K}\left(Z_n,r_n,t_n\right)\right],$$E19

where

$$K\left(r_n,t_n\right)=\left(\begin{array}{cc}0& -\lambda e^{+{\mathit{ikr}}_n}{Z}_n\\ -\lambda e^{-{\mathit{ikr}}_{n-1}}{Z}_n^{\ast }& 0\end{array}\right).$$E20

To integrate, it is necessary to eliminate first the inconvenient terms $e^{\pm \mathit{ikr}}$, which appear in the action with the help of the following change

$${\phi }_n={\phi }_n^{\prime }+{\mathit{kr}}_n.$$E21

The new expression we have to calculate is therefore

$$\begin{array}{l}K\left(f,i;T\right)=\underset{N\to \infty }{lim}{\left(\frac{m}{2\mathit{\pi i\epsilon }}\right)}^{3N/2}\int \prod _{n=1}^N{d}^3{r}_n\prod _{n=1}^{N+1}exp\left\{i\left[\frac{m}{2\epsilon }{\left(r_n-r_{n-1}\right)}^2+\epsilon {\mathit{mgr}}_n\right]\right\}\\ \times \underset{N\to \infty }{lim}\frac{1}{2\mathit{\pi i}}\int \prod _{n=1}^N{\mathit{dZ}}_n^{\ast }{\mathit{dZ}}_n{e}^{-Z_n^{\ast }{Z}_n}\prod _{n=1}^{N+1}exp\left\{\frac{\mathit{i\epsilon }}{\mathrm{\hslash }}\omega \left(Z_n^{\ast }{Z}_{n-1}+\frac{1}{2}\right)\right\}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{d{\phi }^{\prime }dcos\left(\theta \right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n}{2}{e}^{+\frac{i}{2}{\phi }_n^{\prime }},cos\frac{{\theta }_n}{2}{e}^{-\frac{i}{2}{\phi }_n^{\prime }}\right)R_1\left(r_n,t_n\right)\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{\prime }}\\ sin\frac{{\theta }_{n-1}}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{\prime }}\end{array}\right\},\end{array}$$E22

where

$$R_1\left(r_n,t_n\right)=\left(\begin{array}{cc}\left(1-\mathit{i\epsilon }\frac{{\omega }_{\mathit{eg}}}{2}\right)e^{\mathit{ik}\mathrm{\Delta }{r}_n}& -\mathit{i\epsilon \lambda }{Z}_n\\ -\mathit{i\epsilon \lambda }{Z}_n^{\ast }& 1+\mathit{i\epsilon }\frac{{\omega }_{\mathit{eg}}}{2}\end{array}\right)$$E23

$$\text{with},\mathrm{\Delta }{r}_n=r_n-r_{n-1}.$$E24

We use the following identity

$${\int }_{-\infty }^{+\infty }\frac{d^3{p}_n}{{\left(2\pi \right)}^3}exp\left[\frac{-\mathit{i\epsilon }}{2m}{p}_n^2+{\mathit{ip}}_n\mathrm{\Delta }{r}_n\right]={\left(\frac{m}{2\mathit{\pi i\epsilon }}\right)}^{3/2}exp\frac{\mathit{im}}{2\epsilon }{\left(\mathrm{\Delta }{r}_n\right)}^2,$$E25

Thus, propagator (22) can be rewritten as

$$\begin{array}{l}K\left(f,i;T\right)=\underset{N\to \infty }{lim}\int \prod _{n=1}^N{d}^3{r}_n{\int }_{-\infty }^{+\infty }\frac{d^3{p}_n}{{\left(2\pi \right)}^3}exp\sum _{n=1}^{N+1}\left[\frac{-\mathit{i\epsilon }}{2m}{p}_n^2+{\mathit{ip}}_n\mathrm{\Delta }{r}_n+\mathit{i\epsilon }{\mathit{mgr}}_n\right]\\ \times \underset{N\to \infty }{lim}\frac{1}{2\mathit{\pi i}}\int \prod _{n=1}^N{\mathit{dZ}}_n^{\ast }{\mathit{dZ}}_n{e}^{-Z_n^{\ast }{Z}_n}\prod _{n=1}^{N}exp\left\{-\mathit{i\epsilon \omega }\left(Z_n^{\ast }{Z}_{n-1}+\frac{1}{2}\right)\right\}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{d{\phi }^{\prime }dcos\left(\theta \right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n}{2}{e}^{+\frac{i}{2}{\phi }_n^{\prime }},cos\frac{{\theta }_n}{2}{e}^{-\frac{i}{2}{\phi }_n^{\prime }}\right)R_2\left(p_n,t_n\right)\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{\prime }}\\ sin\frac{{\theta }_{n-1}}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{\prime }}\end{array}\right\},\end{array}$$E26

with

$$R_2\left(p_n,t_n\right)=\left(\begin{array}{cc}1-\mathit{i\epsilon }\left(\frac{{\omega }_a}{2}+\frac{k^2}{2m}+\frac{{\mathit{kp}}_n}{m}\right)& -\mathit{i\epsilon \lambda }{Z}_n\\ -\mathit{i\epsilon \lambda }{Z}_n^{\ast }& 1+\mathit{i\epsilon }\frac{{\omega }_a}{2}\end{array}\right).$$E27

By integrating over the $N$ variables $r_n$, we clearly get Dirac functions $\delta \left(\dot{p}-\mathit{mg}\right)$, which means that the particle is only subject to the action of gravitational waves. The atom impulsion is

$$p_n={\mathit{mgt}}_n+p_0\text{where}\left(p_0\text{constant}\right).$$E28

The contribution of the time-linear function in the computation of the propagator has the following result

$$\begin{array}{l}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times \int \frac{{\mathit{dZ}}_n^{\ast }{\mathit{dZ}}_n}{2\mathit{\pi i}}{e}^{-Z_n^{\ast }{Z}_n}exp\left\{i{\int }_0^T\mathit{dt}\left[\frac{i}{2}\left({\dot{Z}}^{\ast }Z-\dot{Z}{Z}^{\ast }\right)-\omega Z^{\ast }Z-\frac{\omega }{2}\right]\right\}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{d{\phi }^{\prime }dcos\left(\theta \right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n}{2}{e}^{+\frac{i}{2}{\phi }_n^{\prime }},cos\frac{{\theta }_n}{2}{e}^{-\frac{i}{2}{\phi }_n^{\prime }}\right)R_3\left(p_0,t_n\right)\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{\prime }}\\ sin\frac{{\theta }_{n-1}}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{\prime }}\end{array}\right\},\end{array}$$E29

with

$$R_3\left(p_0,t_n\right)=\left(\begin{array}{cc}1-\mathit{i\epsilon }\left(\frac{{\omega }_a}{2}+\frac{k^2}{2m}+{\mathit{kgt}}_n+\frac{{\mathit{kp}}_0}{m}\right)& -\mathit{i\epsilon \lambda }{Z}_n\\ -\mathit{i\epsilon \lambda }{Z}_n^{\ast }& 1+\mathit{i\epsilon }\frac{{\omega }_a}{2}\end{array}\right).$$E30

At this level, let us deal with the integration over the Grassmann variables. We shall introduce the following change

$$\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{\prime }}\\ sin\frac{{\theta }_{n-1}}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{\prime }}\end{array}\right\}=e^{-\frac{i}{2}\left(\frac{k^2}{2m}{t}_n+\frac{1}{2}{\mathit{kgt}}_n^2+\frac{{\mathit{kp}}_0}{m}{t}_n\right)}\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{″}}\\ sin\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{″}}\end{array}\right\},$$E31

so the expression of the propagator become

$$\begin{array}{l}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times \int \frac{{\mathit{dZ}}_n^{\ast }{\mathit{dZ}}_n}{2\mathit{\pi i}}{e}^{-Z_n^{\ast }{Z}_n}exp\left\{i{\int }_0^T\mathit{dt}\left[\frac{i}{2}\left({\dot{Z}}^{\ast }Z-\dot{Z}{Z}^{\ast }\right)-\omega Z^{\ast }Z-\frac{\omega }{2}\right]\right\}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{d{\phi }^{″}dcos\left({\theta }^{\prime }\right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_n^{\text{'}\text{'}}},cos\frac{{\theta }_n^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_n^{″}}\right)R_4\left(p_0,{\mathrm{\Omega }}_n\right)\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{″}}\\ sin\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{″}}\end{array}\right\},\end{array}$$E32

where

$$R_4\left(p_0,{\mathrm{\Omega }}_n\right)=\left(\begin{array}{cc}1-\frac{i}{2}\epsilon \mathrm{\Omega }\left(p,g,t\right)& -\mathit{i\epsilon \lambda }{Z}_n\\ -\mathit{i\epsilon \lambda }{Z}_n^{\ast }& 1+\frac{i}{2}\epsilon \mathrm{\Omega }\left(p,g,t\right)\end{array}\right),$$E33

with

$$\mathrm{\Omega }\left(p,g,t\right)={\omega }_a+\frac{k^2}{2m}+{\mathit{kgt}}_n+\frac{{\mathit{kp}}_0}{m}.$$E34

We integrate over all variables $Z_n$, $\left(32\right)$ become

$$\begin{array}{c}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times \underset{N\to \infty }{lim}\int \prod _{n=1}^N\frac{d{\phi }^{″}dcos\left({\theta }^{\prime }\right)}{2\pi }\prod _{n=1}^{N+1}\\ \times \left(cos\frac{{\theta }_n^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_n^{″}},cos\frac{{\theta }_n^{\text{'}}}{2}{e}^{-\frac{i}{2}{\phi }_n^{″}}\right)K\left(p_0,{\mathrm{\Omega }}_n\right)\left\{\begin{array}{c}cos\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{″}}\\ sin\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{″}}\end{array}\right\},\end{array}$$E35

where

$$\begin{array}{l}K\left(p_0,{\mathrm{\Omega }}_n\right)=e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}{e}^{-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)}\right\}\\ \times \left(\begin{array}{cc}1-\mathit{i\epsilon }\mathrm{\Delta }\left(p,g,t\right)& -\mathit{i\epsilon \lambda }\sqrt{j+1}\\ -\mathit{i\epsilon \lambda }\sqrt{j+1}& 1+\mathit{i\epsilon }\mathrm{\Delta }\left(p,g,t\right)\end{array}\right),\end{array}$$E36

where

$$\mathrm{\Delta }\left(t_j\right)=\frac{1}{2}\left[\omega -\left({\omega }_a+3\frac{k^2}{2m}+{\mathit{kgt}}_j+\frac{{\mathit{kp}}_0}{m}\right)\right],$$E37

is the detuning of the atom-field interaction which depends on both the atomic momentum and the gravitational field.

Now, to calculate propagator (35) let us introduce the complex variable $z$ [18].

$$z=tan\frac{{\theta }^{\prime }}{2}{e}^{+i{\phi }^{″}}\text{and}\mid {\theta }^{\prime },{\phi }^{″}⟩=\left(\begin{array}{c}cos\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_{n-1}^{″}}\\ sin\frac{{\theta }_{n-1}^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_{n-1}^{″}}\end{array}\right)=e^{-\frac{i}{2}{\phi }_{n-1}^{″}}\mid z⟩$$E38

The propagator in the $z$ representation is

$$\begin{array}{c}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}exp\left[-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)\right]\right\}\\ \times e^{+\frac{i}{2}{\phi }_f^{″}}{e}^{-\frac{i}{2}{\phi }_i^{″}}\left.⟨z_f\right|exp\left[-i{\int }_{{\mathrm{\Delta }}_0}^{\mathrm{\Delta }\left(T\right)}H\left(p_0,\mathrm{\Delta }\right)\mathit{dS}\right]\mid z_i⟩,\end{array}$$E39

where $H\left(p_0,\mathrm{\Delta }\right)$ belongs to $\mathit{SU}\left(2\right)$ algebra

$$H\left(p_0,\mathrm{\Delta }\right)={\mathrm{\Delta }}_n{S}_0+f_n{S}_{+}+f_n{S}_{-}$$E40

and

$$f=\lambda \sqrt{j+1}.$$E41

The computation of propagator $\left(39\right)$ is readily and the results are given by

$$\begin{array}{c}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}exp\left[-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)\right]\right\}\\ \times e^{+\frac{i}{2}{\phi }_f^{\text{'}\text{'}}}{e}^{-\frac{i}{2}{\phi }_i^{\text{'}\text{'}}}\left.⟨z_f\right|exp\left[\frac{a^{\ast }\left(\mathrm{\Delta }\right)-b^{\ast }\left(\mathrm{\Delta }\right)z_f+b\left(\mathrm{\Delta }\right)z_i^{\ast }+a^{\ast }\left(\mathrm{\Delta }\right)z_i^{\ast }{z}_f}{{\left(1+{\left|z_i\right|}^2\right)}^{\frac{1}{2}}{\left(1+{\left|z_f\right|}^2\right)}^{\frac{1}{2}}}\right]\mid z_i⟩,\end{array}$$E42

where $a\left(\mathrm{\Delta }\right)$, and $b\left(\mathrm{\Delta }\right)$ satisfy

$$\frac{\mathit{da}}{d\mathrm{\Delta }}=-{\mathit{\text{ifbe}}}^{i\mathrm{\Delta }},\frac{\mathit{db}}{d\mathrm{\Delta }}=-{\mathit{\text{ifae}}}^{-i\mathrm{\Delta }}$$E43

with the boundary conditions

$$a\left({\mathrm{\Delta }}_0\right)=1,b\left({\mathrm{\Delta }}_0\right)=0,{\mathrm{\Delta }}_0=\frac{1}{2}\left[\omega -\left({\omega }_a+3\frac{k^2}{2m}+\frac{{\mathit{kp}}_0}{m}\right)\right].$$E44

In terms of the angular variables it becomes

$$\begin{array}{l}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}exp\left[-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)\right]\right\}\\ \times \left(cos\frac{{\theta }_f^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_f^{″}},cos\frac{{\theta }_f^{\text{'}}}{2}{e}^{-\frac{i}{2}{\phi }_f^{″}}\right)\left(\begin{array}{cc}a\left(T\right)& b\left(T\right)\\ -b^{\ast }\left(T\right)& a^{\ast }\left(T\right)\end{array}\right)\left\{\begin{array}{c}cos\frac{{\theta }_i^{\prime }}{2}{e}^{-\frac{i}{2}{\phi }_i^{″}}\\ sin\frac{{\theta }_i^{\prime }}{2}{e}^{+\frac{i}{2}{\phi }_i^{″}}\end{array}\right\},\end{array}$$E45

where

$$a\left(T\right)=e^{i\mathrm{\Delta }T}\left[C\left(1\right)H_{A_j}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j}{2};\frac{1}{2};B^2\left(T\right)\right)\right],$$E46

$$b\left(T\right)=\left[C\left(1\right)H_{A_j+1}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j+1}{2};\frac{1}{2};B^2\left(T\right)\right)\right],$$E47

where $H_n\left(x\right)$ and${}_1{F}_1\left(\alpha ;\beta ;\gamma \right)$ are the Hermite and the confluent hypergeometric functions. Furthermore, we have

$$B\left(t\right)=\left(i+1\right)\left(\frac{\sqrt{2}}{2}\mathit{kgt}-\frac{\sqrt{2}}{4\sqrt{\mathit{kg}}}{\mathrm{\Delta }}_0\right),\text{and}{A}_j=-\left(2+\mathit{i\beta }\right),$$E48

$$\text{with},\beta =\left(\frac{y_j\left(p,g\right)-{\mathrm{\Delta }}_0^2}{2\mathit{kg}}\right),\text{and},y_j\left(p,g\right)=\sqrt{y_j{\left(p,0\right)}^2+2\mathit{ikg}},$$E49

with $y_j{\left(p,0\right)}^2=f^2+{\mathrm{\Delta }}_0^2$ as the gravity-dependent Rabi frequency. And

$$C\left(1\right)=\frac{c_1}{c},C\left(2\right)=\frac{c_2}{c},$$E50

with

$$c_1{=}_1{F}_1\left(-\frac{1}{2}\left(A_j+1\right);\frac{1}{2};B^2\left(0\right)\right),$$E51

and

$$c_2=H_{\left(A_j+1\right)}\left(B\left(0\right)\right),$$E52

and

$$c=H_{A_j}{\left(B\left(0\right)\right)}_1{F}_1\left(-\frac{1}{2}\left(A_j+1\right);\frac{1}{2};B^2\left(0\right)\right)-H_{\left(A_j+1\right)}{\left(B\left(0\right)\right)}_1{F}_1\left(-\frac{A_j}{2};\frac{1}{2};B^2\left(0\right)\right),$$E53

Now we come back to the old Grassmann variables $\left(\theta ,\phi \right)$. So, the exact expression of the propagator concerning to our problem is the following

$$\begin{array}{l}K\left(f,i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}{e}^{-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)}\right\}\\ \times \left(cos\frac{{\theta }_f}{2}{e}^{+\frac{i}{2}{\phi }_f},cos\frac{{\theta }_f}{2}{e}^{-\frac{i}{2}{\phi }_f}\right)S\left(T\right)\left\{\begin{array}{c}cos\frac{{\theta }_i}{2}{e}^{-\frac{i}{2}{\phi }_i}\\ sin\frac{{\theta }_i}{2}{e}^{+\frac{i}{2}{\phi }_i}\end{array}\right\}\end{array}$$E54

where

$$\begin{array}{l}S\left(T\right)=e^{-\frac{i}{2}\left(\frac{k^2}{2m}T+\frac{1}{2}{\mathit{kgT}}^2+\frac{{\mathit{kp}}_0}{m}T\right)}\\ \times \left(\begin{array}{cc}{e}^{{\mathit{ikr}}_f}& 0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}a\left(T\right)& b\left(T\right)\\ -b^{\ast }\left(T\right)& a^{\ast }\left(T\right)\end{array}\right)\left(\begin{array}{cc}{e}^{-{\mathit{ikr}}_i}& 0\\ 0& 1\end{array}\right).\end{array}$$E55

Noting that the angles $\theta ,\phi$ are allowed to vary in the limited domains $\left[0,2\pi \right]$ and $\left[0,4\pi \right]$, our final result for the propagator is thus the following:

$$\begin{array}{c}K\left(f,i;T\right)=\stackrel{+\infty }{\sum _{n=-\infty }}K\left(Z_f,{\theta }_f+2\mathit{n\pi },{\phi }_f+4\mathit{n\pi },Z_i,{\theta }_i,{\phi }_i;T\right)\\ =K\left(Z_f,{\theta }_f,{\phi }_f;Z_i,{\theta }_i,{\phi }_i;T\right).\end{array}$$E56

Our problem is thus solved. We can then determine the corresponding wave functions.

4. Wave functions

Let us now eliminate the coherent states by computing the transition amplitudes between the proper states of the spin. We take as an example the matrix element

$$K_{\uparrow \uparrow }\left(Z_f,Z_i;T\right)=\left.⟨\uparrow \right|K\left(Z_f,Z_i;T\right)\left|\uparrow \right.⟩.$$E57

With the help of the completeness relations, this amplitude becomes

$$\int \frac{dcos\left({\theta }_f\right)d{\phi }_f}{2\pi }\left|{\theta }_f,{\phi }_f\right.⟩\left.⟨{\theta }_f,{\phi }_f\right|=1,\int \frac{dcos\left(\mathit{\theta i}\right)d{\phi }_i}{2\pi }\left|{\theta }_i,{\phi }_i\right.⟩\left.⟨{\theta }_i,{\phi }_i\right|=1$$E58

Then

$$\begin{array}{l}{K}_{\uparrow \uparrow }\left(Z_f,Z_i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}exp\left[-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)\right]\right\}\\ \times \int \frac{dcos\left({\theta }_f\right)d{\phi }_f}{2\pi }\frac{dcos\left({\theta }_i\right)d{\phi }_i}{2\pi }⟨\uparrow |{\theta }_f,{\phi }_f⟩\left(\begin{array}{cc}{S}_{11}\left(T\right)& S_{12}\left(T\right)\\ S_{21}\left(T\right)& S_{22}\left(T\right)\end{array}\right)⟨{\theta }_i,{\phi }_i|\uparrow ⟩\end{array}$$E59

where

$$⟨\uparrow |{\theta }_f,{\phi }_f⟩=cos\frac{{\theta }_f}{2}{e}^{-\frac{i}{2}{\phi }_f},\text{and}⟨{\theta }_i,{\phi }_i|\uparrow ⟩=cos\frac{{\theta }_i}{2}{e}^{+\frac{i}{2}{\phi }_i}$$E60

The integration over polar angles, the propagator matrix element for the up-up states is finally written

$$K_{\uparrow \uparrow }\left(Z_f,Z_i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}{S}_{11}\left(T\right).$$E61

Repeating the calculations and considering all initial and final states of the spin, the propagator takes the following matrix form

$$\begin{array}{l}K\left(m_f,m_i;T\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-\frac{{\left|Z_f\right|}^2+{\left|Z_i\right|}^2}{2}}\sum _{j=0}^{\infty }\left\{\frac{Z_f^{\ast j}{Z}_i^j}{j!}{e}^{-i\left(j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}{\int }_0^T\mathit{dt}\mathrm{\Omega }\left(p,g,t\right)\right)}\right\}\\ \times \left(\begin{array}{cc}{S}_{11}\left(T\right)& S_{12}\left(T\right)\\ S_{21}\left(T\right)& S_{22}\left(T\right)\end{array}\right).\end{array}$$E62

In order to extract the wave functions, it is more convenient to use the basis $\left|n\right.⟩$, where $n$ is the occupancy number related to $\left|Z\right.⟩$ through

$$\left|Z\right.⟩=exp\left(-\frac{{\left|Z\right|}^2}{2}\right)\sum _{n=0}^{\infty }\frac{Z^n}{\sqrt{n!}}\left|n\right.⟩.$$E63

Then the evolution operator is

$$\widehat{U}\left(p,g;T\right)=e^{-\mathit{iHt}}=\left(\begin{array}{cc}{\mathtt{\Lambda }}_{\uparrow \uparrow }& {\mathtt{\Lambda }}_{\uparrow \downarrow }\\ {\mathtt{\Lambda }}_{\downarrow \uparrow }& {\mathtt{\Lambda }}_{\downarrow \downarrow }\end{array}\right),$$E64

which is related to $K\left(f,i;T\right)$ through

$$e^{-\mathit{iHt}}=\int \frac{d^2{Z}_f}{\pi }\int \frac{d^2{Z}_i}{\pi }\left|Z_f\right.⟩K\left(Z_f,Z_i;T\right)\left.⟨Z_i\right|.$$E65

Performing the integrations yields the matrix elements of the evolution operator [19].

$$\begin{array}{c}{\mathtt{\Lambda }}_{\uparrow \uparrow }=f\left(p,g,t\right)e^{\mathit{ik}\left(r_f-r_i\right)}\\ \times \sum _{j=0}^{\infty }{e}^{i\mathrm{\Delta }T}\left[C\left(1\right)H_{A_j}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|j\right.⟩\left.⟨j\right|,\end{array}$$E66

$$\begin{array}{c}{\mathtt{\Lambda }}_{\downarrow \uparrow }=f\left(p,g,t\right)e^{-{\mathit{ikr}}_i}\\ \times \sum _{j=0}^{\infty }\left[C\left(1\right)H_{A_j+1}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j+1}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|j+1\right.⟩\left.⟨j\right|,\end{array}$$E67

$$\begin{array}{c}{\mathtt{\Lambda }}_{\uparrow \downarrow }=f\left(p,g,t\right)e^{{\mathit{ikr}}_f}\\ \times \sum _{j=0}^{\infty }{\left[C\left(1\right)H_{A_j+1}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j+1}{2};\frac{1}{2};B^2\left(T\right)\right)\right]}^{\ast }\left|j\right.⟩\left.⟨j+1\right|,\end{array}$$E68

$$\begin{array}{c}{\mathtt{\Lambda }}_{\downarrow \downarrow }=f\left(p,g,t\right)\\ \times \sum _{j=0}^{\infty }{e}^{-i\mathrm{\Delta }T}{\left[C\left(1\right)H_{A_j}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j}{2};\frac{1}{2};B^2\left(T\right)\right)\right]}^{\ast }\left|j+1\right.⟩\left.⟨j+1\right|,\end{array}$$E69

where

$$\begin{array}{l}f\left(p,g,t\right)=\int \frac{d^3{p}_0}{{\left(2\pi \right)}^3}{e}^{{\left.i\left(\mathit{mgt}+p_0\right)r\right|}_0^T}{e}^{-i\left(\frac{m}{6}{g}^2{T}^3+\frac{p_0^2}{2m}T+\frac{1}{2}{\mathit{gp}}_0{T}^2\right)}\\ \times e^{-i\left[j{\omega }_{c}T+\frac{1}{2}{\omega }_{c}T+\frac{1}{2}\left({\omega }_{\mathit{eg}}T+\frac{k^2}{2m}T+\frac{1}{2}{\mathit{kgT}}^2+\frac{{\mathit{kp}}_0}{m}T\right)\right]}\\ \times e^{-\frac{i}{2}\left(\frac{k^2}{2m}T+\frac{1}{2}{\mathit{kgT}}^2+\frac{{\mathit{kp}}_0}{m}T\right)}.\end{array}$$E70

For the considered atomic system, the wave function is written in the following form

$$\mathrm{\Psi }\left(p,g;T\right)=\int d^{3}p\widehat{U}\left(p,g;T\right)\mathrm{\Psi }\left(p,g,0\right),$$E71

where

$$\mathrm{\Psi }\left(p,g;0\right)=\int d^3\mathit{p\phi }\left(p\right)\sum _{n=0}{\omega }_n\left(\begin{array}{c}\left|n\right.⟩\\ 0\end{array}\right),$$E72

with

$${\omega }_n=\frac{Z^n}{\sqrt{n!}}{e}^{-\frac{{\left|Z\right|}^2}{2}},$$E73

can be deduced, they become equal to

$$\begin{array}{c}\mathrm{\Psi }\left(p,g;T\right)=\sum _{j=0}\sum _{n=0}{\omega }_{n}f\left(p,g,t\right)\\ \times \left(\begin{array}{c}{e}^{\mathit{ik}\left(r_f-r_i\right)}{e}^{i\mathrm{\Delta }T}\left[C\left(1\right)H_{A_j}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|j\right.⟩\left.⟨j\right|\left|n\right.⟩\\ e^{-{\mathit{ikr}}_i}\left[C\left(1\right)H_{A_j+1}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_j+1}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|j+1\right.⟩\left.⟨j\right|\left|n\right.⟩\end{array}\right).\end{array}$$E74

As $\left.⟨j\right|n⟩={\delta }_{j,n}$, we finally obtain an exact and explicit expression for the wave function

$$\begin{array}{c}\mathrm{\Psi }\left(p,g;T\right)=\sum _{n=0}{\omega }_{n}f\left(p,g,t\right)\\ \times \left(\begin{array}{c}{e}^{\mathit{ik}\left(r_f-r_i\right)}{e}^{i\mathrm{\Delta }T}\left[C\left(1\right)H_{A_n}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_n}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|n\right.⟩\\ e^{-{\mathit{ikr}}_i}\left[C\left(1\right)H_{A_n+1}\left(B\left(T\right)\right)+C{\left(2\right)}_1{F}_1\left(-\frac{A_n+1}{2};\frac{1}{2};B^2\left(T\right)\right)\right]\left|n+1\right.⟩\end{array}\right).\end{array}$$E75

This result coincides with that of Ref [20].

5. Conclusions

In this work, we have succeeded in calculating exactly the propagator of the two-level atom interacting with single-mode quantized electromagnetic field and submitted to gravitation using the path integral formalism in the coherent states representation. Thanks to the two angular variables replacing the spin, the propagator has been written, first in the conventional form $\int \mathcal{D}\left(\mathit{\text{path}}\right)exp\left(i/\mathrm{\hslash }\right)S\left(\mathit{\text{path}}\right)$, then determined exactly. The exactness of the results is displayed in the evaluation of the corresponding wave function. The influence of gravitational waves are reflected in our results.