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Dong Kun. Dynamics of two arbitrary qubits strongly coupled to a quantum oscillator. Chinese Physics B, 2016, 25(12): 124202  
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Dynamics of two arbitrary qubits strongly coupled to a quantum oscillator
Dong Kun†,
School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, China

 

† Corresponding author. E-mail: dklovemy@163.com

Project supported by the National Natural Science Foundation of China (Grant No. 10875018).

Abstract
Abstract

Using adiabatic approximation, a two arbitrary qubits Rabi model has been studied in ultra-strong coupling. The analytical expressions of the eigenvalues and the eigenvalues are obtained. They are in accordance with the numerical determined results. The dynamical behavior of the system and the evolution of entanglement have also been discussed. The collapse and revival phenomena has garnered particular attention. The influence of inconsistent coupling strength on them is studied. These results will be applied in quantum information processing.

PACS: 42.50.Md;03.65.Ud;42.50.Pq
Keyword:Rabi model;ultra-strong coupling;adiabatic approximation
1. Introduction

The quantum Rabi model (QRM) is a significant model in quantum optics. It describes the system of a two-level atom (qubit) interacting with a single mode of a quantum harmonic oscillator. This model was initially presented in 1936 by Rabi in order to study the nuclear magnetic spin resonance.[1,2] The QRM is of great interest in circuit quantum electrodynamics (QED)[3–6] and cavity QED,[7–13] both experimentally and theoretically. Though the Rabi model has been studied for over 80 years, analytical solutions for the eigenvalues and eigenfunctions was recently presented by Braak[14,15] and Chen[16] due to the mathematical difficulty. However, there are still many difficulties in the application due to its complicated form.[17] So a number of approximations have been developed to solve QRM. The rotating wave approximation(RWA)[18,19] is the most widely used. However, the RWA relies upon the assumption of near resonance and weak coupling between qubits and oscillator. With the development of circuit QED and cavity QED, it has achieved so-called ultra-strong coupling between qubits and oscillator. Motivated by those experimental developments, another approximation was investigated by Irish et al., called the adiabatic approximation (AA).[20–23] This approximation was shown to work best at Quasi-degenerate qubit and ultra-strong coupling.

The QRM with two qubits has received more attention as it can be used to implement quantum entanglement and quantum information processing (QIP).[24–34] The QRM with two qubits can be realized in several solid devices.[35,36] There have been some previous theoretical discussions to solve the QRM with two qubits.[21,37–44] Chen et al. presented a concise analytic solution to the QRM with two qubits using the method of extended coherent states.[44] However, in some cases,[21,43] the two qubits are not arbitrary qubits, instead identical qubits and the same coupling with the quantum harmonic oscillator. In other cases, the dynamical properties and the evolution of entanglement have not been sufficiently studied. Accordingly, it is very necessary to study of the dynamical properties and the evolution of entanglement between two arbitrary qubits in the ultra-strong coupling strength range.

The organization of this paper is as follows. In the first part, we use the AA to calculate the eigenvalues and eigenfunctions of the QRM with two arbitrary qubits. In the second part, we discuss the evolution of this system. The evolution of entanglement is given in last part.

2. The Spectrum of model Hamiltonian

The Hamiltonian of the QRM with two arbitrary qubits can be written as[38–40]

where and are the usual Pauli matrices in the Hilbert space of the i-th qubit, and and â refer to the creation and the annihilation operators of an interacting mode of a harmonic oscillator, ωi and ω is the frequency of the i-th qubit and quantum harmonic oscillator respectively. βi is the coupling parameter between the i-th qubit and the quantum harmonic oscillator.

The first task is to establish a new set of basis states. The self-energy of two qubits can be treated as the perturbation when ωi ≪ ω. Then the Hamiltonian can be written as

The eigenvalue equation of Ĥ0 is

The eigenstate form of |Φ〉 is

where |m〉 denotes the eigenstate of , and |ϕm〉 are the oscillator eigenstates of Ĥ0 by replacing to |m〉.

The eigenvalues of are m:

In the matrix picture, |m〉, the eigenstate of can be written as:

And |ϕm〉 satisfies the eigenvalue equation

Applying the displacement operator (for real λ) and the inverse operator of displacement to Eq. (5), we obtain

Setting λ = −m, we can obtain

where N = E/ħω + m2. According to the eigenvalue equation of the number operator, we can obtain

Thus, we obtain the eigenstates of Ĥ0

and eigenvalue

These are so-called displaced oscillator basis states.[20,21]

The Hamiltonian can be rewritten in the basis of the displaced oscillator basis states. The transition matrix element between different N is

If HN,N′ is much smaller than the energy gap between different N:

the transitions between different N can be ignored. This approximation is reasonable when ωi ≪ ω. Accordingly, we only preserve the transitions between states with the coincident N. So we can use basis |m〉|Nm〉 to obtain the block diagonal form of the Hamiltonian. This approximation was used by Irish et al.[20] and Agarwal et al.[21] to study a single qubit case or identical qubits case. For each N, in the order of |m1〉|Nm1〉, |m2〉|Nm2〉, |m3〉|Nm3〉, and |m4〉|Nm4〉 for rows and columns, the Hamiltonian has the form

The off-diagonal terms ΩN,1 and ΩN,2 are given by

where LN(x) is the Laguerre polynomial. The normalized eigenfunctions and eigenvalues of ĤN are

We calculated the eigenvalue from Eq. (15) numerically and plot the results in Fig. 1. As shown in Fig. 1, our result can describe the system correctly in full values of β as long as ωi ≪ ω.

Fig. 1. The eigenvalue from adiabatic approximation (dash line) and numerically (solid line) for ω1 = 0.15ω, ω2 = 0.075ω, and β2 = 0.15.

From Eq. (15), we can obtain if ω1 = ω2 and β1 = β2. These conditions are the same conditions for the dark or trapping states[45] by Peng et al.[40] The qubits are maximally entangled, even in the system with dissipation.[45] So our results may be applied in QIP.

3. Dynamical behavior

In order to make sure our results can truly be applied to the actual projects, we need to study the dynamical behavior of the QRM with two arbitrary qubits. Analysis of the dynamical properties of two equivalent qubits can be found in Ref. [21]. In this section we extend the results in Ref. [21] to the case of two arbitrary qubits. We will show the effect of different couplings for the i-th qubit with a quantum harmonic oscillator in the ultra-strong coupling regime. The Fock state and the coherent state will be treated as the initial state for a harmonic oscillator.

First, we consider the initial states in a displaced Fock state, such that |m1 > |Nm1 >. According to Eq. (15), we obtain

Applying the time-evolution operator to act on |Ψm1(0)〉

Using Eq. (15), it is easy to find the probability for the qubits to recur in the initial state:

where

Since four basis states are included, the probability contains six frequencies. This is in contrast to the equivalent qubits case where only three Rabi frequencies control the dynamical behavior.[21] Using the same method for Pm1(N,t), we obtain

Normally, the number state is not an appropriate state for describing the quantum harmonic oscillator in the experiment. A coherent state, by contrast, is desirable to be able to describe the quantum harmonic oscillator. More importantly, the collapse and revival behaviors can be expected.

For convenience of calculations, we make the following reasonable assumptions

So the expressions for Eq. (15) are simplified to

Now we let a coherent-state |α〉 as the initial state for harmonic oscillator

By the same calculation as that of Pm2(N,t), we obtain

The Pm2(α,t) from Eq. (26) and numerical calculations are plotted in Fig. 2. The revival time, cycle, and the heights of the Pm2(α,t) with AA are consistent with the numerical calculations. This is evident in Figs. 2(a) and 2(b). From Figs. 2(b) and 2(c), we can see that there are different revival times, cycles and the heights of the Pm4(α,t) in different conditions, β1 = β2 or β1 ≠ β2. There are four revival sequences for the Pm2(α,t) because it has four Rabi frequencies in the two arbitrary qubits. This is in contrast to the equivalent qubits case where only two revival sequences.[21]

Fig. 2. Collapse and revival dynamics for Pm4(α,t) with ω1 = 0.15ω, ω2 = 0.075ω, α = 3, β1 = 0.16 and β2 = 0.14 [(a) and (b)]; ω1 = 0.15ω, ω2 = 0.075ω, α = 3, and β1 = β2 = 0.14 [(c)].
4. Entanglement dynamics

In this section, the entanglement dynamics of two qubits will be investigated. The influence of β1 ≠ β2 has drawn particular attention.

We consider the case that the qubits and the oscillator are prepared initially in one of the σx(σx|±〉 = ±|±〉) Bell states and in an undisplaced coherent state respectively

When N ≫ βi, we can obtain

So equation (27) becomes

According to Eq. (24), we have

Using Eqs. (24) and (28), we can obtain

Then we use the eigenbasis σz to compact in Eq. (31)

The reduced density matrix for the qubits is obtained by tracing out the oscillator states

where

The concurrence[46] for Eq. (33) is

If |α|2 ≫ 1 we can calculate Eq. (34) approximately (see Appendix A) to obtain

where

In Eq. (37) we have defined

and ΦIm is given in Appendix A.

If we just considered the envelope of S(t), we can obtain the concurrence for ρ1,2(t) as

From Eqs. (36) and (37), we can clearly see that S(t) manifests collapse and revival with S(t). That means the entanglement collapse and revival phenomena may also appear. The exponential part in Eq. (39) dictates the envelope of the revival entanglement. The denumerator term dominates the height of entanglement for each revival. The revival time and the height of C(t) are

From Eq. (39), we can see that the width of the primary revival (k = 0) is

and the width of the k-th revival is

Now, we discuss the effect of β1 ≠ β2 on the collapse and revival phenomena for C(t). From Eqs. (41) and Eq. (43), we need to compare the value of in different situations, β1 = β2 or β1 ≠ β2. The situation becomes more complicated, with no conditionality for βi. When β1 ≠ β2, the value of is bigger if we have an additional restrictive condition for β1 + β2 = constant. That means the width of the i-th revival will increase, and the height of the i-th revival will decrease for β1 ≠ β2. These are evidently shown in Fig. 3.

Fig. 3. The entanglement dynamics between the two qubits for ωi = 0.15ω, α = 3. Here (a) β1 = β2 = 0.16 and (b) β1 = 0.1, β2 = 0.22.

The conclusion mentioned above can be applied in areas of QIP. According to the specific demands, we can extend the duration of revival or enhance the high degree of revival by selecting a suitable nonlinear coupling parameter.

5. Conclusion

In this report we studied the QRM with two arbitrary qubits. The main conclusions are as follows.

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