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arXiv:1805.09860 · Physical review. B./Physical review. B

Single-electron second-order correlation function G(2) at nonzero temperatures

Michael V. Moskalets

Published Sep 11, 2018• 8 citations
arXivHTMLPDF

Abstract

The single-particle state is not expected to demonstrate second-order coherence. Here I analyze the injection of electrons into a conductor and show that, at a nonzero temperature, the underlying Fermi sea causes the single-particle injected state to exhibit second-order coherence. For this purpose, I calculate the second-order correlation function, ${G}^{(2)}$, of electrons injected on top of the Fermi sea. At zero temperature, the function ${G}^{(2)}$ unambiguously demonstrates whether the injected state is a single- or a multiparticle state: ${G}_{}^{(2)}$ vanishes in the former case, while it does not vanish in the latter case. However, at nonzero temperatures, when the injected state is a mixed state, a purely single-particle contribution makes the function ${G}_{}^{(2)}$ nonvanishing even in the case of injection of one electron. The single-particle contribution puts the lower limit on the second-order correlation function and thereby limits its use as a single-particle injection test at nonzero temperatures. The existence of a single-particle contribution to ${G}_{}^{(2)}$ can be verified experimentally by measuring the cross-correlation electrical noise.

DOI
10.1103/physrevb.98.115421

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The single-particle state is not expected to demonstrate second-order coherence. Here I analyze the injection of electrons into a conductor and show that, at a nonzero temperature, the underlying Fermi sea causes the single-particle injected state to exhibit second-order coherence. For this purpose, I calculate the second-order correlation function, G(2){G}^{(2)}G(2), of electrons injected on top of the Fermi sea. At zero temperature, the function G(2){G}^{(2)}G(2) unambiguously demonstrates whether the injected state is a single- or a multiparticle state: G(2){G}_{}^{(2)}G(2)​ vanishes in the former case, while it does not vanish in the latter case. However, at nonzero temperatures, when the injected state is a mixed state, a purely single-particle contribution makes the function G(2){G}_{}^{(2)}G(2)​ nonvanishing even in the case of injection of one electron. The single-particle contribution puts the lower limit on the second-order correlation function and thereby limits its use as a single-particle injection test at nonzero temperatures. The existence of a single-particle contribution to G(2){G}_{}^{(2)}G(2)​ can be verified experimentally by measuring the cross-correlation electrical noise.
The state of a single particle injected onto the surface of the Fermi sea is a pure state if the temperature is zero and is a mixed state if the temperature is finite. Moreover, the state of an injected particle is orthogonal to the state of the Fermi sea at zero temperature, while it is not orthogonal at nonzero temperature. These changes in the quantum state of the injected particles can be detected using the temperature dependence of the shot noise that is generated when the particles one by one pass through a semitransparent quantum point contact. Namely, the shot noise produced by the mixed state is suppressed in comparison with the noise of the pure state. In addition, the correlations between the injected particles and the underlying Fermi sea, present at nonzero temperature, do enhance the shot noise. Furthermore, antibunching of injected particles with possible thermal excitations coming from another input channel of a quantum point contact does suppress shot noise. Here I analyze in detail these three effects, which are responsible for the temperature dependence of the shot noise, and discuss how to distinguish them experimentally.
The state of particles injected onto the surface of the Fermi sea depends essentially on the temperature. The pure state injected at zero temperature becomes a mixed state if injected at finite temperature. Moreover the electron source injecting a single-particle state at zero temperature may excite a multi-particle state if the Fermi sea is at finite temperature. Here I unveil a symmetry of the scattering amplitude of a source, which is sufficient to preserve a single-particle emission regime at finite temperatures if such a regime is achieved at zero temperature. I give an example and analyze the effect of temperature on time-dependent electrical and heat currents carried by a single-particle excitation.
The state of particles injected onto the surface of the Fermi sea depends essentially on the temperature. The pure state injected at zero temperature becomes a mixed state if injected at finite temperature. Moreover the electron source injecting a single-particle state at zero temperature may excite a multi-particle state if the Fermi sea is at finite temperature. Here I unveil a symmetry of the scattering amplitude of a source, which is sufficient to preserve a single-particle emission regime at finite temperatures if such a regime is achieved at zero temperature. I give an example and analyze the effect of temperature on time-dependent electrical and heat currents carried by a single-particle excitation.
Abstract Quantum-optical research on semiconductor single-photon sources puts special emphasis on the measurement of the second-order correlation function g (2) ( τ ), arguing that g (2) (0) < 1/2 implies the source field represents a good single-photon light source. We analyze the gain of information from g (2) (0) with respect to single photons. Any quantum state, for which the second-order correlation function falls below 1/2, has a nonzero projection on the single-photon Fock state. The amplitude p of this projection is arbitrary, independent of g (2) (0). However, one can extract a lower bound on the single-to-multi-photon-projection ratio. A vacuum contribution in the quantum state of light artificially increases the value of g (2) (0), cloaking actual single-photon projection. Thus, we propose an effective second-order correlation function g ˜ ( 2 ) ( 0 ) , which takes the influence of vacuum into account and also yields lower and upper bounds on p . We consider the single-photon purity as a standard figure of merit in experiments, reinterpret it within our results and provide an effective version of that physical quantity. Besides comparing different experimental and theoretical results, we also provide a possible measurement scheme for determining g ˜ ( 2 ) ( 0 ) .
In this review, a detailed discussion of the behaviour of a heavy particle interacting with a Fermi sea is given. Particular emphasis is put on the issue of how strong correlations influence coherence and transport of the particle. First, we investigate the question of whether the heavy particle is a well defined quasiparticle at low temperatures. While in one dimension ( D = 1) and at a van Hove singularity in D = 2 the coherence of the particle is destroyed, the quasiparticle weight is finite in higher dimensions. The most important transport quantity is the diffusion constant or mobility of the heavy particle. We are able to describe both the well known high-temperature properties and the cross-over to the lowest temperatures in a unified approximation scheme based on a self-consistent evaluation of an effective action. Two strong-correlation effects of independent origin are discussed. The first arises if the scattering of the fermions from the heavy particle is nearly resonant, that is, if one of the scattering phase shifts δ is close to π/2. In this regime an anomalous exponent is observed in the temperature dependence of the mobility μ(T). In D = 3, for instance, the mobility is proportional to T-3/2 rather than to T-2. The second effect is a giant mass renormalization in the case of a large particle. In this situation, the low-temperature effective mass M* increases up to an exponentially large value, M* exp[c(r/λF)3], where r is the effective radius of the particle, λF the Fermi wavelength and c a non-universal constant of order one.
The entanglement between occupation numbers of different single particle basis states depends on coupling between different single particle basis states in the second-quantized Hamiltonian. Thus, in principle, interaction is not necessary for occupation-number entanglement to appear. However, in order to characterize quantum correlation caused by interaction , we use the eigenstates of the single-particle Hamiltonian as the single particle basis upon which the occupation-number entanglement is defined. Using this so-called proper single particle basis, if there is no interaction, the many-particle second-quantized Hamiltonian is diagonalized and thus cannot generate entanglement, while its eigenstates can always be chosen to be non-entangled. If there is interaction, entanglement in the proper single particle basis arises in energy eigenstates and can be dynamically generated. Using the proper single particle basis, we discuss occupation-number entanglement in important eigenstates, especially ground states, of systems of many identical particles, in exploring insights the notion of entanglement sheds on many-particle physics. The discussions on Fermi systems start with Fermi gas, the Hartree–Fock approximation and the electron–hole entanglement in excitations. In the ground state of a Fermi liquid, in terms of the Landau quasiparticles, entanglement becomes negligible. The entanglement in a quantum Hall state is quantified as − f ln f − (1 − f )ln(1 − f ), where f is the proper fractional part of the filling factor. For BCS superconductivity, the entanglement is a function of the relative momentum wavefunction of the Cooper pair g k , and is thus directly related to the superconducting energy gap, and vanishes if and only if superconductivity vanishes. For a spinless Bose system, entanglement does not appear in the Hartree–Gross–Pitaevskii approximation, but becomes important in the Bogoliubov theory, as a characterization of two-particle correlation caused by the weak interaction. In these examples, the interaction-induced entanglement as calculated is directly related to the macroscopic physical properties.