Thermal Casimir-Polder forces on a V-type three-level atom

We study the thermal Casimir-Polder (CP) forces on a V-type three-level atom. The competition between the thermal effect and the quantum interference of the two transition dipoles on the force is investigated. To shed light onto the role of the quantum interference, we analyze two kinds of initial states of the atom, i.e., the superradiant state and the subradiant state. Considering the atom being in the thermal reservoir, the resonant CP force arising from the real photon emission dominates in the evolution of the CP force. Under the zero-temperature condition, the quantum interference can effectively modify the amplitude and the evolution of the force, leading to a long-time force or even the cancellation of the force. Our results reveal that in the finite-temperature case, the thermal photons can enhance the amplitude of all force elements, but have no influence on the net resonant CP force in the steady state, which means that the second law of thermodynamics still works. For the ideal degenerate V-type atom with parallel dipoles under the initial subradiant state, the robust destructive quantum interference overrides the thermal fluctuations, leading to the trapping of the atom in the subradiant state and the disappearance of the CP force. However, in terms of a realistic Zeeman atom, the thermal photons play a significant role during the evolution of the CP force. The thermal fluctuations can enhance the amplitude of the initial CP force by increasing the temperature, and weaken the influence of the quantum interference on the evolution of the CP force from the initial superradiant (subradiant) state to the steady state.

Figure 1

(a) Scheme of an atom near the structure made of a LHM slab mounted on a metal substrate. (b) The level scheme of the V-type three-level atom.

Figure 2

(a) The resonant Casimir-Polder forces ${\mathit{F}}^r$ on the ideal V-type atom as a function of the atomic position at different temperatures $T$ and time $t$. The black and pink (gray) curves refer to temperatures $T=500\mathrm{K}$ and 0 K, respectively. The dash, dash-dot, and solid curves refer to the evolution times $t=0$, $1/{\mathbf{\Gamma }}_0$, and $2/{\mathbf{\Gamma }}_0$ respectively. (b) The dispersion Casimir-Polder forces ${\mathit{F}}^{\mathrm{off}}$ on this atom at the initial time as a function of the atomic position. The dashed (blue), dash-dot (orange), and solid (red) curves refer to the temperatures $T=0\mathrm{K},300\mathrm{K},500\mathrm{K}$, respectively. The atom is prepared initially in the superradiant state. ${\mathit{e}}_z$ denotes the unit vector in the $z$ direction. ${\mathbf{\Gamma }}_0$ is the decay rate of a two-level atom with frequency ${\omega }_0$ in free space.

Figure 3

The resonant Casimir-Polder force as a function of the atomic position at different temperatures and evolution time when the Zeeman atom is prepared initially in (a) the subradiant state and (b) the superradiant state. The black and pink (gray) curves refer to temperatures $T=500\mathrm{K}$ and 0 K, respectively. The solid and dashed curves refer to the evolution times $t=0$ and $t=1/{\mathbf{\Gamma }}_0$, respectively.

Figure 4

The evolution of the resonant elements ${\mathit{F}}_{mn}^{\mathrm{r}}({\mathit{r}}_A,t)$ on the Zeeman atom at different temperatures. (a) The Zeeman atom is prepared initially in the subradiant state. (b) The Zeeman atom is prepared initially in the superradiant state. The blue (red) curves with open (full) symbols refer to diagonal terms of ${\mathit{F}}_{11}^{\mathrm{r}}$ and ${\mathit{F}}_{22}^{\mathrm{r}}$ (${\mathit{F}}_{33}^{\mathrm{r}}$). The black curves with half-open symbols refer to the off-diagonal terms ${\mathit{F}}_{12}^{\mathrm{r}}$ and ${\mathit{F}}_{21}^{\mathrm{r}}$. The solid, dashed, and dotted curves refer to temperatures $T=500\mathrm{K},300\mathrm{K},0\mathrm{K}$, respectively. The atom is placed at the focus point $z_A=2\lambda$.

Figure 5

The evolution of the resonant Casimir-Polder force on the Zeeman atom at different temperatures. The solid, dashed, and dotted curves refer to temperatures $T=0\mathrm{K},300\mathrm{K},500\mathrm{K}$, respectively. The black and pink (gray) curves refer to the initial subradiant and the superradiant state, respectively. The Zeeman atom is placed at the focus point $z_A=2\lambda$.