Dissipation control in cavity QED with oscillating mode structures

We demonstrate how a time-dependent dissipative environment may be used as a tool for controlling the quantum state of a two-level atom. In our model system the frequency and coupling strength associated with microscopic reservoir modes are modulated, while the principal features of the reservoir structure remain fixed in time. Physically, this may be achieved by containing a static atom-cavity system inside an oscillating external bath. We show that it is possible to dynamically decouple the atom from its environment, despite the fact that the two remain resonant at all times. This can lead to Markovian dynamics, even for a strong atom-bath coupling, as the atomic decay becomes inhibited into all but a few channels; the reservoir occupation spectrum consequently acquires a sideband structure, with peaks separated by the frequency of the environmental modulation. The reduction in the rate of spontaneous emission using this approach can be significantly greater than could be achieved with an oscillatory atom-bath detuning using the same parameters.

Figure 1

(Color online) This figure corresponds to the model studied in Sec. 3 and shows reservoir mode couplings $g_k\left(t\right)$ [see Eq. (2)] at two different times $t_1$ and $t_2$. The envelope of the couplings remains the same as the reservoir mode frequencies vary in time. Thus the coupling of the atom to an individual mode changes in time. The thick line indicates the same mode $k$ at two different times $t_1$ and $t_2$. Since at time $t_2$ this mode has moved to a different frequency under the static envelope, the coupling has changed accordingly.

Figure 2

(Color online) By rewriting the Hamiltonian (2) in the form (18), the model studied in this paper may be interpreted as describing the decay into a collection of reservoir sidebands, evenly spaced by angular frequency $\Omega$. In this picture, all reservoir mode frequencies are static in time, but the coupling profile of each bath is modulated according to Eq. (13) (vertical arrows). Each bath passes through resonance at a different time, specified by Eq. (19). As discussed in Eqs. (31, 32, 33), the atomic decay rate ${\Gamma }_{\infty }$ can be increased or decreased by tuning the extreme values of $f\left(t\right)$ toward or away from the sideband frequencies $\left|f\left(t\right)\right|\approx n\Omega$.

Figure 3

(Color online) Final reservoir occupation spectrum for the decay process studied in Sec. 3A. The vertical axis shows the discretized form of $S\left(\omega \right)$, i.e., ${\text{lim}}_{t\to \infty }\left\{{\rho }_k{\left|c_k\left(t\right)\right|}^2\right\}$, calculated via a numerical solution to Eqs. (4, 5). The example shown uses $D=\gamma$, $\Omega =20\gamma$, $d=68\gamma$. (a) The final occupation spectrum is a discrete collection of peaks, centered at ${\omega }_0+n\Omega$. This is due to the dynamical resonance effects discussed in Sec. 3. The area bounded by a peak indexed by $n$ is ${\Gamma }_n/{\Gamma }_{\infty }$. (b) The shape of each peak is approximately Lorentzian with a half width ${\Gamma }_{\infty }/2$, as predicted in Sec. 3B.

Figure 4

(Color online) The result of Eq. (30) is plotted as a red line and compared to numerically extracted decay rates, shown as blue crosses. The numerically extracted decay rates were calculated by solving Eqs. (4, 5) and fitting an exponentially decaying curve to the resulting ${\left|c_a\left(t\right)\right|}^2$. The modulation frequency $\Omega =20\gamma$.

Figure 5

(Color online) Final occupation peak weights $S_n$ as a function of the modulation depth $d$ for the first four peaks. The figures (a)–(d) correspond to $n=0,1,2,3$, respectively. The $n$th peak weight is predicted to be ${\Gamma }_n/{\Gamma }_{\infty }$, with ${\Gamma }_n$ given by Eq. (31). As discussed in Sec. 3A, $S_n\sim 0$ for $d<n\Omega$ and $S_n$ peaks at $d\sim n\Omega$.

Figure 6

(Color online) (a) To manipulate the microscopic mode structure of the reservoir (double cavity), the outer-cavity mirrors are moved, which affects the mode frequencies as described by Eq. (14). Since the inner-cavity mirrors are fixed, the macroscopic properties of the reservoir remain invariant which recovers the Hamiltonian of Eq. (1). (b) Modulating the length of the cavity itself acts to dynamically detune the cavity resonance from the atomic transition, resulting in the interaction Hamiltonian (47). (c) Alternatively, variable detuning between the atom and reservoir may be achieved by inducing a time-dependent Stark shift in the atomic transition frequency [37]. This method also results in the Hamiltonian given in Eq. (47).

Figure 7

(Color online) Analytic results (44, 49) (crosses), together with the leading-order terms (46, 50) (solid lines). The modulation frequency $\Omega =20\gamma$. In this limit, the reservoir manipulation studied in our paper (lower curve) is more efficient at suppressing dissipative effects than an oscillatory detuning (upper curve).