1. Measurements of the 1/f noise in Josephson Junctions and the implications for qubits Jan Kycia, Chas Mugford- University of Waterloo Michael Mueck- University of Giessen, Germany John Clarke- University of California, Berkeley Readout SQUID

2. The Group Chas Mugford 1/f noise Jeff Mason SQUID NMR Shuchao Meng SQUID-sSET Lauren Lettress TES sensors Jeff Quilliam Ho: YLiF4 Nat Persaud Liquifier

3. dc-SQUID The most sensitive magnetometer ~1 µ  o / (Hz) 1/2 IbIb IoIo IoIo L V L in

4. Josephson Junctions Superconductor,  1 22 Oxide barrier ICIC The superconducting order parameters are: The phase difference between the superconductors is  =  2 -  1. As the two superconductors are brought closer together, allowing electrons to tunnel, the phases start to interact. Josephson (1962) predicted a phase dependent energy = -E J cos , where E J = =.,  1 = |  1 (x)| e i  1,  2 = |  2 | e i  2 hI co  h  2e (2e) 2 R N I S = I co sin (  ), d  2eV dt h = 22

5. Resistively and Capacitively Shunted Junction Model I = I co sin  + + C V dV R dt hC d 2  h d  2e dt 2 2eR dt + + I co sin  = I Use V =, h d  2e dt Tilted washboard model is the mechanical analog, with a particle of mass ~ C, moving along an axis, , in a potential, U(  ) = -I co cos  - I , with a viscous drag force,. EJEJ C R h d  2eR dt Tilt  I position   velocity  V One period =  o  h  (2e) 2 R N E J = I S = I co sin (  ), d  2eV dt h =

6. I I/2 +J I/2 -J J J 12    L 12   I max LL  The DC SQUID 12   V

7. A flux locked loop using a high frequency flux modulation is used to provide a flux to voltage converter with fixed gain and large dynamic range. 12   V dd dVdV

8. dc SQUID Input coil 2x2µm Junctions Palladium Shunt resistor Magnified Image of DC SQUID Shunts provide required dissipation but also produce noise.

9. SQUID as a near-quantum-limited amplifier at 0.5 GHz M. Mueck, J. B. Kycia, and John Clarke, APL 78, 967 (2001). Wellstood et al found that self heating can be reduced by adding a cooling pad to the shunt resistor. Loss of temperature dependence, at low temperatures, is frequency independent Find “Self Heating” at low temperatures

10. The Hamiltonian, H = -E J cos  1 -E J cos  2 + E c (Q/e) 2 if E J / E C > 1,  is a good quantum number, Q fluctuates. if E J / E C < 1,  fluctuates, Q is a good quantum number. Phase fluctuations allow the particle to diffuse down the washboard; d  d t  0  V  0

11. Transport Measurement Circuit with filters Sample Cu filters RC  filters LC  filters 20 mK 4.2 K 300 K x100 x1000 Lock-in reference input. AC bias  0.1 nA Screened room... Copper powder filters Follow design of Martinis, Devoret, Clarke, Phys. Rev. B, 35, 4682 (1987).

12. G ~ ; ohmic G ~ ; transmission line g T 2 g 1/3 T 5/3 (Wilhelm, Schön, and Zimanyi) dissipation, g = RKRgRKRg Temperature and dissipation dependence of sSET (Ingold, Grabert PRL 1999)

13. H = -E J (  )cos  1 -E J (  ) cos  2 + E c (Q/e) 2 + H(R 2deg ) New configuration Provides in situ control of E J, E c, g and T.

14. 1  m SEM image courtesy of Dan Grupp.

15. Wagenblast, Otterlo, Schon, Zimanyi PRL 1997 Rimberg, Ho, Kurdak, Clarke PRL 1997 Good review: Leggett, Chakravarty, Dorsey, Fisher, Garg, Zwerger Rev Mod Phys (1987).

16. Superconducting Qubits: Charge based qubits: “Cooper pair box” Demonstrated Rabi oscillation: Nakamura et al, Nature 398, 786 (1999). Improved read out scheme, decoherence time ~ 0.5  s (Q = 25,000): Vion et al, Science 296, 886 (2002). Flux based qubits: Demonstrated energy splitting dependence on applied magnetic flux: Friedman et al, Nature 406 43 (2000), van der Wal et al, Science 290, 773 (2000).Coherent Oscillations observed with a dephasing time of 20 ns and a Relaxation time of 900 ns: Chiorescu et al, Science 299, 1869 (2003). Phase based qubits: Exhibited Rabi oscillations between ground state and 1 st excited state of a current biased Josephson junction in its zero-voltage regime: Yu et al, Science 296, 889 (2002), Martinis et al PRL 89, 117901 (2002).

17. Sources of Decoherence: External flux noise Nyquist noise currents in nearby metal objects Noise in the measurement scheme Motion of trapped charge 1/f “flicker” noise in the critical current of the Josephson Junction The goal of our experiment is to measure the level of 1/f noise in the critical current of a resistively shunted Josephson Junction. Once the measurement is made, we can: Measure the temperature, time, material, and fabrication parameter dependence of the 1/f noise level. Estimate the upper limit of the coherence time of superconducting qubits due to these noise sources. Make optimal qubits by selecting the device configuration to minimize the noise sources.

18. -Small loop with three Josephson junctions produces the flux qubit. - Hysteretic DC SQUID is used to read the flux state. Chiorescu et al, Science 299, 1869 (2003). van der Wal et al, Science 290, 773 (2000). Flux Qubit

19. Ramsey Fringes in Flux Qubit I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij, Science 299, 1869 (2003).

20. “Quantronium” D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M. H. Devoret, “Manipulating the Quantum State of an Electrical Circuit”, SCIENCE, 296, 886 (2002).

21. Decoherence in Josephson Phase Qubits from Junction Resonators Simmonds, Lang, Hite, Nam, Pappas, and Martinis, Phys Rev Lett, 93, 077003-1, (2004). “Phase” Qubit

22. a b c d e f Decoherence in Josephson Phase Qubits from Junction Resonators Simmonds, Lang, Hite, Nam, Pappas, and Martinis, Phys Rev Lett, 93, 077003-1, (2004). Resonances Observed -- likely due to defects (fluctuators)

24. 1/f Noise: Dutta-Horn Model Dutta and Horn, Rev Mod Phys, 53, 497 (1981) Random telegraph signal is produced by random transitions between the states of a double potential well. Define 1/  1 and 1/  2 as the probability of a transition from state 1 to 2 and 2 to 1 respectively. If 1/  1/  1 = 1/  2 then the power spectrum is a Lorentzian of the form S(f)   f  2 ] If the transitions are thermally activated then the characteristic time is given by  i =  o exp(U i /k B T), where 1/  o is an attempt frequency. S(f,T) is linear in T because the kernel moves through the distribution of RTS’s as the temperature varies, selecting only those processes that have characteristic frequencies in the window of interest.

25. s.c.2 s.c.1 barrier trap z barrier height I V no electron trapped electron trapped Mechanism Behind 1/ƒ Critical Current Fluctuations in Josephson Junctions The currently accepted picture of the mechanism behind critical current fluctuations involves traps within a Josephson junction. An electron is trapped in the tunnel barrier and is subsequently released. While trapped, the barrier height and hence critical current is modified temporarily. For a junction of area A the change in critical current is modified by the change in area due to an electron  A.  I c =(  A/A)I c

26. Dephasing due to current fluctuations and critical current fluctuations Critical current fluctuations with a l/f spectral density are potentially a limiting source of intrinsic decoherence in superconducting qubits..  = the frequency of oscillation between the +0.5  o and –0.5  o state.

27. Dale Van Harlingen et al, PRB (2004).

28. Methods of Measuring 1/ƒ Noise of the Critical Current of a Josephson Junction Critical current fluctuations have been measured in the non-zero voltage state. Is the 1/f noise the same when the junction is in the zero voltage state? We measured the critical current fluctuations using the same SQUID operated as an RF SQUID in the dispersive regime. Fred Wellstood’s Thesis Berkeley F.C.Wellstood, PhD thesis, University of California, Berkeley 1988. B.Savo, F.C.Wellstood,, and J.Clarke, APL 49, 593 (1986). V.Foglietti et al., APL, 49, 1393 (1986) R.H.Koch, D.J. van Harlingen, and J.Clarke, APL, 41, 197 (1982). F.C. Wellstood, C. Urbina, John Clarke, APL, 5296, 85 (2004).

29. Comparing different junctions: Invariant noise parameter Normalize current noise spectrum to the critical current Choose T = 4.2K and 1Hz. But this does not take into account junction area. For a junction of area A and if the area blocked by a single trap is  A, then change in critical current for a single fluctuator is  Ic = (  A /A)I c If n is the number of traps per area, then the critical current spectrum should scale as: S I 2 ~ n A (  A /A) 2 I c 2 = n  A 2 (I c 2 /A) Van Harlingen et al found that all values of n  A 2 remarkably similar for all measured junctions. S I 2 scales as (I c 2 /A)

30. Scaled quantity invariant quantity (van Harlingen et al. PRB 2004) Wellstood et al. average value of 6 junctions 26 Lukens et al. IEEE 2005 Also see “slower than linear” T dependence

31. Measuring 1/ƒ Noise Due to Critical Current Fluctuations in the Non-Zero Voltage State Readout SQUID DC SQUID and read-out SQUID circuit The sample SQUID is voltage biases. The readout SQUID measured the current running though the 2  resistor. Fluctuation in the critical current leads to a redistribution of the currents flowing through the junction and the resistor.

32. rf tight SMA connectors Readout SQUID Sample SQUID Superconducting lead shield rf tight copper sample container Coaxial µ-metal shields rf tight - low field - superconducting sample container

33. 1/f noise in DC biased junction

34. Current bias reversal eliminates 1/f noise, therefore this 1/f noise is not due to flux noise. Applying Current Bias Reversal DC current bias method

35. Critical current fluctuations due to a single fluctuator Ic = 2.5uA  Ic = 0.65nA This corresponds to a trap radius of ~ 5.6nm

36. Reading out an rf SQUID in the Dispersive Regime - A tank circuit is driven off-resonance with a 360-MHz current of fixed amplitude. - The tank circuit voltage is read out with a low noise amplifier cooled to 4.2K. - Fluctuations in the critical currents of the two junctions modulate the SQUID inductance and thus the resonant frequency of the tank circuit. V rf ƒ mod rf SQUID and FET amplifier circuit

37. Comparing the zero-voltage noise measurement method to the non-zero voltage noise measurement method No difference between the measurements. The 1/f noise is temperature dependent.

38. Annealing Study Annealing lowers critical current and lowers noise

39. Van Harlingen et al. 12 Wellstood et al. 26 Lukens et al. 1.5 Best Sample Comparison of Noise Parameter

40. Conclusion We have demonstrated that the l/f noise in a dc SQUID due to critical current fluctuations has the same magnitude in the zero voltage and non-zero voltage regime. Thus, the levels of critical current l/f noise measured by others in the nonzero voltage state should pertain to qubits operated at zero voltage. Measured noise of different junctions, reduce 1/f noise. Future Experiments Temperature dependence of 1/f noise down to dilution refrigerator temperatures. The dispersive method has no dissipation - best for low temperatures. We can cut away the shunt resistors to see if they are somehow responsible for noise. Continue varying processing parameters. Study dissipation is submicron Josephson junction.

42. New Device will allow the in situ control of E J, E C, and dissipation.

44. Temperature and dissipation dependence of sSET

45. Outline - Describe how Josephson Junctions and SQUIDS work. - Describe how superconducting qubits work. - Explain why 1/f noise is relevant to superconducting qubits. - Present results on 1/f noise measurements.

46. Tunable coupling via curent Bias current: Screening current: Extra flux at constant bias directly increases screening increases γ → indirectly reduces screening B.L.T. Plourde, J. Zhang, K.B. Whaley, F.K.W., T.L. Robertson, T. Hime, S. Linzen, P.A. Reichardt, C.E. Wu, and J. Clarke PRB 70, 140501(R) (2004).

48. rf SQUID Two kinds of behaviour are observed in rf SQUID loops depending of the “SQUID hysteresis parameter”  rf. The difference is seen in the applied flux  e vs the flux threading the loop . The SQUID hysteresis parameter is defined as: If  rf <1 the SQUID is dispersive. Ic is never exceeded If  rf >1 the SQUID is hysteretic or dissipative. 0 1  rf <1 R L ISIS 1 0  rf >1 01 01   e

49. Critical current fluctuations may be a major source of intrinsic decoherence of qubits made from Josephson junctions. We have measured the 1/f noise due to critical current fluctuations in macroscopic ( area  2  2  m 2 ) Josephson junctions. We have exploited two ways for measuring critical current fluctuations, one way where we directly measure changes in the critical current of a voltage biased junction, and a second way in which we measure 1/f flux noise in an rf SQUID running in the dispersive mode. With both methods, we find the magnitude of the critical current fluctuations, at a temperature of 4.2K, to be  I c /I c  10 -5 at a frequency of 1 Hz. Abstract

50. The Bloch sphere Convenient representation of the two-state Hamiltonian and state B eff

51. Tank Circuit Coupled to Josephson Inductance Using the Josephson relations: A Josephson element can be described as a nonlinear inductor by deriving the relationship Where: When a junction is inserted into a superconducting loop it’s behaviour affects the total inductance of the loop. The flux threading the loop is The effective inductance of a SQUID can be approximated by and the circulating supercurrent Combining these three it follows that: Coupling a SQUID loop to a inductor of a tank circuit yields an effective tank circuit resonance modified by the SQUID loop for  rf <<1:

52. The flux qubit Evidence for superposition of macroscopic states C.H. van der Wal, A.C.J. ter Haar, F.K.W., R.N. Schouten, C.J.P.M. Harmans, T.P. Orlando, J.E. Mooij, Science 290, 773 (2000).  φ

53. Measuring 1/ƒ Noise Due to Critical Current Fluctuations in the Zero Voltage State Using an rf SQUID in Dispersive Mode Applying an external flux gives rise to a circulating current which in turn modifies the inductance of the junctions. Fluctuations in the critical current I c appear as equivalent to flux noise. Operating the SQUID in the dispersive regime means that the screening current imposed by an applied flux is always smaller than the critical current I c of a junction.

54. Critical Current Noise Specific Measurement Techniques The spectral density components of low frequency SQUID noise are represented by. S  (f): flux noise due to motion of flux vortices. S I (f): critical current noise – in-phase fluctuations are represented by the second term and the out-of-phase component is represented by the third. AC flux modulation with lock-in detection rejects only the in-phase component of the critical current noise, furthermore it does not affect noise due to flux motion. Reverse bias scheme will eliminate out-of-phase fluctuations in the critical current but does not affect out-of-phase fluctuations due to flux motion. Thus ac modulation with reverse bias will eliminate in-phase and out-of-phase fluctuation due to critical current fluctuations. Therefore if excess noise due to motion of flux vortices exists, the out-of-phase component will still be observed

55. Lab at Waterloo Dilution refrigerator (Base temperature 12 mK) e-beam lithography (for fabrications of sub-micron devices) Optical lithography (for fabrications of large number of micron scale multi-layered devices) Measurement electronics (low noise environment, low 1/f noise)

56. Conclusion of sSET work