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

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Structured light refers to the generation and application of custom light fields. As the tools and technology to create and detect structured light have evolved, steadily the applications have begun to emerge. This roadmap touches on the key fields within structured light from the perspective of experts in those areas, providing insight into the current state and the challenges their respective fields face. Collectively the roadmap outlines the venerable nature of structured light research and the exciting prospects for the future that are yet to be realized.

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Roadmap on structured light

Optical resonance and two-level atoms

We describe a simple technique for generating a cold-atom lattice pierced by a uniform magnetic field. Our method is to extend a one-dimensional optical lattice into the \"dimension\" provided by the internal atomic degrees of freedom, yielding a synthetic two-dimensional lattice. Suitable laser coupling between these internal states leads to a uniform magnetic flux within the two-dimensional lattice. We show that this setup reproduces the main features of magnetic lattice systems, such as the fractal Hofstadter-butterfly spectrum and the chiral edge states of the associated Chern insulating phases.

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Synthetic gauge fields in synthetic dimensions

We observe coherent, cw, 455 nm blue-beam production via frequency upconversion in cesium vapor. Two IR lasers induce strong double excitation in a heated cesium vapor cell, allowing the atoms to undergo a double cascade and produce a coherent, collimated, blue beam copropagating with the two IR pump lasers.

Coherent 455 nm beam production in a cesium vapor

Phonon lasers are mechanical analogues of the ubiquitous optical laser and have been realized in a variety of contexts1–12. However, no such demonstration exists for mesoscopic levitated optomechanical systems, which are emerging as important platforms for conducting fundamental tests of quantum mechanics13–15 and gravity16, as well as for developing sensing modalities that couple mechanical motion to electron spin17–20 and charge21. Inspired by the pioneering work of Arthur Ashkin on optical tweezers22,23, we introduce a mesoscopic, frequency-tunable phonon laser based on the centre-of-mass oscillation of a silica nanosphere levitated in an optical tweezer under vacuum. Unlike previous levitated realizations, our scheme is general enough to be used on single electrons, liquid droplets or even small biological organisms24. Our device thus provides a pathway for a coherent source of phonons on the mesoscale that can be applied to both fundamental problems in quantum mechanics as well as tasks of precision metrology25–27. A phonon laser based on an optically levitated silica nanosphere is demonstrated. A lasing threshold—a phase transition from Brownian motion to coherent oscillation—is observed when the modulation depth of the trapping beam power is increased.

An optical tweezer phonon laser

We observe coherent, continuous wave, 455nm blue beam production via frequency up-conversion in cesium vapor. Two infrared lasers induce strong double-excitation in a heated cesium vapor cell, allowing the atoms to undergo a double cascade and produce a coherent, collimated, blue beam co-propagating with the two infrared pump lasers.

Coherent 455nm beam production in cesium vapor

Spinor Bose–Einstein condensates (BECs) and singular optical systems have both recently served as sandboxes to create and study analogs of phenomena from other fields of physics that are otherwise difficult to create and control experimentally. Interfacing singular optics and spinor BECs allows us to take advantage of and build on the foundations of singular optics to create and describe complex spin textures in BECs that serve as analogs of other systems. Here, the complete BEC wavefunctions are precisely engineered via a two-photon Raman interaction to contain π-symmetric (lemon, star) or 2π-symmetric (saddle, spiral) C-point singularities. The optical Raman beams are singular optical beams that contain these singularities and transfer them to the condensate, thereby creating vector-vortex spin textures—the spinor counterparts to scalar vortices—in pseudo-spin-1/2 BECs. With a version of atom-optic polarimetry, we can measure the Stokes parameters of the atomic cloud and characterize the singularities by the patterns present in their ellipse fields or by the C-point index. In the low density limit, these spin textures are analogs of optical vector vortices and should have dynamics driven by a matter-wave Gouy phase. With precise tuning of Raman beam parameters, we can create full Bloch BECs that contain every possible superposition between two states in the atomic cloud. Full Bloch BECs are similar to topologically stable magnetic skyrmions such as those created in thin metal films and nanowires, which may prove useful for atom-spintronics and topological quantum processes.

Singular atom optics with spinor Bose-Einstein condensates

We demonstrate a waveplate for a pseudo-spin-1/2 Bose–Einstein condensate (BEC) using a two-photon Raman interaction. The angle of the waveplate is set by the relative phase of the optical fields, and the retardance is controlled by the pulse area. The waveplate allows us to image maps of the Stokes parameters of a BEC and thereby measure its relative ground-state phase. We demonstrate the waveplate by measuring the Stokes parameters of a coreless vortex.

Justin T. Schultz

Papers

Coherent 455 nm beam production in a cesium vapor

An optical tweezer phonon laser

Coherent 455nm beam production in cesium vapor

Singular atom optics with spinor Bose-Einstein condensates

A Raman waveplate for spinor Bose–Einstein condensates