The energy level splitting for unharmonic dc-SQUID to be used as phase Q-bit

Physica C 435 (2006) 114–117 www.elsevier.com/locate/physc

The energy level splitting for unharmonic dc-SQUID to be used as phase Q-bit Nikolai V. Klenov a, Victor K. Kornev a

a,*

, Niels F. Pedersen

b

Physics Department, Moscow State University, Leninsky Gory 1, 119992 Moscow, Russia b Oersted-DTU, Technical University of Denmark, Lyngby DK2800, Denmark Available online 21 February 2006

Abstract Dc-SQUID with Josephson junctions characterized by nonsinusoidal current-phase relation is being considered as a basis for phase qubit. It has been shown that the second and third harmonic components each in the current-phase relation are able to provide doublewell potential and the energy level splitting. Threshold condition for the double-well form origin has been determined taking into account the impact of both harmonics. The splitting gap of the ground energy level has been calculated as a function of the harmonic amplitudes for different ratio s of characteristic Josephson energy EC to the Coulomb energy EQ0. It has been shown that the gap value comes to about 7EQ0 with increase of the ratio s. No external field needed, no bias current required and no circular currents are major advantages of such a qubit. Ó 2006 Elsevier B.V. All rights reserved. PACS: 85.25.Dq; 03.67.a Keywords: Nonsinusoidal current-phase relation; SQUID; Energy level splitting

1. Introduction One of the promising types of phase qubits [1] is socalled ‘‘silent’’ qubit based on nonsinusoidal current-phase relation, which is observed in particular for submicron high-TC grain boundary Josephson junctions [2]. The ‘‘silent qubit’’ considered recently in [3,4] is a dc-superconducting quantum interference device (SQUID) with D/D grain boundary junctions (based on d-wave superconductor films), which are characterized by the second harmonic existence in current-phase relation (CPR). The main advantages of the device are (i) protection of the operating point from the fluctuations of the external fields, already on the classical level, (ii) the absence of a state-dependent spontaneous current in the loop of the qubit, and (iii) the uselessness of bias magnetic flux [3,4].

At the same time there is experimental evidence that CPR of some D/D junctions contains both the second and third harmonics [5]. Therefore this paper is aimed at analysis of the role of the third harmonic itself and together with second harmonic. In general case the considered CPR can be written as follows: I j ðuÞ ¼ Aj sin uj  Bj sin 2uj þ C j sin 3uj . . . ;

j ¼ 1; 2 ð1Þ

According to experimental data for conventional Josephson junctions [5] the amplitudes of the first and second harmonics should have opposite signs, while no restriction for C is known. In the case of so-called p-junctions amplitude A is negative. 2. Double-well potential

*

Corresponding author. Tel.: +7 095 939 4351; fax: +7 095 939 3000. E-mail address: [email protected] (V.K. Kornev).

0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.01.027

The qubit inductance L must be as small as possible to eliminate any crosstalk between neighbor qubits. In the

N.V. Klenov et al. / Physica C 435 (2006) 114–117

case of negligibly small normalized inductance l = 2pLI0/ U0  1, the total magnetic flux U through the interferometer loop approaches the applied magnetic flux Ue, and Josephson energy EJ of the system is given by the following expression (constant term is omitted):  u  u  2p þ h þ A2 cos h EJ ¼  A1 cos U0 I 0 2 2  1 1  B1 cosðu þ 2hÞ  B1 cosðu þ 2hÞ þ    ; 2 2 ð2Þ where u = u1  u2, h = (u1 + u2)/2, and u = ue  2pUe/ U0, I0—some normalizing current. Profile of the energy versus phase h is presented in Fig. 1(a). According to (2), when no external flux is applied the energy potential is always a symmetric function of the phase h regardless any asymmetry in the junction parameters. EJ can show double-well form, and the local minimum positions hr and hl may be found from extremum condition   oEJ ðu; hÞ ¼ 0; ð3Þ oh u¼0 If the second harmonic amplitudes are high enough to fulfill threshold relation (see domain in Fig. 1(b)) j2ðB1 þ B2 Þj > jðA1 þ A2 Þj;

ðA1 þ A2 Þ=2ðB1 þ B2 Þ > 0; ð4Þ

the double-well form takes place with the symmetrically located minima at

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hr;l ¼  arccosfðA1 þ A2 Þ=2ðB1 þ B2 Þg þ 2pn; n ¼ 0; 1; 2 . . .

ð5Þ

In the case of identical CPR of the junctions (junction areas may be different) the energy potential remains symmetric regardless to external magnetic field applied. However, if the CPRs do not coincide, the applied magnetic field always breaks symmetry of the potential. In the extreme case of the junction area difference (a1  a2) corresponding to proceeding to the one-junction interferometer, the energy potential becomes always symmetric. Coexistence of the second and third harmonics results in new threshold condition for the double-well form origin instead of condition (4): 4ðC 1 þ C 2 Þ > ðB1 þ B2 Þ þ D1=2 > 4ðC 1 þ C 2 Þ;

ð6aÞ

or 4ðC 1 þ C 2 Þ > ðB1 þ B2 Þ  D1=2 > 4ðC 1 þ C 2 Þ; 2

ð6bÞ

2

where D = (B1 + B2) + 4(C1 + C2)  4(B1 + B2)(A1 + A2), D > 0. One should emphasize that the only third harmonic existence is also able to provide the double-well form, if the following condition is fulfilled (see domain in Fig. 1(b)) 3 < ðA1 þ A2 Þ=ðC 1 þ C 2 Þ < 1.

ð7Þ

3. Energy level splitting If characteristic Josephson energy EC = U0I0/2p is much more than the characteristic Coulomb energy EQ0 = e2/ (2C) (C—the junction capacitances), Josephson-junction phases u1 and u2 (or phases h and u) are well-defined generalized coordinates of the system under consideration. This system is characterized by the following Hamiltonian: H ¼ EQ0

o2 o2  E þ EJ ðu1 ; u2 Þ; Q0 ou21 ou22

ð8Þ

where EJ(u1, u2) is Josephson energy (2). If no external magnetic flux is applied (ue = 0), Schro¨dinger equation can be read in terms of h and u as follows: 2

o2 W þ eJ ðhÞW ¼ eW; oh2

ð9Þ

where W—wave function, e = E/EQ0—normalized eigenvalue of Hamiltonian (8), eJ = EJ/EQ0—normalized Josephson energy. There are two possible approaches to solve the quantum mechanical problem: (i) the use an oscillator model and (ii) the analysis by Mathieu equation. 3.1. Oscillator model Fig. 1. (a) Equivalent scheme of the ‘‘silent’’ qubit proposed and its potential profiles for different values of u at A1 = C1 = I0, B1 = B2 = 0, A2 = C2 = 0.8I0. (b) Domains of the double-well potential existence for the unharmonic SQUID at the presence of the only second harmonic or of the only third harmonic in the current-phase relation of the junctions.

In the frame of this approach the potential (2) should be approximated by parabolic wells near points of local minima by retaining only h-quadratic terms. This allows easy to find discrete set of steady state energy levels Ei. Existence

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Fig. 2. Splitting gap D behavior for three values of the ratio s of characteristic Josephson energy EJ to Coulomb energy EQ at the presence of the only second harmonic (left) or of the only third harmonic (right) in the current-phase relation of the junctions.

of tunneling trough barrier between two wells leads to splitting of the energy levels. Specifically the ground energy level E0 splits also into two levels E± = E0 ± D. The splitting amplitude D is defined by both the height V and the half-width a of the potential barrier between the wells:  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ¼ V exp 2aðE0 Þ ðV  E0 Þ=EQ0 . ð10Þ At C1 = C2 = 0, the ground level E0, barrier height V and barrier half-width a are as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 E0 ¼ 2EQ0 sðB1 þ B2 Þ½1  ðA1 þ A2 Þ =4ðB1 þ B2 Þ =I 0 ; ð11Þ "  # 2   U0 1 A1 þ A 2 A 1 þ A2 V ¼ ðB1 þ B2 Þ  þ1 ; 4 B1 þ B 2 2p B 1 þ B2 a ¼ h ¼ jhr j ¼ jhl j;

ð12Þ

where s = EC/EQ0—the ratio of the characteristic Josephson energy to Coulomb energy. Maximum value of the splitting gap peaks at the total amplitude value (B1 + B2) = (1, 2, . . . , 1, 6)I0 and comes to (3, 5, . . . , 7)EQ0. The maximum position depends on ratio s, as it is shown in Fig. 2. If CPR contains only third harmonic (Bj = 0) the barrier height V and the ground level E0 are described by the following formulas: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( 1 2s 1 A1 þ A2 E0 ¼ EQ0 ð2A1 þ 2A2  18C 1  18C 2 Þ  2 I0 4 4ðC 1 þ C 2 Þ !)1=2 þ6ðC 1 þ C 2  A1  A2 Þ

;

3.2. Mathieu equation method When no external flux is applied (ue = 0) and CPR does not contain third harmonic, Schro¨dinger equation (8) can be read as follows: o2 W  s s e ðA ðB þ þ A Þ cos h  þ B Þ cos 2h þ W ¼ 0. 1 2 1 2 2 4 2 oh2 ð15Þ This equation can be reduced to Mathieu equation in two extreme cases corresponding to negligibly small values of either B-determined term or A-determined term. In the latter case Eq. (15) reduces to the following Mathieu equation:

ð13Þ

(

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 A1 þ A2  ðA1 þ A2  C 1  C 2 Þ 1 4 4ðC 1 þ C 2 Þ 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 191=2  3 = 4 1 A þ A 1 2   1A ; þ ðC 1 þ C 2 Þ@ ; 3 4 4ðC 1 þ C 2 Þ  1=2 1 A1 þ A2  ð14Þ a ¼ arccos 4 4ðC 1 þ C 2 Þ

U0 V ¼ 2p

Fig. 2 presents the splitting gap behavior for the two cases when CPR contains either only second harmonic (Cj = 0) or only third harmonic (Bj = 0).

Fig. 3. Eigenvalues for Mathieu Eq. (16) versus parameter 2q =  s(B1 + B2)/4. The eigenvalues e 0 , e1 correspond to the split energy levels: the ground level E0 and the first level E1. Energy e is normalized by characteristic Coulomb energy EQ0 and counted off from maximum value of the potential barrier between the wells.

N.V. Klenov et al. / Physica C 435 (2006) 114–117

 o2 W  e  2q cos 2h W ¼ 0; ð16Þ þ 2 2 oh where 2q = s(B1 + B2)/4. Using the specified recurrent formulas [6] for this equation, the eigenenergies e 0 (split ground level) and e have been calculated numerically 1 and are presented in Fig. 3. The obtained ground energy level splitting well corresponds to the one calculated in the oscillator model at j(B1 + B2)j  j(A1 + A2)j. 4. Conclusions Existence of MQT of the superconductive phase u in high-TC Josephson junctions has been recently demonstrated [2]. Furthermore, it has been reported that the ratio a of the second harmonic amplitude to the first harmonic amplitude in CPR of YBCO junctions ranges from a  0.1 at T = 900 mK to a saturated value a  0.7 below T = 100 mK [2]. Therefore there are all reasons to believe that the ‘‘silent’’ phase qubit can be built on the base of such high-TC superconductors. There are two possible ways to measure states of the ‘‘silent’’ qubit. Firstly, circular currents in the qubit can appear only in the case of Josephson junctions with different CPR. In such a case, applying a small external flux ux, one can induce small state-dependent currents, and the qubit states can be measured by means of impedance

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measurement technique [7]. At the same time the qubit remains ‘‘silent’’, because the noise caused by the flux is of higher order of vanishing then the currents [4]. Secondly, the possible measurement technique to distinguish the no-current states of the qubit is described in [7]. Acknowledgements This work was supported in part by CRDF RUP1-1493MO-05, ISTC Grant 2369 and Russian Grant for Scientific School 1344.2003.2. References [1] Y. Makhlin, G. Schon, A. Shnirman, Reviews of Modern Physics 73 (2001) 357. [2] T. Bauch, F. Lombardi, F. Tafuri, A. Barone, G. Rotoli, P. Delsing, T. Claeson, Phys. Rev. Lett. 94 (2005) 087003. [3] S.A. Charlebois, T. Lindstrom, A.Ya. Tzalenchuk, M.H.S. Amin, A. Smirnov, A. Zagoskin, T. Claeson, , 2003. [4] M.H.S. Amin, A. Smirnov, A. Zagoskin, T. Lindstrom, S.A. Charlebois, T. Claeson, A.Ya. Tzalenchuk, Phys. Rev. B 73 (2005) 064516. [5] V.K. Kornev, I.I. Soloviev, N.V. Klenov, N.F. Pedersen, I.V. Borisenko, P.B. Mozhaev, G.A. Ovsyannikov, IEEE Trans. Appl. Supercond. 13 (2003) 825. [6] M. Abramovitz, I. Stegun, Appl. Math. Ser. 55 (1964) 532. [7] Ya.S. Greenberg, A. Izmalkov, M. Grajcar, E. Il’ichev, W. Krech, H.-G. Meyer, A. van den Brink, Phys. Rev. B 66 (2002) 214525-1.