Two-electron States Research Articles

Coupling Spin Qubits via Superconductors

We show how superconductors can be used to couple, initialize, and read out spatially separated spin qubits. When two single-electron quantum dots are tunnel coupled to the same superconductor, the singlet component of the two-electron state partially leaks into the superconductor via crossed Andreev reflection. This induces a gate-controlled singlet-triplet splitting which, with an appropriate superconductor geometry, remains large for dot separations within the superconducting coherence length. Furthermore, we show that when two double-dot singlet-triplet qubits are tunnel coupled to a superconductor with finite charging energy, crossed Andreev reflection enables a strong two-qubit coupling over distances much larger than the coherence length.

Observation and spectroscopy of a two-electron Wigner molecule in an ultraclean carbon nanotube

Coulomb interactions can have a decisive effect on the ground state of electronic systems. The simplest system in which interactions can play an interesting role is that of two electrons on a string. In the presence of strong interactions the two electrons are predicted to form a Wigner molecule, separating to the ends of the string due to their mutual repulsion. This spatial structure is believed to be clearly imprinted on the energy spectrum, yet to date a direct measurement of such a spectrum in a controllable one-dimensional setting is still missing. Here we use an ultra-clean suspended carbon nanotube to realize this system in a tunable potential. Using tunneling spectroscopy we measure the excitation spectra of two interacting carriers, electrons or holes, and identify seven low-energy states characterized by their spin and isospin quantum numbers. These states fall into two multiplets according to their exchange symmetries. The formation of a strongly-interacting Wigner molecule is evident from the small energy splitting measured between the two multiplets, that is quenched by an order of magnitude compared to the non-interacting value. Our ability to tune the two-electron state in space and to study it for both electrons and holes provides an unambiguous demonstration of the fundamental Wigner molecule state.

Nondestructive Real-Time Measurement of Charge and Spin Dynamics of Photoelectrons in a Double Quantum Dot

We demonstrate one and two photoelectron trapping and the subsequent dynamics associated with interdot transfer in double quantum dots over a time scale much shorter than the typical spin lifetime. Identification of photoelectron trapping is achieved via resonant interdot tunneling of the photoelectrons in the excited states. The interdot transfer enables detection of single photoelectrons in a nondestructive manner. When two photoelectrons are trapped at almost the same time we observed that the interdot resonant tunneling is strongly affected by the Coulomb interaction between the electrons. Finally the influence of the two-electron singlet-triplet state hybridization has been detected using the interdot tunneling of a photoelectron.

Internal Symmetries and Additional Quantum Numbers for Nanoparticles

Wavefunctions of symmetrical nanoparticles are considered making use of induced representation method. It is shown that when, at the same total symmetry, the order of local symmetry group decreases, additional quantum numbers are required for complete labelling of electron states. It is shown that the labels of irreducible representations of intermediate subgroups can be used for complete classification of states in the case of repeating IRs in symmetry adapted linear combinations. The intermediate symmetry approach is extended to singlet and triplet two-electron states making use of Mackey theorem on symmetrized squares of induced representations.

Band-Edge Electronic Structure and Pre-existing Defects in Remote Plasma Deposited Non-crystalline SiO2 and GeO2

The following topics are addressed: (i) X-ray absorption spectroscopy (XAS) studies of plasma deposited non-crystalline (nc-) SiO2 and GeO2 emphasizing (a) band-edge states and (b) pre-existing bonding defects, and (ii) interpretation of X-ray absorption and photoemission spectra based on many electron theory, including two-electron singlet and triplet electron spin states. The articles also addresses electronic and defect structures in emerging device structures incorporating (GeO2)x(SiO2)1-x alloys, and bonding between Ge and Si substrates and elemental and/or alloy nc-SiO2 and nc-GeO2. The most significant results are identification of local atomic structure of pre-existing defects in the elemental oxides. Defects are created during processing, and are qualitatively different than those produced by X-ray, γ-ray, or high energy electron stressing and detected by second derivative XAS.

Fractional conductance oscillations in quantum rings: wave packet picture of transport in a few-electron system

We study electron transfer across a two-terminal quantum ring using a time-dependent description of the scattering process. For the considered scattering event the quantum ring is initially charged with one or two electrons, with another electron incident to the ring from the input channel. We study the electron transfer probability (T) as a function of the external magnetic field. We determine the periodicity of T for a varied number of electrons confined within the ring. For that purpose we develop a method to describe the wave packet dynamics for a few electrons participating in the scattering process, taking into full account the electron–electron correlations. We find that electron transfer across the quantum ring initially charged by a single electron acquires a distinct periodicity of half of the magnetic flux quantum (Φ0/2), corresponding to the formation of a transient two-electron state inside the ring. In the case of a three-electron scattering problem with two electrons initially occupying the ring, a period of Φ0/3 for T is formed in the limit of thin channels. The effect of disorder present in the confinement potential of the ring is also discussed.

(Invited) Single-Shot Readout of Singlet-Triplet Qubit States in a Si/SiGe Double Quantum Dot

The spin singlet and triplet states of a two-electron double quantum dot can be used to form a logical qubit that combines fast manipulation and a spin readout mechanism. We demonstrate single-shot readout of the two-electron states of a Si/SiGe double quantum dot using a quantum-point-contact charge sensor and spin-to-charge conversion. From the statistics of multiple single-shot measurements, we find the lifetimes of the spin states as a function of in-plane magnetic field. The lifetimes of the singlet and T0 triplet are both ~10ms and insensitive to magnetic field. The T- triplet lifetime increases with magnetic field, reaching ~3s at 1T.

Time-Dependent Configuration Interaction Approach for Multielectron System Confined in Two-Dimensional Quantum Dot

We extend the static multireference description (resonant unrestricted Hartree–Fock) to a dynamical system in order to include the correlation effect dynamically. The resulting time-dependent (TD) Schrödinger equation is simplified into the time-developed rate equation (TD-CI), where the TD external field \\hatH′(t) is taken into account directly in the Hamiltonian without any approximations. This TD-CI approach also has an advantage in that it takes into account the electron correlation by narrowing down the number of employed Slater determinants. We apply our TD-CI approach to the case of two electrons confined in the square quantum dot (QD) having the spin singlet multiplicity, and study theoretically the spatial and temporal fluctuation of the two-electron ground state under photon injection and pulse field application.

Spin-entangled two-particle dark state in quantum transport through coupled quantum dots

We present a transport setup of coupled quantum dots that enables the creation of spatially separated spin-entangled two-electron dark states. We prove the existence of an entangled transport dark state by investigating the system Hamiltonian without coupling to the electronic reservoirs. In the transport regime the entangled dark state which corresponds to a singlet has a strongly enhanced Fano factor compared to the dark state which corresponds to a mixture of the triplet states. Furthermore we calculate the concurrence of the occupying electrons to show the degree of entanglement in the transport regime.

Reverse Engineering Approach to Quantum Electrodynamics

The S matrix of e-e scattering has the structure of a projection operator that projects incoming separable product states onto entangled two-electron states. In this projection operator the empirical value of the fine-structure constant α acts as a normalization factor. When the structure of the two-particle state space is known, a theoretical value of the normalization factor can be calculated. For an irreducible two-particle representation of the Poincaré group, the calculated normalization factor matches Wyler’s semi-empirical formula for the fine-structure constant α. The empirical value of α, therefore, provides experimental evidence that the state space of two interacting electrons belongs to an irreducible two-particle representation of the Poincaré group.

The spectral decomposition of the helium atom two-electron configuration in terms of hydrogenic orbitals

The two-electron configuration in the helium atom is known to very high precision. Yet, we tend to refer to this configuration as a 1s↑1s↓ singlet, where the designations refer to hydrogen orbitals. The high precision calculations utilize basis sets that are suited for high accuracy and ease of calculation, but do not really aid in our understanding of the electron configuration in terms of product states of hydrogen orbitals. Since undergraduate students are generally taught to think of helium, and indeed the rest of the periodic table, in terms of hydrogenic orbitals, we present in this paper a detailed spectral decomposition of the two-electron ground state for helium in terms of these basis states. The 1s↑1s↓ singlet contributes less than 93% to the ground state configuration, with other contributions coming from both bound and continuum hydrogenic states.

Optical control of entangled states in semiconductor quantum wells

We present theory and calculations for coherent high-fidelity quantum control of many-particle states in semiconductor quantum wells. We show that coupling a two-electron double quantum dot to a terahertz optical source enables targeted excitations that are one to two orders of magnitude faster and significantly more accurate than those obtained with electric gates. The optical fields subject to physical constraints are obtained through quantum optimal control theory that we apply in conjunction with the numerically exact solution of the time-dependent Schrodinger equation. Our ability to coherently control arbitrary two-electron states, and to maximize the entanglement, opens up further perspectives in solid-state quantum information.

P-wave Cooper pair splitting.

Background: Splitting of Cooper pairs has recently been realized experimentally for s-wave Cooper pairs. A split Cooper pair represents an entangled two-electron pair state, which has possible application in on-chip quantum computation. Likewise the spin-activity of interfaces in nanoscale tunnel junctions has been investigated theoretically and experimentally in recent years. However, the possible implications of spin-active interfaces in Cooper pair splitters so far have not been investigated.Results: We analyze the current and the cross correlation of currents in a superconductor–ferromagnet beam splitter, including spin-active scattering. Using the Hamiltonian formalism, we calculate the cumulant-generating function of charge transfer. As a first step, we discuss characteristics of the conductance for crossed Andreev reflection in superconductor–ferromagnet beam splitters with s-wave and p-wave superconductors and no spin-active scattering. In a second step, we consider spin-active scattering and show how to realize p-wave splitting using only an s-wave superconductor, through the process of spin-flipped crossed Andreev reflection. We present results for the conductance and cross correlations.Conclusion: Spin-activity of interfaces in Cooper pair splitters allows for new features in ordinary s-wave Cooper pair splitters, that can otherwise only be realized by using p-wave superconductors. In particular, it provides access to Bell states that are different from the typical spin singlet state.

Two-electron dephasing in single Si and GaAs quantum dots

We study the dephasing of two-electron states in a single quantum dot in both GaAs and Si. We investigate dephasing induced by electron-phonon coupling and by charge noise analytically for pure orbital excitations in GaAs and Si, as well as for pure valley excitations in Si. In GaAs, polar optical phonons give rise to the most important contribution, leading to a typical dephasing rate of $\ensuremath{\sim}$5.9 GHz. For Si, intervalley optical phonons lead to a typical dephasing rate of $\ensuremath{\sim}$140 kHz for orbital excitations and $\ensuremath{\sim}$1.1 MHz for valley excitations. For harmonic, disorder-free quantum dots, charge noise is highly suppressed for both orbital and valley excitations, since neither has an appreciable dipole moment to couple to electric field variations from charge fluctuators. However, both anharmonicity and disorder break the symmetry of the system, which can lead to increased dipole moments and therefore faster dephasing rates.

Effective mass theory of interacting electron states in semiconducting carbon nanotube quantum dots

A modified effective mass approach is developed to describe the states of a few interacting electrons in semiconducting carbon nanotube quantum dots, and the accuracy of the approximations used is examined quantitatively. The few-particle states are calculated by exact diagonalization of the modified effective mass Hamiltonian for a range of different nanotube dots. It is shown that the two-electron states are Wigner molecule-like for a large proportion of these dots and the electron correlation is always found to be important. The addition energy is calculated for up to six electrons in the dot and the analysis of experimental addition energy data is discussed.

Nodal Quantum Numbers for Two-Electron States in Solids

The problem of construction and physical labelling of singlet and triplet zero total momentum two-electron states in solids is considered. It is shown that the wavefunctions belonging to repeating multi-dimensional irreducible representations can be labelled making use of theorem of induction transitivity. The intermediate group in this classification can be chosen depending on experimental nodal structure of superconducting order parameter. The application of the results to unconventional superconductors is discussed.

S-Wave Bound- and Resonant States of Two Fermions in Simple Cubic Lattice

Resonant two-electron states are examined in attractive Hubbard model on simple cubic lattice and exact formula for scattering cross section in the limit of low density (empty lattice) is calculated. S-wave pair is considered by means of lattice Green functions (LGF). Analytical form of these functions found by Joyce is used facilitating calculations, which were greatly hindered before by the necessity of using LGF's tabulated values. It is found that the actual peak of scattering cross-section is formed on the lower band boundary in discrepancy with formulae of the theory of scattering in solids.

Two-particle dark-state cooling of a nanomechanical resonator

The steady-state cooling of a nanomechanical resonator interacting with three coupled quantum dots is studied. General conditions for the cooling to the ground state with single and two-electron dark states are obtained. The results show that in the case of the interaction of the resonator with a single-electron dark state, no cooling of the resonator occurs unless the quantum dots are not identical. The steady-state cooling is possible only if the energy state of the quantum dot coupled to the drain electrode is detuned from the energy states of the dots coupled to the electron source electrode. The detuning has the effect of unequal shifting of the effective dressed states of the system that the cooling and heating processes occur at different frequencies. For the case of two electrons injected to the quantum dot system, the creation of a two-particle dark state is established to be possible with spin-antiparallel electrons. The results predict that with the two-particle dark state, an effective cooling can be achieved even with identical quantum dots subject of an asymmetry only in the charging potential energies coupling the injected electrons. It is found that similar to the case of the single-electron dark state, the asymmetries result in the cooling and heating processes to occur at different frequencies. However, an important difference between the single and two-particle dark state cases is that the cooling process occurs at significantly different frequencies. This indicates that the frequency at which the resonator could be cooled to its ground state can be changed by switching from the one-electron to the two-electron Coulomb blockade process.

Triple quantum dots as charge rectifiers

We theoretically analyze electronic spin transport through a triple quantum dot in series, attached to electrical contacts, where the drain contact is coupled to the central dot. We show that current rectification is observed in the device due to current blockade. The current blocking mechanism is originated by a destructive interference of the electronic wavefunction at the drain dot. There, the electrons are coherently trapped in a singlet two-electron dark state, which is a coherent superposition of the electronic wavefunction in the source dot and in the dot isolated from the contacts. Its formation gives rise to zero current and current rectification as the voltage is swept. We analyze this behavior analytically and numerically for both zero and finite magnetic dc fields. On top of that, we include phenomenologically a finite spin relaxation rate and calculate the current numerically. Our results show that triple dots in series can be designed to behave as quantum charge rectifiers.

Exact Wave Functions of Two-Electron Quantum Rings

We demonstrate that the Schrödinger equation for two electrons on a ring, which is the usual paradigm to model quantum rings, is solvable in closed form for particular values of the radius. We show that both polynomial and irrational solutions can be found for any value of the angular momentum and that the singlet and triplet manifolds, which are degenerate, have distinct geometric phases. We also study the nodal structure associated with these two-electron states.