1. Bloch band dynamics of a Josephson junction in an inductive environment Wiebke Guichard Grenoble University –Institut Néel In collaboration with the Josephson junction team: Experiment: Permanent: Nicolas Roch, Olivier Buisson, Cecile Naud PhD students and postdocs: Thomas Weissl, Iulian Matei, Ioan Pop, Etienne Dumur, Bruno Küng, Yuriy Krupko Theory: Permanent: Denis Basko, Frank Hekking PhD students and postdocs: Gianluca Rastelli, Angelo Di Marco, Van Duy Nguyen

2. Outline 1.Ideal Josephson junction and the Josephson effect: Cooper pair tunneling 2.Dual Josephson junction: Quantum phase-slip junction 3.Experiment: Single Josephson junction in an inductive environment 4.Conclusions and outlook

3. Ideal large Josephson junction Josephson Relations

4. Ideal large Josephson junction Josephson Relations Josephson oscillations

5. Ideal large Josephson Junction under microwave irradiation Time Average Phase locking relations Shapiro spikes Quantum Voltage standard J. Kohlmann and R. Behr, Superconductivity - Theory and Applications, chapter 11, edited by Adir Moyses Luiz (2011)

6. Ideal large Josephson Junction Ideal JJ Big C and E J Classical phase dynamics Shapiro spikes

7. Ordinary Josephson junction to Dual Josephson junction “Ordinary” Josephson Junction arrow Dual Josephson Junction or Quantum phase-slip Junction Josephson Relations Coherent Cooper pair tunnelingCoherent quantum phase-slips Dual Josephson Relations - Quantum Complementarity for the Superconducting Condensate and the Resulting Electrodynamic Duality, D. B. Haviland et al, Proc. Nobel Symposium on Coherence and Condensation, Physica Scripta T102, pp. 62 - 68 (2002) -J. E. Mooij and Y. V. Nazarov, Nat. Phys.(2006)

8. Duality Coherent quantum phase-slips Ideal large Josephson junction Quantum phase-slip junction V DC Bloch oscillations Coherent Cooper pair tunneling Josephson oscillations EJEJ U0U0 L I DC

9. Quantum phase-slip junction under microwave irradiation Dual Phase Locking relations Ideal large Josephson Junction Quantum Phase-slip junction Phase Locking relations

10. Quantum phase-slip junctions in experiments Quantum Phase-Slip Junction J.S. Lehtinen et al, Phys. Rev. Lett (2012) Superconducting Nanowires O. V. Astafiev, et al., Nature (2012) C.H. Webster et al Phys. Rev. B (2013), T.T. Hongisto Phys. Rev. Lett. (2012 )

11. Quantum phase-slip junctions in experiments Quantum Phase-Slip Junction Chains of small Josephson junctions Fluxon interference pattern Island charge I. Pop et al, Nature Physics (2010), I. Pop et al, Phys. Rev. B (2012) K.A. Matveev et al. Phys. Rev.Lett. (2002) Current

12. Quantum phase-slip junctions in experiments Quantum Phase-Slip Junction Single small Josephson junction in an inductive environment Fluxonium qubit V. E. Manucharyan, et al., Science (2009) N. A. Masluk et al., Phys. Rev. Lett. (2012) Cooper pair box in an inductive environment M.T. Bell et al, ArXiv 1504-05602 A. Ergül et al, New J. Phys., (2013)

13. Our experiment here Experimental study of the role of the inductance on charge localisation in a Josephson junction in an inductive environment. LU0U0 V bias Single small Josephson junction Josephson junction chain

14. Realisation of the phase-slip non-linearity with a small Josephson junction Averin, Likharev, Zorin (1985) 2π2π For small capacitances quantum phase-slips occur 2π2π-2π q Energy EcEc E J /E C =0.25 E J /E C =1 Energy spectrum of the junction consists of Bloch bands Lowest Bloch band:

15. Experiment: Single Josephson junction in an inductive environment Phase-slip element= Single SQUID with different field dependance Inductance = Josephson junction chain with 9 -109 junctions L= 60nH-654 nH Al/Al 2 O 3 /Al junctions

16. Measurement circuit Experiment: -Base temperature T=50mK -Bias voltage is supplied by a NI-DAQ. -Measurement lines consist of thermocaox and low pass π-Filters. -Output voltage of Femto current to voltage converter is recorded by NI-DAQ. Effective circuit

17. Zero-bias resistance as a function of flux T. Weissl et al, Phys. Rev. B, 2015 Inductance Inductance + single junction

18. Localisation of wave packet in lowest Bloch band when quantum phase-slip rate of quantum phase-slip junction is increased. Charge localisation

19. Three physical phenomena occuring in the system 1) Renormalisation of the Josephson coupling energy of the small junction due to electromagnetic modes propagating along the chain. Effective Bloch band width is larger. 2) Charge diffusion in the lowest Bloch band. 3) The effect of interband transitions (Landau-Zener processes) that dominate the charge dynamics whenever the gap separating the lowest two charge bands becomes too small compared to the characteristic energy of the dynamics of the quasi- charge.

20. Propagating modes in the Josephson junction chain N.A. Masluk et al, Phys. Rev. Lett. (2012) T. Weissl, PhD thesis (2014)

21. Propagating modes in the Josephson junction chain Talk by T. Weissl on Wednesday arXiv 1505.05845

22. In our experiment, in units of temperature, the frequency range between the lowest mode frequency and the plasma frequency corresponds to a range between 300mK and 1K. The equivalent voltage range is between 30  V and 100  V. Influence of the electromagnetic modes on the current voltage caracteristics T. Weissl et al, Phys. Rev. B, 2015

23. Renormalisation of the Bloch band width due to zero point quantum phase-fluctuations induced by the modes EJEJ CJCJ T. Weissl et al, Phys. Rev. B, 2015 Renormalised Josephson energy Effective bandwidth is larger than the bare value

24. Dynamics of wave packet in lowest Bloch band We use Kramers classical result for the escape of a particle from a potential well. Thermal activation is dominant as the temperature is in the same orders of magnitude as the dual plasma frequency  q /2  =4GHz. H. Kramers, Phys Rev B (1940) T. Weissl et al, Phys. Rev. B, (2015)

25. Landau Zener processes Fitting parameters: T. Weissl et al, Phys. Rev. B, 2015

26. Fit of the data taking into account of Landau-Zener tunneling and renormalized bandwidth Landau-Zener processes Bandwidth is smaller than residual noise temperature Fitting parameters: ω x =0,01 E c ω q fit =0,12 E c R Z =170  The frequencies ω x and ω q fit are systematically smaller than the frequency ω q associated to the curvature of the lowest Bloch band. Therefore the charge motion is possibly overdamped. Such overdamped motion could result from a finite quality factor of the electromagnetic modes. T. Weissl et al, Phys. Rev. B, 2015 Charge dynamics in lowest Bloch band

27. Conclusions -The behavior of the zero-bias resistance of a single Josephson junction in series with an inductance can be explained in terms of Bloch band dynamics (coherent quantum phase-slip dynamics). -Charge dynamics in the lowest Bloch band (required for quantum phase-slip junction) occurs only in a small parameter range. Need to increase coherent quantum phase-slip amplitude and inductance to obtain enhanced charge localisation over larger parameter range.

28. Future experiments Realisation of quantum phase-slip element by a chain of small Josephson junctions. Role of off-set charge dynamics on the coherent quantum phase- slip amplitude ? Measurement of coherent quantum phase-slip dynamics in chains of small Josephson junctions via microwave spectroscopy measurements Realisation of larger inductance with longer chains. For larger inductances, i.e. longer chains, the electromagnetic modes appear at lower frequencies. Measurement and analysis of electromagnetic modes in chains of small Josephson junctions. Determination of the quality factor and understanding of dissipation mechanism in Josephson junction chains. PhD or postdoc position available !

29. Fit of the data taking into account of Landau-Zener tunneling

30. Example of fitting for quantum phase-slip junction with N=49 junctions Good fits are achieved for an effective larger Bloch band width. Characteristic charge Frequency is larger than theoretically calculated one. T. Weissl et al, Phys. Rev. B, 2015

31. see A. Di Marco et al, accepted for publication in Phys. Rev. B, Influence of thermal and quantum fluctuations on the Current-voltage characteristics of a Quantum phase-slip junction Requirement of a large environmental inductance and resistance.

32. Zero bias resistance at zero flux frustration Temperature is lower than the plasma frequency  p /2  =25.4GHz therefore thermal activation can be ignored. Use escape formula for underdamped phase dynamics from A.Caldeira and A. Leggett, Ann. Phys.(1983) Experimental fit results in 59  junction.

33. Unperturbed ground-state harmonic oscillator wavefunction in quasi-charge representation Hopping energy (broadening of ground-state energy hbar omega_q/2) Bloch wave function for lowest band (Bloch wave vector k is quasi-phase)

34. Bloch wave vector (phase) Bloch wave vector (phase) L = 300 nH, C = 7 fF, hence rho_q = 0.25 (from Thomas) Rho_q = 0.5 (from figure 1b) Bandwidth is about.14 hbar omega_q Bandwidth is about.04 hbar omega_q

35. Renormalisation of the Josephson coupling

36. Quantized Hamiltonian of the Josephson junction chain denote the charge and the phase of the nth island We diagonlise the Hamiltonian with the help of the following mode expansions for Q and 