Decoherence in Josephson and Spin Qubits. Lecture 3: 1/f noise, two-level systems

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1 Decoherence in Josephson and Spin Qubits Alexander Shnirman University of Innsbruck Lecture 3: 1/f noise, two-level systems 1. Phenomenology of 1/f noise 2. Microscopic models 3. Relation between T1 relaxation and 1/f noise

2 1/f noise charge noise: - charge noise: charged defects in barrier, substrate or surface lead to non- integer induced charge. Static offset, 1/f noise. -critical current noise: neutral/charged defects in barrier. - flux noise: trapped vortices, magnetic domains, magnetic impurities, nuclear spins

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4 1/f noise as sum of many telegraph noises (Dutta-Horn model) E

5 1/f noise, longitudinal linear coupling Golden rule: exponential decay with fails for 1/f noise, where Non-exponential decay of coherence (assuming Gaussian statistics) Cottet et al. (01)

6 G. Ithier et al. PRB 05 Saclay, Charge Flux Qubit F. Yoshihara et al. NEC, Flux Qubit K. Kakuyanagi et al. NTT, Flux Qubit

7 Big puzzle: universality of flux 1/f noise K. Kakuyanagi et al. (NTT) F. Yoshihara et al. (NEC) 7±3x10-6 [ 0 ] SQUID~ m 2 F.C.Wellstood et al. APL50, 772 (1987) 1.5x10-6 [ 0 ] phase qubit ~10000? m 2 (UCSB) ~ 1x10-6 [ 0 ] flux qubit ~1000 m 2 (Berkeley) ~ 1x10-6 [ 0 ] flux qubit ~ 100 m 2 (NTT) Loop size independent?

8 1/f noise, longitudinal linear coupling dephasing of echo signal π/2 π π/2 0 t/2 t

9 Y. Nakamura et al. Dephasing rate reducd by only Bang-bang: frequent flips

10 High-frequency cutoff 500 Spin echo Free decay S Ng 1/ω δ/2π N g -1/2 10 ω 0.5MHz 1/f noise without cutoff would imply Data consistent with hf cutoff at ~ 0.5MHz

11 Symmetry point Vion et al. (Saclay 02) Idea: Go to: Quantronium E C E J less sensitive to charge noise

12 Dephasing at symmetry points Low frequency noise, transverse coupling Adiabatic approximation for quadratic longitudinal noise

13 Linked cluster expansion (analytical) ½ 01 ( t ) / P( t ) f Ramsey ( 0 < < t ) = 1 * i e f Echo ( 0 < < t =2) = 1 & f Echo ( t =2 < < t ) = 1 Yu. Makhlin and A.S., PRL 2004 tr d f ( ) X 2 ( ) + 0 ln P( t ) = X 1 n F n = F n = ( 2i ) n 2 Zt 0 Zt d 1 In Fourier representation 0 d 2 : : : Zt 0 d n f ( 1 ) S X ( 1 2 ) f ( 2 ) : : : S X ( n 1 )

14 Determinant regularization (numerical)

15 Yu. Makhlin, A.S. 04 f» E in general E. Paladino et al. 04 D. Averin et al. 03 1/f spectrum quasistatic even more general

16 Short time decay (static contribution)

17 Driven qubit: Quadratic coupling again Hamiltonian with driving term Hamiltonian in the rotating frame (RWA) at resonance Purely transverse noise 1= p t decay

18 Origins of T1 relaxation Often random as a function of energy splitting

19 Origins of T1 relaxation Losses in dielectric, material dependence Saturation -> Spin (TLS) bath J. Martinis et al. 05

20 Origins of T1 relaxation Coherent two-level systems in JJ J. Martinis et al. 05 Cooper et al. (04) TLS coherence time longer than that of qubit!!!

21 Origins of T1 relaxation Different dielectric materials J. Martinis et al. 05

22 Qubits as spectrometers Bloch-Redfield theory (regular spectra, exponential decay) transverse noise relaxation longitudinal noise dephasing 1/f noise probed in exp s of Astafiev et al. longitudinal noise

23 Experimental data of Astafiev et al. (NEC) S X ( ) 1/f f T Astafiev et al. (PRL 04) Low frequency 1/f noise crosses f quantum noise at h c Η k B T same strength for low- and high-frequency noise

24 Model A. S., G. Schön, I. Martin, Yu. Makhlin, PRL 05 Fluctuations X(t) probed by qubit Source of X(t) is an ensemble of two-level systems (TLS) each TLS is coupled (weakly) to dissipative environment weak relaxation and decoherence TLS TLS Qubit TLS TLS TLS

25 Noise of a single TLS In eigenbasis Correlation function

26 Distribution function for linear -dependence exponential dependence on barrier height overall factor explains observed spectrum S X (ω)

27 Model for P(ε) Faoro, Bergli, Altshuler, Galperin (2004)

28 Microscopic models Paladino, Faoro, Falci, Fazio (2002) +... Galperin, Altshuler, Shantsev (2003) Faoro, Bergli, Altshuler, Galperin (2004) Grishin, Yurkevich, Lerner (2004) de Sousa, Whaley, Wilhelm, von Delft (2005) Faoro, Ioffe (2006,2007)

29 Strong fluctuators 1/f-noise can arise from a set of two-level fluctuators one fluctuator v << γ : weakly coupled, Gaussian approximation works γ << v : strongly coupled, non-gaussian, memory effects Paladino et al., PRL (2002) Galperin et al. PRL (06) Smooth 1/f noise vs. single strong fluctuators? Both observed. How are they related?

30 Strong fluctuators Paladino et al., PRL (02) Galperin et al. PRL (06)

31 Strong fluctuators Many fluctuators Ensemble distribution function Galperin et al. PRL(06) Long tail of very strong fluctuators: central limit theorem does not apply Strong non-gaussian effects, but also no self-averaging Average over ensemble is far from what happens in a single sample

32 Summary 1. Longitudinal and transverse coupling, energy relaxation (T1) and dephasing (T2) 2. 1/f noise dominates dephasing, strongly affected by echo 3. Symmetry point ( sweet spot ), role of quadratic coupling 4. Dissipation still dominated by natural sources, need for better materials 5. Qubits can serve as spectrometers. This allows us to study the noise sources. 6. Relation between high- and low- frequency noise 7. Many puzzles, microscopic origin of noise needs further study