Achilles D. Speliotopoulos

Towards MIGO, the matter-wave interferometric gravitational-wave observatory, and the intersection of quantum mechanics with general relativity

Abstract A dynamical non-Euclidean spacetime geometry in general relativity theory implies the possibility of gravitational radiation. Here we explore novel methods of detecting such radiation from astrophysical sources by means of matter-wave interferometers (MIGOs), using atomic beams emanating from supersonic atomic sources that are further cooled and collimated by means of optical molasses. While the sensitivities of such MIGOs compare favorably with LIGO and LISA, the sizes of MIGOs can be orders of magnitude smaller, and their bandwidths wider. Using a pedagogical approach, we place this problem into the broader context of problems at the intersection of quantum mechanics with general relativity.

The general structure of eigenvalues in nonlinear oscillators

Hilbert spaces of bounded one-dimensional nonlinear oscillators are studied. It is shown that the eigenvalue structure of all such oscillators have the same general form. They depend only on the ground state energy of the system and a single function λ(H) of the Hamiltonian operator H. It is also found that the Hilbert space of the nonlinear oscillator is unitarily inequivalent to the Hilbert space of the simple harmonic oscillator, providing an explicit example of Haags theorem. A number operator for the nonlinear oscillator is constructed and the general form of the partition function and average energy of a nonlinear oscillator in contact with a heat bath is determined. Connection with the WKB result in the semiclassical limit is made. The analysis is then applied to the case of the x4 anharmonic oscillator as an explicit example.

Coupling of Linearized Gravity to Nonrelativistic Test Particles: Dynamics in the General Laboratory Frame

The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of the couplings, and the known results of tidal effects on test particles according to classical general relativity are pointed out. As a step towards resolving these inconsistencies, a General Laboratory Frame fixed on the worldline of an observer is constructed. In this frame, the dynamics of nonrelativistic test particles in the linearized gravity limit is studied, and their Hamiltonian dynamics is derived. It is shown that for stationary metrics this Hamiltonian reduces to the usual Hamiltonian for nonrelativistic particles undergoing geodesic motion. For nonstationary metrics with long-wavelength gravitational waves (GWs) present, it reduces to the Hamiltonian for a nonrelativistic particle undergoing geodesic deviation motion. Arbitrary-wavelength GWs couple to the test particle through a vector-potential-like field Na, the net result of the tidal forces that the GW induces in the system, namely, a local velocity field on the system induced by tidal effects as seen by an observer in the general laboratory frame. Effective electric and magnetic fields, which are related to the electric and magnetic parts of the Weyl tensor, are constructed from Na that obey equations of the same form as Maxwell’s equations . A gedankin gravitational Aharonov-Bohm-type experiment using Na to measure the interference of quantum test particles is presented.

Quantum Interference to Measure Spacetime Curvature: A Proposed Experiment at the Intersection of Quantum Mechanics and General Relativity

An experiment in Low Earth Orbit (LEO) is proposed to measure components of the Riemann curvature tensor using atom interferometry. We show that the difference in the quantum phase Δϕ of an atom that can travel along two intersecting geodesics is given by mR0i0j/ℏ times the spacetime volume contained within the geodesics. Our expression for Δϕ also holds for gravitational waves in the long wavelength limit.

A topological string: the Rasetti Regge Lagrangian, topological quantum field theory and vortices in quantum fluids

The kinetic part of the Rasetti-Regge action I_{RR} for vortex lines is studied and links to string theory are made. It is shown that both I_{RR} and the Polyakov string action I_{Pol} can be constructed with the same field X^mu. Unlike I_{NG}, however, I_{RR} describes a Schwarz-type topological quantum field theory. Using generators of classical Lie algebras, I_{RR} is generalized to higher dimensions. In all dimensions, the momentum 1-form P constructed from the canonical momentum for the vortex belongs to the first cohomology class H^1(M,R^m) of the worldsheet M swept-out by the vortex line. The dynamics of the vortex line thus depend directly on the topology of M. For a vortex ring, the equations of motion reduce to the Serret-Frenet equations in R^3, and in higher dimensions they reduce to the Maurer-Cartan equations for so(m).

A CRYSTAL-BASED MATTER-WAVE INTERFEROMETRIC GRAVITATIONAL-WAVE OBSERVATORY

It is shown that atom interferometry allows for the construction of MIGO, the Matter-wave Interferometric Gravitational-wave Observatory. MIGOs of the same sensitivity as LIGO or LISA are expected to be orders of magnitude smaller than either one. A design for MIGO using crystalline diffraction gratings is introduced, and its sensitivity is calculated.

ON THE KOSTERLITZ–THOULESS TRANSITION ON COMPACT RIEMANN SURFACES

A Lagrangian for the two-dimensional vortex gas is derived from a general microscopic Lagrangian for 4He atoms on an arbitrary compact Riemann Surface without boundary. In the constant density limit the vortex Hamiltonian obtained from this Lagrangian is found to be the same as the Kosterlitz and Thouless Coulombic interaction Hamiltonian. The partition function for the Kosterlitz–Thouless ensemble on the general compact is formulated and mapped into the sine–Gordon field theory.

Observations on the dynamics of the two-dimensional vortex gas on compact Riemann surfaces

The dynamics and symmetries of the two-dimensional vortex gas on compact Riemann surfaces are analysed using Lagrangian dynamics. As the vortex Lagrangian is linear in the canonical momenta, Diracs theory of constraints is then used to form the Hamiltonian dynamics for the system.

On the theory of superfluidity in two dimensions

The superfluid phase transition ofthe general vortex gas, in which the circulatims may be any non-zero integer, is studied. When the net circulation of the system is not zero the absence ofasuperfluid phase is shown. When the net circulation ofthevortices vanishes. the presence of off-diagonal long range order is demonstrated and the existence of an order parameter is proposed. The transition temperature for the general vortex gas is shown to be the Koaterlitz-Thouless temperature. An upper bound for the average vortex number density is established for the general vonex gas and an exact expression is derived for the Kosterlitz-Thouleas ensemble.

The Landau-Ginzberg Theory for the Two-Dimensional Bose Gas

Abstract Using results from sheaf theory combined with the phenomenological theory of the two-dimensional superfluid, the precipitation of quantum vortices is shown to be the genesis of a macroscopic order parameter for a phase transition in two dimensions.