Table Of ContentMany-body spin interactions and the ground state of a dense Rydberg lattice gas
Igor Lesanovsky1
1School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
(Dated: January 25, 2011)
Westudyaone-dimensionalatomiclatticegasinwhichRydbergatomsareexcitedbyalaserand
whose external dynamics is frozen. We identify a parameter regime in which the Hamiltonian is
well-approximated by a spin Hamiltonian with quasi-local many-body interactions which possesses
an exact analytic ground state solution. This state is a superposition of all states of the system
that are compatible with an interaction induced constraint weighted by a fugacity. We perform a
detailedanalysisofthisstatewhichexhibitsacross-overbetweenaparamagneticphasewithshort-
rangedcorrelationsandacrystal. Thisstudyalsoleadsustoaclassofspinmodelswithmany-body
1
1 interactions that permit an analytic ground state solution.
0
2 PACSnumbers: 67.85.-d,32.80.Ee,75.10.Pq,03.67.Bg
n
a Strong interactions competing with non-commuting eters this Hamiltonian is accurately approximated by a
J
single particle terms in a many-body quantum Hamil- spin Hamiltonian which has an exact analytical ground
3
tonian often lead to non-classical ground states. Only statesolution. Weshowthatthisisaconsequenceofthe
2
in exceptional cases analytic or approximately analytic factthattheHamiltonianpossessesamanifoldofapprox-
] results can be found. Paradigm examples are the one- imate Rokhsar-Kivelson points [16] where is assumes a
h
dimensionalxy-modelinatransversefield[1]orthecele- so-called Stochastic Matrix Form [17]. The ground state
p
bratedAffleck-Kennedy-Lieb-Tasakispinmodel[2],both is a coherent superposition of all states compatible with
-
m ofwhichhaveprovenindispensablefortheunderstanding allthelocalconstraintsweightedbyaneffectivefugacity.
o of many-body phenomena in magnetic compounds and Weanalyticallyexplorethepropertiesofthisstatewhich
t valence bond solids. Finding models of experimentally shows a cross-over between a paramagnetic phase with
a
. realizable many-body Hamiltonians with exact solutions short-rangedcorrelationsandacrystallineorderedstate.
s
is hence of fundamental interest. Our study highlights a new perspective for creating and
c
i Models of many-body quantum systems with origin in studying non-classical and entangled states with ultra-
s
y condensedmatterphysicsarecurrentlyimplementedand cold Rydberg gases. It also leads to a class of spin mod-
h studied in ultra cold atomic systems with great success elswithmany-bodyinteractionwhosenon-trivialground
p [3]. Most experiments so far are carried out with ground state solution can be found analytically.
[
state atoms, but very recent efforts exploit the unique Our system consists of a deep one-dimensional lattice
2 properties of atoms excited to Rydberg states. These with L sites evenly spaced at a distance a. For con-
v states offer strong and long-ranged interatomic interac- venience we consider periodic boundary conditions, but
9
tionsinconjunctionwithanextraordinarilylonglifetime this is no necessary requirement. Each site is occupied
4
3 [4, 5]. This enabled remarkable experiments which stud- by a single atom which we treat within the two-level ap-
2 ied the coherence properties in strongly interacting Ryd- proximation where every atom forms a (pseudo)spin 1/2
. berggases[6,7]andeventuallydemonstratedthefeasibil- particle. The atomic ground state |g(cid:105)≡|↓(cid:105) is coupled to
0
1 ity to process quantum information with Rydberg atoms a Rydberg state |r(cid:105)≡|↑(cid:105) by a laser with Rabi frequency
0 [8, 9]. Rydberg gases are moreover an almost ideal ex- Ωanddetuning∆. AtomsinRydbergstatesinteractvia
1 perimental implementation of interacting spin systems a power law interaction with (inverse) exponent γ > 0.
v: such as the aforementioned xy-model [10]. This stimu- The Hamiltonian of this system is (within the rotating
i lated a plethora of theoretical studies investigating the wave approximation for the atom-laser coupling) given
X
real time evolution [11, 12] as well as ground states of by
r
a tmheersiecaslpiwnomrko,dreelvse[a1l3e–d15a].vTarhieetlyatotferin,tperreedsotimnginaqnutalnytnuum- H0 =Ω(cid:88)L σxk+∆(cid:88)L nk+V (cid:88) |kn−mnmk|γ. (1)
phases and studied the creation [13, 14] as well as the k k m>k
melting dynamics [15] of dynamically created crystals. Here V is the interaction strength, σk is a Pauli matrix
x
In this work we provide an analytic study of the non- andn =σkσk istheRydbergnumberoperatorwithσk
k + − ±
trivial entangled many-body ground state of a strongly beingtheraisingandloweringoperatorsofthek-thspin.
interacting one-dimensional Rydberg gas. The strong This Hamiltonian is an Ising-model with long-range in-
interaction among excited atoms gives rise to an effec- teractions in a transverse and a longitudinal field. Most
tive Hamiltonian with a quasi-local three-body interac- experiments up to date use atomic states that interact
tion that effectuates a set of non-commuting local con- viathe van-der-Waalsinteraction, i.e. γ =6. We willfo-
straints. For certain values of the experimental param- cushereonthiscasebutourapproachalsoworksforthe
2
dipole-dipole interacting case with γ = 3. The Hamilto- state solution and H(cid:48) is a perturbation. H is given
3body
nian (1) was employed successfully to describe the exci- by
tation dynamics of driven Rydberg gases and has been
L L
proventoreflectveryaccuratelytheactualexperimental H =Ω(cid:112)ξ−1+ξ(cid:88)h =Ω(cid:88)h†h , (3)
situation [6, 18]. 3body k k k
k k
Theinteractionbetweenexcitedatomsstronglyaffects
the excitation dynamics of the system through a mech- with
anism which is called the Rydberg blockade [19]: The
(cid:114)
large interaction induced energy shift makes it virtually h = 1 P P (cid:2)σk+ξ−1n +ξ(1−n )(cid:3)(.4)
k ξ−1+ξ k−1 k+1 x k k
impossible to excite two nearby atoms simultaneously to
the Rydberg state, i.e. the presence of one excited atom
The term of h which contains σk is proportional to the
blocks the excitation of atoms in its vicinity (we assume k x
three-body interaction term in Hamiltonian (2). The re-
V (cid:29) |∆| and Ω (cid:29) V throughout). In what follows we
maining ones are chosen such that h becomes a self-
will make this blockadeeffect manifest, which willcreate k
adjointoperatorwithpositive-semidefinitespectrumand
an effective three-body spin interaction in the Hamilto- (cid:112)
only one non-zero eigenvalue: ξ−1+ξ. These are the
nian. Owed to the power law decay the strongest in-
terms that were introduced by H . The part that is can-
teraction takes place between nearest neighbors and we ξ
celing them is contained in the perturbation H(cid:48) which
assume that a strict blockade is only present between
we discuss later. At first, we construct the ground state
them. We transform the Hamiltonian (1) into an inter-
|ξ(cid:105) of the Hamiltonian H . This state is annihilated
action picture with respect to the nearest neighbor in- 3body
by all h and has hence energy zero. Note, that oper-
teraction by applying the unitary transformation U = k
(cid:104) (cid:105) ators acting on neighboring sites do not commute, i.e.
exp −itV (cid:80)Lk nknk+1 . The first term of eq. (1) is the [hk,hk±1](cid:54)=0. It is thus not possible to use a local zero
only one that does not commute with U and one obtains eigenstate of each h and then form the total ground
k
U†σkU =(cid:2)P +n eitV(cid:3)σk (cid:2)P +n eitV(cid:3)+h.c. statebyaproductofthesestates. Insteadonefindsthat
x k−1 k−1 + k+1 k+1
whereP ≡1−n . Notethatbothn andP areprojec- the state of the physical subspace that is annihilated by
k k k k
tion operators. Since V (cid:29) Ω one can neglect the terms all h is given by
k
with rapidly oscillating phases which is essentially a ro-
L
tating wave approximation. We then arrive at our work- |ξ(cid:105)= 1 (cid:89)(cid:0)1−ξP σkP (cid:1)|↓↓...↓(cid:105). (5)
ing Hamiltonian (cid:112)Z k−1 + k+1
ξ k
L L
H =Ω(cid:88)P σkP +∆(cid:88)n +V (cid:88) nmnk where Zξ is a normalization constant. This state is a
k−1 x k+1 k |k−m|γ coherent superposition of all states that have no near-
k k m>k+1
(2) est neighbor excitations. The probability of each state is
weighted by the factor (ξ2)n, where n is the total num-
where the first term is the blockade-induced three-body
ber of Rydberg excitations in this state. This state is
interaction: The excitation of an atom to a Rydberg
highly non-classical as it is a coherent superposition of
state on site k can only take place provided that both
all states from the physical subspace and cannot (except
projectors P and P yield a non-zero value. This
k−1 k+1 forξ =0)bewrittenasaproductstate. Theexistenceof
imposes a constraint such that the Hilbert space splits
thisgroundstateisduetothespecialprojectorproperty
into uncoupled blocks each of which is characterized by
of each term of Hamiltonian (3) which is also known as
the number of pairs of neighboring excited atoms. We
Stochastic Matrix Form [17].
will be concerned with the subspace in which there is no
In order to calculate the normalization constant Z
ξ
simultaneous excitation of nearest neighbors (referred to
one has to count the number of all allowed arrangement
as physical subspace).
of excited atoms on the lattice and sum them using the
In the following we will show that for a certain set weights (ξ2)n. Since there is strict nearest neighbor ex-
of parameters (Ω, ∆, V) Hamiltonian (2) possesses an
clusion this sum is equivalent to the partition function
approximate analytical solution. The decisive idea is to
of a lattice gas of hard-core dimers, i.e. hard objects
add the term H = (cid:80) P P [ξ−1n +ξ(1−n )] to
ξ k k−1 k+1 k k that occupy two neighboring lattice sites. In the limit
Hamiltonian (2) where ξ is a real and positive parame- (cid:104) (cid:112) (cid:105)L
L (cid:29) 1 we obtain Z = (1/2)(1+ 1+4ξ2) such
ter and subsequently subtract it. Obviously, adding and ξ
subtracting H does not change H, but regrouping all that we can identify ξ2 as a fugacity. The fugacity sup-
ξ
terms conveniently allows us to rewrite the Hamiltonian presses/enhances the weight of a state with n excited
as H =E +H +H(cid:48) where now each term depends atoms or dimers by (ξ2)n [20]. We emphasize that the
0 3body
on ξ. Here E = −ΩLξ will turn out to be the approxi- correspondence between the quantum problem and the
0
mate ground state energy, H is a spin Hamiltonian dimer gas is solely formal. One striking difference is the
3body
withthree-bodyinteractionsthathasananalyticground range of the interaction: for the classical system only
3
nearest neighbors interact while in the quantum system
also interaction among next-nearest neighbors occur.
The aim is now to find a set of parameters (Ω, ∆, V)
or a whole manifold of them such that H(cid:48) is negligible
compared to H . In this case the Hamiltonian of the
3body
Rydberg gas (1) is be very accurately approximated by
H for which we know the ground state. One finds
3body
that
L
H(cid:48) =(cid:88)(cid:2)∆+Ω(cid:0)3ξ−ξ−1(cid:1)+(cid:0)2−γV −Ωξ(cid:1)n (cid:3)n
k+2 k
k
+V (cid:88) nmnk
|k−m|γ
m>k+2
(cid:88)
−Ω(ξ−ξ−1) n n (2−n ). (6)
k k+1 k+2
k
The first term of H(cid:48) can be eliminated exactly pro-
FIG. 1. Comparison between the numerical results obtained
vided that the conditions (i) ∆ = −Ω(3ξ − ξ−1) and
for a lattice with L = 20 sites and γ = 6 (blue) and the an-
(ii) V =2γΩξ are satisfied. The contribution of the sec- alytical expressions (dashed red). a: Energy per particle in
ond term in eq. (6) is small since it accounts for the the ground state, b: Mean density of Rydberg atoms on the
strongly diminished interaction between excited atoms lattice,c: MandelQ-parameteroftheRydbergnumberdistri-
thatareatleastthreelatticesitesapart. Thethirdterm bution. d: Numericallycalculateddensity-densitycorrelation
function. e: Density-densitycorrelationfunctionobtainedfor
vanishes exactly at ξ = 1, but its contribution is negli-
the state (5). Note that at the same time as ∆ also the po-
gible even away from this point since the probability for
tential is varied according to V =2γΩξ. The values of ξ are
a simultaneous excitation of neighboring atoms is highly
given underneath panel c.
suppressed (even strictly zero in the physical subspace).
These considerations imply that upon meeting con-
dition (ii), i.e. for an interaction strength satisfying way for the experimental characterization of the distri-
V = 2γΩξ, the ground state energy of Hamiltonian (2) bution function is the Mandel Q-factor which quantifies
is given by E = −ξΩL where ξ = (1/6)[−(∆/Ω) + the difference of the distribution p from a Poissonian
0 k
(cid:112)
12+(∆/Ω)2]. The latter relation is obtained directly [21]. This quantity, which is plotted in fig. 1c, evaluates
from condition (i) and yields the conversion between the to Q=((cid:10)N2(cid:11)−(cid:104)N(cid:105)2)/(cid:104)N(cid:105)−1=1/(2(cid:112)1+4ξ2)−(1+
laser parameters and the square root of the fugacity, i.e. 8ξ2)/(2+8ξ2). Except for ξ = 0 it is negative showing
ξ. That this is indeed the case is shown in fig. 1a where apronounced sub-Poissonianbehaviorwhichisexpected
we compare the ground state energy E0 (red curve) with for strongly interacting systems [22].
the numerical result (blue curve) obtained for a lattice A further important quantity characterizing the
withL=20sites. Theexcellentagreementindicatesthat ground state is the connected density-density correla-
conditions (i) and (ii) define a manifold of approximate tion function g (ξ) = (cid:104)n n (cid:105) − (cid:104)n (cid:105)(cid:104)n (cid:105) =
1,1+m 1 1+m 1 1+m
Rokhsar-Kivelson points [16] in the parameter space (Ω, ξ2/(1+4ξ2)[((cid:112)1+4ξ2 −2ξ2 −1)/(2ξ2)]m. It is shown
∆, V) were the Hamiltonian of a gas of interacting Ryd- in fig. 1 in panels d and e together with the numerical
bergatoms(1)allowstheapproximatestochasticmatrix result,bothagaininexcellentagreement. Visiblecorrela-
form decomposition [17] shown in eq. (3) and has the tions build up as soon as ∆<0. They are exponentially
ground state (5). decaying with the interparticle distance and alternating
Wecannowcalculatepropertiesofthegroundstateof in sign, with anti-correlation between nearest neighbors.
the system on this manifold in the same spirit in which The corresponding correlation length is proportional to
weobtainedthenormalizationconstantZξ. Expectation ξa and reaches the system size when −∆/Ω≈3L.
values of classical observables such as the mean number We will now perform an analysis of the coherent prop-
of excited atoms or density-density correlations then re- erties of the system. To this end we study the reduced
duce to the manipulation of the partition function with single particle density matrix ρ (ξ) which allows us to
1
fugacity ξ2. The mean density of Rydberg atoms in the quantify the entanglement of one spin with the rest of
(cid:80)
ground state is given by (cid:104)N(cid:105)/L = k(cid:104)ξ|nk|ξ(cid:105)/L = the system. We find
(cid:112)
[1 − 1/( 1+4ξ2)]/2 which is shown in fig. 1b. We (cid:18) (cid:19)
(cid:104)N(cid:105) −(cid:104)N(cid:105)/ξ
can furthermore obtain the full statistics of the Rydberg ρ (ξ)=(1/L)
1 −(cid:104)N(cid:105)/ξ L−(cid:104)N(cid:105)
number distribution by taking derivatives of the parti-
tion function: The probability p to count k Rydberg which, except for ξ = 0, represent a mixed state. This
k
atomsisgivenbyp =[(k!)−1∂k Z | ]/Z . Acommon indicates entanglement of one atom with the remaining
k ξ2 ξ ξ=0 ξ
4
is indeed possible to find experimental parameters that
achieve that (see Refs. [12–14, 23]).
LetusfinallydiscussthegeneralizationofHamiltonian
(3)tohigherdimensionsandblockaderangesthatcango
beyondthenearestneighbors. Tothisendwereplacethe
product P P by an operator which projects onto
k−1 k+1
(cid:81)
the state |↓(cid:105) . Here G is a set that contains the
q(cid:15)Gk q k
indices of lattice sites that surround the k-th site, i.e.
that are blocked when spin k is excited (see fig. 2c).
ThegroundstateofthisHamiltonianisthenconstructed
analogous to the state (5) with the constraint being that
a simultaneous excitation on site k and on any of the
sites contained in G is forbidden. Calculations of ex-
k
FIG. 2. a: Entanglement entropy S of a single spin with
pectation values here again reduce to the manipulation
the remaining ones. For large negative detuning the ground
of a partition sum of a classical system of hard objects.
statebecomesaGHZstateandS reachesitsmaximumlog2.
It is not immediately evident whether such models actu-
b: Density of excited atoms as a function of the detuning
and the Rabi frequency. The black line represents the set of ally represent an experimentally relevant system. This
parameters where the Rydberg gas ground state is approxi- depends onwhether conditions similar to (i)and (ii) can
mately given by eq. (5). Dashed lines are used as a guide befoundwhichcanceltheunwantedmany-bodytermsin
to the eye delimiting the regions where the Rydberg density H(cid:48). However, the knowledge of the ground state is valu-
is approximately 1/3 and 1/2. c: The spin Hamiltonian (3)
able, e.g. for performing perturbation theory in order to
can be generalized to higher dimensions (here 2d) where the
move away from the exactly solvable situation.
excitation on the k-th site (grey color) blocks the excitation
of all sites contained in the set G . Funding through EPSRC and fruitful discussions with
k
J.P. Garrahan, G. Adesso, B. Olmos, P. Kru¨ger and M.
Mu¨ller are gratefully acknowledged.
others which can be quantified by the entanglement en-
tropy S =−Trρ (ξ) logρ (ξ). This function is shown in
1 1
fig. 2a. Forlargepositivedetuning,i.e. ξ ≈0,theground
(cid:81)
stateisaproductstate|init(cid:105)= |g(cid:105) andhencenoen-
k k
tanglement is present. S increases monotonously with [1] S. Sachdev, Quantum Phase Transitions (Cambridge
University Press, Cambridge, UK, 1999).
ξ and saturates at a value log2 for ξ → ∞ which indi-
[2] I. Affleck et al., Comm. Math. Phys. 115, 477 (1988).
cates maximal entanglement. Here the ground state is
[3] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.
formally given by a GHZ state, which is the coherent su-
80, 885 (2008).
perpositionofthe√twopossibleanti-ferromagneticstates, [4] T. Gallagher, Rydberg Atoms (Cambridge University
i.e. |GHZ(cid:105)=(1/ 2)[|↑↓↑↓...(cid:105)+|↓↑↓↑...(cid:105)] (even number Press, 1984).
of sites assumed). [5] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.
The above considerations indicate that the typical ex- Phys. 82, 2313 (2010).
[6] R.Heidemannetal.,Phys.Rev.Lett.99,163601(2007).
perimentalinitialstate|init(cid:105)(noRydbergatomspresent)
[7] M. Reetz-Lamour et al., Phys. Rev. Lett. 100, 253001
can be adiabatically connected to the fully entangled
(2008).
GHZ state by varying ξ from zero to infinity, i.e. by
[8] E. Urban et al., Nature Phys. 5, 110 (2009).
varying Ω and ∆ in time. Experimentally this is usu- [9] A. Ga¨etan et al., Nature Phys. 5, 115 (2009).
ally done at fixed interaction strength V. The approxi- [10] B. Olmos, R. Gonza´lez-F´erez, and I. Lesanovsky, Phys.
mate manifold of Rokhsar-Kivelson points is then given Rev. Lett. 103, 185302 (2009).
through(2γΩ/V)2−(2γ∆/V)=3whichisobtainedfrom [11] H. Weimer et al., Phys. Rev. Lett. 101, 250601 (2008).
[12] I.Lesanovsky,B.Olmos,andJ.P.Garrahan,Phys.Rev.
(i) and (ii) and shown as the black curve in fig. 2b. The
Lett. 105, 100603 (2010).
GHZ state is obtained by initially choosing a large pos-
[13] T. Pohl, E. Demler, and M. D. Lukin, Phys. Rev. Lett.
itive detuning and following this curve until one reaches
104, 043002 (2010).
∆min = −3/2γV, i.e. Ωmin = 0. Performing this pro- [14] J. Schachenmayer, I. Lesanovsky, and A. Daley, New J.
cess adiabatically becomes increasingly difficult as the Phys. 12 103044 (2010).
number of particles increases due to an ever closing en- [15] H. Weimer and H. P. Bu¨chler, preprint p.
ergy gap. Eventually, this will lead to symmetry break- arXiv:1007.2189 (2010).
[16] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61,
ing which singles out one of the two anti-ferromagnetic
2376 (1988).
states or leads to domain formation. Experiments have
[17] C. Castelnovo et al., Annals of Physics 318, 316 (2005).
tobecarriedoutonatimeshorterthanthelifetimeofthe
[18] R. Heidemann et al., Phys. Rev. Lett. 100, 033601
atomicRydbergstate(typically100µsforRubidiumand (2008).
aprincipalquantumnumberintherangen=40...70). It [19] D. Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000).
5
[20] N. Goldenfield, Lectures on Phase Transitions and the [23] B. Olmos, R. Gonza´lez-F´erez, and I. Lesanovsky, Phys.
Renormalization Group (Westview Press, 1992). Rev. A 81, 023604 (2010).
[21] T.C.Liebischetal.,Phys.Rev.Lett.95,253002(2005).
[22] C. Ates et al., Journal of Physics B 39, L233 (2006).
