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Noise and decoherence in a dc SQUID

CNRS-UJF team (Grenoble). CRTBT - LCMI - LP2MC. PhD : J. Claudon A. Fay A. Ratchov. permanent : W. Guichard F.W.J. Hekking L. Lévy O. Buisson. Noise and decoherence in a dc SQUID. | 4 . | 3 . | 2 . | 1 . | 0 . Introduction. dc - SQUID. artificial atom. =. 10 GHz.

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Noise and decoherence in a dc SQUID

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  1. CNRS-UJF team (Grenoble) CRTBT - LCMI - LP2MC PhD : J. Claudon A. Fay A. Ratchov permanent : W. Guichard F.W.J. Hekking L. Lévy O. Buisson Noise and decoherencein a dc SQUID

  2. |4 |3 |2 |1 |0 Introduction dc - SQUID artificial atom = 10 GHz Low energy sub-space { |0 , |1 } a phase Qubit (cf. J. Martinis et al.) Higher energy sub-space { |0 , |1 ,|2 , ... } more complex dynamics : eg. multilevel coherent oscillations

  3. S1 S2 The Josephson Junction (JJ) A S-I-S junction CJ model Ib Ib   Ox C0 Ic U() U (Ib) |3 |2  dynamics? mechanical analogy |1 |0 • fictitious particle • mass  C0 • position  • potential U() p(Ib) 

  4. The current biased dc SQUID:a tunable artificial atom Ib DU (Ib,Fb) Ls Fb I0 , C0 I0 , C0 JJ1 p (Ib,Fb) shape of the anharmonic well bias point (Ib,Fb)

  5. The current biased dc SQUID:a tunable artificial atom Ib DU (Ib,Fb) Ls F(t) MW Fb I0 , C0 I0 , C0 n,W1 JJ1 p (Ib,Fb) shape of the anharmonic well bias point (Ib,Fb) MW flux F(t) excitation

  6. Outline • I. Experimental setup • II. Finite lifetime of the ground state • MQT measurement and LF noise • III. Quantum dynamics • Multilevel coherent oscillations • Quantum or classical oscillations? • IV. Incoherent processes in the 2 level limit • Low power spectroscopy • Energy relaxation |2 MQT |1 |0 MW |2 |1 |0 |1 Gr |0

  7. Process • e-beam lithography • Al shadow evaporation Sample realization SiO2 200 m MW antenna bias lines 15 m JJs 15 μm² superconducting loop

  8. Vs MW line dc - 20 GHz heavily filtered bias lines dc - 200 kHz • MW excitation • fast measurement of • the quantum state • SQUID bias(Ib,Fb) • voltage readout Experimental setup (simplified) ~ 25 mK Ze() Loc Ib Fhf Ib 50 Ω Vhf Fb Cp coil

  9. Ib-Vs hysteretic characteristic 3.0 1.5 0 -1.5 -3.0 Vs (mV) -500 -250 0 250 500 Voltage states of a hysteretic dc-SQUID Ic Ib (mA) 0-voltage state resistive state |1 |0

  10. Ib-Vs hysteretic characteristic 3.0 3.0 2.5 1.5 2.0 1.5 0 1.0 -1.5 0.5 -3.0 Vs (mV) -500 -250 0 250 500 0.0 -0.5 0 0.5 Bias point (Ib, b) potential shape Voltage states of a hysteretic dc-SQUID “Phase diagram” Ic Ic Ib (mA) Ib (mA) Fb/F0 0-voltage state resistive state Ic =f(b) SQUID electrical parameters I0 = 1.242 A C0= 560 fF LS = 250 pH  = 0.414 |1 |0

  11. Measurement of : principle Ib T=50 μs TA ~ 4000 repetitions (1 kHz) |1 |0 escape Vs MQT t 1 0.9 Pesc 0.5 0.1 ΔI 0 2.38 2.4 2.42 Ib (µA) Escape of the trapped fictitious particle Dependence on the bias current Ib During ΔT, escape probability : ΔI dominant escape mechanism

  12. 30 20 10 -0.4 -0.2 0 0.2 0.4 20 18 16 14 12 10 -0.5 0 0.5 Escape at low temperature ~ 40 mK F. Balestro’s SQUID MQT for all fluxes fit : no free parameters DI(nA) MQT F. Balestro et al, PRL 2003 Fb/F0 ~ 25 mK This sample small deviation from MQT which scales as DI(nA) flux noise MQT Fb/F0

  13. Noise and escape measurements Noise on bias parameters : or • equivalent effects • difference : flux sensitivity is strongly modulated by • Hypothesis : • adiabatic : frequency • gaussian fluctuations Effect on escape curve : depends on noise frequencies compared to Dt

  14. I (t) b Dt ... ... ... t broadening ~ 0 I b 1 0.5 0 1.7 1.72 1.74 1.76 1.78 Noise and escape measurements Low frequency limit (< 1/Dt) MQT & LF noise Pesc MQT Ib (mA)

  15. I (t) I (t) b b Dt Dt ... ... ... ... t t broadening ~ 0 0 I I 1 b b 1 0.5 0.5 0 0 1.7 1.72 1.74 1.76 1.7 1.72 1.74 1.76 1.78 Noise and escape measurements Low frequency limit (< 1/Dt) High frequency limit (>1/Dt) MQT & LF noise Pesc Pesc MQT & HF noise MQT MQT Ib (mA) Ib (mA) shift ~

  16. 20 MQT & LF flux noise 18 16 DI(nA) 14 12 MQT This sample 5.5×10-4 10 -0.5 0 0.5 F. Balestro’s SQUID < 3×10-4 Fb/F0 Delft flux Qubit (loop area normalization) ~ 2×10-4 Interpretation of escape measurements :low frequency flux noise DI measurements : a probe to LF noise [0.2Hz,20kHz] • no significant LF current noise (fit uncertainty) • yes LF flux noise :

  17. |2 MQT |1 |0 Outline • I. Experimental setup • II. Finite lifetime of the ground state • MQT measurement and LF noise • III. Quantum dynamics • Multilevel coherent oscillations • Comparison to classical oscillations • IV. Incoherent processes in the 2 level limit • Low power spectroscopy • Energy relaxation MW |2 |1 |0 |1 Gr |0

  18. Quantum dynamics:typical sketch of experiments 3 3 3 1 2 2 1 Fb is fixed current Ip(t) Ib t 2 (t) Fm Ic flux Fb Fb+Fm bias point : shape of the potential manipulation : (deep well) adiabatic deformation : selective escape of excited states population of excited states MW initial state =|0 O. Buisson etal, Phys. Rev. Lett. 2003

  19. 1 Vhf (V) n = 5 GHz 0.5 t (ns) 0 0 2 4 6 8 Example of flux sequence Voltage signal(entrance of cryostat) excitation MW pulse measurement pulse risetime = 1 ns duration~ 1 to 300 ns risetime = 1.6 ns duration~ 1.5 ns amplitude : ~10-3F0 amplitude : ~10-2F0

  20. 60 40 20 0 Coherent oscillations Bias point : Ib = 2.222 mA , Fb = -0.117 F0 n01 = 8.287 GHz (low power spectroscopy) E/h (GHz) 7 levels 5 2 1 n01-n12 = 160 MHz n01 0 MW excitation • n = n01 • amplitude : W1 • duration TMW : variable

  21. 0.8 0.6 60 0.4 40 0.2 0.8 20 0.6 0 0.4 0.2 0.8 0.6 0.4 0.2 0 20 40 60 80 Coherent oscillations P = -6 dBm W1/2p = 65 MHz Bias point : Ib = 2.222 mA , Fb = -0.117 F0 ncoh = 66 MHz Pesc n01 = 8.287 GHz (low power spectroscopy) E/h (GHz) 7 levels P = 0 dBm W1/2p = 130 MHz 5 Pesc 2 1 n01-n12 = 160 MHz n01 0 ncoh = 122 MHz MW excitation P = +6 dBm W1/2p = 260 MHz • n = n01 • amplitude : W1 • duration TMW : variable Pesc ncoh = 208 MHz Attenuation time ~ 20 ns Tmw (ns)

  22. 250 200 How many levels? 150 anharmonicity excitation amplitude 100 coherent superposition of excited states 50 |0 0 0 50 100 150 200 250 300 Multilevel coherent oscillations solid line = multilevel theory (F. Hekking et al.) J. Claudon etal, Phys. Rev. Lett. 2004 ncoh (MHz) n01 MW amplitude : W1/2p (MHz)

  23. 250 200 150 100 coherent superposition of excited states 50 |0 0 0 50 100 150 200 250 300 Multilevel coherent oscillations solid line = multilevel theory (F. Hekking et al.) 2-level limit 2 J. Claudon etal, Phys. Rev. Lett. 2004 ncoh (MHz) 1 n01 MW amplitude : W1/2p (MHz) W1«n01-n12 Rabi oscillations between |0 and |1 cf. J. Martinis et al.

  24. 250 200 150 100 coherent superposition of excited states 50 1 |0 p1 0 0.5 Populations 0 50 100 150 200 250 300 p2 0 0 5 10 15 20 25 30 Tmw (ns) Multilevel coherent oscillations solid line = multilevel theory (F. Hekking et al.) 2-level limit 2 3 J. Claudon etal, Phys. Rev. Lett. 2004 ncoh (MHz) n01 MW amplitude : W1/2p (MHz) p2 < 12 % close to a 2-level oscillation

  25. 250 200 150 100 coherent superposition of excited states 50 1 |0 0 p1 0.5 Populations 0 50 100 150 200 250 300 p2 p3 0 0 2 4 6 8 10 12 14 16 Tmw (ns) Multilevel coherent oscillations solid line = multilevel theory (F. Hekking et al.) 2-level limit 2 3 J. Claudon etal, Phys. Rev. Lett. 2004 ncoh (MHz) MW amplitude : W1/2p (MHz) p3 < 3 % 3-level oscillation

  26. 250 200 150 100 coherent superposition of excited states 50 |0 0 0 50 100 150 200 250 300 Multilevel coherent oscillations solid line = multilevel theory (F. Hekking et al.) 2-level limit 2 3 J. Claudon etal, Phys. Rev. Lett. 2004 ncoh (MHz) 4 MW amplitude : W1/2p (MHz) W1»n01-n12 # involved levels  1 oscillations still exist classical description ?

  27. Classical Rabi-type oscillations U(j) n, W1 j(t) j(t) ncoh • At t=0, MW on • transient : anharmonicity of the well • modulation of the phased-locked state • energy modulation with frequency  ncoh (a. u.) • small excitation power & n = wp/2p Rabi-type oscillations in a classical Josephson junction N. Grønbech-Jensen and M. Cirillo (2005) cond-mat/0502521 A. Ratchov and F. Faure (LP2MC-Grenoble) unpublished

  28. 250 classical theory with n=wp/2p 200 150 100 50 0 0 50 100 150 200 250 300 Quantum VS classical description # involved levels 2 3 4 quantum theory with n=n01 ncoh (MHz) αΩ12/3 • Low excitation power : • quantum description • Higher excitation power • qualitative agreement • between classical and • quantum descriptions n01-n12 = 160 MHz αΩ1 MW amplitude : W1/2p (MHz)

  29. |2 MQT |1 |0 Outline • I. Experimental setup • II. Finite lifetime of the ground state • MQT measurement and LF noise • III. Quantum dynamics • Multilevel coherent oscillations • Comparison to classical oscillations • IV. Incoherent processes in the 2 level limit • Low power spectroscopy • Energy relaxation MW |2 |1 |0 |1 Gr |0

  30. 12 3 Ib (mA) 2 11 Ic 1 10 0 8 -0.5 0 0.5 9 Fb/F0 6 8 7 4 solid lines : semi-classical theory for X2X3 potential 0.5 1 1.5 2 2.5 2 SQUID electrical parameters 8 8.1 8.2 8.3 8.4 8.5 I0 = 1.242 A C0= 560 fF LS = 250 pH  = 0.414 Low power spectroscopy : resonance frequency n01 (GHz) Ib (mA) n01 ~ gaussian shape Pesc (%) n (GHz)

  31. 12 3 Ib (mA) 2 11 Ic 1 10 0 8 -0.5 0 0.5 9 250 Fb/F0 200 6 8 150 7 4 100 50 2 8 8.1 8.2 8.3 8.4 8.5 0 0.5 1 1.5 2 2.5 Low power spectroscopy : resonance linewidth n01 (GHz) ~ gaussian shape Dn (MHz) Pesc (%) Dn Ic Ic n (GHz) Ib (mA)

  32. 8 12 n01 11 6 Dm 10 120 4 9 100 8 2 80 7 60 0 0 0.1 0.2 0.3 0.4 0.5 0.6 40 0.5 1 1.5 2 2.5 Ib (mA) Energy relaxation Ib = 2.222 mA, Fb = -0.117 F0 1/Gr for different bias points n01 (GHz) Pesc (%) 1 / Gr (ns) Dm (ms) solid line Ic Ic 1 / Gr = 90 ns Qr = 6000

  33. current fluctuations flux fluctuations Transverse and longitudinal coupling SQUID coupling terms environment linear eigenbasis { |0 , |1 } transverse longitudinal relaxation “pure” dephasing

  34. LF flux fluctuators Relevant fluctuations sources Rmcs Rp(w) Loc Lf dI dF Cp Cmsc chip ~ 25mK Heavy filtering and shielding significant fluctuations sources located close to the SQUID quantum fluctuation dissipation theorem • electrical circuit • LF flux fluctuators MQT measurements [0.2Hz,20kHz]

  35. 12 Loc 11 Rp(w) 10 120 9 100 8 80 7 60 40 0.5 1 1.5 2 2.5 Energy relaxation Environment at high frequencies (> 5GHz) gold capacitor n01 (GHz) hypothesis :no high frequency flux noise 1 / Gr (ns) Ic Ic consistent with skin effect estimations Ib (mA) Dissipation is induced by the gold filtering capacitor

  36. STEP 1 Analysis of spectroscopic results reduced density matrix : (interaction frame) NOISE inelastic processes “pure” dephasing hypothesis : • gaussian fluctuations • linear coupling 0.2 Hz 20 kHz flux noise : f neglected (static approximation)

  37. Analysis of spectroscopic results current noise : STEP 2 Linear response NOISE NO MW NOISE MW (linear regime) Fourier transform resonance line shape Dn

  38. 12 11 10 9 250 200 8 150 7 100 50 0 0.5 1 1.5 2 2.5 Low power spectroscopy for various bias points n01 (GHz) • fit without free parameter • Ib Ic : Dn (MHz) longitudinal noise sensitivity and Ic Ic Dn Ib (mA)

  39. MQT & LF flux noise 250 20 DI(nA) MQT 18 200 Fb/F0 16 150 2 3 4 250 14 ncoh (MHz) αΩ12/3 100 200 12 αΩ1 150 50 10 W1/2p (MHz) 100 0 -0.5 0 0.5 50 0 50 100 150 200 250 300 Dn (MHz) 0 0.5 1 1.5 2 2.5 Ic Ic Ib (mA) Conclusion • MQT measurements : a probe to low frequency noise • Multilevel coherent oscillations • Incoherent processes in the 2 level limit

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