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

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

Coherent preparation by laser light of quantum states of atoms and molecules can lead to quantum interference in the amplitudes of optical transitions. In this way the optical properties of a medium can be dramatically modified, leading to electromagnetically induced transparency and related effects, which have placed gas-phase systems at the center of recent advances in the development of media with radically new optical properties. This article reviews these advances and the new possibilities they offer for nonlinear optics and quantum information science. As a basis for the theory of electromagnetically induced transparency the authors consider the atomic dynamics and the optical response of the medium to a continuous-wave laser. They then discuss pulse propagation and the adiabatic evolution of field-coupled states and show how coherently prepared media can be used to improve frequency conversion in nonlinear optical mixing experiments. The extension of these concepts to very weak optical fields in the few-photon limit is then examined. The review concludes with a discussion of future prospects and potential new applications.

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Electromagnetically induced transparency : Optics in coherent media

Techniques that use quantum interference effects are being actively investigated to manipulate the optical properties of quantum systems1. One such example is electromagnetically induced transparency, a quantum effect that permits the propagation of light pulses through an otherwise opaque medium2,3,4,5. Here we report an experimental demonstration of electromagnetically induced transparency in an ultracold gas of sodium atoms, in which the optical pulses propagate at twenty million times slower than the speed of light in a vacuum. The gas is cooled to nanokelvin temperatures by laser and evaporative cooling6,7,8,9,10. The quantum interference controlling the optical properties of the medium is set up by a ‘coupling’ laser beam propagating at a right angle to the pulsed ‘probe’ beam. At nanokelvin temperatures, the variation of refractive index with probe frequency can be made very steep. In conjunction with the high atomic density, this results in the exceptionally low light speeds observed. By cooling the cloud below the transition temperature for Bose–Einstein condensation11,12,13 (causing a macroscopic population of alkali atoms in the quantum ground state of the confining potential), we observe even lower pulse propagation velocities (17?m?s−1) owing to the increased atom density. We report an inferred nonlinear refractive index of 0.18?cm2?W−1 and find that the system shows exceptionally large optical nonlinearities, which are of potential fundamental and technological interest for quantum optics.

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Light speed reduction to 17 metres per second in an ultracold atomic gas

Electromagnetically induced transparency is a technique for eliminating the effect of a medium on a propagating beam of electromagnetic radiation EIT may also be used, but under more limited conditions, to eliminate optical self‐focusing and defocusing and to improve the transmission of laser beams through inhomogeneous refracting gases and metal vapors, as figure 1 illustrates The technique may be used to create large populations of coherently driven uniformly phased atoms, thereby making possible new types of optoelectronic devices

Electromagnetically Induced Transparency

The familiar Rabi model, comprising a two-level system coupled to a quantum harmonic oscillator, continues to produce rich and surprising physics when the coupling strength becomes comparable to the individual subsystem frequencies. We construct approximate solutions for the eigenstates and energies in the regime in which the oscillator frequency is small compared to that of the two-level system and the coupling strength matches or exceeds the oscillator frequency. The resulting oscillator dynamics closely resembles that of a particle tunneling in a classical double-well potential. Relating our calculation to an earlier semiclassical approximation in which coupling to the two-level system creates an effective potential for the oscillator, we examine the extent to which this picture is valid. We find that, for certain parameter regimes, the interpretation of the oscillator dynamics in terms of tunneling holds to a good approximation despite the fundamentally entangled nature of the joint system. We assess the prospects for observation of oscillator tunneling in the context of nano- or micromechanical experiments and find that it should be possible if suitably high coupling strengths can be engineered.

https://absimage.aps.org/image/MAR14/MWS_MAR14-2013-004929.pdf

Oscillator tunneling dynamics in the Rabi model

We use a quantum-trajectory based method (a generalization of earlier work by Mollow) to derive results for the population inversion and atomic dipole of a single atom interacting with a cavity field which is allowed to decay via fluorescence to outside modes. We assume strong coupling to the cavity field (small fluorescence rate), and that the cavity field is initially prepared in a `quasiclassical' state, such as a coherent state with a large (>10) number of photons. We show how a quasiclassical, or `branching' approximation can be introduced under these conditions which simplifies considerably the formal solutions obtained by the quantum-trajectory method and allows one to calculate easily the effect of spontaneous emission on the Jaynes - Cummings revivals and the state preparation effect. Our analytical approximations are compared to the numerical integration of the master equation, with generally good agreement.

Analytic quantum-trajectory results for a fluorescent atom in a lossless cavity

Influence of phase fluctuations on the measurement of the frequency of a laser

We have studied the performance of a geometric phase gate with a quantized driving eld numerically, anddeveloped an analytical approximation that yields some preliminary insight on the way the qubit becomesentangled with the driving eld.Keywords: Quantum computation, adiabatic quantum gates, geometric quantum gates 1. INTRODUCTION It was rst suggested by Zanardi and Rasetti, 1 that the Berry phase (non-abelian holonomy) 24 might in principleprovide a novel way for implementing universal quantum computation. They showed that by encoding quantuminformation in one of the eigenspaces of a degenerate Hamiltonian H one can in principle achieve the fullquantum computational power by using holonomies only. It was then thought that since Berrys phase is apurely geometrical eect, it is resilient to certain errors and may provide a possibility for performing intrinsicallyfault-tolerant quantum gate operations. In a paper by Ekert et.al, 5 a detailed theory behind the implementationof geometric computation was developed and an implementation of a conditional phase gate in NMR was shownby Jones et.al.

Geometric phase gate with a quantized driving field

We present a method to protect the entangled states of distant particles against decoherence due to local (but collective) phase errors, and local exchange-type interactions, by pairing up the entangled particles. The method is based on a four-qubit code which forms a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs. We also show how the scheme can be generalized to protect certain entangled states of more than two particles.

Julio Gea-Banacloche

Papers

Oscillator tunneling dynamics in the Rabi model

Analytic quantum-trajectory results for a fluorescent atom in a lossless cavity

Influence of phase fluctuations on the measurement of the frequency of a laser

Geometric phase gate with a quantized driving field

A method to protect quantum entanglement against certain kinds of phase and exchange errors