Bloch band dynamics of a Josephson junction in

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 Ph. D students and postdocs: Thomas Weissl, Iulian Matei, Ioan Pop, Etienne Dumur, Bruno Küng, Yuriy Krupko Theory: Permanent: Denis Basko, Frank Hekking Ph. D students and postdocs: Gianluca Rastelli, Angelo Di Marco, Van Duy Nguyen

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

Ideal large Josephson junction Josephson Relations

Ideal large Josephson junction Josephson oscillations Josephson Relations

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)

Ideal large Josephson Junction Ideal JJ Big C and EJ Classical phase dynamics Shapiro spikes

Ordinary Josephson junction to Dual Josephson junction “Ordinary” Josephson Junction arrow Coherent Cooper pair tunneling Josephson Relations Dual Josephson Junction or Quantum phase-slip Junction Coherent 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 T 102 , pp. 62 - 68 (2002) -J. E. Mooij and Y. V. Nazarov, Nat. Phys. (2006)

Duality Ideal large Josephson junction VDC EJ Coherent Cooper pair tunneling Josephson oscillations Quantum phase-slip junction IDC L U 0 Coherent quantum phase-slips Bloch oscillations

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

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

Quantum phase-slip junctions in experiments Quantum Phase-Slip Junction Chains of small Josephson junctions Current 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)

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) A. Ergül et al, New J. Phys. , (2013) Cooper pair box in an inductive environment M. T. Bell et al, Ar. Xiv 1504 -05602

Our experiment here Experimental study of the role of the inductance on charge localisation in a Josephson junction in an inductive environment. L U 0 Single small Josephson junction chain Vbias

Realisation of the phase-slip non-linearity with a small Josephson junction -2π 2π For small capacitances quantum phase-slips occur 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)

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

Measurement circuit Experiment: -Base temperature T=50 m. K -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

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

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

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 quasicharge.

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

Propagating modes in the Josephson junction chain Talk by T. Weissl on Wednesday ar. Xiv 1505. 05845

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 300 m. K and 1 K. The equivalent voltage range is between 30 V and 100 V. T. Weissl et al, Phys. Rev. B, 2015

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

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 =4 GHz. H. Kramers, Phys Rev B (1940) T. Weissl et al, Phys. Rev. B, (2015)

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

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

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.

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 phaseslip 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. Ph. D or postdoc position available !

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

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

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,

Zero bias resistance at zero flux frustration Temperature is lower than the plasma frequency p/2 =25. 4 GHz 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.

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)

Rho_q = 0. 5 (from figure 1 b) Blo ch w (p ave ha ve se ct ) or L = 300 n. H, C = 7 f. F, hence rho_q = 0. 25 (from Thomas) Bandwidth is about. 04 hbar omega_q Blo ch w (p ave ha ve se ct ) or Bandwidth is about. 14 hbar omega_q

Renormalisation of the Josephson coupling

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