Fast and robust population transfer with a Josephson qutrit via shortcut to adiabaticity

We propose an effective scheme to implement fast and robust population transfer with a Josephson qutrit via shortcut to adiabaticity. Facilitated by the level-transition rule, a Λ-configuration resonant interaction can be realized between microwave drivings and the qutrit with sufficient level anharmonicity, from which we perform the reversible population transfers via invariant-based shortcut. Compared with the detuned drivings, the utilized resonant drivings shorten the transfer times significantly. Further analysis of the dependence of transfer time on Rabi couplings is helpful to experimental investigations. Thanks to the accelerated process, transfer operation is highly insensitive to noise effects. Thus the protocol could provide a promising avenue to experimentally perform fast and robust quantum operations on Josephson artificial atoms.

Particularly, for speeding up transfer operations, the utilized resonant drivings in our scheme outperform the detuned drivings remarkably. The time dependence of coherent transfer on Rabi couplings could provide the experimental performance with some optimal choices. With the accessible decoherence rates, the protocol possesses high robustness due to a short operation time. So, the proposed QPT with the qutrit driven by two resonant drivings could offer a potential approach to experimentally implement fast and robust transfer operations on Josephson artificial atoms.

System and Model
As schematically depicted in Fig. 1, a Cooper-pair box (CPB) circuit under consideration includes a superconducting box with n extra Cooper pairs, in which the charging energy scale of the system is E c . Through two symmetric Josephson junctions (with the identical coupling energies E J and capacitances C J ), the CPB is linked to a segment of a superconducting ring. In the charge-phase regime 48 , the characteristic system parameter E J has the same order of magnitude as E c , which satisfy Δ ≫ E J ~ E c ≫ k B T, in which the large energy gap Δ prohibits the quasiparticle tunneling, and k B T stands for a low energy of thermal excitation. The CPB is biased by a static voltage V d through a gate capacitance C g . Meanwhile, a static magnetic flux Φ d through the ring aims at adjusting the effective Josephson coupling E Jd . An ac gate voltage ∼ V s is applied to the box through gate capacitance C g as well, and a time-dependent flux Φ ∼ p threads the ring. Here the microwave drivings are used to induce the desired level transitions 46,49 , as mentioned below.
In the absence of the microwave drivings ∼ V s and Φ ∼ p , the static Hamiltonian of the CPB system is given by H 0 = E c (n − n d ) 2 − E Jd cos θ, in which E c = 2e 2 /C t , with C t = 2C J + C g being the total capacitance, and θ denotes the average phase difference of the two junctions, which is canonically conjugate to n, namely, [θ, n] = i. The polarized gate charge induced by V d is is the effective Josephson coupling, in which  where the first term is the charging energy, and the second one represents the Josephson coupling. In light of Eq. (1), we can obtain the eigenlevels and eigenstates of the static charge-phase system. With the Josephson coupling E J = E c , we get E Jd = 1.3E c by adjusting Φ d . And then the first three levels E j versus n d are plotted in Fig. 2, with j = 1, 2, and 3. At a magic point of n d = 0.5, we deal with three eigenstates | 〉 s j , in which each state can be expressed as a superposition of Cooper-pair number states, i.e., | 〉 = ∑ s c n j n jn , with c jn being the superposition coefficients. The quantum states at the magic point are insensitive to the first-order dephasing effect, which thus contributes to prolong the decoherence time of the system 48 . Driven by the considered microwave fields, the first three levels at the magic point can be decoupled from the fourth level due to the prohibition of the level-transition rule 47 3 is selected to constitute an effective qutrit under consideration. It is found that level anharmonicity in the qutrit is enough, leading to energy spacings ω 32 = (E 3 − E 2 )/ℏ and ω 21 = (E 2 − E 1 )/ℏ far away from each other. The sufficient anharmonicity can eliminate the leakage errors induced by the coherent drivings, which is highly beneficial for performing robust population transfer with the qutrit 35,49 .
As shown in Fig. 3, two classical microwave drivings Φ ∼ p = Φ p cos(ω p t) and ∼ V s = V s cos(ω s t), acting as the corresponding pump and Stokes fields, are applied to induce the desired level couplings respectively, where the microwave frequency ω p (ω s ) is resonantly matched with the transition frequency ω 31 (ω 32 ) 47 . Note that the amplitudes Φ p and V s are controllable here. Different from the previous works that induced level transitions only via electrical interactions 35,49 , the present scheme adopts both ac voltage and time-dependent bias flux. Owing to the sufficient level anharmonicity, there exists a large detuning δ s = ω s − ω 21 , and thus the ∼ V s -induced transition between s 1 and s 2 vanishes nearly. Since the amplitude Φ p (V s ) is much smaller than Φ d (V d ) in our scenario, the effects of Φ p and V s on the eigenlevels can be ignored safely.
In our scheme, we treat a Λ-type interaction between the qutrit and the microwave drivings. The magnetic interaction between the CPB system and the bias flux Φ ∼ p reads p Jp n which has a non-diagonal coupling form within the basis n . The amplitude of Φ ∼ p is much smaller than Φ 0 , which    , where n′ and n′′ are Cooper-pair numbers. The interaction Hamiltonian between the microwave pulse ∼ V s and the CPB system takes a diagonal form, s cs n d where  n s = n s cos(ω s t), with n s = C g V s /2e. Here the fast oscillating term  n s 2 with a higher frequency 2ω s has been omitted well under the rotation wave approximation (RWA) 35 2 , the Λ-configuration interaction under the reference frame rotating at frequencies ω p and ω s can be expressed as where the RWA has been adopted and the two-photon resonance is satisfied, i.e., ω 31 Obviously, the Hamiltonian in Eq. (6) has a dark eigenstate with zero eigenenergy, which is a superposition of s 1 and s 2 . Through adiabatically adjusting the Rabi couplings, population transfers can be implemented within the subspace s s { , } 1 2 when the system evolves only along the dark state 1 . However, the adiabatic operations generally need long times, which are undesirable for some artificial systems with short decoherence times.

Fast population transfer via invariant-based STA. The instantaneous eigenstates |ψ
0 where θ = arc tan(Ω p /Ω s ), and the corresponding eigenvalues are E 0 = 0 and  = ± Ω + Ω ± E /2 p s 2 2 , respectively. In ref. 51 , Lewis and Riesenfeld derived a useful relation between the solutions of the Schrödinger equation of a system with time-dependent Hamiltonian and the eigenstates of the corresponding invariant. Based on the invariant-based inverse engineering, we can speed up the population transfer significantly. In the following, we construct a desired dynamical invariant. The Hamiltonian in Eq. (6) can be rewritten as where T x , T y and T z are spin-1 angular momentum operators 52 , The matrices T x , T y , and T z satisfy the commutation relations  where Ω 0 is an arbitrary constant with units of frequency to keep I(t) with dimensions of energy. Consequently, the eigenstates of the invariant I(t), which satisfy I(t)|φ n 〉 = λ n |φ n 〉, can be obtained as sin cos sin cos sin sin cos , (13) whose eigenvalues are λ 0 = 0 and λ ± = ±Ω 0 /2, respectively. According to Lewis-Riesenfeld theory 51,54,55 , the dynamics of the three-level system is generally governed by a superposition of orthonormal dynamical modes 51 , where each C n is a time-independent amplitude and the Lewis-Riesenfeld phases α n are defined as n t n I n 0 In the case of three-level system, we have α 0 = 0 and t p s 0 where the dot represents a time derivative. Because of dI/dt = 0, the time-dependent parameters γ and β are related to Rabi frequencies Ω s,p by the following equations, p Once the appropriate boundary conditions for γ and β are fixed, the Rabi frequencies Ω s and Ω p are determined to perform the desired population transfer from an initial state to a final one 12 .
To keep the state stationary at initial and final times, we set the boundary conditions for β and γ as follows,  (16) and (17), the boundary conditions of γ and β can be given by f Now we design β(t) and γ(t) as polynomial ansatz, As displayed in Fig. 4(a,b), using the polynomial ansatz in Eqs (22) and (23), we have the time-dependent Rabi couplings Ω s,p , by which the target population inversion from s 1 to −s 2 can be accomplished after a duration time t f = 16.8 ns. Even with the slightly diminished couplings π Ω = . /2 0 16 s p m , ( ) GHz 47 , the transfer time is about 19 ns, much shorter than the adiabatic transfer time ~150 ns as discussed in ref. 47 . Here the transferred probability amplitude from the initial state s 1 to target state −s 2 can be formally expressed as is the final state at t = t f . For a chosen ε = 0.02 in our scheme, we get P 1→2 = 99.98%, which is high enough for quantum state engineering. As an important and necessary issue for quantum information processing, the inverse population transfer from an initial state |−s 2 〉 to target state |s 1 〉 has been demonstrated in Fig. 4(d), in which the required frequencies Ω s,p as functions of time are given in Fig. 4(c). As a result, the bidirectional state transfer |s 1 〉 ↔ |−s 2 〉 can be executed flexibly by adjusting the Rabi couplings.
As a distinct advantage, our protocol adopting the resonant two-photon interaction can implement the faster transfer operation, when compared with the largely-detuned drivings. For a two-photon resonance but with a common detuning Δ, the above Λ-type interaction Hamiltonian becomes In the large detuning regime Δ ≫ Ω s,p , as mentioned in ref. 46 , level state |s 3 〉 is scarcely populated during the population transfer |s 1 〉 ↔ |s 2 〉. After an adiabatical elimination of the intermediate state |s 3 〉, a reduced two-level system within the subspace {|s 1 〉, |s 2 〉} can be obtained, whose Hamiltonian can be described by and Rabi coupling Ω e = −Ω p Ω s /(2Δ). Based on the Hamiltonian in Eq. (27), the accelerated population transfers have been studied using the inverse engineering approach 46 . Here our central point of interest is the effect of detuning Δ on the transfer speed. For a state transfer from |s 1 〉 to |−s 2 〉, as indicated in Fig. 5(a), we analyze the dependence of needed time t f on the detuning Δ for the utilized π Ω = . Robustness against decoherence effects. Without dissipation effects, one could obtain an ideal population transfer with conversion probability P id = 1. However, owing to the decoherence effects originating from energy relaxation and dephasing, the system evolution becomes dissipative. By adopting the standard dissipation theory, we next treat the decoherence effects on the population transfer. After tracing out the environmental degrees of freedom 56 , we have the reduced density matrix ρ which is associated with s 1 , s 2 , and s 3 . For a weak coupling between the qutrit and the environment 57 , by taking the Born-Markov approximation, the dynamical evolution of ρ can be characterized by the Lindblad-type master equation [58][59][60] I in which the first term governs the unitary evolution subject to a Λ-type driving, and the second term 47 , in which the energy levels satisfy E k < E l . To quantitatively represent the decoherence effects on the population transfer, we introduce a fidelity as 46

Discussion
The present strategy may have the following characteristics and advantages. (i) Compared with the adiabatic process of population transfer 1 , the shortcut-based operation has been sped up sharply and still insensitive to timing errors and parameter fluctuations, which thus have a variety of potential applications to quantum coherent control and information processing. (ii) Different from the transmon-regime quantum circuit 62 , our considered charge-phase CPB has the sufficient level anharmonicity, and then the driving-induced leakage errors can be neglected safely. The suitable level structure is very beneficial to implement the nonleaky quantum manipulation 63 . (iii) Within the Λ-type qutrit, the two-photon resonant interaction has been constructed in our scheme. By applying the invariant-based shortcut to an effective three-level system directly, we realize the faster quantum operations than that in the case of the reduced two-level system as utilized in 46 . (iv) In contrast to ref. 47 , the invariant-based engineering in the proposed protocol keeps Hamiltonian in its original form, which could provide a more straightforward way to understand the dynamical process. Only two resonant microwave drivings are needed for performing the rapid state transfers, in which the decrease in number of the drivings is highly useful to the experimental implementation. Besides, the operation time has been shortened significantly from 25 ns (with fidelity 99.81%) in ref. 47 to 16.8 ns (with fidelity 99.98%) in the present scheme. (v) Facilitated by the direct magnetic coupling between s 1 and s 3 , the present Λ-type qutrit is addressed at the magic point n d = 0.5, which is different from the previous works 35,50 . Then the first-order dephasing effect can be removed effectively, which thus enhances the system decoherence time greatly. In summary, we propose a promising scheme for speeding up the adiabatic population transfer in a Josephson qutrit by the technique of invariant-based STA. At the magic working point, the three lower levels constitute an effective qutrit with sufficient level anharmonicity. Based on the electric and magnetic couplings, a Λ-type resonant interaction is induced by two microwave drivings. In the resonant regime, we implement the accelerated and reversible population transfer via the invariant-based inverse engineering. As a prominent advantage, our protocol shorten the operation times significantly compared to the largely-detuned drivings. We further analyze the time dependence of state transfer on Rabi couplings, which is helpful to possible realizations. With the accessible decoherence rates, the rapid transfer operation is highly robust against the noise effects. So the protocol could offer an optimal avenue for experimentally investigating fast and robust population transfers with the Josephson artificial atoms.