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arXiv:1712.03896 · Physical Review A

Interferometric sensitivity and entanglement by scanning through quantum phase transitions in spinor Bose-Einstein condensates

Polina Feldmann, Manuel Gessner, Marco Gabbrielli, Carsten Klempt, Luis Santos, Luca Pezzè, Augusto Smerzi

Published Mar 27, 2018• 29 citations
arXivHTMLPDF

Abstract

Recent experiments demonstrated the generation of entanglement by quasiadiabatically driving through quantum phase transitions of a ferromagnetic spin-1 Bose-Einstein condensate in the presence of a tunable quadratic Zeeman shift. We analyze, in terms of the Fisher information, the interferometric value of the entanglement accessible by this approach. In addition to the Twin-Fock phase studied experimentally, we unveil a second regime, in the broken axisymmetry phase, which provides Heisenberg scaling of the quantum Fisher information and can be reached on shorter time scales. We identify optimal unitary transformations and an experimentally feasible optimal measurement prescription that maximize the interferometric sensitivity. We further ascertain that the Fisher information is robust with respect to nonadiabaticity and measurement noise. Finally, we show that the quasiadiabatic entanglement preparation schemes admit higher sensitivities than dynamical methods based on fast quenches.

DOI
10.1103/physreva.97.032339

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Recent experiments demonstrated the generation of entanglement by quasiadiabatically driving through quantum phase transitions of a ferromagnetic spin-1 Bose-Einstein condensate in the presence of a tunable quadratic Zeeman shift. We analyze, in terms of the Fisher information, the interferometric value of the entanglement accessible by this approach. In addition to the Twin-Fock phase studied experimentally, we unveil a second regime, in the broken axisymmetry phase, which provides Heisenberg scaling of the quantum Fisher information and can be reached on shorter time scales. We identify optimal unitary transformations and an experimentally feasible optimal measurement prescription that maximize the interferometric sensitivity. We further ascertain that the Fisher information is robust with respect to nonadiabaticity and measurement noise. Finally, we show that the quasiadiabatic entanglement preparation schemes admit higher sensitivities than dynamical methods based on fast quenches.
Entanglement lies at the core of emergent quantum technologies such as quantum-enhanced metrology, quantum communication and cryptography, and quantum simulation and computing. Spinor Bose-Einstein condensates (BECs) offer a promising platform for the generation and application of entangled states. For example, a spin-1 BEC has served for the proof-of-principle demonstration of a quantum-enhanced atomic clock. Ferromagnetic spin-1 BECs with zero magnetization exhibit three ground-state quantum phases with different entanglement properties. The control parameter can be tuned by a magnetic field or by microwave dressing. As already experimentally demonstrated, an entangled ground state can be reached from a well accessible, non-entangled one by driving the control parameter across quantum phase transitions (QPTs). We investigate which of the entangled ground states afford quantum-enhanced interferometry. The interferometric usefulness is quantified by the quantum Fisher information (QFI), which we analyze throughout all ground-state phases. A large QFI at about half the Heisenberg limit, and thus far above the standard quantum limit, is attained by the well-known Twin-Fock state and by the central broken-axisymmetry (CBA) state. We detail how the CBA state can be used as a probe for quantum-enhanced interferometry. Furthermore, we observe that the large QFI of the CBA state can be traced back to enclosed macroscopic superposition states (MSSs). Measuring the atom number in one out of three modes generates, with high probability and heralded by the measurement outcome, a MSS similar to a NOON state. Our proposal promises NOON-like MSSs of unprecedentedly many atoms. Both proposed applications of the adiabatically prepared CBA state depend only on existent technology. Our numerical results show that they tolerate a reasonably swift quasiadiabatic passage in the presence of atom loss as well as uncertainties of atom counting. Excited-state quantum phase transitions (ESQPTs) extend the concept of QPTs beyond the ground state. While they have been extensively investigated theoretically, there are only few experimental results. From the perspective of quantum-state engineering, it is furthermore surprising how rarely order parameters of ESQPTs are discussed in the literature. Mean-field models for spinor BECs imply ESQPTs, to which some experimental observations on the mean-field dynamics can be attributed. However, so far, neither theoretical nor experimental studies have specifically addressed ESQPTs in spinor BECs. We extend the ground-state phase diagram of ferromagnetic spin-1 BECs with zero magnetization across the spectrum. There are three excited-state phases, corresponding to one ground-state phase each. The ESQPTs are signaled by a diverging density of states. The mean-field phase-space trajectories can be characterized by a winding number that is in one-to-one correspondence to the excited-state phases. We derive a closely related order parameter encoded in the dynamics of coherent states and discuss how this order parameter can be interferometrically measured in current experiments. Remarkably, the mean-field model governing the ESQPTs in spin-1 BECs with zero magnetization is encountered also, e. g., in molecular and nuclear physics. Because of the superior experimental control, spinor BECs can be considered as simulators of the ESQPTs in those systems. Our results contribute to quantum-state engineering and quantum-enhanced interferometry in spinor BECs and to the characterization of excited-state quantum phases. The latter may, in turn, lead on to applications in quantum-state engineering.
Implementation of the quantum interferometry concept to spin-1 atomic Bose–Einstein condensates is analyzed by employing a polar state evolved in time. In order to identify the best interferometric configurations, the quantum Fisher information (QFI) is maximized. Three optimal configurations are identified, among which one was not reported in the literature yet, although it gives the highest value of the QFI in experimentally achievable short time dynamics. Details of the most optimal configurations are investigated based on the error-propagation formula which includes the interaction-based readout protocol to reduce the destructive effect of detection noise. In order to obtain Heisenberg scaling accessible by present day experimental techniques, an efficient measurement and a method for the inversion of dynamics were developed, as necessary for the protocol’s implementation.
We consider an antiferromagnetic Bose-Einstein condensate in a transverse magnetic field with a fixed macroscopic magnetization. The system exhibits two different critical behaviors corresponding to transitions from polar to broken-axisymmetry and from antiferromagnetic to broken-axisymmetry phases, depending on the value of the magnetization. We exploit both types of system criticality as a resource in the precise estimation of the control parameter value. We quantify the achievable precision by the quantum Fisher information. We demonstrate supersensitivity and show that the precision scales with the number of atoms up to N4{N}^{4}N4 around criticality. In addition, we study the precision based on the error-propagation formula, which provides a simple-to-measure signal whose scaling coincides with the quantum Fisher information. Finally, we take into account the effect of nonzero temperature and show that sub-shot-noise sensitivity in the estimation of the control parameter is achievable in the low-temperature limit.
We theoretically investigate estimation of the control parameter in a ferromagnetic Bose-Einstein condensate near second-order quantum phase transitions. We quantify sensitivity by quantum and classical Fisher information and use the error-propagation formula. For these different metrics, we find the same beyond standard quantum limit (SQL) scaling with atom number near critical points and SQL scaling away from critical points. We find that both depletion of the mf=0{m}_{f}=0mf​=0 Zeeman sublevel and transverse magnetization provide signals of sufficient quality to saturate the sensitivity scaling. To explore the effect of experimental imperfections, we study the scaling around criticality at nonzero temperature and with nonzero detection noise. Our results suggest the feasibility of sub-SQL sensing in ferromagnetic condensates with current experimental capabilities.
We propose a method to generate massive entanglement in a spinor Bose-Einstein condensate from an initial product state through an adiabatic sweep of the magnetic field across a quantum phase transition induced by competition between the spin-dependent collision interaction and the quadratic Zeeman effect. The generated many-body entanglement is characterized by the experimentally measurable entanglement depth in the proximity of the Dicke state. We show that the scheme is robust to practical noise and experimental imperfection and under realistic conditions it is possible to generate genuine entanglement for hundreds of atoms.
We have experimentally investigated the quench dynamics of antiferromagnetic spinor Bose-Einstein condensates in the vicinity of a zero temperature quantum phase transition at zero quadratic Zeeman shift qqq. The rate of instability shows good agreement with predictions based upon solutions to the Bogoliubov--de Gennes equations. A key feature of this work was removal of magnetic field inhomogeneities, resulting in a steep change in behavior near the transition point. The quadratic Zeeman shift at the transition point was resolved to 250 mHz uncertainty, equivalent to an energy resolution of ${k}_{B}\phantom{\rule{4pt}{0ex}}\ifmmode\times\else\texttimes\fi{}$ (12 pK). A small (2--3 $\ensuremath{\sigma}$) shift of the transition point was observed, from q=0q=0q=0 to q=+650q=+650q=+650 mHz, whose physical mechanism is currently unknown. In this work, we demonstrate a sub-Hz precision measurement of a phase transition in quantum gases. It paves the way toward observing shifts of the transition point due to finite particle number NNN that scale as 1/N1/N1/N, and also to potential Heisenberg limited spectroscopy with antiferromagnetic spinor gases [L.-N. Wu and L. You, Phys. Rev. A 93, 033608 (2016)].