Speeding up adiabatic population transfer in a Josephson qutrit via counter-diabatic driving

We propose a theoretical scheme to speed up adiabatic population transfer in a Josephson artificial qutrit by transitionless quantum driving. At a magic working point, an effective three-level subsystem can be chosen to constitute our qutrit. With Stokes and pump driving, adiabatic population transfer can be achieved in the qutrit by means of stimulated Raman adiabatic passage. Assisted by a counter-diabatic driving, the adiabatic population transfer can be sped up drastically with accessible parameters. Moreover, the accelerated operation is flexibly reversible and highly robust against decoherence effects. Thanks to these distinctive advantages, the present protocol could offer a promising avenue for optimal coherent operations in Josephson quantum circuits.


Introduction
Superconducting quantum circuits, which behave like artificial atoms, possess many appealing advantages such as an adjustable level structure, flexible controllability and convenient measurement [1][2][3][4]. Coherent control over artificial atoms through varying external bias voltages and magnetic fluxes has attracted considerable attention over recent years [5][6][7][8][9]. The fruitful realization of dynamical behavior is closely related to many (Ξ-, Λ-, V-and Δ-type) configurations of interactions between the artificial atoms and the driving fields [3]. Especially, allowed by the level-transition rule, the novel Δ-type interaction only occurs in artificial atoms [10,11], which has no counterpart in natural atoms. Based on these kinds of interactions, superconducting artificial atoms provide us with an excellent platform from which to explore quantum information science, quantum state engineering and fundamental quantum laws [12][13][14][15].
Quantum population transfer (QPT), acting as a coherent operation, is a critical issue in the context of quantum information processing [16][17][18][19]. The performance of optimal QPT has drawn increasing attention [20][21][22][23]. Based on the powerful method of stimulated Raman adiabatic passage (STIRAP) [16,24], a great deal of effort has been devoted to studying QPTs with Josephson three-level systems both theoretically and experimentally [25][26][27][28][29]. Although adiabatic operations are not sensitive to timing errors and are robust against fluctuations of control parameters, STIRAP processes generally take relatively long times, which makes them undesirable in some practical cases. Therefore, the realization of faster population transfer is highly sought after [22,[30][31][32] and from which the desired quantum operation can be performed within a shorter time and decoherence effects can be greatly reduced. Another key point related to optimal QPT is the reversible transferring quantum state [33][34][35], which is a requisite for the storage and retrieval of quantum information.
A set of attractive techniques called 'shortcut to adiabaticity', consisting of invariant-based inverse engineering [36,37], transitionless quantum driving (TQD) [38][39][40] and the fast-forward approach [41], has been put forward for speeding up adiabatic evolution processes, which can carry out the same target operation as that in the adiabatic process but within a shorter time. Thus the shortcut approach has attracted considerable attention [42][43][44][45]. In two-and three-level systems, the method of inverse engineering has been utilized extensively to speed up adiabatic population transfers [46,47]. By designing counter-diabatic driving, TQD can control and steer the fast evolution of a system mimicking adiabaticity exactly. In recent years, the TQD method has been adopted widely for studying dynamical behaviors and state engineering with different systems [48][49][50][51][52][53]. Particularly, the application of TQD to the rapid generation of entangled states is a robust method of information processing [54,55]. Quite recently, by using the TQD technique, the quantum coherent manipulation of superconducting artificial atoms has been investigated [56,57], which indicated that the shortcut approach is feasible for performing accelerated quantum operations and is more robust against decoherence effects.
In this paper, we develop an effective scheme to speed up adiabatic population transfer in a Josephson qutrit by the TQD method. At a magic bias point, an isolated three-level subsystem of a Josephson quantum circuit constitutes our qutrit. Allowed by the level-transition rule, a Λ-configuration interaction between the qutrit and microwave drivings can be used for adiabatic population transfer by means of STIRAP. We then adopt the TQD technique to speed up the adiabatic QPT. Assisted by counter-diabatic driving, accelerated population transfer is performed in the qutrit with a Δ-configuration interaction. With accessible parameters, we further analyze the flexible reversibility and high robustness of the coherent operation. Through the combination of these advantages, the present scheme could provide an attractive approach to implementing optimal quantum operations on Josephson artificial atoms. This paper is organized as follows. In section 2, we present an artificial qutrit of a Josephson quantum circuit. Section 3 demonstrates the adiabatic population transfer in the qutrit via STIRAP. In section 4, we focus on the accelerated population transfer assisted by a counter-diabatic driving, and then analyze the reversibility and robustness of the population transfer. Finally, a discussion and a conclusion are presented in section 5.

An artificial atom of a Josephson quantum circuit
As schematically depicted in figure 1, the Cooper-pair box (CPB) circuit under consideration contains a superconducting box with n extra Cooper pairs [1], in which the charging energy scale of the system is E c . Through two symmetric Josephson junctions (with identical coupling energies E J and capacitances C J ), the CPB is connected to a segment of a superconducting ring. In the charge-phase regime [58], the characteristic system parameter E J has the same order of magnitude as E c , which satisfy , here the large energy gap Δ prohibits quasiparticle tunneling, and k T B stands for a low energy of thermal excitation. The CPB is biased by a static voltage bias V d through a gate capacitance C d . Meanwhile, a static magnetic flux d F in the ring adjusts the effective Josephson coupling E Jd . Ac gate voltages V s a , are applied to the box through gate capacitances C s a , , and a time-dependent flux p F threads the ring. The microwave drivings are used to induce the desired level transitions [5,59], as mentioned below.
In the absence of the microwave drivings, V s a , and p F , the static Hamiltonian of the CPB system is given by where the first term is the charging energy, and the second one represents the Josephson coupling. In light of equation (1), we can obtain the eigenlevels and eigenstates of the static charge-phase system. With the Josephson coupling E E 1.5 Jd c = , the first four levels E k (with k = g, e, a and f ) versus n d are plotted in figure 2. At a magic point of n 0.5 d = , we deal with four eigenstates kñ | , in which each state can be expressed as a superposition of Cooper-pair number states, namely, k c n n kn ñ = å ñ | | , with c kn being the superposition coefficients. The quantum states at the magic point are insensitive to the first-order dephasing effect, which thus contributes to prolonging the decoherence time of the system [58]. Since the fourth level E f is separated from the three lower levels, we select an effective three-level subspace g e a , , ñ ñ ñ {| | | }to constitute the qutrit under consideration. It is found that level anharmonicity in the qutrit is enough, and thus energy spacings ) are far away from each other. Sufficient anharmonicity can eliminate the leakage errors induced by the coherent drivings, which is beneficial for performing robust population transfer with the qutrit [27,59].

Adiabatic population transfer with the qutrit via stimulated Raman adiabatic passage
Here, E g has been taken as the zero-energy reference.
Assume that the system is in gñ | initially. As plotted in figure 4( , which meet the adiabatic conditions well. Here two partially overlapping pulses are applied in a counterintuitive order, first V s and then p F , the relevant Rabi frequencies s W and p W traverse along a closed loop in the time domain. Therefore, the coherent quantum operation is referred to as the regular STIRAP [16].

Speeding up adiabatic population transfer by a counter-diabatic driving
Generally, based on the STIRAP method, the adiabatic operation for transferring population needs a long time. To speed up the adiabatic population transfer from gñ | to eñ | , we now take the technique of TQD in the present qutrit. Apart from the drivings V s and p F , an auxiliary microwave field V V t e cos a a a i w = b ( ), serving as a counterdiabatic driving, is also applied to the CPB through a gate capacitance C a , as schematically shown in figure 1, where e ib represents an initial phase factor [63] and a w is its microwave frequency. Since the frequency a w is resonantly matched with eg w , the driving V ã causes only coupling between gñ | and eñ | . As a result, there exists a Δ-type interaction between the qutrit and these three drivings, as depicted in figure 5, which is allowed by the level-transition rule.
By applying the driving V ã , a supplementary interaction Hamiltonian H cd in the space g a e , , ñ ñ ñ {| | | }is given by [30,62] In this case, we reproduce the Gaussian-shaped Rabi frequencies as ns. According to equation (7), the dependency of a W on time is explicitly sketched in figure 6(a). With the time-shortened pulses, the population transfer from the initial state gñ | to the target state eñ | can be almost completed within a short time t=25 ns, see figure 6(b). Compared with the adiabatic transfer in figure 4(b), the operation process has been sped up sharply. Numerically, it is found that the accelerated transfer from gñ | to eñ | takes a time t=25 ns to reach an inversion probability P 99.81% in = . However, to achieve the same inversion probability, the adiabatic operation needs a time t = 104.58 ns, much longer than that used in the counter-diabatic protocol. Additionally, there is no Figure 5. Apart from the microwave drivings p F and V s , a counter-diabatic driving V ã with frequency a w is applied to the qutrit to couple gñ | and eñ | . Figure 6. Based on the TQD method, a counter-diabatic driving a W versus time in (a), an accelerated population inversion from gñ | to eñ | in (b). A counter-diabatic driving a W given in (c) for the inverted transfer from eñ | to gñ | in (d).
population of añ | during the accelerated process, which is beneficial to remove the decoherence effects caused by añ | .
As a practical issue related to coherent control, we are concerned with the reversibility of the above population transfer by adjusting the Rabi frequencies s W and p W . After exchanging s t and p t with each other, p W precedes s W in time domain to reach its maximum amplitude. In this way, the Rabi frequencies naturally change GHz and e 2 0.16 for simplicity. By numerically calculating equation (9) with the accessible parameters 2 0.05MHz g p = and 2 0.3MHz g p = j [64], we get a timedependent ρ of the system with an initial condition of 1 gg r = . For the case of the accelerated process, the matrix elements kk r (k = g, e and a) of interest, representing the occupation probabilities of states kñ | , are displayed in figure 7(a). Apparently, even in the presence of decoherence effects, a robust operation with an inversion probability P 96.18% in = can be performed within a duration time t=25 ns by the shortcut approach. Comparatively, we also consider the decoherence effects on the STIRAP-like population transfer, as shown in figure 7(b). Obviously, the decoherence effects on the adiabatic process become much larger as a result of a longer operation time. The occupation probability of eñ | is 82.57% at the final time t=150 ns, and the maximum probability of eñ | during the transfer process is 87.23% at most. Therefore, based on the shortcut method of TQD, the target population transfer can be sped up dramatically, meanwhile, the decoherence effects have been reduced significantly due to a shorter evolution time.

Discussion and conclusion
To verify the experimental feasibility, we now examine the utilized microwave drivings based on the available parameters. It is found that the microwave amplitudes are acceptable within current technology. For example, if the maximum amplitudes of the Gaussian-shaped microwave pulses take max(V s (t))=3.2 μV and max(V a (t))=4. 48 in our scenario, which is very close to that mentioned in [66]. The proposed scheme may have the following characteristics and advantages. (i) The process of STIRAP-like population transfer is typically robust against the timing errors and the fluctuations of the control parameters; however, the operation generally takes a long time. On the other hand, the direct Rabi oscillation between gñ | and eñ | via a resonant driving can be completed quickly, whereas, it is highly sensitive to the operation time and the parameter fluctuations. By combining the advantages of these two methods, the present population transfer is sharply accelerated and still insensitive to timing errors and parameter variations, which thus could have a variety of applications to quantum coherent control and information processing. (ii) Thanks to the prohibition of the parity-symmetry determined transition rule [60], the driving p F can not cause the magnetic coupling between gñ | and f ñ | due to a vanished overlap O 0 gf = . Similarly, the V s -induced electrical dipole interaction between eñ | and f ñ | is also forbidden as a result of O 0 ef = . Although E a and E f are relatively close to each other, the effective qutrit g e a , , ñ ñ ñ {| | | }is well isolated from the effect caused by f ñ | . In this case, adopting RWA to deal with the interactions H I and H cd becomes more viable within the qutrit. (iii) Compared to the transmon-regime quantum circuit [67], the present charge-phase CPB has the sufficient level anharmonicity, and thus the leakage effects induced by the utilized microwave drivings can be neglected safely. The suitable level structure is very helpful to implement the robust quantum manipulation. (iv) Different from the previous works [21,35,60], the present three-level system is selected at the magic point of n 0.5 d = , which contributes to remove the dephasing effect and then to prolong the system decoherence time greatly. (v) During the accelerated and reversible transfer process, the intermediate state añ | is almost not populated within the operation time, which makes the robust inversion insensitive to spontaneous emission caused by añ | . In summary, we propose a promising scheme to speed up the adiabatic population transfer in a Josephson qutrit by using the TQD technique. At the magic working point, the first three levels constitute our qutrit. Allowed by the level-transition rule, a Λ-type interaction is induced by the microwave drivings, from which we implement a STIRAP-like adiabatic population transfer by designing the Rabi couplings. Based on the shortcut approach of TQD, we further apply a counter-diabatic driving to the qutrit for speeding up the adiabatic transfer. In the qutrit with a Δ-type interaction, the target population transfer can be accelerated significantly with the accessible parameters. Moreover, we analyze in detail the reversibility and robustness of the present strategy. Due to the reversible and robust operation in an accelerated manner, the protocol could have many potential applications for experimentally investigating optimal population transfers with the Josephson artificial atoms.