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Motivated by the contemporary advances in quantum implementation of non-unitary operations, we propose a new dilation method based on the biorthogonal representation of the non-unitary operator, mapping it to an isomorphic unitary matrix in the orthonormal computational basis. The proposed method excels in implementing non-unitary operators whose eigenvalues have absolute values exceeding one, when compared to other dilation and decomposition techniques. Unlike the Linear Combination of Unitaries (LCU) method, which becomes less efficient as the number of unitary summands grows, the proposed technique is optimal for small-dimensional non-unitary operators regardless of the number of unitary summands. Thus, it can complement the LCU method for implementing general non-unitary operators arising in positive only open quantum systems and pseudo-Hermitian systems., Comment: 8 Pages, 3 Figures

Electromagnetic waves are an inherent part of all plasmas -- laboratory fusion plasmas or astrophysical plasmas. The conventional methods for studying properties of electromagnetic waves rely on discretization of Maxwell equations suitable for implementing on classical, present day, computers. The traditional methodology is not efficient for quantum computing implementation -- a future computational source offering a tantalizing possibility of enormous speed up and a significant reduction in computational cost. This paper addresses two topics relevant to implementing Maxwell equations on a quantum computer. The first is on formulating a quantum Schrodinger representation of Maxwell equations for wave propagation in a cold, inhomogeneous, magnetized plasma. This representation admits unitary, energy preserving, evolution and conveniently lends itself to appropriate discretization for a quantum computer. Riding on the coattails of these results, the second topic is on developing a sequence of unitary operators which form the basis for a qubit lattice algorithm (QLA). The QLA, suitable for quantum computers, can be implemented and tested on existing classical computers for accuracy as well as scaling of computational time with the number of available processors. In order to illustrate the QLA for Maxwell equations, results are presented from a time evolving, full wave simulation of propagation and scattering of an electromagnetic wave packet by non-dispersive dielectric medium localized in space., Comment: 13 Pages, 4 Figures A complexity analysis on the quantum implementation of QLA has been added along with a relocation of the sub-sections for coherence purposes

It is well known that Maxwell equations can be expressed in a unitary Schrodinger-Dirac representation for homogeneous media. However, difficulties arise when considering inhomogeneous media. A Dyson map points to a unitary field qubit basis, but the standard qubit lattice algorithm of interleaved unitary collision-stream operators must be augmented by some sparse non-unitary potential operators that recover the derivatives on the refractive indices. The effect of the steepness of these derivatives on two dimensional scattering is examined with simulations showing quite complex wavefronts emitted due to transmissions/reflections within the dielectric objects. Maxwell equations are extended to handle dissipation using Kraus operators. Then, our theoretical algorithms are extended to these open quantum systems. A quantum circuit diagram is presented as well as estimates on the required number of quantum gates for implementation on a quantum computer., Comment: 20 pages, 14 figures

In dispersive media, dissipation appears in the Schr\"odinger representation of classical Maxwell equations as a sparse diagonal operator occupying an $r$-dimensional subspace. A first order Suzuki-Trotter approximation for the evolution operator enables us to isolate the non-unitary operators (associated with dissipation) from the unitary operators (associated with lossless media). The unitary operators can be implemented through qubit lattice algorithm (QLA) on $n$ qubits. However, the non-unitary-dissipative part poses a challenge on how it should be implemented on a quantum computer. In this paper, two probabilistic dilation algorithms are considered for handling the dissipative operators. The first algorithm is based on treating the classical dissipation as a linear amplitude damping-type completely positive trace preserving (CPTP) quantum channel where the combined system-environment must undergo unitary evolution in the dilated space. The unspecified environment can be modeled by just one ancillary qubit, resulting in an implementation scaling of $\textit{O}(2^{n-1}n^2)$ elementary gates for the dilated unitary evolution operator. The second algorithm approximates the non-unitary operators by the Linear Combination of Unitaries (LCU). We obtain an optimized representation of the non-unitary part, which requires $\textit{O}(2^{n})$ elementary gates. Applying the LCU method for a simple dielectric medium with homogeneous dissipation rate, the implementation scaling can be further reduced into $\textit{O}[poly(n)]$ basic gates. For the particular case of weak dissipation we show that our proposed post-selective dilation algorithms can efficiently delve into the transient evolution dynamics of dissipative systems by calculating the respective implementation circuit depth. A connection of our results with the non-linear-in-normalization-only (NINO) quantum channels is also presented., Comment: 10 pages, 2 Figures. New material has been added pertinent to non-linear PTP quantum channels and a detailed elaboration on the different algorithmic steps with an emphasis on the role of post-selection and the physical implications

$\bf{Abstract}$: A qubit lattice algorithm (QLA), which consists of a set of interleaved unitary collision-streaming operators, is developed for electromagnetic wave propagation in tensor dielectric media. External potential operators are required to handle gradients in the refractive indices, and these operators are typically non-unitary. A similar problem arises in the QLA for the Korteweg-de Vries equation, as the potential operator that models the KdV nonlinear term is also non-unitary. Several QLAs are presented here that avoid the need of this non-unitary potential operator by perturbing the collision operator. These QLAs are fully unitary., Comment: 10 pages, 0 figures

A reformulation of Maxwell equations for an inhomogeneous, anisotropic, passive and non-dispersive medium results in a quantum-like Dirac equation that admits unitary time evolution. In contrast to other approaches, there is no a-priori introduction of the Riemann-Silberstein-Weber (RSW) vector but the Maxwell equations are considered in their standard fields, with given constitutive relations. From the electromagnetic conservation quantities a pseudo-Hermitian dynamics is found together with a Dyson map that recovers the full Hermicity of the dynamics in an extended Hilbert space that describes the physical notion of unitary evolution. As an example, a uniaxial tensor dielectric medium is considered, with the explicit Dyson map yielding an optimal representation in a set of generalized RSW vectors. In this newly discovered form, a Quantum Computing (QC) implementation for simulation of electromagnetic wave propagation in complex media can be made, with further extension into plasmas., Comment: 12 pages, 5 figures

A quantum lattice algorithm (QLA) is developed for the solution of Maxwell equations in scalar dielectric media using the Riemann-Silberstein representation. For x-dependent and y-dependent inhomogeneities, the corresponding QLA requries 8 qubits/spatial lattice site. This is because the corresponding Pauli spin matrices have off-diagonal components which permit the collisional entanglement of two qubits. However, z-dependent inhomogeneities require a QLA with 16 qubits/lattice site since the Pauli spin matrix $\sigma_z$ is diagonal. QLA simulations are performed for the time evolution of an initial electromagnetic pulse propagating normally to a boundary layer region joining two media of different refractive index. There is excellent agreement between all three representations, as well as very good agreement with nearly all the standard plane wave boundary condition results for reflection and transmission off a dielectric discontinuity. In the QLA simulation, no boundary conditions are imposed at the continuous, but sharply increasing, boundary layer., Comment: 22 pages, 17 figures

Quantum vortex structures and energy cascades are examined for two dimensional quantum turbulence (2D QT) at zero temperature. A special unitary evolution algorithm, the quantum lattice gas (QLG) algorithm, is employed to simulate the Bose-Einstein condensate (BEC) governed by the Gross-Pitaevskii (GP) equation. A parameter regime is uncovered in which, as in 3D QT, there is a short Poincar\'e recurrence time. It is demonstrated that such short recurrence times are destroyed as the nonlinear interaction is strengthened. The similar loss of Poincar\'e recurrence is also reported in 3D QT [1] Energy cascades for 2D QT are considered to examine whether 2D QT exhibits inverse cascades as in 2D classical turbulence. In the parameter regime considered, the spectra analysis reveals no such dual cascades-dual cascades being a hallmark of 2D classical turbulence.

The evolution of the ground state wave function of a zero-temperature Bose-Einstein condensate (BEC) is well described by the Hamiltonian Gross-Pitaevskii (GP) equation. Using a set of appropriately interleaved unitary collision-streaming operators, a quantum lattice gas algorithm is devised which on taking moments recovers the Gross-Pitaevskii (GP) equation in diffusion ordering (time scales as square of length). Unexpectedly, there is a class of initial conditions in which their Poincare recurrence is extremely short. Further it is shown that the Poincare recurrence time scales with diffusion ordering as the the grid is increased. The spectral results of Yepez et.al. [1] for quantum turbulence are corrected and it is found that it is the compressible kinetic energy spectrum that exhibits the 3 cascade regions: a small k classical Kolmogorov k^(-5/3) spectrum, a steep semi-classical cascade region, and a large k quantum Kelvin wave cascade k^(-3) spectrum. The incompressible kinetic energy spectrum exhibits basically a single cascade power law of k^(-3). For winding number 1 linear vortices it is also shown that there is an intermittent loss of Kelvin wave cascade with its signature seen in the time evolution of the kinetic energy, the loss of the k^(-3) spectrum in the incompressible kinetic energy spectrum as well as the minimization of the vortex core isosurfaces that inhibits the Kelvin wave cascade.

The dynamics of vortex solitons in a BEC superfluid is studied. A quantum lattice-gas algorithm (localization-based quantum computation) is employed to examine the dynamical behavior of vortex soliton solutions of the Gross-Pitaevskii equation (phi^4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales a k^{-6} power law emerges, in a classical-quantum transition from vortex filament reconnections to Kelvin wave-acoustic wave coupling. The spontaneous exchange of intermediate vortex rings is observed. Finally, at very small spatial scales a k^{-3} power law emerges, characterizing fluid dynamics occurring within the scale size of the vortex cores themselves, a characteristic Kelvin wave cascade region. Poincare recurrence is studied: in the free non-interacting system, a fast Poincare recurrence occurs for regular arrays of line vortices. The recurrence period is used to demarcate dynamics driving the nonlinear quantum fluid towards turbulence, since fast recurrence is an approximate symmetry of the nonlinear quantum fluid at early times. This class of quantum algorithms is useful for studying BEC superfluid dynamics over a broad range of wave numbers, from quantum flow to a pseudo-classical inviscid flow regime to a Kolmogorov inertial subrange., Comment: 10 pages, 6 figures

Modern research in optical physics has achieved quantum control of strong interactions between a single atom and one photon within the setting of cavity quantum electrodynamics (cQED). However, to move beyond current proof-of-principle experiments involving one or two conventional optical cavities to more complex scalable systems that employ N >> 1 microscopic resonators requires the localization of individual atoms on distance scales < 100 nm from a resonator's surface. In this regime an atom can be strongly coupled to a single intracavity photon while at the same time experiencing significant radiative interactions with the dielectric boundaries of the resonator. Here, we report an initial step into this new regime of cQED by way of real-time detection and high-bandwidth feedback to select and monitor single Cesium atoms localized ~100 nm from the surface of a micro-toroidal optical resonator. We employ strong radiative interactions of atom and cavity field to probe atomic motion through the evanescent field of the resonator. Direct temporal and spectral measurements reveal both the significant role of Casimir-Polder attraction and the manifestly quantum nature of the atom-cavity dynamics. Our work sets the stage for trapping atoms near micro- and nano-scopic optical resonators for applications in quantum information science, including the creation of scalable quantum networks composed of many atom-cavity systems that coherently interact via coherent exchanges of single photons., Comment: 8 pages, 5 figures, Supplemental Information included as ancillary file

The phonon analog of an optical laser has long been a subject of interest. We demonstrate a compound microcavity system, coupled to a radio-frequency mechanical mode, that operates in close analogy to a two-level laser system. An inversion produces gain, causing phonon laser action above a pump power threshold of around 50 $\mu$W. The device features a continuously tunable, gain spectrum to selectively amplify mechanical modes from radio frequency to microwave rates. Viewed as a Brillouin process, the system accesses a regime in which the phonon plays what has traditionally been the role of the Stokes wave. For this reason, it should also be possible to controllably switch between phonon and photon laser regimes. Cooling of the mechanical mode is also possible., Comment: 4 pages, 4 figures

Single photons from a coherent input are efficiently redirected to a separate output by way of a fiber-coupled microtoroidal cavity interacting with individual Cesium atoms. By operating in an overcoupled regime for the input-output to a tapered fiber, our system functions as a quantum router with high efficiency for photon sorting. Single photons are reflected and excess photons transmitted, as confirmed by observations of photon antibunching (bunching) for the reflected (transmitted) light. Our photon router is robust against large variations of atomic position and input power, with the observed photon antibunching persisting for intracavity photon number 0.03 \lesssim n \lesssim 0.7.

Over the past decade, strong interactions of light and matter at the single-photon level have enabled a wide set of scientific advances in quantum optics and quantum information science. This work has been performed principally within the setting of cavity quantum electrodynamics with diverse physical systems, including single atoms in Fabry-Perot resonators, quantum dots coupled to micropillars and photonic bandgap cavities, and Cooper-pairs interacting with superconducting resonators. Experiments with single, localized atoms have been at the forefront of these advances with the use of optical resonators in high-finesse Fabry-Perot configurations. As a result of the extreme technical challenges involved in further improving the multilayer dielectric mirror coatings of these resonators and in scaling to large numbers of devices, there has been increased interest in the development of alternative microcavity systems. Here we show strong coupling between individual Cesium atoms and the fields of a high-quality toroidal microresonator. From observations of transit events for single atoms falling through the resonator's evanescent field, we determine the coherent coupling rate for interactions near the surface of the resonator. We develop a theoretical model to quantify our observations, demonstrating that strong coupling is achieved, with the rate of coherent coupling exceeding the dissipative rates of the atom and the cavity. Our work opens the way for investigations of optical processes with single atoms and photons in lithographically fabricated microresonators. Applications include the implementation of quantum networks, scalable quantum logic with photons, and quantum information processing on atom chips.

We investigate the suitability of toroidal microcavities for strong-coupling cavity quantum electrodynamics (QED). Numerical modeling of the optical modes demonstrate a significant reduction of modal volume with respect to the whispering gallery modes of dielectric spheres, while retaining the high quality factors representative of spherical cavities. The extra degree of freedom of toroid microcavities can be used to achieve improved cavity QED characteristics. Numerical results for atom-cavity coupling strength, critical atom number N_0 and critical photon number n_0 for cesium are calculated and shown to exceed values currently possible using Fabry-Perot cavities. Modeling predicts coupling rates g/(2*pi) exceeding 700 MHz and critical atom numbers approaching 10^{-7} in optimized structures. Furthermore, preliminary experimental measurements of toroidal cavities at a wavelength of 852 nm indicate that quality factors in excess of 100 million can be obtained in a 50 micron principal diameter cavity, which would result in strong coupling values of (g/(2*pi),n_0,N_0)=(86 MHz,4.6*10^{-4},1.0*10^{-3})., Comment: 11 pages, 10 figures