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PHYSICAL REVIEW B 76, 064531 2007Non-Gaussian dephasing in flux qubits due to 1/f noise1,2,3, 4,5 6 6 2Y. M. Galperin, * B. L. Altshuler, J. Bergli, D. Shantsev, and V. Vinokur1Department of Physics and Center of Advanced Materials and Nanotechnology, University of Oslo, P.O. Box 1048 Blindern,0316 Oslo, Norway2Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA3A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia4Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA5NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA6Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, NorwayReceived 26 December 2006; revised manuscript received 30 April 2007; published 27 August 2007Recent experiments by Yoshihara et al. Phys. Rev. Lett. 97, 167001 2006 provided information ondecoherence of the echo signal in Josephson-junction flux qubits at various bias conditions.These results wereinterpreted assuming a Gaussian model for the decoherence due to 1/f noise. Here we revisit this problem onthe basis of the exactly solvable spin-fluctuator model reproducing detailed properties of the 1/f noise inter-acting with a qubit.We consider the time dependence of the echo signal and conclude that the results based onthe Gaussian assumption need essential reconsideration.DOI: 10.1103/PhysRevB.76.064531 PACS ...

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PHYSICAL REVIEW B76, 0645312007 Non-Gaussian dephasing in flux qubits due to 1/fnoise 1,2,3, 4,56 62 Y. M. Galperin, * B. L. Altshuler,D. Shantsev,J. Bergli,and V. Vinokur 1 Department of Physics and Center of Advanced Materials and Nanotechnology, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway 2 Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA 3 A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia 4 Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA 5 NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA 6 Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway Received 26 December 2006; revised manuscript received 30 April 2007; published 27 August 2007  Recent experiments by Yoshiharaet al.Phys. Rev. Lett.97, 1670012006provided information on decoherence of the echo signal in Josephson-junction flux qubits at various bias conditions. These results were interpreted assuming a Gaussian model for the decoherence due to 1/fnoise. Here we revisit this problem on the basis of the exactly solvable spin-fluctuator model reproducing detailed properties of the 1/fnoise inter- acting with a qubit. We consider the time dependence of the echo signal and conclude that the results based on the Gaussian assumption need essential reconsideration. DOI:10.1103/PhysRevB.76.064531PACS numbers: 03.65.Yz,85.25.Cp 1 A remarkable workhas appeared recently, reporting onAssuming that the statistics of the fluctuations oft measurements of the two-pulse echo signal in Josephson- is Gaussian one can express the decoherence rate through junction flux qubits revealing its dependence of the time in- the magnetic noise spectrum,S    terval between the pulses. The authors interpreted their re- =1 /dtcostt0: 0 2 sults in terms of the recent theoryof the decoherence in 2 2  4 qubits caused by 1/fnoise and obtained the qubit dephasing v t sint/4  2  − t/4   t= exp−dS  .2 rate,  , based on the postulation of the Gaussian statistics of 2 the noise. Although this looks like a natural starting point, the importance of the echo signal data for understanding the This expression is common in the theory of spin resonance underlying mechanisms of the qubits, decoherence calls for and allows one to find the decoherence rate from the noise careful examination of the assumptions built into the theoret- spectrum once the coupling, v  , is known. For the 1/fnoise ical description. spectrum In this paper we develop a theory of the time dependence of the echo signal making use of an exactly solvable but S  =A  /,3 experimentally realistic model. We demonstrate that in many 1,4,5 one obtains practical realizations of 1/fnoise the results based on the Gaussian assumption need to be significantly corrected. We  2 −t    .ln 24 t=e,    v  A  show that deviation of noise statistics from the Gaussian changes significantlythe time dependenceof the echo signal. In what follows we compare this approximate result with Using the exactly solvable non-Gaussianspin-fluctuator exact equations that follow from the spin-fluctuator model. modelfor the 1/fnoise we analyze the dependence of the echo signal on both time and the qubit working point, and compare the obtained results with those derived within the I. SPIN-FLUCTUATOR (SF) MODEL FOR 1/fNOISE Gaussian approximation. Let us start with a brief review of the procedure used in One of the most common sources of the low-frequency Ref.1and a similar study reported in Ref.3. In order to noise is the rearrangement of dynamic two-state defects, separate the relaxation due to direct transitions between the fluctuators, see, e.g., Ref.6and references therein. Random energy levels of the qubit, characterized by the timeT 1 , the switching of a fluctuator between its two metastable states1 −t/2T 1 echo signal is expressed aset, wheretis the arrival and 2produces a random telegraph noise. The process is time of the echo signal. The factortdescribes the pure characterized by the switching rates 12 and 21 for the tran- dephasing. sitions 1→2 and 2→1. Only the fluctuators with the energy If dephasing is induced by the noise of the magnetic flux splittingEless than temperature,T, contribute to the dephas- in the superconducting quantum interference device ing since the fluctuators with large level splittings are frozen SQUIDloop, the echo signal is determined by the fluctu- in their ground states. As long asETthe rates 12 and 21 ating part of the magnetic flux through the loop,: are of the same order of magnitude, and without loss of t/2t generality one can assume that 12  21 The random/ 2.         t= exp−i v  −d.1 telegraph process,t, is defined as switching between the 0t/2 values ±1/ 2at random times, the probability to makentran- The strength of the coupling between the qubit and the noise sitions during the timebeing given by the Poisson distri- 1,3 bution. The correlation function of the random telegraph pro- source, v  , depends on the working point of the qubit. 1098-0121/2007/766/0645315The American Physical Society064531-1 ©2007

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− cesses istt+=ewhere/ 4and the corresponding min is theminimalrelaxation rate of the fluctuators. contribution to the noise spectrum is a Lorentzian, In the following we will assume that the number of fluctua- 2 2 / 4+ tors is large, so that . Accordingly, should there be many fluctua- P 0 T1 orN T 2L. After that Eq.5 yields tors coupled to the given qubit via constants v i and having the switching rates i , the dephasing of the noise power spec- 2 22 A 1 ,  0 1 number of effective fluctuators is sufficiently large, the sum  trum is expressed through the sum i v  i /+. If the i i S=AP 0 T.10  2 0 /, 0 , 8 over the fluctuators transforms into an integral over v and We see that the SF model reproduces the 1/fnoise power weighted by the distribution functionP v ,. Upon the as- −1 −2 spectrum3for 0 . The crossover fromtobe- sumption that the coupling constants and switching rates are havior at 0 follows from the existence of a maximal uncorrelated, the distribution density factorizes,P v , switching rate 0 . Below we will see that this crossover =P v  v P  , and the noise spectral function due to fluctua- modifies the time dependence of the echo signal at timest 2 −1 tors reduces to v S, where . 0 The SF model has previously been used for description of 1 P  d 9–14  4  + plied to analysis of decoherence in charge qubits. 2 2 effects of noise in various systemsand was recently ap-  v  d v v P v  v ,S= .5 2 2 7,15–22 Quantum aspects of the model were addressed in Ref.23. The distribution,P  , is determined by the details of the These studies demonstrated, in particular, that the SF model interaction between the fluctuators and their environment, is suitable for the study of non-Gaussian effects and that which causes their switchings. A fluctuator viewed as a two- these may be essential in certain situations. Now we are level tunneling system is characterized by two parameters— ready to analyze consequences of the upper cutoff and the diagonal splitting,, and tunneling coupling,, the distance effect of non-Gaussian noise, and through this identify the 2 2 between the energy levels beingE=  +we follow no- validity region for the prediction of Eq.4. tations of Ref.7where the model is described in detail. The In the following we will express the fluctuation of the environment is usually modeled as a boson bath, which can magnetic flux as a sum of the contributions of the statisti- represent not only the phonon field, but, e.g., electron-hole cally independent fluctuators,t= i b i  i t, whereb i are pairs in the conducting part of the system. The external de- partial amplitudes while i tare random telegraph pro- grees of freedom are coupled with the fluctuator via modu- 2 2   s ¯ v cesses. Consequently, we express the product v AaA lation ofand, modulation of the diagonal splitting 2 2 where ¯ v    v  b. In other words, we include the amplitude being most important. Under this assumption the fluctuator- of the magnetic noise in the effective coupling constant. environment interaction Hamiltonian acquires the form     A. Echo signal in the Gaussian approximation H F-env = cˆ  z − x ,6 E E Substitution of the distribution8into Eq.2yields for where cˆ is an operator depending on the concrete interaction Kt −lnt 2 mechanism. Accordingly, the factor/Eappears in the t/6 ,t1 0 0 interlevel transition rate: 2 2 2  ,ln 2  0 t1 .  g ¯ v 1 Kt=At1 E,=/E 0 E.7 The subscriptgmeans that this result is obtained in the Here the quantity 0 has a meaning of themaximalrelaxation Gaussian approximation. One sees immediately that the rate for fluctuators with the given energy splitting,E. The −1 −2 crossover between theandbehaviors in the noise coupling,, depends exponentially on the smoothly distrib- spectrum does not affect the echo signal at long timest −1 uted tunneling action, leading to the-like distribution of −1 . However, at small times the decay decrement acquires 0 −1 , and consequentlyP  . an extra factor 0 t, which is nothing but the probability for a Since only the fluctuators withETare important and typical fluctuator to change its state during the timet. As a temperatures we are interested in are low as compared to the result, even in the Gaussian approximation at small times relevant energy scale, it is natural to assume the distribution 3 2 K g ttreplacingK g ttbehavior. ofEto be almost constant. Denoting the corresponding den- sity of states in the energy space asP 0 we arrive at the 8 B. Non-Gaussian theory distribution of the relaxation rates as The echo signal given by Eq.1can be calculated exactly P 0 T using the method of stochastic differential equations, if the P  = 0 −.8  fluctuating quantityis a single random telegraph process, see Ref.7and references therein. The method was developed The productP 0 Tdetermines the amplitude of the 1/fnoise. 24 in the contexts of spin resonance,spectral diffusion in Indeed, the integralP  dis nothing but the total number 25–28 glasses, andsingle molecular spectroscopy in disordered of thermally excited fluctuators,N T . Consequently, 29–32 media. Averagingin Eq.1is performed over random P T=N/L,Lln/,9 realizations ofand its initial states and reflects the conven- 0T0 min 064531-2

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1,3 tional experimental procedurewhere the observable signal Data Ref. 1 SF model, v << γ , Eq.15 is accumulated over numerous repetitions of the same se- 0 SF model, v >> γ 16, Eq. quence of inputs. 0 2 2 ρ ~ exp (- Γ t ) For a single fluctuator with switching rate/ 2and cou- 1 20 pling v we find −t /2 2 e 2 v  2  t/2 −t/2 1 +  1 t= 2 e+1 −e− , 2  0.1 where=  1 − v /. In the appropriate limits this can be 2 expanded to give 2 3−1 v t/48,t, v −1 0.01 − ln 1 t v ,  t/2 ,,t v 12  v t/2, v ,t 1 .  2 − 0.1 1 time µ s Note that the last limiting case here is similar to the motional narrowing of spectral lines well-known in physics of spin FIG. 1.Color onlineDephasing component of the echo mea- 24 resonance. surements replotted from Fig. 4aof Ref.1away from the optimal In the case of many fluctuators producing 1/f-like noise, point. The curves show fits to the SF model, Eqs.14and15, and N T 1, the sum of the contributions from individual fluctua- 2 2 −t to the=elawRef.38. The fitting took into account all data tors gives the echo decay decrement points including those that fall outside the range of the plote.g., those with1.   K sf t= −d v P v  v dP  ln 1 t v ,.13 2 3−1  0 A ¯ v t/6 ,t ¯ v ,  4At, t  Let us assume that the distribution of v is a sharp function K sf t 15 −1 centered at some value ¯ v . This reduces integration over v to 0 ¯ v . merely replacing v → ¯ v in the expressions for v ,t. This In this case all fluctuators have v , hence the result at long 2 approximation is valid as long as v is finite. This is seem- times again differs significantly from the Gaussian result Eq. ingly the case, e.g., for magnetic noise induced by tunneling 11. of vortices between different pinning centers within the 2 SQUID loop. The case of divergent v is considered in Refs.7,20, and33. Using then Eq.8for the distribution II. DISCUSSION functionP  and using the appropriate terms from Eq.12in Here we apply the results obtained above to analyze quan- Eq.13we find the time dependence of the logarithm of the titatively the decoherence of the flux qubit. Figure1shows echo signal,K sf t. For ¯ v  0 the time dependence of the echo signal measured in Ref.1 2 3 −1  0 A ¯ v t/6 , t , anda fit based on the SF model. We consider first the case of 0 2 2 −1 −1 ¯ v  0 where the SF model predicts a crossover fromttot 3 K sf t ln 2 ¯ v At, t ¯ v , 14 0  3 −22  ¯ v At, − t,  interpolation formula,K sf t= 0 At/6 ¯ v +t/ 4. Figure1 dependence, Eq.15. We replace the exact result by the 1 ¯ v 2 2 −t where 6. alsoshows the commonly used fit=e, which is repre- At small timest ¯ v we arrive at the same result as in the sented by a straight line with slope 2. The fit to the SF model properly treatedGaussian approach, Eq11 seems at least equally good. From this fit we can extract the . However, at −1 large times,t ¯ v , the exact resultdramatically differsfrom average change in the qubit energy splittingE 01 due to a flip −1 the prediction of the Gaussian approximation. To understand of one fluctuator, ¯ v 3.5s .It allows us to evaluate the change of flux in the qubit loop induced by a fluctuator flip, the reason, notice that forP  1 /, the decoherence is dominated by fluctuators with v . The physical reason for b= ¯ v  , since v  =1 /E 01 /was measured in Ref.1. / v that is clear: very “slow” fluctuators produce slow varying Using the experimental values for all parameters we getb −6 fields, which are effectively refocused in the course of the 3.710 0 , where 0 is the flux quantum while the −3 echo experiment, while the influence of too “fast” fluctuators deviation from the optimal working point was10 0 . is reduced due to the effect of motional narrowing. As shown Hence the flip of one fluctuator changes the qubit energy splitting by 0.4%. The fit also determines the value of the in Ref.20, only the fluctuators with v produce Gaussian ion v we find a noise. Consequently, the noise in this case is essentiallynon- product 0 Aand within our assumpt 0  Gaussian. Only at short timest ¯ v lower estimate for the flux noise amplitude when these most im- A  / 0 1.3 −1 −6 portant fluctuators did not yet have time to switch, and only 10 . the faster fluctuators contribute, is the Gaussian approxima- Similarly, if ¯ v  0 we can fit to Eq.14using the inter- 2 3 −1 tion valid. polation formulaK sf t= ¯ v At/6 0 +t/ ln2. We do not in- For ¯ v  0 we find clude here theK sf tbehavior at largetsince it corresponds 064531-3

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1.5

1.0

0.5

0.0 -0.001

0.000 Φ / Φ

0.001

 FIG. 2.Color onlineSymbols: parameter v A  determined  by fittingtdata at different working points to the SF model, Eq. 14. Line: a fit based on the v  dependence measured in Ref.1 whereis the optimal point and= 0Ais the only fitting  parameter.

−1 toKA1. The fitting gives12which is thens , sf 0  also an upper estimate for ¯ v , andA  / 0 10 . −6 It is not possible to determine from Fig.1which of the cases ¯ v  0 or ¯ v  0 is realized in the experiment. There- fore we applied the same fitting procedure to the whole set of tcurves analyzed in Ref.1measured at different working points. Then it became clear that the formulas for ¯ v  0 give   a better overall fitting. The fitting parameter ¯ v A= v  A  is plotted in Fig.2as a function of the working point. The data can be very well-described using the v  dependence  measured in Ref.1, see the solid line, and we findA/ 0 −6 = 1.1510 .This value is quite close to that obtained in the 1 Gaussian approach.Surprisingly, we found a significant

* [e-mail] 1 F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, and J. S. Tsai, Phys. Rev. Lett.97, 1670012006. 2 A. Cottet, Ph.D. thesis, Université Paris VI, 2002. 3 K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba, H. Takayanagi, F. Deppe, and A. Shnirman, Phys. Rev. Lett.98, 0470042007. 4 A. Shnirman, Y. Makhlin, and G. Schön, Phys. Scr., TT102, 147 2002. 5 Y. Makhlin and A. Shnirman, Phys. Rev. Lett.92, 1783012004. 6 S. Kogan,Electronic Noise and Fluctuations in SolidsCam- bridge University Press, Cambridge, England, 1996. 7 Y. M. Galperin, B. L. Altshuler, and D. V. Shantsev, inFunda- mental Problems of Mesoscopic Physics, edited by I. V. Lerner et al.Kluwer Academic Publishers, The Netherlands, 2004, pp. 141–165. 8 There are other choices ofPbased on the standard for the glassy  system assumption about the smooth distribution of relevant quantities. In particular, assuming wide distribution forone arrives atP  , which readsP  =P T/ 2  1 −/ 0  0 −. 0 This distribution is extensively used in physics of low- temperature properties of glasses where two-level tunneling sys- tems are responsible for low-temperature thermal and transport properties.34–37It leads to the same physical conclusions as the distribution8/since the most important ingredients—1be-

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scatter in the fitting parameter 0 for different working −1 points, most values falling within the range of 3– 20s . Using the maximal value of 0 one can obtain an upper es- timate for the change of flux in the qubit loop as one fluc- −5 tuator flips,b210 0 . Since 0 is expected to grow with temperature, we believe that it would be instructive to perform similar analysis of the echo decay at different tem- peratures.

III. CONCLUSIONS By introducing the spin fluctuator model for 1/fnoise in the qubit level splitting we have determined the time depen- dence of the echo signal. We show that the standard qua- dratic time dependence in the Gaussian approximation Eq. 3 4has a limited range of applicability, andtortdependen- cies are found beyond this range. Fitting to the SF model also allows us to determine the strength ofindividualfluc- tuators, and for the flux qubits reported in Ref.1the change of flux in the qubit loop due to the flip of one fluctuator was −5 found to beb210 0 .

ACKNOWLEDGMENTS

This work was partly supported by the Norwegian Re- search Council, Funmat@UiO, and by the U.S. Department of Energy Office of Science through Contract No. DE-AC02- 06CH11357. We are thankful to Y. Nakamura, F. Yoshihara, and K. Harrabi for helpful discussions and for providing ex- perimental data.

havior at smalland cutoff at=—are present. In particular, 0 both distributions reproduce 1/behavior of the noise spectra. 9 A. Ludviksson, R. Kree, and A. Schmid, Phys. Rev. Lett.52, 950 1984. 10 S. M. Kogan and K. E. Nagaev, Solid State Commun.49, 387 1984. 11 V. I. Kozub, Sov. Phys. JETP59, 13031984. 12 Yu. M. Galperin and V. L. Gurevich, Phys. Rev. B43, 12900 1991. 13 Y. M. Galperin, N. Zou, and K. A. Chao, Phys. Rev. B49, 13728 1994. 14 J. P. Hessling and Y. M. Galperin, Phys. Rev. B52, 50821995. 15 E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett.88, 2283042002. 16 E. Paladino, L. Faoro, A. D’Arrigo, and G. Falci, Physica EAm- sterdam18, 292003. 17 G. Falci, E. Paladino, and R. Fazio, inQuantum Phenomena in Mesoscopic Systems, edited by B. L. Altshuler and V. Tognetti IOS Press, Amsterdam, 2003. 18 G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. A70, 040101R 2004. 19 G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett.94, 1670022005. 20 Y. M. Galperin, B. L. Altshuler, J. Bergli, and D. V. Shantsev, Phys. Rev. Lett.96, 0970092006.

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