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.
We theoretically analyze a Mach-Zehnder interferometer with trapped condensates and find that it is surprisingly stable against the nonlinearity induced by interparticle interactions. The phase sensitivity, which we study for number-squeezed input states, can overcome the shot noise limit and be increased up to the Heisenberg limit provided that a Bayesian or maximum-likelihood phase estimation strategy is used. We finally demonstrate the robustness of the Mach-Zehnder interferometer in the presence of interactions against condensate oscillations and a realistic atom-counting error.
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.
In a gas of identical atoms, each atom is described by a localized de Broglie wave packet. If the gas is trapped and cooled, the atoms move closer together and their de Broglie wavelengths get longer. As the gas is cooled further, the wave packets become larger and eventually start to overlap and interfere with each other. What happens next in these so-called quantum-degenerate gases depends on whether the atoms are bosons or fermions. Quantum gases of both types have proved to be a rich source of new physics in recent years and this trend is continuing with experiments on gases that contain both types of atom, such as the observation of the collapse of a degenerate Fermi gas by researchers at the European Laboratory for Nonlinear Spectroscopy in Florence, Italy (G Modugno et al. 2002 Science 297 2240).
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
N 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∕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∕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.
We compare four-wave mixing in quantum degenerate gases of bosonic and fermionic atoms. We find that matter-wave gratings formed from either bosonic or fermionic atoms can in principle exhibit nearly identical Bragg scattering and four-wave mixing properties. This implies that effects such as coherent matter-wave amplification and superradiance can occur in degenerate Fermi gases. This effect is due to constructive many-particle quantum interferences, which in the boson case are interpreted as "Bose enhancement."