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Modulation of entanglement and quantum discord for circuit cavity QED states Y.H. Ji a,b,∗ , W.D. Li a , S.J. Wen a a b

Department of Physics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, China

a r t i c l e

i n f o

Article history: Received 12 January 2013 Accepted 25 May 2013 Available online xxx PACS: 03.65. Ta 03.65.Ud 03.67.-a Keywords: Coupled qubits Circuit cavity QED Entanglement Quantum discord

a b s t r a c t The paper investigates the dynamic evolution behaviors of entanglement and quantum discord of coupled superconducting qubits in circuit QED system. We put emphasis on the effects of cavity ﬁeld quantum state on quantum entanglement and quantum correlations dynamic behaviors of coupling superconducting qubits. The results show that, (1) generally speaking, the entanglement will appear the death and new birth because of the interaction between qubits and cavity ﬁeld, on the contrary, this phenomenon will not appear in quantum discord. (2) When the cavity ﬁeld is in coherent state, the entanglement survival time is controlled by the average photon number. The more the average photon number is, the longer survival time of entanglement is prolonged. Thus it has the beneﬁt of keeping quantum correlations. (3) When the cavity ﬁeld is in squeezed state, the squeezed amplitude parameters have controlling effects on quantum correlations including entanglement and quantum discord. On the one hand, the increase of squeezed amplitude parameters can prolong the survival time of entanglement, on the other hand, with the increase of squeezed amplitude parameters, the robustness of quantum discord is more and more superior to concurrence and is more advantage to keep the system quantum correlations. The further study results show that the increase of the initial relative phase of coupling superconducting qubits can also keep the quantum correlations. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Atom cavity QED system provides a good platform for quantum information processing experiments, but it has no advantage in integrations [1,2]. In order to meet the requirements of quantum information processing and truly realize the scale and integrated control, the solid quantum system should be considered. Circuit QED system is the realization of cavity QED principle in solid ﬁeld. In circuit QED, superconducting qubits act as the artiﬁcial atoms and the one dimensional superconducting transmission linear resonator acts as the microwave cavity ﬁeld [3–5]. Different to natural atoms, the properties of artiﬁcial atoms can be artiﬁcially designed and controlled. In addition, the strong coupling of superconducting circuit with the cavity also can be realized even if the interference of solid environment is very strong. The newest experimental research results show that the coherent time for superconducting qubits coupled to a microwave cavity can be prolonged to 0.1 ms [6]. The recent research reports also show that superconducting qubits coupled to a microwave cavity can realize the effective quantum feedback control [7]. Through the control, the superconducting

∗ Corresponding author. E-mail address: [email protected] (Y.H. Ji).

qubits will make the coherent oscillation faster, slower or more continuous. The ability that can actively suppress the decoherence will be applied a lot in quantum error correction, quantum state stabilization, entanglement generation and adaptive measurement. Presently, the new research progresses in circuit QED open up a new prospect for quantum state preparation and quantum information processing. Quantum systems exhibit diversiﬁed correlations which have no classical counterparts. It is pointed out recently that quantum entanglement, the most well known measure of quantum correlations which plays essential roles in quantum information processing, cannot describe all the nonclassicality in the correlations. The quantum discord, which can describe quantum correlations in separable states, is an important subject to intensive theoretical studies [8–12]. In the dynamic behavior investigations of quantum and classical correlation in Markovian and non-Markovian process, people ﬁnd that in Markovian environment the quantum discord decays with time in a kind of asymptotic behavior which forms a bright comparison with the possible entanglement sudden death phenomenon appears in entanglement dynamics. Obviously, if the quantum discord is treated as a quantum resource, the asymptotic decay action is more advantageous to realize the quantum information processing.

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Please cite this article in press as: Y.H. Ji, et al., Modulation of entanglement and quantum discord for circuit cavity QED states, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.150

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a

b

1

1

0.8

0.8

C and QD

In this paper, we model a big Josephson junction as coupling element, and we couple two superconducting charge qubits with it [13,14]. Appling the quantization method, we investigate the inﬂuences on the quantum entanglement and quantum correlations dynamics of coupled superconducting qubits when the cavity ﬁeld is in coherent state or squeezed state.

C and QD

2

0.6 0.4 0.2

2. Model

0

0

2

4

6

˚

e

˚0

k=1

−

2

(1)

2

where Ek (Vxk ) = ECk (nk − Ck Vxk /2e) is the electrostatic energy of charge qubits, ECk = 2e2 /(Ck + 2CJk ) and EJk is the charging energy and Josephson energy of kth qubit, and EJ0 is the Josephson energy of the big junction. Considering the situation of quantization microwave ﬁeld, the ﬂux of the change frequency is a quantized ﬂux ˚q a+ + ˚∗q a. After some calculations, the interactive Hamiltonian between quantized magnetic ﬁeld and two superconducting charge qubits can be written as HI = g(a+ 1− 2− + a1+ 2+ ) g=−

2˚q LJ IC1 IC2 sin ˚0

(2)

2˚ e

(3)

˚0

We suppose that the two superconducting qubits are in the Bell state with spins anti-correlated

Q (0)

= cos |gg + eiϕ sin |ee

(4)

and the cavity ﬁeld is initially in the superposition state of number states as

=

F (0)

f (n) |n

(5)

n

f(n) is the probability amplitude of photon number state distriSo the initial state of the system can be bution of quantum ﬁeld. written as ˚(0) = F (0) ⊗ Q (0) . In the interaction picture, we can have the system’s vector at any time t

∞

(t)

xn (t) |e, e, n + yn (t) g, g, n + 1

=

+ f (0)cos g, g, 0

(6)

n=0

with xn (t) = eiϕ f (n)sin cos

yn (t) = eiϕ f (n)sin sin

n + 1gt

− f (n + 1)cos sin

n + 1gt

+ f (n + 1)cos cos

(7)

n + 1gt

n + 1gt

(8)

3. Measure of quantum entanglement and quantum correlation We adopt the concurrence entanglement deﬁned by Wootters to measure the system entanglement. One can calculate that the concurrence for the initial states ˚ is [16]

C(t) = 2

14 (t)41 (t)

(9)

2

4

8

10

6

8

10

d

1

1

0.8

0.8

0.6 0.4 0.2 0

6

gt

0.6 0.4 0.2

0

2

4

6

8

gt

10

0

0

2

4

gt

Fig. 1. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) n¯ √ = 1, (b) n¯ = 2, (c) n¯ = 5, (d) n¯ = 10. The two qubits are initially in the (|gg + |ee) / 2 and the ﬁeld is initially in the coherent state.

A bipartite state may include not only classical correlation but also quantum correlation. We describe the quantum correlation with the quantum discord put forward by Ollivier [17,18] QD (AB ) = I (AB ) − CD (AB )

(10)

where I (AB ) = S (A ) + S (B ) − S (AB ) is the quantum mutual information and CD(AB ) is the classical correlation between the in Refs two subsystems. As discussed and 19

20, the classical correlation CD (AB ) = max S (A ) − S AB B , where {Bk } is a set of {Bk }

von Neumann measurements performed on subsystem B locally,

S AB {Bk } =

pk S (k ) is the quantum conditional entropy,

k

k = (IA ⊗ Bk ) (IA ⊗ Bk ) /Tr (IA ⊗ Bk ) (IA ⊗ Bk ) is the conditional density operator corresponding to the outcome labeled by k, and pk = Tr (IA ⊗ Bk ) (IA ⊗ Bk ) with IA is the identity operator performed on subsystem A. 4. Discussion First, we discuss the quantum correlation dynamics of the qubits when the ﬁeld is in the coherence state. The coefﬁcient f(n) in Eq. (5) can be expressed as n n¯ n) (¯ 2

0

c

cos ϕk − EJ0 cos

0

10

C and QD

Ek (Vxk ) − 3EJk cos

8

gt

C and QD

2

0.4 0.2

We model a big Josephson junction as coupling element and couple two superconducting charge qubits with it. The Hamiltonian for the system is given by [15] H=

0.6

f (n) = exp −

2

√ n!

(11)

n¯ is the average number of photons of the ﬁeld. The concurrence time evolution reﬂects the time evolution of entanglement between two qubits. We can see from Figs. 1–3 that the entanglement between two charge qubits evolves with time in oscillation behaviors, and the time evolution of concurrence appears stronger oscillation behavior with the increase of the average photons number. At some moments, the concurrence of two charge qubits is zero. That is to say at this moment, the entanglement does not exist in two charge qubits so that the entanglement death appears. Under the drive of interactions in qubits and between qubits and transmission linear, the entanglement will appear recovery phenomena. Figs. 1–3 also show that based on the study model of the paper, the time evolution of QD(t) has a big difference to concurrences. On

Please cite this article in press as: Y.H. Ji, et al., Modulation of entanglement and quantum discord for circuit cavity QED states, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.150

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0.6 0.4 0.2

0

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gt

c

0.6 0.4 0.2 0

3

0.6 0.4 0.2

0

2

4

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gt

8

10

0

0

2

4

gt

Fig. 2. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) n¯ = 1, (b) n¯ = 2, (c) n¯ = 5, (d) n¯ = 10.The two qubits are initially in the |gg and the ﬁeld is initially in the coherent state.

Fig. 4. The time evolution of the concurrence C(t) (solid line), QD(t)(dashed line) versus the scaled √ time gt with n¯ = 30. The two qubits are initially in the [|gg + exp(iϕ) |ee] / 2 and the ﬁeld is initially in the coherent state. (a) ϕ = /6, (b) ϕ = /4, (c) ϕ = /3, (d) ϕ = /2.

the one hand, the quantum discord decays obviously slowly than concurrence does, which reﬂects that quantum discord is more robust than concurrence, on the other hand, although the early time evolution of quantum discord is completely similar to the entanglements, the quantum discord being zero won’t appear. In addition, after a period of evolving especially when the average photon number increases, the entanglement death is more difﬁcult to appear and the value of quantum discord is almost equal to 1. It suggests that in this model and under some appropriate conditions, through controlling the average photon number, we can realize the controlling and regulating on the quantum correlations of coupled qubits system, and we can also obtain long time maximum quantum correlations including entanglement and quantum discord.

Fig. 4 shows the relative phase of the initial state of coupled qubits effects on the concurrence and QD(t). We can observe from the ﬁgure that the quantum discord will not appear death, which displays better robustness than entanglement. Although entanglement appears death and rebirth, with the increasing of the relative phase from 0 to /2, the entanglement of coupled charge qubits is not easy to appear death. Therefore, the conclusion is that we can control the quantum correlations and entanglement of two qubits through changing the relative phase, which theoretically provides a method to control quantum correlations.

Fig. 3. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) n¯ = 1, (b) n¯ = 2, (c) n¯ = 5, (d) n¯ = 10.The two qubits are initially in the |ee and the ﬁeld is initially in the coherent state.

Fig. 5. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) √ r = 0.5, (b) r = 1, (c) r = 1.5, (d) r = 2. The two qubits are initially in the (|gg + |ee) / 2 and the ﬁeld is initially in the squeezed vacuum state.

Please cite this article in press as: Y.H. Ji, et al., Modulation of entanglement and quantum discord for circuit cavity QED states, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.150

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Fig. 6. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) r = 0.5, (b) r = 1, (c) r = 1.5, (d) r = 2. The two qubits are initially in the |gg and the ﬁeld is initially in the squeezed vacuum state.

In the squeezed vacuum state, the coefﬁcient f(n) in Eq. (5) can be expressed as f (n) =

(2n)!tan hn r √ n!2n cos hr

(−exp(iϕ))

(12)

where ϕ is the direction angle of compression, for simplicity, we suppose ϕ = 0. r denotes the compressibility factor which shows the degree of compression of the ﬁeld. Figs. 5–8 reﬂects the effects of the squeezed amplitude parameters on the concurrence and quantum discord when cavity ﬁeld is in squeezed vacuum state. In the ﬁgures we choose the most representative time evolution curves of quantum discord and entanglement. In the physical model and initial quantum state of this paper,

Fig. 8. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt with r = 1. (a) ϕ = /6, (b) ϕ = /4, √ (c) ϕ = /3, (d) ϕ = /2. The two qubits are initially in the [|gg + exp(iϕ) |ee] / 2 and the ﬁeld is initially in the squeezed vacuum state.

Figs. 5–8 indicate that entanglement appears death and rebirth, but inversely, quantum discord doesn’t appear death and rebirth. We ﬁnd that the robustness of quantum discord and concurrence of coupled qubits depend on the squeezed amplitude parameters. With the increase of the squeezed amplitude parameters, the time and frequency of entanglement death are reducing and delaying, but the appearance of entanglement death cannot be completely avoided. The robustness of quantum discord is more superior to concurrence at the same time. The above analysis shows that the squeezed amplitude parameters have controlling effects on quantum correlations. The inner parameters of superconducting qubits are hard to regulate, but the quantum state of external circuit cavity is relatively easy to control in experiment, which provides a method to control quantum correlations theoretically. 5. Conclusions

Fig. 7. The time evolution of the concurrence C(t) (solid line), QD(t) (dashed line) versus the scaled time gt: (a) r = 0.5, (b) r = 1, (c) r = 1.5, (d) r = 2. The two qubits are initially in the |ee and the ﬁeld is initially in the squeezed vacuum state.

Through concurrence and quantum discord, the paper explores the dynamic evolution behaviors of entanglement and quantum discord of coupled superconducting qubits in circuit QED system. We separately compare and investigate the controlling and regulating effects of the average number of photons, the squeezed amplitude parameters and the initial relative phase of coupled qubits on the quantum discord and concurrence. The results show that when the cavity ﬁeld is in coherent state, the increasing of the average photon number will be beneﬁt for keeping the quantum correlations and entanglement. When the cavity ﬁeld is in squeezed state, with the increase of the squeezed amplitude parameters, the robustness of quantum discord is more superior to concurrence. In addition, the further investigations on Figs. 1–8 show that the robustness of quantum discord actually also depends on the initial entanglement of quantum state. The higher the initial entanglement degree is, the more robust of quantum discord compared to concurrence. The increasing of the initial relative phase of coupled superconducting qubits can also keep the quantum correlations. People hold different views on the effects of quantum discord in quantum information processing, but in some quantum algorithm, the quantum correlations described by quantum discord is

Please cite this article in press as: Y.H. Ji, et al., Modulation of entanglement and quantum discord for circuit cavity QED states, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.150

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useful than by entanglement. Quantum discord also can be used to improve the efﬁciency of quantum Carnot engine and is better for understanding quantum phase transition and Grover searching. It suggests quantum correlations have potential applications in investigating the basic properties of quantum system. People look forward to the deep analysis on quantum discord dynamical features.

Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 11164009.

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Please cite this article in press as: Y.H. Ji, et al., Modulation of entanglement and quantum discord for circuit cavity QED states, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.05.150

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