Abstract
Studies of nonequilibrium current fluctuations enable assessing correlations involved in quantum transport through nanoscale conductors. They provide additional information to the mean current on charge statistics and the presence of coherence, dissipation, disorder, or entanglement. Shot noise, being a temporal integral of the current autocorrelation function, reveals dynamical information. In particular, it detects presence of nonMarkovian dynamics, i.e., memory, within open systems, which has been subject of many current theoretical studies. We report on lowtemperature shot noise measurements of electronic transport through InAs quantum dots in the Fermiedge singularity regime and show that it exhibits strong memory effects caused by quantum correlations between the dot and fermionic reservoirs. Our work, apart from addressing noise in archetypical strongly correlated system of prime interest, discloses generic quantum dynamical mechanism occurring at interacting resonant Fermi edges.
Introduction
Nonequilibrium electronic shot noise is a powerful diagnostic tool revealing properties of mesoscopic systems inaccessible by the mean current measurements^{1,2,3}. For example, recent noise measurements in the Kondo impurities^{4,5} have brought new insights into strongly correlated transport. Shot noise is sensitive to the presence of nonMarkovian dynamics^{6} intensively studied in broad context ranging from photosynthesis to quantum information^{7,8,9,10,11,12}. However, most theoretical proposals as well as the newest quantumoptical experimental study^{12} rely on extensive engineering and control of the system and/or environment (bath) and a clear observation and identification of the quantum memory effects in “natural”, i.e., routinely fabricated solidstate systems has not been reported yet.
In resonant tunneling, which is ubiquitous in quantum electronic transport, the charge dynamics of the resonant level can be described by a simple Markovian master equation^{3} as long as the relaxation time related to the inverse of the transfer rates is long compared to the characteristic memory time of the fermionic bath (leads) given by the inverse temperature and/or the detuning of the level from the chemical potentials of the leads. Comparable time scales for system relaxation and bath memory break down the conventional description for lowtemperature onresonance transport and indicate^{13} strong nonMarkovian features which, together with many bodyinteractions typical for small nanostructures, influence the lowtemperature width of the resonant steps in the currentvoltage characteristics^{14}, the decay of the level occupations^{15}, or the noise^{6}. For the noise, significant deviations from the conventional master equation description have already been observed^{16}, although their origin has not been identified.
In this work we present new experimental results on the lowtemperature noise measurements in the Fermiedge singularity (FES) regime^{16,17,18,19,20} together with a theoretical analysis clearly revealing the presence of strong quantum memory around the edge. The noisearoundtheedge puzzle^{16} is briefly introduced in Fig. 1e, where the measured points are contrasted with the standard Markovian theory^{3} (black line) showing large deviations of ∼15% in the Fano factor F ≡ S/2eI (a convenient dimensionless measure of the shot noise S), far beyond the experimental uncertainty. Moreover, the measured dip breaks the Markovian lower bound^{3} of 1/2, which is a clear witness of strong memory. The blue line, nicely coinciding with the data, is our new theory accounting for the memory effects.
The FES, a paradigmatic exactly solvable manybody problem^{21,22}, which originates from the Coulomb interaction of conduction electrons with those on a localised discrete level represented by core shell electrons or quantum dot (QD) levels, was first predicted in the Xray spectra of metals^{23}, but its signatures are observed also in resonant tunneling setups as a (truncated) powerlaw singularity of the mean current I around, e.g., the emitter Fermi energy^{17,18,19,20}. The interacting resonant level model describing the FES transport setup has served recently as an important benchmark for novel quantum transport techniques^{24,25,26} including the noise calculation^{27} at its exactly solvable selfdual point (different from our experimental regime).
Results
We first describe crosscorrelation measurements of current shot noise in selfassembled InAs QDs in the FES regime. The experimental setup is depicted in Fig. 1a and explained in more detail in the Methods section. At zero bias voltage the ground state energy level ε_{D} of the InAs dots lies far above the emitter Fermi energy µ_{E} (see Fig. 1b, left). Therefore, a large threshold voltage bias V_{th} ≈ 170 mV applied to the collector lead is required to shift ε_{D} to resonance with the emitter Fermi energy by electrostatic gating with the leverage factor giving the fraction of the bias voltage dropped at the emitterdot junction (see Fig. 1b, right). On resonance, the tunneling current sets in and displays a sharp peak (shaded part of Fig. 1c, solid line) due to the Fermi edge singularity caused by the Coulomb interaction of the occupied dot level with the electrons in the emitter lead (there is no relevant interaction with the collector due to the asymmetry of the setup). Further increase of the bias causes a decrease of the current due to the decrease of the emitter rate induced by the threedimensional density of states (DOS) in the emitter^{19}. Together with the currentvoltage characteristics on a large voltage scale, i.e., far around the edge, Fig. 1c shows the measured shot noise power S (symbols).
Far enough from the edge, i.e., outside of the shaded region of Fig. 1c, we can use the standard Markovian master equation and evaluate the emitter γ_{E} and collector γ_{C} tunneling rates, Fig. 1d, from formulas^{3} I = 2eγ_{E}γ_{C}/(2γ_{E}+γ_{C}) and F = 1 − 4γ_{E}γ_{C}/(2γ_{E} + γ_{C})^{2} (excluded double occupancy due to strong onsite Coulomb interaction implies usage of 2γ_{E} instead of just γ_{E} as for a noninteracting resonant level^{28}). While the collector rate is basically constant, γ_{E} reflects the energy dependence of the emitter DOS^{29} and exhibits an expected asymmetry of the tunneling barriers with γ_{E}/γ_{C} ranging between 0.06 and 0.22. Plausibly assuming constant γ_{C} throughout the resonance we can analogously to Ref. 16 extrapolate the γ_{E} to the resonance (shaded) region in Fig. 1d (solid lines) from the expression for the current. The resulting Markovian prediction based on these extrapolated rates (black curve in Fig. 1e) clearly exhibits substantial deviations from the measurement inexplicable by experimental errors. Obviously, the resonant transport regime calls for a radically new theoretical understanding.
Discussion
Using the procedure briefly described in the Methods section for B = 0, we arrive at a nonMarkovian generalised master equation (GME) for the occupations of the resonant level, where p_{1}(t) is the probability that the level is occupied by an electron, while p_{0}(t) = 1 − p_{1}(t) denotes the probability of the dot being empty,The expressions () for the forward/backward nonMarkovian electron transfer rates across the QD/emitterlead interface involve standard FES Green's functions, whose evaluation is a known result of the FES theory^{21,22,30}. This results in the explicit form of the rates entering equation (1) — in the Laplace space they read and , where is the dimensionless energy/voltage distance from the resonant edge, α is the FES critical exponent, and B(x, y) denotes the betafunction. In the zerotemperature limit the formula simplifies to . When the counting field χ at the emitter junction is included^{6,31} the GME memory kernel corresponding to equation (1) is of the form Using the standard procedure for the cumulant evaluation^{6,31} on this memory kernel, we get, using the abbreviations and , the formulas for the mean current and for the nonMarkovian Fano factor The last term in the Fano factor, proportional to the derivatives, constitutes the nonMarkovian correction. Well above the edge, where both the backflow and the nonMarkovian features can be neglected, we recover the standard master equation result (with ).
In the lowest order in we can write , with magnitude . This implies that for low temperatures and close to the edge the nonMarkovian correction is governed by the collector rate γ_{C} and, thus, it is of the same order as the Markovian correction to the Poissonian noise (with F = 1) due to correlations caused by sequential occupying and emptying of the QD. Being of quantum origin, it vanishes fast with increasing temperature , which kills quantum correlations between the dot and leads responsible for the memory effects. Moreover, it generically assumes both signs — negative above the edge, further suppressing the Fano factor as in Fig. 1e, but also positive below the edge in the purely quantum tunneling regime, where it counteracts the classical term by increasing the Fano factor to potentially superPoissonian values (F > 1). While the noise suppression can be achieved by memory of any origin, quantum or classical, the noise enhancement is a fingerprint of subtle quantum correlations.
We now demonstrate these concepts by more elaborate analysis of the experimental data acquired at various values of the temperature and magnetic field. We start by fitting the experimental data for the mean current (insensitive to memory) around the edge(s) with a straightforward extension of the above theory to the case of two spinsplit levels due to the magneticfield with resulting 3×3 (double occupancy excluded) memory kernel analogous to equation (2) as shown in Fig. 2. In the finite magnetic field case 6 free parameters were fixed by fitting simultaneously curves at various temperatures, namely two independent critical exponents α_{↑} = 0.40, α_{↓} = 0.43 and thresholds V_{th↑,↓} together with an overall prefactor to the emitter rates and the leverage factor η, while in the B = 0 case only 4 parameters due to a single resonance peak were fixed with α = 0.28 in qualitative agreement with indepth investigations^{19,20}.
By this procedure all parameter values are fully determined and the predicted Fano factor curves in the lower panel of Fig. 3 are free of any ambiguity. Considering this, the correspondence between the measurements (points with errorbars) and our nonMarkovian theory (lines) is quite remarkable in all cases encompassing two magnetic field values and various temperatures. We also compare the Markovian, i.e., with the derivative terms in equation (3) omitted (dashed lines), and nonMarkovian (solid lines) predictions in the insets and the detail of Fig. 3 with clear demonstration of the already mentioned nonMarkovian features in the lowtemperature Fano factor, namely, the significantly more pronounced dip on the highvoltage side of the FES and the potentially superPoissonian peak on the lowvoltage side with fast destruction of the nonMarkovian corrections with temperature or distance from the resonant edge.
All these features are clearly seen in the experimental data as well. The superPoissonian Fano factor due to quantum coherence at the lower edge is not reliably confirmed experimentally because of associated large errors resulting from a ratio of very small values of both the current and noise (tunneling regime). Nevertheless, the experimentally observed peak just below the upper edge (see the detail in Fig. 3), although subPoissonian, is caused by the very same mechanism and is thus an indirect confirmation of the purely quantum memory effect. Altogether, the importance of the nonMarkovian corrections due to quantum memory is established both qualitatively and quantitatively.
Methods
Experimental details
The studied InAs QDs are embedded in a GaAsAlAsGaAs resonant tunneling device patterned into pillars with a cross section of 9 × 9 µm sufficiently small to resolve single dot tunneling^{16}. The effective AlAs barrier widths of 4 and 3 nm are slightly asymmetric. The measurements, whose schematic of the electronic setup is shown in Fig. 1a, were performed in a dilution refrigerator at temperatures down to 70 mK and magnetic fields up to 13 T. The DCpart of the source drain current I is measured with a transimpedance amplifier which also biases the sample. Two 4.7 kΩ resistors convert the fluctuating current to voltages which are measured in a crosscorrelation configuration. Together with parasitic capacitances these resistors form RCcircuits which define the bandwidth of our experiment. To increase this bandwidth we use homebuilt coaxial cables thereby lowering the total parasitic capacitance to 20 pF.
The voltage fluctuations are amplified by a twostage low temperature amplifier based on the ATF34143 HEMT with a gain of 22 dB, followed by a room temperature amplifier with a gain of 60 dB. The amplified signal is filtered and digitised, and from the Fourier spectra the crosscorrelation noise power S_{AB} is calculated and averaged over 8 minutes. To retrieve the shot noise power the real part of S_{AB} in the frequency range from 500 kHz to 3 MHz is evaluated. A technical backgroundnoise, largely dominated by thermal noise sources like the first transistor stage of the cryogenic amplifiers and the conversion resistors, is measured at zero current and subtracted. At finite sample impedances the partially correlated thermal background is estimated and also subtracted. The correlation gain parameters are determined by noise thermometry. The error bars in Fig. 1e and Fig. 3 consist of the statistical error, the error of the estimated background and the calibration error.
Theoretical details
Hamiltonian of simplified spinless model of a resonant level tunnelcoupled to two leads (emitter E and collector C) and Coulombcoupled just to the emitter reads H = H_{QD} + H_{E} + H_{C} + H_{T} + V_{X}d ^{†}d with H_{QD} = ε_{D}d ^{†}d, , and , where β = E, C; d and c_{k,β} are QD and lead annihilation operators, ε_{D} and are the energies of the QD level and of the electrons in the leads, respectively, while describe the tunneling between the QD and the leads. The last term describes the scattering of emitter lead electrons on an electron in the QD and is responsible for the FES phenomenon. Since the Fermi level of the collector lead is far below the resonant level and γ_{C} does not depend on energy close to the edge (Fig. 1d) we can use the method by Gurvitz and Prager^{28} to exactly integrate out the collector lead. This leads to the equation of motion for the density operator σ(t; n) of the dot and the emitter resolved with respect to the number n of passed electrons through the emitter/QD interface. After introducing the counting field χ as a conjugate variable to n, one can write the equation of motion for , partly expressed in the block form^{6,31}, σ = (σ_{00}, σ_{11}, σ_{01}, σ_{10})^{T},where is the appropriately modified Hamiltonian of the emitter and QD including the counting field.
The emitter lead can then be handled perturbatively in the tunnel coupling following closely the derivation for dissipative double quantum dot from Ref. 31 by first separating equation (4) into four equations for the elements of σ. Tracing out the electron states of the emitter lead in the two equations for the evolution of the diagonal elements σ_{jj} (j = 0, 1), allows us to find the evolution equations for the generalised QD occupations p_{j}(t; χ) = Tr_{E}σ_{jj}(t; χ) The equation governing the evolution of σ_{01}(t; χ) which enters Eqs. (5) contains the diagonal elements σ_{jj}(t; χ) of the total density matrix. In order to close the equations for p_{j}(t; χ) we perform physically motivated QDstateresolved perturbative decoupling of the density matrix into with being the grandcanonical density matrix of the emitter lead at temperature T and chemical potential μ_{E} when the QD is empty (j = 0) or occupied (j = 1). Thus, after multiplying the forward rate by 2 due to the interplay of spin and Coulomb blockade^{28}, we find equation (2) (reducing to equation (1) for χ = 0) with FES Green's functions reading , .
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Acknowledgements
We thank T. Lüdtke and K. Pierz for device fabrication, C. v. Zobeltitz for valuable discussions, and R. Filip for useful comments on the manuscript. This work was supported by the German Excellence Initiative via QUEST (Hannover), by the Czech Science Foundation via Grant No. 204/11/J042 (T. N.), and the TEAM programme of the Foundation for Polish Science, cofinanced from the European Regional Development Fund (K. R.).
Author information
Affiliations
Institut für Festkörperphysik, Leibniz Universität Hannover, Appelstraβe 2, D30167 Hannover, Germany
 N. Ubbelohde
 , F. Hohls
 , N. Maire
 & R. J. Haug
Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, CZ12116 Prague, Czech Republic
 K. Roszak
 & T. Novotný
Institute of Physics, Wroctaw University of Technology, PL50370 Wroctaw, Poland
 K. Roszak
PhysikalischTechnische Bundesanstalt, D38116 Braunschweig, Germany
 F. Hohls
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Contributions
The experiment was designed by F.H. and R.J.H. and carried out mostly by N.U. and N.M. Theory was designed and developed by K.R. and T.N. N.U. and K.R. jointly analyzed and interpreted the data. Manuscript was written mainly by T.N. with contributions from N.U. and K.R. and with steady input and feedback from F.H. and R.J.H.
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The authors declare no competing financial interests.
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Correspondence to T. Novotný.
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Further reading

Full counting statistics of electronic transport through interacting nanosystems
Journal of Computational Electronics (2013)
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