Fisher information and entanglement of non-Gaussian spin states

Abstract

Entanglement is the key quantum resource for improving measurement sensitivity beyond classical limits. However, the production of entanglement in mesoscopic atomic systems has been limited to squeezed states, described by Gaussian statistics. Here we report on the creation and characterization of non-Gaussian many-body entangled states. We develop a general method to extract the Fisher information, which reveals that the quantum dynamics of a classically unstable system creates quantum states that are not spin squeezed but nevertheless entangled. The extracted Fisher information quantifies metrologically useful entanglement which we confirm by Bayesian phase estimation with sub shot-noise sensitivity. These methods are scalable to large particle numbers and applicable directly to other quantum systems. [14,15]. The production of these fragile states in large systems remains a challenge and efficient methods for characterization are necessary because full state reconstruction becomes intractable. Here, we generate a class of non-Gaussian many-particle entangled states and reveal their quantum properties by studying the distinguishability of experimental probability distributions.A measure of the distinguishability with respect to small phase changes of the state is provided by the Fisher information F [16]. It is related to the highest attainable interferometric phase sensitivity by the Cramer-Rao bound ∆θ CR = 1/ √ F [17]. This limit follows from general statistical arguments for a measurement device with fluctuating output [18]. The Fisher information is limited by quantum fluctuations of the input state as well as the performance of the device. Even in the absence of technical noise, the Fisher information of a classical input state is F ≤ N because of the intrinsic granularity of N independent particles which translates into the shot-noise limit ∆θ ≥ 1/ √ N for phase estimation. This classical bound can be surpassed with a reduction of the input fluctuations by introducing entanglement between the N particles [5]. These states, known as squeezed states, are fully characterized by mean and variance of the observable and already employed in precision measurements [19][20][21]. In contrast, non-Gaussian quantum states can have increased fluctuations of the observable but nevertheless allow surpassing shot-noise limited performance. A textbook example is the Schrödinger cat state characterized by macroscopic fluctuations but achieving the best interferometric performance allowed by quantum mechanics, i.e. at the fundamental Heisenberg limit F = N . The initial coherent spin state (green) ideally evolves into a squeezed state (orange) followed by non-Gaussian states at later evolution times (violet). Edges of shaded areas are contours of the Husimi distribution for N = 380 at 1/e 2 of its maximum. (C) Experimental absorption picture, showing the site-and state-resolved optical lattice after a Stern-Gerlach separation. Shaded boxes indicate the sites with a total atom number in the range 380 ± 15, which are sele...

Geometric structure and information change in phase transitions

“…3 suggest that t c is also close to the merging time t m in Eq. (36). We confirm this by inserting in the parameter values in Eq.…”

“…[30]) and is a generalization of statistical distance [31], where the distance is set by the number of distinguishable states between two PDFs. While the latter was heavily used in equilibrium or near equilibrium of classical and quantum systems [32][33][34][35][36][37][38][39][40], our recent work [25][26][27][28][29] adapted this concept to a nonequilibrium system to elucidate geometric structure of nonequilibrium processes. Specifically, Ref.…”

Entanglement with negative Wigner function of almost 3,000 atoms heralded by one photon

“…Metrologically useful entangled states of large atomic ensembles have been experimentally realized [1][2][3][4][5][6][7][8][9][10], but these states display Gaussian spin distribution functions with a non-negative Wigner function. Non-Gaussian entangled states have been produced in small ensembles of ions [11,12], and very recently in large atomic ensembles [13][14][15]. Here, we generate entanglement in a large atomic ensemble via the interaction with a very weak laser pulse; remarkably, the detection of a single photon prepares several thousand atoms in an entangled state.…”

“…In our system, the maximum atom number of ∼ 3000 is set by the accuracy of the spin rotation, and can be increased by two orders of magnitude by better magnetic-field control [10]. The state purity ρ 11 can probably be further improved by reducing the heralding probability, and a value of ρ 11 > 0.73 would be required for the Fisher information [14] to exceed that of the CSS, and enable metrological gain of up to 3 dB. The detection of two or more photons prepares Schrödinger cat states [20] of the atomic ensemble with more metrological gain.…”

Optimal quantum parameter estimation in a pulsed quantum optomechanical system

“…One of the central ideas of quantum metrology is to beat the shot-noise limit and approach the Heisenberg limit by virtue of quantum resource, such as quantum entanglement or squeezing. There have been many studies on precision of parameter estimation with subshot-noise limit in different physical systems, such as the optical interferometers [7][8][9], Bose-Einstein condensates [10], atomic interferometers [11], and solid-state systems (e.g., the nitrogen-vacancy centres) [12,13]. To the best of our knowledge, only a few papers [14][15][16] have devoted to investigating the quantum metrology in the newly-developed novel quantum optomechanical device .…”