Nonlinear metrology with a quantum interface

We describe nonlinear quantum atom-light interfaces and nonlinear quantum metrology in the collective continuous variable formalism. We develop a nonlinear effective Hamiltonian in terms of spin and polarization collective variables and show that model Hamiltonians of interest for nonlinear quantum metrology can be produced in $^{87}$Rb ensembles. With these Hamiltonians, metrologically relevant atomic properties, e.g. the collective spin, can be measured better than the"Heisenberg limit"$\propto 1/N$. In contrast to other proposed nonlinear metrology systems, the atom-light interface allows both linear and non-linear estimation of the same atomic quantities.


Introduction
In quantum metrology, a quantum state is prepared, evolves under the action of a Hamiltonian containing a parameter x of interest, and is measured. The parameter is estimated from the measurement outcome and knowledge of the system and Hamiltonian. In most problems, the Hamiltonian is assumed to act in the same way on each of N systems (e.g. atoms), and precision scales as δx ∝ N −1/2 for product states, and down to δx ∝ N −1 for entangled states ("Heisenberg limit" scaling) [1]. A number of studies [2,3,4] have considered also nonlinear quantum metrology, in which the Hamiltonian describes a k-system coupling with strength x ′ . Remarkably, the scaling is δx ′ ∝ N −k+1/2 or δx ′ ∝ N −k for independent or entangled states, respectively [2]. Because this improves upon the best possible scaling for the linear case, it has been called "Super-Heisenberg" (SH) scaling [4]. Proposed implementations include scattering in Bose condensates [4], Duffing nonlinearity in nano-mechanical resonators [5], two-pass effective nonlinearity with an atomic ensemble [6], and Kerr nonlinearities [7,8,9].
Here we describe nonlinear metrology applied to measurement of collective spin variables of atomic ensembles. Atomic ensembles with long coherence-time internal degrees of freedom, e.g. nuclear spin, are essential elements of many quantum information and quantum metrology protocols including quantum memory [10], quantum non-demolition measurement [11], spin squeezing [12,13,14], and magnetometry [15].
We follow the approach of collective continuous variables (CCV), in which both light and atoms are described by macroscopic quantum variables. In the case of spinor atoms interacting with polarized light, the N A atoms are described by the collective spin F ≡ i f (i) where f (i) is the spin of the i-th atom. F obeys the commutation relations [F x , F y ] = i F z and cyclic permutations, and can itself be considered a macroscopic spin variable. The light is described by its electric field E = E E E + E E E * , where E E E is the positive-frequency part. The Stokes vector S with components S i = (E * + , E * − )σ i (E + , E − ) T where the subscript indicates plus/minus circular polarization, σ i are the Pauli matrices and σ 0 is the identity. As described by several authors [16,17,18], the electric dipole interaction h int = −E · d, taken in second order perturbation theory, gives rise to an effective (single-atom) Hamiltonian of the form plus terms in S 0 which do not interact with the optical polarization. Here α 1,2 describe the vectorial, and tensorial components of the interaction, respectively, and the atomic collective variable is j z ≡ f z /2 and j 0 ≡ f 2 z /2. The ratio of the α i can be tuned by adjusting the optical frequency ω, giving a variety of Hamiltonians interesting for quantum information tasks [18].
To apply this formalism to nonlinear metrology, we generalize the CCV method to the nonlinear optics regime, i.e., we include higher-order processes in the effective Hamiltonian. For this purpose, naïve application of higher-order perturbation theory fails due to the appearance of vanishing resonance denominators, and degenerate perturbation theory [19] is required. We present the method by way of an example, the D 2 line of 87 Rb, one of the most used transitions for atom-light interactions.

Derivation of the Effective Hamiltonian
We consider the 5 2 S 1/2 → 5 2 P 3/2 transition (the D 2 transition at 780 nm). The ground (3,3). We use these states as a basis, with the ground states preceding the excited states. We calculate the singleatom Hamiltonian h eff , and note that the ensemble Hamiltonian H eff = i h (i) eff is found simply by replacing single-atom operators such as j with collective operators J ≡ i j (i) . The unperturbed Hamiltonian is h 0 = l ω l |l l|, or in matrix notation where ⊕ indicates a direct sum, and I d is the identity matrix of dimension d. We choose the origin of energy such that ω F =1 = 0, and define ∆ ≡ ω F =2 . We work in a frame rotating with the laser frequency ω = ω F ′ =0 + δ. In this frame, the Hamiltonian is In the rotating wave approximation, the single- If E ± are the amplitudes for the sigma-plus/minus components, respectively, of E E E, then with q = m F ′ − m F . Note that q = 0 transitions (π-transitions) are not considered because the z-propagating beam cannot contain this polarization. The dipole matrix elements are related to the "matrix element" J| |er q | |J ′ ≡ D JJ ′ ≈ 3.58410 −29 C · m by angular-momentum addition rules. We follow the conventions given in Steck [20]. In this way, we arrive to the perturbation Hamiltonian To obtain the effective Hamiltonian, we follow Klein [19]. The notation of that work is somewhat obscure, so for ease of understanding we repeat the main results. From Equation (A7) of that work, we have the t-order contribution to the effective Hamiltonian Where k 1 , . . . , k t−1 are non-negative integers, the A are real coefficients, the O, denoted "(k 1 , k 2 , . . . , k t )" by Klein, are operators, and the sum is taken over all {k} satisfying t−1 l=1 k l = t−1. The A are given in Table I of that work and the O are given in Equation (A1) as with P 0 being the projector onto the degenerate subspace and by Equation (II.A.5) where E 0 is the energy of the degenerate subspace. In our case we have chosen E 0 = 0. We can then directly calculate the second-and fourth-order contributions. We are only concerned with h eff as it acts on the F = 1 subspace, that is, with a 3 × 3 matrix, and it is convenient to express it in terms of the pseudo-spin components j 0 , j x , j y , j z and the Stokes components S 0 , S x , S y , S z defined above. Summing the second-order contributions we find H Similarly, the fourth-order contribution is, dropping terms in S 2 0 , H Note that the term in N A arises because h (4) eff contains a self-rotation term of the form β (0) m=0 S 2 Z P m=0 where P m=0 is a projector onto the state |F = 1, m F = 0 . We express this in terms of J 0 and N A using i P (i) , the coefficients, shown graphically in Figure 1, are β (0)

Application to nonlinear metrology
The β terms are nonlinear in S, indicating a photon-photon interaction. We expect these terms to describe polarization effects of fast electronic nonlinearities including saturation and four-wave mixing. As in the linear case, the frequency dependence of the β terms provides considerable flexibility in designing a light-matter interaction. Applied to quantum metrology, these terms produce SH scaling, because they are nonlinear in the S collective variables, while the atomic variables J i , N A play the role of the parameter. The β (0) and β (1) terms are analogous to Hamiltonians considered by Boixo et al. [2]. The β (1) term ∝ S 0 S z , in particular, achieves SH scaling without input or generated entanglement [4]. The β (2) terms describes a nonlinear tensorial contribution, and does not appear to have been considered yet for nonlinear metrology.

Quantum Noise
To understand the quantum noise in this system, we define polarization operatorŝ S i ≡ 1 2 (a † + , a † − )σ i (a + , a − ) T , where a ± are annihilation operators for the ± circular polarizations of a mode defined by the pulse shape. These obey angular momentum commutation relations [Ŝ i ,Ŝ j ] = iε ijkŜk and are related to the Stokes parameters bŷ S i = S i /2γ, where γ ≡ ωZ 0 /2T A is the single-photon intensity, T is the pulse duration, A is the beam area, and Z 0 is the impedance of free space. The total number of photons is N L = 2Ŝ 0 . For a typical input, a coherent state, (Ŝ X ,Ŝ Y ,Ŝ Z ) = (Ŝ 0 , 0, 0) and var(Ŝ i ) =Ŝ 0 /2.
Evolution under this effective Hamiltonian produces, to first order in the interaction time τ ,Ŝ plus terms containingŜ that are negligible for the given input coherent state of the light. This evolution physically corresponds to a paramagnetic Faraday rotation of the input linear polarization. In a metrological scheme one would measure this polarization rotation and from it estimate the atomic variable J Z .
For small rotation, i.e. φ ≡Ŝ We can identify the value of J (in) Z that solves Eq. (20) as the sensitivity, or precision of the estimation, δJ Z . We find Thus the sensitivity will have a transition from shot-noise to SH scaling with increasing N L . As indicated in Figure 1, there are points in the spectrum where either α (1) or β (1) vanish, allowing pure nonlinear or pure linear estimation of the same atomic variable. In another scenario, an unpolarized input state (Ŝ X ,Ŝ Y ,Ŝ Z ) = (0, 0, 0) gives rise to dynamics dominated by the β (0) terms ∝ S 2 Z , sometimes called the "one-axis twisting Hamiltonian." This describes a self-rotation of the optical polarization, and can be used to generate polarization squeezing and also to obtain sensitivity scaling as N N N A , using an entanglement-generating strategy described in reference [4].

Conclusion
We have generalized the formalism of continuous collective variables to the nonlinear regime. The resulting nonlinear effective Hamiltonian includes several distinct nonlinear couplings with strengths widely tunable via the probe light frequency. This allows the production of model Hamiltonians proposed for nonlinear metrology, including both models that generate entanglement and those which achieve super-Heisenberg scaling without entanglement. Similar nonlinear probing techniques could improve optical probing of atomic clocks [21,13] and atomic magnetometers [22,15,23]. Unlike previous proposals, the atomic ensemble system allows both linear and nonlinear estimation of the same atomic variables.