Generation of decoherence-free displaced squeezed states of radiation fields and a squeezed reservoir for atoms in cavity QED

We present a way to engineer an effective anti-Jaynes-Cumming and a Jaynes-Cumming interaction between an atomic system and a single cavity mode and show how to employ it in reservoir engineering processes. To construct the effective Hamiltonian, we analyze the interaction of an atomic system in a $\Lambda$ configuration, driven by classical fields, with a single cavity mode. With this interaction, we first show how to generate a decoherence-free displaced squeezed state for the cavity field. In our scheme, an atomic beam works as a reservoir for the radiation field trapped inside the cavity, as employed recently by S. Pielawa et al. [Phys. Rev. Lett. 98, 240401 (2007)] to generate an Einstein-Podolsky-Rosen entangled radiation state in high-$Q$ resonators. In our scheme, all the atoms have to be prepared in the ground state and, as in the cited article, neither atomic detection nor precise interaction times between the atoms and the cavity mode are required. From this same interaction, we can also generate an ideal squeezed reservoir for atomic systems. For this purpose we have to assume, in addition to the engineered atom-field interaction, a strong decay of the cavity field (i.e., the cavity decay must be much stronger than the effective atom-field coupling). With this scheme, some interesting effects in the dynamics of an atom in a squeezed reservoir could be tested.

Figure 1

(Color online) Atomic and field configuration employed in the interaction engineering process. The ground $\mid g⟩$ and excited $\mid e⟩$ states are coupled to the auxiliary state $\mid i⟩$ through laser fields and the cavity mode.

Figure 2

(Color online) (a) Mean number of photons $⟨n⟩=⟨a^{†}a⟩$ and (b) variance of the cavity field quadratures ${\left(\Delta X_l\right)}^2=⟨X_l^2⟩-{⟨X_l⟩}^2$, $l=1,2$, versus the number of atoms $N_{\mathrm{at}}$ that cross the cavity, each of them interacting during a time $\tau \simeq 3.1∕g$. Squeezing factor $r=1.0$. Solid line is obtained from the exact (numerical) solution of Eq. (3) for a sequence of $N_{\mathrm{at}}$ atoms. The dashed line represents the expected (analytical) value.

Figure 3

Wigner function $\mathcal{W}=\mathcal{W}(x,y)$ of the cavity field state: (a) the initial (vacuum) state and after the passage of (b) 50, (c) 100, and (d) 200 atoms (steady state).

Figure 4

(Color online) Evolution of $⟨{\sigma }_x\left(t\right)⟩$ for a squeezing factor $r=1.5$, $\Gamma =40g$, $\gamma =0$, and three different values of the squeezing angle $\varphi =0$, $\pi ∕2$, and $\pi$. Solid line is obtained from the exact (numerical) solution of Eq. (9) with $H_{\mathrm{eff}}$ given by Eq. (3). The dashed line represents the expected (analytical) value, for an ideal squeezed reservoir.