Conditional displacement interaction in transversal

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Conditional displacement interaction in transversal direction from the quantum Rabi model Gangcheng Wang · Qingyong Wang · Yimin Wang · Jing-Ling Chen · Kang Xue · Chunfeng Wu

Received: date / Accepted: date

Abstract We investigate the realization of conditional displacement interaction in transversal direction from the quantum Rabi model by adjusting parameters of external magnetic fields. The special interaction is derived in the system of qubit(s) coupled to a resonator. We explore the implementation of quantum gates and the generation of superposed coherent states based on the transversal conditional displacement interaction, and consolidate the investigations numerically. We also show the special interaction can be realized by using the quantum Rabi model with qubit-qubit coupling. Keywords Quantum gate · Quantum Rabi model · Schr¨odinger cat states 1 Introduction The quantum Rabi model (QRM) [1] plays a fundamental role in studying lightmatter interaction. Such model is different from the Rabi model [2, 3], which deGangcheng Wang · Qingyong Wang · Kang Xue Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China Yimin Wang College of Communications Engineering, PLA University of Science and Technology, Nanjing 210007, China Beijing Computational Science Research Center, Beijing 100193, China Jing-Ling Chen Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Chunfeng Wu Pillar of Engineering Product Development, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372 E-mail: chunfeng [email protected]

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scribes two-level system (TLS) coupled to classical field. The quantized light in the QRM is described by a single bosonic mode and the quantized matter is described by a TLS. When the coupling strength is much smaller than the frequency of bosonic mode, and meanwhile the TLS and the bosonic mode are near-resonance, the counter-rotating term (CRT) in the interaction term can be neglected and the QRM can be reduced to the so-called Jaynes-Cummings model (JCM) [4]. The JCM has been very successfully applied to understand lots of experimental phenomena, such as vacuum Rabi splitting [5, 6] and quantum Rabi oscillation [7, 8], etc. Recently, the stronger coupling regimes can be reached in a range of experiments based on solid state systems [9–19]. In this case, the rotating wave approximation (RWA) is no longer suitable and the CRT cannot be neglected. This motivated several new potential applications of the QRM in quantum information processing [20–23]. The QRM leads to enhanced fast implementation of quantum gates since the operation time usually depends on the coupling strength between the qubits and the resonator. A stronger coupling is of essential importance in speeding up quantum gate operations, as well as avoiding the concomitant negative effects of decoherence during the evolution of the system. The conditional displacement interaction depicts a quantum resonator conditionally displaced according to qubit(s)’ states. The conditional interaction has played a prominent role in understanding the fundamentals in quantum physics [24–31] and implementing quantum processing protocols [32–41]. Specifically, it has been widely utilized to generate superposed coherent states [25–31] and implement quantum gates [32–41]. It is worth mentioning that the conditional displacement interaction in the literatures has mainly been derived by applying the RWA with respect to qubitresonator coupling on the QRM or JCM [32–39]. Of special note is the effective coupling strength with the RWA that is in general of order 10−3 ∼ 10−2 ωr , and would lead to quantum protocols operating at microseconds or less. On the other hand, Refs. [42,43] discussed a special design of superconducting flux qubits in order to realize the conditional displacement interaction in longitudinal direction in the ultrastrong coupling regime. However, the complicated design of flux qubits makes it difficult to experimentally realize the ultrastrong interaction. In this work, we realize the conditional displacement interaction in transversal direction from the QRM or the QRM with qubit-qubit interaction by resorting to parametric modulation of external magnetic fields. Our scheme paves a promising way to speed up quantum information processing according to the conditional displacement interaction. We then explore the implementation of quantum gates and the generation of Schr¨odinger cat states based on the conditional displacement interaction with numerical results. This paper is organized as follows. In Sec. 2, we present the system Hamiltonian describing the qubits coupled to a resonator and derive an effective Hamiltonian which is just the conditional displacement interaction. We analyse the accuracy of the effective Hamiltonian by numerical calculations. we also show that the conditional displacement interaction can also be realized from the QRM with qubit-qubit interaction. In Sec. 3 and Sec. 4, we investigate the applications of the interaction in implementing quantum gate and generating superposed coherent states, respectively. Finally, we end this paper with discussions and conclusions in Sec. 5.

Conditional displacement interaction in transversal direction from the quantum Rabi model

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2 The realization of transversal conditional displacement interaction In this section, we will show the derivation of transversal conditional displacement interaction from the QRM and from the QRM with qubit-qubit coupling under the frequency modulation field. The validity of the effective Hamiltonian is also studied by means of the numerical approach. 2.1 The transversal conditional displacement interaction from the QRM In our scheme, we consider the two-qubit QRM [44, 45], which describes two-qubit coupling with a bosonic field. The Hamiltonian takes the following form (~ = 1) H=

2 ∑ ωm q m=1

2

† σm z + ωr a a +

2 ∑

gm (a† + a)σmx ,

(1)

m=1

m where ωm q is the energy splitting of m-th qubit, σα (α = x, y, z) is the α component of the m-th Pauli matrix, ωr is the frequency of the bosonic field, a (a† ) is the annihilation (creation) operator, gm is the coupling constant. For the sake of simplicity, we assume that ωm q = ωq = ηωr and gm = g. Apply external magnetic fields defined in the following form on qubits such that it is possible to do parametric modulation [46–48]

ωm q (t) = ωq + εm sin(ωm t − ϕm ) .

(2)

Moving to the rotating frame defined by time-dependent transformation U(t) = U1 (t)U2 (t) with      ∑ ωq m   † U1 (t) = exp −i  σz + ωr a a t , 2 m    ∑ αm  m U2 (t) = exp i cos(ωm t − ϕm )σz  , 2 m

(3a) (3b)

where αm = εm /ωm . For simplification, we set ϕm = ϕ and ωm = ω. In the rotating framework, the transformed Hamiltonian reads ˜ = U † (t)HU(t) − iU † (t)∂t U(t) . H(t)

(4)

Substituting the time-dependent evolution operator U(t) to the Eq. (4), we obtain ˜ =g H(t)

2 [ ∑

] iδ− t iαm cos(ωt−ϕ) iδ+ t −iαm cos(ωt−ϕ) a† σm e + a† σm e + h.c. , −e +e

(5)

m=1 m m where δ± = ωr ± ωq and σm ± ≡ (σ x ± iσy )/2. The exponential term exp(±αm cos(ωt − ϕ)) can be expanded by means of the following Jacobi-Anger Identity [49]

exp (±iαm cos(ωt − ϕ)) =

+∞ ∑

(±i)l Jl (αm ) exp(±il(ωt − ϕ)) .

l=−∞

(6)

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Here Jl (αm ) is the Bessel function of first kind. Substituting Eq. (6) to Eq. (5), we obtain   2  +∞ ∑   † m ∑ l iδ− t il(ωt−ϕ) ˜ H(t) =g i Jl (αm )e e + h.c. a σ− m=1

+g

l=−∞

  +∞   † m ∑ iδ+ t −il′ (ωt−ϕ) l′ + h.c. . (−i) Jl′ (αm )e e a σ+

2  ∑ m=1

(7)

l′ =−∞

By means of frequency modulation field, the terms in Eq. (7) with lower oscillation frequency are selected. Consequently, the terms in Eq. (7) with higher oscillation frequency can be ignored by utilizing the RWA. The oscillation frequency in Eq. (7) are δ− + lω and δ+ − l′ ω. If we set ω = 2ωq and η > 1, the lowest oscillation frequency is |δ− | = |η − 1|ωr . Hence, many higher-order terms in Eq. (7) can be neglected according to the RWA. Then the Eq. (7) can be recast as follows ˜ =g H(t)

2 [ ∑ ] iδ− t iδ− t iϕ a† σm − ia† σm e + h.c. . − J0 (αm )e + J1 (αm )e

(8)

m=1

By choosing αm = 1.4347, we obtain J0 (αm ) = J1 (αm ) = 0.5479. Then the effective coupling geff = 0.5479g. by setting ϕ = π/2, we derive the following effective Hamiltonian as follows (more detailed derivation is presented in appendix A) H˜ eff =

2 ∑

( ) geff a† eiδ− t + ae−iδ− t σmx ,

(9)

m=1

This is just the desired conditional displacement interaction in the transversal direction for two qubits. In the following, we check the validity of the approximation in a numerical way. Choose |ψ(0)⟩ = |gg0c ⟩ as the initial state, where |g⟩ is the ground state of the qubit and |0c ⟩ is the vacuum state of the resonator, and denote |ψ(t)⟩ and |ψ′ (t)⟩ as the evolution states governed by the original Hamiltonian with external magnetic fields and the effective Hamiltonian (9), correspondingly. Let F (t) = |⟨ψ(t)|ψ′ (t)⟩| [50–54] be the fidelity for the states |ψ(t)⟩ and |ψ′ (t)⟩. We numerically find F at T = 2π/|δ− | by choosing α1 = α2 = 1.4347 and the results are shown in Fig. 1, where black solid curve is for ωq = 1.8ωr (i.e. η = 1.8), g = 0.2ωr , and blue dotted curve is for ωq = 2ωr (i.e. η = 2), g = 0.2ωr . It can be inferred from the numerical results that the effective Hamiltonian (9) is approximating the original time-dependent Hamiltonian very well. Moreover, when g is increasing, larger value of η is required to achieve very good approximation according to the conditions mentioned. However, in practical experiments, the required strong driving may not be achievable. As a result, the fidelity will be decreased with currently achievable parameters. We show the numerical results in Fig. 2, where the parameters are chosen as ωq = 3ωr (i.e. η = 3), g = 0.5ωr . As demonstrated in Fig. 2, with comparatively small external driving strength, the fidelity is affected largely when g = 0.5ωr . This tells us in the case that the external driving is not comparatively large, our scheme performs well when g is around 0.2ωr , but becomes less and less desirable when g is increasing.

Conditional displacement interaction in transversal direction from the quantum Rabi model

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1

Fidelity

0.99 0.98 0.97 = 1.8 r, g=0.2 r = 2 r, g=0.2 r q

0.96

q

0.95 0

0.5

1

| -| t/(2 ) Fig. 1 The fidelity F varies as function of the evolution time t for different parameters, where black solid curve is for ωq = 1.8ωr and g = 0.2ωr , and blue dotted curve is for ωq = 2ωr and g = 0.2ωr .

1

Fidelity

0.96 0.92 0.88 0.84 0.8 0

0.5

1

| -| t/(2 ) Fig. 2 The fidelity F varies as function of the evolution time t for the following parameters: ωq = 3ωr and g = 0.5ωr .

2.2 The transversal conditional displacement interaction from the QRM with qubit-qubit coupling In fact, we also can realize the transversal interaction by utilizing the QRM with qubit-qubit coupling as well. Such Hamiltonian can describe flux-qubits couple to LC-resonator [55]. If we consider two frequency modulated flux-qubits coupled with one LC-resonator, the Hamiltonian reads Hq =

2 ∑ ωm q (t) m=1

2

† σm z + ωr a a +

2 ∑ m=1

gm (a† + a)σmx +

2 ∑

Dmn σmx σnx ,

(10)

m,n=1

where Dmn = gm gn /ωr . The definition of other operators and parameters can be found in Sec. 2.1. Applying external magnetic fields defined in the Eq. (2) and moving to

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the rotating frame defined by time-dependent transformation as shown in Eqs. (3), we obtain the transformed Hamiltonian H˜ q = U † (t)Hq U(t) − iU † (t)∂t U(t) with ϕm = ϕ and ωm = ω

H˜ q (t) =g

2 [ ∑

] iδ− t iαm cos(ωt−ϕ) a† σm e + h.c. −e

m=1

+g

2 [ ∑

] iδ+ t −iαm cos(ωt−ϕ) a† σm e + h.c. +e

m=1

+D

2 [ ∑

n i2ωq t −i(αm +αn ) cos(ωt−ϕ) σm e + σ+ e

] + h.c.

(11)

m,n=1

+D

2 [ ∑

] n −i(αm −αn ) cos(ωt−ϕ) σm + h.c. , + σ− e

m,n=1

where δ± = ωr ± ωq . Upon the Jacobi-Anger expansion [49], we find   +∞ 2  ∑ ∑    † m l iδ t il(ωt−ϕ) − a σ− i Jl (αm )e e + h.c. H˜ q (t) =g m=1

l=−∞

  +∞ 2  ∑   † m ∑ l iδ+ t −il(ωt−ϕ) (−i) Jl (αm )e e + h.c. +g a σ+ m=1

+D

2 ∑

l=−∞

  +∞   m n ∑ l i2ω t −il(ωt−ϕ) q σ+ σ+ (−i) Jl (αm + αn )e e + h.c.

m,n=1

l=−∞

m,n=1

l=−∞

(12)

  +∞ 2  ∑   m n ∑ l −il(ωt−ϕ) (−i) Jl (αm − αn )e + h.c. . +D σ+ σ− If we set ωq = ω = ηωr with η > 2, the lowest oscillation frequency is ωr and hence many higher-order terms can be neglected. Take the first line in Eq. (12) as an example, we have   +∞ 2  ∑   † m ∑ l iδ− t il(ωt−ϕ) (1) ˜ i Jl (αm )e e + h.c. . Hq (t) = g (13) a σ− m=1

l=−∞

When l = 0, the phases reduce to e±iδ− t = e±i(ωr −ωq )t . In the case that ωq > 2ωr with proper choice of parameters, it is possible to make |δ− | ≫ g|J0 (αm )| and hence it gives a higher-order oscillating term which can be omitted. When l = 1, the phases are combined as e±i(δ− +ω)t∓iϕ = e±iωr t∓iϕ since ωq = ω. The system is in the ultra-strong coupling regime and when ωr is not much greater than g|J1 (αm )|, it cannot be neglected according to the RWA. When l = −1, the phases are revised as e±i(δ− −ω)t±iϕ = e±i(ωr −2ωq )t±iϕ . By properly choosing parameters, we have |2ωq − ωr | ≫ g|J−1 (αm )|, and this tells us that the term can be neglected too. Following similar reasoning, it can be found that all the terms in H˜ q(1) (t) with l , 1 are

Conditional displacement interaction in transversal direction from the quantum Rabi model

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1

Fidelity

0.99 0.98 0.97 =2.5 r, g=0.2 r =3.5 r, g=0.2 r q

0.96

q

0.95 0

0.5

1

t/(2 ) r Fig. 3 The fidelity Fq varies as function of the evolution time t for different parameters, where black solid curve is for ωq = 2.5ωr and g = 0.2ωr , and blue dotted curve is for ωq = 3.5ωr and g = 0.2ωr .

higher-order oscillating terms. While for the second line in Eq. (12), all the terms with l , 1 are also higher-order oscillating terms based on appropriate parameters. Next let us look the third line in Eq. (12), it is found that most of the terms are higher-order oscillating terms except l = 2. In this case the coefficient is D|J2 (αm + αn )| which is much smaller than g even when D is comparable with g, because J2 (αm + αn ) can be chosen to be much smaller than 1 and therefore the term can be omitted. Lastly for the fourth line in Eq. (12), the only non-higher-order oscillating term is given when l = 0 and the coefficient is DJ0 (αm − αn ) which can be set to be 0 when αm − αn = 2.40483. Finally, ignoring all the higher-order terms, we can obtain effective Hamiltonian as follows, q = H˜ eff

2 ∑

( ) † iωr t gm + ae−iωr t σmx , eff a e

(14)

m=1

where gm eff = gJ1 (αm ), and we select α1 − α2 = 2.40483 and ϕ = π/2. This is nothing but the desired conditional displacement interaction in the transversal direction for m two qubits. The gm eff is tunable and it is possible to realize different geff for individual qubits or g1eff = −g2eff , as long as the condition α1 − α2 = 2.40483 is fulfilled. The effective coupling strength of the conditional interaction reaches its maximum value of gJ1 (1.832) ≈ 0.582g when αm = 1.832. When α1 = −α2 = 1.20242, we find g1eff = −g2eff = gJ1 (1.20242) ≈ 0.499g. Choose |ψq (0)⟩ = |gg0c ⟩ as the initial state, and use |ψq (t)⟩ and |ψ′q (t)⟩ to describe the evolution states governed by the original Hamiltonian with external magnetic fields and the effective Hamiltonian (14), correspondingly. We numerically find the fidelity Fq (t) = |⟨ψq (t)|ψ′q (t)⟩| by choosing α1 = −α2 = 1.20242 and the results are shown in Fig. 3, where black solid curve is for ωq = 2.5ωr (i.e. η = 2.5), g = 0.2ωr , and blue dotted curve is for ωq = 3.5ωr (i.e. η = 3.5), g = 0.2ωr . The numerical results clearly demonstrate that the effective Hamiltonian (14) is really close to the original time-dependent Hamiltonian.

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3 Quantum gate with the transversal conditional interaction We next study the applications of the transversal conditional displacement interaction in implementing two-qubit gates. In the case of two qubits, the evolution operator (the detailed derivation is shown in appendix B) obtained from the transversal interaction is [38–41] ( ) [ ] U(t) = D β(t)J x exp iΦ(t)J x2 , (15) where J x = σ1x + σ2x , β(t) = (geff /δ− )(1 − eiδ− t ) and Φ(t) = (geff /δ− )2 (δ− t − sin(δ− t)), † ∗ and D(β) = eβa −β a . Let the system evolve for a time period of T = 2π/|δ− |, then we obtain β(T ) = 0 and Φ(T ) = −2π(geff /δ− )2 . Then evolution operator can be recast as follows [38–40] ( ) U(T ) = exp −i2π(geff /δ− )2 J x2 . (16) We thus have a tunable quantum phase gate and it can be rewritten as U(T ) = 1 2 e−iθ e−iθσx σx with θ = 4π(geff /δ− )2 . Express the evolution operator in the basis of {|ee⟩, |eg⟩, |ge⟩, |gg⟩}, we obtain   0 0 −i sin θ   cos θ   0 cos θ −i sin θ 0  . U(T ) = e−iθ  (17) 0   0 −i sin θ cos θ −i sin θ 0 0 cos θ The entangling power, which was introduced by Zanardi et.al., describes the capacity of generating entanglement [56, 57]. For the two-qubit case, any unitary operator can be expressed in the following form ] [ (18) V(c1 , c2 , c3 ) = (L1 ⊗ L2 )exp −i(c x σ1x σ2x + cy σ1y σ2y + cz σ1z σ2z ) (L3 ⊗ L4 ), where c x,y,z satisfy π/4 ≥ c x ≥ cy ≥ cz ≥ 0 and Li are local unitary operators. Two local invariants G1 and G2 introduced in Refs. [58, 59] are given as follows ]2 1 [ −2icz e cos 2(c x − cy ) + e2icz cos 2(c x + cy ) , 4 G2 = cos(4c x ) + cos(4cy ) + cos(4cz ).

G1 =

(19)

The entangling power can be recast in terms of G1 as e p (U) = 29 (1 − |G1 |). In our case, c x = θ and cy = cz = 0. Then the entangling power of the evolution operator in Eq. (17) reads e p (U) = 29 sin2 (2θ). Obviously, the evolution operator U(T ) represents non-trivial two-qubit gates when θ , nπ (n = 0, ±1, ±2, · · · ). When θ = π/4, we obtain a quantum gate with maximum quantum entangling power. In this case, the matrix form of the quantum gate reads (the global phase has been omitted)    1 0 0 −i  1  0 1 −i 0   . (20) U(T ) = √  2  0 −i 1 0  −i 0 0 1

Conditional displacement interaction in transversal direction from the quantum Rabi model

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This non-trivial quantum gate is local equivalence to the CNOT gate. We can check CNOT = (u1 ⊗ u2 )U(T )(u3 ⊗ u4 ), where the local unitary operators are as follows ( ) −1 i 1 u1 = √2 , u2 = 1 i ( u3 =

√1 2

) −1 1 , 1 1

( √1 2

(21)

) 1 −i , −i 1

(

) 10 u4 = . 01

(22)

We then study the performance of the quantum gate by generating N = 50 random ( ) two-qubit states of the form |ψ(ai )⟩ = √∑41 2 a1 |ee⟩ + a2 |eg⟩ + a3 |ge⟩ + a4 |gg⟩ ⊗ |0c ⟩ i=1

|ai |

as initial states, where i = 1, 2, 3, 4. We numerically calculate average fidelity defined by F G = |⟨ψideal |U ′ (T )|ψ(ai )⟩|, where |ψideal ⟩ = U(T )|ψ(ai )⟩ and U ′ (T ) is the evolution operator based on Hamiltonian (1) with external magnetic fields. The average fidelity is found to be F G = 0.9965 when ωq = 2ωr (i.e. η = 2), g = 0.2634ωr and α1 = α2 = 1.4347. The numerical results show the excellent performance of our scheme of achieving the two-qubit quantum gate with θ = π/12. Moreover, we also explore the performance of our scheme to achieve the quantum gate with θ = π/8 by choosing ωq = 2ωr (i.e. η = 2), g = 0.3226ωr and α1 = α2 = 1.4347, and the fidelity is 0.9912. Further improvement in the gate fidelity is dependent on the development of experimental techniques to realize very strong external driving. Therefore we achieve to implement non-trivial two-qubit quantum phase gate. The scheme possesses the merit of enhanced fast quantum operation to avoid the detrimental effect of decoherence.

4 The generation of the Schr¨odinger cat states The conditional interaction is also crucial in creating superposed coherent states and hence exploring the superposition rule. To get clear evidence of the quantum superposition, the displacement of the resonator should be maintained rather than destroyed by decoherence. Thus enhanced fast generation of the superposed coherent states is demanded to avoid the negative effect of decoherence. We then investigate the creation of the superposed coherent states with the ultrastrong transversal interaction. Take the case of one qubit coupled to resonator as an example, the evolution operator is represented by Eq. (15) by replacing J x by σ x . Let |Ψ1 (0)⟩ = |g⟩ ⊗ |0c ⟩ be the initial state, and we obtain the final state at time t as follows eiΦ(t) |Ψ1 (t)⟩ = √ (|+⟩ ⊗ |β(t)c ⟩ − |−⟩ ⊗ |−β(t)c ⟩) , 2

(23)

where |±⟩ = √12 (|e⟩ ± |g⟩) and the coherent states |±β(t)c ⟩ = D[±β(t)]|0c ⟩ with coherent-state amplitude ±β(t)c = ±(geff /δ− )(1 − eiδ− t ). Obviously, the spin states

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Gangcheng Wang et al.

|+⟩ and |−⟩ undergo different displacement β(t)c and −β(t)c , respectively. In the basis of {|g⟩, |e⟩}, the final state can be rewritten as [26–31] ) eiΦ(t) ( −1 N+ |g⟩ ⊗ |cat+ ⟩ + N−−1 |e⟩ ⊗ |cat− ⟩ (24) 2 [ ]−1/2 where N± = 2(1 + (−1)±1 exp(−2|β(t)|2 )) and |cat± ⟩ = N±1 (|β(t)c ⟩+(−1)± |−β(t)c ⟩). The superpositions of coherent states |β0 (t)c ⟩ and |β1 (t)c ⟩ are the so-called even and odd Schr¨odinger cat states. Performing projective measurement in the qubit basis, superposed coherent ] states [ ( 1 ±1 2) |cat± ⟩ can be extracted with probability of 2 1 + (−1) exp − 2|β(t)| , respec|Ψ1 (t)⟩ =

tively. The magnitude of the displacement is dependent on evolution time t. When t0 = π/|δ− |, we obtain maximum magnitude of |β(t0 )c | = 2geff /|δ− |. We find the fidelity between ideal state (24) and the final state solved numerically from the original time-dependent Hamiltonian. The parameters are chosen as ωq = 1.6ωr , g = 0.2ωr , and α1 = 1.4347 and the realized displacement is 0.3653. According to our numerical calculations, state (24) and hence the superposed coherent states can be created with a fidelity of 0.9991. The amplitude of the displacement can be enhanced via multi-step system evolution. Specifically, let the initial state be |g0c ⟩, and after the time period t0 , we obtain state |Ψ1 ⟩. Acting the evolution operator on the state |Ψ1 ⟩ for another t0 , we i2Φ(t ) obtain state |Ψ2 ⟩ = e √20 (|+⟩ ⊗ |2β(t0 )c ⟩ − |−⟩ ⊗ |−2β(t0 )c ⟩). Therefore the displacement amplitude is 2|β(t0 )c | = 0.7306 after two steps of evolution with a total evolution time 2π/|δ− |. The corresponding fidelity of generating |Ψ2 ⟩ is 0.9980. Following the multi-step evolution, the amplitude of displacement can be further enhanced. Our results can be extended to the multi-qubit case. For the two-qubit case, let |Ψ1′ ⟩ = √12 (|ee⟩ + |gg⟩) ⊗ |0c ⟩ be the initial state. Acting evolution operator in Eq. (15) on the initial state, we obtain |Ψ1′ (t)⟩ =

e4iΦ(t) √ (|++⟩ ⊗ |2β(t)c ⟩ − |−−⟩ ⊗ |−2β(t)c ⟩) , 2

where |±⟩ = √12 (|e⟩ ± |g⟩) and |±±⟩ = |±⟩ ⊗ |±⟩. Substituting |±⟩ = Eq. (25), we obtain |Ψ1′ (t)⟩ =

√1 (|e⟩ ± |g⟩) 2

(25) to the

] e4iΦ(t) [ ′ −1 √ (N− ) (|ee⟩ + |gg⟩) ⊗ |cat−′ ⟩ + (N+′ )−1 (|eg⟩ + |ge⟩) ⊗ |cat+′ ⟩ . (26) 2 2

where |cat±′ ⟩ = N±′ (|2β(t)c ⟩ ± |−2β(t)c ⟩) with normalization constant N±′ . Performing projective measurement in the qubit basis {|ee⟩, |eg⟩, |ge⟩, |gg⟩}, we can obtain the socalled even and odd cat states. The amplitude of the displacement can be enhanced for the multi-qubit case by comparing the Eq. (24) with the Eq. (26).

5 Discussion and conclusion Based on current experimental techniques of superconducting circuits, it may be possible to realize our scheme in the superconducting system. In our scheme, it is desired

Conditional displacement interaction in transversal direction from the quantum Rabi model

11

that the qubits are ultrastrongly coupled to the resonator with strong driving external magnetic fields. As an example, the required parameters may be ωr ∼ 2π × 1 GHz, ω = 2ωq ∼ 2π × 4 GHz and ε ∼ 2π × 5.7 GHz. The reported strong driving of superconducting qubits in experiments is roughly 2π × 5 GHz [60]. As for the required ultrastrong qubit-resonator coupling strength, there are noticeable achievements in the known literature. In Refs. [9, 10], the authors have experimentally achieved the ultrastrong coupling with flux qubits coupled to a transmission line resonator, and the coupling strength is around 0.1ωr [9, 10]. For the system of superconducting qubits coupled to a resonator, the Semba group has extended the qubit-resonator coupling to the deep strong coupling regime experimentally [11]. It can be inferred that our scheme to achieve the transversal interaction is possibly achievable with the state-ofart techniques. In summary, we have proposed to achieve conditional displacement interaction of qubits and resonator from the QRM. By adjusting external magnetic fields in the system, the conditional displacement interaction is obtained for one or two qubits. We have revealed numerically that the effective Hamiltonian is very close to the original Hamiltonian with external magnetic fields. Based on the transversal interaction, the implementation of two-qubit quantum phase gates and the generation of superposed coherent states are achievable. We have also provided some discussions about the feasibility of the conditional displacement interaction according to the current experimental techniques. Moreover we have shown that the conditional displacement interaction can also be derived from the QRM with qubit-qubit coupling. In this case, the effective coupling strength is controllable dependent on the value of αm . Acknowledgements The work is supported by the NSF of China (Grant Nos. 11405026 and 11575042). Y.M.W. is supported by the NSF of China (Grant No. 11404407), the NSF of Jiangsu (Grant No. BK20140072) and China Postdoctoral Science Foundation (Grant Nos. 2015M580965 and 2016T90028). J.L.C. is supported by the NSF of China (Grant No. 11475089).

A The derivation of effective Hamiltonian in Eq. (9) In this appendix, we give the detailed derivation of Eq. (9) from Eq. (2). For the sake of simplification, we set unified qubit frequency (i.e., ωm q = ωq ) and unified coupling strength (i.e., gm = g). In order to obtain the effective Hamiltonian, we introduce the periodical modulation of the transition frequency in the following form ωm q (t) = ωq + εm sin(ωm t − ϕm ) .

(27)

Then the two-qubit QRM with the frequency modulation field can be recast as follows H(t) =

2 ωm (t) ∑ q m=1

2

† † σm z + ωr a a + g(a + a)

2 ∑

σm x.

(28)

m=1

Moving to the rotating frame defined by time-dependent transformation U(t) = U1 (t)U2 (t) with        ∑ ωq m σz + ωr a† a t , U1 (t) = exp −i  2 m    ∑ αm   cos(ωm t − ϕm )σm U2 (t) = exp i z  , 2 m

(29a) (29b)

12

Gangcheng Wang et al.

where αm = εm /ωm . For simplification, we set ϕm = ϕ and ωm = ω. In the rotating framework, the transformed Hamiltonian reads ˜ = U † (t)HU(t) − iU † (t)∂t U(t) . H(t)

(30)

The first term in Eq. (30) reads   2 ∑    U(t) σm U † (t)HU(t) = U † (t) g(a† + a) x  m=1

2 ( ∑ ) i(ωr −ωq +αm cos(ωt−ϕ))t i(ωr +ωq −αm cos(ωt−ϕ))t =g a† σm + a† σm + h.c. −e +e

(31)

m=1

=g

2 ( ∑ ) iδ− t iαm cos(ωt−ϕ) iδ+ t −iαm cos(ωt−ϕ) a† σm e + a† σm e + h.c. , −e +e m=1

m m where δ± = ωr ± ωq and σm ± ≡ (σ x ± iσy )/2. In order to obtain the second term in Eq. (30), we should calculate the derivative of the operator exponential. Let exp(A(t)) be the operator exponential. If [∂t A(t), A(t)] = 0, the derivative of exp(A(t)) reads

∂t exp(A(t)) = (∂t A(t)) exp(A(t)) = exp(A(t)) (∂t A(t)) .

(32)

By means of the Eq. (32), the derivative of the U(t) reads     ∑ εm  ∑ ωq m    ∂t U(t) = U(t) −i  σz + ωr a† a − i sin(ωm t − ϕm )σm z  . 2 2 m m

(33)

Then the second term in Eq. (30) reads     ∑ εm    ∑ ωq m  −iU † (t)∂t U(t) = −iU † (t)U(t) −i  σz + ωr a† a − i sin(ωm t − ϕm )σm z  2 2 m m   ∑ ωq m  ∑ εm = −  σz + ωr a† a − sin(ωm t − ϕm )σm z 2 2 m m =−

2 ωm (t) ∑ q m=1

2

(34)

† σm z − ωr a a

Substituting Eq. (31) and Eq. (34) to Eq. (30), we obtain ˜ =g H(t)

2 ( ∑

) iδ− t iαm cos(ωt−ϕ) iδ+ t −iαm cos(ωt−ϕ) a† σm e + a† σm e + h.c. , −e +e

(35)

m=1

The exponential term exp(±αm cos(ωt − ϕ)) can be expanded by means of the following Jacobi-Anger Identity [49] exp (±iαm cos(ωt − ϕ)) =

+∞ ∑

(±i)l Jl (αm ) exp(±il(ωt − ϕ)) .

(36)

l=−∞

Here Jl (αm ) is the Bessel function of first kind. Substituting Eq. (36) to Eq. (35), we obtain we obtain ˜ =g H(t)

  2  +∞ +∞ ∑ ∑   † m ∑ l iδ− t il(ωt−ϕ) † m l′ ′ iδ+ t −il′ (ωt−ϕ) a σ− i Jl (αm )e e + a σ+ (−i) Jl (αm )e e + h.c.

m=1

l=−∞

(37)

l′ =−∞

Here Jl (αm ) is the Bessel function of first kind. The oscillation frequency in Eq. (30) are δ− + lω and δ+ − l′ ω. If we set ω = 2ωq and η > 1, the lowest oscillation frequency is |δ− | = |η − 1|ωr . Hence, many

Conditional displacement interaction in transversal direction from the quantum Rabi model

13

higher-order terms in Eq. (37) can be neglected according to the RWA. Taking the first term in Eq. (37) as an example, we have   2  +∞ ∑   † m ∑ (1) l iδ t il(ωt−ϕ) − ˜ a σ− H (t) = g i Jl (αm )e e + h.c. . (38) m=1

l=−∞

−1, the phases are revised as e±i(δ− −ω)t±iϕ . By properly choosing parameters, we have |δ

When l = − − ω| ≫ g|J−1 (αm )|, and this tells us that the term can be omitted. When l = 0, the phases reduce to e±iδ− t = e±i(ωr −ωq )t . With proper choice of parameters, it is possible to make |δ− | not much greater than g|J0 (αm )| and as a result it cannot be neglected according to the RWA. When l = 1, the phases are combined as e±i(δ− +ω)t∓iϕ = e±iδ+ t∓iϕ . It is possible to make |δ+ | ≫ g|J1 (αm )| and hence it gives a higher-order oscillating term which can be omitted too. Similarly it can be found that all the terms in H˜ (1) (t) with l , 0 are higher-order oscillating terms. While for the second term in Eq. (37), all the terms with l′ , 1 are also higher-order oscillating terms based on appropriate parameters. Finally, letting ϕ = π/2 and ignoring all the higher-order terms, we can obtain the following effective Hamiltonian H˜ eff =

2 ∑

( ) geff a† eiδ− t + ae−iδ− t σm x,

(39)

m=1

where geff = 0.5479g and this is because we choose αm = 1.4347 and so J0 (αm ) = J1 (αm ) = 0.5479. This is just the desired conditional displacement interaction in the transversal direction for two qubits.

B The derivation of evolution operator in Eq. (15) In this appendix, we present a detailed derivation of Eq. (15) from Eq. (9). If we set J x = σ1x + σ2x , the Eq. (9) can be recast as follows ) ( H˜ eff (t) = geff a† eiδ− t + ae−iδ− t J x , (40) The evolution operator of effective Hamiltonian in Eq. (40) can be obtained by means of the Magnus expansion. The evolution operator U(t) satisfies the following differential equation, i∂t U(t) = H˜ eff (t)U(t)

(41)

The solution to Eq. (41) can be written as the following Magnus series form, U(t) = exp(Ω(t)) = exp(Ω1 (t) + Ω2 (t) + Ω3 (t) + · · · ), where the first three terms of Ω(t) are given as follows, ∫ t Ω1 (t) = −i H˜ 1 dt1 , Ω2 (t) = Ω3 (t) =

0 (−i)2

2 (−i)3 6

∫

∫

t

t1

dt1 ∫

0

[ ] dt2 H˜ 1 , H˜ 2 ,

0

∫

t

∫

t1

dt1

dt2

0

0

t2

( ) dt3 [H˜ 1 , [H˜ 2 , H˜ 3 ]] + [H˜ 3 , [H˜ 2 , H˜ 1 ]] .

(42)

(43a) (43b) (43c)

0

˜ k ). Substituting Eq. (40) to Eqs. (43), we Here, we have made the notation simplification that H˜ k = H(t obtain Ω1 (t) =

) geff ( −iδ− t a(e − 1) − a† (eiδ− t − 1) J x , δ−

Ω2 (t) = i

g2eff δ2−

Ωk (t) = 0,

(44a)

(δ− t − sin(δ− t)) J x2 ,

(44b)

k ≥ 3.

(44c)

14

Gangcheng Wang et al.

[ ] Here, we used the commutation relation [H˜ i , H˜ j ] = 2ig2eff sin δ− (t j − ti ) J x2 . We also can check the relation [Ω1 (t), Ω2 (t)] = 0. Then the evolution operator U(t) in Eq. (42) can be recast as ( ) U(t) = exp(Ω1 (t))exp(Ω2 (t)) = D[β(t)J x ]exp iΦ(t)J x2 , (45) where J x = σ1x + σ2x , β(t) = (geff /δ− )(1 − eiδ− t ) and Φ(t) = (geff /δ− )2 (δ− t − sin(δ− t)), and D(β) = .

† −β∗ a

eβa

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