Entanglement of Two-Superconducting-Qubit System Coupled with a Fixed Capacitor
aa r X i v : . [ qu a n t - ph ] A p r Entanglement of Two-Superconducting-Qubit System Coupled with a FixedCapacitor
TIAN Li-Jun , , ∗ QIN Li-Guo , , † and ZHANG Hong-Biao Department of Physics, Shanghai University, Shanghai 200444 Shanghai Key Lab for Astrophysics, Shanghai 200234 Institute of Theoretical Physics, Northeast Normal University, Changchun 130024
We study thermal entanglement in a two-superconducting-qubit system in two cases, either iden-tical or distinct. By calculating the concurrence of system, we find that the entangled degree ofthe system is greatly enhanced in the case of very low temperature and Josephson energies for theidentical superconducting qubits, and our result is in a good agreement with the experimental data.PACE: 03.65.Ud, 74.81. − g, 74.50.+rKeywords: Quantum Entanglement; Superconducting qubits; Josephson energy Entanglement has received much attention since itplays a central role in quantum information process-ing and quantum computing.[1] There are several waysto generate entanglement through experiments.[2, 3]However, it is still an open question to generate verygood entangled states. Current interest focuses ongenerating, maintaining and controlling precisely en-tanglement of systems.[4, 5] As is well known, temper-ature and magnetic field can be prepared, control andmaintain entanglement.[6, 7] One may ask a question:are there any other effective ways to control entangle-ment?For micro-systems, entanglement between differ-ent kinds of qubits has been studied, for exam-ple, charge,[8, 9] flux,[10] charge flux,[11] and phasequbits.[12, 13] Of particular importance, the super-conducting qubits[14] take advantage of the two char-acteristic of superconducting and quantum, and there-fore become the most suitable candidates for quan-tum computing,[15] which has been carried out inlaboratory.[16] For superconducting qubits, manipula-tion of quantum states has enabled scientists to gener-ate partly entangled states.[17] However, high quan-tum entangled states are required in such quantumtechnology. In the experimental aspect, entanglementhas been generated for coupled charge qubits[2] andcoupled phase qubits,[3] but the maximally entangledstates are merely in theory.[18] On the other hand, re-cent experiments have observed strong couplings be-tween two superconducting qubits.[13, 19, 20] As aconsequence, they triggers the theoretical research oninvestigating superconducting qubits. We have car-ried out some research on the corresponding rela-tions between the theory and experiment in quantumentanglement.[21]In this Letter, based on experimental study ofchanging the entanglement degree by adjusting the ca-pacitance and LC circuits,[22] thermal entanglementsare studied in two superconducting qubits, either iden-tical or distinct. Different evolutions of the entangle-ment are observed. In the case that the supercon-ducting qubits have the same Josephson energies, weinvestigate the effect of temperature and Josephsonenergies on entanglement. The result exhibits highquantum entangled states at low temperature. In ad- ∗ Electronic address: tianlijun@staff.shu.edu.cn † Electronic address: [email protected] dition, our theoretical results match with the experi-mental data very well, so the entangled qubits, whichare made by making use of our data, should have bet-ter entangled nature. We hope that this would beconfirmed experimentally in the future.The present model is composed of two single cooper-pair box charge qubits, coupled with a fixed ca-pacitor. This model has attracted much attentionand researches.[22–25] The superconducting materi-als act as a superconductor with a suppressed transi-tion temperature T c adjusted by using different ma-terials, which is in mK range for practical opera-tion in efficient and multiplex superconducting cir-cuits. One good example is a superconductive ma-terial made from a superconducting Al/Ti/Au tri-layer with respective thicknesses of 300, 200, and 200˚A, T C = 450 mK.[22, 26] The Hamiltonian of two-superconducting-qubit system is given by[22] H = − { [4 E C ( 12 − n g ) + 2 E m ( 12 − n g )] σ z +[4 E C ( 12 − n g ) + 2 E m ( 12 − n g )] σ z + E J σ x + E J σ x − E m σ zz } , (1)where E Cj and E Jj are respectively the charging andJosephson energies, and E m is the mutual couplingenergy between the two qubits; σ x = σ x ⊗ I , σ x = I ⊗ σ x , σ zz = σ z ⊗ σ z with σ x,z being the normal Paulimatrices and I the identity matrix; n gj = C gj V gj / e is the normalized qubit gate charge with C gj and V gj the control gate capacitance and voltage, respectively.For simplicity, calculations are restricted at the de-generacy point, where n g = n g = 0 .
5, which is thecondition of insensitivity to noise.[24] Under this con-dition, the model Hamiltonian reduces to H = −
12 ( E J σ x + E J σ x − E m σ zz ) , (2)which is independent of charging energy. This re-duced Hamiltonian is applied to study quantum gatestoo,[23] The eigenvalues and eigenvectors of Hamilto-nian can be obtained, H | Ψ i = − √ A | Ψ i , H | Ψ i = 12 √ A | Ψ i ,H | Ψ i = 12 √ B | Ψ i , H | Ψ i = − √ B | Ψ i , (3)where | Ψ i = 1 N [ | i − | i − a ( | i − | i )] , | Ψ i = 1 N [ | i − | i + a ( | i − | i )] , | Ψ i = 1 N [( | i + | i ) + a ( | i + | i )] , | Ψ i = 1 N [( | i + | i ) − a ( | i + | i )] , (4) A = ( E J − E J ) + 4 E m and B = ( E J + E J ) +4 E m . Here a = ( √ A + 2 E m ) / ( E J − E J ), a =( √ A − E m ) / ( E J − E J ), a = ( √ B + 2 E m ) / ( E J + E J ), and a = ( √ B − E m ) / ( E J + E J ). N i is thenormalization coefficient of | Ψ i i ( i = 1 , , , E J E J >
0, the ground state is | Ψ i , and | Ψ i for E J E J <
0. An important observation is that forthe attractive case of E J E J = 0, the degeneracystates in the ground state appear.In order to measure entanglement, concurrence hasbeen proposed, and is defined as[27, 28] C = max { λ − λ − λ − λ , } , (5)where the parameters λ i in decreasing order are thesquare roots of the eigenvalues of the operator ς = ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) , (6)where σ y , are the Pauli spin matrix of the two qubitsand ρ = (1 /Z ) exp( − H/kT ) is the density operatorof the system at the thermal equilibrium, where Z = Tr[exp( − H/kT )] is the partition function. Theconcurrence C ranges from 0 for a separable state to 1for a maximally entangled state. Following the samemethod in the standard basis, the density matrix ofthe system is ρ ( T ) = Z m − m m − m m + m m + m m − m m + m m − m m + m m + m m − m m + m m − m m + m m + m m − m m − m ,where m = (cosh( x A ) + cosh( x B )), m = E m ( sinh( x A ) √ A + sinh( x B ) √ B ), m = E J + E J √ B sinh( x B ), m = E J − E J √ A sinh( x A ), m = (cosh( x B ) − cosh( x A )), m = E m ( sinh( x A ) √ A − sinh( x B ) √ B ) and Z =4 m , with x A = √ A kT and x B = √ B kT . The concurrencecan be easily calculated by Eqs. (5) and (6).For identical superconducting qubits, E J = E J = E J and E C = E C , the model Hamiltonian can berewritten as H = −
12 ( E J σ x + E J σ x − E m σ zz ) , (7)where E J and E m can be adjusted by the experimen-tal multiplexed capacitance in the circuits. Similarmodel was argued elsewhere for the choices of E J asa magnetic field.[29]The eigenvalues and eigenvectors of Hamiltonian Eq. (7) read H | ψ i = − E m | ψ i , H | ψ i = E m | ψ i ,H | ψ i = √ D | ψ i , H | ψ i = −√ D | ψ i (8)with D = E m + E J ; | ψ i = √ ( | i − | i ), | ψ i = √ ( | i − | i ), | ψ i = N + [( | i + | i ) + ξ − ( | i + | i )], and | ψ i = N − [( | i + | i ) + ξ + ( | i + | i )],with ξ ± = E m ±√ DE J and N ± are the normalization co-efficients. Here | ψ i and | ψ i are two of four Bellstates, which are the maximally entangled states.The density matrix can be obtained by the abovesame way in this case. Without lack of generality, themutual coupling energy E m is regarded as the energyunit and k = 1. Thus we can consider E m /k = 1,whose unit is mK. For convenience, we only write itsvalue as the same as E J . By making use of Eqs. (5)and (6), the concurrence can be calculated for theidentical qubits. Especially, for E J = 0 the Hamil-tonian (7) only has the last item whose eigenvectorsare the separable states, so that no thermal entangle-ment is present, namely, C = 0. FIG. 1: (Color online) Two-dimensional plots of the con-currence vs Josephson energy E J for different tempera-tures. Asterisks: the experimental data. Inset: three-dimensional plot of the concurrence as a function of E J and T . In the inset of Fig. 1, we show the concurrenceas functions of Josephson energies and temperature,displaying nonmonotonic behavior for smaller E J andlower temperature. However, in the limit E J → E m ≈ √ D , the degeneracy states are present: ψ and ψ ; ψ and ψ , namely, only two energy levels arepopulated. Thus there is the energy level crossing thepoint E J = 0, namely, the ground state is the degener-ate state of ψ and ψ . With an infinitesimal increaseof E J , the concurrence will increase sharply to a topin accordance with Ref [29]. At the zero tempera-ture, the entanglement primarily depends on | ψ i , i.e.on the ground state, which plays a major role. As T increase, the peaks fall, because the ground statewill mix with excited states in thermodynamic equi-librium and mixing states combine the concurrence ofthe system. To illustrate this feature, Figs. 1 and 2are plotted to show the behavior of C vs E J and C vs T , respectively.Figure 1 clearly shows that no entanglement ispresent for E J = 0. As E J increases, the entan-glement first reaches sharply to the maximum, thendecays rapidly, finally reduces slowly and asymptoti-cally to a stable value. Moreover, lower temperatureand lower Josephson energy will cause the entangle-ment richer. Considering the data of the sample 2for the identical qubits in Ref. [22] and E m as the en-ergy unit, we obtain E J /k = E J /k = 3 . T = 20 mK and E J = 3 . C = 0 . C = 0 .
27 observed experimentally.[22] C E J =0.5E J =2E J =7.2E J =16 FIG. 2: (Color online) Two-dimensional plots of the con-currence as a function of the temperature T for the fourdifferent Josephson energy. In Fig. 2, C vs T for different Josephson energy ispresented. For lower Josephson energy, for example E J = 0 .
5, the concurrence will vary dramatically, butis not so apparent for bigger Josephson energy. Whenthe temperature T = 0, only the ground state | ψ i exists, and C ≈ .
9. With increase of E J , C | ψ i willdecrease, so the intersections of the curve and C -axisdecline. For a fixed smaller E J , as the temperaturerising, the ground state and three excited states mix, C will decrease sharply. On the contrary, for the largerJosephson energy, the change behavior of C becomesvery slow and finally C tends stably at T ≤ T C . Thus,the concurrence is very susceptible to small Josephsonenergy at lower temperature (see Fig. 2).For the distinct superconducting qubits, the con-currence can be calculated through Eqs. (5) and (6).To distinguish different influences of identical and dis-tinct superconducting qubits on the entanglement atthe same temperature, Figure 3 shows the evolutionof the concurrence as functions of E J and T with E J = 17 .
2. Obviously C is smaller than that of theidentical case at low temperature.The contour figure of C is plotted in Fig. 4. It isworth noting that two Josephson energies are smallerand closer, the concurrence decreases more slowly, andthe peak is higher. This proves that C is maximal at E J = E J for the stable temperature. When thevalues of E J stay away from E J , C will decay. Ac-cording to the data in Ref.[22], by taking E J = 13 . E J = 17 .
2, our theoretical result is C = 0 . C = 0 . FIG. 3: (Color online) Three-dimensional plot of the con-currence as functions of temperature T and Josephson en-ergy E J with E J = 17 . E J1 E J FIG. 4: (Color online) Two-dimensional contour plots ofthe concurrence C as a function of the Josephson energies E J and E J at T = 20 mK. imum value of C for E J = E J is much larger than E J = E J . That is to say, choosing the proper super-conducting qubits can enhance the entanglement atlow temperature. Two Josephson energies are smallerand closer, then the maximum value of concurrencewill be larger.In conclusion, we have investigated the effect ofJosephson energies on the thermal entanglement inthe two-superconducting-qubit model. In the twocases, i.e., identical and distinct superconductingqubits, we have presented the evolution of concurrencewith respect to the Josephson energy and tempera-ture. Comparing the results of these two cases, weconclude that the entanglement may be enhanced un-der the identical superconducting qubits for the sametemperature. When the temperature and Josephsonenergies are lower, the entangled degree of the systemis greatly enhanced. Our theoretical prediction is in FIG. 5: (Color online) Two-dimensional plots of the con-currence as a function of Josephson energy E J for T =20 mK. good agreement with the experiments and provides anew way to enhance and to control the entanglementdegree of the system by adjusting the Josephson ener-gies, which can be realized experimentally by changingthe capacitance and LC circuits. Utilizing the resultsof calculation and investigation, we may generate bet-ter entangled and stable states, which could have wideapplications in the quantum communication and phys-ical experiments.This work is partly supported by the NSF ofChina (Grant No. 11075101), by Shanghai LeadingAcademic Discipline Project (Project No. S30105),and by Shanghai Research Foundation (Grant No.07d222020). The authors are grateful to Xin-Jian Xuand Ying Jiang for valuable discussions. [1] Nielsen M and Chuang I 2000 Quantum Computationand Quantum Information (Cambridge: CambridgeUniversity)[2] Pashkin Y A, Yamamoto T, Astafiev O, Nakamura Y,Averin D V and Tsai J S 2003 Nature Fundamen-tals of Quantum Optics and Quantum Information (Berlin: Springer)[16] Mooij H 2005 Science64