Entangling two high-Q microwave resonators assisted by a resonator terminated with SQUIDs

We propose a superconducting circuit for quantum information processing (QIP) on high-quality (high-Q) superconducting resonators (SRs). In the circuit, two high-Q SRs are coupled to a high-frequency SR (acts as a quantum bus) assisted by superconducting quantum interference devices (SQUIDs) terminate in both ends of the high-frequency resonator. Each coupling strength between each high-Q resonator and the high-frequency resonator can be tuned independently from zero to the strong-coupling regime via the external flux threading through the SQUID. In the circuit, the frequencies of the two high-Q resonators are far detuned from the high-frequency resonator. That is, quantum information stored in high-Q resonators cannot be populated in the high-frequency resonator, which lets the bus can be designed to link lots of high-Q resonators for the large-scale QIP. To show the circuit can be used to achieve the QIP, we present a high-fidelity scheme to generate Bell state on the two high-Q resonators. The scheme shows that, to achieve the entanglement operation on high-Q resonators, fast tuning on the coupling is no longer mandatory and the coupling strengths are not required to be turned on or off simultaneously.

Superconducting resonator (SR) has been studied for completing basic tasks in QIP because of its highquality factor and large zero-point electric fields, such as implementing quantum algorithms [29,30], coupling superconducting qubits [31][32][33], entangling remote qubits [34][35][36][37][38], and detecting the motion of a mechanical oscillator near the ground state [39]. In these studies, the SR is fixed with static boundaries, which makes the dynamic of the resonator behaves like a harmonic oscillator. Besides, embedded with Josephson junctions or superconducting quantum interference devices (SQUIDs) to add nonlinearity to the dynamics, the resonator can also exhibit the other characters, such as kerr effect [40] and tunable eigenfrequencies [41][42][43] which can be used as the measurement and the control of the superconducting qubit [44][45][46][47], parameter amplifiers [48][49][50], and parameter converters [51][52][53]. SQUIDs located at the ends of the resonator can also result in tunable boundary conditions controlled by external flux which threads the SQUIDs. This property has been used to create the phenomena like Hawking radiation [54], dynamical Casimir effect [55][56][57][58][59], and twin paradox [60]. Further more, resonators can be nonlinearly coupled to each other by modulating one resonator's boundary with flux created by other resonators to achieve optomechanical or quadratic optomechanical-like coupling between resonators [61,62].
QIP based on SRs has been studied a lot recently both in theory and in experiment, such as the generation of quantum entanglement [63][64][65][66][67], the construction of universal quantum logical gates [25,[68][69][70], and the nondemolition detection of single microwave photon in a SR [71]. To realize large-scale QIP, lots of resonators should be coupled together, which leads to the crosstalk among them. To overcome the crosstalk, one method is to turn the frequencies of resonators to detune with each other largely. Another one is to turn off the coupling between two resonators. There have been many studies attending to achieve the two methods assisted by the flux qubit [72][73][74][75] or the SQUID [76][77][78][79][80].
After embedding with a qubit or a SQUID in a resonator, its quality factor will be reduced largely. This kind of resonators is hard to be treated as a good carrier of quantum information for QIP. Here, we focus on the tunability of the coupling among SRs and propose a three-resonator superconducting circuit to achieve QIP on two high-quality factor resonators. The circuit is composed of a resonator (acts as a quantum bus) inserted with SQUIDs and two resonators with high-quality (high-Q) factors act as carriers of quantum information. These resonators are specially arranged and nonlinearly coupled by using the interaction between the SQUID and the flux produced by the high-Q resonator. The coupling strength between each high-Q resonator and the bus can be tuned from non-coupling to the strong coupling regime independently. To show the circuit is suitable for QIP, we give a simple scheme to generate a high-fidelity Bell state on the two high-Q SRs. The scheme is achieved in the dispersive regime when the frequencies of the high-Q resonators are detuned from the quantum bus, which allows the information not to be populated in the low-quality bus. Furthermore, the tunable coupling strength between the quantum bus and high-Q resonators lets our circuit suitable for the large-scale QIP as the crosstalk can be overcome robustly.
This paper is organized as follows: in section 2, we introduce the configuration of the circuit and change it into an alternative lumped model which can be described in mathematical language. This circuit can be expressed in several Lagrangian formalisms and described by a two-mechanical-oscillator cavity optomechanical-like Hamiltonian after the canonical quantization. The coupling strengths between resonators are tunable and evaluated detailedly. In section 3, we construct a Bell state on the two high-Q resonators with fidelity about 99.2% [81] by considering the dissipation of resonators. Finally, we summarize our result in section 4.

Circuit model
Here, we analyze the circuit composed of a high-frequency resonator (C) coupled to two high-Q resonators (A and B) assisted by two SQUIDs as shown in figure 1(a) with quantum mechanics in detail and derive an effective Hamiltonian of the circuit with large range tunable coupling strength between resonator A (B) and C. First, we decompose the circuit into several lumped-circuit elements in order to write the Lagrangian and boundary conditions of the circuit. Second, with separation of variable method, we derive the mode frequencies of resonators analytically. Finally, we derive the Hamiltonian of the system with canonical quantization procedure.

Lagrangian and boundary conditions
Considering a SR with high quality factor, the capacitive coupling of the resonator to external transmission lines can be neglected i.e. the capacitances located at the two ends of the resonator tend to zero (C st → 0), which means resonators A and B can be considered as open ended resonators. The flux field Φ α (x, t ) at position x (a coordinate of coordinate axis X as shown in figure 1(a)) of the resonator is related to the time-integral of the voltage as α=A and B). The superconducting phase of the macroscopic wave function which describes the superconductors is x t , 2 x t , Assuming that resonators A and B are uniform and the characteristic capacitances and inductances per unit length at any position are constants, i.e. c A (x)=c B (x)=c 0 and l A (x)=l B (x)=l 0 , the Lagrangian density of the resonators can be expressed as [82,83] In the limit C 0 st  , resonators A and B are open ended at both x=0 and x=d (see figure 1(b)), which leads to the boundary conditions of the two resonators as t 0, ) . d is the length of the resonators.
We now consider the resonator C which is placed in the middle of the circuit and terminated with two SQUIDs as shown in figure 1(a). The flux threading SQUIDs generated by high-Q resonators refresh the boundary conditions of the resonator C. We make the assumption that both SQUIDs are identical and symmetric, i.e. In this setup, resonator A and resonator B are coupled to the high frequency resonator C via two SQUIDs. Capacitances C st , located at the ends of resonator A and resonator B, couple the circuit to external transmission lines. The capacitance drawn in the side of the resonator C (not drawn in figure 1(b) for convenience) acts as a port to receive the drive field. (b) The setup is expressed as a lumped-element circuit. Each resonator can be modeled as a chain of LC oscillators with node capacitance c n a , node inductance l n a , and the magnetic node fluxes n F a (α=A, B and C). In the continuum limit, Δs → 0 (s = x or y), the node capacitance c n a , node inductance l n a and node fluxes n F a converge to continuous representation c α (s), l α (s), and Φ α (s) respectively. Each SQUID (the element in purple) consists of two junctions whose capacitance and Josephson energy are C Ji b and E Ji b (β=l, r and i=1, 2), respectively. The flux across each junction is Ji Here, C  , S l  , and S r  are the Lagrangian densities of resonator C, left SQUID, and right SQUID, respectively. The flux field generated by the resonator C at the position y (a coordinate of coordinate axis Y as shown in figure 1 The Lagrangian of the SQUID is written in the form of an effective Josephson junction with Josephson [61,62]. This energy induces an effective inductor: With the method of Euler-Lagrange equation of motion, two effective boundary conditions introduced by the SQUIDs of the resonator C can be expressed as [84]: Here SQUIDs are assumed in the phase regime, i.e. the charge energy is small compared to the Josephson energy and the quantum fluctuations of the phase across the SQUIDs are small enough (2 ,2 1 ). Here we have expanded the cosine function in equation (2) and omitted higher order items.

Modes
From equation (1), equations of motion for flux field obey the one-dimensional massless Klein-Gordon wave equation , 0 can be solved with the separation of variables method by assuming Φ α (x, t)=u α (x)ψ α (t). Then, the wave equation yields two independent ordinary differential equations is the vector number (frequency) of photons. The general solution for u α (x) is a linear combination of sine and cosine functions as where 1 x and ξ 2 are constants determined by boundary conditions. By taking equation ( Here n and m label the discrete modes of resonator α. The normal frequencies of resonator C can be obtained by combining equations (3) and (5), and can be expressed as is the plasma frequency of the left (right) SQUID. We assume that the plasma frequency is larger than the normal mode frequency J l r c n , w w > Equation (14) can be rewritten as Then, the mode frequency ω c,n (equation (17) where d eff is the effective length of resonator C with respect to the static flux and has the form as [61,62]  Here the annihilation operator a n (b n ) destroys a microwave photon with frequency ω A,n (ω B,n ) in resonator A (resonator B) and the creation operator a n † (b n † ) creates one in resonator A (resonator B). These operators satisfy the commutation relations a a , . We assume the external flux r l ext dF ( ) generated by resonator A and resonator B have the form  is determined by the eigenfrequency ω c , the additional length δd, and the geometry factor G 1 . ω c and the length of resonator C have an inverse relationship and can be enhanced by tuning the external flux away from the 0.5Φ 0 [41][42][43] while other parameters are fixed. This property has a positive impact on our scheme for achieving the entanglement on high-Q SRs in section 3. Here ω c is treated as a fixed frequency to analyze the coupling strength g 1 for convenience [61]. The additional length δd is the function of the external flux and can be adjusted continuously. The geometry factor G m depends on the arrangement of the circuit. Here, we only take the geometry factor between resonator B and resonator C as an example because the geometry factor between resonator B and resonator C is the same as the one between resonator A and resonator C. An analytical expression of G m can be obtained by assuming that the magnetic field B(x, r) generated by the resonator B has the form [61,62] where r is the radial distance from the resonator B. μ 0 is the permeability of free space. I B (x) is the current in resonator B at position x. We evaluate the current by using the derivative of the flux field Φ B (x, t) at position x 0 as By comparing the integral magnetic flux through the SQUID area in equation (30), the geometry factor of the fundamental modes of resonators B and C can be expressed as = is the characteristic impedance of the resonator. r 2 and r 1 are the distance from the axle wire of resonator B to the left and right boundary of the right SQUID as shown in figure 1(a). We have set the midpoint of resonator B as x 0 and r 2 /r 1 =2 for convenience.
The coupling strength [61] between the fundamental modes of resonator B and C can be expressed as  figure 2) and the frequency of resonator C increases which enhances the frequency detuning between the quantum bus and high-Q resonators.

Entanglement generation
To generate the entanglement on resonator A and resonator B, we consider the whole system (shown in figure 1) is in the dispersive regime where the eigenfrequency of resonator C is at least 10 times of the eigenfrequencies of the high-Q resonators y ñ = ñ ñ ñ | ( ) | | | , the evolution of the system is given by is the Fock state of resonator α ( A a = , B, C). The excitation of the high-Q resonator can be achieved by coupling a superconducting qubit (not drawn in figure 1 for convenience) to the bus in the dispersive regime and letting the frequency of the qubit equal to the one of the high-Q resonator to exchange the energy from the qubit to the resonator. After an operation time of t 4 c = p , the state of the system will evolve from To show the feasibility of our scheme, we numerically simulate the fidelity of the state and the population of a microwave in resonator α with the Lindblad-Kossakowki master equation Here κ α is the dissipation rate of resonator α. ρ is the density operator of the system. The population of a microwave photon in resonator α and the fidelity of the state are defined as y ñ = ñ ñ ñ | | | | , and t r ( ) is the real density operator of the evolution of the system with the initial state 1 0 0 , β=0.12ω 0 , and ω d =30.8ω 0 . The Rabi-oscillation appears between resonator A and resonator B as expected by equation (42). The maximum value of the photon populated in resonator C is about 0.1, which indicates that the resonator C is only virtually excited. Figure 3 . In our scheme, coupling strength g 1 should equal to g 2 which is described by the black solid line in figure 3(b). Here, the system evolves to the Bell state at 46.17 ns with the fidelity of 99.26%. In experiments, parameters can not meet the requirement of the scheme perfectly. So, in figure 3(b), we give the fidelity of the state varies with δ g =(g 1 −g 2 )/g 2 =±0.1 (g 1 is fixed here) described by the blue dotted line and the red dashed-dotted line with maximal fidelities of 98.62% and 98.65%, respectively. That is, our scheme can work well when the detuning between g 1 and g 2 reaches 0.1g 1 .  high value of 2×10 6 [86]. Figure 3(d) shows that the dissipation rate of the resonator C affect the fidelity of our scheme a little, which means the resonator C can be designed more complex to link lots of high-Q resonator for large-scale QIP. To reduce the influence from the dissipation rate of resonator C further, one can enlarge the frequency detuning between resonators A (B) and C and integrating high-Q resonators with higher quality factor. Figure 3(e) shows that, to keep the fidelity of the entanglement larger than 99%, the frequency detuning between the two high-Q resonators should smaller than 0.5 MHz.

Conclusion
In conclusion, we have proposed a three-resonator circuit to achieve a high fidelity entanglement on two high-Q resonators for QIP. In the circuit, each high-Q resonator is coupled to the high-frequency resonator (acts as a quantum bus) with tunable coupling assisted by a SQUID terminates the bus. The flux-sensitive boundary condition constructed by the SQUID is the key element to couple different resonators. Special arrangement of the circuit leads to the optomechanical like Hamiltonian, in which the coupling strength between each high-Q resonator and the bus can be tuned independently by applying a external-controlled flux through the SQUID. This property helps us to overcome the crosstalk among high-Q resonators in a quantum processor composed of lots of high-Q resonators coupled to a quantum bus. For the achievement of the entanglement, the fast operation on the tuning of the coupling is unnecessary as one can turn on the coupling first and then apply the drive field to achieve the entanglement on high-Q resonators. Moreover, coupling strengths need not to be turned on or off simultaneously.
To show the feasibility of our circuit for the QIP, we propose a high-fidelity scheme to construct Bell state on high-Q resonators. Resonator embedded with the SQUID suffers a decrease on its quality factor and may not suitable for acting a quantum information carrier. So, resonator C embedded with two SQUIDs is chosen as a quantum bus which should be virtually excited as the scheme works in the dispersive regime where the frequency of the resonator C is far detuned from the ones of resonators A and B. The fidelity of the scheme achieves a high fidelity of 99.2% by considering the possible dissipation rates of three resonators.