Simulating Anisotropic quantum Rabi model via frequency modulation

Anisotropic quantum Rabi model is a generalization of quantum Rabi model, which allows its rotating and counter-rotating terms to have two different coupling constants. It provides us with a fundamental model to understand various physical features concerning quantum optics, solid-state physics, and mesoscopic physics. In this paper, we propose an experimental feasible scheme to implement anisotropic quantum Rabi model in a circuit quantum electrodynamics system via periodic frequency modulation. An effective Hamiltonian describing the tunable anisotropic quantum Rabi model can be derived from a qubit-resonator coupling system modulated by two periodic driving fields. All effective parameters of the simulated system can be adjusted by tuning the initial phases, the frequencies and the amplitudes of the driving fields. We show that the periodic driving is able to drive a coupled system in dispersive regime to ultrastrong coupling regime, and even deep-strong coupling regime. The derived effective Hamiltonian allows us to obtain pure rotating term and counter-rotating term. Numerical simulation shows that such effective Hamiltonian is valid in ultrastrong coupling regime, and stronger coupling regime. Moreover, our scheme can be generalized to the multi-qubit case. We also give some applications of the simulated system to the Schrödinger cat states and quantum gate generalization. The presented proposal will pave a way to further study the stronger anisotropic Rabi model whose coupling strength is far away from ultrastrong coupling and deep-strong coupling regimes in quantum optics.


The Derivation of the Effective Hamiltonian
In this section, we consider a qubit coupled to a harmonic oscillator in dispersive regime, and the qubit is modulated by the periodic driving fields. Such setup can be realized in a variety of different physical contexts, such as trapped ions [6][7][8][9] , circuit QED [10][11][12] , cavity QED 13,14 , and so on. Here, we adopt a circuit QED setup to illustrate our proposal (the architecture is depicted in Fig. 1(a)). We consider a tunable transmon qubit, which is comprised of split junctions, is capacitively coupled to a LC resonator. Such split structure allows the qubit to be modulated by the magnetic flux through the pair junctions. The system is described by a time-dependent Hamiltonian as follows (we set ħ = 1)ˆˆˆ= where ε is the transition frequency of the tranmon qubit. σ α is the α-component of the Pauli matrices. ω is the frequency of the LC resonator. â a ( ) † is the annihilation (creation) operator. g is the coupling constant between the qubit and the bosonic field, and Ĥ t ( ) d describes n d periodic driving fields with frequencies Ω j and normalized amplitudes η i . In this work, we consider n d = 2 and the qubit coupled to the resonator in dispersive regime (i.e., g | | |Δ | ±  with Δ ± = ω ± ε). Without periodic driving, the RT and CRT terms can be ignored in dispersive regime. This is because all terms are fast oscillating terms in the rotating framework. If we choose proper www.nature.com/scientificreports www.nature.com/scientificreports/ modulation frequencies and amplitudes such that the near resonant physical transitions are remained and far off resonant transitions can be discarded. Moving to the rotating frame defined by the following unitary operator . Using the following Jacobi-Anger expansion 86,87 with J n (x) being the n-th order Bessel function of the first kind, we obtain n n n n i n n i n n t n n n n i n n i n n t Here, Ω ± (n 1 , n 2 ) = Δ ± + n 1 Ω 1 + n 2 Ω 2 . According to the RWA, only slowly varying terms appearing in α(t) and β(t) will dominate the dynamics. We should choose the suitable driving frequencies to obtain the rotating and counter-rotating interaction terms. We assume there is a small detuning δ 1 (δ 2 ) between Ω 1 (Ω 2 ) and the red (blue) sideband, and the definition of the detunings read The energy levels of the modulated system are shown in Fig. 1 ), one can check that the RT and the CRT will contribute to the dynamics only for lowest oscillating frequencies Ω − (−1, 0) = −δ 1 and Ω + (0, −1) = δ 2 , respectively. When the oscillating frequencies are much larger than the effect couplings, i.e., with (m 1 , m 2 ) ≠ (0, −1) and q q gJ J ( , ) ( 2 ) (2 ) with (q 1 , q 2 ) ≠ (−1, 0), one may safely neglect these fast oscillating terms in Eq. (6). Then the dominant terms in Eq.
where we have used the relation J −n (x) = (−1) n J n (x) for integer n. Then these approximations lead to the following near resonant time-dependent Hamiltonianˆˆˆ † where the effective coupling strengths of RT and CRT are z r c r i i eff where we have set ϕ 1 = 0 and ϕ 2 = θ. The anisotropic parameter λ is the ratio of RT and CRT coupling strengths (i.e., λ =   g g / cr r ). Thus we obtain a controllable AQRM. Below we analyze the parameters in our scheme. In our circuit QED setup, we consider the following realistic parameters 88,89 : the transition frequency of the transmon qubit is ε = 2π × 5.4 GHZ with the decay rate κ = 2π × 0.05 MHz, the resonator frequency is ω = 2π × 2.2 GHz with the loss rate γ = 2π × 0.012 MHz, and the coupling strength of the resonator and qubit is g = 2π × 70 MHz. We can check that the dispersive condition (i.e.,  g | | |Δ | ± ) is fulfilled. The frequency modulation can be implemented by applying proper biasing magnetic fluxes. The modulation parameters Ω i , η i and ϕ i can be chosen on demand by tuning the modulation fields. In circuit QED setups, the modulation frequency and modulation amplitude range from hundreds of megahertz to several gigahertz. It is reasonable to set the modulation amplitude η i Ω i ranges from 0 to 2π × 10 GHz 16 . The detunings δ i can be tuned from 0 to hundreds of megahertz to fulfill the condition

the simulation of QRM and AQRM in UsC and DsC Regimes
To assess the robustness of our proposal in circuit QED system, we should consider the dissipation effects in the following discussions 88 . Considering the zero-temperature Markovian environments and large driven frequencies Ω j , the master equation governing the evolution of the system can be derived as follows 52 are the standard Lindblad super-operators describing the losses of the system. To obtain the master equation in the framework of effective Hamiltonian, we set U(t) = U 2 (t)U 1 (t). Let ρ  t ( ) be the density matrix in the same framework with effective to the master Eq. (13), we obtain the following master equation t is the total system Hamiltonian in the new rotating framework. We show that the Hamiltonian Ĥ eff in Eq. (12) is the approximation of Here we consider the initial phase difference of the driving fields is θ = 0. The parameters η 1,2 and the detuning of first sideband δ 1,2 are tunable parameters. Such tunable parameters determine the parameters in the simulated system in Eq. (12). To verify the validity of the effective Hamiltonian in Eq. (12), we should study the fidelity of the evolution state. Let ψ ∼ (0) be an initial state in the new framework and the corresponding initial density matrix is (12), we obtain the evolution density matrix  ρ t ( ). The ideal case can be obtained by solving the Schördinger equation governed by the effective Hamiltonian (12). We denote the ideal evolution state governed by the effective Hamiltonian (12) with ψ ∼ t ( ) . Then the fidelity of the evolution state reads the simulation of QRM. In this subsection, we will show the performance of the simulated QRM. To obtain equal effective RT and CRT coupling strengths (i.e., λ = 1), we need to adjust the normalized amplitude η i . A simple case is η 1 = η 2 = η. Then the simulated coupling strength g g r c r =   and we denote the simulated coupling strength with . Assuming θ = 0, we can obtain the following tunable QRM The effective frequencies of resonator and qubit are determined by the detunings δ i . One can tuning the ratios of modulation amplitudes and frequencies to obtain different relative coupling strength.
For simplify, we choose the modulation parameters are as follows: Ω 1 = 2π × 3.2 GHz, η 1 Ω 1 = 2π × 2.296 GHz, Ω 2 = 2π × 7.6 GHz, ϕ 1 = ϕ 2 = 0, and the blue sideband modulation amplitude ranges from 0 to 2π × 9.138 GHz. The other parameters are given in Table 1. Then we can obtain δ 1 = δ 2 = 0, η 1 = 0.7173 and θ = 0. The normalized amplitude of blue sideband ranges from 0 to 1.2024. In this case, only interaction terms remain and the effective Hamiltonian reduces to the following degenerate AQRM We check that the simulated Hamiltonian varies from JC model to anti-JC model by tuning the normalized amplitude η 2 . Let g (0) 0 r ψ = ⊗ ∼ be the initial state. We can obtain the dynamics of the evolution states governed by the master equation in Eq. (13). The excitations of qubit and resonator as a function of evolution time and η 2 are shown in Fig. 5. The Fig. 5(a) shows the excitation of qubit σ σ 〈 〉 + −ˆ as a function of evolution time and η 2 . When  η 1 2 , we can check that g cr  approaches to zero and rotating term dominates the dynamics. The qubit and resonator are not excited in the evolution process. If we increase the normalized amplitude η 2 , the effects of the CRT emerge. In this regime, the qubit and resonator are excited in the evolution process. When η 2 = 0.7173 (red dashed line), the ration of the RT and CRT approaches to 1. In this regime, the RT and CRT dominate the dynamics of the evolution. The Fig. 5(b) shows the excitation number of resonator a â † 〈 〉. When η 2 = 0.7173 (red dashed line), the excitation number reaches its maximum value in the evolution process, which originates from the competition of RT and CRT. When η 2 reaches 1.2024, we can check that when  g r approaches to zero, the CRT dominates the evolution. The higher excitation number of the resonator can be excited. The dynamics of the qubit and resonator show the periodic oscillation behavior. The results show that we can drive the system from JC regime to anti-JC regime through quantum Rabi regime (indicated by red dashed line).

Some Applications on the Quantum Information Theory
Our scheme could be utilized as a candidate platform to implement the quantum information and computation device. As an example, we show the generations of Schrödinger cat states and quantum gate. For this purpose, we first generalize our scheme to the multi-qubit case 12 . Considering N qubits coupled to a resonator, we can obtain the simulated anisotropic quantum Dicke model with the same treatment. We assume all the qubits possess the same energy split (i.e., ε i = ε) and the periodic driving fields described in Eq. (2c) act on all the qubits. By means of the same approach, we can obtain the simulated anisotropic quantum Dicke model. The simulated anisotropic quantum Dicke model in the interaction picture reads The parameters are taken as follows: Ω 1 = 2π × 3.2 GHz, η 1 Ω 1 = 2π × 2.296 GHz, Ω 2 = 2π × 7.6 GHz, ϕ 1 = ϕ 2 = 0, and the blue sideband modulation amplitude ranges from 0 to 2π × 9.138 GHz (i.e., η 2 ranges from 0 to 1.2024). The other parameters are given in Table 1. The initial state is chosen as . The red dashed line is plotted for η 2 = 0.7173. The evolution states are governed by the master Eq. (14).
www.nature.com/scientificreports www.nature.com/scientificreports/ˆˆˆˆ= . If we set the detunings of the blue and red sidebands δ 1 = δ 2 = δ, we obtain the degenerate two-level system (i.e.,  ε = 0), and the effective frequency of resonator is  ω δ = . We also can adjust the normalized amplitudes of the driving fields to make = =    g g g r c r . For simplicity, we set amplitudes η 1 = η 2 and initial driving phases ϕ i = 0. In this case, the simulated Hamiltonian in the interaction picture reduces to the following formˆˆˆ † = + .
The evolution operator for the Hamiltonian in Eq. (20), which could be obtained by means of the Magnus expansion, reads 91 Based on the dynamics of this effective Hamiltonian, the Schrödinger cat states and quantum gate can be generated. the generation of schrödinger cat states. Superposition of coherent states plays an important role in quantum theory 79,80,[92][93][94] . In this subsection, we consider how to generate superposition of coherent states for a single-qubit case. Assuming the initial state prepared on ψ = ⊗ ∼ g (0) 0 r , we obtain the evolution state as follows are the coherent states with amplitude ±ξ(t). In the basis e and g , the above state can be rewritten as following form 2 . Performing a projection measurement on the states e and g , we obtain the states  t ( ) + and  − t ( ) , which correspond to the even and odd Schrödinger cat states. The magnitude of the displacement for  t ( ) 2 ( / ) sin( /2) . When the evolution time π ω =  t / 0 , the magnitude of the displacement reaches its maximum value   ω | | g 2 / .
the implementation of quantum gate. In this subsection, we consider two-qubit case. Assuming the evolution time  T 2 / π ω = , we obtain ξ(T) = 0 and   φ π ω = T g ( ) 2 ( / ) 2 . The evolution operator is reduced to the following form . So the evolution operator can be viewed as a nontrival two-qubit quantum gate when θ π ≠ k 2 (k is integer). When π ϑ = /4 (i.e.,   ω = . g / 025), the quantum gate reads Such quantum gate is local equivalent to the control-not (CNOT) gate 97,98 . The equivalent relation reads

Discussion
In conclusion, we have proposed a method to simulate a tunable AQRM, which is achieved by driving the qubit(s) with two-tone periodic driving fields. We have analyzed the parameter conditions under which this proposal works well. By choosing proper modulation frequencies and amplitudes, the coupling constants of RT or CRT can be suppressed to zero, respectively. Consequently, we study the dynamics induced by CRT or RT correspondingly. In addition, we have also discussed the applications of our scheme to the generations of quantum gate and Schrödinger cat states. This proposal provides us with a reliable approach for studying the effects of RT and CRT in different regimes individually. Although we explore the scheme with the circuit QED system, which could be