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.
Abstract The coherent manipulation of a quantum wave is at the core of quantum sensing. For instance, atom interferometers require linear splitting and recombination processes to map the accumulated phase shift into a measurable population signal. Although Bose Einstein condensates (BECs) are the archetype of coherent matter waves, their manipulation between trapped spatial modes has been limited by the strong interparticle collisions. Here, we overcome this problem by using BECs with tunable interaction trapped in an innovative array of double-well potentials and exploiting quantum tunneling to realize linear beam splitting. We operate several Mach-Zehnder interferometers in parallel, canceling common-mode potential instabilities by a differential analysis, thus demonstrating a trapped-atom gradiometer. Furthermore, by applying a spin-echo protocol, we suppress additional decoherence sources and approach unprecedented coherence times of one second. Our interferometer will find applications in precision measurements of forces with a high spatial resolution and in linear manipulation of quantum entangled states for sensing with sub shot-noise sensitivity.
Mach-Zehnder atom interferometry requires hold-time phase squeezing to attain readout accuracy below the standard quantum limit. This increases its sensitivity to phase diffusion, restoring shot-noise scaling of the optimal signal-to-noise ratio in the presence of interactions. The contradiction between the preparations required for readout accuracy and robustness to interactions is removed by monitoring Rabi-Josephson oscillations instead of relative-phase oscillations during signal acquisition. Optimizing the signal-to-noise ratio with a Gaussian squeezed input, we find that hold-time number squeezing satisfies both demands and that sub-shot-noise scaling is retained even for strong interactions.
Many works have stated that nonlinear interactions can improve phase sensitivity beyond the Heisenberg limit scaling of
1/N with
N being the mean photon number. This raises some open questions---among them the conclusive sensitivity limits with respect to single-mode inputs. Namely, when one of two inputs is vacuum, is there a shot-noise-style sensitivity bound on a nonlinear Mach-Zehnder interferometer? Within the reach of second-order nonlinear phase shifts, we make an attempt to provide an answer to this question. Based upon phase-averaging approach, this puzzle is partially resolved with careful calculations of the quantum Fisher information regarding three kinds of common inputs: Gaussian states, squeezed number states, and Schrödinger cat states. The results suggest that shot-noise-style sensitivity limit is no longer available, and the ideal candidate is squeezed vacuum.
Mach-Zehnder interferometer, a powerful tool for a wide variety of measurements, has been realized with Bose-Einstein condensates in recent experiments. In this report, we propose and analyze a realizable scheme for performing a Heisenberg-limited Mach-Zehnder interferometry with dipolar spin-1 condensate. Based upon adiabatic processes of sweeping the transverse magnetic field, we demonstrate a perfect phase transition, which accomplishes the beam splitter, phase shifter and recombiner as for a Mach-Zehnder interferometer. The attractive dipolar interaction ensures the existence of a path-entangled state which enhances the phase measurement precision to the Heisenberg limit. We also discuss the spin-
1 squeezing induced in the adiabatic passage and show that the squeezing parameter attains its minimal value near the point of saturation field.
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 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
r and the amplitude of the coherent mode $\ensuremath{\alpha}$ by using a Bayesian phase inference protocol.