Recently, condensed matter and atomic experiments have reached a length-scale and temperature regime where new quantum collective phenomena emerge. Finding such physics in systems of photons, however, is problematic, as photons typically do not interact with each other and can be created or destroyed at will. Here, we introduce a physical system of photons that exhibits strongly correlated dynamics on a meso-scale. By adding photons to a two-dimensional array of coupled optical cavities each containing a single two-level atom in the photon-blockade regime, we form dressed states, or polaritons, that are both long-lived and strongly interacting. Our zero temperature results predict that this photonic system will undergo a characteristic Mott insulator (excitations localised on each site) to superfluid (excitations delocalised across the lattice) quantum phase transition. Each cavity's impressive photon out-coupling potential may lead to actual devices based on these quantum manybody effects, as well as observable, tunable quantum simulators.The Jaynes-Cummings [1] model is arguably the most important model for understanding light-matter interactions. It describes the interaction of a single, quasiresonant optical cavity field with a two-level atom. The coupling between the atom and the photons leads to optical nonlinearities and an effective photon-photon repulsion. Perhaps the most extreme demonstration of this photonic repulsion is photon blockade, demonstrated recently by Birnbaum et al. [2], where photonic repulsion prevents more than one photon from being in the cavity at any one time. Photon blockade was initially theoretically described with a four-state system [3], with multiplication of the weak Kerr nonlinearity effected by placing a large number of atoms within each cavity. However, it was quickly realised that the photonic blockade mechanism does not persist in the limit of many atoms [4], rapidly degrading as the number of atoms per cavity is increased [5]. Later Rebic et al. showed that the nonlinear interaction afforded by placing a single two-level atom inside a cavity would suffice for realising photon blockade [6]. This observation was highly significant as it allowed the full weight of the Jaynes-Cummings model to be used to attack and understand this problem.To create an atom-photon system whose dynamics mirror those traditionally associated with strongly interacting condensed matter systems, we consider a twodimensional array of photonic bandgap cavities. Each cavity contains a single two-level atom, quasi-resonant with the cavity mode. Evanescent coupling between the cavities due to their proximity allows inter-cavity photon hopping. This configuration is depicted schematically in Fig. 1(a), where we have explicitly chosen three nearest neighbours per cavity (coordination number z = 3), for reasons explained below. Because we are considering small cavities, with volumes of order λ 3 where λ is the wavelength of the light, there will be strong atom-photon couplings that will dominate over the...

Although silicon is a promising material for quantum computation, the degeneracy of the conduction band minima (valleys) must be lifted with a splitting sufficient to ensure the formation of well-defined and long-lived spin qubits. Here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metal-oxidesemiconductor quantum dot, providing splittings spanning 0.3-0.8 meV. The splitting varies linearly with applied electric field, with a ratio in agreement with atomistic tight-binding predictions. We demonstrate single-shot spin read-out and measure the spin relaxation for different valley configurations and dot occupancies, finding one-electron lifetimes exceeding 2 s. Spin relaxation occurs via phonon emission due to spin-orbit coupling between the valley states, a process not previously anticipated for silicon quantum dots. An analytical theory describes the magnetic field dependence of the relaxation rate, including the presence of a dramatic rate enhancement (or hot-spot) when Zeeman and valley splittings coincide.

Silicon has many attractive properties for quantum computing, and the quantum dot architecture is appealing because of its controllability and scalability. However, the multiple valleys in the silicon conduction band are potentially a serious source of decoherence for spin-based quantum dot qubits. Only when these valleys are split by a large energy does one obtain well-defined and long-lived spin states appropriate for quantum computing. Here we show that the small valley splittings observed in previous experiments on Si/SiGe heterostructures result from atomic steps at the quantum well interface. Lateral confinement in a quantum point contact limits the electron wavefunctions to several steps, and enhances the valley splitting substantially, up to 1.5 meV. The combination of electronic and magnetic confinement produces a valley splitting larger than the spin splitting, which is controllable over a wide range. These results improve the outlook for realizing spin qubits with long coherence times in silicon-based devices.The fundamental unit of quantum information is the qubit. Qubits can be constructed from the quantum states of physical objects like atomic ions [1], quantum dots [2,3,4,5,6,7] or superconducting Josephson junctions [8]. A key requirement is that these quantum states should be well-defined and isolated from their environment. An assemblage of many qubits into a register and the construction of a universal set of operations, including initialization, measurement, and single and multi-qubit gates, would enable a quantum computer to execute algorithms for certain difficult computational problems like prime factorization and database search far faster than any conventional computer [9].The solid state affords special benefits and challenges for qubit operation and quantum computation. State-ofthe-art fabrication techniques enable the positioning of electrostatic gates with a resolution of several nanometers, paving the way for large scale implementations. On the other hand, the solid state environment provides numerous pathways for decoherence to degrade the computation [10]. Spins in silicon offer a special resilience against decoherence because of two desirable materials properties [11,12]: a small spin-orbit coupling and predominately spin-zero nuclei. Isotopic purification could essentially eliminate all nuclear decoherence mechanisms.Silicon, however, also has a property that potentially can increase decoherence. Silicon has multiple conduction band minima or valleys at the same energy. Unless this degeneracy is lifted, coherence and qubit operation will be threatened. In strained silicon quantum wells there are two such degenerate valleys [13] whose quantum numbers and energy scales compete directly with the spin degrees of freedom. In principle, sharp confinement potentials, like the quantum well interfaces, couple these two valleys and lift the degeneracy, providing a unique ground state if the coupling is strong enough [14,15]. Theoretical analyses for noninteracting electrons in perfectly f...

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Spin qubits composed of either one or three electrons are realized in a quantum dot formed at a Si/SiO2 interface in isotopically enriched silicon. Using pulsed electron spin resonance, we perform coherent control of both types of qubits, addressing them via an electric field dependent g-factor. We perform randomized benchmarking and find that both qubits can be operated with high fidelity. Surprisingly, we find that the gfactors of the one-electron and three-electron qubits have an approximately linear but opposite dependence as a function of the applied dc electric field. We develop a theory to explain this g-factor behavior based on the spinvalley coupling that results from the sharp interface. The outer "shell" electron in the three-electron qubit exists in the higher of the two available conduction-band valley states, in contrast with the one-electron case, where the electron is in the lower valley. We formulate a modified effective mass theory and propose that inter-valley spin-flip tunneling dominates over intra-valley spin-flips in this system, leading to a direct correlation between the spin-orbit coupling parameters and the g-factors in the two valleys. In addition to offering all-electrical tuning for single-qubit gates, the g-factor physics revealed here for one-electron and three-electron qubits offers potential opportunities for new qubit control approaches.

Direct phonon spin-lattice relaxation of an electron qubit bound by a donor impurity or quantum dot in SiGe heterostructures is investigated. The aim is to evaluate the importance of decoherence from this mechanism in several important solid-state quantum computer designs operating at low temperatures. We calculate the relaxation rate 1/T1 as a function of [100] uniaxial strain, temperature, magnetic field, and silicon/germanium content for Si:P bound electrons. The quantum dot potential is much smoother, leading to smaller splittings of the valley degeneracies. We have estimated these splittings in order to obtain upper bounds for the relaxation rate. In general, we find that the relaxation rate is strongly decreased by uniaxial compressive strain in a SiGe-Si-SiGe quantum well, making this strain an important positive design feature. Ge in high concentrations (particularly over 85%) increases the rate, making Si-rich materials preferable. We conclude that SiGe bound electron qubits must meet certain conditions to minimize decoherence but that spin-phonon relaxation does not rule out the solid-state implementation of error-tolerant quantum computing.

We introduce an always-on, exchange-only qubit made up of three localized semiconductor spins that offers a true "sweet spot" to fluctuations of the quantum dot energy levels. Both single-and two-qubit gate operations can be performed using only exchange pulses while maintaining this sweet spot. We show how to interconvert this qubit to other three-spin encoded qubits as a new resource for quantum computation and communication.Semiconductor qubits [1,2] are a leading candidate technology for quantum information processing [3]. Spins can have extremely long quantum coherence due to a decoupling of spin information from charge noise in many materials, and they are small, enabling high density. But these strengths pose a challenge for control as microwave pulses generally result in slow gates with significant potential for crosstalk to nearby qubits. The exchange interaction on the other hand provides a natural and fast method for entangling semiconductor qubits: it can be used to perform two-spin entangling operations with a finite-length voltage pulse or to couple spins with a constant interaction. Exchange also provides a solution to the control problem by allowing a two-level system to be encoded into the greater Hilbert space of multiple physical spins. Following work on decoherence free subspaces and subsystems (DFS) [4][5][6], many multi-spinbased qubits have been proposed and demonstrated with various desirable properties for quantum computing: as examples, 2-DFS (a.k.a. "singlet-triplet") [7][8][9][10], 3-DFS (a.k.a. "exchange-only") [11][12][13][14], or 4-DFS qubits [15] of various implementations are possible. The DFS gives some immunity to global field fluctuations, but more importantly it allows for gate operations via a sequence of pair-wise exchange interactions between spins with fast, baseband voltage control on the metallic top-gates, obviating the need for RF pulses. However, charge noise likely limits gate fidelity [16,17] as charge and spin are coupled while spins undergo exchange.The effects of charge (or other) noise on memory or gate fidelity can be suppressed to a certain extent by taking advantage of natural or engineered "sweet" spots: a spot in parameter space where critical system properties are minimally effected by certain environmental changes. Sweet spots have been an effective tool to increase the coherence of superconducting qubits [18,19] and more recently have been applied to exchange-only qubits [20][21][22][23][24]. For example, in the "resonant exchange" (RX) qubit-an encoded qubit made out of 3 quantum dot qubits with "always-on" exchange interactions and a much higher chemical potential for the middle dot than the outer dots-a partial sweet spot is maintained while microwave control allows for single qubit operations [20,21] "resonant" with the gap of the 3-spin system. The first derivative of the RX qubit frequency vanishes for one of the two detuning parameters that are affected by charge noise. For two qubit gates, the RX qubit offers a relatively large transition dipole...

We expand on previous work that treats relaxation physics of low-lying excited states in ideal, single electron, silicon quantum dots in the context of quantum computing. These states are of three types: orbital, valley, and spin. The relaxation times depend sensitively on system parameters such as the dot size and the external magnetic field. Generally, however, orbital relaxation times are short in strained silicon (10 −7 to 10 −12 s), spin relaxation times are long, (10 −6 to 1 s), while valley relaxation times are expected to lie in between. The focus is on relaxation due to emission or absorption of phonons, but for spin relaxation we also consider competing mechanisms such as charge noise. Where appropriate, comparison is made to reference systems such as quantum dots in III-V materials and silicon donor states. The phonon bottleneck effect is shown to be rather small in the silicon dots of interest. We compare the theoretical predictions to some recent spin relaxation experiments and comment on the possible effects of non-ideal dots.