Quantum Brownian motion at strong dissipation probed by superconducting tunnel junctions

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by  Berthold Jäck

QuantumBrownianmotionatstrongdissipationprobedbysuperconductingtunneljunctions
BertholdJa¨ck,1,∗ JacobSenkpiel,1 MarkusEtzkorn,1 JoachimAnkerhold,2 ChristianR.Ast,1 andKlausKern1,3
1Max-Planck-Institut fu¨r Festko¨rperforschung, 70569 Stuttgart, Germany
2Institut fu¨r Komplexe Quantensysteme and IQST, Universita¨t Ulm, 89069 Ulm, Germany
3InstitutdePhysiquedelaMatie`reCondense´e,EcolePolytechniqueFe´de´raledeLausanne,1015Lausanne,Switzerland
(Dated:January17,2017)
Wehavestudiedthetemporalevolutionofaquantumsystemsubjectedtostrongdissipationatultra-lowtem-
peratureswherethesystem-bathinteractionrepresentstheleadingenergyscale. Inthisregime,theorypredicts
thetimeevolutionofthesystemtofollowageneralizationoftheclassicalSmoluchowskidescription,thequan-
tumSmoluchowskiequation,thus,exhibitingquantumBrownianmotioncharacteristics. Forthispurpose,we
haveinvestigatedthephasedynamicsofasuperconductingtunneljunctioninthepresenceofhighdamping.
We performed current-biased measurements on the small-capacitance Josephson junction of a scanning tun-
7 nelingmicroscopeplacedinalowimpedanceenvironmentatmilli-Kelvintemperatures. Wecandescribeour
1 experimentalfindingsbyaquantumdiffusionmodelwithhighaccuracyinagreementwiththeoreticalpredica-
0 tionsbasedonthequantumSmoluchowskiequation. Inthiswayweexperimentallydemonstratethatquantum
2 systemssubjectedtostrongdissipationfollowquasi-classicaldynamicswithsignificantquantumeffectsasthe
leadingcorrections.
n
a
PACSnumbers:74.50.+r,74.55.+v,05.30.-d
J

5
1 Introduction.–Brownianmotion–thatisthefateofaheavy the realm of classical physics is then defined by the relation

particleimmersedinafluidoflighterparticles–istheproto- ¯hγ,¯hω (cid:28) k T, the predominantly explored quantum do-
] 0 B
h typeofadissipativesystemcoupledtoathermalbath[1]. Its main of weak system-bath interaction obeys ¯hγ (cid:28) k T (cid:28)
B
p quantummechanicalanaloguecanbefoundinopenquantum ¯hω with the bare level spacing exceeding all other energy
0
-
t systems,whichhavereceivedconsiderableattentioninthelast scales.Thisisthegenericsituationforcavityandcircuitquan-
n
decade[2]. Thisismainlyduetotheexperimentalprogressin tumelectrodynamicalset-ups[2].
a
fabricating quantum devices on ever growing scales with the
u
q intention to control their quantum properties to an unprece-
[ dentedaccuracy. Effortshavethusfocusedtotametheimpact

ofdecoherenceandnoiseinordertopreservefragilefeatures
1
v suchasentanglementaspossibleresourcesfortechnological
4 applications[3].
8
0 The regime, where dissipation cannot be seen as a pertur-
4 bation,buttendstocompletelydominatethesystemdynamics
0 hasreceivedmuchlessattention. Thisisinsharpcontrastto
.
1 classicalnon-equilibriumdynamics,wheretheso-calledover-
0 damped regime, also known as the classical Smoluchowski
7 regime(cSM),playsapivotalrolefordiffusionphenomenain FIG.1:(a)Parameterdiagramforthedynamicsofoverdampedphase
1 abroadvarietyofrealizations[1]. Theoretically,accordingto diffusioninasuperconductingtunneljunction. Theregimeofover-
v: Smoluchowski, this domain is characterized by a separation damped quantum diffusion (qSM, blue) emerges at lower temper-
i atures from the classical Smoluchowski domain (cSM, grey) when
X oftimescalesbetweentherelaxationofmomentum(fast)and the dimensionless friction η sufficiently exceeds the dimensionless
the relaxation of position (much slower) implying that on a
r inversetemperatureΘ. (b)Thephasedynamicscorrespondstothe
a coarselygrainedtimescalethelatterconstitutestheonlyrele- dissipativequantumdynamicsofaparticleinawashboardpotential
vantdegreeoffreedom. Thesituationinquantummechanics (grey solid), which appears as macroscopic quantum tunneling for
is more subtle though. Position and momentum are bound weakfriction,i.e. thetunnelingofthephasewavefunctionthrough
togetherbyHeisenberg’suncertaintyrelation,whichisdetri- the potential wall (red wavepackage), and as quantum diffusion in
theqSMdomain,forwhichquantumfluctuationseffectivelyreduce
mental to the tendency of strong dissipation to induce local-
thebarrierheight(bluesolid).Classicalphasediffusionisillustrated
izationandevendissipativephasetransitions[4,5].
as the thermally activated escape of a particle (green dot) over the
Roughlyspeaking,adissipativequantumsystemischarac- potentialbarrier.
terized by three typical energy scales, namely, an excitation
energy ¯hω , where ω denotes some specific energyscale of Thequantumrangeofstrongdissipationiscomplementary,
0 0
the bare system and ¯h the reduced Planck’s constant, a cou- i.e.k T (cid:28) ¯hω (cid:28) ¯hγ. Ithasbeenpredictedbytheorythat
B 0
pling energy to the environment ¯hγ, where γ denotes the in this domain a separation of time scales and thus a quan-
coupling parameter, and the thermal energy k T with k tum Smoluchowski regime (qSM) exists indeed, cf. Fig.1(a)
B B
as Boltzmann’s constant and T as the temperature. While [6]. Quantum Brownianmotionin theqSMis almostclassi-
2
cal,however,substantiallyinfluencedbyquantumfluctuations
yielding quantum diffusion characteristics. Consequently, in
all processes sensitive to these fluctuations the dynamics is
predictedtodeviatestronglyfromtheclassicalone. Hence,to
exploretheqSMisnotonlyoffundamentalinterest,butalso
ofdirectrelevanceforstronglycondensedphasesystems,yet,
itsexperimentalobservationhasbeenelusivesofar.
TheaimofthisLetteristoclosethisgap. Forthispurpose,
westudythedynamicsofthephaseφasacontinuouscollec-
tive degree of freedom in a superconducting tunnel junction
in the qSM regime. In fact, in the past superconducting cir-
cuits have proven to serve as ideal testbeds to explore quan-
tumdissipativephenomenaase.g.macroscopicquantumtun-
neling (MQT), transitions from quantum to classical, or de-
phasinganddecoherence[7–11]. Here,weaccessthehitherto
untouched qSM domain by investigating the current-voltage
characteristics(IVC)ofcurrentbiasedsmallcapacitancetun-
nel junctions in an ultra-low temperature scanning tunneling
microscope(STM)[12,13]. Itsphasedynamicsisequivalent
toquantumdiffusionalongatiltedwashboardpotentialunder
strongdamping,cf.Fig.1(b),referringtoalowimpedanceen- FIG.2:(a)Leftside:Circuitdiagramoftheexperimentalsetupwith
vironmentwithohmic-resistanceRDC muchsmallerthanthe IBasthebiascurrentsource,RBasthesourceimpedance,R0andC0
quantumresistanceR = h/4e2,i.e.ρ ≡ R /R (cid:28) 1(e astheload-lineresistorandshuntcapacitor,respectively. Thejunc-
Q DC Q
tionanditsdirectenvironmentishighlightedbythebluebox,where
denotestheelementarycharge)[14,15]. AccessingtheqSM
I andC denotethejunctionelementandcapacitor,respectivelyand
regime where quantum diffusion can be observed further re- 0 J
Zdenotestheenvironmentalimpedance.Rightside:Schematicrep-
quires that the dimensionless friction η = E /(2π2ρ2E ),
C J resentationoftheSTMtipontopofthereconstructedV(100)surface
with charging energy EC, tunnel coupling EJ, and dimen- [16].(b)IVCofthesuperconductingSTMtunneljunctionmeasured
sionless impedance ρ sufficiently exceeds the dimensionless at0.004G .(c)Simulatedrealpart(cid:60)[Z]ofthefrequency-dependent
0
inverse temperature Θ = βE /(2π2ρ) with β = 1/k T. environmentalimpedance[18]. (d)In-gapregionofanIVCfroma
C B
Intermsofexperimentallyaccessiblecircuitparameters,this current-biasedmeasurementatGN =0.074G0.
conditioncorrespondstoE (cid:29) E foralowimpedanceen-
C J
vironmentρ(cid:28)1andmilli-Kelvintemperatures[16].
Theory.–The supercurrent through a superconducting tun- cal Smoluchowski-type of treatment for thermal phase dif-
nel junction is determined by its phase dynamics according fusion, the Ivancheko Z’ilberman approach [28], by replac-
to the first Josephson relation, I = I sin(φ) with the crit- ing the bare tunnel coupling with its renormalized value E∗.
J 0 J
ical current I0 = 2eEJ/¯h. In the qSM regime and for a TheexplicitdependenceofEJ∗onECinEq.2reflectstheim-
pureDCenvironmentthediffusionofthephaseφoccursina pact of charge fluctuations on the phase dynamics and thus,
washboardpotentialtiltedbyabiascurrentI asdepictedin inthemechanicalanalogue,thepresenceofmomentumfluc-
B
Fig.1(b)[14,15]. BelowtheswitchingcurrentI , thephase tuationsintheoverdampeddiffusionofposition. Thesimul-
S
diffusesfromonewelltoanotherduetotheinterplayofther- taneous interplay of classical and quantum diffusion leading
malandquantumfluctuations,thiswayacquiringafiniteve- toEq.1,therefore,correspondstoquantumBrownianmotion
locityφ˙ (cid:54)= 0,cf.1(b). AccordingtoJosephson’ssecondrela- dynamics. Thephysicalinterpretationisthatquantumfluctu-
tion,(cid:104)φ˙(cid:105) = (h¯/2e)V,thisvelocityisrelatedtoameasurable ations of the phase close to the top of the washboard poten-
voltage drop across the junction. The corresponding Cooper tial barrier depicted in Fig.1(b), effectively reduce its height
paircurrentthentakesthecompactform considerably [15]. We note, that one may see this effect as
complementary to MQT, occurring at low temperatures and
eρβπ βeV
IqSM = (E∗)2 (1) weakdampingintheoppositedomainEJ (cid:29) EC,[9]. Ashas
J ¯h J (βeV)2+π2ρ2 been demonstrated theoretically, quantum Brownian motion
dynamicssubstantiallyreducesthemaximumsupercurrentI ,
witharenormalizedJosephsonenergy S
inthefollowingreferredtoasswitchingcurrent,muchbelow
(cid:18)βE (cid:19)−ρ itscriticalcurrentI0[14,15].Wewillemploythiseffectasan
E∗ =E ρρ C e−ρc0, (2) experimentalprobeforthedetectionofoverdampedquantum
J J 2π2
phasediffusionandtodistinguishitfromitsclassicalcounter-
where c = 0.5772... denotes Euler’s constant [14, 15]. It part. Inthefollowing,wewillprovideconvincingexperimen-
0
turns out that the above expression can also be understood taldataforthevalidityofEq.1intheqSMregime.
as the quantum generalization of the corresponding classi- Superconductingcircuit.–Thecircuitdiagramoftheexper-
3
imentalsetupisdepictedinFig.2(a). Theatomic-scaletunnel pacitor C = 3nF and a load-line resistor of R = 3.5kΩ
0 0
junction appears between a superconducting vanadium STM [17]. Owing to the large output impedance of our constant
tipandavanadium(100)singlecrystalsurface,havingasmall current source, R = 1.33GΩ, the circuit features a hor-
B
junctioncapacitanceC ofafewfemtofarads[16]. Thejunc- izontal load-line. Concerning the electromagnetic environ-
J
tion and its direct electromagnetic environment are thermal- mentofthetunneljunction,weobtainavirtualDCimpedance
izedatthebasetemperatureofourdilutionrefrigeratorSTM for frequencies up to the low GHz regime by choosing an
of T = 15mK [12], so that one easily arrives at Θ (cid:29) 1 STMtipofadequatelength,movingthetipresonancemodes
(seeFig.1(a)). AtypicalIVCmeasuredoverabroadvoltage in the environmental impedance Z(ν) to higher frequencies
rangeatalownormalstateconductivityG =0.004G with [18]. This is illustrated by the frequency-dependent part of
N 0
G = 1/(2R )beingtheconductancequantum, isshownin the simulated system impedance in Fig.2(c), where the tip
0 Q
Fig.2(b)andexhibitsawell-developedsuperconductinggap. resonancemodesappearassharppeaks[18]. Inthelowfre-
We determine the normal state conductivity by a linear fit to quency limit ν → 0, the environmental impedance Z(0) is,
the metallic part of the spectrum with a relative deviation of therefore,dominatedbythecouplingofthetunnelingCooper
5%; detailed analysis of the superconducting properties of pairs to the electromagnetic vacuum in the gap between tip
STMtipandsample,thepreparationandmeasurementproce- and sample (tunnel barrier) with the STM being operated in
dureisdescribedingreatdetailinRef.[16]. Sincetheset-up ultra-high vacuum conditions. Thus, the vacuum impedance
is operated in the deep tunneling regime at G (cid:28) G with R =Z(ν →0)=377ΩdeterminestheDCimpedancein
N 0 DC
E directly proportional to G , one has E (cid:28) E and the the vicinity of the tunnel junction, whereas the resistance of
J N J C
condition1(cid:28)Θ(cid:28)ηisessentiallyalwaysfulfilled. theleads(transmissionlines)isnegligiblysmall. Inthisway,
we realize the required low impedance environment, ρ (cid:28) 1
inducing the required high damping to the phase dynamics
andthequantumphasediffusionmodelofEq.1becomesap-
plicabletoourexperimentatsmallvoltagesaroundzerobias
[19].
Results.–Wehavemeasuredcurrent-biasedIVCsatatotal
of 14 different values of the normal state conductivity from
0.05G < G < 0.3G bychangingthevacuumgapwidth
0 N 0
between STM tip and the sample surface. In Fig.2(d), we
plotoneexamplemeasuredatG =0.073G toillustratethe
N 0
general properties of our experimental setup. For increasing
bias current starting from zero, the current follows a phase-
diffusion branch at small voltages V (cid:54)= 0, before switching
toadissipativein-gapcurrent. Duetothehorizontalload-line
ofourcircuit,wecanonlyaccessregionsofpositivedifferen-
tialconductance,yieldingahysteresisintheIVCasshownin
Fig.2(d). FordecreasingbiascurrentsstartingfromI >I ,
B S
thecurrentisfixedtothein-gapcurrentbeforeswitchingback
into the phase-diffusion branch. At our experimental condi-
tions, thein-gapcurrentoriginatesfromquasiparticleexcita-
tions due to life-time effects of Cooper pairs in the STM tip
andtunnelingofindividualCooperpairsviainteractionwith
resonancemodesintheenvironment[18,21,22].
FIG.3:(a)ZoomintothelowvoltageregimeofIVCsfromcurrent- Consideringtheexperimentalcurrentamplitudeofthedata
biased measurements (red dots) at different indicated values of showninFig.2(d), wefindtheswitchingcurrentI = 49±
S
GN/G0.ThedashedlinesdisplaythecorrespondingtheoreticalIVCs 1pA reduced by approximately two orders of magnitude in
calculatedbyusingtheQPDmodel. Thequantitativedeviationbe-
comparison to the calculated critical current I = 4.80 ±
0
tweenexperimentandcalculationishighlightedviathegreybarsfor
0.24nA[16,23]. Thermalenergybeingcomparativelysmall
each measurement. (b) Upper part: Calculated critical Josephson
E β ≥ 5 (see Refs.[16, 24] how to determine the E val-
currentI (reddots)andlinearfit(black,dashedline)asafunction J J
0
ofG /G (errorbarsarecontainedwithinthesymbols).Lowerpart: ues),classicalphasediffusioncannotbethedominantprocess
N 0
Experimentalswitchingcurrent,IS (redcrosses),calculatedswitch- here. Instead, this strong reduction of IS indicates a major
ingcurrent,IqSM (blackcrosses),aswellasaquadraticfit(black, relevance of quantum fluctuations. To better understand this
J,max
dashedline)asafunctionofGN/G0. observation,wewillinthefollowingpresentaseriesofIVCs
measuredatdifferentvaluesofG /G andquantitativelyan-
N 0
Inordertominimizetheinfluenceofthebiasingcircuitonto alyze these curves by applying the quantum diffusion model
the phase dynamics, we separate the time-scales of the junc- ofEq.1. Weemphasizethatvaryingthenormalstateconduc-
tion phase and the biasing circuit by using a large shunt ca- tivityatfixedtemperatureprovidesuswithanidealhandleto
4
tune the η ∝ E−1 ∝ G−1 value along a vertical axis in the
J N
phasediagraminFig.1(a).
InFig.3(a),wepresentaseriesofIVCsmeasuredatdiffer-
ent values of G /G and take a closer look at the low volt-
N 0
age regime V < 30µV for positive bias currents. Towards
higher normal state conductivities, we observe a strong and
non-linear increase of the switching current I . This obser-
S
vation is illustrated in Fig.3(b), where we plot the extracted
I values as a function of G /G and fit a quadratic depen-
S N 0
dence I ∝ G2, as it is predicted by Eq.1. We additionally
S N
findI stronglyreducedincomparisontothecalculatedcrit-
S
ical Josephson current I ∝ G also shown in Fig.3(b) for
0 N
all measurements, as we have mentioned before. In combi-
FIG.4: (a)Relativedeviationbetweentheoreticalandexperimental
nation, these findings reveal the major relevance of quantum currentmaxima∆I aswellastheqSMregimeboundaryη/Θas
qSM
fluctuationsinthelimitE (cid:28)E [20,24]. afunctionofthenormalizednormalstateconductivity. (b)Compar-
J C
To quantitatively analyze the experimental IVCs in ison between experimental data (red, solid) and an IVC calculated
Fig.3(a),wecalculatethecorrespondingtheoreticalIVCsby (black,dashed)fromthepurelyclassicalIZmodelatanormalstate
conductivityofG =0.051G .
usingEq.1foreachG /G valueandplotthemalongsidethe N 0
N 0
experimentaldata. Weobtainthenecessaryexperimentalval-
uesofE ,E andT byperformingadditionalvoltage-biased
C J
3%<∆I <9%. Bycontrast,wefind∆I >12%at
measurementsonthesametunneljunctionatthesameG /G qSM qSM
N 0
highconductivitieswheretheconditionforqSMdynamicsis
values.FittingtheresultingIVCswithP(E)-theoryallowsus
violatedη/Θ≈1,yieldinganoverallconsistentpicture.
toextracttheseparameterswithhighprecision(seeRefs.[16]
and[24]formoredetails)[25,26]. Owingtothesmalljunc- Additionally, we compare our experimental data with the
tioncapacitanceintheabsenceofalargeshuntcapacitorinthe classical IZ model (IZ) in which phase diffusion only oc-
junctionvicinity,thermallyinducedcapacitivenoisebroaden cursviathermallyactivatedescapeoverthewashboardpoten-
the measured IVCs. We account for this effect by convolv- tial barrier displaying classical Brownian motion dynamics,
ingthecalculatedIVCfromEq.1withanormalizedGaussian cf. Fig.1(b) [27, 28]. We plot an experimental IVC together
functionP (E)ofwidthσ =(cid:112)2E /β [13,16]. withatheoreticalIVCcalculatedusingtheIZmodelandthe
N C
ComparingexperimentalandtheoreticalIVCsinFig.3(a), sameparametersetfromP(E)-theoryasbeforeforanormal
we find excellent quantitative agreement in the low voltage stateconductivity of GN = 0.051G0 inFig.4(b). Ascan be
regime V < 30µV for small normal state conductivity val- seen,theorybasedonclassicaldiffusioncannotreproduceour
ues. We recall that our experimental setup can only access experimental data and largely overestimates the IVC ampli-
regionsofpositivedifferentialresistance. Atthesesmallvolt- tude. This observation clearly underlines the high relevance
ages,weattributetheslightlydifferentslopeofthephasedif- ofquantumfluctuations,significantlyreducingthewashboard
fusionbranchbetweenexperimentandtheorytosmalldevia- potentialbarrierheightand,thustheswitchingcurrentincom-
tionsofourenvironmentalimpedancefromaperfectDCbe- parisontoclassicalphasediffusiondynamics. Formally, this
havior as required for the derivation of Eq.1. By contrast, effectyieldstherenormalizedJosephsoncouplingenergyEJ∗
considering the IVCs measured at high GN values shown in inEq.2,ascomparedforthebareEJintheclassicalregime.
Fig3(a),weobservesignificantquantitativedeviationsinthe Together, the accurate agreement between experiment and
current amplitude between theory and experiment. We will theoryatlowconductivitiesdemonstratesthattheoverdamped
analyze these deviations in the following in more detail and quantum phase dynamics of our tunnel junction can be de-
alsocompareourexperimentaldatawithaclassicaldiffusion scribed in the framework of the qSM equation [15]. At high
model. conductivities,however,theconditionforaseparationoftime
Discussion.– We start our discussion by comparing the scales of position and momentum dynamics, η (cid:29) Θ, is vi-
maximum of the measured IVC, I , with the maximum of olated and, accordingly, the qSM equation is not applicable
S
the calculated IVC, IqSM , as a measure for the agreement anymore. Hence,ourexperimentalstudyrevealsthatthetime
J,max
betweenexperimentandtheory. InFig.4(a),weplottheirrel- evolution of an overdamped quantum system follows quasi-
ativedeviation∆I = |IqSM −I |/IqSM asafunction classicaldynamicswithsignificantquantum-mechanicalcor-
qSM J,max S J,max
of G /G . Furthermore, we include the conditions for the rectionsinleadingorderthatisquantumBrownianmotion[6].
N 0
qSMregime, η (cid:29) Θ (cid:29) 1byusingthesamesetofparame- Basedonitsfundamentalcharacterintheframeworkofquan-
tersfromtheP(E)-fitsasbefore. WeobtainΘ = 122forall tumstatistics,ourstudyshouldbeofgeneralrelevanceforthe
measurementsandplotthedependenceofη/ΘonG /G in fieldsofsuperconductingquantumcircuits[3],quantumgases
N 0
Fig.4(a).Inthelowconductivityregime,theconditionforthe [29]andnano-mechanicaloscillators[30],forinstance.
observation of qSM dynamics is fulfilled η/Θ > 4, explain- Conclusion.– We have studied the temporal evolution of a
ingtheverysmalldeviationsbetweentheoryandexperiment quantumsystemsubjectedtostrongdissipationintheframe-
5
work of the qSM equation. We experimentally investigated [10] Bertet, P., Chiorescu, I., Burkard, G., Semba, K., Harmans,
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∗ Corresponding author; electronic address: [22] Eltschka, M., Ja¨ck, B., Assig, M., Kondrashov, O. V.,
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