PHYSICAL REVIEW A 93, 013415 (2016)

Nonperturbative calculation of phonon effects on spin squeezing

D. Dylewsky,1 J. K. Freericks,1 M. L. Wall,2 A. M. Rey,2 and M. Foss-Feig3

1Department of Physics, Georgetown University, Washington, DC 20057, USA2JILA, NIST, and University of Colorado, Boulder, Colorado 80309, USA

3Joint Quantum Institute and the Joint Center for Quantum Information and Computer Science, NIST,University of Maryland, College Park, Maryland 20742, USA

(Received 13 October 2015; published 15 January 2016)

Theoretical models of spins coupled to bosons provide a simple setting for studying a broad range of importantphenomena in many-body physics, from virtually mediated interactions to decoherence and thermalization. Inmany atomic, molecular, and optical systems, such models also underlie the most successful attempts to engineerstrong, long-ranged interactions for the purpose of entanglement generation. Especially when the couplingbetween the spins and bosons is strong, such that it cannot be treated perturbatively, the properties of suchmodels are extremely challenging to calculate theoretically. Here, exact analytical expressions for nonequilibriumspin-spin correlation functions are derived for a specific model of spins coupled to bosons. The spatial structureof the coupling between spins and bosons is completely arbitrary, and thus the solution can be applied to systemsin any number of dimensions. The explicit and nonperturbative inclusion of the bosons enables the study ofentanglement generation (in the form of spin squeezing) even when the bosons are driven strongly and nearresonantly, and thus provides a quantitative view of the breakdown of adiabatic elimination that inevitably occursas one pushes towards the fastest entanglement generation possible. The solution also helps elucidate the effectof finite temperature on spin squeezing. The model considered is relevant to a variety of atomic, molecular, andoptical systems, such as atoms in cavities or trapped ions. As an explicit example, the results are used to quantifyphonon effects in trapped ion quantum simulators, which are expected to become increasingly important as theseexperiments push towards larger numbers of ions.

DOI: 10.1103/PhysRevA.93.013415

I. INTRODUCTION

Spin-spin interactions play a crucial role in the generationof entanglement for applications in quantum information andmetrology [1]. In atomic, molecular, and optical (AMO) sys-tems, intrinsic spin-spin couplings are often extremely weak,and generating entanglement much faster than decoherencetime scales remains an important and challenging task. Onestrategy to realize strong, long-range spin couplings, whichis routinely employed in both trapped-ion systems and cavityQED, is to mediate them via a collection of auxiliary bosonicdegrees of freedom (e.g., phonons in the case of trappedions [2] and photons in cavity QED [3]). If these bosonicmodes are far off resonance and the temperature is sufficientlylow, they are only virtually occupied and can be (pertur-batively) adiabatically eliminated [4]. This procedure yieldsapproximate spin-only models that are generally easier totreat theoretically and often more desirable experimentally. Forexample, if ω is the characteristic energy input needed to createa boson and g is the characteristic coupling strength betweenthe spins and bosons, spin-spin interactions of strength ∼g2/ω

can be generated (see Fig. 1). However, the limit in which thisprocedure is quantitatively valid (ω � g) is directly at oddswith the limit in which the spin dynamics is fastest (largeg2/ω). In order to overcome intrinsic time-scale limitations,experiments are often forced to operate in parameter regimeswhere perturbative adiabatic elimination is not quantitativelyjustified, and a simple spin-only picture is questionable.

More generally, coupled spin-boson models play a centralrole in our understanding of quantum systems in contactwith an environment, and have been studied extensivelyin both the condensed-matter and AMO communities for

decades. Even in the case of a single spin coupled to manynoninteracting bosons [5] or many spins coupled to a singlebosonic mode [6,7], remarkably rich and complex behavioremerges. The general problem of many spins coupled tomany bosons has very few analytically tractable limits andis extremely difficult to study numerically, especially out ofequilibrium and in more than one spatial dimension. As such,exact solutions, even of the simplest nontrivial models, canplay an important role in extending our understanding of theseintricate coupled quantum systems.

Here we provide an exact solution for the far-from-equilibrium dynamics of a collection of spins (with S = 1/2)coupled uniaxially to a collection of noninteracting bosonicmodes (see Fig. 1). The solution is valid for arbitrary spatialstructure of the bosonic modes, and therefore applies to

∼ ω . . . . . .∼ g

g2/ω

FIG. 1. (a) Schematic of the model, with spins coupled (atcharacteristic coupling strength g) to a collection of noninteractingbosonic modes (at characteristic energy ω). When the phonons arefar-off-resonance, ω � g, one can generally derive an approximatespin-only description of the system, which has direct spin-spincouplings of order ∼g2/ω.

2469-9926/2016/93(1)/013415(12) 013415-1 ©2016 American Physical Society

DYLEWSKY, FREERICKS, WALL, REY, AND FOSS-FEIG PHYSICAL REVIEW A 93, 013415 (2016)

systems in any number of spatial dimensions. To a highdegree of approximation, this model describes the dynamicsof trapped-ion crystals when they are perturbed by a spin-dependent force [8]. When the coupled spin-phonon systemis driven far off (phonon) resonance, the phonons can beadiabatically eliminated and the dynamics is governed byan Ising Hamiltonian acting only on the spins. Severalexperimental groups have exploited this result to engineerspin-entangled states of trapped ions [9]. However, as systemsizes increase, it is crucial to characterize and understand thediscrepancies from this idealized situation that arise from finitepopulation of the phonons, either due to their nonzero initialtemperature or due to deviations from the far-off-resonancelimit. As a demonstration of its utility, the solution is usedto calculate the spin squeezing generated dynamically byinitializing the system in a product state of the spin and phonondegrees of freedom that is far from equilibrium. The solutionyields expressions that are efficient to evaluate numerically,enabling the calculation of dynamics for most experimentallyachievable system sizes (N � 103 spins and phonons).

The organization of the paper is as follows. In Sec. II wepresent the model and review its realization with trapped ionsin a simple context. In Sec. III A we explain the formalismused to derive the exact results for this system. Details for howto explicitly calculate correlation functions are presented inSec. III B and numerical results for spin squeezing follow inSec. IV. In Sec. V we discuss several interesting directions forfuture research.

II. THE MODEL AND ITS REALIZATION WITH IONS

The model we solve consists of a collection of Nb

bosonic modes coupled uniaxially to Ns spins. The spin-bosoncouplings can be time dependent, and it is useful to breakthe Hamiltonian up into static and time-varying parts asH(t) = H0 + V(t) [10], with

H0 =Nb∑α=1

ωαa†αaα,

V(t) =Ns∑j=1

Nb∑α=1

σ zj

[gα

j (t)a†α + gα

j (t)aα

]. (1)

Here a†α (aα) creates (annihilates) a boson in a particular mode

α, and σ rj are the (r = x,y,z) Pauli spin matrices for the

j th spin. The boson energies ωα in H0 are arbitrary, as arethe coupling constants gα

j (t) in V(t) [the overbar on gαj (t)

denotes complex conjugation]. A coupling to longitudinalfields ∼∑

j hj σzj could also be included in H0, but such a

term can be removed by working in a suitably rotating frame,and so it is ignored from the outset. Also, note that termscoupling to spin directions other than z are not included andin general prohibit an exact solution.

For time-independent couplings (or, alternatively, at fixedt), the eigenstates of the above Hamiltonian are product statesbetween all spins and suitably displaced vacuum states of thebosonic modes, and hence equilibrium properties of the modelare essentially classical. However, we are concerned with theresponse of the system when driven out of equilibrium; theensuing relaxation dynamics is highly nontrivial, generically

being accompanied by entanglement growth between the spinsand bosons. If the couplings gα

j are independent of time andif the interactions are weak (gα

j � ωα), then V can be treatedperturbatively. For a system initialized in the boson vacuum,the bosons will only be populated virtually in the dynamics,which can therefore be described by an effective time-independent spin-only Hamiltonian. For example, working tosecond order in V , one obtains

Hspineff =

Ns∑j,k=1

Jjkσzj σ z

k , Jjk =Nb∑α=1

gαj gα

k

ωα

. (2)

Even at this level of approximation, the spin dynamics isnontrivial, and exact time-dependent correlation functionswere only recently obtained for general coupling constantsJjk [11–13]. If the Jjk do not depend on space, then Eq. (2) re-duces to the single-axis-twisting model [14], which is a specialcase of the more general Lipkin-Meshkov-Glick model [15]. Inthis case, the analysis is greatly simplified because the squareof the total spin becomes a good quantum number, and themodel can be solved in terms of collective spin variables [6].

Before solving for the time dependence of correlationfunctions induced by H(t), recall that Eq. (1) appears naturallyin the description of various AMO systems. For example,in ion traps the spin is realized by some internal structureof an ionized atom and the bosons are excitations of thevibrational modes of the crystalized ions (phonons). In cavityQED, where identical [16] or closely related [17] models canbe realized, the spins are two-level neutral atoms and thebosons are photons in long-lived cavity modes. In the contextof trapped ions, the model can emerge in several differentways [8], the conceptually simplest of which is through theapplication of a spin-dependent optical force to a crystalof ions [18–20] (though see Refs. [21,22] for a commonalternative realization). For example, in the spirit of Ref. [23],the ions can be driven by two lasers with difference frequencyμ and relative wave vector krel, as in Fig. 2(a). Each ion isassumed to possess two long-lived hyperfine states labeled |↑〉and |↓〉; they will represent the spin degree of freedom. If the

|

|

μ

(b)(a)

2π

krel

Δ

|eΔ/2

Δ/2

μ

Ω0

Ω0

FIG. 2. Realization of Eq. (1) with trapped ions. (a) Ions driventransversely via stimulated Raman transitions. The bosonic modes arerealized as the normal modes of oscillation of the crystal around itsequilibrium configuration (here shown as a one-dimensional chain).(b) Simplified level diagram illustrating the essential ingredients forgenerating spin-phonon couplings in trapped ions. Here, �0 denotesthe strength of the coupling between the states |↑〉,|↓〉 and theoptically excited state |e〉 (figure not drawn to scale).

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NONPERTURBATIVE CALCULATION OF PHONON . . . PHYSICAL REVIEW A 93, 013415 (2016)

energy splitting between these two states is �, then the ionHamiltonian in the absence of driving is

Hion =Nb∑α=1

ωαa†αaα + �

2

Ns∑j=1

σ zj . (3)

For simplicity, we assume that the crystal possesses a directionalong which a single set of decoupled normal modes oscillate,the z direction, and that krel points along this direction; theindex α in Eq. (3) enumerates this set of modes. We alsoassume that the laser couples both spin states to a singleoptically excited state |e〉, and choose the laser frequencyso that the detunings from the optical transition are equalin magnitude and opposite in sign (±�/2) for the two spinstates [see Fig. 2(b)]. If the single-photon Rabi frequency �0

(assumed to be the same for both the |↑〉,|↓〉 ↔ |e〉 transitions)is small compared to the single-photon detuning �/2, theelectronic excited state can be adiabatically eliminated, leavingbehind an ac Stark shift for each spin state that oscillatesin time at the difference frequency μ and in space at thedifference wave vector krel. Combined with a rotating-waveapproximation (i.e., ignoring all terms with optical-frequencytime dependences) and a frame transformation to remove theenergy splitting �, adiabatic elimination of |e〉 yields

Hiondriven(t) =

Nb∑α=1

ωαa†αaα + �

Ns∑j=1

cos(krelzj − μt)σ zj . (4)

Here � ≡ 4�20/� is the characteristic strength of the spin-

dependent ac Stark shift experienced by the states |↑〉 and |↓〉,and krel = |krel|. The position operator along the z directionfor the j th ion, denoted by zj , can be expanded in termsof creation and annihilation operators for the normal modesof the crystal as krelzj = ∑

α ηαbαj (a†

α + aα). Here, bαj is

the orthogonal normal-mode transformation matrix and ηα =krel

√�/2mωα (restoring � temporarily) parametrizes how

small the characteristic ion displacements in the ground stateof the mode α are compared to the length scale k−1

rel over whichthe applied spin-dependent potential changes appreciably. Inthe Lamb-Dicke limit, ηα � 1 for all α, and working to lowestorder in ηα , Eq. (4) becomes

Hiondriven(t) ≈

Nb∑α=1

ωαa†αaα

+� sin(μt)Ns∑j=1

Nb∑α=1

σ zj ηαbα

j (a†α + aα), (5)

which is Eq. (1) with gαj (t) = gα

j (t) = � sin(μt)ηαbαj . Having

motivated the general form of the Hamiltonian in Eq. (1),we now proceed to compute correlation functions evolvingunder it. With the formal solution in hand, however, we willeventually return to the context of trapped ions and Eq. (5)when discussing the application of our results to computingspin squeezing in Sec. IV.

III. SOLUTION FOR CORRELATION FUNCTIONS

The following section includes technical derivations thatare not essential for following most of the discussion in

Sec. IV; readers wishing to skip these details can proceeddirectly to that section. Because V(t) in Eq. (1) is explicitlytime dependent, the time-evolution operator correspondingto H(t) must be written as a time-ordered product, whichcomplicates the calculation of observables. The first step inobtaining closed forms for correlation functions, therefore,is to obtain an explicit form of the time-evolution operatorthat does not require time ordering. It is well known (seeRefs. [24,25]) that this can be accomplished via appropriatefactorizations of the time-evolution operator. However, in theinterest of maintaining a self-contained solution of the model,this procedure is briefly reviewed in Sec. III A. With an explicitform of the time-evolution operator in hand, we then move onto our main formal results in Sec. III B, obtaining closed-formexpressions for spin-spin correlation functions.

A. Explicit form for the time-evolution operator

The time-evolution operator satisfies the equation of motioni∂U(t,t0)/∂t = H(t)U(t,t0) with respect to the full Hamil-tonian H(t) defined in Eq. (1), and can be written as atime-ordered product,

U(t,t0) = Tt exp

(−i

∫ t

t0

dτ H(τ )

), (6)

with Tt the time-ordering operator. The first step in rewritingthe time-evolution operator without the need for time orderingis to move to the interaction picture with respect to H0.Defining the perturbation in the interaction picture

VI (t,t0) = eiH0(t−t0)V(t)e−iH0(t−t0)

=Ns∑j=1

Nb∑α=1

σ zj

[gα

j (t)eiωα(t−t0)a†α + gα

j (t)e−iωα(t−t0)aα

],

(7)

we can write U(t,t0) = e−iH0(t−t0)UI (t,t0), with

UI (t,t0) ≡ Tt exp

(−i

∫ t

t0

dτ VI (τ,t0)

). (8)

Here H0(t − t0) is the product of the unperturbed HamiltonianH0 and the time difference t − t0; the latter should not beconfused as an argument of H0, which is manifestly timeindependent.

The next step follows the textbook problem of drivenharmonic oscillators [26,27]. Defining the operator

W(t,t0) ≡∫ t

t0

dτ VI (τ,t0), (9)

which satisfies dW(t,t0)/dt = VI (t,t0), we further factorizeUI (t,t0) = e−iW(t,t0)U(t,t0), with

U(t,t0) ≡ eiW(t,t0)UI (t,t0). (10)

The benefit of this factorization becomes immediately clearupon differentiating U(t,t0) with respect to t . We take thederivative with the help of the equality

deiW(t,t0)

dt=

(d

dλei[W(t,t0)+λVI (t,t0)]

)∣∣∣∣λ=0

, (11)

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DYLEWSKY, FREERICKS, WALL, REY, AND FOSS-FEIG PHYSICAL REVIEW A 93, 013415 (2016)

which can be verified by Taylor expanding both sides.Crucially, the commutator of W(t2,t0) with VI (t1,t0) dependsonly on the operators σ z

j , and hence commutes with both Wand VI at all other times,

[W(t3,t0),[W(t2,t0),VI (t1,t0)]] = 0, (12)

[VI (t3,t0),[W(t2,t0),VI (t1,t0)]] = 0. (13)

As an immediate consequence of Eqs. (12) and (13), one canmake the replacement

ei[W(t,t0)+λVI (t,t0)] = eiW(t,t0)eiλVI (t,t0)eλ[W(t,t0),VI (t,t0)]/2

in Eq. (11), evaluate the right-hand side, and thereby obtain

d

dtU(t,t0) = 1

2[W(t,t0),VI (t,t0)]U(t,t0). (14)

Because Eqs. (12) and (13) hold for all times, Eq. (14) can beintegrated without regard for time ordering, yielding U(t,t0) =exp{∫ t

t0dτ [W(τ,t0),VI (τ,t0)]/2}. The full time-evolution op-

erator can now be written as

U(t,t0) = e−iH0(t−t0)e−iW(t,t0)

× exp

(1

2

∫ t

t0

dτ [W(τ,t0),VI (τ,t0)]

). (15)

At this point we have reduced the evaluation of a time-orderedproduct to the evaluation of the product of three different time-evolution terms, each generated by an operator that commuteswith itself at different times [but note that, in general, theorder of the three exponential factors in Eq. (15) must bemaintained]. It turns out that for the choice of Hamiltonian inEq. (1), only the first and second factors do not commute witheach other and need to maintain their relative ordering.

The operator W(t,t0) can be written explicitly as

W(t,t0) = i

Ns∑j=1

Nb∑α=1

[Aα

j (t,t0)a†α − Aα

j (t,t0)aα

]σ z

j , (16)

with

Aαj (t,t0) = −i

∫ t

t0

dτ gαj (τ )eiωα(τ−t0). (17)

Taking the commutator in the third factor of Eq. (15) andintegrating yields

U(t,t0) = exp[−iH0(t − t0)] exp[−iW(t,t0)]

× exp

(− i

Ns∑j,k=1

Sjk(t,t0)σ zj σ z

k

), (18)

with

Sjk(t,t0) = ImNb∑α=1

∫ t

t0

dτ

∫ τ

t0

dτ ′

×(

gαj (τ ′)gα

k (τ ) + gαk (τ ′)gα

j (τ )

2

)eiωα(τ ′−τ ). (19)

Note that we have utilized [σ zj ,σ z

k ] = 0 to write the Isingcoefficients in an explicitly symmetric form, so Sjk = Skj .

B. Calculating time-dependent expectation valuesof spin operators

To simplify the notation in what follows, we set t0 = 0 andsuppress its appearance, in which case the time-dependentexpectation value of an operator O is given by O(t) =〈ψ0|U†(t)OU(t)|ψ0〉, with |ψ0〉 the initial wave function att = 0 and the evolution operator given in Eq. (18). We will onlyconsider expectation values of spin operators; because theycommute with the bosonic Hamiltonian H0, the boson-onlypart of the evolution operator always cancels out, leaving

O(t) = 〈ψ(t)|O|ψ(t)〉,

|ψ(t)〉 = e−iW(t) exp

⎛⎝−i

Ns∑j,k=1

Sjk(t)σ zj σ z

k

⎞⎠ |ψ0〉. (20)

The results that follow can easily be worked out for arbitraryproduct states between all of the degrees of freedom (spinand boson). However, in order to simplify the discussion wepresent results only for initial states where all spins initiallypoint along the x axis,

|ψ0〉 = 2−Ns/2∑

σ1,...,σNs=↑,↓

|σ1〉 ⊗ · · · ⊗ |σNs〉 ⊗ |ϕ1〉 ⊗ · · · ⊗ |ϕNb〉.

(21)

The bosons are taken to be in a product state betweenthe different bosonic modes, but we allow the state of anyparticular mode, |ϕα〉, to be arbitrary. While Eq. (21) mayseem restrictive, it is a natural choice for the generation ofspin squeezing, as will be made clear in Sec. IV.

To characterize the spin dynamics of the system, wecompute a number of time-dependent expectation values ofproducts of spin operators. Defining spin raising (+) andlowering (−) operators σ±

j = (σ xj ± iσ

y

j )/2 for each spin j ,we compute

⟨σ a

m

⟩ = 〈ψ(t)| σ am |ψ(t)〉 , (22)⟨

σ amσ z

n

⟩ = 〈ψ(t)| σ amσ z

n |ψ(t)〉 , (23)⟨σ a

mσ bn

⟩ = 〈ψ(t)| σ amσ b

n |ψ(t)〉 , (24)

with a,b = ± and m �= n. Note that all nontrivial expectationvalues of one or two spin operators can be obtained as linearcombinations of those in Eqs. (22)–(24) (correlation functionsinvolving only the σ z

j are independent of time, as [H(t),σ zj ] =

0).In order to calculate these expectation values, we make

a few observations about Eq. (20). First, the final factor ofthe time-evolution operator, which involves only the Isingspin operators, can be further factorized into a product ofexponentials (instead of an exponential of a sum of terms),because each term in the exponent commutes with every otherterm in the exponent. Second, we can factorize the exponentialof W(t) into factors that have fixed α (but still sum over j ),because those operators also commute with each other. Eachone of these factors that commutes with the operator O can bemoved from the right, through O, and then cancels against aninverse term on the left coming from the Hermitian conjugateof the time-evolution operator.

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NONPERTURBATIVE CALCULATION OF PHONON . . . PHYSICAL REVIEW A 93, 013415 (2016)

Using the identity

eiλσ zm σ±

m e−iλσ zm = σ±

m e±2iλ, (25)

valid for [λ,σ zm] = [λ,σ±

m ] = 0, together with

eiW(t,t0)σ±m e−iW(t,t0)

= eiW(t,t0) exp

(∓2

Nb∑α=1

[Aα

m(t)a†α − Aα

m(t)aα

])

× e−iW(t,t0)σ±m

= exp

⎡⎣∓2

Nb∑α=1

⎛⎝[

Aαm(t)a†

α − Aαm(t)aα

]

−Ns∑

j �=m

Im[Aα

m(t)Aαj (t)

]σ z

j

⎞⎠

⎤⎦σ±

m , (26)

which follow from standard operator identities, we cansimplify the expectation values of the operator averages thatwe are calculating. After some additional algebra, and definingthe displacement operators Dα(ϑ) = exp(ϑa†

α − ϑ aα) andmodified spin-spin couplings

Smn(t) = Smn(t) + 1

2Im

Nb∑β=1

Aβm(t)Aβ

n (t), (27)

the final results are as follows:

⟨σ a

m

⟩ = 1

2

Nb∏α=1

〈ϕα|D†α

[2aAα

m(t)]|ϕα〉

Ns∏j �=m

cos[4Smj (t)], (28)

⟨σ a

mσ zn

⟩ = ai

2

Nb∏α=1

〈ϕα|D†α

[2aAα

m(t)]|ϕα〉

× sin[4Smn(t)]Ns∏

j �=m,n

cos[4Smj (t)], (29)

⟨σ a

mσ bn

⟩ = 1

4

Nb∏α=1

〈ϕα| D†α

[2aAα

m(t) + 2bAαn(t)

] |ϕα〉

×Ns∏

j �=m,n

cos[4aSmj (t) + 4bSnj (t)]. (30)

From Eq. (17) we see that Im[Aαm(t)Aα

n(t)] = 0, and thereforeSmn(t) = Smn(t), whenever gα

m(t1)gαn (t2) = gα

n (t1)gαm(t2). This

situation is realized for the normal modes of ions in linearPaul traps, and also for the axial modes of Penning traps(though not the in-plane modes) [28]. There, gα

j (t) = gαj (t) =

� sin(μt)ηαbαj , since the normal-mode transformation matrix

bαj can always be chosen to be real.

C. Evaluation of the boson matrix elements

A boson matrix element of the form 〈ϕα|Dα(ϑ)|ϕα〉 caneasily be evaluated for an arbitrary state |ϕα〉 = ∑∞

n=0 cαn |n〉α ,

where |n〉α = 1√n!

(a†α)n|0〉α are normalized Fock states

of the αth mode. Writing the displacement operator as

Dα(ϑ) = e−|ϑ |2/2eϑa†α e−ϑ aα , straightforward algebra leads to

〈ϕα|Dα(ϑ)|ϕα〉 =∞∑

n,p,q=0

cαn+pcα

n+qϑpϑq

√(n + q)!(n + p)!

p!q!n!e|ϑ |2/2.

(31)

The complete boson matrix elements in Eqs. (28)–(30) followby taking the product of Eq. (31) over all of the Nb modeslabeled by α. Expectation values such as Eq. (31) can easilybe generalized to deal with finite-temperature states of thebosonic modes. For example, in what follows we consider asituation where all bosonic modes are thermally populated atan inverse temperature β, such that the initial boson densitymatrix is �(β) = ⊗

α ρα(β), with

ρα(β) = Zα(β)−1∞∑

nα=0

|nα〉〈nα| e−βωαnα , (32)

Zα(β) =∞∑

nα=0

e−βωαnα = 1

1 − e−βωα. (33)

Expectation values of the form 〈ϕα|Dα(ϑα)|ϕα〉 appearingin Eqs. (28)–(30) should be replaced with Dα(β,ϑα) ≡Tr[ρα(β)Dα(ϑα)]. Inserting Eq. (32) into this trace andutilizing Eq. (31), straightforward algebra leads to

Dα(β,ϑα) = exp[− 1

2 |ϑα|2 coth(βωα/2)]. (34)

IV. SPIN SQUEEZING

It is also possible to obtain closed-form expressions forhigher-order (n-spin) versions of the two-spin correlationfunctions derived above. However, already at the level oftwo-spin correlation functions, we can learn a great dealabout the time evolution of the system, and the nature of theentanglement that develops. For example, from the correlationfunctions in Eqs. (28)–(30), we can compute the variance ofthe spin distribution, which enables us to characterize spinsqueezing [14,29]. Spin squeezing is just one of many mea-sures of entanglement in a many-body system, which has thevirtue of quantifying the potential enhancement in precisionobtainable in Ramsey spectroscopy (as compared to the caseof unentangled spins) [30]. Moreover, it establishes a lowerbound on the depth of entanglement, i.e., the minimum numberof simultaneously entangled particles in the system [31].

If the bosonic modes are initially cooled to a vacuum state,driven weakly so that they can be adiabatically eliminated,and if the resulting effective spin-spin coupling strength J

is independent of the spatial distance between the spins, thedynamics is governed by the single-axis-twisting HamiltonianHsat = 4J SzSz, where Sz ≡ (1/2)

∑j σ z

j [14]. In this model,spin squeezing is generated by first polarizing the collectivespin vector along the x axis, and then letting it evolve underHsat. Making a mean-field approximation, Hsat ≈ 8J 〈Sz〉Sz −4J 〈Sz〉2, the dynamics can be understood as a precession aboutthe z axis in a direction determined by the mean z componentof the spin; the initial spin state has quantum fluctuations aboveand below the equator and therefore this dynamics causes it toget sheared and elongated, as in Fig. 3. Uncertainty along one

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DYLEWSKY, FREERICKS, WALL, REY, AND FOSS-FEIG PHYSICAL REVIEW A 93, 013415 (2016)

(π/2)y

timeH(t) (squeezing)

|ψ0

ΔSmin

x

y

z

θ

(θmin)x

FIG. 3. Illustration of an experimental protocol employed togenerate spin squeezing. Spins are forced into a nonequilibrium statepolarized along the x axis. The Hamiltonian then acts for some periodof time, causing squeezing of the initially Gaussian spin state, afterwhich the minimal variance of the spin distribution is mapped ontothe z axis and measured. The coordinate system shown correspondsto that used in Eq. (35).

axis is reduced (squeezed), while uncertainty in an orthogonaldirection is increased.

The extent of squeezing along a particular direction in theyz plane can be quantified by the parameter

ξ (θ ) = N 1/2s

�Sθ

|〈Sx〉|, (35)

where Sθ ≡ Sz cos θ+Sy sin θ and �Sθ = (〈S2θ 〉−〈Sθ 〉2)1/2.

The spin-squeezing parameter is then defined by minimiz-ing the standard deviation, �Smin = minθ �Sθ , such thatξ = minθ ξ (θ ) = N 1/2

s �Smin/|〈Sx〉|. Straightforward algebraenables the optimal angle to be expressed explicitly in termsof spin-spin correlation functions [29],

θ = 1

2arctan

(〈SzSy〉 + 〈Sy Sz〉〈SzSz〉 − 〈Sy Sy〉

). (36)

A. Connection to trapped ions

Equations (28)–(30) are very general, enabling a completedescription of spin-spin correlations in a variety of differentphysical systems, but they cannot be further simplified or eval-uated without choosing explicit forms for the boson spectrumωα and spin-boson couplings gα

j (t). In the remainder of Sec. IVwe compute the squeezing induced by the Hamiltonian inEq. (5) using parameters relevant for ions in a linear Paultrap, though many of the qualitative features discussed beloware insensitive to the detailed trap geometry and would besimilar for ions in a Penning trap if the axial modes werebeing driven [23]. Specifically, we examine the dynamics of 20ions, and assume that the wave-vector difference of the drivinglasers, krel, points along one of the two transverse principalaxes of the trap. In this configuration, the spins only coupleto normal modes oscillating along a single spatial directionand therefore the number of (coupled) phonon modes is equalto the number of spins; N ≡ Ns = Nb = 20. To calculate the

normal-mode eigenvectors and associated frequencies, we setthe ratio of longitudinal to transverse trap frequencies equalto 0.1. The mass of the ions enters the calculation only as anoverall scaling of the normal-mode frequencies and can beignored if all energies (times) are measured in units of thecenter-of-mass frequency ωc.m. (1/ωc.m.). The phonon modesare calculated in the standard fashion assuming a pseudopoten-tial approximation holds [28,32,33], which neglects any effectsdue to micromotion. We first find the equilibrium positions ofthe N ions in the trap for the given trap curvatures in thedifferent spatial directions and the mutual Coulomb repulsionof the ions. Then we determine the spring constant matricesby expanding the Coulomb interaction to quadratic orderabout equilibrium. Diagonalizing these dynamical matricesyields both the normal-mode oscillation frequencies ωα andthe associated orthonormal eigenvectors bα

j . Because thespring-constant matrices are real and symmetric, the bα

j arereal, and the spin-phonon coupling constants satisfy gα

j (t) =gα

j (t) = � sin(μt)ηαbαj .

Substituting this expression for gαj (t) into Eqs. (17) and (19)

and performing the integrals, we obtain

Aαj (t) = i�ηαbα

j

ω2α − μ2

(μ − eiωαt [μ cos(μt) − iωα sin(μt)]),

Sjk(t) = − �2N∑

α=1

η2αbα

j bαk

ω2α − μ2

(ωαt

2− ωα

4μsin(2μt)

− μ2 cos(μt) sin(ωαt) − μωα sin(μt) cos(ωαt)

ω2α − μ2

).

(37)

Here Aαj (t) is proportional to the interaction-picture phase-

space displacement of the j th spin as a result of periodicallydriving the mode α. Because this driving is periodic, thedisplacement amplitudes Aα

j (t) have a simple structure. Inparticular, for a single mode driven near resonance (|δα| � ωα ,with δα ≡ μ − ωα), it is straightforward to show that

Aαj (t) ≈ i

2

�ηαbαj

δα

(e−iδα t − 1). (38)

This amplitude traces a closed circle, vanishing at times suchthat δαt = 2nπ [Fig. 4(a)], with n an integer, at which thephonon matrix elements of the mode α in Eqs. (28)–(30)all become equal to unity. At these times, the mode α

becomes unentangled with the spins, regardless of its initialstate. Physically, the independence on initial state reflects theindependence of the period of a harmonic oscillator on itsdisplacement and implies that the periodic disentanglementof a given phonon mode occurs even at finite temperature,which was pointed out originally in Ref. [21]. Formally, it isimmediately apparent from Eq. (34) that finite-temperaturephonon expectation values of the form Dα(β,Aα

j (t)) willreturn to unity periodically for any inverse temperature β

[see Fig. 4(b)]. However, when multiple modes participatein the dynamics, they do not all disentangle at the sametime, with further off-resonance modes exhibiting phase-spaceexcursions with smaller radii but larger frequencies, as inFig. 4(c).

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NONPERTURBATIVE CALCULATION OF PHONON . . . PHYSICAL REVIEW A 93, 013415 (2016)

zα ω1ω2ω3ω4

δαt/2π

(a) (b) (c)

pα

zα

pαt = 2π/δα

0

ln α β, Aαj (t)

FIG. 4. Schematic phase-space dynamics of the phonon modes.(a) When driven close to resonance, a single mode periodically returnsto its original position in phase space (here pα = Im[Aα

j (t)] and zα =Re[Aα

j (t)]). (b) The phonon modes periodically disentangle at finitetemperature, as can be seen from the expectation values Dα(β,Aα

j (t))returning to unity. The overlap of a given mode with its initial value,however, decreases more sharply away from the recurrence times athigher temperature (here kBT = �ωα × {0,2,5,10}, from darker tolighter curves). (c) When multiple modes are driven, those furtherfrom resonance make smaller but faster excursions through phasespace and in general the different modes do not simultaneously returnto their initial states.

In Sec. IV B, we produce plots of the squeezing parameter ξ

as a function of time using the exact solution and also using twouseful approximations. In the first approximation, we evolveour initial state with the spin-only time evolution operator

Uspin(t) = exp

⎛⎝−i

N∑j,k=1

Sjk(t)σ zj σ z

k

⎞⎠, (39)

which amounts to ignoring spin-phonon entanglement. Thisevolution is achieved by replacing Aα

j (t) → 0 in Eqs. (28)–(30)while treating the Sjm(t) [which in reality depend implicitlyon the Aα

j (t)] as independent parameters and then evaluatingthe Sjk(t) using Eq. (37). In the second approximation, spin-motion entanglement is ignored and the spin-spin couplingsare replaced with their time averages, yielding time evolutionunder a time-independent Ising spin model

U avgspin(t) = exp

⎛⎝−it

N∑j,k=1

Savgjk σ z

j σ zk

⎞⎠, (40)

with time-averaged coupling constants

Savgjk = lim

t→∞Sjk(t)

t= −�2

2

N∑α=1

η2αbα

j bαk ωα

ω2α − μ2

. (41)

We produce plots by varying �, which controls the spin-phonon coupling strength, δ ≡ μ − ωc.m., which is the dif-ference between the two-photon detuning of the Raman lasersand the center-of-mass mode frequency, and also the phonontemperature T , which controls the initial number distributionin the phonon modes. For the rest of the paper we measureall energies in units of the transverse center-of-mass modefrequency ωc.m. (� = 1), and all temperatures in units ofωc.m./kB , with kB Boltzmann’s constant.

B. Phonon effects at T = 0

Trapped ion experiments aimed at quantum simulationsgenerally cool the phonon modes to temperatures on the orderof, or ideally lower than, typical phonon energies. However,in general, and especially for two-dimensional or large one-dimensional crystals, these temperatures are decidedly notzero for all of the relevant modes. Nevertheless, the phononsare often assumed to be cooled well enough that one canapproximate them as being at zero temperature; we willaddress the zero-temperature situation first, and return laterto analyze the consequences of having thermal phonons in theinitial state. At zero temperature, the qualitative structure of thedynamics is determined by two independent considerations.First, the coupling strength to a given phonon mode, measuredrelative to its detuning from the drive frequency μ, determineshow strongly that mode is driven. Because the vibrations ofthe particles in the trap are transverse, all vibrational modeshave frequency ωα � ωc.m.. As a result, by choosing δ � 0(so that no modes other than the center-of-mass mode canbe resonant), we ensure that �/δ is a suitable measure of howdeeply in the perturbative limit the system is. When this ratio issmall, all phonon modes are weakly populated in the dynamicsand we expect spin-phonon entanglement to be unimportant.Conversely, when this ratio is large, spin-phonon entanglementwill be important, and we do not expect the approximationsdescribed above to agree well with the exact solution. Next, wemust consider the absolute size of δ and � relative to the modebandwidth, denoted by D, which controls the relative extentto which different phonon modes participate in the dynamics.For example, when δ and � are small compared to the typicalmode spacing, δ,� � D/N , spin dynamics occurs primarilydue to coupling to the center-of-mass mode (more generally,if the modes are not evenly spaced, we must assume δ,�

much smaller than the gap between the center-of-mass modeand the mode nearest to it in energy). By increasing δ frommuch smaller than the typical mode spacing to much largerthan the mode bandwidth, all the while keeping � � δ, onecan navigate between two extreme limits. (a) For δ � D/N ,the center-of-mass mode dominates the mediation of spin-spininteractions, which are therefore independent of the distancebetween two spins (since the center-of-mass mode is spatiallyhomogeneous). (b) For δ � D, all modes participate equallyin mediating the spin-spin interactions, which fall off roughlyas the cube of the distance between two spins [4]. In betweenthese extremes, it is common to approximate the spin-spininteraction to decay as a power law, Savg

jk ∝ 1/rζ , with r thedistance between ions j and k and 0 < ζ < 3. All of theseconsiderations are summarized in Fig. 5, where a guide to theparameter space explored in the rest of this section can also befound.

Figure 6 illustrates the behavior of the squeezing parameterξ at T = 0, with the detuning chosen close to the center-of-mass mode (δ � D/N ) and marginally in the perturbativelimit (�/δ = 1/4). The squeezing parameter is normalizedsuch that log(ξ ) = 0 for a coherent state, so the region of theplot where the squeezing parameter dips below the horizontalaxis denotes a period of improved uncertainty with respectto the standard quantum limit. Under these conditions, theexact solution and the two approximations exhibit fairly similar

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DYLEWSKY, FREERICKS, WALL, REY, AND FOSS-FEIG PHYSICAL REVIEW A 93, 013415 (2016)

Perturbative

Non-Perturbative

7C,8A

7B

7A

8B

8C

ζ ≈ 0

ζ ≈ 3

δ=

Dδ

=D

/N

Ω = D/N

6

lnΩ

lnδ

FIG. 5. Parameter space at zero temperature. On this logarith-mic scale, the ratio of �/δ, which controls the extent to whichthe system is in the perturbative limit, is given by the distanceto the diagonal dashed line. The boxed area on the lower left indicatesthe parameter space in which the center-of-mass mode dominatesthe dynamics. Many trapped-ion experiments attempt to operatein the perturbative limit, shown here as a white arrow. Along thisline, the decay of interactions can be tuned from 1/r0 (at δ � D/N )to 1/r3 (at δ � D). The white dots with numeric labels indicate theparameters investigated in the numbered figures that follow.

behavior. The smooth red curve produced by the time-averagedspin-coupling approximation correctly captures the overalltrend of squeezing observed in the exact solution. As shownby the black curve, the time dependence of the spin couplings

Time 1 com

Sque

ezin

gdB

Time-averaged couplings

No spin-phonon entanglement

Exact solution

Time (units of 1/ωc.m.)

FIG. 6. Plots of the squeezing parameter ξ as a function oftime at zero temperature and with μ tuned very close to ωc.m.. Theparameters used here are {T = 0, δ = 10−3, and � = 2.5 × 10−4}.The three curves show the results of the three different approximationsdescribed in Sec. IV A. The blue curve is the exact solution, the blackcurve ignores spin-phonon entanglement but retains the full timedependence of the spin couplings Sjk(t), and the red curve uses thetime-averaged spin couplings Savg

jk .

produces small-amplitude high-frequency oscillations. Theseoscillations are amplified by spin-phonon entanglement, asindicated by the exact solution (blue curve), showing thateven at zero temperature and nominally in the perturbativelimit, the creation of phonons during the dynamics cansignificantly affect the spin squeezing. We note that veryslight improvements in squeezing over the time-averaged spin-coupling approximation do periodically occur. They appearto be due to the time dependence of the true spin couplingsSjk(t), rather than the spin-phonon entanglement, as they occurin both the blue and black curves (and the latter bounds theformer from below).

To better understand the role that dynamical phononcreation plays in spin squeezing, we change the detuning fromthe center-of-mass mode, δ, while holding � constant. Figure 7illustrates a series of results in which δ is increased but keptsmall compared to the detuning from all other modes (i.e.,δ � D/N ). These plots therefore primarily reflect changes inbehavior caused by variations in the strength with which thecenter-of-mass mode is driven relative to its detuning, i.e., theratio �/δ. In the time-averaged spin-coupling approximation(red curve), the dynamics is completely insensitive to this ratioup to an overall time scale, and upon scaling the maximumtime by δ/�, we observe nearly the same squeezing behaviorin all three panels. In the first plot, δ/� is relatively small; thecenter-of-mass mode is strongly driven, resulting in stronglyoscillatory spin couplings and large spin-phonon entangle-ment. In this limit, neither approximation accurately capturesthe exact dynamics, nor do they agree with each other. As δ

is increased, the oscillation amplitudes of the time-dependentcouplings Sjk(t) diminish compared to their time-averagedvalues, so the two approximations begin to agree better witheach other. At the same time, population in the center-of-mass mode becomes suppressed, spin-motion entanglementdiminishes, and the exact result begins to converge to bothapproximations. As discussed in Sec. IV A, when consideringcoupling only to the center-of-mass mode, the phonon degreeof freedom periodically becomes unentangled from the spins,even at strong driving. This behavior is reflected in the periodicagreement between the exact solution and the approximationignoring spin-phonon entanglement.

In all panels of Fig. 7, both δ and � are kept smallcompared to the mode spacing, and therefore the variationof �/δ is the dominant factor contributing to the changesin behavior. However, these plots provide little insight intothe other important effect of increasing the detuning δ: theincreased importance of modes other than the center-of-massmode. In order to isolate the latter effect, in Fig. 8 we againvary δ but now keep the ratio �/δ = 1/4 fixed. This keeps thecoupling to any given mode in the (barely) perturbative limit,thus counteracting the dominant role that varying spin-phononentanglement played in the qualitative trends observed inFig. 7. As δ is increased, the time-averaged spin-couplingapproximation correctly captures an overall trend of the exactsolution: The time of optimal spin squeezing is becomingshorter and the extent of squeezing that occurs at that timeis diminishing. The former effect is simply the result ofincreasing the spin-phonon coupling, which increases theoverall rate of dynamics. The reduction of squeezing achievedat the optimal time reflects the diminishing spatial range of

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NONPERTURBATIVE CALCULATION OF PHONON . . . PHYSICAL REVIEW A 93, 013415 (2016)

Time 1 com

Sque

ezin

gdB

δ = 0.005

(a)

Time (units of 1/ωc.m.) Time 1 com

δ = 0.007

(b)

Time (units of 1/ωc.m.) Time 1 com

δ = 0.01

(c)

Time (units of 1/ωc.m.)

FIG. 7. At zero temperature, the detuning from the center-of-mass mode is varied at constant � = 2.5 × 10−3. Note that the time-axisscaling is different in each plot and the optimal squeezing occurs progressively later as δ is increased. The color code here is the same as thatused in Fig. 6 (in order of smallest to largest oscillation amplitudes, the curves correspond to the time-averaged spin-coupling approximation,the approximation of ignoring spin-phonon entanglement while keeping the time-dependent spin couplings, and the exact solution).

the spin-spin couplings (approaching 1/r3 for large δ) asmore phonon modes participate in mediating them. With orwithout the inclusion of spin-phonon entanglement (i.e., in theblue or black curves), the squeezing exhibits high-frequencyoscillations arising from the coupling to phonon modes otherthan the center-of-mass mode. As multiple phonon modes atdifferent frequencies become entangled with the spins, it is nolonger possible for all of those modes to become disentangledfrom the spins simultaneously. This is strongly reflected in thethird panel, where the exact solution no longer agrees witheither approximation at regular intervals.

C. Effects of finite temperature

In Sec. IV B, we were primarily interested in the effectsof dynamical phonon generation starting from the phononvacuum. In many experiments, especially those using largenumbers of ions, this starting state is not realistic. For example,in Ref. [23], the Doppler-cooling limit of T ∼ 1 mK �10(�ωc.m./kB) leads to � 10 phonons per transverse mode. Asexplained in Sec. III C, spin-spin correlation functions can alsobe computed at finite temperatures, and the above analysis cantherefore be extended to capture the consequences of nonzeroinitial motional temperature on spin squeezing.

With the addition of temperature, there is a large parameterspace to be explored; here we focus our attention on thebarely perturbative limit at fixed �/δ = 1/4 (the same ratioused in Fig. 8), and consider both variations of δ and T (seeFig. 9 for a guide to the parameter space explored). We firstexamine the case of near detuning from the center-of-massmode (δ,� � D/N ), but now taking the phonon modes to beat a temperature T at time t = 0 (Fig. 10). Here we plot just theexact solution and the time-averaged spin-coupling approxi-mation (both the time-averaged spin-coupling approximationand the approximation of ignoring spin-phonon entanglementare insensitive to the phonon distribution, so neither varieswith T ). The primary effect of increasing temperature on thesqueezing is that the amplitude of oscillations above the curveobtained from the time-averaged spin-coupling approximationincreases. Nevertheless, the exact solution continues to agreewith the approximation at regular intervals. As discussed inSec. IV A, this behavior can be understood as the insensitivityof a harmonic oscillator’s period to its state of excitation;at finite temperature many Fock states of the center-of-massmode are occupied, but as they are driven periodically theyall return to their initial point in phase space simultaneously,at which point they are unentangled from the spins. AsT becomes larger, the system spends most of its time in

Time 1 com

Sque

ezin

gdB

(a)

δ = 0.01

Time (units of 1/ωc.m.) Time 1 com

(b)

δ = 0.1

Time (units of 1/ωc.m.) Time 1 com

(c)

δ = 1.0

Time (units of 1/ωc.m.)

FIG. 8. At zero temperature, the detuning from the center-of-mass mode is varied while keeping �/δ = 1/4, thereby controlling the relativecontribution of the different phonon modes without greatly affecting the extent to which the individual modes are in the perturbative limit [i.e.,�/(μ − ωα) is not changing very much]. Note that as δ grows larger, the time of optimal squeezing becomes shorter and the total amount ofsqueezing obtained at that time is shrinking. The color code here is the same as that used in Fig. 6 (in order of smallest to largest oscillationamplitudes, the curves correspond to the time-averaged spin-coupling approximation, the approximation of ignoring spin-phonon entanglementwhile keeping the time-dependent spin couplings, and the exact solution).

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DYLEWSKY, FREERICKS, WALL, REY, AND FOSS-FEIG PHYSICAL REVIEW A 93, 013415 (2016)

Fig.10

Fig. 11

lnΩ

ln δ

lnT

FIG. 9. Schematic illustration of a two-dimensional cross sectionof the full three-dimensional parameter space spanned by �, δ, and T .Working just barely in the perturbative limit (and at a fixed value of�/δ), we explore the effects of temperature for multiple values of δ,which controls the relative participation of the various phonon modesin mediating spin-spin interactions. The parameter space explored inFigs. 10 and 11 are indicated as vertical arrows.

states with large uncertainty (poor squeezing), but preciselytimed measurements of the system could nevertheless yield asignificantly improved resolution over the standard quantumlimit. Interestingly, at times short compared to 2π/δ, thespin distribution is always antisqueezed, and the degree ofantisqueezing could in principle be used to perform in situtemperature measurements of the phonon modes.

In Fig. 11, we explore the behavior of the system detunedaway from the center-of-mass frequency and at finite temper-ature. The behavior exhibited in Fig. 11 reflects the generaltrends observed in the previous plots: Increasing δ inducesadditional high-frequency structure in the exact solution and

Time 1 com

Sque

ezin

gdB

T = 0

T = 2

T = 5 T = 10

Time (units of 1/ωc.m.)

FIG. 10. For a system driven very close to the center-of-massfrequency (δ = 0.001 and � = 2.5 × 10−4), spin squeezing is veryrobust against large initial temperatures. Here the temperature isvaried from zero up to 10ωc.m. (≈0.1 mK) and to a very goodapproximation the squeezing obtains its T = 0 value at regularintervals.

Time 1 com

Sque

ezin

gdB

(Inset) T = 0T = 2

T = 10

Time 1 com

Sque

ezin

gdB

Time (units of 1/ωc.m.)

Time (units of 1/ωc.m.)

FIG. 11. For a system driven further off-resonance from thecenter-of-mass frequency (δ = 0.1 and � = 0.025), finite temper-ature has a much more detrimental effect on the spin squeezing,eventually preventing any squeezing from occurring beyond a criticaltemperature (here T ≈ 10). Inset: Note that, due to the relativelylarge population of many modes, there is significant high-frequencystructure that is hidden on the time scale over which squeezingoccurs.

increasing T produces an overall growth in the amplitude ofoscillations in the spin squeezing. Unlike in Fig. 10, however,here the exact solution reveals that increasing temperaturecauses the local squeezing minima to become increasinglydisplaced from the curve calculated in the time-averaged spin-coupling approximation. Even though at T = 0 the squeezingnearly agrees with this approximation throughout the entiretime region plotted, spin squeezing is completely destroyedwhen the temperature reaches T ≈ 10.

Indeed, for a given detuning there will always be sometemperature threshold above which no squeezing takes placeat all (ξ � 1 for all t). By varying T at evenly spacedintervals of ln δ, in Fig. 12 we produce a phase diagramdemarcating this boundary in the parameter space. Thequalitative downward trend of the boundary can be understoodas an increased sensitivity of the spin dynamics to the initial

T

Squeezed

Not Squeezed

Fig. 11

FIG. 12. Squeezing phase diagram in the δ-T plane, showing thatno squeezing occurs above a critical temperature (note that �/δ isbeing held constant as δ is changed; see Fig. 9).

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NONPERTURBATIVE CALCULATION OF PHONON . . . PHYSICAL REVIEW A 93, 013415 (2016)

phonon temperature as more phonon modes participate inthe dynamics. The additional modes prevent the spins frombecoming periodically unentangled with the phonons; theconsequences of this residual spin-motion entanglement onspin squeezing are exacerbated at finite temperature becausethe occupation of a given phonon mode affects the extent towhich that mode remains entangled with the spins at a giventime [see Fig. 4(b)]. At small δ, squeezing persists even fortemperatures on the order of 100 times the center-of-mass-mode energy. However, note that at these high temperaturesthe approximation that the phonons are noninteracting (orequivalently that the potential experienced by an individual ionis harmonic) is likely to break down and the system may wellbe outside the Lamb-Dicke limit used to justify a descriptionin terms of Eq. (1).

V. DISCUSSION AND FUTURE DIRECTIONS

While Eq. (1) is quite general and can describe a diversearray of practically relevant experiments, there is certainly asense in which it is extremely constrained: It has a large numberof locally conserved quantities. Indeed, a careful inspection ofour calculations reveals that the local conservation of σ z

j is, atseveral different levels, directly responsible for the solvabilityof the model. Nevertheless, our solutions form a usefulbenchmark for powerful (but computationally expensive) toolscapable of numerically solving the nonequilibrium dynamicsof more general models, such as time-dependent versions ofmatrix-product-state algorithms [34]. Despite the existence ofstructure not present in more general, nonintegrable models,bipartite entanglement entropy does grow with time afterthe quench considered, posing similar challenges for matrix-product-state algorithms as more general models would.Especially in two dimensions, but generally even in onedimension given the long-range nature of the spin-couplingsinduced by the (often) delocalized bosonic modes, it remainsunclear to what extent there are any general purpose andcontrolled numerical tools for studying this dynamics. We alsopoint out that questions about equilibration and thermalization,which require studies of long-time dynamics, are difficult evenin one dimension for all but the smallest systems.

The exact solution will be useful in testing a variety of ex-perimental idealizations that are frequently made but generallynot quantitatively justified. For example, many trapped-ionexperiments employ a spin-echo pulse in order to obtain acoherence time on the order of the spin-spin interaction time.If the spins are unentangled with the phonons, then a spinecho completely removes the effects of an inhomogeneous

magnetic field (∼ ∑j hj σ

zj ) added to the Hamiltonian in

Eq. (1). However, the existence of spin-phonon entanglementat the time of the echo pulse invalidates this picture [35] andthe consequences on spin-spin correlation functions could beexplored using the solution developed in this paper.

There are also many interesting purely theoretical questionsabout the present model, many of which we believe thetools developed here are well suited to begin answering.For example, while spin-squeezing calculations reveal theimpact of spin-boson entanglement on attempts to generatestrictly spin-spin entanglement, it should also be possible toanalyze spin-boson entanglement more directly by calculatingcorrelation functions involving both spin and boson operators.It would also be interesting to explore to what extent thesolvability of this model can be extended to calculating moregeneral quantities than low-order correlation functions. Whileour solution enables efficient calculation of arbitrary-ordercorrelation functions of the form given in Eqs. (22)–(24),calculating the full counting statistics along any spin directionorthogonal to z is still very difficult. In particular, computationof the full counting statistics is equivalent to the computationof ∼N th-order correlation functions of Pauli (x,y,z) matrices,which involves the summation of an exponentially large(in N ) number of high-order correlation functions of theform in Eqs. (22)–(24). It seems very plausible that thisdifficulty is related to computational hardness results forclassically sampling spin distributions following dynamicsunder commuting spin Hamiltonians [36]. The present modeladds an interesting twist, in that it builds bosonic degreesof freedom into a commuting spin Hamiltonian in a waythat preserves its solvability (in the sense of calculatinglow-order correlation functions); the bosons alone, despitebeing noninteracting and therefore solvable in the same senseas the model studied here, are thought to be hard to simulateclassically in a precise sense [37].

ACKNOWLEDGMENTS

We acknowledge helpful conversations with John Bollinger,Joe Britton, Zhexuan Gong, Bryce Yoshimura, Alexey Gor-shkov, Bill Fefferman, and Arghavan Safavi-Naini. J.K.F.acknowledges support from the National Science Foundationunder Grant No. PHY-1314295 and from the McDevittbequest at Georgetown University. A.M.R. acknowledgesfunding from the NSF (Grants No. PHY-1521080 and No.PFC-1125844), AFOSR, AFOSR-MURI, NIST, and AROindividual investigator awards. M.L.W. and M.F.-F. thank theNRC postdoctoral fellowship program for support.

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