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Physical Review A

Using the ground state of an antiferromagnetic spin-1 atomic condensate for Heisenberg-limited metrology

Ling-Na Wu, Li You

Published Mar 3, 2016• 11 citations
Publisher

Abstract

We show that the ground state of a spin-1 atomic condensate with antiferromagnetic interactions constitutes a useful resource for quantum metrology upon approaching the Heisenberg limit. Unlike a ferromagnetic condensate state where individual atomic spins are aligned in the same direction, the antiferromagnetic ground-state condensate is a condensate of spin-singlet atom pairs. The inherent correlation between paired atoms allows for parameter estimation at precisions beyond the standard quantum limit (SQL) for uncorrelated atoms. The degree of improvement over the SQL is measured by the scaled quantum Fisher information (QFI), whose dependence on the ratio of linear Zeeman shift $p$ to spin-dependent atomic interaction $c$ is studied. At a typical value of $p=0.4c$, which corresponds to a magnetic field of $28.6\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{G}$ for $c=50\mathrm{h}$ Hz (for $^{23}\mathrm{Na}$ atom condensate in the $F=1$ state at a typical density of $\ensuremath{\sim}{10}^{14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$), the scaled QFI can reach $\ensuremath{\sim}0.48N$, which approaches the limit of $0.5N$ for the twin-Fock state ${|N/2\ensuremath{\rangle}}_{+}{|N/2\ensuremath{\rangle}}_{\ensuremath{-}}$. Our work encourages experimental efforts to reach the ground state of an antiferromagnetic condensate at a extremely low magnetic field.

DOI
10.1103/physreva.93.033608

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We show that the ground state of a spin-1 atomic condensate with antiferromagnetic interactions constitutes a useful resource for quantum metrology upon approaching the Heisenberg limit. Unlike a ferromagnetic condensate state where individual atomic spins are aligned in the same direction, the antiferromagnetic ground-state condensate is a condensate of spin-singlet atom pairs. The inherent correlation between paired atoms allows for parameter estimation at precisions beyond the standard quantum limit (SQL) for uncorrelated atoms. The degree of improvement over the SQL is measured by the scaled quantum Fisher information (QFI), whose dependence on the ratio of linear Zeeman shift ppp to spin-dependent atomic interaction ccc is studied. At a typical value of p=0.4cp=0.4cp=0.4c, which corresponds to a magnetic field of 28.6\ensuremathμG28.6\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{G}28.6\ensuremathμG for c=50hc=50\mathrm{h}c=50h Hz (for 23Na^{23}\mathrm{Na}23Na atom condensate in the F=1F=1F=1 state at a typical density of \ensuremath∼1014cm\ensuremath−3\ensuremath{\sim}{10}^{14}\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}\ensuremath∼1014cm\ensuremath−3), the scaled QFI can reach \ensuremath∼0.48N\ensuremath{\sim}0.48N\ensuremath∼0.48N, which approaches the limit of 0.5N0.5N0.5N for the twin-Fock state ∣N/2\ensuremath⟩+∣N/2\ensuremath⟩\ensuremath−{|N/2\ensuremath{\rangle}}_{+}{|N/2\ensuremath{\rangle}}_{\ensuremath{-}}∣N/2\ensuremath⟩+​∣N/2\ensuremath⟩\ensuremath−​. Our work encourages experimental efforts to reach the ground state of an antiferromagnetic condensate at a extremely low magnetic field.
We study theoretically the usefulness of spin-1 Bose condensates with macroscopic magnetization in a homogeneous magnetic field for quantum metrology. We demonstrate Heisenberg scaling of the quantum Fisher information for states in thermal equilibrium. The scaling applies to both antiferromagnetic and ferromagnetic interactions. The effect preserves as long as fluctuations of magnetization are sufficiently small. Scaling of the quantum Fisher information with the total particle number is derived within the mean-field approach in the zero-temperature limit and exactly in the high-magnetic-field limit for any temperature. The precision gain is intuitively explained owing to subtle features of the quasidistribution function in the phase space.
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.
Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. An interferometer using N independent particles to measure a parameter χ can achieve at best the "standard quantum limit" (SQL) of sensitivity δχ ∝ N-1/2. The same interferometer using N entangled particles can achieve in principle the "Heisenberg limit" δχ ∝ N-1, using exotic states, e.g., NooN states. Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity δχ ∝ N-kwith appropriate entangled states and δχ ∝ N-(k-1/2)even without entanglement. Here we demonstrate this "super-Heisenberg" scaling in a nonlinear, non-destructive measurement of the magnetisation of an atomic ensemble.
Many-body kkk-local ground states as probes for unitary quantum metrology
Multipartite quantum states saturating the Heisenberg limit of sensitivity typically require full-body correlators to be prepared. On the other hand, experimentally practical Hamiltonians often involve few-body correlators only. Here, we study the metrological performances under this constraint, using tools derived from the quantum Fisher information. Our work applies to any encoding generator, also including a dependence on the parameter. We find that typical random symmetric ground states of kkk-body permutation-invariant Hamiltonians exhibit Heisenberg scaling. Finally, we establish a tradeoff between the Hamiltonian's gap, which quantifies preparation hardness, and the quantum Fisher information of the corresponding ground state.
We study in this work the metrology precision of a probe system coupled to an ancillary qubit. Restricting the probe-qubit coupling along only one or two directions is found to be a sufficient criterion for the effective dynamical generator to achieve the Heisenberg limit in precision. Under the criterion, the quantum Fisher information (QFI) about the to-be-estimated parameter becomes the expectation value of mean square of the effective generator with respect to the initial state of the composite system. Our criterion is justified in two distinct systems. For a bosonic probe, QFI about the displacement estimation is found to be proportional to the mean excitation number of probe. It renders a counterintuitive result that quantum metrology sensitivity can be enhanced by increasing the temperature of the probe system. For a spin-ensemble probe, QFI about both rotation-phase and magnetic-field estimation exhibit a quadratic dependence on the probe-spin number. It is found that even when the spin-ensemble is prepared as a finite-temperature state, faraway from the so-called resource states, e.g., the squeezed state or the Greenberger-Horne-Zeilinger state, QFI can still manifest a Heisenberg scaling behavior.
We study the spin entanglement in a spin-1 Bose-Einstein condensate with antiferromagnetic atomic interactions using the Hartree-Fock approach. Based on the isomorphism between symmetric NNN-qubit states and spin-j=N/2j=N/2j=N/2 states, we analyze the negativity of the spin-1 ground state of the condensate viewed as a two spins 1/21/21/2, and explore its dependence with the temperature and external Zeeman fields. The scope of this type of entanglement is highlighted and contrasted with other types of entanglement, as the mode and particle entanglement. It is shown that, at finite temperatures, there is a strong dependence of the negativity with respect to the strengths of quadratic Zeeman fields and the spin-spin interactions of the condensate. Interestingly, in the antiferromagnetic ground-state phase, the negativity and the linear Zeeman field are connected quadratically through the equation of a simple circle, in which its radius depends on the temperature. On the other hand, for the polar phase, being the phase that exhibits the highest degree of entanglement, we were able to identify a clear dependency on the spin-spin interactions and the Zeeman fields that can be expressed analytically in a closed form as a function of the temperature. This results might be relevant for applications in quantum information and metrology.