Experimental and theoretical progress toward quantum computation with spins in quantum dots (QDs) is reviewed, with particular focus on QDs formed in GaAs heterostructures, on nanowire-based QDs, and on self-assembled QDs. We report on a remarkable evolution of the field, where decoherence—one of the main challenges for realizing quantum computers—no longer seems to be the stumbling block it had originally been considered. General concepts, relevant quantities, and basic requirements for spin-based quantum computing are explained; opportunities and challenges of spin-orbit interaction and nuclear spins are reviewed. We discuss recent achievements, present current theoretical proposals, and make several suggestions for further experiments.

Prospects for Spin-Based Quantum Computing in Quantum Dots

Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries have uncovered new phenomena that completely change our understanding of transport in these devices, especially the role of the spin and valley degrees of freedom. This review describes the modern understanding of transport through nanotube devices. Unlike in conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and valley freedom. The interplay between the two is the focus of this review. The energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are explained, together with their consequences for transport measurements through nanotube quantum dots. In double quantum dots, the combination of quantum numbers modifies the selection rules of Pauli blockade. This can be exploited to read out spin and valley qubits and to measure the decay of these states through coupling to nuclear spins and phonons. A second unique property of carbon nanotubes is that the combination of valley freedom and electron-electron interactions in one dimension strongly modifies their transport behavior. Interaction between electrons inside and outside a quantum dot is manifested in SU(4) Kondo behavior and level renormalization. Interaction within a dot leads to Wigner molecules and more complex correlated states. This review takes an experimental perspective informed by recent advances in theory. As well as the well-understood overall picture, open questions for the field are also clearly stated. These advances position nanotubes as a leading system for the study of spin and valley physics in one dimension where electronic disorder and hyperfine interaction can both be reduced to a low level.

/pdf/quantum-transport-in-carbon-nanotubes-482jlbi7aa.pdf

Quantum transport in carbon nanotubes

We propose a tune-free scheme to realize Kramers pairs of Majorana bound states in recently discovered higher-order topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an s-wave superconductor, the competition between local and crossed Andreev pairing leads to the formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electron-electron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions compared with nonhelical nanowires.

Majorana Kramers Pairs in Higher-Order Topological Insulators

We study normal transport and the subgap spectrum of superconductor-normal-superconductor (SNS) junctions made of semiconducting nanowires with strong Rashba spin-orbit coupling. We focus, in particular, on the role of confinement effects in long ballistic junctions. In the normal regime, scattering at the two contacts gives rise to two distinct features in conductance: Fabry-Perot resonances and Fano dips. The latter arise in the presence of a strong Zeeman field $B$ that removes a spin sector in the leads (helical leads), but not in the central region. Conversely, a helical central region between nonhelical leads exhibits helical gaps of half-quantum conductance, with superimposed helical Fabry-Perot oscillations. These normal features translate into distinct subgap states when the leads become superconducting. In particular, Fabry-Perot resonances within the helical gap become parity-protected zero-energy states (parity crossings), well below the critical field ${B}_{c}$ at which the superconducting leads become topological. As a function of Zeeman field or Fermi energy, these zero modes oscillate around zero energy, forming characteristic loops, which evolve continuously into Majorana bound states as $B$ exceeds ${B}_{c}$. The relation with the physics of parity crossings of Yu-Shiba-Rusinov bound states is discussed.

/pdf/sns-junctions-in-nanowires-with-spin-orbit-coupling-role-of-2jsqduqpm4.pdf

SNS junctions in nanowires with spin-orbit coupling: Role of confinement and helicity on the subgap spectrum

Hybrid semiconductor-superconductor systems are interesting melting pots where various fundamental effects in condensed-matter physics coexist. For example, when a quantum dot is coupled to a superconducting electrode two very distinct phenomena, superconductivity and the Kondo effect, compete. As a result of this competition, the system undergoes a quantum phase transition when the superconducting gap $\ensuremath{\Delta}$ is of the order of the Kondo temperature ${T}_{K}$. The underlying physics behind such transition ultimately relies on the physics of the Anderson model where the standard metallic host is replaced by a superconducting one, namely the physics of a (quantum) magnetic impurity in a superconductor. A characteristic feature of this hybrid system is the emergence of subgap bound states, the so-called Yu-Shiba-Rusinov (YSR) states, which cross zero energy across the quantum phase transition, signaling a switching of the fermion parity and spin (doublet or singlet) of the ground state. Interestingly, similar hybrid devices based on semiconducting nanowires with spin-orbit coupling may host exotic zero-energy bound states with Majorana character. Both parity crossings and Majorana bound states (MBSs) are experimentally marked by zero-bias anomalies in transport, which are detected by coupling the hybrid device with an extra normal contact. We here demonstrate theoretically that this extra contact, usually considered as a nonperturbing tunneling weak probe, leads to nontrivial effects. This conclusion is supported by numerical renormalization-group calculations of the phase diagram of an Anderson impurity coupled to both superconducting and normal-state leads. We obtain this phase diagram for an arbitrary ratio $\frac{\ensuremath{\Delta}}{{T}_{K}}$, which allows us to analyze relevant experimental scenarios, such as parity crossings as well as Kondo features induced by the normal lead, as this ratio changes. Spectral functions at finite temperatures and magnetic fields, which can be directly linked to experimental tunneling transport characteristics, show zero-energy anomalies irrespective of whether the system is in the doublet or singlet regime. We also derive the analytical condition for the occurrence of Zeeman-induced fermion-parity switches in the presence of interactions which bears unexpected similarities with the condition for emergent MBSs in nanowires.

/pdf/shiba-states-and-zero-bias-anomalies-in-the-hybrid-normal-3dqz25ry27.pdf

Shiba states and zero-bias anomalies in the hybrid normal-superconductor Anderson model

The two electrons of a Cooper pair in a conventional superconductor form a spin singlet and therefore a maximally entangled state. Recently, it was demonstrated that the two particles can be extracted from the superconductor into two spatially separated contacts via two quantum dots in a process called Cooper pair splitting (CPS). Competing transport processes, however, limit the efficiency of this process. Here we demonstrate efficiencies up to 90%, significantly larger than required to demonstrate interaction-dominated CPS, and on the right order to test Bell's inequality with electrons. We compare the CPS currents through both quantum dots, for which large apparent discrepancies are possible. The latter we explain intuitively and in a semiclassical master equation model. Large efficiencies are required to detect electron entanglement and for prospective electronics-based quantum information technologies.

Near-Unity Cooper Pair Splitting Efficiency

We experimentally investigate Andreev bound states (ABSs) in a carbon nanotube quantum dot (QD) connected to a superconducting Nb lead (S) A weakly coupled normal metal contact acts as a tunnel probe that measures the energy dispersion of the ABSs Moreover, we study the response of the ABS to nonlocal transport processes, namely, Cooper pair splitting and elastic co-tunnelling, which are enabled by a second QD fabricated on the same nanotube on the opposite side of S We find an appreciable nonlocal conductance with a rich structure, including a sign reversal at the ground-state transition from the ABS singlet to a degenerate magnetic doublet We describe our device by a simple rate equation model that captures the key features of our observations and demonstrates that the sign of the nonlocal conductance is a measure for the charge distribution of the ABS, given by the respective Bogoliubov-de Gennes amplitudes $u$ and $v$

Nonlocal spectroscopy of Andreev bound states

We demonstrate that ultraclean single, double, and triple quantum dots (QDs) can be formed reliably in a carbon nanotube (CNT) by a straightforward fabrication technique. The QDs are electrostatically defined in the CNT by closely spaced metallic bottom gates deposited in trenches in SiO2 by sputter deposition of Re. The carbon nanotubes are then grown by chemical vapor deposition (CVD) across the trenches and contacted using conventional resist-based electron beam lithography. Unlike in previous work, the devices exhibit reproducibly the characteristics of ultraclean QDs behavior even after the subsequent electron beam lithography and chemical processing steps. We specifically demonstrate the high quality using CNT devices with two narrow bottom gates and one global back gate. Tunable by the gate voltages, the device can be operated in four different regimes: (i) fully p-type with ballistic transport between the outermost contacts (over a length of 700 nm), (ii) clean n-type single QD behavior where a QD can be induced by either the left or the right bottom gate, (iii) n-type double QD, and (iv) triple bipolar QD where the middle QD has opposite doping (p-type). Our simple fabrication scheme opens up a route to more complex devices based on ultraclean CNTs, since it allows for postgrowth processing.

Ultraclean Single, Double, and Triple Carbon Nanotube Quantum Dots with Recessed Re Bottom Gates

/pdf/ultraclean-single-double-and-triple-carbon-nanotube-quantum-488b6xp6jw.pdf

Ultraclean single, double, and triple carbon nanotube quantum dots with recessed Re bottom gates

We show frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements of liquid and glassy glycerol. This allows a direct comparison of both quantities determined independently. After cooling the glycerol sample to 179 K with a cooling rate of $0.85\text{ }\text{ }\mathrm{K}/\mathrm{min}$ we studied the aging time dependence of the dielectric permittivity and the polarization fluctuations using the identical glycerol sample. A cross-correlation technique allows measurements of noise levels below the amplifier noise. In the frequency range between 0.3 and 300 Hz we find the measured data to be in agreement with the fluctuation-dissipation theorem for the liquid and glassy state not depending on the aging time.

Fluctuation-dissipation theorem in liquid and glassy glycerol: frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements.

Jens Schindele

Papers

Near-Unity Cooper Pair Splitting Efficiency

Nonlocal spectroscopy of Andreev bound states

Ultraclean Single, Double, and Triple Carbon Nanotube Quantum Dots with Recessed Re Bottom Gates

Ultraclean single, double, and triple carbon nanotube quantum dots with recessed Re bottom gates

Fluctuation-dissipation theorem in liquid and glassy glycerol: frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements.