The counterfeit coin problem requires us to find all false coins from a given bunch of coins using a balance scale. We assume that the balance scale gives us only ``balanced'' or ``tilted'' information and that we know the number k of false coins in advance. The balance scale can be modeled by a certain type of oracle and its query complexity is a measure for the cost of weighing algorithms (the number of weighings). In this paper, we study the quantum query complexity for this problem. Let Q(k,N) be the quantum query complexity of finding all k false coins from the N given coins. We show that for any k and N such that k < N/2, Q(k,N)=O(k^{1/4}), contrasting with the classical query complexity, \Omega(k\log(N/k)), that depends on N. So our quantum algorithm achieves a quartic speed-up for this problem. We do not have a matching lower bound, but we show some evidence that the upper bound is tight: any algorithm, including our algorithm, that satisfies certain properties needs \Omega(k^{1/4}) queries.

We investigate whether size imposes a fundamental constraint on the efficiency of small thermal machines. We analyse in detail a model of a small self-contained refrigerator consisting of three qubits. We show that this system can reach the Carnot efficiency, and thus demonstrate that there exists no complementarity between size and efficiency.

We prove that the family of embezzlement states defined by van Dam and Hayden [vanDamHayden2002] is universal for both quantum and classical entangled two-prover non-local games with an arbitrary number of rounds. More precisely, we show that for each $\epsilon>0$ and each strategy for a k-round two-prover non-local game which uses a bipartite shared state on 2m qubits and makes the provers win with probability $\omega$, there exists a strategy for the same game which uses an embezzlement state on $2m + 2m/\epsilon$ qubits and makes the provers win with probability $\omega-\sqrt{2\epsilon}$. Since the value of a game can be defined as the limit of the value of a maximal 2m-qubit strategy as m goes to infinity, our result implies that the classes QMIP*_{c,s}[2,k] and MIP*_{c,s}[2,k] remain invariant if we allow the provers to share only embezzlement states, for any completeness value c in [0,1] and any soundness value s < c. Finally we notice that the circuits applied by each prover may be put into a very simple universal form.

A generic quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. Here, the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multi-mode bosonic Gaussian channel is analyzed both for mixed and pure environment corresponding to the Stinespring representation. In particular, for the case of pure environment we compute this quantity and present an explicit unitary dilation for arbitrary bosonic Gaussian channel. These results considerably simplify the characterization of these continuous-variable maps and can be applied to address some open issues concerning the transmission of information encoded in bosonic systems.

Two-dimensional crystals of trapped ions are a promising system with which to implement quantum simulations of challenging problems such as spin frustration. Here, we present a design for a surface-electrode elliptical ion trap which produces a 2-D ion crystal and is amenable to microfabrication, which would enable higher simulated coupling rates, as well as interactions based on magnetic forces generated by on-chip currents. Working in an 11 K cryogenic environment, we experimentally verify to within 5% a numerical model of the structure of ion crystals in the trap. We also explore the possibility of implementing quantum simulation using magnetic forces, and calculate J-coupling rates on the order of 10^3 / s for an ion crystal height of 10 microns, using a current of 1 A.

In this paper, we present a loss-tolerant quantum strong coin flipping protocol with bias 0.359. This is an improvement over Berlin etal's protocol [BBBG08] which achieves a bias of 0.4. To achieve this, we extend Berlin et al.'s protocol by adding an encryption step that hides some information to Bob until he confirms that he successfully measured.

Various authors have considered schemes for {\it quantum tagging}, that is, authenticating the classical location of a classical tagging device by sending and receiving quantum signals from suitably located distant sites, in an environment controlled by an adversary whose quantum information processing and transmitting power is potentially unbounded. This task raises some interesting new questions about cryptographic security assumptions, as relatively subtle details in the security model can dramatically affect the security attainable. We consider here the case in which the tag is cryptographically secure, and show how to implement tagging securely within this model.

Using convex optimization, we propose entanglement-assisted quantum error correction procedures that are optimized for given noise channels. We demonstrate through numerical examples that such an optimized error correction method achieves higher channel fidelities than existing methods. This improved performance, which leads to perfect error correction for a larger class of error channels, is interpreted in at least some cases by quantum teleportation, but for general channels this interpretation does not hold.

Entanglement is the fundamental characteristic of quantum physics. Large experimental efforts are devoted to harness entanglement between various physical systems. In particular, entanglement between light and material systems is interesting due to their prospective roles as "flying" and stationary qubits in future quantum information technologies, such as quantum repeaters and quantum networks. Here we report the first demonstration of entanglement between a photon at telecommunication wavelength and a single collective atomic excitation stored in a crystal. One photon from an energy-time entangled pair is mapped onto a crystal and then released into a well-defined spatial mode after a predetermined storage time. The other photon is at telecommunication wavelength and is sent directly through a 50 m fiber link to an analyzer. Successful transfer of entanglement to the crystal and back is proven by a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by almost three standard deviations (S=2.64+/-0.23). These results represent an important step towards quantum communication technologies based on solid-state devices. In particular, our resources pave the way for building efficient multiplexed quantum repeaters for long-distance quantum networks.

In a recent paper, H. Mueller, A. Peters and S. Chu [A precision measurement of the gravitational redshift by the interference of matter waves, Nature 463, 926-929 (2010)] argued that atom interferometry experiments published a decade ago did in fact measure the gravitational redshift on the quantum clock operating at the very high Compton frequency associated with the rest mass of the Caesium atom. In the present Communication we show that this interpretation is incorrect.

Optical detection of single defect centers in the solid state is a key element of novel quantum technologies. This includes the generation of single photons and quantum information processing. Unfortunately the brightness of such atomic emitters is limited. Therefore we experimentally demonstrate a novel and simple approach that uses off-the-shelf optical elements. The key component is a solid immersion lens made of diamond, the host material for single color centers. We improve the excitation and detection of single emitters by one order of magnitude, as predicted by theory.

We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions.

The reversible transfer of quantum states of light in and out of matter constitutes an important building block for future applications of quantum communication: it allows synchronizing quantum information, and enables one to build quantum repeaters and quantum networks. Much effort has been devoted worldwide over the past years to develop memories suitable for the storage of quantum states. Of central importance to this task is the preservation of entanglement, a quantum mechanical phenomenon whose counter intuitive properties have occupied philosophers, physicists and computer scientists since the early days of quantum physics. Here we report, for the first time, the reversible transfer of photon-photon entanglement into entanglement between a photon and collective atomic excitation in a solid-state device. Towards this end, we employ a thulium-doped lithium niobate waveguide in conjunction with a photon-echo quantum memory protocol, and increase the spectral acceptance from the current maximum of 100 MHz to 5 GHz. The entanglement-preserving nature of our storage device is assessed by comparing the amount of entanglement contained in the detected photon pairs before and after the reversible transfer, showing, within statistical error, a perfect mapping process. Our integrated, broadband quantum memory complements the family of robust, integrated lithium niobate devices. It renders frequency matching of light with matter interfaces in advanced applications of quantum communication trivial and institutes several key properties in the quest to unleash the full potential of quantum communication.

It is well-known that von Neumann entropy is nonmonotonic unlike Shannon entropy (which is monotonically nondecreasing). Consequently, it is difficult to relate the entropies of the subsystems of a given quantum state. In this paper, we show that if we consider quantum secret sharing states arising from a class of monotone span programs, then we can partially recover the monotonicity of entropy for the so-called unauthorized sets. Furthermore, we can show for these quantum states the entropy of the authorized sets is monotonically nonincreasing.

Spin-echo experiments are often said to constitute an instant of anti-thermodynamic behavior in a concrete physical system that violates the second law of thermodynamics. We argue that a proper thermodynamic treatment of the effect should take into account the correlations between the spin and translational degrees of freedom of the molecules. To this end, we construct an entropy functional using Boltzmann macrostates that incorporates both spin and translational degrees of freedom. With this definition there is nothing special in the thermodynamics of spin echoes: dephasing corresponds to Hamiltonian evolution and leaves the entropy unchanged; dissipation increases the entropy. In particular, there is no phase of entropy decrease in the echo. We also discuss the definition of macrostates from the underlying quantum theory and we show that the decay of net magnetization provides a faithful measure of entropy change.

We present a general scheme for treating particle beams, including stationary beams, as many particle systems. This includes the full counting statistics and the requirements of Bose/Fermi symmetry. We treat in detail a model of a source, creating particles in a fixed state, which then evolve under the free time evolution, and we determine the resulting stationary beam in the far field. In comparison to the one-particle picture we obtain a correction from Bose/Fermi statistics, which depends on the emission rate.

Recent proposals using heterostructures of superconducting and either topologically insulating or semiconducting layers have been put forth as possible platforms for topological quantum computation. These systems are predicted to contain Ising anyons and share the feature of having only neutral edge excitations. In this note, we show that these proposals can be combined with the recently proposed ``sack geometry'' for implementation of a phase gate in order to conduct robust universal quantum computation. In addition, we propose a general method for adjusting edge tunneling rates in such systems, which is necessary for the control of interferometric devices. The error rate for the phase gate in neutral Ising systems is parametrically smaller than for a similar geometry in which the edge modes carry charge: it goes as $T^3$ rather than $T$ at low temperatures. At zero temperature, the phase variance becomes constant at long times rather than carrying a logarithmic divergence.

Long decay times were previously observed in samples such as 29Si, C60,Y2O3 by applying multipulse nuclear magnetic resonance sequences to measure decoherence times. They are originated in stimulated echoes caused by the pulse angle distributions predictable for inhomogeneously broadened lines. In the present work, a detailed analysis describing how the stimulated echoes can be exploited as quantum coherence memories is presented. We introduce a method based on field gradients to storage coherences as polarization in a controlled way in homogeneous samples. The possibility to keep a coherent state frozen while another part of the sample is subjected to quantum operations opens new perspectives in the field of quantum information. Upon recovery of the storaged coherences, interactions among the whole system can be turned on. However, in order to perform quantum computation, the knowledge of the true coherence time is necessary. We applied the proposed method to demonstrate under the stimulated echo formalism, the appropriate experimental scheme that enables a quenching of the coherence storage, thus rendering a measurement of the coherence decay time T2.

Remarks on Michael Frayn's play "Copenhagen".

We show how the momentum distribution of gaseous Bose-Einstein condensates can be shaped by applying a sequence of standing--wave laser pulses. We present a theory of the effect of such a pulse sequence on the condensate wave function in momentum space. We generalize the previous result to the case of N pulses of arbitrary intensity separated by arbitrary intervals and show how these parameters can be engineered to produce a desired final momentum distribution. We find that several momentum distributions, important in atom interferometry applications, can be engineered with high fidelity with two or three pulses.

Two proof-of-principle experiments towards T1-limited magnetic resonance imaging with NV centers in diamond are demonstrated. First, a large number of Rabi oscillations is measured and it is demonstrated that the hyperfine interaction due to the NV's 14N can be extracted from the beating oscillations. Second, the Rabi beats under V-type microwave excitation of the three hyperfine manifolds is studied experimentally and described theoretically.

We study the single particle dynamics of a mobile non-Abelian anyon hopping around many pinned anyons on a surface. The dynamics is modelled by a discrete time quantum walk and the spatial degree of freedom of the mobile anyon becomes entangled with the fusion degrees of freedom of the collective system. Each quantum trajectory makes a closed braid on the world lines of the particles establishing a direct connection between statistical dynamics and quantum link invariants. We find that asymptotically a mobile Ising anyon becomes so entangled with its environment that its statistical dynamics reduces to a classical random walk with linear dispersion in contrast to particles with Abelian statistics which have quadratic dispersion.

We present an efficient arbitrary polarization qubit transmission scheme against channel noise by utilizing frequency degree of freedom, which is more stable in transmission surroundings. The information of quantum state is encoded in frequency state during the transmission and transferred to polarization state later. Both the fidelity of quantum state transmitted and the success probability of this scheme are 1 in principle.

Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Markovian master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation. This representation may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Markovianity have been presented. Their common underlying definition of non-Markovianity is whether negative decoherence rates may appear in the Lindblad-like form of the master equation. We therefore propose to use the negative decoherence rates themselves, as they appear in the unique canonical form of the master equation, as a primary measure to more completely characterize non-Markovianity. The advantages of this are especially apparent when many decoherence channels are present.

In the present paper we investigate the set $\Sigma_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-RJ$. This enables one to construct the collection of operators $C_{\chi,\omega}$ realizing the property of stable $C$-symmetry for extensions $A\in\Sigma_J$ directly in terms of ${\mathcal C}l_2(J,R)$ and to parameterize the corresponding subset of extensions with stable $C$-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.

We have measured the isotope shift of the narrow quadrupole-allowed 5 2S1/2 - 4 2D5/2 transition in 86Sr+ relative to the most abundant isotope 88Sr+. This was accomplished using high-resolution laser spectroscopy of individual trapped ions, and the measured shift is Delta-nu_meas^(88,86) = 570.281(4) MHz. We have also tested a recently developed and successful method for ab-initio calculation of isotope shifts in alkali-like atomic systems against this measurement, and our initial result of Delta-nu_calc^(88,86) = 457(28) MHz is also presented. To our knowledge, this is the first high precision measurement and calculation of that isotope shift. While the measurement and the calculation are in broad agreement, there is a clear discrepancy between them, and we believe that the specific mass shift was underestimated in our calculation. Our measurement provides a stringent test for further refinements of theoretical isotope shift calculation methods for atomic systems with a single valence electron.

The basic principles of classical and semi-classical theories of molecular optical activity are discussed. These theories are valid for dilute solutions of optically active organic molecules. It is shown that all phenomena known in the classical theory of molecular optical activity can be described with the use of one pseudo-scalar which is a uniform function of the incident light frequency $\omega$. The relation between optical rotation and circular dichroism is derived from the basic Kramers-Kronig relations. In our discussion of the general theory of molecular optical activity we introduce the tensor of molecular optical activity. It is shown that to evaluate the optical rotation and circular dichroism at arbitrary frequencies one needs to know only nine (3 + 6) molecular tensors. The quantum (or semi-classical) theory of molecular optical activity is also briefly discussed. We also raise the possibility of measuring the optical rotation and circular dichroism at wavelengths which correspond to the vacuum ultraviolet region, i.e. for $\lambda \le 150$ $nm$.

The correct form of the Schmidt decomposition of the stationary wave functions for a system of two interacting particles trapped in a two-dimensional harmonic potential is given

Oligomers of the organic semiconductor PTCDA are studied by means of helium nanodroplet isolation (HENDI) spectroscopy. In contrast to the monomer absorption spectrum, which exhibits clearly separated, very sharp absorption lines, it is found that the oligomer spectrum consists of three main peaks having an apparent width orders of magnitude larger than the width of the monomer lines. Using a simple theoretical model for the oligomer, in which a Frenkel exciton couples to internal vibrational modes of the monomers, these experimental findings are nicely reproduced. The three peaks present in the oligomer spectrum can already be obtained taking only one effective vibrational mode of the PTCDA molecule into account. The inclusion of more vibrational modes leads to quasi continuous spectra, resembling the broad oligomer spectra.

We solve analytically the Schr\"odinger equation for the N-dimensional inverse square potential in quantum mechanics with a minimal length in terms of Heun's functions. We apply our results to the problem of a dipole in a cosmic string background. We find that a bound state exists only if the angle between the dipole moment and the string is larger than {\pi}/4. We compare our results with recent conflicting conclusions in the literature. The minimal length may be interpreted as a radius of the cosmic string.

A momentum representation treatment of the hydrogen atom problem with a generalized uncertainty relation,which leads to a minimal length ({\Delta}X_{i})_{min}=ℏ√(3{\beta}+{\beta}′), is presented. We show that the distance squared operator can be factorized in the case {\beta}′=2{\beta}. We analytically solve the s-wave bound-state equation. The leading correction to the energy spectrum caused by the minimal length depends on √{\beta}. An upper bound for the minimal length is found to be about 10⁻⁹ fm.

Based on the stochastic theory developed by Kubo and Anderson, we present an exact result of the decoherence function of a qubit in telegraph-like noises under dynamical decoupling control. We prove that for telegraph-like noises, the decoherence can be suppressed at most to the third order of the time and the periodic Carr-Purcell-Merboom-Gill sequences are the most efficient scheme in protecting the qubit coherence in the short-time limit.

By identifying non-local effects in systems of identical Bosonic qubits through correlations of their commuting observables, we show that entanglement is not necessary to violate certain squeezing inequalities that hold for distinguishable qubits and that spin squeezing may not be necessary to achieve sub-shot noise accuracies in ultra-cold atom interferometry.

We propose to turn two distant resonant cavities effectively into one by coupling them via an optical fiber which is coated with two-level atoms [Franson et al., Phys. Rev. A 70, 062302 (2004)]. The length of the fiber should be such that it supports a small frequency range of standing waves which includes the optical frequency of the cavities. The purpose of the atoms is to measure their evanescent field destructively on a time scale which is long compared to the time it takes a photon to travel from one side to the other. In fact, the fiber should provide an additional reservoir for one common cavity field mode but not for the other. If the corresponding decay rate is sufficiently large, this mode decouples effectively from the system dynamics due to overdamping of its population.

In practical quantum key distribution (QKD) system, the state preparation and measurement are imperfect comparing with the ideal BB84 protocol, which are always state-dependent in practical realizations. If the state-dependent imperfections can not be regarded as an unitary transformation, it should not be considered as part of quantum channel noise introduced by the eavesdropper, the commonly used secret key rate formula GLLP can not be applied correspondingly. In this paper, the unconditional security of quantum key distribution with state-dependent imperfection has been analyzed by estimating the upper bound of the phase error rate about the quantum channel.

We discuss the coupler system of two nonlinear oscillators excited by an external coherent field prepared in a maximally entangled state (Bell-like state). We show that as a result of the coupler interaction of the system with external broadband squeezed vacuum bath, entanglement decay dynamics can be considerably affected. Besides the phenomena of sudden entanglement death and its rebirth, a shortening (or lengthening) of the total disentanglement time {\tau}D can be observed, depending on the squeezing parameters. Moreover, on the example of one of the reborn entanglement cases it is shown that by changing the values of these parameters the maximal values of the negativity for the 3 \otimes 2 system discussed can be tailored.

We show that stochastic phase-space methods within the truncated Wigner approximation can be used to solve non-equilibrium dynamics of bosonic atoms in 1d traps. We consider systems both with and without an optical lattice and address different approximations in stochastic synthesization of quantum statistical correlations of the initial atomic field operator. We also present a numerically efficient projection method for analyzing correlation functions of the simulation results. Physical examples demonstrate non-equilibrium quantum dynamics of solitons and atom number squeezing in optical lattices in which case we, e.g., numerically track the soliton coordinates and calculate quantum mechanical expectation values and uncertainties for the position of the soliton.

We analyze the simultaneous time-optimal control of two-spin systems. The two non coupled spins which differ in the value of their chemical offsets are controlled by the same magnetic fields. Using an appropriate rotating frame, we restrict the study to the case of opposite shifts. We then show that the optimal solution of the inversion problem in a rotating frame is composed of a pulse sequence of maximum intensity and is similar to the optimal solution for inverting only one spin by using a non-resonant control field in the laboratory frame. An example is implemented experimentally using techniques of Nuclear Magnetic Resonance.

We predict the existence of exchange broadening of optical lineshapes in disordered molecular aggregates and a nonuniversal disorder scaling of the localization characteristics of the collective electronic excitations (excitons). These phenomena occur for heavy-tailed L\'evy disorder distributions with divergent second moments - distributions that play a role in many branches of physics. Our results sharply contrast with aggregate models commonly analyzed, where the second moment is finite. They bear a relevance for other types of collective excitations as well.

Given a pure state vector |x> and a density matrix rho, the function p(x|rho)=<x|rho|x> defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher-Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived.

In relativistic quantum mechanics wave functions of particles satisfy field equations that have initial data on a space--like hypersurface. We propose a dual field theory of ``wavicles'' that have their initial data on a time--like worldline. Propagation of such fields is superluminal, even though the Hilbert space of the solutions carries a unitary representation of the Poincare group of mass zero. We call the objects described by these field equations ``Kairons''. The paper builds the field equations in a general relativistic framework, allowing for a torsion. Kairon fields are section of a vector bundle over space-time. The bundle has infinite--dimensional fibres.

We find analytically an approximate Bloch-Messiah reduction of a noncollinear parametric amplifier pumped with a focused monochromatic beam. We consider type I phase matching. The results are obtained using a perturbative expansion and scaled to high gain regime. They allow a straightforward maximization of the signal gain and minimization of the parametric fluorescence noise. We find fundamental mode of the amplifier which is an elliptic Gaussian defining optimal seed beam shape. We conclude that the output of the amplifier should be stripped of higher order modes, which are approximately Hermite-Gaussian beams. Alternatively, the pump waist can be adjusted such that the amount of noise produced in the higher order modes is minimized.

We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path \gamma connecting points q_0 and q_1 in time t, we define a formal power series V_\gamma(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_\gamma) satisfies Schr\"odinger's equation, and explain in what sense the t\to 0 limit approaches the \delta distribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to Schr\"odinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.

We study the entanglement spectra of bilayer quantum Hall systems at total filling factor nu=1. In the interlayer-coherent phase at layer separations smaller than a critical value, the entanglement spectra show a striking similarity to the energy spectra of the corresponding monolayer systems around half filling. The transition to the incoherent phase can be followed in terms of low-lying entanglement levels, constituting a link between the entanglement spectrum and a quantum phase transition. Finally, we describe the relation between those two types of spectra in terms of effective thermodynamic quantities.

There are many formalisms to describe quantum decoherence. However, many of them give a non general and ad hoc definition of "pointer basis" or "moving preferred basis", and this fact is a problem for the decoherence program. In this paper we will consider quantum systems under a general theoretical framework for decoherence and present a tentative very general definition of the moving preferred basis, in the "random" case, In addition, this definition and another one for a non-random case, are implemented in a well known model. The obtained decoherence and the relaxation times are defined and compared within this model.

We systematically describe and classify 1-dimensional Schr\"odinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable Schr\"odinger equations that can be reduced to the Hermite equation.

The understanding of disordered quantum systems is still far from being complete, despite many decades of research on a variety of physical systems. In this review we discuss how Bose-Einstein condensates of ultracold atoms in disordered potentials have opened a new window for studying fundamental phenomena related to disorder. In particular, we point our attention to recent experimental studies on Anderson localization and on the interplay of disorder and weak interactions. These realize a very promising starting point for a deeper understanding of the complex behaviour of interacting, disordered systems.

This paper proposes a robust control method based on sliding mode design for two-level quantum systems with bounded uncertainties. An eigenstate of the two-level quantum system is identified as a sliding mode. The objective is to design a control law to steer the system's state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system Hamiltonian. We propose a controller design method using the Lyapunov methodology and periodic projective measurements. In particular, we give conditions for designing such a control law, which can guarantee the desired robustness in the presence of the uncertainties. The sliding mode control method has potential applications to quantum information processing with uncertainties.

We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase of entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long term fluctuations which become smaller when quantum chaos sets in.

We use the extended Lifshitz theory to study the behavior of the Casimir forces between finite-thickness functional slabs. We first study the interaction between a semi-infinite Drude metal and a finite-thickness magnetic slab with or without substrate. For no substrate, the large distance $d$ dependence of the force is repulsive and goes as $1/d^5$; for the Drude metal substrate, a stable equilibrium point appears at an intermediate distance which can be tuned by the thickness of the slab. We then study the interaction between two identical chiral metamaterial slabs with and without substrate. For no substrate, the finite thickness of the slabs $D$ does not influence significantly the repulsive character of the force at short distances, while the attractive character at large distances becomes weaker and behaves as $1/d^6$; for the Drude metal substrate, the finite thickness of the slabs $D$ does not influence the repulsive force too much at short distances until $D=0.05\lambda_0$.

By harnessing aspects of quantum mechanics, communication and information processing could be radically transformed. Promising forms of quantum information technology include optical quantum cryptographic systems and computing using photons for quantum logic operations. As with current information processing systems, some form of memory will be required. Quantum repeaters, which are required for long distance quantum key distribution, require optical memory as do deterministic logic gates for optical quantum computing. In this paper we present results from a coherent optical memory based on warm rubidium vapour and show 87% efficient recall of light pulses, the highest efficiency measured to date for any coherent optical memory. We also show storage recall of up to 20 pulses from our system. These results show that simple warm atomic vapour systems have clear potential as a platform for quantum memory.

The teleportation of an unknown polarization state of one of the photons in a system of identical particles has been considered. It has been shown that spatial degrees of freedom, which are various directions of the momentum of three photons, are of significant importance for teleportation in the system of identical particles. The inclusion of the spatial degrees of freedom increases the dimension of single-particle state space. In view of this increase, a four-dimensional subspace of two-particle states, which is similar to the state space spanned by the Bell states in a system of two distinguishable qubits, can be separated in the experimental configuration.

We investigate the possibility to form high fidelity atomic Fock states by gradual reduction of a quasi one dimensional trap containing spin polarized fermions or strongly interacting bosons in the Tonk-Girardeau regime. Making the trap shallower and simultaneously squeezing it can lead to the preparation of an ideal atomic Fock state as one approaches either the sudden or the adiabatic limits. Nonetheless, the fidelity of the resulting state is shown to exhibit a non-monotonic behaviour with the time scale in which the trapping potential is changed.

We consider an infinite one dimensional anisotropic XY spin chain with a nearest neighbor time-dependent Heisenberg coupling J(t) between the spins in presence of a time-dependent magnetic field h(t). We discuss a general solution for the system and present an exact solution for particular choice of J and h of practical interest. We investigate the dynamics of entanglement for different degrees of anisotropy of the system and at both zero and finite temperatures. We find that the time evolution of entanglement in the system show non-ergodic and critical behavior at zero and finite temperatures and different degrees of anisotropy. The asymptotic behavior of entanglement at the infinite time limit at zero temperature and constant J and h depends only the parameter lambda=J/h rather than the individual values of J and h for all degrees of anisotropy but changes for nonzero temperature. Furthermore, the asymptotic behavior is very sensitive to the initial values of J and h and for particular choices we may create finite asymptotic entanglement regardless of the final values of J and h. The persistence of quantum effects in the system as it evolves and as the temperature is raised is studied by monitoring the entanglement. We find that the quantum effects dominates within certain regions of the kT-lambda space that vary significantly depending on the degree of the anisotropy of the system. Particularly, the quantum effects in the Ising model case persists in the vicinity of both its critical phase transition point and zero temperature as it evolves in time. Moreover, the interplay between the different system parameters to tune and control the entanglement evolution is explored.

Continuous center-of-mass position measurements performed on an interacting harmonically trapped Bose-gas are considered. Using both semi-analytical mean-field approach and completely quantum numerical technique based on positive P-representation, it is demonstrated that the atomic delocalization due to the measurement back action is smaller for a strongly interacting gas. The numerically calculated second-order correlation functions demonstrate appearance of atomic bunching as a result of the center-of-mass measurement. Though being rather small the bunching is present also for strongly interacting gas which is in contrast with the case of unperturbed gas. The performed analysis allows to speculate that for relatively strong interactions the size of atomic bunches can become smaller than the initial cloud size resulting in a sort of squeezing effect.

Resonance modes in single crystal sapphire ($\alpha$-Al$_2$O$_3$) exhibit extremely high electrical and mechanical Q-factors ($\approx 10^9$ at 4K), which are important characteristics for electromechanical experiments at the quantum limit. We report the first cooldown of a bulk sapphire sample below superfluid liquid helium temperature (1.6K) to as low as 25mK. The electromagnetic properties were characterised at microwave frequencies, and we report the first observation of electromagnetically induced thermal bistability in whispering gallery modes due to the material $T^3$ dependence on thermal conductivity and the ultra-low dielectric loss tangent. We identify "magic temperatures" between 80 to 2100 mK, the lowest ever measured, at which the onset of bistability is suppressed and the frequency-temperature dependence is annulled. These phenomena at low temperatures make sapphire suitable for quantum metrology and ultra-stable clock applications, including the possible realization of the first quantum limited sapphire clock.

It is argued that Bell's nonlocality is a particular case of nonlocality at detection, which appears already in single-particle interference experiments. The unity of nonlocality and local causality is crucial to provide a consistent description of the world.

The Loschmidt echo is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential decay with a perturbation independent decay rate given by the classical Lyapunov exponent. However, a non-uniform decay - instead of the Lyapunov regime - has been reported in several studies. In this work we show that this behavior arises from the so-called non-diagonal contribution of the semiclassical expansion of the LE. Moreover, we analytically compute the decay rate of this contribution. The interplay between the diagonal and non-diagonal contributions is discussed in detail for completely hyperbolic quantum maps.

Generalized PT symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT-symmetric Hamiltonian. There is a corresponding class of PT-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. For both quantum and classical models, the class of models with generalized PT symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT symmetry is unbroken, the sign problem can always be solved in principle. Moreover, there are models with PT symmetry which can be simulated for all parameter values, including cases where PT symmetry is broken.

We report on the use of a single NV center to probe fluctuating AC magnetic fields. Using engineered currents to induce random changes in the field amplitude and phase, we show that stochastic fluctuations reduce the NV center sensitivity and, in general, make the NV response field-dependent. We also introduce two modalities to determine the field spectral composition, unknown a priori in a practical application. One strategy capitalizes on the generation of AC-field-induced coherence 'revivals', while the other approach uses the time-tagged fluorescence intensity record from successive NV observations to reconstruct the AC field spectral density. These studies are relevant for magnetic sensing in scenarios where the field of interest has a non-trivial, stochastic behavior, such as sensing unpolarized nuclear spin ensembles at low static magnetic fields.

In this paper we review Castagnino's contributions to the foundations of quantum mechanics. First, we recall his work on quantum decoherence in closed systems, and the proposal of a general framework for decoherence from which the phenomenon acquires a conceptually clear meaning. Then, we introduce his contribution to the hard field of the interpretation of quantum mechanics: the modal-Hamiltonian interpretation solves many of the interpretive problems of the theory, and manifests its physical relevance in its application to many traditional models of the practice of physics. In the third part of this work we describe the ontological picture of the quantum world that emerges from the modal-Hamiltonian interpretation, stressing the philosophical step toward a deep understanding of the reference of the theory.

Future quantum technologies rely heavily on the possibility of high-efficiency protection of quantum entanglement against environment-induced decoherence. A recent study showed that an extension of Uhrig's dynamical decoupling (UDD) sequence can lock an arbitrary but known two-qubit entangled state to the Nth order using a sequence of N control pulses [Mukhtar et al., Phys. Rev. A 81, 012331 (2010)]. By nesting three layers of explicitly constructed UDD sequences, here we first consider the protection of unknown two-qubit states as superposition of two known basis states, without making assumptions of the system-environment coupling. It is found that the obtained decoherence suppression can be highly sensitive to the ordering of the three UDD layers and can be remarkably effective with the correct ordering. Our detailed results are useful for general understanding of the nature of controlled quantum dynamics under nested UDD. As an extension of our three-layer UDD, it is finally pointed out that a completely unknown two-qubit state can be protected by nesting four layers of UDD sequences. Our results show that when UDD is applicable (e.g., when environment has a sharp frequency cut-off and when control pulses can be taken as instantaneous pulses), dynamical decoupling using nested UDD sequences is a powerful approach for entanglement protection.

The Floquet theorom and decomposition of operator will be generalized to calculate the non-abelian cyclic geometric phase. The general formula is achieved. Furthermore, the methods is applied to calculate a concret system named LMG.

We show that when photons in NOON states undergo Bloch oscillations, they exhibit a periodic transition between spatially bunched and antibunched states. The period of the bunching/antibunching oscillation is $N$ times faster than the period of the oscillation of the photon density, manifesting the unique coherence properties of NOON states. The transition occurs even when the photons are well separated in space.

We review the fictitious integrable system approach which predicts dynamical tunneling rates from regular states to the chaotic region in systems with a mixed phase space. It is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the regular island. We focus on the direct regular-to-chaotic tunneling process which dominates, if nonlinear resonances within the regular island are not relevant. For quantum maps, billiard systems, and optical microcavities we find excellent agreement with numerical rates for all regular states.

We study the Hamiltonian that is not at first hermitian. Requirement that a measurement shall not change one Hamiltonian eigenstate into another one with a different eigenvalue imposes that an inner product must be defined so as to make the Hamiltonian normal with regard to it. After a long time development with the non-hermitian Hamiltonian, only a subspace of possible states will effectively survive. On this subspace the effect of the anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian becomes hermitian. Thus hermiticity emerges automatically, and we have no reason to maintain that at the fundamental level the Hamiltonian should be hermitian. We also point out a possible misestimation of a past state by extrapolating back in time with the hermitian Hamiltonian. It is a seeming past state, not a true one.

This course is aimed at graduate students in physics in mathematics and designed to give a comprehensive introduction to Weyl quantization and semiclassics via Egorov's theorem. Chapter 2 gives a quick overview of classical and quantum mechanics on R^d. Some mathematical preliminaries concerning Hilbert space theory, operator theory and tempered distributions are detailed in Chapters 3-5. Weyl quantization and semiclassics are the content of Chapters 6 and 7. Finally, an application of Weyl calculus to Born-Oppenheimer systems is discussed in Chapter 8.

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.

We explore the possibility of using quantum walks on graphs to find structural anomalies, such as extra edges or loops, on a graph. We focus our attention on star graphs, whose edges are like spokes coming out of a central hub. If there are $N$ spokes, we show that a quantum walk can find an extra edge connecting two of the spokes or a spoke with a loop on it in $O(\sqrt{N})$ steps. We initially find that if all of the spokes have loops except one, the walk will not find the spoke without a loop, but this can be fixed if we choose the phase with which the particle is reflected from the vertex without the loop. Consequently, quantum walks can, under some circumstances, be used to find structural anomalies in graphs.

The relation between high-harmonic spectra and the geometry of the molecular orbitals in position and momentum space is investigated. In particular we choose two isoelectronic pairs of homonuclear and heteronuclear molecules, such that the highest occupied molecular orbital of the former exhibit at least one nodal plane. The imprint of such planes is a strong suppression in the harmonic spectra, for particular alignment angles. We are able to identify two distinct types of nodal planes. If the nodal planes are determined by the atomic wavefunctions only, the angle for which the yield is suppressed will remain the same for both types of molecules. In contrast, if they are determined by the linear combination of atomic orbitals at different centers in the molecule, there will be a shift in the angle at which the suppression occurs for the heteronuclear molecules, with regard to their homonuclear counterpart. This shows that, in principle, molecular imaging, which uses the homonuclear molecule as a reference and enables one to observe the wavefunction distortions in its heteronuclear counterpart, is possible.

We report measurements of the optical properties of the 1042 nm transition of negatively-charged Nitrogen-Vacancy (NV) centers in type 1b diamond. The results indicate that the upper level of this transition couples to the m_s=+/-1 sublevels of the {^3}E excited state and is short-lived, with a lifetime <~ 1 ns. The lower level is shown to have a temperature-dependent lifetime of 462(10) ns at 4.4 K and 219(3) ns at 295 K. The light-polarization dependence of 1042 nm absorption confirms that the transition is between orbitals of A_1 and E character. The results shed new light on the NV level structure and optical pumping mechanism.

Parametric two-electron reduced-density-matrix (p-2RDM) methods have enjoyed much success in recent years; the methods have been shown to exhibit accuracies greater than coupled cluster with single and double substitutions (CCSD) for both closed- and open-shell ground-state energies, properties, geometric parameters, and harmonic frequencies. The class of methods is herein discussed within the context of the coupled electron pair approximation (CEPA), and several CEPA-like topological factors are presented for use within the p-2RDM framework. The resulting p-2RDM/n methods can be viewed as a density-based generalization of CEPA/n family that are numerically very similar to traditional CEPA methodologies. We cite the important distinction that the obtained energies represent stationary points, facilitating the efficient evaluation of properties and geometric derivatives. The p-2RDM/n formalism is generalized for an equal treatment of exclusion-principle-violating (EPV) diagrams that occur in the occupied and virtual spaces. One of these general topological factors is shown to be identical to that proposed by Kollmar [C. Kollmar, J. Chem. Phys. 125, 084108 (2006)], derived in an effort to approximately enforce the D, Q, and G conditions for N-representability in his size-extensive density matrix functional.

For the last decade quantum memories have been intensively studied for potential applications to quantum information and communications using atomic and ionic ensembles. With the importance of a multimode storage capability in quantum memories, on-demand control of reversible inhomogeneous broadening of an optical medium has been broadly investigated recently. However, the photon storage time in these researches is still too short to apply for long-distance quantum communications. In this paper, we demonstrate new physics of spin population decay dependent ultralong photon storage method, where spin population decay time is several orders of magnitude longer than the conventional constraint of spin phase decay time.

A Wigner crystal formed with trapped ion can undergo structural phase transition, which is determined only by the mechanical conditions on a classical level. Instead of this classical result, we show that through consideration of quantum and thermal fluctuation, a structural phase transition can be solely driven by change of the system's temperature. We determine a finite-temperature phase diagram for trapped ions using the renormalization group method and the path integral formalism, and propose an experimental scheme to observe the predicted temperature-driven structural phase transition, which is well within the reach of the current ion trap technology.

Results of the numerical Monte-Carlo simulations for the Stark-tuned F\"orster resonance and dipole blockade between 2 to 5 cold rubidium Rydberg atoms in various spatial configurations are presented. Effect of the atom spatial uncertainties on the resonance amplitude and spectrum is investigated. Feasibility to observe coherent Rabi-like population oscillations at a F\"orster resonance between two cold Rydberg atoms is analyzed. Spectra and fidelity of the Rydberg dipole blockade are calculated for various experimental conditions, including nonzero detuning from the F\"orster resonance and finite laser line width. The results are discussed in the context of quantum information processing with Rydberg atoms.

In this paper, the realignment criterion and the RCCN criterion of separability for states in infinite-dimensional bipartite quantum systems are established. Let $H_A$ and $H_B$ be complex Hilbert spaces with $\dim H_A\otimes H_B=+\infty$. Let $\rho$ be a state on $H_A\otimes H_B$ and $\{\delta_k\}$ be the Schmidt coefficients of $\rho$ as a vector in the Hilbert space ${\mathcal C}_2(H_A)\otimes{\mathcal C}_2(H_B)$. We introduce the realignment operation $\rho^R$ and the computable cross norm $\|\rho\|_{\rm CCN}$ of $\rho$ and show that, if $\rho$ is separable, then $\|\rho^{R}\|_{\rm Tr}=\|\rho\|_{\rm CCN}=\sum\limits_k\delta_k\leq1.$ In particular, if $\rho$ is a pure state, then $\rho$ is separable if and only if $\|\rho^{R}\|_{\rm Tr}=\|\rho\|_{\rm CCN}=\sum\limits_k\delta_k=1$.

The paper review and develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative $\star$-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The formalism of the phase space quantum mechanics is presented in a very systematic way for the case of Hamiltonian systems without any constrains and for a very wide class of deformations. The considered class of deformations and the corresponding $\star$-products contains all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of $\star$-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the $\star$-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. In addition, examples of a free particle and a simple harmonic oscillator illustrating the formalism of the deformation quantization and its classical limit are given.

The phenomenon of vacuum decay, i.e. electron-positron pair production due to the instability of the quantum electrodynamics vacuum in an external field, is a remarkable prediction of Dirac theory whose experimental observation is still lacking. Here a classic wave optics analogue of vacuum decay, based on light propagation in curved waveguide superlattices, is proposed. Our photonic analogue enables a simple and experimentally-accessible visualization in space of the process of pair production as break up of an initially negative-energy Gaussian wave packet, representing an electron in the Dirac sea, under the influence of an oscillating electric field.

We analyze in detail the heating of bosonic atoms in an optical lattice due to incoherent scattering of light from the lasers forming the lattice. Because atoms scattered into higher bands do not thermalize on the timescale of typical experiments, this process cannot be described by the total energy increase in the system alone (which is determined by single-particle effects). The heating instead involves an important interplay between the atomic physics of the heating process and the many-body physics of the state. We characterize the effects on many-body states for various system parameters, where we observe important differences in the heating for strongly and weakly interacting regimes, as well as a strong dependence on the sign of the laser detuning from the excited atomic state. We compute heating rates and changes to characteristic correlation functions based both on perturbation theory calculations, and a time-dependent calculation of the dissipative many-body dynamics. The latter is made possible for 1D systems by combining time-dependent density matrix renormalization group (t-DMRG) methods with quantum trajectory techniques.

One of the key features of quantum mechanics is the interference of probability amplitudes. The reason for the appearance of interference is mathematically very simple. It is the linear structure of the Hilbert space which is used for the description of quantum systems. In terms of physics we usually talk about the superposition principle valid for individual and composed quantum objects. So, while the source of interference is understandable it leads in fact to many counter-intuitive physical phenomena which puzzle physicists for almost hundred years. The present thesis studies interference in two seemingly disjoint fields of physics. However, both have strong links to quantum information processing and hence are related. In the first part we study the intriguing properties of quantum walks. In the second part we analyze a sophisticated application of wave packet dynamics in atoms and molecules for factorization of integers. The main body of the thesis is based on the original contributions listed separately at the end of the thesis. The more technical aspects and brief summaries of used methods are left for appendices.

We apply the notion of asymptotic iteration method (AIM) to determine eigenvalues of the bosonic Hamiltonians that include a wide class of quantum optical models. We consider solutions of the Hamiltonians, which are even polynomials of the fourth order with the respect to Boson operators. We also demonstrate applicability of the method for obtaining eigenvalues of the simple Lie algebraic structures. Eigenvalues of the multi-boson Hamiltonians have been obtained by transforming in the form of the single boson Hamiltonian in the framework of AIM.

An approximate method is proposed to solve position dependent mass Schr\"odinger equation. The procedure suggested here leads to the solution of the PDM Schr\"odinger equation without transforming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schr\"odinger equation. The results show that PDM and constant mass Schr\"odinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.

For decades, light has served as a useful tool in condensed matter physics, yet rarely has light itself been studied in this same framework. The reason that light has been relegated to a tool of condensed matter physics, rather than a subject, is that photons do not interact, and even mediated interactions are weak. Recently, several proposals have been set forth to study strongly correlated macroscopic systems with interacting photons or polaritons in arrays of cavities coupled to atoms or qubits. Here, we demonstrate a mediated photon-photon interaction that results in a non-resonant photon blockade using a single element of these lattices, a cavity coupled to a qubit. The blockade is characterized by measuring the total transmitted power in a fixed measurement bandwidth while varying the energy spectrum of the photons incident on the cavity. A staircase with four distinct steps emerges, which can be understood in analogy with electron transport and the Coulomb blockade in quantum dots. This work differs from previous efforts in that the cavity-qubit excitations retain a photonic nature rather than a hybridization of qubit and photon.

Non-Gaussian states represent a powerful resource for quantum information protocols in the continuous variables regime. Cat states, in particular, have been produced in the motional degree of freedom of trapped ions by controlled displacements dependent on the ionic internal state. An alternative method harnesses the Kerr nonlinearity naturally existent in this kind of system. We present detailed calculations confirming its feasibility for typical experimental conditions. Additionally, this method permits the generation of complex non-Gaussian states with negative Wigner functions. Especially, superpositions of many coherent states are achieved at a fraction of the time necessary to produce the cat state.

The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.

The Caldeira-Leggett Hamiltonian (Eq. (1) below) describes the interaction of a discrete harmonic oscillator with a continuous bath of harmonic oscillators. This system is a standard model of dissipation in macroscopic low temperature physics, and has applications to superconductors, quantum computing, and macroscopic quantum tunneling. The similarities between the Caldeira-Leggett model and the linearized Vlasov-Poisson equation are analyzed, and it is shown that the damping in the Caldeira-Leggett model is analogous to that of Landau damping in plasmas [1]. An invertible linear transformation [2, 3] is presented that converts solutions of the Caldeira-Leggett model into solutions of the linearized Vlasov-Poisson system.

We present exact formulas for the entanglement and R\'{e}nyi entropies generated at a quantum point contact (QPC) in terms of the statistics of charge fluctuations, which we illustrate with examples from both equilibrium and non-equilibrium transport. The formulas are also applicable to groundstate entanglement in systems described by non-interacting fermions in any dimension, which in one dimension includes the critical spin-1/2 XX and Ising models where conformal field theory predictions for the entanglement and R\'{e}nyi entropies are reproduced from the full counting statistics. These results may play a crucial role in the experimental detection of many-body entanglement in mesoscopic structures and cold atoms in optical lattices.

With an eye on developing a quantum theory of gravity, many physicists have recently searched for quantum challenges to the equivalence principle of general relativity. However, as historians and philosophers of science are well aware, the principle of equivalence is not so clear. When clarified, we think quantum tests of the equivalence principle won't yield much. The problem is that the clash/not-clash is either already evident or guaranteed not to exist. Nonetheless, this work does help teach us what it means for a theory to be geometric.

We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction nu=5/2. We complete the program started in Nucl. Phys. B 506, 685 (1997) and show that the degenerate four-quasihole and six-quasihole wavefunctions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c=1+1/2 conformal field theory. As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wavefunctions. Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators. We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wavefunctions for an arbitrary number of quasiholes. Our method also applies to the anti-Pfaffian wavefunction and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.

We study the excitation dynamics of an inhomogeneously broadened spin ensemble coupled to a single cavity mode. The collective mode coupled most strongly to the cavity acquires an energy shift which may be large enough to prevent its dephasing due to the inhomogeneity in the ensemble, while other collective modes evolve in a non-trivial manner due to the joint effect of the inhomogeneity and the coupling to the cavity. Rather than identifying stationary eigenmodes we define `bare time' modes, for which the dephasing due to inhomogeneities is described exactly as a linear translation. Interaction with the cavity mode `freezes' this translation of the strongly coupled spin mode, while other collective modes experience an additional translational shift as they propagate around the frozen mode. The result is relevant for multi-mode quantum memories where qubits are encoded in different spin waves.

Dynamical systems associated with a q-deformed two dimensional phase space are studied as effective dynamical systems described by ordinary variables. In quantum theory, the momentum operator in such a deformed phase space becomes a difference operator instead of the differential operator. Then, using the path integral representation for such a dynamical system, we derive an effective short-time action, which contains interaction terms even for a free particle with q-deformed phase space. Analysis is also made on the eigenvalue problem for a particle with q-deformed phase space confined in a compact space. Under some boundary conditions of the compact space, there arises fairly different structures from $q=1$ case in the energy spectrum of the particle and in the corresponding eigenspace .

We show how two qubits encoded in the orbital states of two quantum dots can be entangled or disentangled in a controlled way through their interaction with a weak electron current. The transmission/reflection spectrum of each scattered electron, acting as an entanglement mediator between the dots, shows a signature of the dot-dot entangled state. Strikingly, while few scattered carriers produce decoherence of the whole two-dots system, a larger number of electrons injected from one lead with proper energy is able to recover its quantum coherence. Our numerical simulations are based on a real-space solution of the three-particle Schroedinger equation with open boundaries. The computed transmission amplitudes are inserted in the analytical expression of the system density matrix in order to evaluate the entanglement.

We investigate the entanglement and nonlocality properties of one- and two-mode combination squeezed vacuum state (OTCSS, with two-parameter lamda and gamma) by analyzing the logarithmic negativity and the Bell's inequality. It is found that this state exhibits larger entanglement than that of the usual two-mode squeezed vacuum state (TSVS), and that in a certain regime of lamda, the violation of Bell's inequality becomes more obvious, which indicates that the nonlocality of OTCSS can be stronger than that of TSVS. As an application of OTCSS, the quantum teleportaion is examined, which shows that there is a region spanned by lamda and gamma in which the fidelity of OTCSS channel is larger than that of TSVS.

{\small We investigate nonclassical properties of the field states generated by subtracting any number photon from the squeezed thermal state (STS). It is found that the normalization factor of photon-subtracted STS (PSSTS) is a Legendre polynomial of squeezing parameter }${\small r}${\small \ and average photon number }$\bar{n}$ {\small of thermal state. Expressions of several quasi-probability distributions of PSSTS are derived analytically. Furthermore, the nonclassicality is discussed in terms of the negativity of Wigner function (WF). It is shown that the WF of single PSSTS always has negative values if }$\bar{n}<\sinh^{2}r${\small \ at the phase space center. The decoherence effect on PSSTS is then included by analytically deriving the time evolution of WF. The results show that the WF of single PSSTS has negative value if }$2\kappa t<\ln\{1-(2\bar{n}+1)(\bar{n}-\sinh^{2}% r)${\small }$[(2\mathfrak{N}+1)(\bar{n}\cosh2r+\sinh^{2}r)]\}${\small, which is dependent not only on average number }$\mathfrak{N}${\small \ of environment, but also on }$\bar{n}$ {\small and }$r${\small . }

The problems associated with practical implementation of the scheme proposed for preparation of arbitrary states of polarization ququarts based on biphotons are discussed. The influence of frequency dispersion effects are considered, and the necessity of group velocities dispersion compensation in the frequency non-degenerate case even for continuous pumping is demonstrated. A method for this compensation is proposed and implemented experimentally. Physical restrictions on the quality of prepared two-photon states are revealed.

In a recent work, Y.D. Chong et al. [Phys. Rev. Lett. {\bf 105}, 053901 (2010)] proposed the idea of a coherent perfect absorber (CPA) as the time-reversed counterpart of a laser, in which a purely incoming radiation pattern is completely absorbed by a lossy medium. The optical medium that realizes CPA is obtained by reversing the gain with absorption, and thus it generally differs from the lasing medium. Here it is shown that a laser with an optical medium that satisfies the parity-time $(\mathcal{PT})$ symmetry condition $\epsilon(-\mathbf{r})=\epsilon^*(\mathbf{r})$ for the dielectric constant behaves simultaneously as a laser oscillator (i.e. it can emit outgoing coherent waves) and as a CPA (i.e. it can fully absorb incoming coherent waves with appropriate amplitudes and phases). Such a device can be thus referred to as a $\mathcal{PT}$-symmetric CPA-laser. The general amplification/absorption features of the $\mathcal{PT}$ CPA-laser below lasing threshold driven by two fields are determined.

Light propagation in distributed feedback optical structures with gain/loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and related to well-known physical phenomena like resonance narrowing and laser oscillation.

Reflectionless defects in Hermitian tight-binding lattices, synthesized by the intertwining operator technique of supersymmetric quantum mechanics, are generally not invisible and time-of-flight measurements could reveal the existence of the defects. Here it is shown that, in a certain class of non-Hermitian tight-binding lattices with complex hopping amplitudes, defects in the lattice can appear fully invisible to an outside observer. The synthesized non-Hermitian lattices with invisible defects possess a real-valued energy spectrum, however they lack of parity-time (PT) symmetry, which does not play any role in the present work.

Entanglement charge is an operational measure to quantify nonlocalities in ensembles consisting of bipartite quantum states. Here we generalize this nonlocality measure to single bipartite quantum states. As an example, we analyze the entanglement charges of some thermal states of two-qubit systems and show how they depend on the temperature and the system parameters in an analytical way.

Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.

We compare the quantisation of linear systems of bosons and fermions. After stating the existing facts on bosons, we discuss the pre-quantisation and quantisation of fermions using calculus of fermionic variables. We then define a natural connection on the bundle of Hilbert spaces and show that it is projectively flat. This identifies, up to a phase, constructions of the spinor representation under various polarisations. We introduce the concept of metaplectic correction for fermions and show that the bundle of corrected Hilbert spaces is naturally flat. We then show that the parallel transport in the bundle of Hilbert spaces along a geodesic is the rescaled projection or the Bogoliubov transformation provided the geodesic lies within the complement of a cut locus. The decomposition of the bundle of Hilbert spaces when there is a symmetry is also studied.

The ability of fully reconstructing quantum maps is a fundamental task of quantum information, in particular when coupling with the environment and experimental imperfections of devices are taken into account. In this context we carry out a quantum process tomography (QPT) approach for a set of non trace-preserving maps. We introduce an operator $\OO$ to characterize the state dependent probability of success for the process under investigation. We also evaluate the result of approximating the process with a trace-preserving one.

The geometric measure, the logarithmic robustness and The relative entropy of entanglement are proved to be equal for a stabilizer quantum codeword. The entanglement upper and lower bounds are determined with the generators. The entanglement of self-dual CSS codes and Gottesman codes are given. An iterative algorithm is developed in order to determine the exact value of the entanglement.

We consider a generalization of the 2-dimensional (2D) quantum-Hall insulator to a non-compact, non-Abelian gauge group, the Heisenberg-Weyl group. We show that this kind of insulator is actually a layered 3D insulator with nontrivial topology. We further show that nontrivial combinations of quantized transverse conductivities can be engineered with the help of a staggered potential. We investigate the robustness and topological nature of this conductivity and connect it to the surface modes of the system. We also propose a very simple experimental realization with ultracold atoms in 3D confined to a 2D square lattice with the third dimension being mapped to a gauge coordinate.

When a neutral atom moves in a properly designed laser field, its center-of-mass motion may mimic the dynamics of a charged particle in a magnetic field, with the emergence of a Lorentz-like force. In this Colloquium we present the physical principles at the basis of this artificial (synthetic) magnetism and relate the corresponding Aharonov-Bohm phase to the Berry's phase that emerges when the atom follows adiabatically one of its dressed states. We also discuss some manifestations of artificial magnetism for a cold quantum gas, in particular in terms of vortex nucleation. We then generalise our analysis to the simulation of non-Abelian gauge potentials and present some striking consequences, such as the emergence of an effective spin-orbit coupling. We address both the case of bulk gases and discrete systems, where atoms are trapped in an optical lattice.

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on $SU(3)$ and its dimension is $8$, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.