Coherent control of two Jaynes-Cummings cavities

Abstract

We uncover new features on the study of a two-level atom interacting with one of two cavities in a coherent superposition. The James-Cummings model is used to describe the atom-field interaction and to study the effects of quantum indefiniteness on such an interaction. We show that coherent control of the two cavities in an undefined manner allows novel possibilities to manipulate the atomic dynamics on demand which are not achievable in the conventional way. In addition, it is shown that the coherent control of the atom creates highly entangled states of the cavity fields taking a Bell-like or Schrödinger-cat-like state form. Our results are a step forward to understand and harness quantum systems in a coherent control, and open a new research avenue in the study of atom-field interaction exploiting quantum indefiniteness.

Figure 1

The figure depicts the system under consideration. At time $t=0$ the state of the system is prepared and, in the first stage, the atom moves with constant velocity reaching the cavities at the time $t=T_0$. (a) In the first situation we consider that the atom interacts with both cavity fields during a certain time interval. (b) In the second scheme an additional stage is considered: the atom exits the cavities.

Figure 2

The atomic population inversion (Eq. 27) time-evolution when both cavity fields start out in the vacuum state, i.e., $n=0$ as function of some relevant parameters. (a) The effect of manipulating the control parameter $\theta$ with fixed value $t_m=\pi /(2g)$. (b) The dependence of the population inversion on the measuring time $t_m$ for the fixed control parameter value $\theta =\pi /4$.

Figure 3

Time-evolution of the atomic population inversion (Eq. 27) when the field in each cavity contains initially (a) one photon and (b) five photons (solid blue lines) with $t_m=\pi /g$. In addition, the dashed line corresponds to the single cavity atomic population inversion with the same photon number.

Figure 4

Average number of photons $⟨a_j^{†}{a}_j⟩$ for $n_0=n_1=n$, $g_0=g_1=g$, $g{t}_m=\pi /2$, and different values of the control parameter $\theta$ using equation (28). Figures (a,c) illustrate $⟨a_0^{†}{a}_0⟩$, while figure (b,d) show $⟨a_1^{†}{a}_1⟩$. Figures (a,b) depict the case $n=0$, while figures (c,d) illustrate the case $n=10$.

Figure 5

The average number of photons $⟨a_j^{†}{a}_j⟩$ at maximum indefiniteness as function of the non-dimensional measurement time $g{t}_m$ and gt for $n_0=1$ and $n_1=1$.

Figure 6

Total probability ${\mathcal{P}}_i$ to interchange one photon between two cavities as a function of $g{t}_m$ for two different cases. Case 1: figure (a) has $n_0=1$ and $n_1=0$, while figure (b) has $n_0=10$ and $n_1=0$. Case 2: figure (c) has $n_0=n_1=1$ and figure (d) has $n_0=n_1=10$. All plots were done for a given time $gt=64\pi /5$ and $g_1=g_2=g$.

Figure 7

Total probability ${\mathcal{P}}_i$, calculated from Eq. (33), to interchange one photon between two cavities at maximum indefiniteness as function of the non-dimensional measurement time $g{t}_m$ and gt for $n_0=1$ and $n_1=1$.

Figure 8

The figure illustrates a contour plot of the probability in Eq. (45) to find both cavity fields in a Schrödinger cat state as a function of $\mathrm{\Theta }$ and the expected number of photons ${|\alpha |}^2$ in the coherent state ${|\alpha ⟩}_k$ $(k=0,1)$.