Dicke coupling by feasible local measurements at the superradiant quantum phase transition

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by  M. Bina

Dicke coupling by feasible local measurements at the superradiant quantum phase
transition
M. Bina,1 I. Amelio,1 and M. G. A. Paris1,2,3,∗
1Dipartimento di Fisica, Universita` degli Studi di Milano, I-20133 Milano, Italy
2CNISM, UdR Milano Statale, I-20133 Milano, Italy
3INFN, Sezione di Milano, I-20133 Milano, Italy
(Dated: January 18, 2016)
We address characterization of many-body superradiant systems and establish a fundamental
connection between quantum criticality and the possibility of locally estimating the coupling con-
stant, i.e extracting its value by probing only a portion of the whole system. In particular, we
consider Dicke-like superradiant systems made of an ensmble of two-level atoms interacting with a
single-moderadiationfieldatzeroeffectivetemperature,andaddressestimationofthecouplingby
6 measurements performed only on radiation. At first, we obtain analytically the Quantum Fisher
1 Information (QFI) and show that optimal estimation of the coupling may be achieved by tuning
0 the frequency of the radiation field to drive the system towards criticality. The scaling behavior
2 of the QFI at the critical point is obtained explicitly upon exploiting the symplectic formalism for
Gaussian states. We then analyze the performances of feasible detection schemes performed only
n
on the radiation subsystem, namely homodyne detection and photon counting, and show that the
a
J corresponding Fisher Informations (FIs) approach the global QFI in the critical region. We thus
conclude that criticality is a twofold resource. On the one hand, global QFI diverges at the critical
5
point, i.e. the coupling may be estimated with the arbitrary precision. On the other hand, the FIs
1
offeasiblelocalmeasurements,(whicharegenerallysmallerthantheQFIoutofthecriticalregion),

] show the same scaling of the global QFI, i.e. optimal estimation of coupling may be achieved by
h locally probing the system, despite its strongly interacting nature.
p
-
t
n I. INTRODUCTION ultimate quantum bound to precision for any inference
a strategy aimed at estimating a given parameter.
u
q Quantumphasetransitions(QPTs)occuratzerotem- In the recent years, the connection between quan-
[ perature and demarcate two statistically distinguishable tum criticality and parameter estimation has been ad-

groundstatescorrespondingtodifferentquantumphases dressed from differente perspectives [6–12], showing that
1
of the system [1]. In the proximity of the critical point, the QFI is indeed (substantially) enhanced in correspon-
v
5 small variations of a parameter driving the QPT cause denceofthecriticalpoint[2]. Thefundamentalinterpre-
1 abrupt changes in the ground state of the system. Criti- tation of this relationship lies in the geometrical theory
0 cality is, thus, a resource for precision measurements [2] of quantum estimation, for which Hilbert distances be-
4 since driving the system to the critical region makes it tweenstatesaretranslatedintomodificationsofthephys-
0 extremely sensitive to perturbations, either affecting an ical parameters [6–8]. Criticality as a resource for quan-
.
1 internal parameter such as its coupling constant, or due tum metrology has been investigated in several critical
0 to fluctuations of environmental parameters, e.g. tem- systems [9–12]. Nonetheless, finding an optimal observ-
6 perature fluctuations. ablewhichalsocorrespondstoafeasibledetectionscheme
1 is usually challenging, especially for strongly interacting
: It is often the case that those parameters are not di-
v systems where the entangled nature of the ground state
rectly measurable. In these cases, the determination of
i usually leads to an inseparable optimal observable.
X their values should be pursued exploiting indirect ob-
servations and the technique of parameter estimation. In this paper we consider the superradiant QPT oc-
r
a In this situations, the maximum information extractable curring in the Dicke model, which describes the strong
from an indirect estimation of the parameters is the so- interaction of a single-mode electromagnetic field and an
called Fisher Information (FI), which itself determines ensemble of two-level atoms [13]. The radiation mode
the best precision of the estimation strategy via the in the superradiant phase acquires macroscopic occupa-
Cramer-Rao theorem [3]. Upon optimizing over all the tion as a consequence of cooperative excitation of the
possible quantum measurements one obtains the Quan- atoms in the strong coupling regime. The Dicke QPT
tum Fisher Information (QFI), which depends only on has been extensively studied in the past years consid-
the family of states (density operators) describing the ering also generalizations of the original work of Dicke
groundstateoftheconsideredsystem[4,5]asafunction [14–16], or focussing on the quantum-cahotic properties
of the parameter of interest. In turn, the QFI sets the ofthesystem[17,18]. Recenttheoreticalstudiesconcern-
ing entanglement and squeezing of the Dicke QPT have
been carried on [19], also in relation to the QFI of radia-
tionandatomicsubsystemsseparately[20]. Someimple-
∗Electronicaddress: matteo.paris@fisica.unimi.it mentations in cavity-QED [21] and circuit-QED [22, 23]
2
systems, together with computing applications via mul- istic function, defined as χ[(cid:37)ˆ](α(cid:126)) ≡ Tr[(cid:37)ˆDˆ(α(cid:126))], where
timodal disordered couplings [24], have been proposed. Dˆ(α(cid:126)) =(cid:78)M exp{α aˆ† −α∗ aˆ } is the displacement
m=1 m m m m
Eventually, recent experimental realizations of the Dicke operator and α(cid:126) = {α ,...,α }, with complex coeffi-
1 √M
QPTinvolvingBose-Einsteincondensatesinopticalcav- cients α = (α(r) +iα(i))/ 2 and {α(r),α(i)} ∈ R. It
ities [25], cavity-assisted Raman transitions with Rb87 m m m m m
is responsible for rigid translations of states in the phase
atoms [26] or NV-centers in diamond coupled to super-
space, allowing to express any coherent state as a dis-
conductingmicrowavecavities[27],havebeenperformed. placed vacuum state |α(cid:126)(cid:105)=Dˆ(α(cid:126))|0(cid:105). Equivalently, in the
Motivated by the renewed experimental and theoreti-
cartesian representation, the displacement operator can
cal interests in the Dicke QPT, we address the charac- be written in the compact form Dˆ(Λ(cid:126)) = exp{−iΛ(cid:126)TΩR(cid:126)},
terization of its coupling constant and analyze in details
with Λ(cid:126) = {α(r),α(i),...,α(r),α(i)}, acting on the vector
whetheroptimalestimationispossibleusingonlyfeasible 1 1 M M
local measurements, i.e. whether the ultimate precision of quadratures as Dˆ†(Λ(cid:126))R(cid:126)Dˆ(Λ(cid:126))=R(cid:126) +Λ(cid:126).
allowedbyquantummechanicsmaybeachievedbyprob- A density operator (cid:37)ˆdescribing the state of a system
ing only a portion of the whole system. of M bosonic modes, is called Gaussian when its charac-
The paper is structured as follows. In Sec. II we teristic function χ[(cid:37)ˆ](Λ(cid:126))≡Tr[(cid:37)ˆDˆ(Λ(cid:126))] is Gaussian in the
introduce the properties of Gaussian states and sym- cartesian coordinates Λ(cid:126) and reads
plectic transformations, together with some elements (cid:26) (cid:27)
1
of quantum estimation theory (QET) in the Gaussian χ[(cid:37)ˆ](Λ(cid:126))=exp − Λ(cid:126)TΩσΩTΛ(cid:126) −iΛ(cid:126)TΩ(cid:104)R(cid:126)(cid:105) , (2)
2
continuous-variable formalism. In Sec. III, we briefly
review the Dicke model at zero temperature, establish
or,equivalently,whentheassociatedWignerfunctionhas
notation and find the Gaussian ground states of the two
the Gaussian form
phases of the system. In Sec. IIIB we evaluate the QFI
(cid:110) (cid:111)
as a function of the radiation-atoms coupling parameter exp −1(X(cid:126) −(cid:104)R(cid:126)(cid:105))Tσ−1(X(cid:126) −(cid:104)R(cid:126)(cid:105))
and discuss its properties. Eventually, in Sec. IV we W[(cid:37)ˆ](X(cid:126))= 2 (cid:112) , (3)
πM Det[σ]
present our main results concerning the analysis of the
FI associated to two locally feasible observables, homo- the two being related by the Fourier transform
dyne detection and photon counting. We will show that
these feasible measurements allow to achieve optimal es- W[(cid:37)ˆ](X(cid:126))= 1 (cid:90) d2MΛ(cid:126) exp{iΛ(cid:126)TΩX(cid:126)}χ[(cid:37)ˆ](Λ(cid:126)). (4)
timation of the coupling parameter by probing only the (2π)2M
radiation part of the system.
A Gaussian state is completely determined by the first-
moments vector (cid:104)R(cid:126)(cid:105) and the second moments encoded
in the covariance matrix (CM) σ, of elements
II. TOOLS OF QUANTUM ESTIMATION
THEORY FOR GAUSSIAN STATES 1
σ = (cid:104)R R +R R (cid:105)−(cid:104)R (cid:105)(cid:104)R (cid:105), (5)
ij 2 i j j i i j
In this section we briefly introduce the formalism of
whichallowstowritetheHeisenberguncertaintyrelation
Gaussian states for continuous-variable bosonic systems
as σ + iΩ ≥ 0. The purity µ = Tr[(cid:37)ˆ2] of a Gaussian
and of symplectic diagonalization of quadratic Hamilto- 2
state is expressed in terms of the CM by the relation
nians [28–30]. (cid:112)
µ=(2M Det[σ])−1.
A property of Gaussian states, which will reveal to be
useful in the following calculations, is that the reduced
A. Gaussian states and symplectic transformations
density matrix, obtained by means of the partial trace
operation over the degrees of freedom of a subsystem,
A system composed by M bosonic modes is described
keeps its Gaussian character [31]. For instance, exploit-
by quantized fields aˆ satisfying the commutation re-
m ingtheGlauberrepresentationofadensityoperatorofa
lation [aˆ ,aˆ†] = δ . An equivalent description is
m l m,l bipartitestate,withcartesiancoordinatesΛ(cid:126) =(Λ(cid:126) ,Λ(cid:126) )
provided, through the Cartesian decomposition of field a1 a2
matoodrsesxˆ,min=te(ramˆms +ofaˆp†mo)s/it√io2n-anadndpˆmmo=mein(aˆtu†mm−-liaˆkme)o/p√e2r-. (cid:37)ˆa1a2 = (2π1)2 (cid:90)R4d4Λ(cid:126) χ[(cid:37)ˆa1a2](Λ(cid:126))Dˆ†(Λ(cid:126)), (6)
Introducing the vector of ordered quadratures R(cid:126) =
(xˆ1,pˆ1,··· ,xˆM,pˆM)T and the symplectic matrix together with Tr[Dˆ(Λ(cid:126)a1)] = (2π)δ(2)(Λ(cid:126)a1), then the re-
duced density operator (cid:37)ˆ is a Gaussian state with
a1
Ω= (cid:77)M ω , ω =(cid:18) 0 1(cid:19), (1) an associated characteristic function χ[(cid:37)ˆa1a2](Λ(cid:126)a1,0) =
m m −1 0 Tr[(cid:37)ˆ Dˆ(Λ(cid:126) )].
m=1 a1 a1
To become more familiar with these concepts, we
the commutation relations become [R ,R ] = iΩ . The list here some examples of Gaussian states. A single-
i j ij
state (cid:37)ˆ of a system of M bosonic modes can be de- mode system in a equilibrium with a thermal environ-
scribed in the phase space by means of the character- ment is described by the density operator νˆ (n¯) =
th
3
(cid:80)∞ n¯k(1 + n¯)−(1+k)|k(cid:105)(cid:104)k| expressed on the Fock ba- from which we can straightforwardly rewrite the uncer-
k=0
sis {|k(cid:105)}∞. The corresponding covariance matrix is tainty relation as d ≥ 1/2. Pure Gaussian states have
0 −
σ = (1+n¯)/2, with n¯ the number of average thermal I = 1/16 and I +I +2I = 1/2. The separability of
th 4 1 2 3
photons. Other examples include the classes of coher- thetwosubsystemsisformalizedintermsofthecriterion
ent states and squeezed states, for which the uncertainty of positivity under partial transpose (ppt) [32], which
relation σ + iΩ ≥ 0 is saturated with (cid:104)∆xˆ2(cid:105)(cid:104)∆pˆ2(cid:105) = can be written in terms of the symplectic invariants as
1/4. All cohe2rent states have (cid:104)∆xˆ2(cid:105) = (cid:104)∆pˆ2(cid:105) = 1/2, d˜− ≥1/2, where
whereas squeezed states possess a covariance matrix of
(cid:115)
tahnedkrind∈σRsqi=s a21Dreiaalg(seq2ur,eee−zi2nrg),pwahraemreet(cid:104)e∆r.xˆ2(cid:105)A=(cid:54) g(cid:104)e∆nepˆr2ic(cid:105) d˜± = I1+I2−2I3±(cid:112)(I21+I2−2I3)2−4I4 (9)
squeezed state is obtained from the vacuum by applying
theunitaryoperatorSˆ(ξ)=exp{(ξ(aˆ†)2−ξ∗aˆ2)/2},with are the symplectic eigenvalues of the CM of the partially
complex squeezing parameter ξ = reiψ. The most gen- transposed density operator describing a bipartite Gaus-
eral single-mode Gaussian state is a displaced squeezed sian state. A measure of entanglement is, thus, provided
thermal state (DSTS) described by the density operator by the logarithmic negativity [33]
(cid:37)ˆ = Dˆ(γ)Sˆ(r)νˆ (n¯)Sˆ†(r)Dˆ†(γ). For two-mode systems
th E (σ)=max{0,−ln2d˜ }, (10)
suchageneralformdoesnotexist,butarelevantsubclass N −
ofbipartiteGaussianstatesisgivenbythesqueezedther- which quantifies monotonically the amount of violation
mal states (cid:37)ˆa1a2 = Sˆ2(ξ)νˆth(n¯1) ⊗ νˆth(n¯2)Sˆ2†(ξ), where of the ppt-criterion.
Sˆ (ξ)=exp{ξaˆ†aˆ† −ξ∗aˆ aˆ } is the two-mode squeezing
2 1 2 1 2
operator.
An important property of Gaussian states is related B. Local QET
to transformations induced by quadratic Hamiltonians.
Gaussian states preserve their Gaussian character under Whenever a parameter of a physical system is not di-
symplectic transformations of coordinates R(cid:126) → FR(cid:126) +d(cid:126), rectly accessible by an observable, it is always possible
where d(cid:126) is a vector of real numbers, leaving unchanged to infer its average value by means of classical estima-
the Hamilton equations of motion and fulfilling the sym- tion theory inspecting the set of data {x} of an indirect
plectic condition FΩFT = Ω. Thus, the first-moment measurement. Let us suppose that an observable Xˆ is
vector and the CM of a Gaussian state follow the trans- measured on the considered physical system described
formation rules by a parameter-dependent density operator (cid:37)ˆ . A set
λ
of data {x ,...,x }, corresponding to the possible out-
1 m
(cid:104)R(cid:126)(cid:105)→F(cid:104)R(cid:126)(cid:105)+d(cid:126), σ →FσFT. (7) comesofXˆ,isthencollectedaccordingtothedistribution
p(x|λ) = Tr[(cid:37)ˆ Xˆ] provided by the Born rule, which de-
λ
Moreover, symplectic transformations originate from scribes the conditional probability to obtain an outcome
Hamiltonians at most bilinear in the field modes x given the value of the parameter λ. The value of the
(quadratic) and the diagonalization process of these parameter λ is then inferred from the statistics of an es-
Hamiltonians goes under the name of symplectic diago- timator λ¯ = λ¯(x ,...,x ), evaluating its average value
1 m
nalization, which transforms the coordinates by preserv- E[λ¯] and variance Var = E[λ¯2]−E[λ¯]2 (valid for any
λ
ing canonical commutation relations. Symplectic trans- unbiased estimator E[λ¯] = λ). From classical estima-
formations possess the property of unitary determinant tiontheory,optimalestimatorssaturatetheCram´er-Rao
Det[F] = 1. As an example, consider a thermal state bound
νˆth(n¯)evolvingunderthesingle-moderealsqueezerSˆ(r). 1
The associated symplectic matrix is F = Diag(er,e−r) Varλ ≥ mF (λ), (11)
and, according to Eq. (7), the CM transforms as σ = Xˆ
1(1+2n¯)Diag(e2r,e−2r), which is the CM of a squeezed where the FI F (λ) is the maximum information ex-
2 Xˆ
thermal state. tractable from a measurement of the observable Xˆ and
Inthelightofthepropertiesofsymplectictransforma- reads
tions and writing the CM of Gaussian bipartite states in (cid:90)
(cid:18) A C(cid:19) F (λ)= dxp(x|λ) (cid:0)∂ (cid:2)lnp(x|λ)(cid:3)(cid:1)2. (12)
the most general way as σ = , it is possible to Xˆ λ
CT B R
identify four symplectic invariants given by I1 =Det[A], The ultimate limit to the precision in an estimation pro-
I2 = Det[B], I3 = Det[C] and I4 = Det[σ]. The sym- cess is given the quantum Cram´er-Rao bound
plectic eigenvalues of a CM can be expressed in terms of
1
these invariants as Var ≥ , (13)
λ mH(λ)
(cid:115)
(cid:112)
I +I +2I ± (I +I +2I )2−4I where the QFI H(λ) does not depend on measurements
1 2 3 1 2 3 4
d = , (8)
± 2 but only on the probe state (cid:37)ˆλ. The QFI is the result of
4
a maximization over all the possible observables on the with transition frequency ω , assumed to be equal for
0
physical system and it is such that H(λ) ≥ F (λ). The all the spins, and a single radiation mode (bosonic field)
Xˆ
QFI is analytically computable as H(λ)=Tr[(cid:37)ˆ Lˆ2], i.e. of frequency ω, which is characterized in terms of anni-
in terms of the hermitean operator Lˆλ called syλmmλetric hilation and creation operators, aˆ1 and aˆ†1 respectively.
logarithmic derivative (SLD), implicitly defined as The coupling between the two quantum systems is suit-
ably described within the dipole approximation, where
Lˆ (cid:37)ˆ +(cid:37)ˆ Lˆ each atom couples to the electric field of radiation with
∂ (cid:37)ˆ ≡ λ λ λ λ. (14)
λ λ 2 a coupling strength λ:
The SLD operator represents the optimal positive-
λ
operator valued measurement (POVM) saturating the Hˆ =ω Jˆ +ωaˆ†aˆ + √ (aˆ† +aˆ )(Jˆ +Jˆ ). (17)
(1,2) 0 z 1 1 1 1 + −
N
Cram´er-Rao bound (13).
CriticalityataQPTisaresourceforquantumestima-
Overall,theatomicsubsystemcanbedescribedasapseu-
tionasasmallchangeintheparameterλyieldsadrastic
dospin of length N/2 by the collective spin operators
changeinthegroundstateattheboundaryofthecritical
Jˆ = 1(cid:80)N σˆ(i) and Jˆ = (cid:80)N σˆ(i), where {σˆ(i),σˆ(i)}
parameter, thus allowing the QFI to diverge. It is, thus, z 2 i=1 z ± i=1 ± z ±
is the set of Pauli matrices that completely characterize
desirable to find an optimal observable maximizing the
single two-level systems.
FI to the values of the QFI in order to achieve the best
precision in the parameter estimation. The diagonalization of Hamiltonian (17) is performed
InthecontextofGaussianstatesitispossibletoderive employing the Holstein-Primakoff (H-P) representation
analytical expressions for the QFI and the SLD operator of the atomic spin operators [35, 36], namely Jˆ =
+
(cid:113) (cid:113)
[34],whichdependonthephysicalparameterscharacter- aˆ† N −aˆ†aˆ , Jˆ = N −aˆ†aˆ aˆ and Jˆ =aˆ†aˆ − N,
izingthestateofthesystem. Exploitingthenotionsout- 2 2 2 − 2 2 2 z 2 2 2
lined in Sect. IIA and redefining the partial derivation where aˆ2 and aˆ†2 are bosonic fields satisfying [aˆ2,aˆ†2]=1.
as ∂λ(f) ≡ f˙, the QFI and SLD for a generic Gaussian As will become soon clearer, the bosonic fields {aˆ1,aˆ2}
state read are allowed to have ma√croscopic occupations √in such a
way that aˆ → aˆ −α N and aˆ → aˆ +β N with
H(λ)=Tr(cid:2)ΩTσ˙ΩΦ(cid:3)+(cid:104)R(cid:126)˙ (cid:105)Tσ−1(cid:104)R(cid:126)˙(cid:105) (15) {α,β}∈R.1Now,1we consider the t2hermo2dynamic limit,
for which the ratio N/V is constant as N,V →∞, being
L =R(cid:126)TΦR(cid:126) +R(cid:126)Tζ(cid:126)−ν, (16)
λ N the number of atoms and V the corresponding occu-
piedvolume,andexpandtheH-Prepresentationkeeping
where ν = Tr[ΩTσΩΦ] is related to the property of the √
only the terms proportional to N. Applying stabil-
SLD (14) to have zero-mean value Tr[(cid:37)ˆ Lˆ ] = 0. For
λ λ ityconsiderations,forwhichlineartermsproportionalto
√
pure Gaussian states all the quantities in Eqs. (15)-
N must vanish [36], we obtain the expression for the
(16) are easy to compute and read Φ = −σ˙ and ζ(cid:126) =
displacing parameters
ΩTσ−1(cid:104)R(cid:126)˙(cid:105).
 √
In the following we will apply these tools to the Dicke α=±λ 1−k2
ω
model, in order to exploit the predicted QPT, and the (cid:113) (18)
corresponding ground states, for the estimation of the β =± 1−2k
coupling parameter λ.
whereweintroducedthedimensionlesscriticalparameter
III. DICKE QUANTUM PHASE TRANSITION (cid:26)1 for λ<λ
c
FOR QUANTUM ESTIMATION k ≡ λλ22c for λ>λc (19)
InthissectionwedescribetheDickeQPTintheGaus- √
and the critical coupling strength λ ≡ ωω /2. As
sian formalism, suitable for establishing a tight connec- c 0
it is now clear, the macroscopic occupation of the two
tion with local estimation theory performed with mea-
subsystems individuates the phase transition between a
surements typical of quantum optics. In particular we
normal phase for λ < λ and a superradiant phase for
derive the Gaussian ground states corresponding to the c
λ > λ . The Hamiltonian of this system can be cast
normal and superradiant phases, computing the amount c
in diagonal form (see Ref. [18]), by introducing a new
of entanglement and the scaling behaviors of the associ-
couple of bosonic modes {ˆb ,ˆb } which satisfy the com-
ated QFI and SLD. − +
mutation relations [ˆb ,ˆb† ] = [ˆb ,ˆb†] = 1, and describe
− − + +
two independent harmonic oscillators
A. The superradiant QPT
Hˆ = ε ˆb†ˆb +ε ˆb†ˆb +
(−,+) − − − + + +
TheDickemodel[13]describestheinteractionbetween 1(cid:16) ω (cid:17) ω (1+k2)N (20)
a dense collection of N two-level atoms (spin objects) + 2 ε−+ε+−ω− k0 − 0 2k 2 ,
5
where the eigenfrequencies are given by �
���
�

2ε2 =ω2+ ω02 ±(cid:115)(cid:20)ω02 −ω2(cid:21)2+16λ2ωω k . (21) � � �-
± k2 k2 0
� �
��� ��� ��� ��� ���
λ
The diagonalization in Ref. [18] can be obtained by per-
forming the symplectic transformation F =F ◦F ◦F :
3 2 1
FIG. 1: (Color online) Plot of the logarithmic negativity
E (λ) (blue solid curve) and of the lowest symplectic eigen-
N
value d˜ (λ) (red dashed curve) of the partially transposed
−
(cid:18) 1 √ 1 √ (cid:19) CM. The dashed gray line at d˜− =0.5 represents the thresh-
F =Diag √ , ω,√ , ω˜ old of separability, under which the state is entangled. Pa-
1 ω ω˜ rameters: ω = ω = 1 and λ = 0.5 (gray vertical line), in
0 c
(cid:18)cosθI −sinθI (cid:19) units of ω0.
F = 2 2 (22)
2 sinθI cosθI
2 2
(cid:18) (cid:19)
√ 1 √ 1
F =Diag ε ,√ , ε ,√ . where
3 − ε + ε
− +
ω (cid:18)cos2θ sin2θ(cid:19)
σ = +
11 2 ε ε
− +
Symplectic matrix F corresponds to a local squeez- σ = 1 (cid:0)ε cos2θ+ε sin2θ(cid:1)
1√ √ 22 2ω − +
ing Sˆl(o1c) = Sˆ(−log( ω)) ⊗ Sˆ(−log( ω˜)) applied to ω˜ (cid:18)cos2θ sin2θ(cid:19)
the quadratures of the atomic and photonic subsystems, σ = + (25)
with ω˜ = ω (1 + k)/2k. Then the rotation Uˆ(θ) = 33 2 ε+ ε−
0
exp{−iθ(xˆ1pˆ2 − xˆ2pˆ1)} is associated to the symplectic σ = 1 (cid:0)ε cos2θ+ε sin2θ(cid:1)
matrix F (I is a 2×2 identity matrix) and allows to 44 2ω˜ + −
2 2 √
eliminate the interaction term in the H√amiltonian upon ωω˜sin2θ (cid:18) 1 1 (cid:19)
the choice of the angle 2θ = tan−1(cid:2)4λ ωω0kk2/(ω02− σ13 =σ31 = 4 ε+ − ε−
k2ω2)(cid:3). Eventually, a second local squeezing Sˆ(2) = sin2θ
Sˆ(log(√ε−)⊗Sˆ(−log(√ε+)), related to the symplolecctic σ24 =σ42 =−4√ωω˜ (ε−−ε+).
transformation F , completes the diagonalization. The
3
ground state of the diagonalized Hamiltonian Hˆ(−,+) is The ground states |Ψ(cid:105) describing the two phases, are
thevacuumstate|ψ(cid:105)≡|0(cid:105)−⊗|0(cid:105)+, withCMσψ√=I4/2. nowcompletelycharacterizedasGaussianstatesbytheir
Accounting for the displacement Dˆ ≡ Dˆ (α N) ⊗ Wignerfunction(3),andthecorrespondingCM(23)and
√ 12 1
Dˆ (−β N) responsible for the macroscopic occupation first-moment vector (24) are now expressed in terms of
2
of the original modes {aˆ ,aˆ } in the superradiant phase, the physical parameters {λ,ω,ω ,N}. We notice that
1 2 0
the form of the Gaussian ground state |Ψ(cid:105) is straightfor- thedependenceonthesizeN oftheatomicsubsystemis
wardly obtained by means of the transformation |Ψ(cid:105) = contained only in the first-moment vector (24).
Dˆ Uˆ |ψ(cid:105), whereUˆ ≡Sˆ(1)Uˆ(θ)Sˆ(2) istheunitaryevo- In both phases the ground state is a pure Gaussian
12 F F loc loc
lution of the modes associated to the symplectic trans- state (µ=1), with d± =1/2, since it has been obtained
formation F. The corresponding CM σ ≡ σ and first- by a symplectic transformation of the vacuum state |ψ(cid:105).
Ψ
moment vector (cid:104)R(cid:126)(cid:105) are derived using Eqs. (7): When the coupling λ between the two subsystems gets
stronger, the two become increasingly entangled, as wit-
nessed by the logarithmic negativity (10), which quanti-
fies in a monotonic way the violation of ppt-criterion for
theseparabilityofabipartitestate. AsitisshowninFig.
σ 0 σ 0 
11 13 1, the atomic and radiation subsystems get increasingly
0 σ 0 σ
σ =Fσ FT = 22 24 (23) entangled as their coupling approaches the critical value
ψ σ 0 σ 0 
31 33 λ . Thealreadyestablishedresultthatentanglementen-
0 σ 0 σ c
42 44 hances the precision of a measurement [37, 38] will be
√ √ confirmedinthefollowing,wherewewilladopttheQET
(cid:104)R(cid:126)(cid:105)=(α 2N,0,−β 2N,0)T, (24)
approach to the considered critical system.
6
B. QFI and SLD
TABLE I: Limiting behaviors of the QFI in the normal and
superradiant phases at λ→λ±, λ→0 and λ→∞.
c
Once the ground states in the two phases are known,
itispossibletostudythebehavioroftheQFI,asafunc- Normal phase Superradiant phase
tion of the coupling parameter λ driving the QPT and
the tunable radiation frequency ω, which sets the criti-
(cid:104) (cid:105) (cid:104) (cid:105)
cal point λc. We point out that in our model λ and ω λ→λc 8(λ−1λc)2 +O |λ−1λc| 8(λ−1λc)2 +O |λ−1λc|
are considered independent on each other, for the sake
λ→0 4 +O[λ2] —
of simplicity, but that in some experimental realizations (ω+ω0)2
(see, e.g., Ref. [25]) they may be related to the tunable λ→∞ — 4N +O(cid:2)λ−4(cid:3)
parameters of an external pumping. ω2
ReferringtoEq. (15),itispossibletoanalyticallyeval-
uate the QFI in the two phases, but we report here only
|λ−λ |−3/2, namely
the limiting behaviors in proximity of the critical value c
λ . Inparticular,theleadingtermintheseriesexpansion √
ofctheQFIapproachingthecriticalparameterfromboth R(cid:126)T(−σ˙)R(cid:126) ∼ √ 4ωω0 1 (xˆ(cid:48) −xˆ(cid:48))2, (26)
the two phases, is H(λ)∼[2√2(λ−λc)]−2, whereas the 8 2(cid:112)ω2+ω02|λ−λc|3/2 1 2
main limiting cases are displayed in Table I. At the crit-
ical point the QFI for the whole radiation-atoms system whereR(cid:126)(cid:48) =F1R(cid:126) isthevectorofquadraturestransformed
diverges with a second-order singularity, thus highlight- accordingtothelocalsqueezingemployedintheHamilto-
ing the possibility to estimate the parameter λ (in the nian diagonalization (22). In the superradiant phase the
ideal thermodynamic limit) with infinite precision. By linear term of the SLD, dependent also on the number
tuningλ withω,itispossibletoobtainthehighestpre- of atoms N, is constant very close to the critical point,
c
cisionforeveryvalueofthecouplingparameterλ,asthe namely
behavioroftheQFIatλ isleftunvaried(seeFig. 2). We
c (cid:115)
pointoutthatthesecondterminEq. (15)isnon-zeroin R(cid:126)Tζ(cid:126)∼ 32N (cid:0)ω2pˆ(cid:48) −ω2pˆ(cid:48)(cid:1), (27)
the superradiant phase, in particular the QFI behaves in ω3ω2(ω2+ω2) 0 1 2
0 0
the thermodynamic limit as a linear increasing function
of N, with finite-size corrections of the order N−1/2, for in such a way that, ultimately, the SLD diverges at λ
c
everyvalueofthecouplingλ. Nonetheless,atthecritical as in Eq. (26), but still more slowly than the QFI (see
point λ the dominant contribution to H(λ) is ruled by Table I for comparison). Since the SLD is associated to
c
the coupling parameter (see Table I). theoptimalPOVMsaturatingthequantumCram´er-Rao
bound (13), we note that Eq. (26) contains a combina-
Now we compute the SLD operator in the two phases
tion of position quadratures relative to both the atomic
and analyze the asymptotic behaviors, with respect to
and radiation subsystems, confirming the highly entan-
λ, at the critical point. In the normal phase the second
gled nature of the two (see Fig. 1).
term of Eq. (16) is null, since the amplitudes of the
In the next sectionwe willshowthat itis stillpossible
displacements (18) are zero. In both phases ν = 0
tooptimallyestimatetheparameterλaroundthecritical
and the main term of the SLD has the same dependence
point, by means of locally feasible measurements.
�(λ)(����) IV. OPTIMAL LOCAL MEASUREMENTS
�
The main results of this work are examined in depth
inthissectionandconcernthepossibilitytoprobeoneof
thetwosubsystems(radiationmodeoratomicensemble)
� with local and handy measurements, in order to retrieve
the optimal FI. In particular, we address the two most
known and employed optical techniques for measuring
and characterizing a single-mode radiation, namely ho-
� λ
��� ��� ��� modyne detection and photon counting.
FIG. 2: (Color online) Plot of the QFI as a function of λ,
whereallthequantitiesarecomputedinunitsofω andN = A. Homodyne detection
0
100. Resonancecondition: ω =ω=1(solidbluecurve)with
0
λc = 0.5. Off-resonance condition: ω = 0.25 (dashed orange Since all the information about the radiation mode is
curve) with λ =0.25.
c encoded in its Wigner function, it is possible to recon-
7
TABLEII:LimitingbehaviorsoftheFIforhomodyne-likedetectionofbothradiationF (λ)andatomicF (λ)subsystems
xˆ(φ) yˆ(φ)
(with respect to QFI), in the normal and superradiant phases at λ→λ±, λ→0 and λ→∞.
c
Normal phase Superradiant phase
λ→0 λ→λ− λ→λ+ λ→∞
c c
F (λ)/H(λ) 2[ω+(ω+ω0)cos(2φ)]2λ2+O[λ3] 1+O[(cid:112)|λ−λ |] 1+O[(cid:112)|λ−λ |] cos2φ+O[λ−4]
xˆ(φ) ω2(ω+ω0)2 c c
F (λ)/H(λ) 2[ω0+(ω+ω0)cos(2φ)]2λ2+O[λ3] 1+O[(cid:112)|λ−λ |] 1+O[(cid:112)|λ−λ |] O[λ−6]
yˆ(φ) ω02(ω+ω0)2 c c
struct the corresponding Gaussian state (cid:37)ˆusing the ho- are listed in Table II for both the normal and superradi-
modyne tomography technique, i.e. repeatedly measur- antphases. Itisremarkablethatameasurementonlyon
ing the field mode quadratures according to the set of a part of the system, namely the radiation mode subsys-
observables tem, providestheoptimalvalueoftheFIinproximityof
the critical point. Homodyne detection results to be an
aˆe−iφ+aˆ†eiφ
xˆ(φ)= √ ≡Uˆ†(φ)xˆUˆ(φ), (28) optimallocalmeasurement, easilyfeasiblewithstandard
2 opticaltechniques,abletoprovidethebestperformances
inparameterestimationandtocapturethequantumcrit-
where Uˆ(φ) ≡ e−iφaˆ†aˆ is a phase-shift operator. The
icality. In Fig. 4 we show that for different values of the
probability distribution of the possible outcomes of
angle of the measured quadrature xˆ(φ), at the critical
a quadrature-measurement px(φ) = (cid:104)x|Uˆ(φ)(cid:37)ˆUˆ†(φ)|x(cid:105), point λ → λ± FI diverges with the very same scaling
c
corresponds to the marginal distribution
behavior of QFI, saturating the quantum Cram´er-Rao
(cid:90) bound(13). Theonlyexception, whichdonotinvalidate
px(φ) = dpW[(cid:37)ˆ](xcosφ−psinφ,xsinφ+pcosφ), the homodyne measurement, is that exactly at φ = π/2
R the FI is no longer optimal at λ , even though its di-
(29) c
verging character (see the insets in Fig. 4) represents
where the Wigner function W[(cid:37)ˆ](x,p) of the reduced
a high precision measurement according to the classical
state of the radiation mode (see Sec. IIA) (cid:37)ˆ =
Cram´er-Rao bound (11). Besides, we point out that the
Tr [|Ψ(cid:105)(cid:104)Ψ|] is Gaussian with second and first moments
2 FI, in the superradiant phase and in the thermodynamic
given by
limit,scalesasalinearfunctionofN,withfinite-sizecor-
(cid:32) (cid:33)
σ 0
σ = 11 (30)
0 σ
22
√
(cid:104)R(cid:126)(cid:105)=(α 2N,0)T. (31)
In Fig. 3 we show the Wigner function associated to
the radiation subsystem, together with the marginal dis-
tributions corresponding to homodyne measurements of
the position xˆ(0) and momentum xˆ(π/2). From the se-
quenceofframesatdifferentvaluesofλ, theQPTisevi-
dent,wherethefieldmodeessentiallyundergoesastrong
squeezingaroundλ andthenadisplacementforλ>λ .
c c
We now evaluate the FI associated to the homodyne
measurement probing the Gaussian ground state of the
radiation mode, as a function of the parameter λ driv-
ing the QPT. Since the probability distribution (29) has
Gaussianformwithmeanvalue(cid:104)xˆ(φ)(cid:105)=cosφ(cid:104)xˆ(0)(cid:105)and
variance σ(φ) = cos2φσ +sin2φσ , it is straightfor-
11 22
ward to derive a general expression for the FI (12) valid FIG.3: (Coloronline)WignerfunctionW[(cid:37)ˆ](x,p)oftheradi-
ation mode state at different values of λ=0.3 (a), λ=0.499
for both the normal and superradiant phases
(b), λ=0.6 (c) and λ=1.5 (d). Marginal distributions, cor-
2σ(φ)(cid:104)xˆ(φ)(cid:105)2+σ˙2(φ) respondingtotheprobabilityforthepositionandmomentum
Fxˆ(φ)(λ)= 2σ2(φ) . (32) quadratures, respectively px(0) and px(π/2), are also shown.
The values of the chosen parameters are ω = ω = 1 and
0
λ =0.5(inunitsofω ),inthesuperradiantphaseN =100.
The scaling behaviors of the FI, compared to the QFI, c 0
8
��(ϕ)/� B. Photon counting
���
��� ��(ϕ)(λ)(����)
��
� Another typical observable used to probe the electro-
��� � magnetic field is the photon number operator
�
�
��� λ
���� ���� ���� ���� ����
��� ∞
ω�=ω=� Nˆ ≡aˆ†aˆ = (cid:88)n|n(cid:105)(cid:104)n|
λ 1 1 1
��� ��� ��� ��� n=0
(33)
�����(ϕ)/� (cid:88)∞ np(n)=Tr[(cid:37)ˆNˆ ],
1
��(ϕ)(λ)(����) n=0
���
��
�
��� �
�
� where p(n)=(cid:104)n|(cid:37)ˆ|n(cid:105) is the probability to detect a pho-
��� λ
����� ���� ����� ���� ton in the Fock state |n(cid:105). Photon counters capable of
��� discriminating among the number of incoming photons
ω�=��ω=���� arecommonlyemployedinquantumopticalexperiments
λ
��� ��� ��� ��� ��� [25, 39, 40]. As we mentioned in Sec. IIA, the partial
traceofaGaussianbipartitestate,isasingle-modeGaus-
FIG.4: (Coloronline)PlotoftheratiobetweenFIforhomo- sianstatewhichcanbecastinthegeneralformofaDSTS
dyne detection and QFI, as a function of λ. The insets show (cid:37)ˆ = Dˆ(γ)Sˆ(r)νˆ (n¯)Sˆ†(r)Dˆ†(γ). The analytic and gen-
th
the behavior of the FI (solid curves) and the QFI (dashed
eral expression for the photon number probabilities [41],
curve), both diverging at the critical parameter λ . The ar-
c applied to the state of the radiation subsystem with CM
rows indicate the increasing values of the quadrature angle
andfirst-momentvector givenbyEq. (30)and Eq. (31),
φ = 0,π/3,π/2 (solid curves). Upper panel: resonance con-
dition with ω = ω = 1 and λ = 0.5. Lower panel: off-
0 c
resonance condition with ω =1, ω=0.25 and λ =0.25. In
0 c
bothcasesthesetofparametersisinunitsofω andN =100.
0
�(�)
� (�)
��-� λ=����
��-�
rections of the order N−1. Thus, the ratio between QFI λ=���
and FI plotted in Fig. 4 is essentially independent on N,
�
for every value of the coupling λ. � � � � � ��
�(�)
���� (�)
Analogously, a homodyne-like detection of the atomic
���� λ=����
subspace, corresponding to measure the generic compo-
nent Jˆ(φ) ≡ Jˆ cosφ+Jˆ sinφ of the collective atomic ����
x y
spininthe{x,y}-plane,resultstobeoptimalatthecrit-
����
icalcouplingλc. Theonlydifferencesare: (i)inthelimit λ=���
λ → 0, the atomic and radiation frequencies, ω and ω, ����
0
areinterchangedand(ii)inthelimitλ→∞,theFIgoes
�
to zero (see Table II). �� �� �� �� �� �� ��
FIG. 5: (Color online) Logarithmic plot of photon number
probability distributions (34) of the radiation ground state
Interestingly,theelectromagneticfieldquadraturesap- in the normal phase (a), with ω =ω =1 and λ=0.3,0.49.
0
pearinthelimitingexpressionfortheSLD(26),thuscon- Photonnumberprobabilitydistributions(34)oftheradiation
firming the optimal character of the chosen homodyne- groundstateinthesuperradiantphase(b),withω0 =ω=1,
type detection employed to probe just one of the two N =100 and λ=0.55,0.7. The mean values of the distribu-
tions, Eq. (36), are specified with dashed vertical lines.
subsystems.
9

〈��〉 ���/�
���
�
��� ���/�
�
��
���
�
��� ���
�
���
λ
����� ��� �����
λ
λ
���� ���� ���� ���� ��� ��� ��� ��� ���
FIG. 6: (Color online) Plot of the mean energy of the ra- FIG. 7: (Color online) Plot of the ratio between FI for a
diation mode subsystem (dot-dashed curve) as a function photon-countmeasurementandQFI,asafunctionofλ. The
of the coupling parameter λ. Three contributions to (cid:104)Nˆ (cid:105) insetshowsamagnificationaroundthecriticalparameterλc,
1 showing in a clearer way that the observable Nˆ is optimal.
are showed: mean thermal photons n¯ (solid curve), mean 1
Thevaluesoftheparameters(inunitsofω )areω =ω=1,
squeezed photons n (dashed curve) and mean coherent en- 0 0
ergy |α|2N (dotted csurve). λc =0.5 and N =100 in the superradiant phase.
n = sinh2r, with r = Log((cid:112)4 σ /σ ). The extensive
respectively, reads s 11 22
contribution is provided by the amplitude of displace-
√
p(n)=R (−1)n2−2n(A˜+|B˜|)n× ment γ =α N, depending on the number of atoms. As
00
plotted in Fig. 6, it is evident how, in proximity of the
(cid:88)n H2k(0)H2n−2k(cid:18)iC˜(cid:104)A˜+|B˜|(cid:105)−12(cid:19)(cid:34)A˜−|B˜|(cid:35)k pinhcarseeastersandsuietioton,athsteromnegandepghreoetoonf snquumebezeirngdraanmdaatichailglhy
× ,
k!(n−k)! A˜+|B˜| thermal component. Only in the superradiant phase the
k=0 extensive contribution |α|2N dominates far away of the
(34)
critical parameter, due to an increasing coherent state
component(seealsoFig.3(d)). Wepointoutthatevenin
whereH (x)areHermitepolynomials. Allthequantities
m the thermodynamic limit, although in the normal phase
appearing in Eq. (34) depend only on first- and second-
theextensivecontributionisnotpresent,intheproximity
moments as follows:
of the critical point a non-negligible fraction of squeezed
2exp{− (cid:104)xˆ1(cid:105)2 } thermalphotonsshouldbemeasuredbyaphotodetector.
R = 1+2σ11 The FI information associated to the observable (33)
00 (cid:112)
(1+2σ11)(1+2σ22) is given by Eq. (12) expressed in discrete form
4σ σ −1
A˜= (1+2σ1111)2(21+2σ22) (35) F (λ)= (cid:88)∞ [∂λp(n)]2. (37)
2(σ −σ ) Nˆ1 p(n)
B˜ = 22 11 n=0
(1+2σ )(1+2σ )
11 22
√ In Fig. 7 we show the behavior of the FI associated
2(cid:104)xˆ (cid:105)
C˜ = 1 . to a photon-count measurement compared to the QFI.
1+2σ11 Even though numerical simulations necessarily imply a
cut-off value of the dimensionality of the Fock space in
In Fig. 5 we plot the probability distributions for the
evaluating the series in Eq. (37), making the numerical
photon number characterizing the ground states of the
calculationsawkwardaroundλ ,itisevidentthattheob-
two phases. In the normal phase, the reduced ground c
servable Nˆ tends to be optimal at the critical coupling.
state for the radiation subsystem is a squeezed thermal 1
We can, thus, strengthen our main result, according to
state with typical photon number distribution peaked in
which optimal parameter estimation around the region
n = 0, whereas in the superradiant phase it acquires
ofcriticalitycanbeachievedevenbyprobingonlyapart
macroscopicoccupationduetothenon-zerodisplacement
of the composite system.
amplitude(18). Thegeneralexpressionofthemeanpho-
tonnumberofagenericsingle-modeGaussianstateinthe
DSTS form is
V. CONCLUSIONS
(cid:104)Nˆ (cid:105)=n +n¯(1+2n )+|γ|2. (36)
1 s s
We have analyzed the superradiant QPT occurring in
It is possible to identify an intensive contribution to theDickemodelintermsofGaussiangroundstateswith
(cid:104)Nˆ (cid:105) given by the mean number of thermal photons the help of the symplectic formalism. In this framework,
1 √
n¯ = σ σ −1/2 and the fraction of squeezed photons wehaveaddressedtheproblemofestimatingthecoupling
11 22
10
parameter, investigating whether and to which extent ests, recently arisen in connection to the realization of
criticality is a resource to enhance precision. In partic- exotic matter phases, we believe that a quantum estima-
ular, we have obtained analytic expressions and limiting tion approach, as the one outlined in this work, can be
behaviors for the QFI, showing explicitly its divergence profitably employed in quantum critical systems. The
at critical point. Upon tuning the radiation frequency gain is twofold, since (i) criticality is a resource for the
wemayalsotunethecriticalregionand,inturn,achieve estimation of unaccessible Hamiltonian parameters and
optimal estimation for any value of the radiation-atoms (ii) the search for optimal observable providing high-
coupling. precision measurements allows a fine-tuning detection of
Besides, we studied two feasible measurements to be the QPT itself. The analysis may be also extended to fi-
performed only onto a part of the whole bipartite sys- nite temperature and to systems at thermal equilibrium.
tem,homodyne-likedetectionandphotoncounting. The Work along these lines is in progress and results will be
remarkable result is that by probing just one of the two reported elsewhere.
subsystems, namely the radiation mode or the atomic
ensemble, it is possible to achieve the optimal estima-
tion imposed by the quantum Cram´er-Rao bound. No-
tice that this is a relevant feature of the system, in view
Acknowledgments
ofitsstronglyinteractingnatureandofthehighdegreeof
entanglementofthetwosubsystemsatthecriticalpoint.
The possibility of probing the system accessing only the ThisworkhasbeensupportedbyEUthroughtheCol-
radiation part is of course a remarkable feature for prac- laborativeProjectsQuProCS(GrantAgreement641277)
tical applications. and by UniMI through the H2020 Transition Grant 14-
Motivated by relevant and fruitful experimental inter- 6-3008000-625.
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