Dynamical Casimir–Polder force on a two-level atom with superposition state in a cavity comprising a dielectric

We study the dynamical Casimir–Polder force on a two-level atom with different initial states in the one-dimensional dielectric cavity with output coupling, and obtain the analytical expression of the expectation value of dynamical Casimir–Polder force. Results show that the expectation values of dynamical Casimir–Polder force may be affected by the initial states of the atom. Moreover, the expectation value of Casimir–Polder force may vanish at some special atomic positions by properly selecting the initial state of the system. The effects of different relative dielectric constants and the cavity size on the expectation value of Casimir–Polder force are also discussed.


Model and Hamiltonian
We consider a one-dimensional cavity which is filled with a dielectric (Region 1 shown in Fig. 1). The wall at x = −l is an ideal-conductor plate. The other wall at x = 0 is no coating and allows the environment (Region 2 shown in Fig. 1) couples to the cavity. A two-level atom is located at x 0 in the cavity. The Hamiltonian for the system within the dipole approximation can be written as [32][33][34] We can defined the parameter g j in Eq. (3) where µ is the dipole operator component in the polarization of the field. "1" and "0" are the dielectric constants of the dielectric and vacuum, respectively. S z , S + and S − are the operators of the atomic system, where S z = 1/2(|↑��↑| − |↓��↓|) , S + = |↑��↓| and S − = |↓��↑| . |↑� and �↓| are the excited and ground states of the atom. b † j and b j are the creation and annihilation operators for the jth mode of the field with the frequency ω 1j = c 1 k 1j where c 1 is the speed of the light in the dielctric. The superscript i = 1 or 2 represents to the region 1 or region 2, respectively. δ 0,n is the Kronecker delta.

Calculation of the expectation value of dynamical Casimir-Polder force
In the above section, we have analyzed the electromagnetic field in the cavity and the Hamiltonian for the system. In the following, we will calculate the expectation value of dynamical Casimir-Polder force F(x, t) via the expectation value of second-order interaction-energy shift �E (2) of the system. The Casimir-Polder force is the negative derivative of the second-order energy shift with respect to x 0 : In order to get the expectation value of second-order energy shift �E (2) , we will first obtain the Heisenberg equations of the field and atomic operators, and then solve the equations at the zeroth and first orders. According to perturbation theory 31,35 , the equations of b j (t) and S + (t) are obtained where ζ (x, t) is defined as ∂x 0 . www.nature.com/scientificreports/ The expectation value of second-order energy shift �E (2) can be derived by using the perturbation theory 31 .
Here |ϕ� is the superposition state of the system, and That is, the atom may be in the excited state or ground state at t=0, and the probability of being in the ground state is cos 2 θ , while the probability of being in the excited state is sin 2 θ . We evaluate the expectation value of second-order energy shift �E (2) by substituting Eqs. (6) and (7) into Eq. (3). H (2) int (t) is expressed as Then the expectation value of second-order energy shift �E (2) is obtained We can calculate �E (2) by using the method of Ref. 31,32 , and obtain the expectation value of Casimir-Polder force by Eq. (5). The expression of the expectation value of Casimir-Polder force can be obtained as follows: where we define the notations as follows: Herein, we define τ = 2� √ ε r /c, c = 3 × 10 8 m/s, z 0 = τ (x 0 + l), z 1 = τ (x 0 + l + nl) and z 2 = −τ (x 0 + l − nl) .
Si(z) and Ci(z) represent the sine integral function and the cosine integral function, respectively.

Dynamical casimir-polder force on an initially general superposition state atom
In this section, we focus on finding the characters of the expectation value of Casimir-Polder force on an initially general superposition state atom. Figure 2a shows that the expectation value of Casimir-Polder force is a function of θ . As is seen in Fig. 2,the expectation value of Casimir-Polder force increases with θ , which shows that the www.nature.com/scientificreports/ expectation values of dynamical Casimir-Polder force may be affected by the initial state of the atom. In Fig. 2b, for θ ≈ 0.0104π , we can see that the expectation value of Casimir-Polder force shows oscillations and it reaches a steady zero value for a long time. That is, at special atomic positions, the expectation value of Casimir-Polder force may change from negative value to positive value with different values of θ.
To further investigate the effect of the relative dielectric constant and the cavity length on the expectation value of Casimir-Polder force, we describe the plots of the Casimir-Polder force with different values of the relative dielectric constant and the size of cavity versus θ , as shown in Figs. 3 and 4, respectively. In Fig. 3 reveals the effect of different relative dielectric constants on the Casimir-Polder force. The green curve, the blue curve and the red curve indicate the case where the relative dielectric constant is 1.25, 1.55 and 1.85, respectively. The intersection of the red curve ( ε r = 1.85 ) and the green curve ( ε r = 1.25 ) with the zero line is on the right side of the blue curve ( ε r = 1.55 ), which indicates that the initial state which can make the expectation value of the Casimir-Polder force zero may be affected by the relative permittivity of the dielectric. Figure 4 investigates the effect of different cavity lengths on the expectation of the Casimir-Polder force. The green curve, the blue curve and the red curve indicate the case that the cavity length is l = 3.5 × 10 −7 m, l = 4.5 × 10 −7 m and l = 5.5 × 10 −7 m , respectively. The intersection of the red curve ( l = 5.5 × 10 −7 m ) and the green curve ( l = 3.5 × 10 −7 m ) with the zero line is on the right side of the blue curve ( l = 4.5 × 10 −7 m ), which indicates that the initial state which can make the expectation value of the Casimir-Polder force zero may be affected by the size of cavity.  www.nature.com/scientificreports/

conclusions
In this paper, we have calculated the expectation value of dynamical Casimir-Polder force on a two-level atom starting from the different initial states in the one-dimensional cavity with output coupling by using the perturbation theory, and have obtained the analytical expression of the expectation value of dynamical Casimir-Polder force.
We have observed the relationship between the expectation value of Casimir-Polder force and the initial state of the atom, that is, the expectation values of dynamical Casimir-Polder force may be affected by the initial state of the atom. By selecting the proper initial state, the expectation value of Casimir-Polder force may vanish at some special atomic positions. The relative permittivity and the size of the cavity may also affect the expectation value of the Casimir-Polder force.