Thermodynamic uncertainty relation in atomic-scale quantum conductors

The thermodynamic uncertainty relation (TUR) is expected to hold in nanoscale electronic conductors, when the electron transport process is quantum coherent and the transmission probability is constant (energy and voltage independent). We present measurements of the electron current and its noise in gold atomic-scale junctions and confirm the validity of the TUR for electron transport in realistic quantum coherent conductors. Furthermore, we show that it is beneficial to present the current and its noise as a TUR ratio in order to identify deviations from noninteracting-electron coherent dynamics.


I. INTRODUCTION
The thermodynamic uncertainty relation (TUR), a cost-precision trade-off relationship, has been of great interest recently in classical statistical physics. While it was originally conjectured for continuous time, discrete state Markov processes in steady-state 1 , it was later proved based on the large deviation technique 2,3 . An incomplete list of studies on the TUR includes its generalizations to finite-time statistics [3][4][5][6] , Langevin dynamics 4,7-10 , periodic dynamics 11,12 , broken time reversal symmetry systems [13][14][15] , as well as derivations of trade-off relations for heat engines 16 . Generalized versions of the TUR, which are based on the fundamental fluctuation relation were recently derived in Refs. 17,18 . Furthermore, a TUR bound for quantum systems in nonequilibrium steady-state was obtained in Ref. 19 using quantum information theoretic concepts.
For a two-terminal single-affinity system, the TUR connects the steady-state current j , its variance j 2 = j 2 − j 2 , and the average entropy production rate σ in a nonequilibrium process 1 , Here, k B is the Boltzmann constant. This relation, which was originally derived based on Markovian dynamics, reduces to an equality in linear response. Away from equilibrium, Eq. (1) describes the trade-off between precision and dissipation: A precise process with little noise is realized with high thermodynamic (entropic) cost. Systems that obey this inequality satisfy the TUR. TUR violations correspond to situations in which the left hand side of Eq. (1) is smaller than 2. We refer to this special bound as the TUR-contrasting it to generalized TUR bounds-and we highlight that it is not universal and that it may be violated for certain processes 20 . Violations of the bound (1) were theoretically predicted in Refs. [20][21][22] for charge and energy transport prob-lems in single and double quantum dot junctions in certain parameter regimes, when the transmission function was structured in the bias window. The first experimental interrogation of the bound (1) was recently reported in Ref. 23 by probing energy exchange between qubitsalbeit in the transient regime. It was demonstrated in Ref. 23 that this bound could be violated by tuning the energy exchange parameters (qubit-qubit coupling), in line with theoretical predictions, and while satisfying the looser, generalized TUR bounds [17][18][19] .
In what follows, we focus on the bound (1), rather than on its generalized forms 17,18 or the looser quantum bound 19 , since it is expected to be valid for quantum transport junctions with a constant transmission probability 20 . Considering (single-affinity) steady-state charge transport under an applied bias voltage V , dissipation is given by Joule's heating, σ = j V T , with T the temperature of the electronic system. The inequality (1) then simplifies to with β = (k B T ) −1 . For convenience, we introduce the combination Q ≡ βV j 2 j , which is a function of voltage and temperature. We refer to Q as the TUR ratio.
The TUR allows understanding of the trade-off between current fluctuations and entropy production. Furthermore, verifying or violating Eq. (2) provides insight into the underlying charge transport statistics as was discussed in Ref. 20 . Atomic-scale junctions offer a rich playground for studying steady-state quantum transport at the nanoscale 24 . It was pointed out in Ref. 20 that in junctions with a constant transmission probability, Eq.
(2) should be valid. However, an experimental verification for this prediction is missing. Furthermore, beyond the fundamental interest in thermodynamical bounds, it might be useful to examine the behavior of the TUR ratio. We therefore ask here the following question: Does the measure Q reveal useful, additional information about the transport process beyond what is separately contained in the current and its fluctuations?
The objective of this work is to study the TUR in charge-conducting atomic-scale junctions and use this compound measure to learn about charge transport mechanisms in real systems. Different realizations of gold atomic scale junctions depict distinct differential conductance traces 24 . Furthermore, corresponding shot noise measurements display pronounced anomalous characteristics at high voltage [25][26][27][28] . Here, we confirm the validity of the TUR [Eq. (2)] in steady-state using experimental data, in accord with theoretical predictions 20 . Furthermore, we argue that the TUR ratio can distill underlying transport mechanisms in atomic-scale junctions, which may be convoluted at the level of the current and its noise. The ratio Q begins at the equilibrium value of 2. We show that its linear behavior in voltage indicates the shared underlying quantum coherent dynamics. In contrast, a quadratic term in voltage distinguishes nonlinear contributions beyond the constant transmission limit.
Altogether, this study (i) validates and verifies the TUR in atomic-scale junctions and (ii) illustrates that the TUR can assist in diagnosing transport regimes. Yet more broadly, this study bridges a gap between quantum transport junctions 24 and stochastic thermodynamics 29,30 , illustrating that thermodynamical bounds can be tested in nanoscale systems, in the quantum domain, down to the level of atomic-scale electronic conductors.

A. TUR for normal shot noise
We consider nanoscale conductors in the quantum coherent limit with a constant transmission function (ohmic conductors), τ = i τ i , collecting the contribution of independent transmission channels. The electrical current and its noise, under the chemical potential ∆µ = eV , are given by 24,31,32 We identify the electrical conductance, G = G 0 i τ i , and the constant Fano factor F = i τ i (1 − τ i )/ i τ i . G 0 = 2e 2 /h is the quantum of conductance. Note that the zero frequency spectral density of the noise, commonly denoted by S(ω = 0), is defined a factor of 2 greater than the second cumulant of the noise. The expression for the current noise combines the (zero voltage) thermal noise and the (zero temperature) shot noise.
From these expressions we prepare the TUR ratio, Since x coth x ≥ 1, the bound (2) is satisfied in the constant transmission limit, independent of voltage and temperature. In the high bias limit, ∆µ ≫ k B T , this relation reduces to Note that in this limit the current noise is j 2 = e| j |F , which is the quantum shot noise with the suppression factor F . In contrast, in the limit of very low voltage we expand Eq. (3) and get To study the latter expansion, one would need to inspect the current and its noise close to equilibrium. For T=7 K, β ≈ 1500 (eV) −1 and |βV | < 1 requires scanning the noise for fine bias voltage V < 1 mV. In the experiments reported below the voltage was scanned between 10 and 1000 mV, focusing on the examination of Eq. (5). Comparing Eq. (3) to Eq. (4), we note that these expressions are closely related. However, we argue that the TUR ratio, Eq. (4), provides a beneficial representation of the scaled noise since (i) its development from the equilibrium value of 2 to the high voltage regime can be clearly observed, and (ii) it brings the data together onto a universal curve. In Sec. III we demonstrate these points on measured data.

B. TUR for anomalous shot noise
Measurements of shot noise in Au atomic-scale contacts reveal anomalous (nonlinear) characteristics at high voltage [25][26][27][28] . These observations were interpreted in Ref. 28 based on a coherent quantum transport model with two elements: The transmission function for electrons was assumed to be energy dependent, and the voltage drop on the electrodes was allowed to be asymmetric.
For the Au atomic-scale junctions analyzed in this work, G ≈ 1G 0 , and it is therefore sufficient to consider two channels 24,28 : a primary channel, which is almost fully open, and a secondary channel with a low transmission probability. We model the transmission function of the dominant channel by the low order (linear) Taylor where τ 1 is the constant value of the transmission function at the Fermi energy and τ ′ 1 (µ) is the derivative of this function, evaluated at the Fermi energy. The contribution of the secondary channel is minor, and for its transmission function we use the constant approximation, τ 2 (ǫ) ∼ τ 2 ≪ τ 1 . The partition of the bias voltage is quantified by the parameter α, with µ L = µ + α∆µ and µ R = µ − (1 − α)∆µ; 0 ≤ α ≤ 1; the bias voltage is symmetrically divided at the electrodes when α = 1/2. It should be highlighted that the linear approximation for the transmission function describes only a certain class of measurements, while other atomic-scale junctions display more complicated trends 28 .
The determinant for the energy (and possibly voltage) dependent transmission function in Au atomic-scale junctions could be quantum interference of electron waves with randomly-placed defects in the metal contacts 27 ; our analysis does not presume the root for the functional form of τ (ǫ). The origin of the bias voltage asymmetry could be structural differences in the contact region at the left and right sides. The mean-field parameter α emerges due to underlying many-body effects, that is, the response of electrons in the junction to the applied electric field.
Based on these ingredients, expressions for the current (divided here already by voltage) and its noise were derived in Ref. 28 , The complete expression for the noise is included in the Appendix; here we already took the low temperature limit eV ≫ k B T and assumed that |τ ′ ∆µ| < 1 and α = 1/2. For simplicity, we define a combined nonlinear coefficient γ ≡ (α − 1/2)τ ′ 1 (µ), which has the physical dimension of inverse energy. This coefficient conjoins the two elements that are responsible for anomalous behavior: Many body electron-electron effects (phenomenologically captured by α) and an energy-dependent transmission probability.
We now write down the TUR ratio in the limit eV ≫ k B T to the lowest order in |γ∆µ| < 1, for the positive voltage branch, We can simplify this expression by noting that in our experiments (see Sec. III) βF ≫ |γ| and that τ 2 ≪ τ 1 . We get which simplifies Eq. (8) to This result is remarkable since the second, nonlinear term in voltage distills the nonlinear contribution γ. Since γ could be positive or negative, the TUR ratio may show The dark (light) regions at the left and right sides represent electron occupation (empty states). The atomic configuration at the junction varies, and the junction is not necessarily spatially symmetric. As a result, the applied voltage may be partitioned unevenly across the atomic-scale junction, quantified by the parameter α.
either a suppression or an enhancement from the linear normal shot noise term. Further, γ∆µ could be comparable to the constant Fano factor F , therefore the contribution of the quadratic term could be substantial. In fact, the TUR could be violated at high voltage once F < γ∆µ.
Altogether, we argue that presenting the noise as a TUR ratio is beneficial for elucidating transport processes. According to Equation (10): (i) the constant term, 2, represents the equilibrium value. (ii) The linear term in voltage describes quantum suppressed-Poissonian dynamics and it identifies the corresponding Fano factor. This term emerges from a quantum coherent transport process with a constant transmission coefficient. (iii) The nonlinear term includes deviations from the constant transmission limit and it reflects the departure from the simple quantum coherent picture, with the involvement of many body effects (γ).
Equation (10) is valid at high voltage, or correspondingly, low temperature, ∆µ ≫ k B T . In the Appendix we discuss the behavior of the TUR ratio for anomalous shot noise at high temperature.

III. ANALYSIS OF ATOMIC-SCALE GOLD JUNCTIONS
We test the theoretical expressions for the TUR ratio with measurements of the current and current fluctuations in Au atomic-scale junctions. The atomic junction and scattering processes are illustrated in Fig. 1. We use the mechanically-controllable break junction technique at cryogenic conditions to form an ensemble of atomicscale junctions 33 . By repeatedly breaking and reforming the junction, realizations with somewhat different atomic configurations are generated, supporting a range of conductance values of G ≈ 1G 0 . Differential conductance measurements are performed for each junction, as well as current-voltage traces and shot noise measurements. For gold atomic-scale junctions, proximity-induced superconductivity measurements 34 and shot-noise analysis 35,36 suggest that a single channel dominates the conduction with a nearly perfect coupling to the metals 24 .
The data that we analyze: the differential conductance, current-voltage characteristics and the shot noise for different atomic-scale junctions is shown in Figures  2 and 3. Based on Fig. 3, we classify three regimes in the current noise: (i) The close-to-equilibrium, or the low voltage regime. In this case, eV is lesser or equal to the thermal energy and the thermal noise is prominent; if T < 10 K, V < 5 mV. The TUR in this regime [see Eq. (6) and the Appendix] is not probed in our work. (ii) Normal shot noise regime, 10 mV< V 75 mV. In this region the current noise follows the standard-normal shot noise expression, and it is linear in voltage. (iii) Anomalous shot noise regime, around V > 75 mV. In this region the shot noise displays anomalous trends as it is no longer linear in voltage.
We extract the zero-voltage electrical conductance,  (4) before the dashed line, which marks the bias voltage 75 mV, beyond which deviations from the universal function show. The temperature, calculated from the equilibrium noise, is T = 6.0 K. G = G0 τi was obtained from the zero-voltage differential conductance. The constant Fano factor was deduced from the normal shot noise regime, based on the behavior of the differential conductance and the noise. Not all voltage points in the noise trace were measured in the electrical current. However, since the current is highly linear in voltage, we performed a linear interpolation for the current-voltage curve and added the few missing points in between; we only interpolate for voltages higher than 10 mV. G = G 0 i τ i , from the differential conductance at the zero voltage. The temperature is verified from the equilibrium noise j 2 = 2k B T G, and is in the range of T = 5 − 10 K. The constant Fano factor F is obtained by fitting the shot noise at low voltage (typically, V < 20 mV) to Eq. (3) and dividing by the zero-voltage electrical conductance.
In Fig. 2, we display the differential conductance of 10 representative junctions. While in some cases the differential conductance is about constant with voltage, other junctions explicate a more significant variability of d j /dV with voltage, indicating deviations from Eq.
(3). We use different colors for different junctions roughly grouped according to their constant Fano factor. In Fig.  3(a), we display the currents for these 10 junctions, which are nearly ohmic throughout. We further present the current noise in Fig. 3(b), which largely deviates from a linear behavior beyond V ∼ 75 mV 28 . Recall that j 2 ∝ |V | for ohmic conductors. Deviations from this trend are referred to as the "anomalous shot noise". This effect was the focus of Ref. 28 .
To test Equation (4), we select junctions 6 and 9 that display approximately a constant differential conductance (thus a constant transmission function), see Fig. 2 (we exclude junction 8 since its noise measurements were missing values in the normal shot noise voltage regime). To contrast it, we also analyze junction 1 for which the differential conductance varies more substantially with voltage. While the temperature was similar for the three junctions (T = 6.0± 0.1 K), the Fano factor was quite distinct, varying between 0.1 to 0.01.
We present the TUR ratio (after subtracting the equilibrium value), Q − 2 of the three junctions in Fig. 4(a). As expected, Q ≥ 2 throughout, Furthermore, by plotting the ratio (Q − 2)/F in Fig. 4(b) we demonstrate that the measurements collapse on the universal function βV coth(βV /2) − 2 up to around 75 mV for junctions 6 and 9. In fact, when T ∼ 6 K (β ∼ 2000 1/(eV)), | coth β∆µ/2| ≈ 1 beyond 5 mV, thus arriving at Eq. (5) with (Q − 2)/F ≈ β|∆µ|. The fact that the data agrees with Eq. (4) is not surprising, since F was extracted from the noise formula (3). However, it is advantageous to present the data in this manner: The combination (Q− 2)/F illustrates the common quantum coherent transport mechanism underlying the shot noise in these junctions for V 75 mV. We test Eq. (10), which describes deviations from the universal form by studying junctions 2 and 3. In these two cases, the differential conductance approximately follows a linear line-corresponding to the theoretical model behind Eq. (7). Junction 4, which was analyzed in Ref. 28 suffers from more significant 1/f noise contribution, and we therefore do not include it.
The slope of the differential conductance provides the coefficient 2γ, which is adopted in Eq. (10) to calculate the TUR ratio. Results are displayed in Fig. 5. The TUR ratio agrees with the constant transmission expression (4) up to ≈ 75 mV. However, as we increase the voltage, the nonlinear expression (10) provides a better description of the curved TUR function with the quadratic coefficient βγ. We retrieve γ ∼ 0.13 (eV) −1 for junction 2. Given that F = 0.06 for this junction, the TUR could be violated at high voltage, V > 0.6 V. However, at this bias voltage, one would need to consider higher order γ∆µ terms in the expansion (10). For junction 3, we obtain γ ∼ −0.095 (eV) −1 , indicating the enhancement of the shot noise relative to the normal shot noise regime. The lowest-voltage TUR ratio for junction 3 seemingly violates the TUR. However, this point suffers from large relative error (since both the current and the noise are small), and we cannot draw conclusions based on this single observation. Careful measurements of the current and its noise at low voltage, V < 10 mV, would allow the analysis of the TUR close to equilibrium, as discussed in the Appendix.

IV. CONCLUSIONS
Cost-precision entropy-fluctuation trade-off relationships are fundamental to understanding nonequilibrium processes. In this work, we focused on steady-state charge transport in atomic-scale junctions, a process which is essentially quantum coherent. Based on experimental data and theoretical derivations, we show that the TUR (1) is satisfied in this system, even when the noninteracting electron picture is corrected to include many-body effects (in a mean-field form). The generalized quantum TUR 19 is factor of 2 looser that this bound, and is obviously satisfied in our system. Our work illustrates that the TUR bound is advantageous for exploring the fundamentals of transport processes: The TUR ratio is developed from the equilibrium value, and it therefore identifies far-from-equilibrium effects. Indeed, organizing the current noise as a TUR ratio is beneficial to understanding the charge transport problem. This is clearly observed by the evolution of the noise from the linear-universal behavior at intermediate voltage to the anomalous regime at high voltage.
More broadly, our study illustrates that atomic-scale junctions offer a rich test bed for studying theoretical results in stochastic thermodynamics-while extending these predictions to the quantum domain. As such, our combined theory-experiment analysis presents a step into consolidating the quantum transport and statistical thermodynamics research endeavors.
The TUR for charge transport in steady-state, Eq. (2), can be violated once the transmission function is structured with sharp resonances of width smaller than the thermal energy 20 . This situation might be realized in molecular junctions at room temperature and at low voltage. Specifically, quantum dot structures offer a rich playground for studying the suppression of electronic noise in nanodevices. Experiments that directly probe the behavior of high order moments of the current [37][38][39] could be used to examine thermodynamical bounds 20,22 . Future work will be focused on the behavior of the current noise and the associated TUR in many body systems such as transport junctions with pronounced electronvibration coupling. Equations (5) and (10), which we used to explain exponential data, were derived in the limit of high bias voltage, eV ≫ k B T . Here we study the complementary limit of high temperature (or low voltage) and discuss the possible violation of the TUR in this regime.
The electric current and its noise can be formally expanded in orders of the applied bias voltage as Here, G 1 is the linear conductance, G 2 , G 3 ,... are the nonlinear coefficients in the current-voltage expansion. Similarly, S 0 is the equilibrium (Johnson Nyquist) noise, and S 1 , S 2 , are the nonequilibrium noise terms. We sub-stitute these expansions into Eq. (2) and get 20 , We make use of the fluctuation-dissipation (Green-Kubo) relation, S 0 = 2k B T G 1 , and the first of the Saito-Utsumi relationships 40 , S 1 = k B T G 2 , both resulting from the fluctuation relation 41 , and reduce Eq. (A2) to We now introduce the expansion of the TUR ratio around equilibrium, Here, Q 2 , Q 3 ,... are coefficients of the TUR ratio, and they depend on internal parameters and temperature. Note that Q 1 is missing in this expansion 20 . A negative Q 2 identifies TUR violation in the second order of voltage.
We now specify this analysis to the gold atomic-scale junctions with G ∼ 1G 0 , as described in Sec. III and in Ref. 28 . To model quantum coherent transport in Au junctions we assume that: (i) The transmission function, which describes the probability for electrons to cross the junction, is linear in energy. (ii) The bias voltage is divided asymmetrically at the contacts, quantified by the parameter α. (iii) Two channels contribute to the transmission, a dominant one which is almost fully open and a secondary channel with a small transmission coefficient τ 2 ≪ τ 1 . (iv) We take into account the variation of the transmission function with energy for the dominant channel only. Using this setup, the charge current is given by 28 The linear conductance and the nonlinear coefficients can be collected as The shot noise is given by 28 with the first three coefficients (A1), . (A8) We can now verify the fluctuation-dissipation relation, S 0 = 2k B T G 1 , as well as the first of the Saito-Utsumi relations, S 1 = k B T G 2 . Indeed, though α (phenomenologically) builds on many-body effects, we can still write down the Levitov-Lesovik formula for the cumulant generating function and show that it satisfies the exchange steady-state fluctuation symmetry 41 . Since G 3 = 0, the series for the TUR ratio, Eq. (A4), reduces to Substituting S 0 and S 2 we get βV j 2 j = 2 + (eV ) 2 6 β 2 F − (τ ′ 1 ) 2 (π 2 − 6) This expansion is valid only close to equilibrium, and as such it is complementary to Eq. (8), which was derived at high voltage.
Altogether, the TUR is satisfied in atomic-scale junctions given that the transmission coefficient is constant (energy independent). Furthermore, as discussed in Ref. 20 , while the single resonance level model can only display very weak TUR violations at high temperature, double-dot models could break the TUR quite substantially depending on the inter-site coupling and the metaldots hybridization energy.