There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

Physical Review B

Although known since the late 19th century, organic–inorganic perovskites have recently received extraordinary research community attention because of their unique physical properties, which make them promising candidates for application in photovoltaic (PV) and related optoelectronic devices. This review will explore beyond the current focus on three-dimensional (3-D) lead(II) halide perovskites, to highlight the great chemical flexibility and outstanding potential of the broader class of 3-D and lower dimensional organic-based perovskite family for electronic, optical, and energy-based applications as well as fundamental research. The concept of a multifunctional organic–inorganic hybrid, in which the organic and inorganic structural components provide intentional, unique, and hopefully synergistic features to the compound, represents an important contemporary target.

Organic–Inorganic Perovskites: Structural Versatility for Functional Materials Design

Recently a new type of system exhibiting spontaneous coherence has emerged—the exciton–polariton condensate. Exciton–polaritons (or polaritons for short) are bosonic quasiparticles that exist inside semiconductor microcavities, consisting of a superposition of an exciton and a cavity photon. Above a threshold density the polaritons macroscopically occupy the same quantum state, forming a condensate. The polaritons have a lifetime that is typically comparable to or shorter than thermalization times, giving them an inherently non-equilibrium nature. Nevertheless, they exhibit many of the features that would be expected of equilibrium Bose–Einstein condensates (BECs). The non-equilibrium nature of the system raises fundamental questions as to what it means for a system to be a BEC, and introduces new physics beyond that seen in other macroscopically coherent systems. In this review we focus on several physical phenomena exhibited by exciton–polariton condensates. In particular, we examine topics such as the difference between a polariton BEC, a polariton laser and a photon laser, as well as physical phenomena such as superfluidity, vortex formation, and Berezinskii–Kosterlitz–Thouless and Bardeen–Cooper–Schrieffer physics. We also discuss the physics and applications of engineered polariton structures. Exciton–polaritons, resulting from the light–matter coupling between an exciton and a photon in a cavity, form Bose–Einstein-like condensates above a critical density. Various aspects of the physics of exciton–polariton condensates are now reviewed.

Exciton–polariton condensates

A microcavity structure to which a periodic potential is applied has been designed, which effectively creates an array of weakly-coupled condensates. This allows the observation of fundamental dynamic behaviour, namely the build-up of certain superfluid-like states, which has been predicted for arrays of atomic Bose–Einstein condensates, but not yet observed. The effect of quantum statistics in quantum gases and liquids results in observable collective properties among many-particle systems. One prime example is Bose–Einstein condensation, whose onset in a quantum liquid leads to phenomena such as superfluidity and superconductivity. A Bose–Einstein condensate is generally defined as a macroscopic occupation of a single-particle quantum state, a phenomenon technically referred to as off-diagonal long-range order due to non-vanishing off-diagonal components of the single-particle density matrix1,2,3. The wavefunction of the condensate is an order parameter whose phase is essential in characterizing the coherence and superfluid phenomena4,5,6,7,8,9,10,11. The long-range spatial coherence leads to the existence of phase-locked multiple condensates in an array of superfluid helium12, superconducting Josephson junctions13,14,15 or atomic Bose–Einstein condensates15,16,17,18. Under certain circumstances, a quantum phase difference of π is predicted to develop among weakly coupled Josephson junctions19. Such a meta-stable π-state was discovered in a weak link of superfluid 3He, which is characterized by a ‘p-wave’ order parameter20. The possible existence of such a π-state in weakly coupled atomic Bose–Einstein condensates has also been proposed21, but remains undiscovered. Here we report the observation of spontaneous build-up of in-phase (‘zero-state’) and antiphase (‘π-state’) ‘superfluid’ states in a solid-state system; an array of exciton–polariton condensates connected by weak periodic potential barriers within a semiconductor microcavity. These in-phase and antiphase states reflect the band structure of the one-dimensional polariton array and the dynamic characteristics of metastable exciton–polariton condensates.

Coherent zero-state and π-state in an exciton–polariton condensate array

We examine the problem of simulating lattice gauge theories on a universal quantum computer. The basic strategy of our approach is to transcribe lattice gauge theories in the Hamiltonian formulation into a Hamiltonian involving only Pauli spin operators such that the simulation can be performed on a quantum computer using only one- and two-qubit manipulations. We examine three models, the U(1), SU(2), and SU(3) lattice gauge theories, which are transcribed into a spin Hamiltonian up to a cutoff in the Hilbert space of the gauge fields on the lattice. The number of qubits required for storing a particular state is found to have a linear dependence on the total number of lattice sites. The number of qubit operations required for performing the time evolution corresponding to the Hamiltonian is found to be between a linear to quadratic function of the number of lattice sites, depending on the arrangement of qubits in the quantum computer. We remark that our results may also be easily generalized to higher $\mathrm{SU}(N)$ gauge theories.

/pdf/simulating-lattice-gauge-theories-on-a-quantum-computer-45hpxd3h50.pdf

Simulating lattice gauge theories on a quantum computer

The massive Schwinger model is studied using a density matrix renormalization group approach to the staggered lattice Hamiltonian version of the model. Lattice sizes up to 256 sites are calculated, and the estimates in the continuum limit are almost two orders of magnitude more accurate than previous calculations. Coleman's picture of ``half-asymptotic'' particles at a background field $\ensuremath{\theta}=\ensuremath{\pi}$ is confirmed. The predicted phase transition at finite fermion mass $(m/g)$ is accurately located and demonstrated to belong in the 2D Ising universality class.

Density matrix renormalization group approach to the massive Schwinger model

We analyze quantum computation using quantum information stored on two component Bose-Einstein condensates (BECs). We construct a general framework for quantum algorithms to be executed using the collective states of the BECs. The use of BECs allows for an increase of energy scales via bosonic enhancement, resulting in two-qubit gate operations that can be performed at a time reduced by a factor of $N$, where $N$ is the number of bosons in the BEC. We illustrate the scheme by an application to Grover's algorithm, and discuss possible experimental implementations and decoherence effects.

Tim Byrnes

Papers

Exciton–polariton condensates

Coherent zero-state and π-state in an exciton–polariton condensate array

Simulating lattice gauge theories on a quantum computer

Density matrix renormalization group approach to the massive Schwinger model

Macroscopic quantum computation using Bose-Einstein condensates