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arXiv:1603.03896 · Physical review. B./Physical review. B

Optimal noninvasive measurement of full counting statistics by a single qubit

A. V. Lebedev, Gordey B. Lesovik, G. Blatter

Published Mar 24, 2016• 6 citations
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Abstract

The complete characterization of the charge transport in a mesoscopic device is provided by the full counting statistics (FCS) ${P}_{t}(m)$, describing the amount of charge $Q=me$ transmitted during the time $t$. Although numerous systems have been theoretically characterized by their FCS, the experimental measurement of the distribution function ${P}_{t}(m)$ or its moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ is rare and often plagued by strong back-action. Here, we present a strategy for the measurement of the FCS, more specifically its characteristic function $\ensuremath{\chi}(\ensuremath{\lambda})$ and moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$, by a qubit with a set of different couplings ${\ensuremath{\lambda}}_{j}$, $j=1,\ensuremath{\cdots},k,\ensuremath{\cdots}k+p$, $k=\ensuremath{\lceil}n/2\ensuremath{\rceil}$, $p\ensuremath{\ge}0$, to the mesoscopic conductor. The scheme involves multiple readings of Ramsey sequences at the different coupling strengths ${\ensuremath{\lambda}}_{j}$, and we find the optimal distribution for these couplings ${\ensuremath{\lambda}}_{j}$ as well as the optimal distribution ${N}_{j}$ of $N=\ensuremath{\sum}{N}_{j}$ measurements among the different couplings ${\ensuremath{\lambda}}_{j}$. We determine the precision scaling for the moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ with the number $N$ of invested resources and show that the standard quantum limit can be approached when many additional couplings $p\ensuremath{\gg}1$ are included in the measurement scheme.

DOI
10.1103/physrevb.93.115140

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Traditionally, (charge) transport phenomena describe the average current flow in response to a constant applied driving force. Additional information is provided by current-current correlations or noise, telling about the granularity, the statistical properties, and the interactions between the particles. The complete characterization of transport is given by the full counting statistics, quantifying the probability that n particles pass the device during a time window t. The desire for a complete characterization of transport in a mesoscopic device goes together with the development of new single-particle sources and the availability of quantum bits as measurement devices. In this thesis, we will describe how to combine these three elements, full counting statistics, single-particle sources, and qubit measurement devices and how these concepts relate to the description of charge transport of noninteracting electrons in a mesoscopic device. In the work on full counting statistics, we develop a new wave-packet formalism that provides access to an entire class of interesting results. The wavepacket formalism is derived from an insight regarding the equivalence between the fidelity of a quantum system and the generating function of full counting statistics; furthermore, we show how qubits can be used to measure both quantities, fidelity and full counting statistics, exploiting the induced ‘decoherence’ as a signal; thereby, we transfer the measurement of the fidelity and of the full counting statistics from the realm of a ‘Gedanken’ experiment to a practical proposal realizable with today’s qubit technology. Subsequently, we study the full counting statistics for the transmission of two identical particles with positive or negative symmetry under exchange, for the situation where the scattering depends on energy. We make use of our firstquantized wave-packet formalism and find interesting results. While it is expected that the noise as well as higher-order correlators will be sensitive to the exchange symmetry, we find the astounding result that exchange can hugely enhance (or suppress) the average charge already. The effect is a consequence of the symmetry-induced reshuffling of weight in the momentum distribution of the
We present a scalable protocol for measuring full counting statistics (FCS) in experiments or tensor-network simulations. In this method, an ancilla in the middle of the system acts as a turnstile, with its phase keeping track of the time-integrated particle flux. Unlike quantum gas microscopy, the turnstile protocol faithfully captures FCS starting from number-indefinite initial states or in the presence of noisy dynamics. In addition, by mapping the FCS onto a single-body observable, it allows for stable numerical calculations of FCS using approximate tensor-network methods. We demonstrate the wide-ranging utility of this approach by computing the FCS of the transferred magnetization in a Floquet Heisenberg spin chain, as studied in a recent experiment with superconducting qubits, as well as the FCS of charge transfer in random circuits.
Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.
We analyze the full counting statistics (FCS) of the charge transport through the Anderson impurity model (AIM) and similar systems with a single conducting channel. The object of principal interest is the generating function for the cumulants of charge current distribution. We derive an exact analytic formula relating the FCS generating function to the self energy of the system in the presence of the measuring field. We first check that our approach reproduces correctly known results in simple limits, such as the FCS of the resonant level system (AIM without Coulomb interaction). We then proceed to study the FCS for the AIM both perturbatively in the Coulomb interaction and in the Kondo regime at the Toulouse point (we also study a related model of a spinless single-site quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid phase at a special interaction strength). At zero temperature the FCS turns out to be binomial for small voltages. For the generic case of arbitrary energy scales the FCS is shown to be captured very well by generalizations of the Levitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a presence of coherent electron pair tunnelling in addition to conventional single-particle processes. By means of perturbative expansions around the Toulouse point we succeeded in showing the universality of the binomial FCS at zero temperature in linear response. Based on our general formula for the FCS we then argue for a more general binomial theorem stating that the linear response zero-temperature FCS for any interacting single-channel setup is always binomial.
Performing quantum measurements produces not only the expectation value of a physical observable OOO but also the probability distribution P(o)P(o)P(o) of all possible outcomes ooo. The full counting statistics (FCS) $Z(\ensuremath{\phi},O)\ensuremath{\equiv}{\ensuremath{\sum}}_{o}{e}^{i\ensuremath{\phi}o}P(o)$, a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of QA{Q}_{A}QA​, the charge operator in subsystem AAA, for one-dimensional systems described by non-Hermitian Sachdev-Ye-Kitaev-like models, which are solvable in the large-NNN limit. In both the volume-law entangled phase for interacting systems and the critical phase for noninteracting systems, the conformal symmetry emerges, which gives $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\equiv}lnZ(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}{\ensuremath{\phi}}^{2}ln|A|$. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as $F(\ensuremath{\phi},{Q}_{A})\ensuremath{\sim}(1\ensuremath{-}cos\ensuremath{\phi})|\ensuremath{\partial}A|$ for $\ensuremath{\zeta}\ensuremath{\gg}J$, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequences of discrete symmetry, long-range hopping, and generalizations to higher dimensions.
Performing quantum measurements produces not only the expectation value of a physical observable OOO but also the probability distribution P(o)P(o)P(o) of all possible outcomes ooo. The full counting statistics (FCS) Z(ϕ,O)≡∑oeiϕoP(o)Z(ϕ, O)\equiv \sum_o e^{iϕo}P(o)Z(ϕ,O)≡∑o​eiϕoP(o), a Fourier transform of this distribution, contains the complete information of the measurement outcome. In this work, we study the FCS of QAQ_AQA​, the charge operator in subsystem AAA, for 1D systems described by non-Hermitian SYK-like models, which are solvable in the large-NNN limit. In both the volume-law entangled phase for interacting systems and the critical phase for non-interacting systems, the conformal symmetry emerges, which gives F(ϕ,QA)≡log⁡Z(ϕ,QA)∼ϕ2log⁡∣A∣F(ϕ, Q_A)\equiv \log Z(ϕ, Q_A)\sim ϕ^2\log |A|F(ϕ,QA​)≡logZ(ϕ,QA​)∼ϕ2log∣A∣. In short-range entangled phases, the FCS shows area-law behavior which can be approximated as F(ϕ,QA)∼(1−cos⁡ϕ)∣∂A∣F(ϕ, Q_A)\sim (1-\cosϕ) |\partial A|F(ϕ,QA​)∼(1−cosϕ)∣∂A∣ for ζ≫Jζ\gg Jζ≫J, regardless of the presence of interactions. Our results suggest the FCS is a universal probe of entanglement phase transitions in non-Hermitian systems with conserved charges, which does not require the introduction of multiple replicas. We also discuss the consequence of discrete symmetry, long-range hopping, and generalizations to higher dimensions.