Quantum Brownian motion at strong dissipation probed by superconducting tunnel junctions
Berthold Jäck, Jacob Senkpiel, Markus Etzkorn, Joachim Ankerhold, Christian R. Ast, Klaus Kern
QQuantum Brownian motion at strong dissipation probed by superconducting tunnel junctions
Berthold J¨ack, ∗ Jacob Senkpiel, Markus Etzkorn, Joachim Ankerhold, Christian R. Ast, and Klaus Kern
1, 3 Max-Planck-Institut f¨ur Festk¨orperforschung, 70569 Stuttgart, Germany Institut f¨ur Komplexe Quantensysteme and IQST, Universit¨at Ulm, 89069 Ulm, Germany Institut de Physique de la Mati`ere Condens´ee, Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland (Dated: October 9, 2018)We have studied the temporal evolution of a quantum system subjected to strong dissipation at ultra-low tem-peratures where the system-bath interaction represents the leading energy scale. In this regime, theory predictsthe time evolution of the system to follow a generalization of the classical Smoluchowski description, the quan-tum Smoluchowski equation, thus, exhibiting quantum Brownian motion characteristics. For this purpose, wehave investigated the phase dynamics of a superconducting tunnel junction in the presence of high damping.We performed current-biased measurements on the small-capacitance Josephson junction of a scanning tun-neling microscope placed in a low impedance environment at milli-Kelvin temperatures. We can describe ourexperimental findings by a quantum diffusion model with high accuracy in agreement with theoretical predica-tions based on the quantum Smoluchowski equation. In this way we experimentally demonstrate that quantumsystems subjected to strong dissipation follow quasi-classical dynamics with significant quantum effects as theleading corrections.
PACS numbers: 74.50.+r, 74.55.+v, 05.30.-d
Introduction.–
Brownian motion – that is the fate of a heavyparticle immersed in a fluid of lighter particles – is the proto-type of a dissipative system coupled to a thermal bath [1]. Itsquantum mechanical analogue can be found in open quantumsystems, which have received considerable attention in the lastdecade [2]. This is mainly due to the experimental progress infabricating quantum devices on ever growing scales with theintention to control their quantum properties to an unprece-dented accuracy. Efforts have thus focused to tame the impactof decoherence and noise in order to preserve fragile featuressuch as entanglement as possible resources for technologicalapplications [3].The regime, where dissipation cannot be seen as a pertur-bation, but tends to completely dominate the system dynamicshas received much less attention. This is in sharp contrast toclassical non-equilibrium dynamics, where the so-called over-damped regime, also known as the classical Smoluchowskiregime (cSM), plays a pivotal role for diffusion phenomena ina broad variety of realizations [1]. Theoretically, according toSmoluchowski, this domain is characterized by a separationof time scales between the relaxation of momentum (fast) andthe relaxation of position (much slower) implying that on acoarsely grained time scale the latter constitutes the only rele-vant degree of freedom. The situation in quantum mechanicsis more subtle though. Position and momentum are boundtogether by Heisenberg’s uncertainty relation, which is detri-mental to the tendency of strong dissipation to induce local-ization and even dissipative phase transitions [4, 5].Roughly speaking, a dissipative quantum system is charac-terized by three typical energy scales, namely, an excitationenergy ¯ hω , where ω denotes some specific energy scale ofthe bare system and ¯ h the reduced Planck’s constant, a cou-pling energy to the environment ¯ hγ , where γ denotes thecoupling parameter, and the thermal energy k B T with k B as Boltzmann’s constant and T as the temperature. While the realm of classical physics is then defined by the relation ¯ hγ, ¯ hω (cid:28) k B T , the predominantly explored quantum do-main of weak system-bath interaction obeys ¯ hγ (cid:28) k B T (cid:28) ¯ hω with the bare level spacing exceeding all other energyscales. This is the generic situation for cavity and circuit quan-tum electrodynamical set-ups [2]. FIG. 1: (a) Parameter diagram for the dynamics of overdamped phasediffusion in a superconducting tunnel junction. The regime of over-damped quantum diffusion (qSM, blue) emerges at lower temper-atures from the classical Smoluchowski domain (cSM, grey) whenthe dimensionless friction η sufficiently exceeds the dimensionlessinverse temperature Θ . (b) The phase dynamics corresponds to thedissipative quantum dynamics of a particle in a washboard potential(grey solid), which appears as macroscopic quantum tunneling forweak friction, i. e. the tunneling of the phase wavefunction throughthe potential wall (red wavepackage), and as quantum diffusion inthe qSM domain, for which quantum fluctuations effectively reducethe barrier height (blue solid). Classical phase diffusion is illustratedas the thermally activated escape of a particle (green dot) over thepotential barrier. The quantum range of strong dissipation is complementary,i. e. k B T (cid:28) ¯ hω (cid:28) ¯ hγ . It has been predicted by theory thatin this domain a separation of time scales and thus a quan-tum Smoluchowski regime (qSM) exists indeed, cf. Fig. 1(a)[6]. Quantum Brownian motion in the qSM is almost classi- a r X i v : . [ qu a n t - ph ] J a n cal, however, substantially influenced by quantum fluctuationsyielding quantum diffusion characteristics. Consequently, inall processes sensitive to these fluctuations the dynamics ispredicted to deviate strongly from the classical one. Hence, toexplore the qSM is not only of fundamental interest, but alsoof direct relevance for strongly condensed phase systems, yet,its experimental observation has been elusive so far.The aim of this Letter is to close this gap. For this purpose,we study the dynamics of the phase φ as a continuous collec-tive degree of freedom in a superconducting tunnel junctionin the qSM regime. In fact, in the past superconducting cir-cuits have proven to serve as ideal testbeds to explore quan-tum dissipative phenomena as e.g. macroscopic quantum tun-neling (MQT), transitions from quantum to classical, or de-phasing and decoherence [7–11]. Here, we access the hithertountouched qSM domain by investigating the current-voltagecharacteristics (IVC) of current biased small capacitance tun-nel junctions in an ultra-low temperature scanning tunnelingmicroscope (STM) [12, 13]. Its phase dynamics is equivalentto quantum diffusion along a tilted washboard potential understrong damping, cf. Fig. 1(b), referring to a low impedance en-vironment with ohmic-resistance R DC much smaller than thequantum resistance R Q = h/ e , i.e. ρ ≡ R DC /R Q (cid:28) ( e denotes the elementary charge) [14, 15]. Accessing the qSMregime where quantum diffusion can be observed further re-quires that the dimensionless friction η = E C / (2 π ρ E J ) ,with charging energy E C , tunnel coupling E J , and dimen-sionless impedance ρ sufficiently exceeds the dimensionlessinverse temperature Θ = βE C / (2 π ρ ) with β = 1 /k B T .In terms of experimentally accessible circuit parameters, thiscondition corresponds to E C (cid:29) E J for a low impedance en-vironment ρ (cid:28) and milli-Kelvin temperatures [16]. Theory.–
The supercurrent through a superconducting tun-nel junction is determined by its phase dynamics accordingto the first Josephson relation, I J = I sin( φ ) with the crit-ical current I = 2 eE J / ¯ h . In the qSM regime and for apure DC environment the diffusion of the phase φ occurs in awashboard potential tilted by a bias current I B as depicted inFig. 1(b) [14, 15]. Below the switching current I S , the phasediffuses from one well to another due to the interplay of ther-mal and quantum fluctuations, this way acquiring a finite ve-locity ˙ φ (cid:54) = 0 , cf. 1(b). According to Josephson’s second rela-tion, (cid:104) ˙ φ (cid:105) = (¯ h/ e ) V , this velocity is related to a measurablevoltage drop across the junction. The corresponding Cooperpair current then takes the compact form I qSM J = eρβπ ¯ h ( E ∗ J ) βeV ( βeV ) + π ρ (1)with a renormalized Josephson energy E ∗ J = E J ρ ρ (cid:18) βE C π (cid:19) − ρ e − ρc , (2)where c = 0 . . . . denotes Euler’s constant [14, 15]. Itturns out that the above expression can also be understoodas the quantum generalization of the corresponding classi- FIG. 2: (a) Left side: Circuit diagram of the experimental setup with I B as the bias current source, R B as the source impedance, R and C as the load-line resistor and shunt capacitor, respectively. The junc-tion and its direct environment is highlighted by the blue box, where I and C J denote the junction element and capacitor, respectively and Z denotes the environmental impedance. Right side: Schematic rep-resentation of the STM tip on top of the reconstructed V(100) surface[16]. (b) IVC of the superconducting STM tunnel junction measuredat . G . (c) Simulated real part (cid:60) [ Z ] of the frequency-dependentenvironmental impedance [18]. (d) In-gap region of an IVC from acurrent-biased measurement at G N = 0 . G . cal Smoluchowski-type of treatment for thermal phase dif-fusion, the Ivancheko Z’ilberman approach [28], by replac-ing the bare tunnel coupling with its renormalized value E ∗ J .The explicit dependence of E ∗ J on E C in Eq. 2 reflects the im-pact of charge fluctuations on the phase dynamics and thus,in the mechanical analogue, the presence of momentum fluc-tuations in the overdamped diffusion of position. The simul-taneous interplay of classical and quantum diffusion leadingto Eq. 1, therefore, corresponds to quantum Brownian motiondynamics. The physical interpretation is that quantum fluctu-ations of the phase close to the top of the washboard poten-tial barrier depicted in Fig. 1(b), effectively reduce its heightconsiderably [15]. We note, that one may see this effect ascomplementary to MQT, occurring at low temperatures andweak damping in the opposite domain E J (cid:29) E C , [9]. As hasbeen demonstrated theoretically, quantum Brownian motiondynamics substantially reduces the maximum supercurrent I S ,in the following referred to as switching current, much belowits critical current I [14, 15]. We will employ this effect as anexperimental probe for the detection of overdamped quantumphase diffusion and to distinguish it from its classical counter-part. In the following, we will provide convincing experimen-tal data for the validity of Eq. 1 in the qSM regime. Superconducting circuit.–
The circuit diagram of the exper-imental setup is depicted in Fig. 2(a). The atomic-scale tunneljunction appears between a superconducting vanadium STMtip and a vanadium (100) single crystal surface, having a smalljunction capacitance C J of a few femtofarads [16]. The junc-tion and its direct electromagnetic environment are thermal-ized at the base temperature of our dilution refrigerator STMof T = 15 mK [12], so that one easily arrives at Θ (cid:29) (see Fig. 1(a)). A typical IVC measured over a broad voltagerange at a low normal state conductivity G N = 0 . G with G = 1 / (2 R Q ) being the conductance quantum, is shown inFig. 2(b) and exhibits a well-developed superconducting gap.We determine the normal state conductivity by a linear fit tothe metallic part of the spectrum with a relative deviation of ; detailed analysis of the superconducting properties ofSTM tip and sample, the preparation and measurement proce-dure is described in great detail in Ref. [16]. Since the set-upis operated in the deep tunneling regime at G N (cid:28) G with E J directly proportional to G N , one has E J (cid:28) E C and thecondition (cid:28) Θ (cid:28) η is essentially always fulfilled. FIG. 3: (a) Zoom into the low voltage regime of IVCs from current-biased measurements (red dots) at different indicated values of G N /G . The dashed lines display the corresponding theoretical IVCscalculated by using the QPD model. The quantitative deviation be-tween experiment and calculation is highlighted via the grey bars foreach measurement. (b) Upper part: Calculated critical Josephsoncurrent I (red dots) and linear fit (black, dashed line) as a functionof G N /G (error bars are contained within the symbols). Lower part:Experimental switching current, I S (red crosses), calculated switch-ing current, I qSM J, max (black crosses), as well as a quadratic fit (black,dashed line) as a function of G N /G . In order to minimize the influence of the biasing circuit ontothe phase dynamics, we separate the time-scales of the junc-tion phase and the biasing circuit by using a large shunt ca- pacitor C = 3 nF and a load-line resistor of R = 3 . k Ω [17]. Owing to the large output impedance of our constantcurrent source, R B = 1 . G Ω , the circuit features a hor-izontal load-line. Concerning the electromagnetic environ-ment of the tunnel junction, we obtain a virtual DC impedancefor frequencies up to the low GHz regime by choosing anSTM tip of adequate length, moving the tip resonance modesin the environmental impedance Z ( ν ) to higher frequencies[18]. This is illustrated by the frequency-dependent part ofthe simulated system impedance in Fig. 2(c), where the tipresonance modes appear as sharp peaks [18]. In the low fre-quency limit ν → , the environmental impedance Z (0) is,therefore, dominated by the coupling of the tunneling Cooperpairs to the electromagnetic vacuum in the gap between tipand sample (tunnel barrier) with the STM being operated inultra-high vacuum conditions. Thus, the vacuum impedance R DC = Z ( ν →
0) = 377 Ω determines the DC impedance inthe vicinity of the tunnel junction, whereas the resistance ofthe leads (transmission lines) is negligibly small. In this way,we realize the required low impedance environment, ρ (cid:28) inducing the required high damping to the phase dynamicsand the quantum phase diffusion model of Eq. 1 becomes ap-plicable to our experiment at small voltages around zero bias[19]. Results.–
We have measured current-biased IVCs at a totalof 14 different values of the normal state conductivity from . G < G N < . G by changing the vacuum gap widthbetween STM tip and the sample surface. In Fig. 2(d), weplot one example measured at G N = 0 . G to illustrate thegeneral properties of our experimental setup. For increasingbias current starting from zero, the current follows a phase-diffusion branch at small voltages V (cid:54) = 0 , before switchingto a dissipative in-gap current. Due to the horizontal load-lineof our circuit, we can only access regions of positive differen-tial conductance, yielding a hysteresis in the IVC as shown inFig. 2(d). For decreasing bias currents starting from I B > I S ,the current is fixed to the in-gap current before switching backinto the phase-diffusion branch. At our experimental condi-tions, the in-gap current originates from quasiparticle excita-tions due to life-time effects of Cooper pairs in the STM tipand tunneling of individual Cooper pairs via interaction withresonance modes in the environment [18, 21, 22].Considering the experimental current amplitude of the datashown in Fig. 2(d), we find the switching current I S = 49 ± pA reduced by approximately two orders of magnitude incomparison to the calculated critical current I = 4 . ± . nA [16, 23]. Thermal energy being comparatively small E J β ≥ (see Refs. [16, 24] how to determine the E J val-ues), classical phase diffusion cannot be the dominant processhere. Instead, this strong reduction of I S indicates a majorrelevance of quantum fluctuations. To better understand thisobservation, we will in the following present a series of IVCsmeasured at different values of G N /G and quantitatively an-alyze these curves by applying the quantum diffusion modelof Eq. 1. We emphasize that varying the normal state conduc-tivity at fixed temperature provides us with an ideal handle totune the η ∝ E − J ∝ G − N value along a vertical axis in thephase diagram in Fig. 1(a).In Fig. 3(a), we present a series of IVCs measured at differ-ent values of G N /G and take a closer look at the low volt-age regime V < µ V for positive bias currents. Towardshigher normal state conductivities, we observe a strong andnon-linear increase of the switching current I S . This obser-vation is illustrated in Fig. 3(b), where we plot the extracted I S values as a function of G N /G and fit a quadratic depen-dence I S ∝ G N , as it is predicted by Eq. 1. We additionallyfind I S strongly reduced in comparison to the calculated crit-ical Josephson current I ∝ G N also shown in Fig. 3(b) forall measurements, as we have mentioned before. In combi-nation, these findings reveal the major relevance of quantumfluctuations in the limit E J (cid:28) E C [20, 24].To quantitatively analyze the experimental IVCs inFig. 3(a), we calculate the corresponding theoretical IVCs byusing Eq. 1 for each G N /G value and plot them alongside theexperimental data. We obtain the necessary experimental val-ues of E C , E J and T by performing additional voltage-biasedmeasurements on the same tunnel junction at the same G N /G values. Fitting the resulting IVCs with P ( E ) -theory allows usto extract these parameters with high precision (see Refs. [16]and [24] for more details) [25, 26]. Owing to the small junc-tion capacitance in the absence of a large shunt capacitor in thejunction vicinity, thermally induced capacitive noise broadenthe measured IVCs. We account for this effect by convolv-ing the calculated IVC from Eq. 1 with a normalized Gaussianfunction P N ( E ) of width σ = (cid:112) E C /β [13, 16].Comparing experimental and theoretical IVCs in Fig. 3(a),we find excellent quantitative agreement in the low voltageregime V < µ V for small normal state conductivity val-ues. We recall that our experimental setup can only accessregions of positive differential resistance. At these small volt-ages, we attribute the slightly different slope of the phase dif-fusion branch between experiment and theory to small devia-tions of our environmental impedance from a perfect DC be-havior as required for the derivation of Eq. 1. By contrast,considering the IVCs measured at high G N values shown inFig 3(a), we observe significant quantitative deviations in thecurrent amplitude between theory and experiment. We willanalyze these deviations in the following in more detail andalso compare our experimental data with a classical diffusionmodel. Discussion.–
We start our discussion by comparing themaximum of the measured IVC, I S , with the maximum ofthe calculated IVC, I qSM J, max , as a measure for the agreementbetween experiment and theory. In Fig. 4(a), we plot their rel-ative deviation ∆ I qSM = | I qSM J, max − I S | /I qSM J, max as a functionof G N /G . Furthermore, we include the conditions for theqSM regime, η (cid:29) Θ (cid:29) by using the same set of parame-ters from the P ( E ) -fits as before. We obtain Θ = 122 for allmeasurements and plot the dependence of η/ Θ on G N /G inFig. 4(a). In the low conductivity regime, the condition for theobservation of qSM dynamics is fulfilled η/ Θ > , explain-ing the very small deviations between theory and experiment FIG. 4: (a) Relative deviation between theoretical and experimentalcurrent maxima ∆ I qSM as well as the qSM regime boundary η/ Θ asa function of the normalized normal state conductivity. (b) Compar-ison between experimental data (red, solid) and an IVC calculated(black, dashed) from the purely classical IZ model at a normal stateconductivity of G N = 0 . G . < ∆ I qSM < . By contrast, we find ∆ I qSM >
12 % athigh conductivities where the condition for qSM dynamics isviolated η/ Θ ≈ , yielding an overall consistent picture.Additionally, we compare our experimental data with the classical IZ model (IZ) in which phase diffusion only oc-curs via thermally activated escape over the washboard poten-tial barrier displaying classical Brownian motion dynamics,cf. Fig. 1(b) [27, 28]. We plot an experimental IVC togetherwith a theoretical IVC calculated using the IZ model and thesame parameter set from P ( E ) -theory as before for a normalstate conductivity of G N = 0 . G in Fig. 4(b). As can beseen, theory based on classical diffusion cannot reproduce ourexperimental data and largely overestimates the IVC ampli-tude. This observation clearly underlines the high relevanceof quantum fluctuations, significantly reducing the washboardpotential barrier height and, thus the switching current in com-parison to classical phase diffusion dynamics. Formally, thiseffect yields the renormalized Josephson coupling energy E ∗ J in Eq. 2, as compared for the bare E J in the classical regime.Together, the accurate agreement between experiment andtheory at low conductivities demonstrates that the overdampedquantum phase dynamics of our tunnel junction can be de-scribed in the framework of the qSM equation [15]. At highconductivities, however, the condition for a separation of timescales of position and momentum dynamics, η (cid:29) Θ , is vi-olated and, accordingly, the qSM equation is not applicableanymore. Hence, our experimental study reveals that the timeevolution of an overdamped quantum system follows quasi-classical dynamics with significant quantum-mechanical cor-rections in leading order that is quantum Brownian motion [6].Based on its fundamental character in the framework of quan-tum statistics, our study should be of general relevance for thefields of superconducting quantum circuits [3], quantum gases[29] and nano-mechanical oscillators [30], for instance. Conclusion.–
We have studied the temporal evolution of aquantum system subjected to strong dissipation in the frame-work of the qSM equation. We experimentally investigatedthe phase dynamics of a small-capacitance Josephson junc-tion in an STM placed in a low-impedance environment atultra-low temperatures by means of current-biased measure-ments. We can theoretically describe the measured IVCsin the low voltage regime by using a quantum phase diffu-sion model derived from the qSM equation with high accu-racy. In this way, our study reveals that the dynamics of anoverdamped quantum system corresponds to quantum Brow-nian motion, which is quasi-classical dynamics with signifi-cant quantum-mechanical corrections. In addition, our studydemonstrates the unique potential of ultra-low temperatureSTM to address questions in the field of mesoscopic trans-port and quantum statistics. We envision that future experi-ments on the temperature dependent phase dynamics in super-conducting tunnel junctions should reveal the transition fromquantum-mechanical to classical Smoluchowski dynamics.It is our pleasure to acknowledge inspiring discussions withP. W. Anderson, D. Esteve, J. Pekola, F. Tafuri, M. Ternes, S.Kochen and A. Yazdani. ∗ Corresponding author; electronic address:[email protected]fl.ch; Present address: PrincetonUniversity, Joseph Henry Laboratory at the Department ofPhysics, Princeton, NJ 08544, USA[1] Risken, H.
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