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arXiv:cond-mat/0307437 · Physical Review A

Optical generation and quantitative characterizations of electron-hole entanglement

Yu Shi

Published Mar 31, 2004• 3 citations
arXivHTMLPDF

Abstract

Using a method of characterizing entanglement in the framework of quantum field theory, we investigate the optical generation and quantitative characterizations of quantum entanglement in an electron-hole system, in presence of spin-orbit coupling, and especially make a theoretical analysis of a recent experimental result. Basically, such entanglement should be considered as between occupation numbers of single-particle basis states, and is essentially generated by coupling between different single-particle basis states in the second-quantized Hamiltonian. Interaction with two resonant light modes of different circular polarizations generically leads to a superposition of ground state and two heavy-hole excitonic states. When and only when the state is a superposition of only the two excitonic eigenstates, the entanglement reduces to that between two distinguishable particles, each with two degrees of freedom, namely, band index, as characterized by angular momentum, and orbit, as characterized by position or momentum. The band-index state, obtained by tracing over the orbital degree of freedom, is found to be a pure state, hence the band index and orbital degrees of freedom are separated in this state. We propose some basic ideas on spatially separating the electron and the hole, so that the entanglement of band indices, or angular momenta, is between spatially separated electron and hole.

DOI
10.1103/physreva.69.032318

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Using a method of characterizing entanglement in the framework of quantum field theory, we investigate the optical generation and quantitative characterizations of quantum entanglement in an electron-hole system, in presence of spin-orbit coupling, and especially make a theoretical analysis of a recent experimental result. Basically, such entanglement should be considered as between occupation numbers of single-particle basis states, and is essentially generated by coupling between different single-particle basis states in the second-quantized Hamiltonian. Interaction with two resonant light modes of different circular polarizations generically leads to a superposition of ground state and two heavy-hole excitonic states. When and only when the state is a superposition of only the two excitonic eigenstates, the entanglement reduces to that between two distinguishable particles, each with two degrees of freedom, namely, band index, as characterized by angular momentum, and orbit, as characterized by position or momentum. The band-index state, obtained by tracing over the orbital degree of freedom, is found to be a pure state, hence the band index and orbital degrees of freedom are separated in this state. We propose some basic ideas on spatially separating the electron and the hole, so that the entanglement of band indices, or angular momenta, is between spatially separated electron and hole.
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.
We investigate entanglement transfer from a system of two spin-entangled electron-hole pairs, each placed in a separate single mode cavity, to the photons emitted due to cavity leakage. Dipole selection rules and a splitting between the light hole and the heavy hole subbands are the crucial ingredients establishing a one-to-one correspondence between electron spins and circular photon polarizations. To account for the measurement of the photons as well as dephasing effects, we choose a stochastic Schrödinger equation and a conditional master equation approach, respectively. The influence of interactions with the environment as well as asymmetries in the coherent couplings on the photon entanglement is analysed for two concrete measurement schemes. The first one is designed to violate the Clauser-Horne-Shimony-Holt (CHSH) inequality, while the second one employs the visibility of interference fringes to prove the entanglement of the photons. Because of the spatial separation of the entangled electronic system over two quantum dots, a successful verification of entangled photons emitted by this system would imply the detection of nonlocal spin entanglement of massive particles in a solid state structure.
We report on the study of quantum beats in the time-integrated four-wave mixing signal in reflection geometry from a pair of excitons with different valence orbitals in GaN and ZnSe. We observe a \ensuremath{\pi}-phase shift between quantum beats in the co- and cross-linear polarization configuration, and nearly 100% modulation of the signal for both materials. In the co-circular polarization configuration, the observed phases of the quantum beats at the frequency of the A exciton in GaN and heavy-hole exciton in ZnSe are 0.2\ensuremath{\pi} and 0.1\ensuremath{\pi}, respectively. The phases of the quantum beats change sign for co- and counter-circular polarization configurations and also for A- (heavy-hole) and B-(light-hole) excitons in GaN(ZnSe). We describe the third-order coherent optical response of the exciton pair with different valence orbitals by taking into account the finite memory depth of the four-particle correlation. In particular, our experimental findings indicate that excitons with different valence orbitals and equal angular momentum attract each other similar to excitons with the same valence orbitals but opposite angular momentum. Excitons with different orbitals and opposite angular momenta repel one another. The comparison between experimental and theoretical results allows us to develop a quantitative analysis of the four-particle correlation in the presence of an exciton pair with different valence orbitals. We show that the observed \ensuremath{\pi}-phase shift and nearly 100% modulation of the signal in the co- and cross-linear polarization configuration impose restraints on the memory functions, which describe the exciton-exciton interaction. These restraints imply, in particular, that electron spins play a more important role in the exciton-exciton interaction in comparison to hole spins. We show that a striking similarity, which we observe in the quantum beat signal from GaN and ZnSe, originates from the strong four-particle correlation contribution to the third-order excitonic nonlinearity.
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.
Optically induced entanglement is identified by the spectrum of the phase-sensitive homodyne-detected coherent nonlinear optical response in a single gallium arsenide quantum dot. The electron-hole entanglement involves two magneto-excitonic states differing in transition energy and polarization. The strong coupling needed for entanglement is provided through the Coulomb interaction involving the electrons and holes. The result presents a first step toward the optical realization of quantum logic operations using two or more quantum dots.
We propose a minimal quantum-optical scheme for generating hybrid entanglement between photon polarization and electronic orbital angular momentum in a semiconductor quantum disk. A spin--orbit structured two-photon state excites two channels in the same disk: a radiatively recombining zero-orbital-angular-momentum channel and a finite-orbital-angular-momentum channel that stores the electronic orbital qubit. An effective coherent mapping prepares a selected two-excitation state, followed by emission of a photon whose polarization is entangled with the residual electronic orbital state. A master-equation analysis shows that the heralded state conditioned on single-photon occupation of the selected output mode approaches the target entangled state in the ideal coherent limit. We also discuss orbital relaxation and perturbative validity conditions for branch-dependent Coulomb shifts, orbital-angular-momentum mixing, and finite-orbital-angular-momentum radiative leakage. This proof-of-principle effective model suggests a route toward structured-light-mediated photon--electron hybrid entanglement in semiconductor nanostructures.