Effects of motion in cavity QED

We consider effects of motion in cavity quantum electrodynamics experiments where single cold atoms can now be observed inside the cavity for many Rabi cycles. We discuss the timescales involved in the problem and the need for good control of the atomic motion, particularly the heating due to exchange of excitation between the atom and the cavity, in order to realize nearly unitary dynamics of the internal atomic states and the cavity mode which is required for several schemes of current interest such as quantum computing. Using a simple model we establish ultimate effects of the external atomic degrees of freedom on the action of quantum gates. The perfomance of the gate is characterized by a measure based on the entanglement fidelity and the motional excitation caused by the action of the gate is calculated. We find that schemes which rely on adiabatic passage, and are not therefore critically dependent on laser pulse areas, are very much more robust against interaction with the external degrees of freedom of atoms in the quantum gate.Comment: 10 pages, 5 figures, REVTeX, to be published in Walls Symposium Special Issue of Journal of Optics

Experimental Observation of Quantum Correlations in Modular Variables

We experimentally detect entanglement in modular position and momentum variables of photon pairs which have passed through $D$-slit apertures. We first employ an entanglement criteria recently proposed in [Phys. Rev. Lett. {\bf 106}, 210501 (2011)], using variances of the modular variables. We then propose an entanglement witness for modular variables based on the Shannon entropy, and test it experimentally. Finally, we derive criteria for Einstein-Podolsky-Rosen-Steering correlations using variances and entropy functions. In both cases, the entropic criteria are more successful at identifying quantum correlations in our data.Comment: 7 pages, 4 figures, comments welcom

Complementarity and quantum walks

We show that quantum walks interpolate between a coherent `wave walk' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a trade-off between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measurement.Comment: 7 pages, RevTex, 2 figures; v2 adds references; v3 updated to incorporate feedback and updated references; v4 substantially expanded to clarify presentatio

Hitting time for quantum walks on the hypercube

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary ``coin'' is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.Comment: 8 pages in RevTeX format, four figures in EPS forma

Quantum walks on quotient graphs

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. If this unitary evolution operator has an associated group of symmetries, then for certain initial states the walk will be confined to a subspace of the original Hilbert space. Symmetries of the original graph, given by its automorphism group, can be inherited by the evolution operator. We show that a quantum walk confined to the subspace corresponding to this symmetry group can be seen as a different quantum walk on a smaller quotient graph. We give an explicit construction of the quotient graph for any subgroup of the automorphism group and illustrate it with examples. The automorphisms of the quotient graph which are inherited from the original graph are the original automorphism group modulo the subgroup used to construct it. We then analyze the behavior of hitting times on quotient graphs. Hitting time is the average time it takes a walk to reach a given final vertex from a given initial vertex. It has been shown in earlier work [Phys. Rev. A {\bf 74}, 042334 (2006)] that the hitting time can be infinite. We give a condition which determines whether the quotient graph has infinite hitting times given that they exist in the original graph. We apply this condition for the examples discussed and determine which quotient graphs have infinite hitting times. All known examples of quantum walks with fast hitting times correspond to systems with quotient graphs much smaller than the original graph; we conjecture that the existence of a small quotient graph with finite hitting times is necessary for a walk to exhibit a quantum speed-up.Comment: 18 pages, 7 figures in EPS forma

Qudit surface codes and gauge theory with finite cyclic groups

Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2 (qubit) lattices, such ground states have been proposed as topologically protected memory for qubits. We extend these constructions to lattices or more generally cell complexes with qudits, either of prime level or of level $d^\ell$ for $d$ prime and $\ell \geq 0$, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact $\mathbb{Z}_d\cong\mathbb{Z}/d\mathbb{Z}$ gauge theory whose excitations correspond to abelian anyons. We provide protocols for qudit storage and retrieval and propose an interferometric verification of topological order by measuring quasi-particle statistics.Comment: 26 pages, 5 figure

Quantum walks with infinite hitting times

Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well-defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.Comment: 28 pages, 3 figures in EPS forma

Hitting time for the continuous quantum walk

We define the hitting (or absorbing) time for the case of continuous quantum walks by measuring the walk at random times, according to a Poisson process with measurement rate $\lambda$. From this definition we derive an explicit formula for the hitting time, and explore its dependence on the measurement rate. As the measurement rate goes to either 0 or infinity the hitting time diverges; the first divergence reflects the weakness of the measurement, while the second limit results from the Quantum Zeno effect. Continuous-time quantum walks, like discrete-time quantum walks but unlike classical random walks, can have infinite hitting times. We present several conditions for existence of infinite hitting times, and discuss the connection between infinite hitting times and graph symmetry.Comment: 12 pages, 1figur

Quantum random walks in optical lattices

We propose an experimental realization of discrete quantum random walks using neutral atoms trapped in optical lattices. The random walk is taking place in position space and experimental implementation with present day technology --even using existing set-ups-- seems feasible. We analyze the influence of possible imperfections in the experiment and investigate the transition from a quantum random walk to the classical random walk for increasing errors and decoherence.Comment: 8 pages, 4 figure

Controlling discrete quantum walks: coins and intitial states

In discrete time, coined quantum walks, the coin degrees of freedom offer the potential for a wider range of controls over the evolution of the walk than are available in the continuous time quantum walk. This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.Comment: 10 (+epsilon) pages, 10 embedded eps figures, typos corrected, references added and updated, corresponds to published version (except figs 5-9 optimised for b&w printing here