Session G40: Focus Session: Materials for Quantum Computing I

Single-Atom Indexing of Quantum State Superpositions

The ultimate miniaturization of electronic devices will likely require the local control of single-electron wavefunctions. One system where this may be accomplished consists of two-dimensional metallic electron states confined within atomically engineered nanostructures. Here we describe experiments showing that an individual atom inside a 44-atom quantum corral can index arbitrary coherent superpositions of spatial quantum states. We demonstrate how the quantum mirage effect can be harnessed to image the resulting quantum superposition. We also present a straightforward method for determining the appropriate atom location for any desired superposition. The atom provides a real-space handle for an abstract Hilbert space, providing a simple, novel technique for coherently manipulating quantum states.   

Valley Splitting in a Silicon Two-Dimensional Electron Gas

We have performed low-temperature microwave transport spectroscopy of low-lying valley states in a silicon two-dimensional electron gas. The magnitude of this splitting determines whether the ground state is degenerate for purposes of quantum computing with spins. The valley splitting varies linearly with magnetic field from 0.3 to 3~T, reaching 75~$\mu$eV, with no sign of saturation. We unambiguously identify the observed resonance as a valley excitation by comparing with Shubnikov-de Haas oscillations. The origin of the splitting is the coupling of the two $z$ conduction valleys in the silicon band structure, due to quantum well confinement. Previous theory suggests that the valley splitting can be of order 1~meV. However, we present a theory incorporating atomic steps, which are present in experimental systems. The theory leads to small valley splittings at zero magnetic field, and a linearly increasing splitting at nonzero fields, as observed in experiments.   

Valley Splitting Theory for Silicon 2DEGs Grown on a Vicinal Surface

We develop a theory for the energy splitting of the two-fold degenerate conduction valleys of a silicon 2DEG. We assume that the quantum well is not perfectly aligned with the crystallographic axes, as consistent with typical experimental conditions. Under these general conditions, the valley splitting can be suppressed by many orders of magnitude from its theoretical upper bound. However, the confined electrons are able to recover some of their valley splitting, and thus lower their total energy, by a variety of means. We discuss two recovery methods, which apply to the cases of zero and nonzero magnetic fields, respectively. The results show a linear dependence of the valley splitting on the magnetic field, as consistent with the experimental data.   

Quantum theory of decoherence in solid-state spin quantum computing architectures

Decoherence is the adversary of any proposed quantum computer. In order to overcome it, we must understand it. Nuclear induced spectral diffusion, a type of electron spin decoherence caused by interaction with a nuclear spin bath, is a predominant source of information loss for solid-state spin quantum computing architectures. All previous theories for this 50-year-old problem have used phenomenological, semi-classical models. This talk presents our cluster expansion method which provides the first fully microscopic and quantum mechanical solution to this problem. With our method it becomes possible to ascertain the effectiveness of pulse sequences designed to enhance spin coherence.   

Effect of exchange interaction on spin dephasing in a double quantum dot

We use a pulsed-gate technique to study mixing of singlet and triplet spin states in a two-electron double quantum dot with a tunable exchange interaction. Spin dynamics in this system are governed by the interplay of exchange (which tends to preserve spin correlations) and hyperfine interaction with the lattice nuclei (which tends to destroy them.) When the two interaction strengths are comparable, we observe saturation of dephasing and damped temporal oscillations, persisting well beyond the hyperfine dephasing time, of the spin correlator. Both features of the data show good agreement with predictions from a quasistatic model of the hyperfine field.   

Exchange in singlet-triplet qubits: spin funnel and magnetic field structure

We employ density functional (DF) calculated eigenstates as a basis for exact diagonalization studies of lateral semiconductor double quantum dots through the transition from the symmetric bias regime to the regime where both electrons occupy the same dot. The DF basis allows us to maintain the geometric fidelity of the device in the calculation while still capturing all of the many-body effects. Recent experiments by Petta \textit{et al.} [Science \textbf{309}, 2184 (2005)] have shown the existence of a ``spin funnel'' in the behavior of the singlet-triplet splitting (the exchange coupling) as a function of bias detuning J($\varepsilon )$ in the vicinity of the crossover from the (1,1) to the (0,2) honeycomb stability cells. Here we calculate the spin funnel and explain its origin and functional form. For an applied magnetic field B we predict the existence of local minima where dJ($\varepsilon $,B)/d$\varepsilon $ =0, and suppression of voltage noise can be expected.   

Competition between Coulomb localization and barrier modulation of the exchange energy in two-electron laterally coupled quantum dots

We present calculations of the singlet-triplet energy separation (exchange energy) in the two-electron system confined in the double quantum dot system used in recent experiments on coherent manipulation of coupled electron qubits. The hybrid multiscale approach where the confined electrons are described by the direct diagonalization of the Schr\"odinger equation within the full quantum dot device environment was used to perform the calculations. We analyze the behavior of the exchange in the magnetic fields and find large changes from meV to sub-$\mu$eV value as the confinement gate biases (effective barrier) are varied, while the singlet-triplet transition magnetic field remains at about 1 T independently of the gate biases. The small values of the exchange in this structure are attributed to the large separation between the electrons leading to an almost classical description of the system due to the dominance of the Coulomb repulsion.   

Entanglement of two strongly correlated electrons in a lateral quantum dot

Exact-diagonalization calculations for two electrons in an elliptic lateral quantum dot show that the electrons can localize and form a molecular dimer even for screened interelectron repulsion. The calculated singlet-triplet splitting ($J$) as a function of the magnetic field ($B$) agrees with cotunneling measurements;\footnote{D.M. Zumb\"{u}hl {\it et al.\/}, Phys. Rev. Lett. {\bf 93}, 256801 (2004).} its behavior reflects the effective dissociation\footnote{C. Yannouleas and U. Landman, Int. J. Quantum Chem. {\bf 90}, 699 (2002)} of the electron dimer for large $B$. Knowledge of the dot shape and of $J(B)$ allows determination of two measures of entanglement (concurrence and von Neumann entropy for {\it indistinguishable\/} fermions), whose behavior correlates also with the dissociation of the dimer. The theoretical value for the concurrence at $B=0$ agrees with the experimental estimates.   

Detection and measurement of the Dzyaloshinskii-Moriya interaction in double quantum dot systems

Spins in quantum dots can act as qubits for quantum computation. In this context we point out that spins on neighboring dots will experience the Dzyaloshinskii-Moriya interaction, which mixes the spin singlet and triplet states. This will have a strong influence on spin-dependent tunneling. We show that the effects of this interaction depend strongly on the direction of the external field, and demonstrate how to detect and measure the effect.   

Engineering Double Quantum Dots in Si/SiGe using Shottky Top Gates

Quantum information processing in silicon based semiconductors have gained importance lately due to its inherent advantages like low spin orbit coupling and thus very large spin coherence times, as compared to competing III-V devices. Previous work done on silicon based low dimensional devices include successful fabrication of etch defined quantum dots in Si/SiGe quantum wells, whose potentials are modulated by lateral side gates. Another way of forming quantum dots is by using shottky top gates to deplete the underlying two dimensional electron gas (2DEG) in selected areas. Quantum dots formed by shottky top gates are preferred over etched ones to achieve better control over the tuning of tunnel barriers and to minimize etch induced depletion of the 2DEG. Unlike III-V materials, shottky gates formed in SiGe/Si quantum wells have been found to be very leaky. In this talk we discuss the issues of gate leakage, formation of tunnel junctions and engineering double quantum dots (Qubits) in SiGe/Si quantum wells with the help of an additional back-gate.   

Stability Diagram of a Few Electron Triatom

A lateral few electron triple quantum dot system has been studied and the stability diagram mapped out using charged detection techniques. All three quantum dots are coupled together. The device could be tuned to observe quadruple points including the fundamental ones for quantum information applications, associated with the confinement of one, two and three electrons. The stability diagram includes a series of novel elements, including the cloning of charge transfer lines. The main results are successfully modeled by both capacitive and Hubbard models.   

External field control of donor electrons at the Si-SiO$_2$ interface

Prospects for the quantum control of electrons in the silicon quantum computer architecture are considered theoretically. In particular, we investigate the feasibility of shuttling donor-bound electrons forth and back between the impurity in the bulk and the Si-SiO$_2$ interface by tuning an external electric field. We calculate the shuttling time to range from sub-picoseconds to nanoseconds depending on the distance ($\sim$ 10-50 nm) of the donor from the interface. For a certain range of parameters, the state at the interface is localized in all three dimensions, which allows to take the electron back to the donor. The size of the wave-function at the interface can be manipulated by applying a perpendicular magnetic field. Our results establish that quantum control in such nanostructure architectures should be achievable.   

Experimental Stark tuning the donor electron spin resonance in silicon

We measure the hyperfine and spin-orbit shifts due to electric fields applied to bound electrons in silicon. The ability to electrically tune resonances allows for selective single qubit operations on electron spins without localized magnetic fields. We study $^{31}$P donors in natural silicon epilayers and $^{121}$Sb donors implanted into isotopically purified $^{28}$Si. The E-fields ($\sim $kV/cm) are applied by lithographically-patterned, interdigitated metal top-gates. Small shifts in resonant energies are measured by pulsed electron spin resonance using a modified two-pulse (Hahn) spin echo experiment with an electrical pulse applied to the gates between microwave pulses. At B$_{0}\approx $0.35T, we find the 2$^{nd}$ order hyperfine and spin-orbit Stark shifts to be comparable in magnitude. Though the literature has heretofore focused on the hyperfine Stark shift, we predict that the spin-orbit Stark shift will be the dominant tuning parameter at higher magnetic fields where a future quantum computer is likely to operate.   

Nuclear Spin of Phosphorus Donors in Silicon: Spin Relaxation Times and Environmental Decoupling

All shallow donors in silicon (and their various isotopes) have non-zero nuclear spins and thus, both the electron and nuclear spins of neutral donors have been proposed for coding, manipulating and storing quantum information. We have recently demonstrated that the spin of electrons bound to donors have extremely long coherence times of at least T$_{2e}$= 60ms at liquid helium temperatures which permits 10$^6$ single-qubit operations before the electron spin decoheres [1]. Here we extend this work and demonstrate that spin states of both the electron and nucleus of a $^{31}$P donor can be accurately controlled using resonant microwave and RF pulses in pulsed electron nuclear double resonance (ENDOR) experiments. We measure the spin relaxation times of the $^{31}$P nuclear spin and observe long longitudinal relaxation times T$_{1n}$= 70s at 6K, limited by hyperfine interaction with the electron spin residing on the donor. We implement a recently proposed bang-bang strategy which decouples the nuclear spin from a decohering environment, through repeated manipulation of the coupled electron spin [2]. This highlights the potential benefits of physical ‘qubit’ systems beyond the simple 2-level structure. [1] A. M. Tyryshkin et al. PRB, 68, 193207 (2003); [2] J. J. L. Morton et al. Nature Physics in press (2005)   

Experimental realization of single electron confinement in an InAs quantum dot

We demonstrate an enhancement-mode lateral single electron transistor (SET). In contrast to the depletion-mode SETs that reach one-electron regime by expelling electrons from multi-electron QDs, our SET structure uses a single top gate to create two symmetric tunnel barriers and make electrons tunnel into this empty quantum dot one at a time. The sample used in this work to demonstrate this novel SETs concept is an InAs/GaSb composite quantum well, whose bandgap is tunable by the thicknesses of the two QWs and is $\sim $100meV in our case. Using a gated Hall bar geometry, we show that the device can undergo a transition from hole accumulation to the inversion of electrons. As the widths of the conducting channel and the top gate are reduced to sub-micron, the conductance displays single electron tunneling. The data indicate a 15meV Coulomb charging energy and a 20meV orbital energy spacing, which imply a quantum dot of 20nm in diameter. Combining with the inherent advantage of a large electron g* factor in InAs, our demonstration is significant for solid state implementation of scalable quantum computing.