Fast Navigation in a Large Hilbert Space Using Quantum Optimal Control

The precise engineering of quantum states, a basic prerequisite for technologies such as quantum-enhanced sensing or quantum computing, becomes more challenging with increasing dimension of the system Hilbert space. Standard preparation techniques then require a large number of operations or slow adiabatic evolution and give access to only a limited set of states. Here, we use quantum optimal control theory to overcome this problem and derive shaped radio-frequency pulses to experimentally navigate the Stark manifold of a Rydberg atom. We demonstrate that optimal control, beyond improving the fidelity of an existing protocol, also enables us to accurately generate a nonclassical superposition state that cannot be prepared with reasonable fidelity using standard techniques. Optimal control thus substantially enlarges the range of accessible states. Our joint experimental and theoretical work establishes quantum optimal control as a key tool for quantum engineering in complex Hilbert spaces.

Popular Summary

Quantum technologies harness the quantum features of light and matter to improve communications, computing, and sensing. Doing so requires the ability to precisely navigate the landscape of available quantum states, a challenging prospect in large, complex systems. A strategy known as quantum optimal control offers a solution to this challenge by allowing one to design the time evolution of experimental control knobs, such as electromagnetic fields, to accomplish a given task most efficiently. Here, we go beyond typical quantum control experiments, which focus on relatively simple systems, and experimentally demonstrate how quantum optimal control can navigate a large landscape of quantum states and reach an arbitrary state for which no intuitive preparation method is known.

We focus on the manipulation of Rydberg atoms, atoms with one of their electrons in an exceptionally excited state that offer attractive abilities for quantum technologies. In particular, we prepare a rubidium atom in a long-lived circular Rydberg state (where the excited electron lives in a circular orbit) from a state with much lower angular momentum, as well as a so-called cat state, a superposition of states with different classical properties. The control knobs are the amplitude and phase of a radio-frequency field tuned to induce atomic transitions in the presence of an electric field. Within about 20 oscillations of the radio-frequency field, we prepare the target state with high fidelity.

Our study exhibits the power of quantum control to navigate safely in a complex space of quantum states, an enabling step for a variety of quantum network, computing, and sensing applications.

Figure 1

Stark levels. Scheme of the energy levels for $F\sim 2.5\mathrm{V}/\mathrm{cm}$, sorted by $m$ values for $m\ge 0$. The circular state is labeled $c$. The levels of the lower diagonal are labeled $m_j$ for low-$m$ states and $e_j$ for high-$m$ states. Finally, $e_1^{\prime }$ is the Stark level with $m=50$ above the circular state. The laser excitation prepares the $m_2$ state. The atom is then driven by a ${\sigma }^{+}$ rf field at frequency ${\omega }_{\mathrm{rf}}={\omega }_n$ (blue arrows). Since a ${\sigma }^{+}$ field only couples levels with $\mathrm{\Delta }m=+1$, the dynamics of the atom remains in the subspace of the manifold lower diagonal (levels represented as thick lines).

Figure 2

Scheme of the experiment. A rubidium thermal beam (dark blue arrow), effusing from an oven, crosses a structure made of two plane-parallel electrodes $A$ and $B$ (blue) creating the vertical directing electric field $\mathbf{F}$. They are surrounded by four ring electrodes (yellow, two of them are not represented). At the center $O$ of the structure, the atoms interact with three laser beams (780 and 776 nm, ${\sigma }^{+}$ polarized, red arrow; 1258 nm, $\pi$ polarized, green arrow) that cross at 90° in the horizontal plane. At the end of the sequence, the atoms are detected in the state-selective field-ionization detector $D$.

Figure 3

Circular state preparation with a square pulse. Evolution of the population $P_j(\tau )$ of the levels labeled by $j$ ($j=m_1$, $m_2$, $m_3$, $e_2$, $e_1$, and $c$, where the labels and the associated color code are defined in Fig. 1) as a function of the duration $\tau$ of the rf pulse. The points are experimental with statistical error bars. The shaded areas (purple and orange) shows the error range for levels $m_1$ and $m_3$ due to the uncertainty on the MW probe pulses calibration (see Appendix pp3). The solid lines correspond to the numerical simulation of the experiment.

Figure 4

Optimized circularization pulse. Amplitude of the two quadratures (red and black lines) of the optimized rf pulse (a) before and (b) after postprocessing (flattening of the oscillations that result from the constraint on the spectral bandwidth of the pulse). The gray area indicates the pulse envelope. (c) Evolution of the population of each Stark level of the lower diagonal as a function of time for the rf pulse of (b). The inset presents the histogram of the populations at $\tau =40\mathrm{ns}$ (red bars). (d) Snapshot of the $Q$ function of the atomic state represented as a large angular momentum on a generalized Bloch sphere for $\tau =0\mathrm{ns}$ (i), $\tau =15\mathrm{ns}$ (ii), and $\tau =40\mathrm{ns}$ (iii). The first 40 ns of the pulse transfer the $52m_2$ level (i) into a spin coherent state (iii).

Figure 5

Circular state preparation with an optimized pulse. The voltage $V_r$ and $V_i$ for the optimized pulse are shown in the upper panel. The main frame shows the evolution of the population $P_j(\tau )$ of the levels $j$ as a function of the duration $\tau$, after which the rf pulse is interrupted, for $j=m_1,m_2,m_3$ ($\tau \le 52\mathrm{ns}$) and $j=e_2,e_1,c$ ($\tau \ge 80\mathrm{ns}$). The points are experimental with statistical error bars, using the color code of Fig. 1. The shaded area (purple and orange) shows the error range for levels $m_1$ and $m_3$ due to the uncertainty on the MW probe pulses calibration. The solid lines correspond to the numerical simulation of the experiment. The discrepancy between the data and the simulation at the beginning of the optimized pulse is probably due to the finite bandwidth of the rf circuit. When we set $V_r$ and $V_i$ to zero at time $\tau$, the rf has a finite ring-down time, which affects the measured population. The effect of this ring down is not visible at the end of the pulse, as it corresponds there to a small additional rotation of the spin, which is automatically compensated by the optimization procedure. Finally, we find $P_c=96.2(3)\%$, $P_{e_1}=0.20(1)\%$, $P_{e_2}=0.74(4)\%$. We also measure $P_{e_3}=0.15(7)\%$, $P_{e_4}=0.13(8)\%$, $P_{m_3}=0.08(3)\%$, $P_{m_2}=0.37(4)\%$, $P_{m_1}=1.16(8)\%$.

Figure 6

Coherence test. (a) Evolution of the control voltages $V_r(\tau )$ (black line) and $V_i(\tau )$ (red line) in the sequence that tests the coherence of the optimized circularization pulse. We first apply a time-reversed optimized pulse followed by a circularization pulse similar to that of Fig. 2. The flat parts of the pulses ($0\le \tau \le 73$ and $163\le \tau \le 236\mathrm{ns}$) are now divided in two steps, ensuring that the reference $50c$ level is transferred back into itself by each pulse. (b) Probability to finally detect the atoms in $52c$ as a function of the relative phase between the MW Ramsey pulses. The points are experimental with statistical error bars, the solid line is a sine fit. The visibility [89.1(6)%] is excellent, limited mostly by the technical electric field noise in the experiment.

Figure 7

Optimized cat state preparation pulse. (a) Amplitude of the two quadratures of the rf pulse given by the Krotov optimization. Time evolution of the population of each Stark level of the lower diagonal (b) and of the instantaneous eigenstates of the atom-rf field Hamiltonian in the rotating frame (c), denoted as “dressed states” in the main text. The vertical position of the lines in (c) reflects the instantaneous energy of the corresponding state, and the color of a line shows its population at time $\tau$. The inset in (b) presents the computed population histogram for the different $m$ levels at $\tau =60\mathrm{ns}$, showing that the pulse prepares, during the first few tens of nanoseconds, a superposition of $m=1$ and a spin coherent state made up of levels with $m>2$.

Figure 8

Cat state preparation with the optimized pulse. (a) Control voltages $V_r(\tau )$ (black line) and $V_i(\tau )$ (red line), applied to produce the superposition of $52m_1$ and $52c$, as a function of time. (b) Time evolution of the population $P_j(\tau )$ of level $j$ ($j=m_1$, $m_2$, $m_3$, $e_2$, $e_1$, and $c$). The points are experimental with statistical error bars, using the color code of Fig. 1. The solid lines correspond to the theoretical prediction for the optimized pulse. The final target state population (averaged over the last six points) is $P_c=48(1)\%$ and $P_{m_1}=48.4(8)\%$. (c) Probability to find the atoms in the $52m_2$ state after the pulse (a) followed by the time-reversed optimized pulse that recombines $52m_1$ and $52c$ (Appendix pp6) as a function of the delay $\mathrm{\Delta }t$ between the pulses. The points are experimental with statistical error bars, the solid line is a sine fit. The large fringe visibility, mostly limited by the technical electric field noise, demonstrates the coherence of the cat state prepared by the first pulse.

Figure 9

Preparation of the $52m_1$ state. To normalize the population of $52m_1$ in the superposition state, we modulate the two quadratures of the 250 MHz carrier at 59.5 MHz in order to generate a 190.5 MHz rf field that is resonant with the $52m_2-52m_1$ transition. The points represent the population $P_{m_2}$ remaining in the $52m_2$ state as a function of duration $\tau$ of the pulse. For $\tau =88\mathrm{ns}$, we measure $P_{m_2}\approx 0.3\%$. The amplitude of the rf is chosen low enough so that the atoms cannot be transferred into $52m_3$ (numerical simulations estimate that $P_{m_3}<0.5\%$). The pulse thus prepares $51m_1$ with more than 99% efficiency.

Figure 10

Ionization signals. Number of detected atoms per sequence as function of the ionization field applied in the field-ionization detector for the laser-excited $52m_2$ state (red line) and for the $52c$ state prepared by the adiabatic passage (black line). The difference of height of the two peaks results from the shorter lifetime of the $52m_2$ state. The residual peak in the black signal at the position of the red one probably results from Rydberg atoms spuriously prepared in the state $m=-2$ due to laser polarization imperfections.

Figure 11

Radio-frequency circuit. The outputs of an eight-channel synthesizer (light red), labeled $S_1^{r,0}$, $S_2^{r,0}$, $S_3^{r,0}$, $S_4^{r,0}$, $S_1^{i,0}$, $S_2^{i,0}$, $S_3^{i,0}$, and $S_4^{i,0}$, are sent to eight sets of three successive mixers that are controlled either by the voltage $V_r(t)$ (black line, for the outputs $S_1^{r,0}$, $S_2^{r,0}$, $S_3^{r,0}$, $S_4^{r,0}$) or by $V_i(t)$ (red line, for the outputs $S_1^{i,0}$, $S_2^{i,0}$, $S_3^{i,0}$, $S_4^{i,0}$). The resulting signals are combined two by two using 3-dB couplers and sent to amplifiers, which feed the four ring electrodes surrounding the experiment (in gray).

Figure 12

dc response of the mixers. Left: Measurement of the ratio $S_k^r(V)/S_k^r(V_r^0)$, where $S_k^r(V)$ is the amplitude of the signal $S_k^r(t)$ when the voltage $V_r(t)$ is set to $V$ for $k=1$ (black squares), $k=2$ (orange circles), $k=3$ (blue upward triangles), and $k=4$ (green downward triangles). Right: Measurement of the ratio $S_k^i(V)/S_k^i(V_i^0)$, where $S_k^i(V)$ is the amplitude of the signal $S_k^i(t)$ when the voltage $V_i(t)$ is set to $V$ (same color code). We choose $k=3$ to determine $f_r(V)$ and $f_i(V)$.

Figure 13

Scaling factor optimization. Populations of the $52c$ (black), $52e_1$ (red), and $52e_2$ (green) states obtained after optimizing the parameter of the rotation part of the circularization pulse, for different values of the scaling factor $\lambda$.

Figure 14

Optimization of the first lobe of the cat state pulse. Populations of the level $m_1$, $m_2$, $m_3$, $e_2$, $e_1$, and $c$ as a function of the scaling factor $\alpha$ with which we amplify the first 4 ns of the pulse. Increasing $\alpha$ increases the final population of $52m_1$ and decrease the final population of $52c$ (until eventually the SCS part of the wave function no longer reaches the north pole at the end of the pulse, leading to an increase of $P_{e_1}$).

Figure 15

Visibility as a function of the field sensitivity. Visibility of the fringes measuring the coherence of (a) $|50c⟩+|52m_2⟩$ and (b) $|52c⟩+|52m_1⟩$, as function of the phase sensitivity of the fringes to the electric field $\xi$. The points are experimental (with statistical error bars), the line is a Gaussian fit of the form $V={\mathcal{V}}_0\exp (-{\xi }^2{\sigma }_F^2/2)$, where ${\mathcal{V}}_0$ is the intrinsic visibility that the fringes would have in the absence of electric field noise.