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Physical Review A

Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope

Jonathan P. Dowling

Published Jun 1, 1998• 317 citations
Publisher

Abstract

I derive the quantum phase-noise limit to the sensitivity of a Mach-Zehnder interferometer in which the incident quantum particles enter via both input ports. I show that if the incident particles are entangled and correlated properly, then the phase sensitivity scales asymptotically like the Heisenberg-limited $\ensuremath{\Delta}\ensuremath{\varphi}=O(1/N),$ for large $N,$ where $N$ is the number of particles incident per unit time. (In a one-input-port device, the sensitivity can be at best $\ensuremath{\Delta}\ensuremath{\varphi}=1/\sqrt{N}.$) My calculation applies to bosons or fermions of arbitrary integer or half-integer spin. Applications to optical, atom-beam, and atom-laser gyroscopes are discussed---in particular, an atom-laser can be used to obtain the required entanglements for achieving this Heisenberg-limited sensitivity with atomic matter waves.

DOI
10.1103/physreva.57.4736

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I derive the quantum phase-noise limit to the sensitivity of a Mach-Zehnder interferometer in which the incident quantum particles enter via both input ports. I show that if the incident particles are entangled and correlated properly, then the phase sensitivity scales asymptotically like the Heisenberg-limited \ensuremathΔ\ensuremathφ=O(1/N),\ensuremath{\Delta}\ensuremath{\varphi}=O(1/N),\ensuremathΔ\ensuremathφ=O(1/N), for large N,N,N, where NNN is the number of particles incident per unit time. (In a one-input-port device, the sensitivity can be at best \ensuremathΔ\ensuremathφ=1/N.\ensuremath{\Delta}\ensuremath{\varphi}=1/\sqrt{N}.\ensuremathΔ\ensuremathφ=1/N​.) My calculation applies to bosons or fermions of arbitrary integer or half-integer spin. Applications to optical, atom-beam, and atom-laser gyroscopes are discussed---in particular, an atom-laser can be used to obtain the required entanglements for achieving this Heisenberg-limited sensitivity with atomic matter waves.
We derive the quantum limits for an atomic interferometer from a second-quantized theory in which the atoms obey either Bose-Einstein or Fermi-Dirac statistics. It is found that the limiting quantum noise is due to the uncertainty associated with the particle sorting between the two branches of the interferometer, and that this noise can be reduced in a sufficiently dense atomic beam by using fermions as opposed to bosons. As an example, the quantum-limited sensitivity of a generic matter-wave gyroscope is calculated and compared with that of a laser gyroscope.
The field of atom interferometry1 is interesting since the potential sensitivity of matter-wave interferometers far exceeds that of their light-wave antecedents.2 As was emphasized at the recent Solvay conference on quantum optics, there is at present no paradigm available for calculating the quantum noise limits to matter-wave interferometers.3 We show that the phase uncertainty is inversely proportional to the square root of the total number, rather than the mean number, of atoms in the interferometert.4 We find that a matter-wave interferometer used as a gyroscope can be expected to be more sensitive to rotation by some four orders of magnitude than present laser gyro devices.
We show that the phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ of a Mach-Zehnder interferometer illuminated by a coherent state in one input port and a squeezed-vacuum state in the other port is (i) independent of the true value of the phase shift and (ii) can reach the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1/{N}_{T}$, where NT{N}_{T}NT​ is the average number of input particles. We also demonstrate that the Cramer-Rao lower bound of phase sensitivity, $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}\phantom{\rule{0ex}{0ex}}1/\sqrt{|\ensuremath{\alpha}{|}^{2}{e}^{2r}+{sinh}^{2}r}$, can be saturated for arbitrary values of the squeezing parameter rrr and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.
Bose-Einstein condensates (BEC), with their coherence properties, have attracted wide interest for their possible application to ultraprecise interferometry and ultraweak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\ensuremath{\Delta}\ensuremath{\theta}$ scaling with the total number of condensate particles NNN as (i) the standard quantum limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{1∕2}$, (ii) the sub shot-noise $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$, and the (iii) the Heisenberg limit $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕N$. However, in a realistic dynamical BEC splitting, the 1∕N1∕N1∕N limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub-shot-noise sensitivity $\ensuremath{\Delta}\ensuremath{\theta}\ensuremath{\sim}1∕{N}^{3∕4}$ can be reached in a realistic experimental setting. We also show that the 1∕N3∕41∕{N}^{3∕4}1∕N3∕4 scaling is a rigorous upper bound in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.
Quantum entanglement has the potential to revolutionize the entire field of interferometric sensing by providing many orders of magnitude improvement in interferometer sensitivity. The quantum-entangled particle interferometer approach is very general and applies to many types of interferometers. In particular, without nonlocal entanglement, a generic classical interferometer has a statistical-sampling shot-noise limited sensitivity that scales like 1/N1/\sqrt{N}1/N​, where NNN is the number of particles passing through the interferometer per unit time. However, if carefully prepared quantum correlations are engineered between the particles, then the interferometer sensitivity improves by a factor of N\sqrt{N}N​ to scale like 1/N, which is the limit imposed by the Heisenberg Uncertainty Principle. For optical interferometers operating at milliwatts of optical power, this quantum sensitivity boost corresponds to an eight-order-of-magnitude improvement of signal to noise. This effect can translate into a tremendous science pay-off for space missions. For example, one application of this new effect is to fiber optical gyroscopes for deep-space inertial guidance and tests of General Relativity (Gravity Probe B). Another application is to ground and orbiting optical interferometers for gravity wave detection, Laser Interferometer Gravity Observatory (LIGO) and the European Laser Interferometer Space Antenna (LISA), respectively. Other applications are to Satellite-to-Satellite laser Interferometry (SSI) proposed for the next generation Gravity Recovery And Climate Experiment (GRACE II).
Quantum entanglement has the potential to revolutionize the entire field of interferometric sensing by providing many orders of magnitude improvement in interferometer sensitivity. The quantum-entangled particle interferometer approach is very general and applies to many types of interferometers. In particular, without nonlocal entanglement, a generic classical interferometer has a statistical-sampling shot-noise limited sensitivity that scales like 1/√N N, where N is the number of particles passing through the interferometer per unit time. However, if carefully prepared quantum correlations are engineered between the particles, then the interferometer sensitivity improves by a factor of √N to scale like 1/N, which is the limit imposed by the Heisenberg Uncertainty Principle. For optical interferometers operating at milliwatts of optical power, this quantum sensitivity boost corresponds to an eight-order-of-magnitude improvement of signal to noise. This effect can translate into a tremendous science pay-off for space missions. For example, one application of this new effect is to fiber optical gyroscopes for deep-space inertial guidance and tests of General Relativity (Gravity Probe B). Another application is to ground and orbiting optical interferometers for gravity wave detection, Laser Interferometer Gravity Observatory (LIGO) and the European Laser Interferometer Space Antenna (LISA), respectively. Other applications are to Satellite-to-Satellite laser Interferometry (SSI) proposed for the next generation Gravity Recovery And Climate Experiment (GRACE II).