Bloch band dynamics of a Josephson junction in an inductive environment

1. 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 Bloch band dynamics of a Josephson junction in an inductive environment

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

3. Ideal large Josephson junction Josephson Relations

4. Ideal large Josephson junction Josephson oscillations Josephson Relations

5. Ideal large Josephson Junction under microwave irradiation Shapiro spikes Phase locking relations Time Average 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 EJ 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 Coherent Cooper pair tunneling Coherent quantum phase-slips Dual Josephson Relations 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 Quantum phase-slip junction Ideal large Josephson junction VDC EJ IDC L U0 Coherent Cooper pair tunneling Coherent quantum phase-slips Josephson oscillations Bloch oscillations

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

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

12. Quantum phase-slip junctions in experiments Quantum Phase-Slip Junction Single small Josephson junction in an inductive environment Cooper pair box in an inductive environment Fluxonium qubit M.T. Bell et al, ArXiv 1504-05602 V. E. Manucharyan, et al., Science (2009) N. A. Masluk et al., Phys. Rev. Lett. (2012) 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. L U0 Single small Josephson junction Josephson junction chain Vbias

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

15. Experiment: Single Josephson junction in an inductive environment Al/Al2O3/Al junctions Inductance = Josephson junction chain with 9 -109 junctions L= 60nH-654 nH Phase-slip element= Single SQUID with different field dependance

16. Measurement circuit • Experiment: • -Base temperature T=50mK • -Biasvoltage issupplied by a NI-DAQ. • -Measurementlinesconsist of thermocaox and lowpassπ-Filters. • -Output voltage of Femto current to voltage converterisrecordedby NI-DAQ. Effective circuit

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

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

19. Three physical phenomena occuring in the system • 1) Renormalisationof the Josephson couplingenergy of the smalljunction due to • electromagnetic modes propagatingalong the chain. Effective Bloch band widthislarger. • 2) Charge diffusion in the lowest Bloch band. • 3) The effect of interband transitions (Landau-Zenerprocesses) thatdominate the • charge dynamicswhenever the gap separating the lowesttwo charge bands • becomestoosmallcompared to the characteristicenergy 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. Influence of the electromagnetic modes on the current voltage caracteristics 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. 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 EJ CJ Renormalised Josephson energy Effective bandwidth is larger than the bare value T. Weissl et al, Phys. Rev. B, 2015

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 Bandwidth is smaller than residual noise temperature Charge dynamics in lowest Bloch band Landau-Zener processes Fitting parameters: ωx=0,01 Ec ωqfit=0,12 Ec RZ=170  The frequencies ωx and ωqfit are systematically smaller than the frequency ωq 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

27. Conclusions • The behavior of the zero-biasresistance of a single Josephson junction in series • with an inductance canbeexplained in terms of Bloch band dynamics (coherent • quantum phase-slip dynamics). • Charge dynamics in the lowest Bloch band (required for quantum phase-slip junction) • occursonly in a smallparameter range. Need to increase coherent quantum phase-slip amplitude and inductance to obtain enhanced charge localisation over larger parameter range.

28. Future experiments • Realisationof 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 microwavespectroscopymeasurements • Realisationof larger inductance with longer chains. For larger inductances, • i.e. longer chains, the electromagnetic modes appear at lowerfrequencies. • Measurement and analysis of electromagnetic modes in chains of small • Josephson junctions. • Determination of the quality factor and understanding of dissipation mechanism in • Josephson junctionchains. 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. 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. see A. Di Marco et al, accepted for publication in Phys. Rev. B,

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