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Effects of linear modulation of qubits–resonator coupling on quantum entanglement in circuit QED
Optik 127 (2016) 3950–3954
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effects of linear modulation of qubits–resonator coupling on quantum entanglement in circuit QED Juju Hu a,b,∗ , Qiang Ke a,b a b
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, PR China
a r t i c l e
i n f o
Article history: Received 17 November 2015 Accepted 11 January 2016 Keywords: Circuit quantum electrodynamics Quantum entanglement Linear modulation Fock state
a b s t r a c t We investigate to efficiently maintain the quantum entanglement between qubits through linearly modulating the coupling coefficient between qubits and field in circuit quantum electrodynamics system. We find that the maintenance time of entanglement between coupling qubits is closely related with the modulation type. Compared to the case of constant coupling coefficients, the first disappearance of entanglement between qubits is largely delayed when the coupling coefficients are linearly modulated with a small ramp. However, it is unfavorable for the maintenance of entanglement even leads to fast disappearance if the coupling coefficients are modulated with a large slope. The disappearance time of entanglement not only relies on the quantum state of coupling qubits, but also depends on the average photon number. The more the average photon number is, the more disadvantageous quantum entanglement maintains. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Quantum entanglement is the most interesting quantum phenomenon in many-body systems. As a physical resource, entangled state plays an important role in many aspects of quantum information such as quantum teleportation, quantum key distribution and quantum computation [1–3]. However, pure entangled state will degenerate to mixed state in real physical world because of environmental decoherent effect. Quantum communication and quantum computation with such mixed entangled state lead to the information distortion [4–6]. Otherwise, the unavoidable interaction between the environment and single qubit leads to exponential decay of the coherences of the qubit while the entanglement between two or more qubits terminates its value in a finite time; a phenomenon termed entanglement sudden death (ESD) [7,8]. Indeed, decoherence is the most important limit for quantum computation since the quantum superposition decays into statistical mixtures [9,10]. It is known that decoherence and disentanglement are closely connected. Therefore, there is an
increasing interest in studying the quantum mechanism of the entanglement dynamics in quantum computation and quantum information. How to generate and transfer entanglement is a problem of fundamental interest in quantum information processing [11], and the greatest barrier is decoherence and disentanglement induced by environment. In order to overcome the influence of decoherence, one of the tactics proposed by the electrical engineers and researchers is to improve the coherent time of the qubits by optimizing the system [12]. The existing research results indicate that quantum entanglement can be manipulated by quantum correction and coherent feedback, and the entanglement sudden death can be effectively delayed [13]. For convenience in experiment operation, manipulating entanglement by classical driving field is an effective method. Recently, Ref. [14] manipulated the coupling coefficient between atom-field in vacuum state and prolonged the entanglement between atoms effectively. Vacuum state is a special case of Fock state. As the most fundamental and important state in quantum optics, the photon number state |n obeys sub-Poisson distribution and constitutes a com-
|n n| = 1 that arbitrary photon state can spread
plete set
n
∗ Corresponding author at: College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China Tel.: +86 18926419186.. E-mail address: [email protected] (J. Hu). http://dx.doi.org/10.1016/j.ijleo.2016.01.081 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
accordingly and the spread is exclusive. In the past few decades, people paid much attention on how to prepare Fock state. Most recently, the method of atom-cavity interactions has been applied in solid-state circuit QED to prepare multi-photon Fock states
J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
3951
In the interaction picture, with the resonant interaction considered, the effective Hamiltonian of the system is expressed as ( = 1)
g aj+ + a+ j−
˜ eff = − H
+ gAB 1+ 2− + 1− 2+
(1)
j=A,B
Fig. 1. Schematic of coupling qubits circuit quantum electrodynamics. A superconducting charge qubit and a flux qubit containing a dc-SQUID are coupled to a transmission line of length l. The charge qubit couples to the transmission line through the gate capacity and the biased flux ˚1 , and the flux qubit couples to the transmission line through the biased flux ˚2 and ˚3 .
and generalized binomial states [15]. The high-Q one-dimensional superconducting transmission line resonator (TLR), similar to the optical resonator, can be used as quantum bus to realize the controllable coupling of several qubits. In Ref. [16], using TLR as data bus, we proposed a scheme to realize multi-photon Fock state in circuit QED. In the proposed scheme, each qubit passes in turn through the circuit cavity via externally biasing flux, and then we implement quantum measurement on the ground state after the qubit passes through the circuit cavity. It effectively avoids the influence of decoherence since each qubit is operated alone. In this paper, we further discuss how to manipulate the coupling between superconducting qubit and field by external classical driving field to effectively manipulate the quantum entanglement in circuit-QED. The paper is organized as follows. In Section 2, we show the model and present the scheme of preparing Fock states in circuit QED. The dynamical behaviors of quantum entanglement for the coupled superconducting qubits in circuit QED are investigated in Section 3. Finally, in Section 4 the obtained results are summarized and discussed.
2. Modulation of the coupling coefficient between qubit and cavity field In this section, we firstly introduce the manipulation of coupling coefficient briefly. The coupled circuit QED system is shown in Fig. 1 [17]. The system consists of a charge qubit, a flux qubit, and the TTR. The TLR can be modeled as a simple harmonic oscillator near its resonance frequency. The charge qubit and the flux qubit are separately placed at l/2 and l/4 of the transmission line, where maximum quantization voltage and current can be achieved. The charge qubit couples to the transmission line through gate capacity on which the coupling strength depends, whereas the flux qubit couples to the transmission line through the external flux. There are two steps in practical operation. First, prepare the high-Q one-dimensional superconducting transmission line resonator (TLR) in Fock state |n. The basic operation is as follows: we completely shut down the external flux ˚1 (that is to shut down the charge qubit), and control the external flux ˚3 by pulse counter and realize the preparation of multi-photon Fock states. Second, place the charge qubit and flux qubit in the resonator as shown in Fig. 1 after the preparation of proper Fock state |n. The charge qubit couples to the transmission line through the gate capacity and the biased flux ˚1 , and the flux qubit couples to the transmission line through the biased flux ˚2 and ˚3 .
where a and a+ are respectively the annihilation and creation operator, k± (k = A, B) represents the Pauli matrix for the qubit, and gAB the interacting strength between two qubits. Both the detailed deduction and assumptions (or conditions) of above-mentioned Hamiltonian is given in Ref. [18]. Different from Ref. [18], we consider the interaction between qubits for the purpose of generality. Particularly, the coupling coefficient g between TLR and qubits (for convenience, we assume gA = gB = g) in the proposed scheme is dependent on the resonant frequency ω, length l, capacitance C and inductance L of the transmission line, as well as the gate capacitance Cg of the charge qubit, which makes the manipulation of quantum correlation more convenient. If we control one parameter, such as the gate capacitance Cg , with an external function f(t), the coupling coefficient g becomes time dependent. Then, utilizing external control function f(t) to simultaneously manipulate the coupling between each superconducting qubit and field g, we rewrite Eq. (1) as: Heff (t) = −
g f (t) aj+ + a+ j− +gAB 1+ 2− +1− 2+ .
(2)
j=A,B
Following Ref. [19], we consider that the cavity field is switched on by a linear ramp described by the function
f (t) =
kt
0≤t≤T
0,
others
,
(3)
where the parameters k and T determine the slope of the linear ramp. As the value of k changes over a fixed T, the slope of the linear ramp changes, too. Note that the equation above incorporates two limiting cases corresponding to sudden and adiabatic changes which can be achieved by only changing the value of k for a fixed T. Utilizing spontaneous parametric down-conversion (SPDC) and cavity quantum electrodynamics system, two-particle Bell state has been prepared experimentally [20,21]. We first suppose that the coupled superconducting qubits are in the Bell state, written as [22]
= cos ↑↓ + sin | ↑↓ · (0 ≤ ≤ ) .
Q (0) I
(4)
By using the Schrödinger equation, the vector of the system can be easily obtained at any time t
I (t) = x1 ↑↑ n − 1 + x2 |↑↓ n + x3 |↓↑ n + x4 ↓↓ n + 1 , (5) where x1 () =
cos + sin
ng
˛
x2 () =
cos + sin
+
x3 () =
√
e−i(+ +− ) ,
2
−
(+ ei+ − − ei− )
cos − sin
2˛
(7)
(+ ei + − − ei− )
cos − sin 2
(6)
2˛
cos + sin
(ei − − ei + ) ,
e−i(+ +− ) ,
(8)
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J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
Fig. 2. Time evolutions of the concurrence versus the scaled time gt with = /6, gAB = 0.1g, n = 1. Dash line k/T = 0.1; dot line k/T = 1; and dash dot line k/T = 5. For comparison, solid line corresponds to the case without external manipulation function f(t) and the coupling between qubit and cavity is constant.
x4 () =
cos + sin
√
n + 1g
˛
1.0
(ei− − ei+ ) ,
(9)
0
0
0
0 0
x4 (n, t)2
⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠ (10)
Where * denotes complex conjugate. Second, we suppose that the coupled superconducting qubits are in the Bell state as follows,
Q (0) II
= cos |↑↑ + sin |↓↓
(11)
By using the Schrödinger equation, the vector of the system can be
II (t)
= y1 (t) |↑↑, n + y2 (t) ↑↓, n + 1 + y3 (t) ↓↑, n + 1
where y1 (n, ) =
+ y4 (t) ↓↓, n + 2 + y5 (t) |↓↓, n ,
(12)
0.4 0.2 0.0 0.0
0.4
0.8
gt
1.2
1.6
2.0
Fig. 3. Time evolutions of the concurrence versus the scaled time gt with = /12, gAB = 0.1g, n = 1. Dash line k/T = 0.1; dot line k/T = 1; and dash dot line k/T = 5. For comparison, solid line corresponds to the case without external manipulation function f(t) and the coupling between qubit and cavity is constant.
II =
⎛ y1 (n, t)2 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎝ y1∗ (n, t) sin
⎞
0
y1 (n, t) sin
y2 (n, t)y3∗ (n, t)
y2∗ (n, t)y3 (n, t)
y3 (n, t)2
0
0
0
y4 (n, t)2 + sin2
0
y2 (n, t)2
0
⎟ ⎟ ⎟. ⎟ ⎟ ⎠
2(2n + 3)
(ei +
(n + 1)gAB i + + ei − ) + − ei − ) (e 2(2n + 3)ˇ 3. Manipulation of quantum entanglement
n+2 cos , 2n + 3
(13)
√ g n + 1 i + y2 (n, ) = y3 (n, ) = − ei − ) cos , (e ˇ
(14)
y4 (n, ) =
0.6
(17)
n+1 +
Concurrence
⎞
0
(t) state
II
0.8
with
2 + 8(2n + 1)g 2 , = −gAB ±˛ , = k t 2 . gAB ˛= ± T 2 According to Eqs. (5)–(9), we can easily obtain the density of the hybrid qubits by tracing the quantum field of TLR:
⎛ x1 (n, t)2 0 0 ⎜ 2 ⎜ x2 (n, t) x2 (n, t)x3∗ (n, t) 0 ⎜ I = ⎜ ⎜ x3 (n, t)2 ⎜ 0 x2∗ (n, t)x3 (n, t) ⎝
in the
(n + 1)(n + 2) 2(3 + 2n)
[(ei + + ei − )
+gAB (ei + − ei − ) − 1] · cos ,
y5 (n, ) = sin , with
2 + 8(2n + 3)g 2 , = −gAB ±ˇ . ˇ= gAB ± 2 According to Eqs. (12)–(16), we get
(15)
(16)
We adopt the concurrence entanglement defined by Wootters to measure the system entanglement [23]. Figs. 2–5 display the entanglement evolution when two qubits are initially in the entangled state. The solid line corresponds to the constant qubit-field coupling, and the other lines exhibit the entanglement evolution under linear modulation of the qubit-field coupling. For constant coupling, the entanglement between two qubits in the quantum state I (t) periodically disappears and takes a while to recover. From Fig. 2, we can see that manipulation of the qubit-field coupling greatly influences the entanglement and effectively modulates its survival time. The first disappearance of entanglement will be delayed significantly if the ramp slope of the coupling coefficient is small, indicating the linear modulation effect is obvious. Increase the ramp slope (the dot line for k/T = 1), the first disappearance of entanglement is still delayed evidently, but lesser than the case of k/T = 0.1. With the increasing of k/T, however, the
1.0
1.0
0.8
0.8
Concurrence
Concurrence
J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
0.6 0.4 0.2 0.0 0.0
3953
0.6 0.4 0.2
0.2
0.4
gt
0.6
0.8
0.0 0.0
1.0
0.2
0.4
gt
0.6
0.8
1.0
1.0
1.0
0.8
0.8
Concurrence
Concurrence
Fig. 4. Time evolution of the concurrence versus the scaled time gt with = /6, gAB = 0, k/T = 1. Solid line n = 0; dash line n = 1; and dot line n = 3.
0.6 0.4
0.4 0.2
0.2 0.0 0.0
0.6
0.4
0.8
gt
1.2
1.6
2.0
0.0 0.0
0.4
0.8
gt
1.2
1.6
2.0
Fig. 5. Time evolution of the concurrence versus the scaled time gt with = /6, n = 1, k/T = 0.5. Solid line gAB /g = 0; dash line gAB /g = 0.5; dot line gAB /g = 1; and dash dot line gAB /g = 5.
first disappearance of entanglement will be advanced. It not only may be shifted to the time of entanglement disappearing under constant coupling, but also probably lead to entanglement death between qubits. The results suggest that too large a value of k/T would be counterproductive although it is effective to adjust the quantum entanglement between qubits by linearly modulating the coupling between qubit and field. Above analysis show that the influence of the time-dependent qubit-field coupling coefficient is nonlinear and relates with the initial quantum state, photon number in the field, and interaction between qubits. It is known that the evolution of entanglement is closely related with the initial quantum state. Entanglement sudden death is entirely avoidable via proper choice of initial state. Based on the parameters in Fig. 2 and the initial state Q (0) = cos(/6) |↑↑ + II sin(/6) |↓↓, ESD absolutely does not occur to the entanglement between qubits. However, if the initial state is changed, for example, the coupling qubits are initially in Q (0) while the initial II degree of entanglement is changed, such as = /12, then ESD shown in Fig. 3 will occur to the entanglement between qubits as time evolves. The entanglement time is effectively delayed through linearly modulating the qubit-field coupling coefficient. From Figs. 2 and 3, we find that the entanglement between qubits under linear modulation no longer evolves periodically. Eqs. (5) and (11) show that the degree of entanglement between qubits is related to the square of time(periodic function of time under constant coupling coefficient), which accounts for the aperiodic evolution of entanglement between qubits when the coupling coefficient is modulated by the external function. In the Fock state field, collapse and revival occurs to the time evolution of the concurrence, and more intense oscillations appear with the increasing of the photon number. Therefore, from Fig. 4, entanglement sudden death appears as time evolves. The photons in the cavity shorten the evolution period of the entanglement between qubits and make them in disentangled state for a long time. When there are photons in the cavity and each coefficient
is nonzero, with time evolving, some coefficients can be neglected since they are very small, and then the disentangled state or maximum entangled state is obtained. It may be explained as follows, n plays the role of coupling strength between two qubits and the circuit cavity, large n is equivalent to strong coupling between two subsystems. Once we only consider two qubits by tracing out the circuit cavity as the environment, the large value of n means that two qubits suffer strong dissipation. It indicates that decreasing the average photons number will be beneficial for keeping the quantum correlation between qubits. For similar reasons, the photon number in cavity greatly weakens the linear manipulation effect. Instead, it can be seen from Fig. 5 that the interaction between qubits is conductive to strengthen the linear manipulation of external function on the quantum entanglement. 4. Conclusions We investigate the quantum entanglement dynamics of circuitQED in Fock state, and focus on the dynamical behavior of quantum entanglement when the coupling coefficient between qubit and cavity field is linearly modulated. The following results are obtained: (1) in the case of constant coupling between qubit and cavity, collapse and revival occurs to the quantum entanglement between qubits; (2) through linearly modulating the coupling coefficient between qubits by an external linear manipulation function of small amplitude, the coupling coefficient between qubit and cavity is effectively reduced so that the entanglement effect between qubits is greatly improved and the first disappearance time of entanglement is delayed. Further investigations also indicate that the initial disappearing time of entanglement is related to the average photon number. The more average photons there are in the cavity, the more disadvantageous to maintain quantum entanglement. It should reinforce that classical noise is inevitably introduced in the process of implementing external modulation, and quantum correlation will recover in this environment under certain conditions, which has been verified by experiment [24].
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J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
Therefore, classical noise can be used to manipulate quantum correlation. Our next work is to explore the effect of the noise in modulating signal on the behavior of entanglement. Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant no. 11404150. References [1] S. Bose, Quantum communication through an unmodulated spin chain, Phys. Rev. Lett. 91 (2003) 207901. [2] G.F. Zhang, Thermal entanglement and teleportation in a two-qubit Heisenberg chain with Dzyaloshinski–Moriya anisotropic antisymmetric interaction, Phys. Rev. A 75 (2007) 034304. [3] J. Allcock, N. Linden, Quantum communication beyond the localization length in disordered spin chains, Phys. Rev. Lett. 102 (2009) 110501. [4] T. Yu, J.H. Eberly, Sudden death of entanglement, Science 323 (2009) 598. [5] Y.H. Ji, Dynamics of super-quantum discord and direct control with weak measurement in open quantum system, Chin. Phys. B 24 (2015) 120302. [6] J.J. Hu, Y.H. Ji, Dynamics control of geometric quantum discord for two coupling qubits in a squeezed vacuum reservoir, Optik 125 (2014) 1337. [7] J.P. Paz, A.J. Roncaglia, Dynamics of the entanglement between two oscillators in the same environment, Phys. Rev. Lett. 100 (2008) 220401. [8] M. Scala, B. Militello, A. Messina, J. Piiol, S. Maniscalco, Microscopic derivation of the Jaynes–Cummings model with cavity losses, Phys. Rev. A 75 (2007) 013811. [9] J. Ma, Z. Sun, X.G. Wang, F. Nori, Entanglement dynamics of two qubits in a common bath, Phys. Rev. A 85 (2012) 062323.
[10] Y.H. Ji, Y.M. Liu, Regulation of entanglement and geometric quantum discord of hybrid superconducting qubits for circuit QED, Int. J. Theor. Phys. 52 (2013) 3220. [11] J.G. Li, J. Zou, B. Shao, Entanglement backflow under the composite effect of two non-Markovian reservoirs, Phys. Lett. A 376 (2012) 1020. [12] M. Salerno, V. Popkov, Optimal transport and von Neumann entropy in a Heisenberg XXZ chain out of equilibrium, Phys. Rev. E 87 (2013) 022108. [13] Y.H. Ji, Y.M. Liu, Investigations into quantum correlation of coupled qubits in a squeezed vacuum reservoir, Chin. Phys. B 22 (2013) 020305. [14] J.J. Hu, S. Li, Influence of composite effect from quantum channels on entanglement, Commun. Theor. Phys. 62 (2014) 183. [15] R.L. Franco, G. Compagno, A. Messina, A. Napoli, Efficient generation of Nphoton binomial states and their use in quantum gates in cavity QED, Phys. Lett. A 374 (2007) 2235. [16] Y.H. Ji, J.J. Hu, Preparation of multi-photon Fock states and quantum entanglement properties in circuit QED, Chin. Phys. B 23 (2014) 040307. [17] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin, R.J. Selloelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation, Phys. Rev. A 69 (2004) 062320. [18] Q.Q. Wu, J.Q. Liao, L.M. Kuang, Quantum state transfer between charge and flux qubits in circuit-QED, Chin. Phys. Lett. 25 (2008) 1179. [19] A. Joshi, S.V. Lawande, Generalized Jaynes–Cummings models with a timedependent atom-field coupling, Phys. Rev. A 48 (1993) 2276. [20] J.W. Pan, A. Zeilinger, Greenberger–Horne–Zeilinger-state analyzer, Phys. Rev. A 57 (1998) 2208. [21] G.P. Guo, C.F. Li, G.C. Guo, Scheme for the preparation of multiparticle entanglement in cavity QED, Phys. Rev. A 65 (2002) 042102. [22] A. Joshi, R.R. Puri, S.V. Lawande, Effect of dipole interaction and phaseinterrupting collisions on the collapse-and-revival phenomenon in the Jaynes–Cummings model, Phys. Rev. A 44 (1991) 2135. [23] W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80 (1998) 2245. [24] J.S. Xu, K. Sun, C.F. Li, X.Y. Xu, G.C. Guo, Nat. Commun. 4 (2013) 2851.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effects of linear modulation of qubits–resonator coupling on quantum entanglement in circuit QED Juju Hu a,b,∗ , Qiang Ke a,b a b
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China Key Laboratory of Photoelectronics and Telecommunication of Jiangxi Province, Nanchang, Jiangxi 330022, PR China
a r t i c l e
i n f o
Article history: Received 17 November 2015 Accepted 11 January 2016 Keywords: Circuit quantum electrodynamics Quantum entanglement Linear modulation Fock state
a b s t r a c t We investigate to efficiently maintain the quantum entanglement between qubits through linearly modulating the coupling coefficient between qubits and field in circuit quantum electrodynamics system. We find that the maintenance time of entanglement between coupling qubits is closely related with the modulation type. Compared to the case of constant coupling coefficients, the first disappearance of entanglement between qubits is largely delayed when the coupling coefficients are linearly modulated with a small ramp. However, it is unfavorable for the maintenance of entanglement even leads to fast disappearance if the coupling coefficients are modulated with a large slope. The disappearance time of entanglement not only relies on the quantum state of coupling qubits, but also depends on the average photon number. The more the average photon number is, the more disadvantageous quantum entanglement maintains. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction Quantum entanglement is the most interesting quantum phenomenon in many-body systems. As a physical resource, entangled state plays an important role in many aspects of quantum information such as quantum teleportation, quantum key distribution and quantum computation [1–3]. However, pure entangled state will degenerate to mixed state in real physical world because of environmental decoherent effect. Quantum communication and quantum computation with such mixed entangled state lead to the information distortion [4–6]. Otherwise, the unavoidable interaction between the environment and single qubit leads to exponential decay of the coherences of the qubit while the entanglement between two or more qubits terminates its value in a finite time; a phenomenon termed entanglement sudden death (ESD) [7,8]. Indeed, decoherence is the most important limit for quantum computation since the quantum superposition decays into statistical mixtures [9,10]. It is known that decoherence and disentanglement are closely connected. Therefore, there is an
increasing interest in studying the quantum mechanism of the entanglement dynamics in quantum computation and quantum information. How to generate and transfer entanglement is a problem of fundamental interest in quantum information processing [11], and the greatest barrier is decoherence and disentanglement induced by environment. In order to overcome the influence of decoherence, one of the tactics proposed by the electrical engineers and researchers is to improve the coherent time of the qubits by optimizing the system [12]. The existing research results indicate that quantum entanglement can be manipulated by quantum correction and coherent feedback, and the entanglement sudden death can be effectively delayed [13]. For convenience in experiment operation, manipulating entanglement by classical driving field is an effective method. Recently, Ref. [14] manipulated the coupling coefficient between atom-field in vacuum state and prolonged the entanglement between atoms effectively. Vacuum state is a special case of Fock state. As the most fundamental and important state in quantum optics, the photon number state |n obeys sub-Poisson distribution and constitutes a com-
|n n| = 1 that arbitrary photon state can spread
plete set
n
∗ Corresponding author at: College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China Tel.: +86 18926419186.. E-mail address: [email protected] (J. Hu). http://dx.doi.org/10.1016/j.ijleo.2016.01.081 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
accordingly and the spread is exclusive. In the past few decades, people paid much attention on how to prepare Fock state. Most recently, the method of atom-cavity interactions has been applied in solid-state circuit QED to prepare multi-photon Fock states
J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
3951
In the interaction picture, with the resonant interaction considered, the effective Hamiltonian of the system is expressed as ( = 1)
g aj+ + a+ j−
˜ eff = − H
+ gAB 1+ 2− + 1− 2+
(1)
j=A,B
Fig. 1. Schematic of coupling qubits circuit quantum electrodynamics. A superconducting charge qubit and a flux qubit containing a dc-SQUID are coupled to a transmission line of length l. The charge qubit couples to the transmission line through the gate capacity and the biased flux ˚1 , and the flux qubit couples to the transmission line through the biased flux ˚2 and ˚3 .
and generalized binomial states [15]. The high-Q one-dimensional superconducting transmission line resonator (TLR), similar to the optical resonator, can be used as quantum bus to realize the controllable coupling of several qubits. In Ref. [16], using TLR as data bus, we proposed a scheme to realize multi-photon Fock state in circuit QED. In the proposed scheme, each qubit passes in turn through the circuit cavity via externally biasing flux, and then we implement quantum measurement on the ground state after the qubit passes through the circuit cavity. It effectively avoids the influence of decoherence since each qubit is operated alone. In this paper, we further discuss how to manipulate the coupling between superconducting qubit and field by external classical driving field to effectively manipulate the quantum entanglement in circuit-QED. The paper is organized as follows. In Section 2, we show the model and present the scheme of preparing Fock states in circuit QED. The dynamical behaviors of quantum entanglement for the coupled superconducting qubits in circuit QED are investigated in Section 3. Finally, in Section 4 the obtained results are summarized and discussed.
2. Modulation of the coupling coefficient between qubit and cavity field In this section, we firstly introduce the manipulation of coupling coefficient briefly. The coupled circuit QED system is shown in Fig. 1 [17]. The system consists of a charge qubit, a flux qubit, and the TTR. The TLR can be modeled as a simple harmonic oscillator near its resonance frequency. The charge qubit and the flux qubit are separately placed at l/2 and l/4 of the transmission line, where maximum quantization voltage and current can be achieved. The charge qubit couples to the transmission line through gate capacity on which the coupling strength depends, whereas the flux qubit couples to the transmission line through the external flux. There are two steps in practical operation. First, prepare the high-Q one-dimensional superconducting transmission line resonator (TLR) in Fock state |n. The basic operation is as follows: we completely shut down the external flux ˚1 (that is to shut down the charge qubit), and control the external flux ˚3 by pulse counter and realize the preparation of multi-photon Fock states. Second, place the charge qubit and flux qubit in the resonator as shown in Fig. 1 after the preparation of proper Fock state |n. The charge qubit couples to the transmission line through the gate capacity and the biased flux ˚1 , and the flux qubit couples to the transmission line through the biased flux ˚2 and ˚3 .
where a and a+ are respectively the annihilation and creation operator, k± (k = A, B) represents the Pauli matrix for the qubit, and gAB the interacting strength between two qubits. Both the detailed deduction and assumptions (or conditions) of above-mentioned Hamiltonian is given in Ref. [18]. Different from Ref. [18], we consider the interaction between qubits for the purpose of generality. Particularly, the coupling coefficient g between TLR and qubits (for convenience, we assume gA = gB = g) in the proposed scheme is dependent on the resonant frequency ω, length l, capacitance C and inductance L of the transmission line, as well as the gate capacitance Cg of the charge qubit, which makes the manipulation of quantum correlation more convenient. If we control one parameter, such as the gate capacitance Cg , with an external function f(t), the coupling coefficient g becomes time dependent. Then, utilizing external control function f(t) to simultaneously manipulate the coupling between each superconducting qubit and field g, we rewrite Eq. (1) as: Heff (t) = −
g f (t) aj+ + a+ j− +gAB 1+ 2− +1− 2+ .
(2)
j=A,B
Following Ref. [19], we consider that the cavity field is switched on by a linear ramp described by the function
f (t) =
kt
0≤t≤T
0,
others
,
(3)
where the parameters k and T determine the slope of the linear ramp. As the value of k changes over a fixed T, the slope of the linear ramp changes, too. Note that the equation above incorporates two limiting cases corresponding to sudden and adiabatic changes which can be achieved by only changing the value of k for a fixed T. Utilizing spontaneous parametric down-conversion (SPDC) and cavity quantum electrodynamics system, two-particle Bell state has been prepared experimentally [20,21]. We first suppose that the coupled superconducting qubits are in the Bell state, written as [22]
= cos ↑↓ + sin | ↑↓ · (0 ≤ ≤ ) .
Q (0) I
(4)
By using the Schrödinger equation, the vector of the system can be easily obtained at any time t
I (t) = x1 ↑↑ n − 1 + x2 |↑↓ n + x3 |↓↑ n + x4 ↓↓ n + 1 , (5) where x1 () =
cos + sin
ng
˛
x2 () =
cos + sin
+
x3 () =
√
e−i(+ +− ) ,
2
−
(+ ei+ − − ei− )
cos − sin
2˛
(7)
(+ ei + − − ei− )
cos − sin 2
(6)
2˛
cos + sin
(ei − − ei + ) ,
e−i(+ +− ) ,
(8)
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J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
Fig. 2. Time evolutions of the concurrence versus the scaled time gt with = /6, gAB = 0.1g, n = 1. Dash line k/T = 0.1; dot line k/T = 1; and dash dot line k/T = 5. For comparison, solid line corresponds to the case without external manipulation function f(t) and the coupling between qubit and cavity is constant.
x4 () =
cos + sin
√
n + 1g
˛
1.0
(ei− − ei+ ) ,
(9)
0
0
0
0 0
x4 (n, t)2
⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠ (10)
Where * denotes complex conjugate. Second, we suppose that the coupled superconducting qubits are in the Bell state as follows,
Q (0) II
= cos |↑↑ + sin |↓↓
(11)
By using the Schrödinger equation, the vector of the system can be
II (t)
= y1 (t) |↑↑, n + y2 (t) ↑↓, n + 1 + y3 (t) ↓↑, n + 1
where y1 (n, ) =
+ y4 (t) ↓↓, n + 2 + y5 (t) |↓↓, n ,
(12)
0.4 0.2 0.0 0.0
0.4
0.8
gt
1.2
1.6
2.0
Fig. 3. Time evolutions of the concurrence versus the scaled time gt with = /12, gAB = 0.1g, n = 1. Dash line k/T = 0.1; dot line k/T = 1; and dash dot line k/T = 5. For comparison, solid line corresponds to the case without external manipulation function f(t) and the coupling between qubit and cavity is constant.
II =
⎛ y1 (n, t)2 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎝ y1∗ (n, t) sin
⎞
0
y1 (n, t) sin
y2 (n, t)y3∗ (n, t)
y2∗ (n, t)y3 (n, t)
y3 (n, t)2
0
0
0
y4 (n, t)2 + sin2
0
y2 (n, t)2
0
⎟ ⎟ ⎟. ⎟ ⎟ ⎠
2(2n + 3)
(ei +
(n + 1)gAB i + + ei − ) + − ei − ) (e 2(2n + 3)ˇ 3. Manipulation of quantum entanglement
n+2 cos , 2n + 3
(13)
√ g n + 1 i + y2 (n, ) = y3 (n, ) = − ei − ) cos , (e ˇ
(14)
y4 (n, ) =
0.6
(17)
n+1 +
Concurrence
⎞
0
(t) state
II
0.8
with
2 + 8(2n + 1)g 2 , = −gAB ±˛ , = k t 2 . gAB ˛= ± T 2 According to Eqs. (5)–(9), we can easily obtain the density of the hybrid qubits by tracing the quantum field of TLR:
⎛ x1 (n, t)2 0 0 ⎜ 2 ⎜ x2 (n, t) x2 (n, t)x3∗ (n, t) 0 ⎜ I = ⎜ ⎜ x3 (n, t)2 ⎜ 0 x2∗ (n, t)x3 (n, t) ⎝
in the
(n + 1)(n + 2) 2(3 + 2n)
[(ei + + ei − )
+gAB (ei + − ei − ) − 1] · cos ,
y5 (n, ) = sin , with
2 + 8(2n + 3)g 2 , = −gAB ±ˇ . ˇ= gAB ± 2 According to Eqs. (12)–(16), we get
(15)
(16)
We adopt the concurrence entanglement defined by Wootters to measure the system entanglement [23]. Figs. 2–5 display the entanglement evolution when two qubits are initially in the entangled state. The solid line corresponds to the constant qubit-field coupling, and the other lines exhibit the entanglement evolution under linear modulation of the qubit-field coupling. For constant coupling, the entanglement between two qubits in the quantum state I (t) periodically disappears and takes a while to recover. From Fig. 2, we can see that manipulation of the qubit-field coupling greatly influences the entanglement and effectively modulates its survival time. The first disappearance of entanglement will be delayed significantly if the ramp slope of the coupling coefficient is small, indicating the linear modulation effect is obvious. Increase the ramp slope (the dot line for k/T = 1), the first disappearance of entanglement is still delayed evidently, but lesser than the case of k/T = 0.1. With the increasing of k/T, however, the
1.0
1.0
0.8
0.8
Concurrence
Concurrence
J. Hu, Q. Ke / Optik 127 (2016) 3950–3954
0.6 0.4 0.2 0.0 0.0
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0.6 0.4 0.2
0.2
0.4
gt
0.6
0.8
0.0 0.0
1.0
0.2
0.4
gt
0.6
0.8
1.0
1.0
1.0
0.8
0.8
Concurrence
Concurrence
Fig. 4. Time evolution of the concurrence versus the scaled time gt with = /6, gAB = 0, k/T = 1. Solid line n = 0; dash line n = 1; and dot line n = 3.
0.6 0.4
0.4 0.2
0.2 0.0 0.0
0.6
0.4
0.8
gt
1.2
1.6
2.0
0.0 0.0
0.4
0.8
gt
1.2
1.6
2.0
Fig. 5. Time evolution of the concurrence versus the scaled time gt with = /6, n = 1, k/T = 0.5. Solid line gAB /g = 0; dash line gAB /g = 0.5; dot line gAB /g = 1; and dash dot line gAB /g = 5.
first disappearance of entanglement will be advanced. It not only may be shifted to the time of entanglement disappearing under constant coupling, but also probably lead to entanglement death between qubits. The results suggest that too large a value of k/T would be counterproductive although it is effective to adjust the quantum entanglement between qubits by linearly modulating the coupling between qubit and field. Above analysis show that the influence of the time-dependent qubit-field coupling coefficient is nonlinear and relates with the initial quantum state, photon number in the field, and interaction between qubits. It is known that the evolution of entanglement is closely related with the initial quantum state. Entanglement sudden death is entirely avoidable via proper choice of initial state. Based on the parameters in Fig. 2 and the initial state Q (0) = cos(/6) |↑↑ + II sin(/6) |↓↓, ESD absolutely does not occur to the entanglement between qubits. However, if the initial state is changed, for example, the coupling qubits are initially in Q (0) while the initial II degree of entanglement is changed, such as = /12, then ESD shown in Fig. 3 will occur to the entanglement between qubits as time evolves. The entanglement time is effectively delayed through linearly modulating the qubit-field coupling coefficient. From Figs. 2 and 3, we find that the entanglement between qubits under linear modulation no longer evolves periodically. Eqs. (5) and (11) show that the degree of entanglement between qubits is related to the square of time(periodic function of time under constant coupling coefficient), which accounts for the aperiodic evolution of entanglement between qubits when the coupling coefficient is modulated by the external function. In the Fock state field, collapse and revival occurs to the time evolution of the concurrence, and more intense oscillations appear with the increasing of the photon number. Therefore, from Fig. 4, entanglement sudden death appears as time evolves. The photons in the cavity shorten the evolution period of the entanglement between qubits and make them in disentangled state for a long time. When there are photons in the cavity and each coefficient
is nonzero, with time evolving, some coefficients can be neglected since they are very small, and then the disentangled state or maximum entangled state is obtained. It may be explained as follows, n plays the role of coupling strength between two qubits and the circuit cavity, large n is equivalent to strong coupling between two subsystems. Once we only consider two qubits by tracing out the circuit cavity as the environment, the large value of n means that two qubits suffer strong dissipation. It indicates that decreasing the average photons number will be beneficial for keeping the quantum correlation between qubits. For similar reasons, the photon number in cavity greatly weakens the linear manipulation effect. Instead, it can be seen from Fig. 5 that the interaction between qubits is conductive to strengthen the linear manipulation of external function on the quantum entanglement. 4. Conclusions We investigate the quantum entanglement dynamics of circuitQED in Fock state, and focus on the dynamical behavior of quantum entanglement when the coupling coefficient between qubit and cavity field is linearly modulated. The following results are obtained: (1) in the case of constant coupling between qubit and cavity, collapse and revival occurs to the quantum entanglement between qubits; (2) through linearly modulating the coupling coefficient between qubits by an external linear manipulation function of small amplitude, the coupling coefficient between qubit and cavity is effectively reduced so that the entanglement effect between qubits is greatly improved and the first disappearance time of entanglement is delayed. Further investigations also indicate that the initial disappearing time of entanglement is related to the average photon number. The more average photons there are in the cavity, the more disadvantageous to maintain quantum entanglement. It should reinforce that classical noise is inevitably introduced in the process of implementing external modulation, and quantum correlation will recover in this environment under certain conditions, which has been verified by experiment [24].
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