Exact solution for many-body Hamiltonian of interacting particles with linear spectrum

The exact solution of the Schr\"odinger equation for the one-dimensional system of interacting particles with the linear dispersion law in an arbitrary external field is found. The solution is reduced to two groups of particles moving with constant velocities in the opposite directions with a fixed distance between the particles in each group. The problem is applied to the edge states of the 2D topological insulator.Comment: 4 page

Dual-Resonator Speed Meter for a Free Test Mass

A description and analysis are given of a ``speed meter'' for monitoring a classical force that acts on a test mass. This speed meter is based on two microwave resonators (``dual resonators''), one of which couples evanescently to the position of the test mass. The sloshing of the resulting signal between the resonators, and a wise choice of where to place the resonators' output waveguide, produce a signal in the waveguide that (for sufficiently low frequencies) is proportional to the test-mass velocity (speed) rather than its position. This permits the speed meter to achieve force-measurement sensitivities better than the standard quantum limit (SQL), both when operating in a narrow-band mode and a wide-band mode. A scrutiny of experimental issues shows that it is feasible, with current technology, to construct a demonstration speed meter that beats the wide-band SQL by a factor 2. A concept is sketched for an adaptation of this speed meter to optical frequencies; this adaptation forms the basis for a possible LIGO-III interferometer that could beat the gravitational-wave standard quantum limit h_SQL, but perhaps only by a factor 1/xi = h_SQL/h ~ 3 (constrained by losses in the optics) and at the price of a very high circulating optical power --- larger by 1/xi^2 than that required to reach the SQL.Comment: RevTex: 13 pages with 4 embedded figures (two .eps format and two drawn in TeX); Submitted to Physical Review

Observation of opto-mechanical multistability in a high Q torsion balance oscillator

We observe the opto-mechanical multistability of a macroscopic torsion balance oscillator. The torsion oscillator forms the moving mirror of a hemi-spherical laser light cavity. When a laser beam is coupled into this cavity, the radiation pressure force of the intra-cavity beam adds to the torsion wire's restoring force, forming an opto-mechanical potential. In the absence of optical damping, up to 23 stable trapping regions were observed due to local light potential minima over a range of 4 micrometer oscillator displacement. Each of these trapping positions exhibits optical spring properties. Hysteresis behavior between neighboring trapping positions is also observed. We discuss the prospect of observing opto-mechanical stochastic resonance, aiming at enhancing the signal-to-noise ratio (SNR) in gravity experiments.Comment: 4 pages, 5 figure

Sensitivity limitations in optical speed meter topology of gravitational-wave antennae

The possible design of QND gravitational-wave detector based on speed meter principle is considered with respect to optical losses. The detailed analysis of speed meter interferometer is performed and the ultimate sensitivity that can be achieved is calculated. It is shown that unlike the position meter signal-recycling can hardly be implemented in speed meter topology to replace the arm cavities as it is done in signal-recycled detectors, such as GEO 600. It is also shown that speed meter can beat the Standard Quantum Limit (SQL) by the factor of $\sim 3$ in relatively wide frequency band, and by the factor of $\sim 10$ in narrow band. For wide band detection speed meter requires quite reasonable amount of circulating power $\sim 1$ MW. The advantage of the considered scheme is that it can be implemented with minimal changes in the current optical layout of LIGO interferometer.Comment: 20 pages, 12 figure

The noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization

It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output (the photon number flux N(t) entering the final photodetector) commutes with itself at different times in the Heisenberg Picture, [N(t), N(t')] = 0, and thus can be regarded as classical. (ii) This number flux is linear in the test-mass initial position and momentum operators x_o and p_o, and those operators influence the measured photon flux N(t) in manners that can easily be removed by filtering -- e.g., in most interferometers, by discarding data near the test masses' 1 Hz swinging freqency. The test-mass operators x_o and p_o contained in the unfiltered output N(t) make a nonzero contribution to the commutator [N(t), N(t')]. That contribution is cancelled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in N(t). This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations. These conclusions are true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.Comment: Submitted to Physical Review D; Revtex, no figures, prints to 14 pages. Second Revision 1 December 2002: minor rewording for clarity, especially in Sec. II.B.3; new footnote 3 and passages before Eq. (2.35) and at end of Sec. III.B.

Energetic Quantum Limit in Large-Scale Interferometers

For each optical topology of an interferometric gravitational wave detector, quantum mechanics dictates a minimum optical power (the ``energetic quantum limit'') to achieve a given sensitivity. For standard topologies, when one seeks to beat the standard quantum limit by a substantial factor, the energetic quantum limit becomes impossibly large. Intracavity readout schemes may do so with manageable optical powers.Comment: Revised version; to be published in Proceedings of the 1999 Edoardo Amaldi Conference on Gravitational Waves; 11 pages including figures; manuscript is RevTex; figures are .eps; an AIP style file is include

To the practical design of the optical lever intracavity topology of gravitational-wave detectors

The QND intracavity topologies of gravitational-wave detectors proposed several years ago allow, in principle, to obtain sensitivity significantly better than the Standard Quantum Limit using relatively small anount of optical pumping power. In this article we consider an improved more ``practical'' version of the optical lever intracavity scheme. It differs from the original version by the symmetry which allows to suppress influence of the input light amplitude fluctuation. In addition, it provides the means to inject optical pumping inside the scheme without increase of optical losses. We consider also sensitivity limitations imposed by the local meter which is the key element of the intracavity topologies. Two variants of the local meter are analyzed, which are based on the spectral variation measurement and on the Discrete Sampling Variation Measurement, correspondingly. The former one, while can not be considered as a candidate for a practical implementation, allows, in principle, to obtain the best sensitivity and thus can be considered as an ideal ``asymptotic case'' for all other schemes. The DSVM-based local meter can be considered as a realistic scheme but its sensitivity, unfortunately, is by far not so good just due to a couple of peculiar numeric factors specific for this scheme. From our point of view search of new methods of mechanical QND measurements probably based on improved DSVM scheme or which combine the local meter with the pondermotive squeezing technique, is necessary.Comment: 27 pages, 6 figure

Minimum Length from Quantum Mechanics and Classical General Relativity

We derive fundamental limits on measurements of position, arising from quantum mechanics and classical general relativity. First, we show that any primitive probe or target used in an experiment must be larger than the Planck length, $l_P$. This suggests a Planck-size {\it minimum ball} of uncertainty in any measurement. Next, we study interferometers (such as LIGO) whose precision is much finer than the size of any individual components and hence are not obviously limited by the minimum ball. Nevertheless, we deduce a fundamental limit on their accuracy of order $l_P$. Our results imply a {\it device independent} limit on possible position measurements.Comment: 8 pages, latex, to appear in the Physical Review Letter