Fast generation of W states of superconducting qubits with multiple Schrödinger dynamics

In this paper, we present a protocol to generate a W state of three superconducting qubits (SQs) by using multiple Schrödinger dynamics. The three SQs are respective embedded in three different coplanar waveguide resonators (CPWRs), which are coupled to a superconducting coupler (SCC) qubit at the center of the setups. With the multiple Schrödinger dynamics, we build a shortcuts to adiabaticity (STA), which greatly accelerates the evolution of the system. The Rabi frequencies of the laser pulses being designed can be expressed by the superpositions of Gaussian functions via the curves fitting, so that they can be realized easily in experiments. What is more, numerical simulation result shows that the protocol is robust against control parameters variations and decoherence mechanisms, such as the dissipations from the CPWRs and the energy relaxation. In addition, the influences of the dephasing are also resisted on account of the accelerating for the dynamics. Thus, the performance of the protocol is much better than that with the conventional adiabatic passage techniques when the dephasing is taken into account. We hope the protocol could be implemented easily in experiments with current technology.


The multiple Schrödinger dynamics
In this section, we would like to review the multiple Schrödinger dynamics [104][105][106] firstly. Assume that the original Hamiltonian of the system is H 0 (t). We perform a picture transformation as ψ ψ  . To forbid the transitions between {|n 1 (t)〉 } and diagonalize the Hamiltonian in the 2-nd interaction picture, H cd (1) can be calculated as . Repeating the processing as the 1-st and the 2-nd iterations, according to the Hamiltonian in the j-th ( ∈ … j N {1, 2, 3, , }) interaction picture (H j (t)), one can obtain the j-th modified Schrödinger Hamiltonian as

Fast generation of W states of superconducting qubits with multiple Schrödinger dynamics
In this section, we will show how to generate a W state of three SQs with multiple Schrödinger dynamics. Consider a system composed of a superconducting coupler (SCC) qubit and three CPWRs (CPWR 1 , CPWR 2 and CPWR 3 ). As shown in Fig. 1(a), the SCC qubit in the center of the devices is coupled to CPWR k through capacitor C k (k ∈ {1, 2, 3}). There is a SQ named SQ k in the CPWR k , which has an excited state |e〉 k and two ground states |g〉 k and |f〉 k . As shown in Fig. 1(b) the transition |e〉 k ↔ |f〉 k is driven by the laser pulse with Rabi frequency Ω k (t), and the transition |e〉 k ↔ |g〉 k is coupled to CPWR k with coupling constant λ k . As for the SCC qubit, it has an excited state |e〉 c and two ground states |g〉 c and |f〉 c , which has similar structure as the three SQs. The transition |e〉 c ↔ |f〉 c is driven by the laser pulse with Rabi frequency Ω c (t). Different from the three SQs, the transition |e〉 c ↔ |g〉 c may couple to three CPWRs with different coupling constants. We assume that the coupling constant for the transition |e〉 c ↔ |g〉 c coupled to CPWR k is ν k . Therefore, in the interaction picture, the Hamiltonian for the system can be written by For simplicity of calculations, we adopt λ k = λ and ν k = ν in the following. Assuming that the initial state of the system is Ψ = f g g g (0) 00 0 r  1  2  3  1  2  3 , where, |0〉 k and |1〉 k are the vacuum state and one-photon state of the cavity mode in k-th CPWR, respectively. The excited number operator of the system is defined by f g g g e g g g g g g g g g g g g g g g g e g g g g e g g g g e g f g g g g f g g g g f Moreover, the eigenstates of H c can be described as with corresponding eigenvalues ε 0 = 0, For simplicity, we set , ν, we can obtain the effective Hamiltonian of the system as  . Then, the system's effective Hamiltonian can written by Afterwards, the instantaneous eigenstates of H eff (t) can be solved as with corresponding eigenvalues ϵ 0 = 0, ϵ + = Ω and ϵ − = − Ω, respectively. Therefore, the picture transformation If we add H cd (0) to modify the Hamiltonian H eff (t) in Eq. (6), the structure of the system is also required to be adjusted. Therefore, we consider the 2-nd iteration picture to find another shortcut. Then, the Hamiltonian in the 1-st iteration picture can be solved in basis can be given by Submitting j = 2, Eqs (6) and (12) into Eq. (1), one can obtain the 2-nd modified Hamiltonian for . We find that H t ( ) eff (2) has the same form as H eff (t). Therefore, using H t ( ) eff (2) instead of H eff (t), we only need to adjust the Rabi frequencies Ω b (t) and Ω a (t). Now, let us design the frequencies Ω b (t) and Ω a (t) so that the system governed by H t ( ) eff (2) can be driven from its initial state ϕ Ψ = (0) 1 to the target state |W〉 . Firstly, when the system is governed by H t ( ) of H eff (t) at t = 0 and t = T. Therefore, we adopt the boundary condition θ θ = =   T (0) ( ) 0, and we set θ(0) = 0 and θ(T) = π/2. Then, we will have the following results . After the boundary conditions of θ and θ  are set, in the second step, let us design the Rabi frequencies of the laser pulses. To satisfy the boundary conditions of θ and θ  mentioned above, we firstly design the Ω b and Ω a via STIRAP. Ω b and Ω a can be expressed as where Ω 0 is the pulse amplitude, t 0 = 0.16T and t c = 0.25T are two related parameters.  (2) . However, the forms of Ω ∼ b and Ω ∼ a are too complex to be realized in experiments. For the sake of making the protocol more feasible in experiments, the Rabi frequencies of laser pulses should be expressed by some frequently used functions (e.g. Gaussian functions and sine function), or their superpositions. Fortunately, by using curves fitting, Ω ∼ b and Ω ∼ a can be replaced respectively with Ω b and Ω a , whose expressions can be written by ζ ζ ζ ζ = . = . = . = .
T T  T  T   T  T  T  T   T  T  T  T   7   when Ω 0 = 8/T. As a comparison, we plot Fig. 2(a) and and Ω t ( ) a versus t/T in Fig. 2(b). As shown in Fig. 2 ). In the next section, we will show that the laser pulses with Rabi frequencies Ω can drive the system from its initial state with a high fidelity, so, the replacements here for the Rabi frequencies of the laser pulses are effective.

Numerical Simulations and Discussions
In this section, various numerical simulations will be performed to demonstrate the effective of the present protocol. The fidelity of the target state |W〉 is defined as is the density operator of the system. Firstly, let us choose suitable coupling constants λ and ν. As we adopted α = π/4, the relation between λ and ν is λ ν = 3 . And the Rabi frequencies of laser pulses satisfy Ω = Ω We plot the final fidelity F(T) versus λ in Fig. 3. As shown in Fig. 3, F(T) is near 1 around λ = 10/T. Moreover, F(t) is close to 1 when λ > 20/T. One can easily find that even when the condition Ω t ( ) , ν is not satisfied well, the target state |W〉 can also be obtained. This can also easily be understood, as the evolution of the system, between the initial state ϕ 1 and the target state |W〉 , may move along different medium states, and it is not governed by the effective Hamiltonian H eff (t), which guides the system moving through the dark state φ 0 of H c as the only medium state. However, when the condition Ω t ( ) , ν is broken, the system may move through a medium state with higher energy. That will make the evolution of the system suffers more from dissipations. On the other hand, for a relative higher evolution speed, the value of λT should not be too large, as λ has a upper limit  in a real experiment. Therefore, to make the protocol with both high speed and robustness against dissipations, we choose λ = 35/T, slightly larger than Ω 0 ( λ Ω ≈ . / 03 0 ). Secondly, since we have adopted a suitable value of the coupling constant λ, we would like to examine the fidelity F(t) and the population ϕ ρ ϕ 1, 2, , 11) of state ϕ m during the evolution. The fidelity F(t) versus t/T is plotted in Fig. 4(a). And the the population P m of each state is shown in Fig. 4(b). As shown in Fig. 4(a), the fidelity F(t) keeps steady during time intervals [0, 0.3T] and [0.8T, T], and increases rapidly to approach 1 during time interval [0.3T, 0.8T]. As shown in Fig. 4(b), P 1 falls from 1 to 0 during evolution; P 9 , P 10 and P 11 are initial 0 and final 1/3 at time t = T as our expectation.
Thirdly, to show that the present protocol is faster than the adiabatic protocol, we plot the fidelity of the target state |W〉 with different methods versus t/T in Fig. 5. The Rabi frequencies of laser pulses for the STIPAP method can be set as where Ω a (t) and Ω b (t) are shown in Eq. (15). And it is easy to obtain that Ω′ = Ω Ω Ω Ω = Ω   STIRAP method, the present protocol to obtain a W state is much faster by using multiple Schrödinger dynamics. In addition, it is also been shown in ref. 31 that, to obtain a W state with the adiabatic passage with a fidelity larger that 0.99, the authors should chose λ > 100/T and Ω 0 = 0.35λ. That supports the discussion here as well.
Fourthly, since the dissipations caused by decoherence mechanisms are ineluctable in real experiments, it is worthwhile to discuss the fidelity F(t) when different kinds of decoherence factors are considered. In the present protocol, the decay of the cavity mode in each CPWR, the energy relaxation and the dephasing of every SQ play the major roles. The evolution of the system can be described by a master equation in Lindblad form as following   in which γ ks and γ φks (k = 1, 2, 3, s = f, g) are the energy relaxation rate and dephasing rate of the k-th SQ for decay path |e〉 k → |s〉 k , respectively. And γ cs and γ φcs (s = f, g) are the energy relaxation rate and dephasing rate of the SCC qubit for decay path |e〉 c → |s〉 c , respectively. κ k (k = 1, 2, 3) is the decay rate of the k-th cavity mode in CPWR k . We suppose γ ks = γ, γ γ = φ φ ks (k = 1, 2, 3, s = f, g) and κ κ = k (k = 1, 2, 3) for simplicity. We plot the final fidelity F(T) versus κ λ / and γ/λ in Fig. 6(a), versus κ λ / and γ λ φ / in Fig. 6(b) and versus γ/λ and γ/λ in Fig. 6(c). And we also examine some samples of the final fidelities F(T) with corresponding κ λ / , γ/λ and γ λ φ / and give them in Table 1. As shown in Fig. 6 and Table 1, we can obtain following results. (i) F(T) is insensitive to decays of the cavity modes in CPWRs. This is easy to be understood by seeing Fig. 4(b). Because the populations of ϕ 3 , ϕ 4 and ϕ 5 are all almost zero during the whole evolution, the influences from decays of the cavity modes in CPWRs will be greatly resisted. (ii) The fidelity suffers more influence from the energy relaxations of SQs comparing with the influences from decays of the cavity modes in CPWRs. However, F(T) is 0.9502 when γ/λ = 0.01, γ λ = φ / 0 and κ λ = / 0, i.e., the decreasing of F(T) caused by the increasing of γ is only about 0.05. Therefore, the present protocol is also robust against the energy relaxations of SQs. (iii) The dephasing plays a significant role here. When γ λ φ / increases from 0 to 0.001, F(T) decreases from 1 to 0.9824. However, in ref. 31, with the adiabatic passages, the fidelity of the target W state deceases from 1 to 0.85 when γ φ increases only from 0 to 0.0001. This shows that the present protocol is more robust against the dephasing comparing with the adiabatic passages. According to ref. 107, in experiments, parameters λ = 2π × 300 MHz, γ = 6π MHZ, κ π = 6 MHz and γ = φ 10 kHz can be realized. By submitting these parameters, we have F(T) = 0.9484.
Fifthly, since most of the parameters are hard to faultlessly achieve in experiments, it is necessary to investigate the variations of the parameters caused by the experimental imperfection. Here, we discuss the variation δT of the evolution time T, the variation δΩ 0 of the laser amplitude Ω 0 and the variation δλ of the coupling constant λ. We plot F(T′ ) versus δT/T and δλ/λ in Fig. 7(a), F(T′ ) versus δT/T and δΩ Ω / 0 0 in Fig. 7(b) and F(T) versus δλ/λ and δΩ Ω / 0 0 in Fig. 7(c), where T′ = T + δT is the real evolution time when the variation of the evolution time is taken into account. Seen from Fig. 7(a,c), the final fidelity is quite insensitive to the variation δλ. This results has also been announced in Fig. 3. Moreover, according to Fig. 7(a,b), the final fidelity F(T′ ) is very robust against the variation δT. The final fidelity almost unchanged when both δT/T, δλ/λ ≤ 10%. As shown in Fig. 7(b,c), variation δΩ 0 influences the final fidelity mainly. However, even when δΩ Ω = / 10% 0 0 , the final fidelity is still higher than 0.95. Therefore, we conclude that the present protocol for generating a W state of three SQs is robust against the variations δT, δΩ 0 and δλ.
Sixthly, in experiments, the protocol can be realized in charge qubits and CPWR coupling system. In other words, all the superconducting qubits including the SCC qubit can be chosen to be charge qubits. The structure of the a charge qubit is shown in Fig. 8. As shown in Fig. 8, the charge qubit contains a gate capacitance and two Josephson junctions with Josephson energy E J . The charge qubit can be manipulated by controlling the gate voltage V g and the magnetic flux Φ threading the loop. It was pointed out in previous protocols 108, 109 that, for a charge qubit with energy structure as Fig. 1(b), when an external applied magnetic flux Φ x of a pulse threads the ring, it can driven the transition between |e〉 ↔ |f〉 , and the Rabi frequency can be given by where, B g (r) is the magnetic components of the cavity mode 109,110 . For the SCC qubits placed in the center of the devices, it can couple capacitively to three different CPWR directly. This kind of directly coupling has been shown in many previous protocols both in theory 38,111 . For example, Yang et al. 38 have used these kind of coupling to generate entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting  qubit. Moreover, to improve the efficiency of the coupling between SCC qubit and each CPWR, one can chose SCC qubit to be a transmon 112 or a phase qubit 113 as well.

Conclusions
In conclusion, we have proposed a protocol to generate a W state of three SQs by using multiple Schrödinger dynamics to construct a shortcut to adiabaticity, so that the evolution of the system has been greatly accelerated. Interestingly, the form of the Hamiltonian being designed by the multiple Schrödinger dynamics was the same as that of the system's original Hamiltonian. Therefore, we only need to adjust the Rabi frequencies of laser pulses. In this protocol, the Rabi frequencies of the laser pulses can be expressed by the superpositions of Gaussian functions via the curves fitting. So, the laser pulses can be realized easily in experiments. One the other hand, numerical simulations results have demonstrated that the protocol is robust against different kinds of control parameters  variations and decoherence mechanisms. Notably, the present protocol is more robust against the dephasing, comparing with adiabatic passages. Therefore, we hope the protocol could be controlled and implemented easily in experiments based on a circuit quantum electrodynamics system.