Quantum criticality of quantum speed limit for a two-qubit system in the spin chain with the Dzyaloshinsky–Moriya interaction

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Quantum criticality of quantum speed limit for a two-qubit system in the spin chain with the Dzyaloshinsky–Moriya interaction Shaoying Yin a,b , Jie Song a , Shutian Liu a,∗ a b

Department of Physics, Harbin Institute of Technology, Harbin 150001, China Department of Physics, Harbin University, Harbin 150086, China

a r t i c l e

i n f o

Article history: Received 30 July 2018 Received in revised form 29 September 2018 Accepted 22 October 2018 Available online xxxx Communicated by M.G.A. Paris Keywords: Open quantum system Quantum speed limit Spin chain Dzyaloshinsky–Moriya interaction

a b s t r a c t We study the quantum speed limit (QSL) time of a two-qubit system coupled to a spin–chain model with the Dzyaloshinsky–Moriya (DM) interaction. For the Bell state coupled to the Ising model or anisotropic XY model, we find that there is a prominent corresponding relationship between the QSL time and quantum phase transition in a spin–chain environment with larger scale, and the DM interaction can strongly enhance or suppress the response relation. Remarkably, when the surrounding environment is set to the XX model, the DM interaction makes it possible for us to witness the quantum phase transition by the local anomalous enhancement of the QSL time near the critical point. In addition, our analyses indicate that the entanglement can speed-up the system evolution in many-body environment. © 2018 Elsevier B.V. All rights reserved.

1. Introduction A kind of phase transitions, driven by quantum fluctuations due to the Heisenberg uncertainty principle, generally occur at absolute zero temperature, which is the so-called quantum phase transitions [1] (QPTs). QPTs can be occurred by changing environmental parameters or coupling constant, and which lead to qualitatively distinct properties of the matter. An investigation on QPTs can contribute to improving the understanding of the physical properties of various matters from the quantum perspective. Hence, further study on the QPTs is a significant research direction in condensed matter physics. In view of the QPTs stemming from quantum uncertainty, many researchers in the field of quantum information science have payed more attention to it. Several physical quantities used extensively in the field of quantum information science are successfully used to deepen the understanding the QPTs, such as quantum correlations [2–9], quantum coherence [10–16], non-Markovianity [17,18]. Quantum speed limit (QSL) time, depended on the Heisenberg uncertainty relation of energy and time, is the minimum evolution time which a system needs to evolve from an initial state to its final state. The determination of the minimal duration of a process is utmost importance for quantum information science and technology, such as quantum optimal control protocols [19,20],

*

Corresponding author. E-mail address: [email protected] (S. Liu).

https://doi.org/10.1016/j.physleta.2018.10.027 0375-9601/© 2018 Elsevier B.V. All rights reserved.

quantum metrology [21], quantum communication [22], quantum computation [23], and many other fields of quantum physics. Recently, the relation between QPTs and quantum speed limit (QSL) has also drawn considerable interest from researchers [18,24–27]. Several researchers have found that the QSL time has some abnormal behaviors in QPT point for a central qubit coupled to an XY spin–chain environment [18] or a Lipkin–Meshkov–Glick bath [25]. Therefore, to understand clearly the quantum critical behavior of the QSL time is very important for developing the quantum technology in many-body environment. On the other hand, the antisymmetric Dzyaloshinsky–Moria (DM) interaction is often present in the models of many low-dimensional magnetic materials, such as the Heisenberg XY spin–chain. The DM exchange interaction was introduced by Dzyaloshinsky [28] to indicate the weak ferromagnetic behavior in the anti-ferromagnetic crystals and was later concluded theoretically by Moriya [29]. The DM interaction arising from spin–orbit coupling can generate many spectacular influences, especially in spin model [15,16,30–33]. In this work, we investigate the impact of DM interaction on the quantum criticality of the QSL time for a two-qubit system in spin chain environment. We find that the DM interaction can effectively modulate the quantum critical behavior of the QSL time, and hence the QSL time can signal clearly the critical point of QPT in one-dimensional planar spin models based on different anisotropy parameters. Such investigation can help us to study quantum phase transition in a wider class of models with the spin–orbit coupling instead of the spin–spin interaction.

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The paper is organized as follows. In Section 2, we introduce the physical model and derive the reduce density matrix of the central two-qubit system. Whereafter, we present a bound of the QSL time for the minimal evolution time of an open quantum system. In Section 3, we investigate the impact of DM interaction on the quantum criticality of the QSL time for two-qubit system in XY spin environment. Our conclusions are given in Section 4. 2. Physical model and quantum speed limit 2.1. Physical model

N   1+γ

2

l

σlx σlx+1 +

1−γ 2

y σl y σl+ + λσlz 1

  y y + D σlx σl+1 − σl σlx+1 , HI = −

g 2

σ Az + σ Bz



N 

σlz ,

λμ

HE =

λμ

Ek

†

bk,λ bk,λμ − μ

k

1 2 λμ

where the energy spectrum E k

γ 2 sin2

2π k 1/ 2 N



(2)

,

  λμ k 2 reads E k = 2 λμ − cos 2π + N

+ 4D sin 2πN k . The initial density matrix of two-

qubit system and the environmental spin chain can be expressed by the product state as ρ (0) = ρ E (0) ⊗ ρ S (0), where ρ E (0) = |φ E (0) φ E (0)| and ρ S (0) = |φ S (0) φ S (0)| are the initial density matrix of the environment and the central system, respectively. The subsequent time evolution of the total system is determined by ρ (t ) = U (t ) ρ (0)U † (t ), where U (t ) is the time evolution operator. Then, the reduced density matrix of the two-qubit system can be obtained by tracing over the environment, which can be obtained with the following equation 

λμ

ρ S (t ) = φ E | U †E(λν ) (t ) U E (λμ )

where U E





(t ) |φ E  ρ S (0) , λ

op

τ D

op ,tr ,hs

,

1

1

, hs tr τ D τ D







sin2 L(ρ0 , ρτ D ) ,

(4)

τ

where τ D = (1/τ D ) 0 D dt  L t (ρt ) op ,tr ,hs ,  L t (ρt ) op ,tr ,hs represent the operator norm, trace norm, and Hilbert–Schmidt norm, respectively. This formula is convenient for obtaining the quantum speed limit time for a fixed initial time with different dynamical time τ D .

(1)

l



τ Q S L = max

1

3. The effect of DM interaction on the quantum criticality of QSL time

where H E denotes the Hamiltonian of the environment, and H I represents the interaction Hamiltonian between the two-qubit system and the spin–chain environment. σlα (α = x, y , z) are the Pauli matrices denoting the l-th site of the chain, σ Az ( B ) is the spin operator along z direction for two-qubit system. The factor D denotes the intensity of DM interaction along z direction, which typically varies from −1 to 1. γ is the anisotropy of exchange interaction in the xy plane. By means of the anisotropy parameter, we can divide the spin–chain model into three well-known models. They are the Ising spin chain with γ = 1, the XX chain with γ = 0, and the general anisotropic spin chain with 0 < γ < 1. λ is the strength of the external magnetic field, N is the total number of sites, and g describes the coupling strength between the system and surrounding chain. Based on the Eq. (1), we can diagonalize the environmental Hamiltonian H E by use of the Jordan–Wigner, Fourier, and Bogoliubov transformations in sequence, the final diagonalized Hamiltonian is expressed by



In the following, we introduce a unified expression of the quantum speed limit time for generic time-dependent (positive) dynamics of open quantum system. Based on the Bures angle L(ρ0 , ρτ D ) = arccos ψ0 |ρτ D |ψ0 , von Neumann trace inequality, and Cauchy–Schwarz inequality, Deffner et al. [35] obtain the minimal evolution time from an initial state to final state by combining the results of Mandelstam–Tamm [36] and Margolus–Levitin [37]. The specific expression is



The model under consideration is a two-qubit system coupled to a Heisenberg XY chain with DM interaction in an external magnetic field. The total Hamiltonian [3,15,30,34] is H = H E + H I with

HE = −

2.2. Quantum speed limit

(t ) = exp −i H Eμ t



(3)

is the effective time evolution †(λν )

operator. We define F μν (t ) = φ E |U E 1, 2, 3, 4) as the decoherence factor.

(λμ )

(t )U E

(t )|φ E  (μ, ν =

In the section, we will mainly study the impact of DM interaction on the quantum critical phenomenon of QSL time for a two-qubit system coupled to the XY spin–chain model. We assume that the initial state of the √ central two-qubit system is the Bell state (|φ = (|00 + |11)/ 2 ). According to Eq. (4), the QSL time of the central system can be expressed as

τQ S L =

1 2





∗ (τ ) 2 − F 14 (τ D ) − F 14 D

 τD 1 ˙ τ D 0 | F 14 (t ) |dt

,

(5)

where F 14 (t ) is the decoherence factor of central two-qubit density matrix, F 14 (τ D ) is the decoherence factor at t = τ D . In order to better investigate the effect of DM interaction, we divide the spin chain model into three cases by the anisotropy parameter γ , which are Ising model with γ = 1, anisotropic case with 0 < γ < 1, and isotropic case with γ = 0. 3.1. Ising model with γ = 1 When the anisotropy parameter is set to unity, the spin–spin interaction is only in the σ x direction. Thus, the spin chain becomes Ising model in this case. For one-dimensional Ising model, scientists have obtained the rigorous analytical solution. In the following, we study the impact of DM interaction on the quantum criticality of QSL time for two-qubit system in the Ising model. In Figs. 1(a), (b), and (c), we plot the QSL time as a function of magnetic field λ in the weak coupling regime with different DM interactions. In Fig. 1(a), without DM interaction, the QSL time decreases strongly in an oscillatory way with the increasing magnetic field, but there is a local anomalous enhancement of the QSL time near the critical point λc = 1. Remarkably, in Fig. 1(b), when we set the DM interaction to 0.8, this corresponding relationship is strongly influenced by DM interaction from two aspects, (i ) the QSL time is magnified from 0.063 to 0.104 at the critical point (nearly 1.6 times), (ii) the waveform of anomalous enhancement becomes sharper. In a word, the DM interaction makes the corresponding relationship more obvious, and the abnormal increase of QSL time near λc = 1 can be used to witness the QPT of spin chain. However, in Fig. 1(c), when the DM interaction is set to −0.8, the anomalous enhancement of QSL time near λc is suppressed, which makes the corresponding relationship disappear. In order to understand clearly the impact of DM interaction on the quantum

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Fig. 2. The QSL time as a function of the external magnetic field strength λ with the DM interaction D = 0.8. The other parameters are set to N = 101, γ = 1, g = 0.05, and τ D = 10.

Fig. 1. The QSL time as a function of the external magnetic field strength λ with different DM interactions (a) D = 0.0, (b) D = 0.8, and (c) D = −0.8, where these insets show more detailed information of the QSL time near the λc = 1. (d) The QSL time as a function of the DM interaction with the magnetic field strength λc = 1. The other parameters are set to N = 601, γ = 1, g = 0.05, and τ D = 10.

criticality of QSL time, we plot the QSL time as a function of DM interaction in Fig. 1(d). When DM interaction D ∈ (−0.7, 1.0), the critical value of QSL time near λc almost increases linearly with the increasing DM interaction. It is well known that entanglement can enhance the evolution speed in some particular situations [38–40]. The effect of the DM interaction on the critical behavior of QSL time can be explained by means of the entanglement. In quantum critical point of the spin–chain environment, the entanglement of two-qubit system will become smaller with the increasing DM interaction along the z direction ( D > 0). However, when the DM interaction is imposed along the opposite z direction ( D < 0), the entanglement of the central system will become larger with the increasing of the value of | D |, and when | D | approaches the range of (0.7, 1.0), the entanglement reaches to a maximum value and remains unchanged [30]. Thus, the DM interaction indirectly affects the quantum critical behavior of QSL time by the entanglement of two-qubit system. The demonstration about entanglement can accelerate the dynamical evolution of the system in many-body environment can see the Fig. 4. The size of spin chain N is an important parameter, which has a remarkable effect on the dynamical evolution of quantum correlations and coherence. Thus, we study the quantum criticality of QSL time of the two-qubit system in the Ising model with a smaller size. The results indicate that there is no obvious corresponding relationship between the QSL time and QPT except the stronger oscillation, which is not enough to detect exactly a QPT. In order to enhance the quantum critical phenomenon of the QSL time, the DM interaction is taken into consideration, and we plot the QSL time as a function of the magnetic field as the size N = 101 in Fig. 2. Although we add the spin–orbit interaction ( D = 0.8), there is still not evident quantum critical phenomenon near the QPT point. In consequence, the relationship between the QSL time and the QPT will become more statistically significant for larger size of spin chain. 3.2. Anisotropic case with 0 < γ < 1 When the anisotropy parameter γ ∈ (0, 1), the XY spin chain becomes anisotropic case. Without loss of generality, we set the anisotropy parameter γ to 0.5, and plot the QSL time as a function of magnetic field in the Fig. 3 (a). We can see that, from Fig. 3 (a),

Fig. 3. (a) The QSL time as a function of the external magnetic field strength λ without the DM interaction, the inset shows more detailed information of the QSL time near the λc = 1. (b) The QSL time as a function of the DM interaction with the magnetic field strength λc = 1. The other parameters are set to N = 601, γ = 0.5, g = 0.05, and τ D = 10.

Fig. 4. (a) The QSL time as a function of the anisotropic parameter γ . (b) The entanglement of two-qubit system as a function of the anisotropic parameter γ . The other parameters are set to N = 601, λc = 1, g = 0.05, τ D = 10, and D = 0.0.

the QSL time becomes shorter in an oscillatory way with the increasing magnetic field, there is still an anomalous enhancement of the QSL time near the critical point λc = 1. The corresponding relation between the QSL time and QPT still exists. The QSL time oscillates strongly far from the critical point when λ ∈ (0, 1), and while λ ∈ (1, 2), the oscillation becomes weaker. It is natural to ask how the DM interaction affect the quantum criticality of QSL time. Fig. 3 (b) shows that the QSL time as a function of DM interaction in the QPT point, surprisingly, there is a minimal value of QSL time as the DM interaction approaches to −0.4. In fact, the impact of the DM interaction on the critical behavior of QSL time can still be explained by means of entanglement. In a word, when DM interaction is smaller than 0, it suppresses the corresponding relationship between the QSL time and QPT in quantum critical point, while DM interaction is larger than 0, it will strengthen this relationship. For the anisotropic spin–chain, the anisotropy parameter γ ∈ (0, 1). Above analyses only concentrate on γ = 0.5, the subsequent work is to investigate the impact of anisotropy parameter γ on the quantum critical behavior of QSL time in spin chain environment. We plot the QSL time as a function of the anisotropy parameter in QPT point in Fig. 4 (a). The QSL time firstly undergoes a rapid increase and a following slow decline, it exists a maximum value as

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Fig. 5. The QSL time as a function of the external magnetic field strength λ with the γ = 0 (the XX model). The curves in (a) and (b) correspond to different DM interaction (a) D = 0.0 and (b) D = 0.8, the insets show more detailed information of the QSL time near the λc = 1. The other parameters are set to N = 601, γ = 0, g = 0.05, and τ D = 10.

the anisotropy parameter approaches to γ = 0.125. In order to find out the reason for such change, we employ the concurrence [41] to measure entanglement for two-particle system in spin chain environment, and plot the entanglement as a function of the anisotropy parameter in the Fig. 4 (b). We find that the entanglement firstly has a rapid decay and a following slow growth, whose change is highly negatively correlated with that of QSL time. This significant corresponding relationship indicates that entanglement can reduce the QSL time for two-particle state in spin environment. 3.3. The XX chain with γ = 0 When the anisotropy parameter γ = 0, the Heisenberg XY model becomes an isotropic model, which is so-called XX model. In this model, we will study the variation of QSL time with the increasing external magnetic field, and the effect of DM interaction on the QSL time in quantum critical point. In Fig. 5 (a), we plot the QSL time as a function of the magnetic field. There are two notable features, (i ) the oscillatory way for QSL time near the critical point is the same as that of QSL time around the critical point, i.e., we cannot detect the QPT by the distinctive oscillatory way for QSL time near the critical point. (ii) the QSL time becomes constant as the γ > 1.05. The above analyses in the Ising model and the anisotropic cases show that the DM interaction is helpful for using QSL time to detect QPT, and hence we investigate the impact of DM interaction on the quantum-critical behavior of QSL time. Interestingly, we find in Fig. 5 (b) that the amplitude of the oscillation abruptly becomes larger near the QPT point, and becomes largest at the QPT point. Thus, in XX model, the quantum correlations, the Loschmidt echo and the non-Markovianity [18] are incapable of signaling the QPT, while the DM interaction makes it possible for QSL time to witness the QPT. 4. Conclusion We study the significant effect of DM interaction on the quantum criticality of the QSL time in a composite system, which includes a Bell state and its surrounding spin–chain environment. When we set the Ising model with γ = 1 and the anisotropic XY model with 0 < γ < 1 as the surrounding environment, we find that the QSL time is a potential candidate for a identification of QPT in spin–chain model with larger size. Remarkably, the corresponding relation between QSL time and QPT will become more prominent or disappear in view of the different direction of the anti-symmetric DM interaction along z axis. However, for the spin model with smaller scale, there is no an obvious critical behavior for QSL time in the QPT point in spite of considering the DM interaction. We also study the impact of the anisotropy parameter γ on the QSL time and entanglement of Bell state in quantum critical point. Base on their quantum critical behaviors, we find that

the entanglement of a central two-qubit system can speed-up the system evolution in many-body environment. On the other hand, when the surrounding environment is set to be the XX spin–chain model with γ = 0, the QSL time of the Bell state is incapable of signaling the QPT point. However, the amplitude of QSL time has an anomalous enhancement near the critical point and a local maximum at QPT point as we consider the DM interaction. In a word, our results will deepen our understanding for the quantum criticality of the QSL time in a many-body environment, which are important for us to study the development of quantum speed limits and the quantum properties in condensed matter physics. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant Nos. 61575055, 61377016, 11104049 and 10974039, and Specialized Research Fund for the Doctoral Program of Higher education (Grant 20102302120009), and the Program for New Century Excellent Talents in University (NCET12-0148), and Special fund project of Harbin Science and Technology innovation talents research (RC2017QN017004). References [1] S. Sachdev, Quantum Phase Transition, Cambridge University Press, Cambridge, UK, 1999. [2] A. Osterloh, L. Amico, G. Falci, R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416 (2002) 608–610. [3] Z.G. Yuan, P. Zhang, S.S. Li, Disentanglement of two qubits coupled to an XY spin chain: role of quantum phase transition, Phys. Rev. A 76 (2007) 042118. [4] Z.G. Yuan, P. Zhang, S.S. Li, Loschmidt echo and Berry phase of a quantum system coupled to an XY spin chain: proximity to a quantum phase transition, Phys. Rev. A 75 (2007) 012102. [5] S.J. Gu, S.S. Deng, Y.Q. Li, H.Q. Lin, Entanglement and quantum phase transition in the extended Hubbard model, Phys. Rev. Lett. 93 (2004) 086402. [6] T. Werlang, C. Trippe, G.A.P. Ribeiro, G. Rigolin, Quantum correlations in spin chains at finite temperatures and quantum phase transitions, Phys. Rev. Lett. 105 (2010) 095702. [7] Y.C. Li, H.Q. Lin, Thermal quantum and classical correlations and entanglement in the XY spin model with three-spin interaction, Phys. Rev. A 83 (2011) 052323. [8] S.Y. Liu, Y.R. Zhang, L.M. Zhao, W.L. Yang, H. Fan, General monogamy property of global quantum discord and the application, Ann. Phys. 348 (2014) 256–269. [9] S.Y. Yin, Q.X. Liu, J. Song, X.X. Xu, K.Y. Zhou, S.T. Liu, Quantum correlations dynamics of three-qubit states coupled to an XY spin chain: role of coupling strengths, Chin. Phys. B 26 (2017) 100501. [10] G. Karpat, B. Cakmak, F.F. Fanchini, Quantum coherence and uncertainty in the anisotropic XY chain, Phys. Rev. B 90 (2014) 104431. [11] W.W. Cheng, Z.Z. Du, L.Y. Gong, S.M. Zhao, J.M. Liu, Signature of topological quantum phase transitions via Wigner–Yanase skew information, Europhys. Lett. 108 (2014) 46003. [12] A.L. Malvezzi, G. Karpat, B. Cakmak, F.F. Fanchini, T. Debarba, R.O. Vianna, Quantum correlations and coherence in spin-1 Heisenberg chains, Phys. Rev. B 93 (2016) 184428. [13] Y.C. Li, H.Q. Lin, Quantum coherence and quantum phase transitions, Sci. Rep. 6 (2016) 26365. [14] N.J. Hui, Y.Y. Xu, J.C. Wang, Y.X. Zhang, Z.D. Hu, Quantum coherence and quantum phase transition in the XY model with staggered Dzyaloshinsky–Moriya interaction, Physica B 510 (2017) 7–12. [15] C. Radhakrishnan, I. Ermakov, T. Byrnes, Quantum coherence of planar spin models with Dzyaloshinsky–Moriya interaction, Phys. Rev. A 96 (2017) 012341. [16] Z.M. Huang, H.Z. Situ, C. Zhang, Quantum coherence and correlation in spin models with Dzyaloshinskii–Moriya interaction, Int. J. Theor. Phys. 56 (2017) 2178–2191. [17] P. Haikka, J. Goold, S. McEndoo, F. Plastina, S. Maniscalco, Non-Markovianity, Loschmidt echo, and criticality: a unified picture, Phys. Rev. A 85 (2012) 060101. [18] Y.B. Wei, J. Zou, Z.M. Wang, B. Shao, Quantum speed limit and a signal of quantum criticality, Sci. Rep. 6 (2016) 19308. [19] G.C. Hegerfeldt, High-speed driving of a two-level system, Phys. Rev. A 90 (2014) 032110. [20] A.D. Cimmarusti, Z. Yan, B.D. Patterson, L.P. Corcos, L.A. Orozco, S. Deffner, Environment-assisted speed-up of the field evolution in cavity quantum electrodynamics, Phys. Rev. Lett. 114 (2015) 233602.

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