Abstract Methods of charge projectors using special gauge transformations for tagging particles are presented. Such engineering of a many‐body wave function allows extracting information regarding properties of a physical system beyond average values. The method is first used to establish under which circumstances the properties of particle currents can be understood as discrete transfers of particles from one region to another. Next, the method is extended to show that in a tunnel junction coupled to a two‐level system, the transmission of electrical noise causes decoherence and thereby a projective measurement of the two‐level system as encoded by the amount of tunnelled charge. Finally the method is extended to deal with the exceptional circumstance where a measurement on a system with a large number of particles gives macroscopically distinguishable outcomes that are unpredictable, the observed interference pattern of interfering BEC's.
Abstract Methods of charge projectors using special gauge transformations for tagging particles are presented. Such engineering of a many‐body wave function allows extracting information regarding properties of a physical system beyond average values. The method is first used to establish under which circumstances the properties of particle currents can be understood as discrete transfers of particles from one region to another. Next, the method is extended to show that in a tunnel junction coupled to a two‐level system, the transmission of electrical noise causes decoherence and thereby a projective measurement of the two‐level system as encoded by the amount of tunnelled charge. Finally the method is extended to deal with the exceptional circumstance where a measurement on a system with a large number of particles gives macroscopically distinguishable outcomes that are unpredictable, the observed interference pattern of interfering BEC's.
A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the “gauge” field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.
Charge transfer statistics of quantum particles is obtained by analysing the time evolution of the many-body wave function. Exploiting properly chosen gauge transformations, we construct the probabilities for transfers of a discrete number of particles. Generally, the derived formula for counting statistics differs from the one previously obtained by Levitov et al. ( J. Math. Phys. , 37 (1996) 4845). The two formulae agree only if the initial state is prohibited from being a superposition of different charge states. Their difference is illustrated for cases of a single particle and a tunnel junction, and the role of charge coherence is demonstrated.
Counting statistics of charge transfers in a point contact interacting with an arbitrary quantum system is studied. The theory for the charge specific density matrix is developed, allowing the evaluation of the probability of the outcome of any joint measurement of the state of the quantum system and the transferred charge. Applying the method of charge projectors, the master equation for the charge specific density matrix is derived in the tunneling Hamiltonian model of the point contact. As an example, the theory is applied to a quantum measurement of a two-state system: The evolution of the charge specific density matrix in the presence of Nyquist or Schottky noise is studied and the conditions for the realization of a projective measurement are established.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
The electrical noise of mesoscopic devices can be strongly influenced by the quantum motion of electrons. To probe this effect, we have measured the current fluctuations at high frequency (5 to 90 gigahertz) using a superconductor-insulator-superconductor tunnel junction as an on-chip spectrum analyzer. By coupling this frequency-resolved noise detector to a quantum device, we can measure the high-frequency, nonsymmetrized noise as demonstrated for a Josephson junction. The same scheme is used to detect the current fluctuations arising from coherent charge oscillations in a two-level system, a superconducting charge qubit. A narrow band peak is observed in the spectral noise density at the frequency of the coherent charge oscillations.
Determining the quantum state of a given quantum-mechanical system is a fundamental task in physics. Quantum-state tomography has been pivotal for establishing quantum optics [1-4] and for revealing the properties of bound charges in materials [5-7]. An emerging other object for studying and utilizing quantum effects are free electrons, elementary particles that are central to high-resolution microscopy [8,9], electron-based quantum optics [10-17], ul-trafast electron microscopy [18-24] and particle accelerators [25-27]. However, free electrons are intrinsically incoherent, and we lack a broadly applicable method to measure and control their quantum state beyond special cases with discrete energy sidebands [28,29]. Here, we report a universal approach to measure arbitrary free-electron quantum states in continuous variables. Two monochromatic but spectrally shifted laser waves produce interfering quan-tum paths that directly reveal the density matrix and thus all essential properties of the pure wavepackets, the ensemble, and their interlinks. As a first application, we show how the quantum state of a single electron is modified by many-body Coulomb interactions of a sur-rounding electron gas. The reported concepts and results provide insight into otherwise hid-den correlations in electron beams and enable the controlled optimization of exceptional quantum states for free-electron quantum optics or quantum electron microscopy.