Spin Squeezing as a Probe of Emergent Quantum Orders Ilija K. Nikolov1 Stephen Carr12 Adrian G. Del Maestro34 Chandrasekhar

2025-05-03 0 0 8.54MB 7 页 10玖币

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Spin Squeezing as a Probe of Emergent Quantum

Orders

Ilija K. Nikolov1, Stephen Carr1,2, Adrian G. Del Maestro3,4, Chandrasekhar

Ramanathan5, and Vesna F. Mitrovi´

1Department of Physics, Brown University, Providence, RI 02912, USA

2Brown Theoretical Physics Center, Brown University, Providence, Rhode Island

02912-1843, USA

3Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996,

USA

4Min H. Kao Department of Electrical Engineering and Computer Science, University of

Tennessee, Knoxville, TN 37996, USA

5Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA

E-mail: vemi@brown.edu

(Received August 4, 2022)

Nuclear magnetic resonance (NMR) experiments can reveal local properties in materials, but

are often limited by the low signal-to-noise ratio. Spin squeezed states have an improved res-

olution below the Heisenberg limit in one of the spin components, and have been extensively

used to improve the sensitivity of atomic clocks, for example [1]. Interacting and entangled

spin ensembles with non-linear coupling are a natural candidate for implementing squeez-

ing. Here, we propose measurement of the spin-squeezing parameter that itself can act as a

local probe of emergent orders in quantum materials. In particular, we demonstrate how to

investigate an anisotropic electric ﬁeld gradient via its coupling to the nuclear quadrupole

moment. While squeezed spin states are pure, the squeezing parameter can be estimated for

both pure and mixed states. We evaluate the range of ﬁelds and temperatures for which a

thermal-equilibrium state is suﬃcient to improve the resolution in an NMR experiment and

probe relevant parameters of the quadrupole Hamiltonian, including its anisotropy.

KEYWORDS: Spin squeezing, Quadrupole moment, Electric ﬁeld gradient, NMR

1. Introduction

Developing local probes of matter is relevant not only for basic research in quantum ma-

terials but also for generating and exploiting speciﬁc properties. Nuclear magnetic resonance

(NMR) is one of the forefront spectroscopic techniques for microscopic study of magnetic

systems and investigation of quantum phases of matter [2–5]. Nevertheless, because much

of the NMR technique relies on a clear resolution of the spectral lines, complex and un-

resolved lines become a real obstacle. An important example is Ba2NaOsO6(BNOO), a

Mott insulator with strong spin-orbit coupling (SOC), which is believed to host a complex

multipolar-ordered phase, that in the intermediate temperatures is characterized by broad-

ening of the NMR spectra [6–11], preventing a determination of the exact microscopic nature

of this exotic phase of matter.

In high-precision quantum metrology, measurement resolution is increased through squeezed

states generated by non-linear operations, and thus they are sensitive to rotations [12–14].

Here, we propose an enhanced NMR probe using squeezing techniques, depicted in Fig. 1a.

Speciﬁcally, we show how spin squeezing can enable probing of the microscopic nature of

arXiv:2210.03697v1 [quant-ph] 7 Oct 2022

complex emergent orders, even when no speciﬁc features can be resolved in traditional NMR

spectroscopic measurements. Previous work has detailed either the quadrupole coupling in

NMR [15], or the squeezing parameter of nuclei in electric ﬁeld gradients [16], but the two

were not put together in the context of probing complex orders by NMR as presented here.

We evaluated the performance of our proposed technique using PULSEE [17].

2. Squeezing as an enhanced NMR probe

We will now examine how much squeezing diﬀerent initial spin states produce under the

quadrupole Hamiltonian.

2.1 Coherent spin states and the squeezing Hamiltonian

Coherent spin states (CSS) are an eigenstate of the spin momentum operator in a given

direction (θ, φ) that saturate the Heisenberg uncertainty relation [14]. In terms of the eigen-

states of ˆ

Iz, CSS are deﬁned as

|ζ(θ0, φ0)i=

m=−I2I

I+m1

cos(θ0/2)I+msin(θ0/2)I−mei(I−m)φ0|I, mi,

where Iis the nuclear spin number. The CSS can also be written as |αi ∝ eαI−|I, Iifor

α= tan(θ0

2)eiφ0. A spin squeezed state (SSS) has a correlated variance that is smaller than

the Heisenberg limit in one spin component, at the expense of another non-commuting spin

component [14]. The Husimi Qfunction is used to illustrate the diﬀerence between CSS and

SSS in Fig. 1b. Any single spin-1/2 system is a one-elementary-spin CSS, and thus cannot be

correlated and squeezed [12,14]. However, quadrupole nuclei, I > 1/2, are a natural candidate

for producing SSS because a nonlinear spin interaction gives nontrivial quantum correlations

between neighboring nuclear spins.

Fig. 1. (a) Schematic of the squeezing NMR probe.

While coherent spin states (CSS) give best results, here,

we show that a rotated thermal-equilibrium state is suf-

ﬁcient at certain ﬁelds and temperatures. (b) Husimi Q

function for the initial thermal-equilibrium state ˆρ0of

Eq. 2, CSS |ζ(0,0)i, and spin squeezed state (SSS), by

the Hamiltonian Eq. 1 (η= 1) at time t= 0.5ω−1

Q. The

Qfunction is obtained by taking hα|ρ|αifor α=x+iy.

Local symmetry breaking, often-

times caused by crystalline lattice dis-

tortions and complex multipolar or-

der, may remove rotational symmetry

and induce a non-symmetric electronic

charge distribution. Such a non-zero

electric ﬁeld gradient (EFG) couples

to a nuclear quadruple moment, thus

aﬀecting an NMR observable, and be-

coming a sensitive local probe. The

quadrupole coupling in the principal

axes (PAS) of the EFG is given by

HQ=ωQ

2h(3ˆ

z−ˆ

1)+η(ˆ

x−ˆ

y)i,(1)

where ωQis the coupling strength, and

the NMR splitting between peaks is

νQ= 3ωQ/2πfor I= 3/2 & η= 0.

2.2 Thermal-equilibrium states

In order to extract useful informa-

tion about the microscopic nature of the material via spin squeezing protocols, the initial

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SpinSqueezingasaProbeofEmergentQuantumOrdersIlijaK.Nikolov1,StephenCarr1;2,AdrianG.DelMaestro3;4,ChandrasekharRamanathan5,andVesnaF.Mitrovic11DepartmentofPhysics,BrownUniversity,Providence,RI02912,USA2BrownTheoreticalPhysicsCenter,BrownUniversity,Providence,RhodeIsland02912-1843,USA3DepartmentofPhy...

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Spin Squeezing as a Probe of Emergent Quantum Orders Ilija K. Nikolov1 Stephen Carr12 Adrian G. Del Maestro34 Chandrasekhar

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