Experimental Demonstration of Swift Analytical Universal Control over Nearby Transitions

Despite frequent utilization of two-level systems for many quantum enhanced applications, most practical systems are intrinsically of many levels nature, whose energy separations come from couplings of internal degrees of freedom or from splittings by external fields. Besides, for applications with quantum many-body system, such as, high-sensitivity precision measurements [1], large-scale quantum computation [2] and simulation of quantum many-body system [3], either more qubits or larger internal complexities must be involved, and then the energy or associated frequency spectrum would become crowded. In these cases, spectrally resolving a transition with an external drive field for high-fidelity manipulation would become difficult, which may limit the scalability of quantum systems, and may hinder large-scale quantum tasks where precise quantum control over individual transition is required.

A straightforward solution for this limitation is to narrow the range of Fourier frequency components from the pulse of the drive field, for example to apply smooth-shaped pulses [4, 5, 7, 6, 8] and using narrow linewidth transitions [9]. However, in general cases, a desired transition rate determines the minimum temporal integral of pulse strength, i.e. “pulse area”, and the lower field strength in turn results in longer pulse duration, leaving the control vulnerable to frequency drifts, decoherence or limiting the number of operations in given finite coherence times. Meanwhile, insufficiently weak pulse strength would create unwanted transition in the adjacent spectral line, giving rise to a population leakage or cross-talk error in the control.

To tackle with the problem of spectral crowding while keeping the operation fast, one idea is to abandon the assumption that the unwanted nearby transitions remain without being driven, but rather allowing a controlled evolution where the overall desired operation is reached at the end of the drive pulse. By numerically searching for desired pulse shape with optimal control techniques [10, 11, 12, 13, 14, 15], an overall operation close enough to the target one can be obtained. However, the numerical pulse shaping can only be done in a case-by-case way, and only one certain shape can be obtained under a particular set of conditions. Besides, due to the piecewise constant control ansatz, the complex numerical shape may be inaccurate in cases of bounded bandwidth.

On the other hand, for two-level quantum systems, analytic solution for detuned single or multiple axis driving become available [16, 17, 18]. This brings an opportunity to better understand the underlying physics, and converting a full numerical optimization to a parametric boundary problem, where a series of solutions can be obtained when the boundary conditions are met. Thus, one has the additional freedom to incorporate various pulse-shaping techniques, and find experimental-friendly simple pulse solutions. We note that this analytic solution can be extended to drive desired transitions in multi-level quantum systems, while leaving adjacent transitions intact. Therefore, such a technique can go beyond addressing a certain transition, and can be tailored to simultaneously drive nearby transitions. The combination of individual and simultaneous control would lead to the capability of arbitrary manipulations in the nearby transitions, and may assist in quantum logic operations on multi-level systems [19, 20, 7, 21, 22, 23, 24, 25]. Remarkably, as this technique removes the weak drive limitation, it might be also useful for spectroscopy, where a faster probe to obtain more information might be possible.

Here, we extend these analytical based swift quantum control techniques to a four-level system, where individual or simultaneous quantum control over the two pair of levels can be effectively obtained. We then experimentally demonstrate individual operation where a transition is driven while the other is intact, and simultaneous operations where quantum phase and Hadamard gate operations are performed on both transitions. We achieve operation fidelities ranging from 99.2(3)% to 99.6(3)%, limited by decoherence and experimental imperfections.

We firstly extend previous two-level schemes [16, 17, 18] to the case of four levels $\{\ket{1},\ket{2},\ket{3},\ket{4}\}$ forming two pairs of nearby transitions, as shown in Fig. 1(a), to implement a fast, high-fidelity universal control over the two transitions. Labeling the transition frequency between $\ket{i}\leftrightarrow\ket{j}$ as $\omega_{ij}$, when the transition with $\omega_{34}$ is driven resonantly by an external field with time-varied Rabi rate $\Omega$, the transition with $\omega_{12}$ is driven simultaneously by the same field with Rabi rate $\Omega^{\prime}$ with detuning $\Delta$ from resonance. Here we consider the case that $\Omega\sim\Omega^{\prime}$ and with a fixed ratio; the transition frequencies across the two pairs $\{\ket{1},\ket{2}\}\leftrightarrow\{\ket{3},\ket{4}\}$ are much larger than $\omega_{12}$ or $\omega_{34}$, thus we can limit our discussion within each pair. In the rotating framework, we obtain the interacting Hamiltonian as

where $\varphi$ is the phase of drive field. Thus, the nearby small detuned transition will be excited unintentionally, leading to phase and leakage errors, i.e., the so-called spectral crowding problem. The trivial way to deal with the problem is to drive the target transitions with a lower Rabi rate, i.e., max$\{\Omega,\Omega^{\prime}\}/\Delta\ll 1$, typically with max$\{\Omega,\Omega^{\prime}\}/\Delta\lesssim 0.1$ for a general control. However, lower Rabi rates lead to longer control time and less number of control pulses within the coherence times. As we analyze and experimentally demonstrate below, the time-dependent control in this work would enable individual or simultaneous control of both transitions even with max$\{\Omega,\Omega^{\prime}\}/\Delta\sim 1$, thus giving approximately an order of magnitude speed up.

To illustrate the idea for swift analytic-based control, we consider the general evolution operator of Hamiltonian in Eq. (33) with a duration $[0,T]$ as

where $U(t)$ and $U^{\prime}(t)$ are the evolution operator for the resonant and detuned subspace, respectively. To obtain precise control over the transitions, we need to obtain target dynamics for both subspaces, preferably beyond the weak drive limit. While the evolution in the resonant subspace is related to the pulse area and phase given by $\Omega(t)e^{i\varphi}$ in Eq. (33) in a straightforward way, the complex dynamics of the detuned subspace can be obtained, by extending Refs. [16, 17, 18], as [26]

where $U_{R}$ is

and $U_{C}(t)=U_{0}(t)\cdot U_{0}^{\dagger}(0)$ with

which is parameterized by $\zeta(t)$, and $\xi_{\pm}(t)=\frac{1}{2}\int^{t}_{0}\Delta\sqrt{1-\frac{(2\dot{\zeta}(t))^{2}}{\Delta^{2}}}\csc{[2\zeta(t)]}dt^{\prime}\pm\frac{1}{2}\arcsin{\frac{2\dot{\zeta}(t)}{\Delta}}$. This analytic solution allows us to achieve any SU(2) matrix, and the required $\Omega^{\prime}(t)$ is determined by $\zeta(t)$ and $\Delta$ as

Thus it is possible to design $\zeta(t)$ for arbitrary control in the detuned subspace. For the resonant subspace, given a fixed value of $\Omega/\Omega^{\prime}$, Eq. (10) leads to a waveform of $\Omega(t)$ for a controlled rotation; concatenated segments with varied $\varphi$ give arbitrary control [26]. To obtain $\zeta(t)$, we can further decompose $\zeta(t)$ into a weighted sum of analytic functions, e.g., sinusoidal functions, with a period equal to the total operation time $T$, and we can numerically find proper weights to obtain a required overall operation at $T$, under further constrains of $\dot{\zeta}(t)^{2}/\Delta^{2}<1$ and $\cot[2\zeta(t)]$ being finite. In practice, we set $T\sim{1}/{\Delta}$, then $\dot{\zeta}(t)$ and $\ddot{\zeta}(t)$ are on the order of $\Delta$ and $\Delta^{2}$, respectively. Thus in general the two terms in Eq. (10) are both on the order of $\Delta$ and are not canceled with each other, so that the average of Rabi rates can be roughly on the order of $\Delta$, and the operation time is much shorter than the weak driven case.

In our experiment, we trap a ${}^{9}\rm{Be}^{+}$ ion in a linear Paul trap [28] as shown in Fig. 1(b), and the desired Hilbert space is defined within the $2s~{}^{2}S_{1/2}$ hyperfine states $\ket{F,m_{F}}$ of the ion, with $\ket{1}\equiv\ket{1,1}$, $\ket{2}\equiv\ket{1,0}$, $\ket{3}\equiv\ket{2,1}$ and $\ket{4}\equiv\ket{2,0}$. Applying an ambient magnetic field at 6.7798 mT, we have $\omega_{12}=2\pi\times 48.798$ MHz, $\omega_{34}=2\pi\times 48.879$ MHz, $\Delta\equiv\omega_{34}-\omega_{12}=2\pi\times 81$ kHz, and $\omega_{13}=2\pi\times 1.16~{}\rm{GHz}$. We apply radio-frequency (RF) fields driving transitions $\ket{1}\leftrightarrow\ket{2}$ and $\ket{3}\leftrightarrow\ket{4}$. RFs are generated by an arbitrary-wave-generator (AWG) with a power amplifier [26], and thus can be fully controlled by programming the AWG. When we drive the transition $\ket{3}\leftrightarrow\ket{4}$ on resonance with Rabi rate $\Omega$, the transition $\ket{1}\leftrightarrow\ket{2}$ is also driven with Rabi rate $\Omega^{\prime}$ and a detuning $\Delta$, thus implementing the Hamiltonian in Eq. (33). The Rabi rate ratio $\Omega/\Omega^{\prime}$ is experimentally measured to be 1.7, and remains fixed throughout the experiment since it depends on the coupling constants. To assist state preparation and detection, we also apply microwave fields to a separate antenna, as shown in Fig. 1(b), sourced by a power amplified frequency-doubled direct-digital-synthesizer.

As a demonstration of individual control over one of the two closely spaced transitions, while leaving the other intact, we perform a simultaneous operation on both pairs of transitions with $U(T)=X$ and $U^{\prime}(T)=I$ ($\varphi=0$), where $X$ is a Pauli operator and $I$ is an identity operation. According to Eq. (10), we can obtain $\Omega(t)$ from $\zeta(t)$ and $\Delta$, where we set

with $\phi_{Z}=\pi/4$, leading to $\dot{\zeta}(0)=\dot{\zeta}(T)=\ddot{\zeta}(0)=\ddot{\zeta}(T)=0$, thus $\xi_{+}(0)=\xi_{-}(0)=0$, $\xi_{+}(T)=\xi_{-}(T)$ and $\Omega(0)=\Omega(T)=0$. When we design a specific waveform, we can set the boundary conditions and find parameters by numerical optimization with trust-region-dogleg method [29]. By setting $T=8.88$ $\mu$s for $\Delta=2\pi\times$81 kHz and with numerical optimization, we obtain $A_{3,4,5}=\{-0.793,0.464,-0.085\}$ respectively, giving $\xi_{+}(T)=\xi_{-}(T)=2\pi$. Thus the average of Rabi rates $\bar{\Omega}$=$2\pi\times$56 kHz and $\bar{\Omega}^{\prime}$ = $2\pi\times$33 kHz, comparable to $\Delta$. The quantum dynamics is shown in Fig. 2 (a), where the solid lines are obtained by theoretical simulation while the points are the experimental results. The quantum process can be described by $\rho_{out}=\sum_{j,k}E_{j}\rho_{in}E^{\dagger}_{k}\chi_{jk}$, where $\rho_{in}$ is the density matrix of the initial state, $\rho_{out}$ is the density matrix of the state after the process, $E_{j,k}\in\{I,X,Y,Z\}$ are basis operators, $I$ is the identity operation, and $X$, $Y$ and $Z$ represent Pauli operators. Thus the matrix $\chi$ uniquely represents the process. To estimate the components of $\chi$ for the operation, we perform process tomography [30, 31, 32], where we initialize the ion to various states, apply the operation, and measure the outcome. For example, to measure components of $U$, we first apply Doppler cooling, optical pumping with 313 nm laser pulses [26] to prepare $\ket{2,2}$ state, and apply resonant microwave $\pi$ or $\pi/2$ pulses to transfer to states of $\ket{3}$, $\ket{4}$, $(\ket{3}+\ket{4})/\sqrt{2}$ and $(\ket{3}+i\ket{4})/\sqrt{2}$, respectively. After applying the operation, we map one state in the pair to $\ket{1,-1}$ and the other to $\ket{2,2}$ by a series of microwave pulses, followed by resonant fluorescent measurement by driving $\ket{2,2}$ to $2p~{}^{2}P_{3/2}$ for 400 $\rm{\mu s}$, while the $\ket{1,-1}$ state is out of resonance and appears dark. We typically collect on average 25 counts for $\ket{2,2}$ and less than 1 count for dark states. We perform each measurement for 1000 trials to reconstruct $\rho_{out}$ by maximum likelihood state tomography [33, 6, 34], and from which we further obtain the process matrix components, as shown in Fig. 2 (b). Similar procedure is repeated to measure components of $U^{\prime}(T)$. We obtain process fidelity [30, 35] $F_{U}=99.5(4)\%$ and $F_{U^{\prime}}=99.6(3)\%$, respectively, through the definition of

where $\chi_{\rm ideal}$ is the ideal process matrix for the corresponding operation and $\chi_{\rm exp}$ is the measured one. In our experiment, the measured coherence times for the resonant and detuned subspaces are approximately 1.9 ms and 1.1 ms, respectively, with Ramsey type experiment, which will contribute up to approximately 0.2% operation infidelity. Experimentally, we also observe fast drifts of the ambient magnetic field after each calibration, which, together with other unknown ones, we attribute as other possible error sources. Finally, to compare with the case of weak square pulse, we performed a numerical simulation [26]. Given a realistic decoherence of $\Delta/40$ and $\Omega/\Omega^{\prime}=1.7$ as in the experiment and without further imperfections, the optimal control fidelity with square pulse is reached at $\Omega/\Delta\sim 0.08$, while for our implementation a similar fidelity can be achieved at $\bar{\Omega}/\Delta\sim 0.7$, and thus decrease the required operation time by approximately an order of magnitude.

We next demonstrate simultaneous quantum control over both pairs of transitions, by implementing $U(\tau)=U^{\prime}(\tau)$ as quantum logic gates of S gate, T gate and Hadamard (H) gate [26], in a two-step way with their corresponding manipulation times being $\tau_{1}$ and $\tau_{2}$, respectively, so that the total gate-time is $\tau=\tau_{1}+\tau_{2}$. In the phase gate cases, a rotation $R_{z}(\phi)$ around the $Z$ axis with angle $\phi$ is applied in both detuned and resonant subspaces, where $\phi_{S}=\pi/4$, $\phi_{T}=\pi/8$ for S and T gates, respectively, by setting $\xi_{+}^{\phi}(\tau_{1,2})=\xi_{-}^{\phi}(\tau_{1,2})=\phi_{S,T}/2$. According to Eq. (11), we can obtain specific parameters for $\Omega^{\prime}(t)$ in Eq. (10). Then, we can apply two waveforms similar to Eq. (11) with different initial phase $\phi=\varphi_{1}-\varphi_{2}$ for the two steps to construct a desired phase gate, leading the first step to $U_{1}^{{}^{\prime}}(\tau_{1})=R_{z}(\phi/2)$ and $U_{1}(\tau_{1})=X$ with the initial phase of the pulse being $\varphi_{1}=0$, and for the second step $U_{2}^{{}^{\prime}}(\tau_{2})=R_{z}(\phi/2)$ and $U_{2}(\tau_{2})=R_{z}(\phi)X$ with the initial phase being $\varphi_{2}=-\phi$. In this way, we can get $U^{{}^{\prime}}(\tau)=U(\tau)=R_{z}(\phi)$. As for H gate, in the first step, we set $\varphi_{1}=\pi/2$, $\phi_{H}=3\pi/8$ in Eq. (11) and optimize $\{A_{n}\}$ to obtain $\xi_{+}^{\phi}(\tau_{1})=\xi_{-}^{\phi}(\tau_{1})=\pi/2$, which lead to $U_{1}^{{}^{\prime}H}(\tau_{1})=H$ and $U_{1}^{H}(\tau_{1})=\sqrt{Y}$, and for the second step, we can apply a waveform that is the same as the individual control in Eq. (11) with $\varphi_{2}=0$, to get $U_{2}^{{}^{\prime}H}(\tau_{2})=I$ and $U_{2}^{H}(\tau_{2})=X$. The two operations form $U^{{}^{\prime}}(\tau)=U(\tau)=H$. We design waveforms with gate time $\tau_{S,T,H}=\{20.77,22.29,25.37\}$ $\rm{\mu}$s respectively for each gate, similar to the process above and perform process tomography, with details in supplemental material [26]. The process matrices are shown in Fig. 3, with process fidelity $F_{S}^{\prime}$= 99.4(3)%, $F_{S}$= 99.4(3)%, $F_{T}^{\prime}$= 99.3(3)%, $F_{T}$= 99.4(4)%, $F_{H}^{\prime}$= 99.2(2)% and $F_{H}$= 99.2(3)%, where $F_{S/T/H}^{\prime}$ and $F_{S/T/H}$ correspond to S, T and H gates on the detuned and resonant transitions, respectively.The decoherence affects the measured fidelity with a contribution up to approximate 0.3%, 0.3% and 0.4% for S, T and H gates, respectively.

In conclusion, we experimentally demonstrate quantum control over two pairs of nearby transitions with time-dependent pulses, where the waveform parameters can be readily obtained with the help of an analytical model. We demonstrate individual and simultaneous control of each pair of transitions, reaching a process fidelity up to 99.6(3)%. Our work can be extended to a range of systems. For example in recent experiments with trapped molecular ions, adjacent transitions can be spaced on the order of kHz, rendering the transition pulse duration on the order of ms with smooth edges [7]; and for long ion chain, coupled motional modes can space below 10 kHz [19, 25] and superconducting transmon qubits with weak anharmonic levels [20]. For future research, it might be interesting to enhance the robustness for the designed shaped pulses or even incorporate optimization requirements into the process of finding a target solution [36]. In addition, as more levels may be involved for many-body or many-degree-of-freedom quantum systems, extending the current mathematical model to cover more levels that are closely spaced may be also of practical interest. On the other hand, our demonstration involves four levels orthogonal to each other, thus some extension would be needed when the transitions involve a shared energy level, such as $\Lambda$-, V-, and ladder-type systems. Regarding quantum sensing and metrology, this demonstration might be beneficial to vector magnetometry utilizing nitrogen-vacancy centers [37] to produce a fast probe driving nearby transitions corresponding to a few crystal orientations.