Dissipative preparation of large $W$ states in optical cavities

Two schemes are proposed for the dissipative preparation of large $W$ states, of the order of ten qubits, within the context of cavity QED. By utilizing properties of the irreducible representations of su(3), we are able to construct protocols in which it is possible to restrict the open-system dynamics to a fully symmetric irreducible subspace of the total Hilbert space and hence obtain analytic solutions for effective ground-state dynamics of arbitrary-size ensembles of $\Lambda$ atoms within an optical cavity. In the proposed schemes, the natural decay processes of spontaneous emission and photon loss are no longer undesirable, but essential to the required dynamics. All aspects of the proposed schemes relevant to implementation in currently available optical cavities are explored, especially with respect to increasing system size.

Figure 1

Cavity QED setup for a single atom. The $|0\rangle -|e\rangle$ transition is driven by a coherent laser with a resonant Rabi frequency of $\Omega$ and a detuning of $\Delta$, while levels $|1\rangle$ and $|e\rangle$ are coupled via the cavity field, with an atom-cavity interaction strength of $g$. The entire setup consists of $n$ identical atoms within a single cavity.

Figure 2

Action of shift operators in the $(T_3,Y)$ plane. The units are scaled such that one unit on the $Y$ axis is $\sqrt{3/4}$ times a single unit on the $T_3$ axis.

Figure 3

Construction of the multiplet $(3,0)$ in the $(T_3,Y)$ plane. The top row of the multiplet consists of ground states, while the second row consists of single-atomic-excitation states.

Figure 4

Construction of the total basis for the symmetric single-excitation subspace of $n$ $\Lambda$ atoms in a single optical cavity. The top row consists of single-cavity-excitation states, the second row of ground states, and the third row of single-atomic-excitation states. The notation is as per Eqs. (48, 49, 50).

Figure 5

Partitioned matrix form of ${\widehat{H}}_{NH}$.

Figure 6

Summary of effective ground-state processes (85, 86, 87, 88) omitting effective loop processes of ${\widehat{L}}_{\mathrm{eff}}^2$ and ${\widehat{H}}_{\mathrm{eff}}$. Each effective process consists of a coherent excitation, an intermediate propagation within the single-excitation subspace (facilitated by ${\widehat{H}}_{NH}^{-1}$), and a deexcitation via coherent driving or dissipation.

Figure 7

(a)–(c) Plots of protocol benchmarks against cooperativity $C$ for different values of $n$. For these plots $\left(\widetilde{\Omega ,\stackrel{̃}{\gamma }}\right)=(1/5,1/2)$ such that the protocol is within the weak driving regime and the cooperativity is varied through $\tilde{\kappa }$. (d) Plot of preparation time $T_p$ as a function of coherent driving strength $\Omega$ for different values of $n$. For this plot $\left(\widetilde{\kappa ,\stackrel{̃}{\gamma }}\right)=(1/2,1/2)$ such that $C=400$. (a)–(d) For all plots $(\tilde{\delta },\tilde{\Delta })=(8/11,11/8)$, the numerically optimized choice within the restriction $\tilde{\Delta }\tilde{\delta }=1$, while $z=0.85$. Furthermore, typical values of $g$ are on the order of 10 MHz [23] such that preparation times are on the order of $\mu \text{s}$ and $\Omega$ values are on the order of MHz.

Figure 8

Evolution of populations with time, from initial state $|0\rangle$, for different values of cooperativity and different system sizes. The lower plot corresponds to $C=200$ with $(\tilde{\gamma },\tilde{\kappa })=(1/2,1)$, while the upper plot corresponds to $C=600$ with $(\tilde{\gamma },\tilde{\kappa })=(1/2,1/3)$. Both plots are at $(\tilde{\Omega },\tilde{\delta },\tilde{\Delta })=(1/5,8/11,11/8)$. Note that typical values of $g$ are on the order of 10 MHz [23] such that preparation times are on the order of $\mu \text{s}$. Populations in the upper plot have been increased by one for display.

Figure 9

Cavity QED setup for a single atom in a bimodal cavity. The entire setup consists of $n$ identical atoms within a single cavity.

Figure 10

Partitioned matrix form of ${\widehat{H}}_{NH}$ for the bimodal scheme.

Figure 11

(a)–(c) Plots of protocol benchmarks against cooperativity values $C_{\left(1\right)}$ and $C_{\left(2\right)}$ for fixed system size with $n=3$ and for the case ${\Omega }_2=0$. For these plots $\left(\widetilde{\Omega ,\stackrel{̃}{\gamma }}\right)=(1/5,1/2)$ such that the protocol is within the weak driving regime and cooperativities are varied through $\widetilde{{\kappa }_1}$ and $\widetilde{{\kappa }_2}$. For all plots $({\delta }_1,{\delta }_2,{\Delta }_1,g_1)$ are at numerically optimized values, as per Eq. (188), within the restrictions set by Eqs. (184) and (185). The threshold value has been set strictly, with $z=0.85$, and the initial state of the system is assumed as $|0\rangle$. Furthermore, typical values of $g$ are on the order of 10 MHz [23] such that preparation times are on the order of $\mu \text{s}$.

Figure 12

(a) and (b) Plots of protocol benchmarks against system size at different values of cooperativity $C_{\left(2\right)}$ for the case ${\Omega }_2=0$. For these plots $(\widetilde{\Omega ,\stackrel{̃}{\gamma }},\widetilde{{\kappa }_1})=(1/5,1/2,1)$ such that the protocol is within the weak driving regime and $C_2$ is varied through $\widetilde{{\kappa }_2}$ while $C_1=100$ remains fixed. For both plots $({\delta }_1,{\delta }_2,{\Delta }_1,g_1)$ are at numerically optimized values, as per Eq. (188), within the restrictions set by Eqs. (184) and (185). The threshold value has been set strictly, with $z=0.85$, and the system is assumed to be in the initial state $|0\rangle$.

Figure 13

(a) Plot of preparation time against cooperativities $C_{\left(1\right)}$ and $C_{\left(2\right)}$ with a fixed system size of $n=10$. Coherent driving is set within the regime of weak driving with ${\Omega }_2=1/2$, while $\tilde{\gamma }=1$ such that cooperativities are varied through $\widetilde{{\kappa }_1}$ and $\widetilde{{\kappa }_2}$. (b) Plot of preparation time against ${\Omega }_2$ with fixed system size $n=5$. (c) Plot of preparation time against system size with ${\Omega }_2=1/2$. (b) and (c) Cooperativities are set with $(\tilde{\gamma },\widetilde{{\kappa }_1},\widetilde{{\kappa }_2})=(1,2,1/4)$. (a)–(c) For all plots ${\Omega }_1=0$ and $({\delta }_1,{\delta }_2,{\Delta }_1,g_1)$ are at numerically optimized values, as per Eq. (188), within the restrictions set by Eqs. (184) and (185), necessary for the second phase of the protocol. The threshold value has been set strictly with $z=0.95$ and the system is assumed to be in the initial state $|3\rangle$. Furthermore, typical values of $g$ are on the order of 10 MHz [23] such that preparation times are on the order of $\mu \text{s}$ and $\Omega$ values are on the order of MHz.

Figure 14

Evolution of populations with time for a system of size $n=8$. Coherent driving is set with ${\Omega }_1=0$ and ${\Omega }_2=1/2$, while $({\delta }_1,{\delta }_2,{\Delta }_1,g_1)$ are at numerically optimized values, as per Eq. (188), within the restrictions set by Eqs. (184) and (185), necessary for the second phase of the protocol. The threshold value has been set strictly with $z=0.95$ and the system is assumed to be in the initial state $|3\rangle$. Note that typical values of $g$ are on the order of 10 MHz [23] such that preparation times are on the order of $\mu \text{s}$.