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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, 0645312007 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, 1670012006provided 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 numbers: 03.65.Yz,85.25.Cp 1 A remarkable workhas appeared recently, reporting onAssuming that the statistics of the fluctuations oft 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 /dtcostt0: 0 2 sults in terms of the recent theoryof the decoherence in 2 24 qubits caused by 1/fnoise and obtained the qubit dephasingvt sint/4 2−t/4 t= exp−dS.2 rate,, based on the postulation of the Gaussian statistics of2 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. We2 −t .ln 24 t=e, vA 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 timeT1, the switching of a fluctuator between its two metastable states1 −t/2T 1 echo signal is expressed aset, wheretis the arrival and 2produces a random telegraph noise. The process is time of the echo signal. The factortdescribes the pure characterized by the switching rates12and21for 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 SQUIDloop, the echo signal is determined by the fluctu-in their ground states. As long asETthe rates12and21 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 that1221The random/ 2. t= exp−iv−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 noisesitions during the timebeing 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. 1098-0121/2007/766/0645315The American Physical Society064531-1 ©2007
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− cesses istt+=ewhere/ 4and the correspondingminis 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-P0T1 orNT2L. After that Eq.5 yields tors coupled to the given qubit via constantsviand having the switching ratesi, the dephasing of the noise power spec-2 22A1 ,01 number of effective fluctuators is sufficiently large, the sum trum is expressed through the sumivi/+. If the i iS=AP0T.10 20/,0, 8 over the fluctuators transforms into an integral overvand We see that the SF model reproduces the 1/fnoise power weighted by the distribution functionPv,. Upon the as-−1 −2 spectrum3for0. The crossover fromtobe-sumption that the coupling constants and switching rates are havior at0follows from the existence of a maximal uncorrelated, the distribution density factorizes,Pv, switching rate0. Below we will see that this crossover =PvvP, and the noise spectral function due to fluctua-modifies the time dependence of the echo signal at timest 2 −1 tors reduces tovS, where . 0 The SF model has previously been used for description of 1Pd9–14 4+plied to analysis of decoherence in charge qubits. 2 2effects of noise in various systemsand was recently ap-v dv vPvv,S= .5 2 27,15–22 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=ibiit, wherebiare pairs in the conducting part of the system. The external de-partial amplitudes whileitare random telegraph pro-grees of freedom are coupled with the fluctuator via modu-2 2 s¯v cesses. Consequently, we express the productvAaA lation ofand, modulation of the diagonal splitting 2 2 where¯vvb. 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 HF-env=cˆz−x,6 E E Substitution of the distribution8into Eq.2yields for wherecˆis an operator depending on the concrete interactionKt −lnt 2 mechanism. Accordingly, the factor/Eappears in the t/6 ,t1 0 0 interlevel transition rate:2 2 2,ln 20t1 . g¯v1 Kt=At1 E,=/E0E.7 The subscriptgmeans that this result is obtained in the Here the quantity0has 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 theandbehaviors 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 factor0t, which is nothing but the probability for a Since only the fluctuators withETare 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 distribution3 2 KgttreplacingKgttbehavior. ofEto be almost constant. Denoting the corresponding den-sity of states in the energy space asP0we arrive at the 8 B. Non-Gaussian theory distribution of the relaxation rates as The echo signal given by Eq.1can be calculated exactly P0T using the method of stochastic differential equations, if the P=0−.8 fluctuating quantityis a single random telegraph process, see Ref.7and references therein. The method was developed The productP0Tdetermines the amplitude of the 1/fnoise. 24 in the contexts of spin resonance,spectral diffusion in Indeed, the integralPdis nothing but the total number 25–28 glasses, andsingle molecular spectroscopy in disordered of thermally excited fluctuators,NT. Consequently, 29–32 media. Averagingin Eq.1is performed over random P T=N/L,Lln/,9realizations ofand its initial states and reflects the conven-0T0 min 064531-2
NON-GAUSSIAN DEPHASING IN FLUX QUBITS DUE TO…
PHYSICAL REVIEW B76, 0645312007
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>>γ16, Eq. quence of inputs. 0 2 2 ρ~ exp (-Γt ) For a single fluctuator with switching rate/ 2and cou-1 20 plingvwe find −t/2 2 e2v 2 t/2 −t/2 1 + 1t=2e+1 −e− , 2 0.1 where=1 −v/. In the appropriate limits this can be 2 expanded to give 2 3−1 vt/48,t,v −1 0.01 − ln1tv, t/2 ,,tv12 vt/2,v,t1. 2 − 0.1 1 timeµs 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 onlineDephasing component of the echo mea-24 resonance. surements replotted from Fig. 4aof 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.14and15, and NT1, the sum of the contributions from individual fluctua-2 2 −t to the=elawRef.38. The fitting took into account all data tors gives the echo decay decrement points including those that fall outside the range of the plote.g., those with1. Ksft= −dvPvvdPln1tv,.13 2 3−1 0A¯vt/6 ,t¯v, 4At,t Let us assume that the distribution ofvis a sharp function Ksft 15 −1 centered at some value¯v. This reduces integration overvto0¯v. merely replacingv→¯vin the expressions forv,t. This In this case all fluctuators havev, hence the result at long 2 approximation is valid as long asvis 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 divergentvis considered in Refs.7,20, and33. Using then Eq.8for the distributionII. DISCUSSION functionPand using the appropriate terms from Eq.12in Here we apply the results obtained above to analyze quan-Eq.13we find the time dependence of the logarithm of the titatively the decoherence of the flux qubit. Figure1shows echo signal,Ksft. For¯v0 the time dependence of the echo signal measured in Ref.1 2 3−1 0A¯vt/6 ,t, anda fit based on the SF model. We consider first the case of 0 2 2−1 −1¯v0where the SF model predicts a crossover fromttot 3 Ksft ln 2¯vAt,t¯v,14 0 3 −22 ¯vAt,−t,interpolation formula,Ksft=0At/6¯v+t/ 4. Figure1 dependence, Eq.15. We replace the exact result by the 1 ¯v 2 2 −t where6. alsoshows the commonly used fit=e, which is repre-At small timest¯vwe 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, Eq11seems 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 splittingE01due to a flip −1 the prediction of the Gaussian approximation. To understandof one fluctuator,¯v3.5s .It allows us to evaluate the change of flux in the qubit loop induced by a fluctuator flip, the reason, notice that forP1 /, the decoherence is dominated by fluctuators withv. The physical reason forb=¯v, sincev=1 /E01/was measured in Ref.1. /v that is clear: very “slow” fluctuators produce slow varyingUsing the experimental values for all parameters we getb −6 fields, which are effectively refocused in the course of the3.7100, where0is the flux quantum while the −3 echo experiment, while the influence of too “fast” fluctuatorsdeviation from the optimal working point was100. is reduced due to the effect of motional narrowing. As shownHence 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 withvproduce Gaussian ionvwe find a noise. Consequently, the noise in this case is essentiallynon-product0Aand within our assumpt0 Gaussian. Only at short timest¯vlower estimate for the flux noise amplitudewhen these most im-A/01.3 −1 −6 portant fluctuators did not yet have time to switch, and only10 . the faster fluctuators contribute, is the Gaussian approxima-Similarly, if¯v0we can fit to Eq.14using the inter-2 3−1 tion valid.polation formulaKsft=¯vAt/60+t/ ln2. We do not in-For¯v0we findclude here theKsftbehavior 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 onlineSymbols: parametervAdetermined by fittingtdata at different working points to the SF model, Eq. 14. Line: a fit based on thevdependence measured in Ref.1 whereis the optimal point and= 0Ais the only fitting parameter.
−1 toKA1. The fitting gives12which is thens , sf 0 also an upper estimate for¯v, andA/010 . −6 It is not possible to determine from Fig.1which of the cases¯v0or¯v0is realized in the experiment. There-fore we applied the same fitting procedure to the whole set of tcurves analyzed in Ref.1measured at different working points. Then it became clear that the formulas for¯v0give a better overall fitting. The fitting parameter¯vA=vAis plotted in Fig.2as a function of the working point. The data can be very well-described using thevdependence measured in Ref.1, see the solid line, and we findA/ 0 −6 = 1.1510 .This value is quite close to that obtained in the 1 Gaussian approach.Surprisingly, we found a significant
*iouri.galperine@fys.uio.no 1 F. Yoshihara, K. Harrabi, A. O. Niskanen, Y. Nakamura, and J. S. Tsai, Phys. Rev. Lett.97, 1670012006. 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, 0470042007. 4 A. Shnirman, Y. Makhlin, and G. Schön, Phys. Scr., TT102, 147 2002. 5 Y. Makhlin and A. Shnirman, Phys. Rev. Lett.92, 1783012004. 6 S. Kogan,Electronic Noise and Fluctuations in SolidsCam-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 forone arrives atP, which readsP=P T/ 21 −/00−. 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 distribution8/since the most important ingredients—1be-
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scatter in the fitting parameter0for different working −1 points, most values falling within the range of 3– 20s . Using the maximal value of0one can obtain an upper es-timate for the change of flux in the qubit loop as one fluc-−5 tuator flips,b2100. Since0is 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 4has 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 beb2100.
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 smalland 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, 13031984. 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, 50821995. 15 E. Paladino, L. Faoro, G. Falci, and R. Fazio, Phys. Rev. Lett.88, 2283042002. 16 E. Paladino, L. Faoro, A. D’Arrigo, and G. Falci, Physica EAm-sterdam18, 292003. 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, 040101R 2004. 19 G. Falci, A. D’Arrigo, A. Mastellone, and E. Paladino, Phys. Rev. Lett.94, 1670022005. 20 Y. M. Galperin, B. L. Altshuler, J. Bergli, and D. V. Shantsev, Phys. Rev. Lett.96, 0970092006.
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