Bloch band dynamics of a Josephson junction in an inductive environment Wiebke Guichard Grenoble...

Bloch band dynamics of a Josephson junction in an inductive environment

Wiebke GuichardGrenoble 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

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 Relations

tVe

ItI DCcJ 2

sin)( Josephson oscillations

Ideal large Josephson Junction under microwave irradiation

TimeAverage

0sin

22sin

t

VetV

eItI mw

mw

mwDCcJ

0max

,

sin2

2

mw

mwncn

mwnDC

eVJII

enV

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)

Autore

the situation becomes more interesting when the DC voltage biased Josephson junction is irradiated with microwavesthe total potential drop is now the sum of the DC part and the oscillating oneusing Josephson relations it turns out that in this case the phase and the current through the junction depend on an a term proportional to the intensity of the microwaves and oscillating with the frequency of the microwavesthe measured time average of the current is now different from zero each time the DC bias voltage is a multiple of omega microwave and its amplitude is given by the Bessel function of the intensity of the microwaves

Ideal large Josephson JunctionIdeal JJ

Big C and EJ

Classical phase dynamics

Shapiro spikes

Autore

So far I showed you the importance of the phase of the junction in the determination of its IV characteristic.Another important quantity which determines the properties of the JJ is the charge Q localized across the capacitor formed by the junction itself, which is the conjugate variable of phiIndeed the JJ is equivalent to a non linear element characterized by the Josephson energy in parallel to the capacitance of the junctionusing this picture the hamiltonian of the system can be easily written as the sum of the charging energy and the non linear term cos phithe latter describes the cooper pair tunneling and it is responsible for the appearance of the Shapiro’s stepsthis hamiltonian can be interpreted as the Hamiltonian of a particle of mass C with momentum Q moving in a periodic potential. you see that as long as C and Ej are big, the JJ can be considered as a classical systemand just in this classical limit Shapiro’s steps are usually observed experimentally

Ordinary Josephson junction to Dual Josephson junction

“Ordinary” Josephson Junctionarrow

Dual Josephson Junction orQuantum phase-slip Junction

Josephson Relations

Coherent Cooper pair tunneling 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 T102 , pp. 62 - 68 (2002)-J. E. Mooij and Y. V. Nazarov, Nat. Phys.(2006)

Autore

for a harmonic oscillator nothing occurs if one swaps position and momentum

Autore

so, to sum up, an ordinary JJ is a device described by the hamiltonian of a non linear component in parallel with a capacitor, where charge tunneling occurs and whose IV characteristic is determined by the Josephson relationsLet me now consider the dual situation where the role of the charge and phase is interchangedthen in the hamiltonian of the dual JJ the phase and the charge would play the role of the momentum and the position of a particle of mass L respectively. Here L is an inductanceit follows that a dual JJ would be equivalent to a non linear element with energy scale U0 in series with an inductancein this device phase tunneling would occur and the its IV characteristic would be determined by the dual Josephson relations written in terms of the charge variable rather than the phase with the role of V and I interchanged

Duality

Coherent quantum phase-slips

Ideal large Josephson junction Quantum phase-slip junction

VDC

Bloch oscillations

Coherent Cooper pair tunneling

Josephson oscillations

EJ U0LIDC

Autore

as a result, the zero voltage peak in the IV plot of an ordinary JJ would become a zero current plateau, in other words, the dual device would show Coulomb blockadeDual applications

Quantum phase-slip junction under microwave irradiation

Dual Phase Locking relations

Ideal large Josephson Junction Quantum Phase-slip junction

0max

,

sin2

2

mw

mwncn

mwnDC

eVJII

enV

Phase Locking relations

0max

,

sin

mw

mwncn

mwnDC

e

IJVV

enI

Autore

and consequently under microwave irradiation, the IV curve of the dual JJ would have current plateaus whose position and amplitude would be determined by the dual phase locking relations

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)

Autore

the answer is yes, the device that behaves like a dual JJ is the so-called QPSJtwo of the most promising candidatesthis component can be realized either using a small JJ embedded in a highly inductive environment or by means of extremely thin superconducting nanowire



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)

Curr

ent

Autore




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)

Autore


Our experiment here

Experimental study of the role of the inductance on charge localisation in a Josephson junction in an inductive environment.

L U0

Vbias

Single small Josephson junctionJosephson junction chain

Realisation of the phase-slip non-linearity with a small Josephson junctioncosJE

Averin, Likharev, Zorin (1985)

2π

For small capacitances quantum phase-slips occur

2π-2π

q

Energy

Ec

EJ/EC=0.25

EJ/EC=1

Energy spectrum of the junctionconsists of Bloch bands

1

0 )/ˆcos()ˆ(k

k eqkUqE

Lowest Bloch band:

Experiment: Single Josephson junction in an inductive environment

biask

k qVeqkL

H

1

2

)/ˆcos(2

ˆˆ

Phase-slip element= Single SQUID with different field dependance

Inductance =Josephson junction chain with 9 -109 junctions L= 60nH-654 nH

Al/Al2O3/Al junctions

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

Zero-bias resistance as a function of flux

T. Weissl et al, Phys. Rev. B, 2015

Inductance Inductance + single junction0/

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

GHzhE

GHz

GHzh

C

q

8.2/

42/

4.2/0

Charge localisation

0

20

21

1

q

q

q

q

q

UC

LC

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.

Propagating modes in the Josephson junction chain

N.A. Masluk et al, Phys. Rev. Lett. (2012)T. Weissl, PhD thesis (2014)

Propagating modes in the Josephson junction chain

Talk by T. Weissl on WednesdayarXiv 1505.05845

)(0

0 tieV )(tit

teV

In our experiment , in units of temperature, the frequency range between the lowestmode 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


kJJ kkC

Le

NEE

]cos1)[(2

1exp

2*

Renormalisation of the Bloch band width due to zero pointquantum phase-fluctuations induced by the modes

0EJ

CJ


Renormalised Josephson energy

Effective bandwidth is larger than thebare value

*0

0

46.0/ 0

Dynamics of wave packet in lowest Bloch band

TkeVq Be /)(/

0

2

Tk

q

BQ

BeTk

RR /0

0

)(2 eI

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 magnitudeas the dual plasma frequency q/2=4GHz.

biasqVU 0

eV0 eV0

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

Landau Zener processes

Fitting parameters: fitqxzR ,,

Tkfit

q

BQ

BeTk

RR /0

*0

ZZZm RPRPR 0)1(

xC

gapZ E

EP

22

4exp


eV0 eV0

Fit of the data taking into account of Landau-Zener tunneling and renormalized bandwidth

Landau-Zener processes

Bandwidthis smaller than residual noise temperature

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

Charge dynamics in lowest Bloch band

ZZZm RPRPR 0)1(

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 inductanceto 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 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 !

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

Tkfitq

BQ

BeTk

RR /0

0

Example of fitting for quantum phase-slip junction with N=49 junctions

Good fits are achieved for an effective larger Bloch band width. Characteristic chargeFrequency is larger than theoretically calculated one.


see A. Di Marco et al, accepted for publication in Phys. Rev. B,

Influence of thermal and quantum fluctuations on theCurrent-voltage characteristics of a Quantum phase-slip junction

Requirement of a large environmental inductance and resistance.

Zero bias resistance at zero flux frustration

eIEU

UB

UA

Ae

pJ

p

p

p

Bqps

22/2

5

36

,2

612

)(22

qpsqpse

h

dt

d

eV

105

360

p

qpsQRR

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 in59 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)

Bloch

wav

e vec

tor

(pha

se)

Bloch

wav

e vec

tor

(pha

se)

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

chch HJHJ EEU 00 coscos)cos()(

2*

2

coschH

cheEEE JHJJ

kJJ kkC

Le

NEE

]cos1)[(2

1exp

2*

46.0f

48N

Renormalisation of the Josephson coupling

0

Quantized Hamiltonian of the Josephson junction chain

kkkkch aaH )

2

1(

C

Ck

kk

2)cos(1

)cos(1

00

k

iknkk

kn

ikn

kkk

kn

eaae

kC

N

ieQ

eaakC

e

N

)(2

)(

)()(

21

2

2

mnmnmnnm

nnn

chmnm

mnnch

CCCCC

eLQCQH

,1,1,0

2

1

21

,

2

2

1

nnQ , 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

Bloch band dynamics of a Josephson junction in an inductive environment Wiebke Guichard Grenoble...

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