Quantized Fields for Optimal Control in the Strong Coupling Regime

We tailor the quantum statistics of a bosonic field to deterministically drive a quantum system into a target state. Experimentally accessible states of the field achieve good control of multilevel or multiqubit systems, notably also at coupling strengths beyond the rotating-wave approximation. This extends optimal control theory to the realm of fully quantized, strongly coupled control and target degrees of freedom.

Figure 1

Optimal control problem under consideration. Given a quantum system (black atoms) initially prepared in state $|i⟩$ and interacting with a bosonic field (red shaded cavity), which is the optimal initial state $|{\varphi }_{\mathrm{opt}}⟩$ of the field such that, after an interaction time $T$, the probability to prepare the sample in a predefined target state $|f⟩$ (green atoms) independent of the cavity’s final state is maximal?

Figure 2

Optimal control of a single resonant two-level system via a bosonic mode. (a) Ratio ${\mathcal{F}}_{\mathrm{coh}}/\mathcal{F}$ between the fidelities ${\mathcal{F}}_{\mathrm{coh}}$ and $\mathcal{F}$ achieved by a field mode prepared in the optimally classical (coherent) state and in the optimal state $|{\varphi }_{\mathrm{opt}}⟩$ given by the eigenvector(s) with largest eigenvalue of (1), respectively, as a function of the coupling strength $g$ and of the target state’s Bloch angle $\theta$ ($\phi =0$), for an interaction time $T=25/{\omega }_c$. The solid black line indicates $2\sqrt{N_{\max }}gT=\theta$. Notice the two different scales on the $x$ axis, separated by the blank space. (b),(c) Number statistics $|{\varphi }_n{|}^2=|⟨n|{\varphi }_{\mathrm{opt}}⟩{|}^2$ of the two optimal states [associated with the two largest eigenvalues of (1)], for different $g$ (top to bottom), $T=25/{\omega }_c$, and final states with $\theta =\pi$, $\pi /2$, $\varphi =0$ in (b) and (c), respectively. The black dashed lines correspond to Poissonian distributions (multiplied by magnification factors to ease the comparison of the distribution’s width) with average photon numbers $n_{\mathrm{av}}$ equal to those of the optimal states. (d)–(f) Wigner functions [59] of the optimal states for (d) $T=25/{\omega }_c$, $\theta =\pi /2$, $g=0.03{\omega }_c$, (e) $T=25/{\omega }_c$, $\theta =\pi$, $g=0.2{\omega }_c$, and (f) $T=6.8/{\omega }_c$, $\theta =\pi /2$, $g=0.2{\omega }_c$. In (d) the black solid and dashed lines correspond to circles around the phase space origin, with radius $n_{\mathrm{av}}$ and $n_{\mathrm{av}}\pm \sqrt{n_{\mathrm{av}}}$, respectively, highlighting the sub-Poissonian nature of the state in radial direction.

Figure 3

Control of a multilevel system. (a) Target level scheme under consideration, with arrows indicating the allowed transitions with $g_{kl}^{(1)}\ne 0$ in (2). (b) Fidelities $\mathcal{F}$ achieved for three different parameter ranges: Jaynes-Cummings (JC; $g=0.001{\omega }_c$, ${\omega }_{E_1}=0.86{\omega }_c$, ${\omega }_{E_2}=0.92{\omega }_c$, ${\omega }_{E_3}=1.1{\omega }_c$, ${\omega }_{E_4}=1.2{\omega }_c$, ${\omega }_F=2{\omega }_c$), resonant strong coupling (RSC; $g=0.1{\omega }_c$, spectrum as for JC), and diabatic (DR; $g=0.5{\omega }_c$, ${\omega }_{E_1}=0.32{\omega }_c$, ${\omega }_{E_2}=0.35{\omega }_c$, ${\omega }_{E_3}=0.4{\omega }_c$, ${\omega }_{E_4}=0.42{\omega }_c$, ${\omega }_F=0.5{\omega }_c$). Wigner function (c) of the optimal state $|{\varphi }_{\mathrm{opt}}⟩$ for $T=0.02\times 2\pi /{\omega }_c$ and (d) population of the target state $|F⟩$, as a function of time, respectively, in the DR. In (d), the control field is prepared either in $|{\varphi }_{\mathrm{opt}}⟩$ or in the coherent state $|\alpha ⟩$ with $\alpha =⟨{\varphi }_{\mathrm{opt}}|\widehat{X}+i\widehat{P}|{\varphi }_{\mathrm{opt}}⟩$, and the vertical purple line indicates the control time $T$. (e),(f) Same as in (c) and (d), respectively, but for RSC, with $T=0.01\times 2\pi /{\omega }_c$. $N_{\max }=80$, in all cases.

Figure 4

Control of multiple qubits. Table: quantum statistics of the optimal state $|{\varphi }_{\mathrm{opt}}⟩$ driving multiple qubits from their ground state to different final states $|f⟩$ (see main text). (a) Target fidelity $\mathcal{F}$ vs control time $T$, for $g=0.01{\omega }_c$ and $N_{\mathrm{A}}=3$ qubits. (b) Largest target fidelity ${\mathcal{F}}_{\mathrm{opt}}$ achievable within $0\le gT\le 8$ vs register length $N_A$ (same color code as on the left; target fidelities of all but the GHZ state essentially coincide).