Simultaneous tracking of spin angle and amplitude beyond classical limits

Abstract

Measurement of spin precession is central to extreme sensing in physics,1,2 geophysics,3 chemistry,4 nanotechnology5 and neuroscience,6 and underlies powerful magnetic resonance spectroscopies.7 Because there is no spin-angle operator, any measurement of spin precession is necessarily indirect, e.g., inferred from spin projectors Fα at different times. Such projectors do not commute, and thus quantum measurement back-action (QMBA) necessarily enters the spin measurement record, introducing errors and limiting sensitivity. Here we show how to reduce this disturbance below δFα∼N, the classical limit for N spins, by directing the QMBA almost entirely into an unmeasured spin component. This generates a planar squeezed state8 which, because spins obey non-Heisenberg uncertainty relations,9,10 allows simultaneous precise knowledge of spin angle and amplitude. We use high-dynamic-range optical quantum non-demolition measurements11–13 applied to a precessing magnetic spin ensemble, to demonstrate spin tracking with steady-state angular sensitivity 2.9 dB beyond the standard quantum limit, simultaneous with amplitude sensitivity 7.0 dB beyond Poisson statistics.14 This method for the first time surpasses classical limits in non-commuting observables, and enables orders-of-magnitude sensitivity boosts for state-of-the-art sensing15–18 and spectroscopy.19,20

State-independent uncertainty relations from eigenvalue minimization

“…For n M ax = 30 we have that the maximum Schmidt coefficient λ M ax ≈ 0.99931 such that one is led to consider the corresponding state |λ M ax |λ M ax as a fairly good approximation of the ground state. Indeed ε 0 gs,x |λ M ax |λ M ax ≈ 0.99931 and therefore |ψ sat = |λ M ax in this case is a good candidate for the minimization of (14). This is confirmed by the value V xn (|λ M ax ) ≈ 0.415139 such that the relative error of the approximation V xn (|λ M ax ) − ε 0 gs,x /ε 0 gs,x ≈ 0.5% is excellent.…”

Continuous Faraday measurement of spin precession without light shifts

“…Used with warm atomic vapors, this 'Faraday interface' has squeezed spins [24], squeezed light [25], entangled states of collective atomic ensembles [26], encoded light into quantum memories [27] and teleported states of light to atoms [28]. Applied to ultracold (but not degenerate) atoms [8], the Faraday interface has created macroscopic singlet states [4], squeezed two spin projections simultaneously [12], cooled by feedback [10] and made macroscopic quantum non-demolition measurements [11].…”

“…Faraday measurements have opened new perspectives in quantum metrology [1], quantum information [2], non-linear mean-field [3] and many-body [4] systems; and have potential for probing strongly-correlated systems [5]. The Faraday interface has been applied to progressively colder systems: magneto-optical traps [6], dark- [7] and bright-optical dipole traps [4,[8][9][10][11][12], and Bose-Einstein condensates (BEC) [3,13]. However, the classical backaction of the Faraday probe perturbs atomic motional and spin degrees of freedom, limiting and confounding measurements of emergent phenomena or weak external fields at ultracold temperatures.…”

Squeezing on Momentum States for Atom Interferometry

“…This limit can be overcome by introducing correlations between the individual particles, thereby producing squeezed atomic states, potentially reaching the Heisenberg limit ∆φ H = 1/N [9, 10]. Many schemes have been studied both theoretically [11,12] and experimentally [13][14][15][16][17][18][19][20][21] with about 20 dB noise reduction compared to the SQL [22,23]. The key feature of most of these schemes is the enhanced atom-light interaction in an optical resonator.…”