Session X3: Invited Session: Full Counting Statistics, Fluctuation Theorems, and Many-Body Entanglement

Towards Measuring the Many-Body Entanglement from Fluctuations

The degree of entanglement in a many-body quantum system is often characterized using the bipartite entanglement entropy. We propose that bipartite fluctuations are also an effective tool for studying many-body physics [1] particularly its entanglement properties, in the same way that noise and full counting statistics have been used in mesoscopic transport and cold atoms. We apply some concepts underlying the field of full counting statistics to the study of the ground states of many-body Hamiltonians, with the boundary introduced by the bipartition playing the role of the scattering or interacting region. For systems that are equivalent to non-interacting fermions, we show that fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy [1-3]. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time [2] then would constitute a strong indication of many-body entanglement [1]. Here, the measurability of the entanglement entropy, while suggestive, is particular to the nature of non-interacting particles [4,5]. \\[4pt] [1] H. Francis Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie and K. Le Hur, arXiv:1109.1001. 30 pages + 25 pages supplementary information.\\[0pt] [2] I. Klich and L. Levitov, Phys. Rev. Lett. 102, 100502 (2009).\\[0pt] [3] H. F. Song, C. Flindt, S. Rachel, I. Klich and K. Le Hur, Phys. Rev. B 83, 161408R (2011).\\[0pt] [4] B. Hsu, E. Grosfeld and E. Fradkin, Phys. Rev. B 80, 235412 (2009).\\[0pt] [5] H. Francis Song, Stephan Rachel and Karyn Le Hur, Phys. Rev. B 82, 012405 (2010).   

Charge sensing and real-time electron counting in quantum dots

The charge state of a Coulomb blockaded quantum dot can be sensed at the single-electron level using charge sensors integrated on-chip [1]. Quantum point contacts or quantum dots have proven to work as reliable charge sensors. Coupling between the quantum dot's charge state and the sensor is provided by Coulomb interactions between electrons in the two systems. The technique is therefore independent of the material of quantum dot and sensor, and works for diverse systems such as GaAs [1] and graphene [2]. Different materials, such as InAs and GaAs have even been combined. Time-resolved charge sensing is of fundamental interest for studying the statistical properties of charge flow, like the full counting statistics. Single-shot charge and spin read-out schemes are also needed for the measurement of qubits. Some of our recent work has focused on the exploration and optimization of the read-out bandwidth limits. Towards this goal, we learned how to influence the dot-detector coupling [3], and how to employ high-frequency techniques in the range of a few hundred megahertz [4], or even up to a few gigahertz for improved performance. New experiments using the slower conventional sensing schemes have allowed us to explore non-equilibrium statistics and the fluctuation theorem. In our measurements, rare events can be detected, where electrons flow against the applied source-drain bias direction thereby consuming entropy in agreement with theoretical predictions. \\[4pt] [1] T. Ihn, S. Gustavsson, U. Gasser, R. Leturcq, I. Shorubalko, and K. Ensslin, Physica E: Low-dimensional Systems and Nanostructures {\bf 42}, 803-808 (2010).\\[0pt] [2] J. G\"uttinger, J. Seif, C. Stampfer, A. Capelli, K. Ensslin, and T. Ihn, Phys. Rev. B {\bf 83}, 165445 (2011).\\[0pt] [3] C. R\"ossler, B. K\"ung, S. Dr\"oscher, T. Choi, T. Ihn, K. Ensslin, and M. Beck, Appl. Phys. Lett. {\bf 97}, 152109 (2010).\\[0pt] [4] T. M\"uller, B. K\"ung, S. Hellm\"uller, P. Studerus, K. Ensslin, T. Ihn, M. Reinwald, and W. Wegscheider, Appl. Phys. Lett. {\bf 97}, 202104 (2010).   

High-order counting statistics and interactions

Full counting statistics concerns the stochastic transport of electrons in mesoscopic structures [1]. Recently it has been shown that the charge transport statistics for noninteracting electrons in a two-terminal system is always generalized binomial: it can be decomposed into independent single-particle events, and the zeros of the generating function are real and negative [2]. In this talk I show how the zeros of the generating function move into the complex plane due to interactions and demonstrate how the positions of the zeros can be detected using high-order factorial cumulants [3]. As an illustrative example I discuss electron transport through a Coulomb blockade quantum dot for which the interactions on the quantum dot are clearly visible in the high-order factorial cumulants. These findings are important for understanding the influence of interactions on counting statistics, and the characterization in terms of zeros of the generating function provides a simple interpretation of recent experiments, where high-order statistics have been measured [4]. \\[4pt] [1] Yu. V. Nazarov, ed., Quantum Noise in Mesoscopic Physics, NATO Science Series, Vol. 97 (Kluwer, Dordrecht, 2003) \newline [2] A. G. Abanov and D. A. Ivanov, Phys. Rev. Lett. 100, 086602 (2008), Phys. Rev. B 79, 205315 (2009) \newline [3] D. Kambly, C. Flindt, and M. B\"{u}ttiker, Phys. Rev. B 83, 075432 (2011) -- Editors' Suggestion \newline [4] C. Flindt, C. Fricke, F. Hohls, T. Novotn\'{y}, K. Netocn\'{y}, T. Brandes, and R. J. Haug, Proc. Natl. Acad. Sci. USA 106, 10116 (2009)   

Hidden correlations in the zero-point motion of electrons revealed by non-Gaussian quantum noise

We have performed the measurement of the correlation between the low frequency voltage fluctuations and the power fluctuations of the high frequency electromagnetic field generated by a tunnel junction at very low temperature. Our experiment provides the first observation of the correlation between the electron transport at low frequency and the photon field at high frequency, both when real photons are emitted (\textit{eV$>$hf}, with $V$ the dc voltage and $f$ the frequency) and when the electromagnetic field is solely due to vacuum fluctuations. In terms of electrons only, our observations indicate that the intrinsic current fluctuations in a tunnel junction are given by $S_{3}(0,f)=e^{2}I$, regardless of the frequency $f$. Despite its classical look, this result expresses that the high frequency current fluctuations, caused by the zero-point motion of electrons, are correlated with their low frequency counterpart, associated with ``real'' motion of electrons. In terms of photons only, we observe that the electromagnetic field exhibits a discontinuity at \textit{eV=hf}, not in the amplitude of the fluctuations but in their relative phases.   

Noise and fluctuation statistics in mesoscopic heat transport

Fluctuations play important role in thermodynamics of small systems. In the talk, I will discuss two recent results on fluctuations in mesosopic heat transport. One is the demonstration [1] that the fluctuation-dissipation theorem for thermal conductance of a mesocopic junction is not valid at non-zero frequencies $\omega $. Finite relaxation energy creates fluctuations of the energy flux in the junction even at vanishing temperature, T=0, when the conductance vanishes. This suggest that in contract to electrical conductance, there is no ``Kubo-Green formula'' for equilibrium thermal conductance at $\omega $ $\ne $0. Non-equilibrium heat transfer satisfies general ``fluctuation relations'' of non-equilibrium thermodynamics. Recently, we have established the conditions of applicability of these relations to single-electron tunneling (SET), and calculated explicitly the statistics of dissipated energy in driven SET transitions [2], which gives an example of general statistics of energy dissipation in reversible information processing. An interesting consequence of this statistics is the possibility of implementing the electronic version of Maxwell's demon in the SET structures [3]. \\[4pt] [1] D.V. Averin and J.P. Pekola, Phys. Rev. Lett. \textbf{104}, 220601 (2010). \\[0pt] [2] D.V. Averin and J.P. Pekola, arXiv:1105.041. \\[0pt] [3] D.V. Averin, M. Mottonen, and J.P. Pekola, arXiv:1108.5435.