Experimental realization of phase-controlled dynamics with hybrid digital-analog approach

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2][3] While in principle this can be done for an arbitrary quantum system, 4 it often requires an intimidating number of gate operations with high precision.Analog approaches exploiting the continuous nature of quantum evolutions may often be more efficient, 5-8 but usually must be designed on an ad hoc basis.0][11][12] The flexibility in engineering and assembling digital and analog modules generates abundant possibilities for quantum simulation that are hardly available otherwise.For example, in a simulation of the quantum Rabi model, 13 a deep-strong coupling that is inaccessible to pure analog or digital approaches could be realized via a hybrid method. 14his work, we show that by employing a hybrid method, one can perform quantum simulations that otherwise cannot be implemented on a given platform.In particular, we demonstrate phase-controlled quantum dynamics and related phenomena via closed-contour interaction (CCI) in superconducting quantum circuits, which was originally forbidden by certain symmetryimposed selection rules.The simplest realization of C-CI involves a three-level system.Such systems with two of the three possible transitions being coherently driven have been widely researched for both fundamental interest and promising applications in areas such as quantum sensing 15,16 and quantum information processing. 17By opening the third transition, the three levels form a loop with a CCI, which leads to fundamentally new quantum phenomena, including phase-dependent coherent popula- |g ↔ |e and |e ↔ |f .The effective Hamiltonian of the system under rotating-wave approximation is given by where Ω A,B and ∆ A,B are the amplitudes and detunings, respectively, of the two external driving fields (see Fig.
If the system assumes a restrictive symmetry, then the third transition |g ↔ |f of the same type is forbidden.Even in systems of less restrictive symmetry (e.g., artificial atoms such as superconducting qubits), the amplitude of such transitions is usually vanishingly small. 28 Previously, a third driving of a different type or of the same type but of higher order was used to close the loop to form a CCI. 25,26We take a different approach.By combining an analog module corresponding to the evolution driven by H 0 with two digital modules that are unitary operators constructed from standard quantum gates, we effectively transform the original Hamiltonian H 0 to the following form (see Methods for details): To arrive at the above form, we set the am- For a three-level system subjected to a pumping drive It has been shown that by adding a counterdiabatic driving Ω q (t) (|1 ↔ |3 ) to close the loop, the resultant dynamics of the population become dependent on the handedness of the system. 30,31In particular, with the same driving fields, the Hamiltonian of the sys- For example, with carefully chosen values of the pulse areas, the handedness can be efficiently determined by measuring P 3 alone, where P 3 = 1(P 3 = 0) for L(R) 198 handedness. 27We note that such a counterdiabatic driv- Consider the four-level system formed by two Xmon superconducting qubits with a coupling strength of J (see The hybrid digital-analog approach used here is essential to our work, since on the one hand the above symmetry constraints forbid an inherent CCI that would manifest in the analog evolutions of the systems, and on the other hand, a pure digital approach is practically in-

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FIG. 1 .
FIG. 1. Realization of CCI with a hybrid digital-analog approach.(a) A three-level system (a qutrit) driven by two detuned external fields (described by a Hamiltonian of H0), when combined with specially designed digital modules (the T block) constructed from discrete quantum gates, can be used to realize a new Hamiltonian H hosting an inherent C-CI: T e −iH 0 t T † ≡ e −iHt , with a gauge-invariant phase φ.For consistency with the literature, we relabel the states of the qutrit as |g , |e , |f = 1, 2, 3. (b) In a similar way, combining the natural evolution of two resonant qutrits driven by two external fields (with identical amplitudes and phases, indicated by different colors) with certain digital modules can result in a CCI in a subspace of the system.Here 1, 2, and 3 correspond to |gg , |ge , and |eg , respectively.The gray sphere beside the state of 3 represents a dark state that is decoupled from the evolution of the system (see Methods).
FIG. 2. Phase-controlled quantum dynamics resulting from CCI in a single qutrit (see Fig. 1(a)).(a) Upper part: flowchart of the experiment, including a block for initialization (Xg,e), a digital module of T † composed of three gate operations, an analog module of the natural evolution driven by H0, and another digital module T , followed by projection measurements that yield the three populations of P1,2,3.Lower part: P1,2,3 as functions of the timespan of the intermediate natural evolution for three values of the gauge-invariant parameter φ.(b) Energy spectrum of the Hamiltonian of H in Eq. (2), obtained via discrete Fourier transform of the measured populations.It is shown in the form of |Em −En|, where E k are the eigenenergies of H, and m, n ∈ {1, 2, 3}.Dashed white lines represent theoretical predictions.

93FIG. 4 .
Fig.1(a)).Therefore, inherent CCI dynamics can be expected for such a Hamiltonian.In the case of equal and constant magnitudes, Ω p,q,s ≡ Ω, the population dynamics are strongly dependent on the phases φ p,s,q of the driving fields, through a gauge-invariant global phase φ = φ p + φ s − φ q .We will show an experimental demonstration of such CCI dynamics.We used Xmon-type superconducting qutrits in our experimental work.In this kind of artificial atom, the transitions of |g ↔ |e and |e ↔ |f are electric-dipole allowed, whereas the transition |g ↔ |f of the same type has a vanishingly small amplitude.28Two external microwave driving fields in the forms described above (Ω A,B ) are applied to the qutrit, with Ω p,q,s ≡ Ω and three independently adjustable phases φ p,q,s .Details of the experimental setup can be found in the Supplemental Materials.CCI dynamics.We first study the CCI dynamics of the system by measuring its time evolution at different values of φ.Figure2(a) shows the temporal sequence of operations.The system is initialized in the first excited state of |ψ(t = 0) = |e by a standard X gate.A digital module containing three quantum gates is applied to the qutrit, followed by an analog evolution driven by H 0 with two control parameters: the time span and the gauge-invariant phase φ.Another digital module, which is the Hermitian conjugate of the first digital module, is applied, followed by projection measurements that yield populations of all three states.As discussed previously, the combined effect of the middle three blocks is to subject the system to evolve under a new Hamiltonian H as in Eq. (2): e −iHt/ ≡ T e −iH0t/ T † .The gauge-invariant phase φ assumes a role as the flux of a synthetic magnetic field, which controls the dynamics of the system.At φ = 0, the populations evolve in time with a symmetric pattern without a preferred direction of circulation (middle panel, Fig.2(a)).Such symmetry in the circulation pattern is not observed for values of φ that are not integers of π.Two examples corresponding to φ = ±π/2 are shown in Fig.2(a).In each case, a circulation of certain chirality is observed: clockwise for φ = −π/2 and counterclockwise for φ = π/2.Such differences are rooted in the symmetry of the system upon time reversal.An examination of the time-reversal symmetry (TRS) in a strict sense requires reversing the flow of time, which is of course not experimentally feasible.However, the periodicity presented in the evolutions shown in Fig.2(a) allows for a practical definition of the TRS: ψ(t) = ψ(T 0 − t), where T 0 is the period of a given evolution 5 .By comparing the evolutions from t = 0 for- |2 ) and Stokes drive Ω s (t) (|2 ↔ |3 ) (see Fig. 3(a); for consistency with the literature, here we label the three states as |1 , |2 , and |3 ), the three eigenenergies and corresponding eigenstates are λ ± = ± Ω 2 p + Ω 2 s , λ 0 = 0, and |χ ± = (sin θ|0 ±|2 +cos θ|3 ), |χ 0 = cos θ|1 − sin θ|3 , with tan θ(t) = Ω p (t)/Ω s (t).In the celebrated technique of stimulated Raman adiabatic passage, 29 the two pulses are arranged in a counterintuitive order with the Stokes pulse coming first, and the eigenstate |χ 0 evolves adiabatically from |1 to -|3 as θ varies from 0 to π/2, thus accomplishing a nearly perfect state transfer coherently.

(
Fig.3(a)), where the +(−) sign is for L(R) handedness, and H.c is the Hermitian conjugate.Such a sign difference will result in the same counterdiabatic driving doubling or canceling the nonadiabatic coupling presented in the system, depending on its handedness.If φ is set to −π/2, then the populations of the final state, P 3 , of the enantiomers with L and R handedness are different.

199
ing was originally proposed to accelerate various adiabat-200 ic processes, but here its major effect is to differentiate 201 the L and R handedness.202We use pump and Stokes pulses of a Gaussian form 203 in our experiment: Ω p (t) = Ω 0 e −(t−τ /2) 2 /τ 2 , Ω s (t) =204Ω 0 e −(t+τ /2) 2 /τ 2 .Both pulses have a width of τ and are 205 delayed by the same amount.A third pulse in the form 206 of Ω q (t) = ±2 θ(t) is applied, where the +(−) sign corre-207 sponds to L(R) handedness.We prepare the system in 208 an initial state of |χ 0 .As discussed above, for L hand-209 edness, the nonadiabatic transition is canceled by Ω q (t) 210 and the system remains in the state |χ 0 , inducing a per-211 fect population transfer from |1 to |3 with P 1→3 = 1 212 as θ(t) evolves from 0 to π/2.Conversely, for R handed-213 ness, the nonadiabatic transition doubles, which enables 214 |χ 0 → |χ ± and P 1→3 < 1. Figure 3(c) shows the time 215 evolution of P 3 with different pulse areas Aπ, which is 216 defined as Ω p,s dt = Ω 0 τ √ π ≡ Aπ.The driving fields 217 Ω p,s,q in Fig. 3(b) result in a population transfer |1 → |3 218 for L handedness with P 1→3 = 0.986, and a suppression 219 of the same transfer for R handedness with P 1→3 = 0.003 220 when A ≈ 1.23 (Fig. 3(d)).221 Entanglement generation with CCI.Next, we ex-222 tend the generation of CCI via pure microwave drivings 223 to a more complex system of two coupled qubits, and 224 further demonstrate a new mechanism of entangling two qubits based on CCI, different from existing schemes that are widely used in quantum information processing with superconducting quantum circuits.

Fig. 1 (
Fig. 1(b)).We apply two transverse resonant driving fields to the two qubits, with an identical amplitude of J/ √ 2 and a phase difference of φ a −φ b = φ.Similar to the single-qubit case discussed above, we combine the natural evolution of such a driven system (an analog module) and a unitary operation T ′ (two digital modules implemented via standard gate operations) to realize an effective Hamiltonian for a three-state system {|eg , |ge , |gg } that can host CCI (see Fig. 1(b) and Methods).Furthermore, we can generate entangled states of the two qubits by removing the unitary operation T ′ , since it transforms the entangled state |gg + e iφ |ee to the ground state |gg , and the special form of T ′ e −iHt T ′ † used in this work mathematically corresponds to a linear transformation in the Hilbert space.Specifically, the two-qubit system can be directly transferred from the non-entangled state |eg or |ge| to the maximum entangled states of (|gg ± i|ee ) / √ 2 (Fig. 4(a) and (b)), within a time of t b = 2π/(3 √ 3J), under the condition of maximum TRS breaking at φ = ±π/2.The density matrices ρ ± of the entangled states |ψ ± characterized by quantum state tomography are given in Fig. 4(c)-(f), with fidelities of F + = 0.963 ± 0.026 and F − = 0.923 ± 0.029.The analytical form of the nontrivial two-qubit unitary operator e −iHt b is given in the Supplementary Information.This new mechanism to generate entanglement based on chiral CCI dynamics is different from the previous constructions of iSWAP 32,33 and controlled-Z gates, 34-36 formed by the subspace {|ge , |eg } or {|ee , |f g } in superconducting qubits.Discussion.We have proposed and experimentally demonstrated an effective realization of CCI in genuine three-level systems that do not host CCI inherently due to certain symmetry constraints.By assembling an analog module of the natural evolution governed by their original Hamiltonians with carefully designed digital modules, we can effectively bypass such constraints and establish a CCI without auxiliary driving signals that are technically challenging to implement.Based on such a CCI, we can demonstrate a variety of interesting related phenomena such as a phase-controlled chiral dynamics, chiral separation, and a new mechanism to generate entangled states.

Figure 4
Figure 4 feasible, as too many quantum gate operations would be 298ing two qubits, they are coupled via an ancillary qubit that can fine 299 tune the effective coupling strength38.Further details of the sam-300 ples and measurement setup can be found in the Supplementary 301 Information.302EffectiveHamiltonian of the three-level system.The ef-303 fective Hamiltonian of the microwave-driven qutrit in a rotat-304 ing frame described by the operator U = |g g| + |e e|e iω A t + 305 |f f |e i(ω A t+ω B t) and under the rotating-wave approximation is 306 given by Eq. 1.The unitary operator T that serves as a digital