Quantum-enhanced parameter estimation with entangled spin-waves

We describe a quantum-enhanced parameter estimation protocol whereby a phase shift on a single ensemble iof the quadripartite statei ∈ {a, b, c, d}can be detected with efficiency beyond that for any separable state.

Specifically, we consider aπ-phase shiftUˆπ,i = exp(iπnˆi)applied on an unknown spin-wave component i ∈ {a, b, c, d}(nˆi = ˆS†iSˆi) of the atomic stateρˆ

(A)

W , or on a spatial field modeγ2i ∈ {a2, b2, c2, d2}

of the photonic stateρˆ(Wγ) (ˆnγ2i = ˆa†γ2iˆaγ2i). Our goal is to find theπ-phase shifted ensemblei (optical modeγ2i), in asingle-measurementunder the condition that an average of one spin-wave is populated in

total; i.e., P_iTr(ˆniρˆ (A) W ) = 1 (or P iTr(ˆnγ2iρˆ (γ)

W ) = 1 for optical modes). As a quantum benchmark,

we consider an average success probabilityPs = 1₄PiTr( ˆΠ (u)

i Uˆπ,† iρˆ (A)

W Uˆπ,i)(failure probabilityPf =

1₋Ps) for distinguishing the phase shifted ensemblei(modeγ2i) among the four possibilities{a, b, c, d}

First, we consider an idealW-state_|W_io=|WiA(or|Wiγ2) with atomic phasesφi ∈ {φ1, φ2, φ3}(pho- tonic phasesφ0

i ∈ {φ01, φ02, φ03}). In this case, theπ-phase shifted entangledW-states|Wiif ∈ {|W (π)

a if,

|W_b(π)_if,|Wc(π)if,|Wd(π)if}can be detected deterministically, because|W(iπ)if = ˆUπ,i|Wioforms an or- thonormal complete set that spans the state-spaceρ1ˆ , resulting from the underlying symmetry of_|W_iowith

respect to any rotationUˆπ,i on a generalized Bloch sphere. Operationally, we set the verification phases β1,2−φ01,2 = 0 andβ3 −φ03 = π. Then, theπ-phase shifted ensemble i can be unambiguously dis-

criminated because the otherwise balanced output photon probabilities~pv ={p1000, p0100, p0010, p0001} =

{0.25,0.25,0.25,0.25}of the verification interferometer will be transformed to~pv = {1,0,0,0}for aπ-

phase induced on ensemblea, to~pv ={0,1,0,0}on ensembleb, to~pv={0,0,1,0}on ensemblec, and to

~

pv={0,0,0,1}on ensembled, each with success probabilityPs(ent)= 1.

For fully separable states_|Ψ_io = |ψaia|ψbib|ψcic|ψdid with|ψiii = P∞

n=0c (n)

i |nii, we displace the resultingπ-phase shifted state_|Ψ(πi)if = ˆUπ,i|Ψio with a local unitary transformationVˆi|ψiii = |0ii. The overall processVˆaVˆbVˆcVˆdUˆπ,i maps the initial product state|Ψio intoVˆaUˆπ,a|ψaia|0ib|0ic|0id (phase shift on ensemble a),_|0iaVˆbUˆπ,b|ψbib|0ic|0id (ensemble b),|0ia|0ibVˆcUˆπ,c|ψcic|0id(ensemble c),

and_|0ia|0ib|0icVˆdUˆπ,d|ψdid(ensemble d), with only oneicontaininghnˆii>0excitations. Thus, we can unambiguouslyidentify the phase shifted ensembleigiven a photodetection, albeit with a failure probability

Pf = 1₄Pi|ih0|VˆiUˆπ,i|ψiii| 2 ₌ 1

4 P

i|ihψi|Uˆπ,i|ψiii|

2 _{arising from inconclusive null events} (i.e.,_|0000_ih0000_|). We derive the maximum success probabilityPs(max)= 1−P_f(min)and the optimal state

|Ψ_io=|Ψioptby minimizingPf(min)over all possible realizations ofc

(n)

i satisfying P

ihψi|nˆi|ψiii = 1. Specifically, we find that an optimal (pure) separable state_|Ψ_iopt =Qi(

p

3/4_|0_ii + p

1/4_|1_ii)can be used for the parameter estimation protocol to inferi withPs(max) = 0.75. Similarly, maximum success

probabilityPs(coh) can be derived for multimode coherent states Qi|αiii, giving a classical bound of Ps(coh)= 1−1/e.

Finally, we consider the upper bound Ps(max) for mixed separable states ρˆ(osep) with pure state de-

compositions ρˆ(osep) = P_mpm|ΨmiohΨm|. Generally, the transformations Vˆi, as discussed above, do not exist forρˆ(osep), excluding the possibility of an unambiguous state discrimination. Thus, the success

P

mpmPs(|ΨmiohΨm|)≤Ps(max)= 0.75. Importantly, the maximum success probabilityPs(max)= 0.75,

attainable for anyρˆ(osep), is less thanPs(ent)= 1for entangled states|Wio. Thus, the entangled spin-waves

in the experiment can be applied for sensing an atomic phase shift beyond the limit for any unentangled state. Comprehensive analysis of our protocol including experimental imperfections (e.g., detection efficiency) as well as other measurement strategies will be discussed elsewhere.

Chapter 3

Atom-light interactions: waveguide

QED

3.1

Introduction

In this chapter, we describe the underlying physics of atom-photon interactions along a 1D waveguide. First, we describe photon transport properties in the presence of a single photon, including reflection transmission spectra, saturation behavior, and photon correlation. Then we discuss the collective effects that emerge with two atoms coupled to the waveguide mode. Depending on the distance between atoms, the collective effects change from superradiance to dipole-dipole interaction. We further considerNatom coupled to the 1D PCW, which can separate these two collective effects across the band edge of the PCW and introduce finite-range dipole-dipole interaction inside the band gap of the PCW.