Physica B 414 (2013) 26–29

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Physica B

0921-45

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n Tel.:

E-m

journal homepage: www.elsevier.com/locate/physb

Collapse and revival of entanglement of two-qubit in superconductingquantum dot lattice with magnetic flux and inhomogeneous gate voltage

Sujit Sarkar n

PoornaPrajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 5600 80, India

a r t i c l e i n f o

Article history:

Received 18 July 2012

Received in revised form

4 December 2012

Accepted 21 December 2012Available online 15 January 2013

Keywords:

Superconducting qubits

Quantum entanglement physics

Concurrence

26/$ - see front matter & 2013 Elsevier B.V. A

x.doi.org/10.1016/j.physb.2012.12.041

þ91 80 23617465.

ail address: sujit.tifr@gmail.com

a b s t r a c t

We study the entanglement of a two-qubit system in a superconducting quantum dot (SQD) lattice in

the presence of magnetic flux and gate voltage. The ground state is always in a maximally entangled

Bell state for homogeneous gate voltage. In the presence of inhomogeneous gate voltage, the half-

integer magnetic flux quantum, completely washes out the entanglement of the system at zero

temperature. The entanglement is much higher for the Mott insulating phase. At finite temperature,

collapse of entanglement occurs for wider region of magnetic flux.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

A very intense research activities have started in recent yearsinvolving an interaction between the subjects of quantum infor-mation science and quantum many body physics [1–5]. Quantumentanglement is the pivot of quantum information science, a trulyunique feature of quantum mechanical systems, it has no classicalanalogue. Entanglement implies non-local correlations betweenparticles in quantum many body systems. The quantum phasetransition is one of the elegant phenomena of quantum manybody system associate with different physical phenomena of thatsystem. It has been observed recently that the quantum phasetransition of the system is also associated with the quantumentanglement physics of the system and it has been observed thatthe entanglement of the quantum system extends over themacroscopic distances as ordinary correlation do. For example,let us consider a many body interacting quantum spin system.The ground state wave function changes qualitatively in aquantum phase transition. Therefore, one should be curious tosee how the quantum entanglement changes as the transitionpoint is traversed. The fidelity is an important concept to measurethe quantum entanglement in the system. The fidelity typicallydrops in an abrupt manner at a quantum critical point indicatinga dramatic change in the nature of the ground state wave function[6–11].

We have been motivated by the extensive studies of entangle-ment physics and interesting results in spin system [12–25]. We

ll rights reserved.

decided to apply this concept of study in the different disciplinesof quantum condensed matter many-body system. Here we studythe entanglement physics of two qubit in superconducting quan-tum dot lattice. Before we proceed further, we would like todiscuss the basic aspects of a superconducting quantum dotlattice: Superconducting quantum dot lattice consists of array ofsuperconducting grains. The superconducting grains are of nanos-cale size, and different states of the superconducting grain arecontrolled by the ratio between charging energy and Josephsonenergy, while the average charge of the dot is controlled by thegate voltage [26–32] lattice. Superconducting circuits (qubitlattice) have attracted considerable interest in the recent yearsowing to the interesting physical properties. Superconductingcircuits are macroscopic in size but have generic quantumproperties such as quantized energy levels, superposition of statesand entanglement. Superconducting circuits are generally hun-dreds of nanometer wide and contains trillions of electrons butthey posses quantum nature. The quantum nature of this circuit isobservable because they can be represented by a single degree offreedom. This most attractive feature of these superconductingcircuits has opened a new area of fundamental science and theother part is the long term potential for quantum computing [5].There are three types of superconducting qubits, namely, charge,flux and phase qubits. These qubits are operating based on twofundamental properties of superconductors such as flux quantiza-tion and Josephson tunneling [33–36].

One can consider the Cooper pair of SQD as a charged boson.The Physics of SQD can therefore be described in terms ofinteracting bosons. Bosonic physics is more interesting and hardto understand than the fermionic physics. At the same timequantum phase diagram of this SQD lattice is very rich with

S. Sarkar / Physica B 414 (2013) 26–29 27

different quantum phases. Therefore the study of the entangle-ment physics for the SQD lattice for both zero and finitetemperature is interesting in its own right [25–27].

It is well known from our previous studies that mesoscopicSQUID array can also be treated as the superconducting quantumdot lattice with modulated Josephson junction [25,26]. Theauthors of Ref. [29] have found the magnetic flux induced super-conducting Coulomb blocked in mesoscopic SQUID array and alsothe magnetic flux induced superconductor–insulator quantumphase transition. Experimentally and also theoretically it revealsthat the applied magnetic flux has an important effect in the SQDlattice system. We will see in due course of our study that theinhomogeneity of the gate voltage plays an important role inentanglement to disentanglement (product state) transition.Therefore we are motivated to study entanglement physics ofSQD lattice in the presence of applied magnetic flux and inho-mogeneity of gate voltage. Here we consider the two qubits ofSQD lattice which is sufficient to predict entanglement to disen-tanglement transition of our system. At the same time ourapproach is completely analytical and there is no approximationsto study this two qubits problem. We will see that this minimalmodel contains many important results. In the previous literature,there are few studies for two qubits for different physical systems[4,24,25]. The plan of the paper is as follows. We present themodel Hamiltonian and entanglement physics in Section 2 of thispaper. We present summary and conclusion in Section 3 of thepaper.

2. Model Hamiltonian for inhomogeneous superconductingquantum dot lattice and the study of entanglement physics

2.1. Model Hamiltonian and ground state analysis

At first we write down the model Hamiltonian of SQD latticesystem with Josephson couplings having on-site charging ener-gies and inter-site interactions in the presence of gate voltage andexternal magnetic flux. We also consider the inhomogeneity inthe applied gate voltage. The Hamiltonian is written as

H¼HJ1þHEC0þHEC1: ð1Þ

We recast different parts of the Hamiltonian in quantum phasemodel as

HJ1 ¼�EJ1 cos p FF0

� ���������X

i

cosðfiþ1�fiÞ,

where fi and fiþ1 are quantal phase of the SQD at the point i andiþ1 respectively, as

HEC0 ¼EC0

2

Xi

�i@

@fi

�Ni

2

� �2

,

where EC0 is the on-site charging energy. Now

HEC1 ¼ EZ1

Xi

niniþ1,

where EZ1 is the NN charging energies between the dots respec-tively. In the phase representation, ð�i@=@fiÞ is the operatorrepresenting the number of Cooper pairs at the ith dot, and thusit takes only the integer values (ni). Here, Hamiltonian HEC0

accounts for the influence of gate voltage ðeN� VgÞ, where eN isthe average dot charge induced by the gate voltage. When theratio EJ1=EC0-0, the SQD array is in the insulating state having agap of the width � EC0, since it costs an energy � EC0 to changethe number of pairs at any dot. The exceptions are the discretepoints at N¼ ð2nþ1Þ, where a dot with charge 2ne and 2ðnþ1Þehas the same energy because the gate charge compensates the

charges of extra Cooper pair in the dot. On this degeneracy point,a small amount of Josephson coupling leads the system to thesuperconducting state.

Here we recast our basic Hamiltonians in the spin language,where each site of the dot is either empty or singly occupied.During this process we follow Refs. [26,30]. Now

HJ1 ¼�2EJ1 cos p FF0

� ���������X

i

ðSyi S�iþ1þh:c:Þ,

and

HEC0 ¼EC0

2

Xi

ð2SZi �hÞ2,

HEC0 ¼�2EC0

Xi

hiSZi :

Here hi ¼ ðNi�2n�1Þ=2 allows the tuning of the system aroundthe degeneracy point by means of gate voltage. We can tune thegate voltage in such a way that we can generate inhomogeneity inon-site charging energy. Without loss of generality we can alsowrite the model Hamiltonian as

HEC0 ¼X

i

ðEC0þdVgÞSzi þX

i

ðEC0�dVgÞSziþ1: ð2Þ

where dVg is the variation of gate voltage around the lattice sites.HEC1 ¼ EZ1

PiS

zi Sz

iþ1.The total Hamiltonian of the system is

H¼ 2EJ1 cos p FF0

� ���������X

i

ðSyi S�iþ1þh:cÞþEZ1

Xi

Szi Sz

iþ1

þX

i

ðEC0þdVgÞSzi þX

i

ðEC0�dVgÞSziþ1: ð3Þ

Now we consider the Hamiltonian for N¼2 case. We would like towrite the Hamiltonian in the standard basis, 91,1S, 91,0S, 90,1S,90,0S

H¼

EZ1þEC0 0 0 0

0 �EZ1þdVg B 0

0 B �EZ1�dVg 0

0 0 0 EZ1�EC0

0BBBB@

1CCCCA

B¼ 2EJ19cosðpF=F0Þ9. The eigenstates of these two Hamiltonian

sites are 9c1S¼ 90,0S, 9c1S¼ 91,1S, 9c3S¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ð1�c2

1Þ

qðc19

1,0Sþ90,1SÞ, 9c4S¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=ð1þc2

2Þ

qðc291,0S�90,1SÞ. Where c1 ¼

ðdVg�AÞ=B, A¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidV2

gþ4E2J 9cosðpF=F0Þ9

2q

, c2 ¼ ðdVgþAÞ=B. E1 ¼12

ð2EZ1�2EC0Þ, E2 ¼12 ð2EZ1þ2EC0Þ, E3 ¼�EZ1�A E4 ¼�EZ1þA. If we

consider the homogeneous system, i.e., there is no variation of

gate voltage over the lattice sites. The two states 9c3S and 9c4S

are the maximally entangled Bell states, i.e., ð1=ffiffiffi2pÞð90,1S�91,0SÞ,

ð1=ffiffiffi2pÞð90,1Sþ91,0SÞ. As we see from our analytical expression

that ground state depends on the value of EC0, EZ1 and A. Groundstate is in the disentangle state (product state) when the groundstate energy is either E1 or E2, otherwise the system is in theentangle state. Thus for this superconducting quantum dot latticesystem there is a transition between the disentangle state toentangle state due to the variation of the system parameters. Wewill see in due course of our study that the magnetic flux plays animportant role in the transition between the disentangled state tothe entangle state.

2.2. Entanglement study for zero and finite temperature

Now we calculate the thermal entanglement of two arbitraryqubits (N¼2) of the superconducting quantum dot lattice. The

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Magnetic Flux

Con

curr

ence

Fig. 1. Concurrence vs. magnetic flux at zero temperature for different values of

dVg ¼ 0:0, 0:2, 0:5 for the dashed bullet and þ signs respectively. Here

EJ1 ¼ 1:0, EC0 ¼ 1:0 and EZ1 ¼ 0:4.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Magnetic Flux

Con

curr

ence

Ej1=1Ej2=2Ej3=3

Fig. 2. Concurrence vs. magnetic flux at zero temperature for different values of

Josephson couplings. Here EJ1 ¼ 1:0, dVg ¼ 0:5 and EZ1 ¼ 0:5.

S. Sarkar / Physica B 414 (2013) 26–2928

density matrix of superconducting quantum dot lattice in equili-brium at temperature, T is r¼ ð1=ZÞexpð�H=kBTÞ, where H is theHamiltonian of the system, Z is the partition function of thesystem, kB is the Boltzmann constant. We calculate the concur-rence to measure the entangle of two qubit system of super-conducting quantum dot lattice following the reference ofWootters’ formula [14,15]. The analytical expression for theconcurrence is

C ¼max 0,2 max li�X

i

li

!, ð4Þ

where li is the square roots of the eigenvalues of the matrixR¼ rðsy

1 � sy2Þr

nðsy1 � sy

2Þ. rn is the complex conjugate of r. The

system is maximally entangled when C ¼ 1 and the system is indisentangle state when C¼0.

The density matrix of the system is

r¼

A1 0 0 0

0 B2 C2 0

0 C2 C3 0

0 0 0 D4

0BBBB@

1CCCCA:

The square roots of the eigenvalues of the matrix R are l1 ¼

l2 ¼ffiffiffiffiffiffiffiffiffiffiffiA1D4

p, l3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2C3þC2

pand l4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2C3�C2

p, where A1 ¼

ð1=ZÞe�EZ1=T ½coshðEC0=TÞ�sinhðEC0=TÞ D4 ¼ ð1=ZÞe�EZ1=T ½cosh ðEC0=TÞþ

sinhðEC0=TÞ� C2 ¼�ðB=ZAÞeEZ1=T sinhðA=TÞ

B2 ¼ ð1=ZÞeEZ1=T ½coshðA=TÞ�ðdVg=AÞsinhðA=TÞ�;C3 ¼ ð1=ZÞeEZ1=T ½coshðA=TÞþðdVg=AÞsinhðA=TÞ�. In this derivationwe consider kB ¼ 1.

Finally we obtain the formula for concurrence by using therelation equation (5),

C ¼2

ZeEZ1=T 4EJ sinh

A

T

� ��e�2Ez1=T

� �: ð5Þ

Now we calculate, the concurrence of the system by using theabove relation and the final expression for the concurrence atT¼0 is

C ¼ 9B=A9: ð6Þ

Now we analyze the above equation of concurrence in differentlimits. If we consider the gate voltage in the lattice is homo-geneous, i.e., dVg ¼ 0 that implies the concurrence CðT ¼ 0Þ ¼ 1.The system is always in maximally entangled state. For thissituation magnetic flux has no effect to make a transition fromentangled state to disentangled state.

If we consider the presence of inhomogeneous gate voltage,dVg a0 then the system has finite concurrence but it has notmaximally entangled for any arbitrary magnetic flux. We observethat at half-integer values of magnetic flux quantum, the con-currence is zero.

The analytical expression which we derive for the concurrenceis obtained satisfies the following relation:ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidV2

gþ4E2J cos

pff0

� ���������

s 2

ZEC0�2EZ1: ð7Þ

From the analysis of the above equation we get the analyticalexpression of magnetic flux which yield a finite concurrence ofthe system as

fZf0

pcos�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEC0�2EZ1�dVgÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðEC0�2EZ1þdVg

pÞ

2EJ: ð8Þ

Now we present our results and physical explanations. Fig. 1,shows the variation of concurrence with magnetic flux fordifferent values of dVg . We observe from the figure that thesystem is maximally entangled Bell state when there is no

inhomogeneity in the applied voltage in the lattice. This maxi-mally entangled Bell state is independent of the applied magneticflux. We also observe from this figure that a finite inhomogeneityin the applied gate voltage introduces a transition from entangledstate to disentangled state. In the entanglement phase system isalmost maximally entangled state. Therefore it reveals from ourstudy that the inhomogeneity of gate voltage plays an importantrole for introducing a transition from entangled state to disen-tangled state. At the same time another important observation ofour study is that the presence of on-site and inter-site chargingenergies are not sufficient to protect the entangle state of thesystem. One can naively think for EJ¼0 (at f¼f0=2), where thereis no tunneling between the SQD lattice points. As a result of it,the system is in the disentangled state but this is not the case forhomogeneous gate voltage.

Fig. 2 shows the variation of concurrence with magnetic fluxfor different values of Josephson couplings. We observe from ourstudy that concurrence is large for the smaller values of ratiobetween the Josephson coupling to on-site Coulomb chargingenergies. As we have obtained from our previous studies that thislarge ratio of EJ=EC0 favors the Luttinger liquid and superconduct-ing phase of the system and the smaller one favors the Mottinsulating density wave phases of the system [25,26]. Thus it isclear from our study that entanglement of the system is large forMott-insulating phase of the system where the Cooper pairs are

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Magnetic Flux

Con

curr

ence

EJ = 1.0,δ Vg =0.5

T=3T=5T=8

Fig. 3. Concurrence vs. magnetic flux at different finite temperatures. Here

dVg ¼ 0:5, EC0 ¼ 1:0 and Ez1 ¼ 0:5.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Magnetic Flux

Con

curr

ence

δ Vg = 0.0

δ Vg = 0.3

δ Vg = 0.5

Fig. 4. Concurrence vs. magnetic flux for different inhomogeneities of gate voltage

in SQD lattice. Here EJ1 ¼ 1:0, EC0 ¼ 1:0, Ez1 ¼ 0:5 and T¼4.

S. Sarkar / Physica B 414 (2013) 26–29 29

localized. But the collapse of entanglement at the half-integermagnetic flux quantum has the common feature for all values ofJosephson couplings for inhomogeneous gate voltage.

Fig. 3 shows the variation of concurrence with magnetic flux atdifferent temperatures. It reveals from our study that the con-currence is smaller for the higher values of temperature and atthe same time the system is in the disentangled state for widerregion of magnetic flux. In this wider region of magnetic flux,EJ a0, therefore it is clear from our study that the system is in thedisentangled state even for finite tunneling of Cooper pair acrossthe SQD lattice. The region of disentanglement state is larger forthe higher values of temperature. The solid lines in the figures arefor the considerations of superconducting Coulomb blocked effectinduced co-tunneling effect in the system. We observe that co-tunneling effect has no drastic effect to change entangle transitionof the system at the quantitative level. For ground state entangle-ment, the effect of co-tunneling effect is not explicit in theanalytical expression (Eq. (8)).

Fig. 4 shows the variation of concurrence with magnetic fluxfor different values of dVg . Here we observe that the presence andabsence of gate voltage inhomogeneity has no appreciable for thetransition from entangle state to disentangle state. This behavior

at finite temperature is in contrast with the behavior at zerotemperature where the system is always entangled state forhomogeneous gate voltage.

3. Summary and conclusions

We have studied the entanglement of a two qubit of a super-conducting quantum dot lattice. We have observed the magneticflux dependent entangled state to disentangled state transition atzero and finite temperature. We have found an important con-straint of magnetic flux during the study of ground state entan-glement. We have observed that at zero temperature that thesystem is always maximally entangled state for homogeneousgate voltage in the gate voltage in the lattice sites. We have alsoobserved that concurrence is larger for the Mott insulating phasecompare to the superconducting phase of this system.

Acknowledgments

The author would like to acknowledge the CCMT of the PhysicsDepartment of IISc for extended facility use. The author wouldlike to acknowledge Dr. R. Srikanth for reading the manuscriptcritically. Finally, Mr. Manjunath of RRI Library for providingmany scientific documents.

References

[1] L. Amico, R. Fazio, A. Osterloh, V. Vedral, Rev. Mod. Phys. 80 (2008) 517.[2] M. Lewenstein, et al., Adv. Phys. 56 (2007) 2.[3] R. Fazio, H. van der Zant, Phys. Rep. 355 (2001) 235.[4] I. Bose, A. Tribedi, Lecture Notes in Physics, vol. 802, Springer-Verlag,

Heidelberg, 2010, pp. 177–200.[5] T.J. Osborne, M.A. Nielsen, Phys. Rev. A 66 (2002) 032110.[6] S.J. Gu, arXiv:0811.3127.[7] H.T. Quan, et al., Phys. Rev. Lett. 96 (2006) 140604.[8] P. Zanardi, N. Paunkovic, Phys. Rev. E 74 (2006) 031123.[9] M. Cozzini, R. Ionicioiu, P. Zanardi, Phys. Rev. B 76 (2007) 104420.

[10] H.Q. Zhou, e-print arXiv:0704.2945.[11] S. Chen, L. wang, S.J. Gu, Y. Wang, Phys. Rev. E 76 (2007) 061108.[12] M.A. Nielsen, Phys. Rev. A 63 (2001) 022114.[13] X. Wang, Phys. Rev. A 64 (2001) 012313.[14] G.L. Kamta, A.F. Starace, Phys. Rev. Lett. 88 (2002) 107901.[15] K.M. O’Connor, W.K. Wootters, Phys. Rev. A 63 (2001) 052302.[16] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.[17] Y. Sun, Y. Chen, H. Chen, Phys. Rev. A 68 (2003) 044301.[18] Y. Yeo, Phys. Rev. A 66 (2002) 062312.[19] D.V. Khveshchenko, Phys. Rev. B 68 (2003) 193307.[20] G. Zhang, S. Li, J. Liang, Opt. Commun. 245 (2005) 457.[21] S. Bose, Phys. Rev. Lett 91 (2003) 207901.[22] M. Christandl, N. Datta, A. Ekert, A.J. Landahl, Phys. Rev. Lett. 92 (1996).[23] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120.[24] Guo-Feng Zhang, Shu-Shen Li, Phys. Rev. A 72 (2005) 034302.[25] G.H. Yang, W.B. Gao, L. Zhou, H.S. Song, arXiv:quant-ph/0602051.[26] S. Sarkar, Phys. Rev. B 75 (2007) 014528.[27] S. Sarkar, Eur. Phys. J. B 67 (2009) 559 arxiv0805.4713.[28] S. Sarkar, Renormalization group study of mesoscopic SQUID array, in:

Quantum Phase Transition and Dynamics: Quenching, Annealing and Quan-tum Computation, Springer, 2010.

[29] E. Chow, et al., Phys. Rev. Lett. 81 (1998) 204.[30] L.I. Glazman, A.I. Larkin, Phys. Rev. Lett. 79 (1997) 3786.[31] H. Jaeger, D. Haviland, B. Orr, A. Goldman, Phys. Rev. B 40 (1989) 182.[32] B.G. Orr, H.M. Jaeger, A.M. Goldman, C.G. Kuper, Phys. Rev. Lett. 56 (1986)

378.[33] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York,

1996.[34] M.H. Devoret, J.M. Martinis, J. Clarke, Phys. Rev. Lett. 55 (1985) 1908.[35] Y. Nakamura, Y.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786.[36] J. Clarke, F.K. Wilhelm, Nature 453 (2008) 1031.