Characterization of the quantum phase transition in a two-mode Dicke model for different cooperation numbers

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by  L. F. Quezada

Characterization of the quantum phase transition in a two-mode Dicke model for
different cooperation numbers
L. F. Quezada∗ and E. Nahmad-Achar†
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Apartado Postal 70-543, 04510 Ciudad de México, México.
We show how the use of variational states to approximate the ground state of a system can
be employed to study a multi-mode Dicke model. One of the main contributions of this work is
the introduction of a not very commonly used quantity, the cooperation number, and the study
of its influence on the behavior of the system, paying particular attention to the quantum phase
transitionsandtheaccuracyoftheusedapproximations. Wealsoshowhowthesephasetransitions
affect the dependence of the expectation values of some of the observables relevant to the system
and the entropy of entanglement with respect to the energy difference between atomic states and
7 thecouplingstrengthbetweenmatterandradiation,thuscharacterizingthetransitionsindifferent
1 ways.
0
2
Introduction Notice that in the previous definition, for the sake of
n
generality,wedidnotconsiderthethermodynamiclimit,
a
J Quantum phase transitions (QPTs) are informally seen asithasbeenshownthatinterestingphenomenaregard-
6 as sudden, drastic changes in the physical properties of ing QPTs occur even for a finite number of particles
2 the ground state of a system at zero temperature due [11, 12].
to the variation of some parameter involved in the mod-

] eling Hamiltonian. One model of particular interest for
h Modeling Hamiltonian
the study of such phenomena is the Dicke model [1], as
p
- it describes, in a simplified way (electric dipole approxi- The Hamiltonian (Dicke’s Hamiltonian) describing the
nt mation),theinteractionbetweenmatterandelectromag- interaction, in a dipolar approximation, between N two-
a neticradiation. In1973,HeppandLieb[2,3],andWang
levelidenticalatoms(sameenergydifferencebetweenthe
u and Hioe [4] first theoretically proved the existence of a
two levels) and one-mode of an electromagnetic field in
[q second-order QPT in the Dicke model. Wang and Hioe an ideal cavity, has the expression ((cid:126)=1)
also treated the multi-mode radiation case, where they

1 reduce it to a single-mode case by using an effective cou-
v pling constant. To date, this QPT has been experimen- H =ω J +Ωa†a− √γ (J +J )(cid:0)a+a†(cid:1). (1)
D A z − +
8 tally observed in a Bose-Einstein Condensate coupled to N
5
an optical cavity [5, 6] and it has been shown to be rel-
8 evant to quantum information and quantum computing Here, ωA is the energy difference between the atomic
7 levels, Ω is the frequency of the field’s mode, γ is the
[7–10].
0 dipolar coupling constant, J , J , J are the collective
1. Even though the formal definition of a QPT requires spinoperatorsanda,a† aretzhea−nnih+ilationandcreation
us to compute the ground state’s energy as a function
0 operators of the harmonic oscillator. The multi-mode
of any desired parameter in order to find its transition
7 Hamiltonian is obtained summing over the number k of
1 values, one of the main contributions of this work is to modes [13], and has the expression
: show how the QPT in the Dicke model influences the
v
behavior of other quantities relevant to the system, thus
i
X characterizing the transition in different, simpler ways. k k
H =ω J +(cid:88)Ω a†a −√1 (cid:88)γ (J +J )(cid:0)a +a†(cid:1).
r A z ı ı ı ı − + ı ı
a N
ı=1 ı=1
Quantum Phase Transitions
(2)
Thek modesoftheelectromagneticfieldaredescribed
The formal definition of the concept of “quantum phase”
in terms of annihilation and creation operators for each
that we will be using throughout this paper is that of
mode a , a†, acting on the tensor product of k copies
an open region R ⊆ R(cid:96) where the ground state’s energy ı ı
k
(cid:78)
E ,asafunctionof(cid:96)parametersinvolvedinthemodeling of the Fock space F and satisfying the commutation
0 ı
Hamiltonian,isanalytic. ThusaQPTisidentifiedbythe ı=1
boundary∂Roftheregionatwhich ∂nE0 isdiscontinuous relations
∂xn
for some n (known as the order of the transition).
(cid:2)a ,a†(cid:3)=δ , [a ,a ]=(cid:2)a†,a†(cid:3)=0. (3)
ı  ı ı  ı 
∗ [email protected] Atwo-levelatomisdescribedusingthe 1-spinmatrices
2
† [email protected] S = 1σ , S = 1(σ ±iσ ) (σ , σ and σ being the
z 2 z ± 2 x y x y z
2
j(cid:61)1 ue j(cid:61)5 ue j(cid:61)9 ue
E(cid:45)2 E(cid:45)10 E(cid:45)18
(cid:45)3 (cid:45)11.5 (cid:45)20.5
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
(cid:45)13
(cid:45)4 (cid:45)23
Γ u Γ u Γ u
(cid:45)6 (cid:45)3 3 6 2 e (cid:45)2 (cid:45)1 1 2 2 e (cid:45)1 (cid:45)0.5 0.5 1 2 e
Figure 1. Energy of the ground state as a function of γ obtained using CS (dark gray dashed / red online), SAS with CS’s
2
minima(lightgraydashed/orangeonline),SASminimizednumerically(darkgray/blueonline)andquantumsolution(light
gray / cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light gray
/ cyan online), SAS minimized num(cid:72)eri(cid:76)cally (dark gray / blue online) and SAS(cid:72)wi(cid:76)th CS’s minima (light gray dashed / ora(cid:72)ng(cid:76)e
online). Left: j=1, center: j=5, right: j=9. Assuming k = 2 and using N = 18, ω = Ω = Ω = 2u , γ = 1u , where u
A 1 2 e 1 2 e e
stands for any energy unit ((cid:126)=1).
j(cid:61)1 ue j(cid:61)5 ue j(cid:61)9 ue
E E E
(cid:45)0.2 (cid:45)4
(cid:45)15
(cid:45)0.5 (cid:45)6.5
(cid:45)20
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
(cid:45)0.8
(cid:45)9 (cid:45)25
Ω u Ω u Ω u
0.2 0.6 1 A e 1 2 A e 1.5 3 A e
Figure 2. Energy of the ground state as a function of ω obtained using CS (dark gray dashed / red online), SAS with CS’s
A
minima(lightgraydashed/orangeonline),SASminimizednumerically(darkgray/blueonline)andquantumsolution(light
gray / cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light gray
/ cyan online), SAS minimized num(cid:72)eri(cid:76)cally (dark gray / blue online) and SAS(cid:72)wi(cid:76)th CS’s minima (light gray dashed / ora(cid:72)ng(cid:76)e
online). Left: j=1, center: j=5, right: j=9. Assuming k=2 and using N =18, Ω =Ω =2u , γ = 1u , γ =1u , where u
1 2 e 1 2 e 2 e e
stands for any energy unit ((cid:126)=1).
Paulimatrices),whichactonatwo-dimensionalcomplex To overcome this issue, we must use the common set of
Hilbert space C2 and satisfy the commutation relations eigenvectors {|j,m(cid:105)} of the two commuting observables
J and J2 = 1(J J +J J )+J2, where the label j
z 2 + − − + z
is limited to the values j ∈ (cid:8)r,r+1,...,N(cid:9) (r = 0 for
[S ,S ]=2S , [S ,S ]=±S . (4) 2
+ − z z ± ± evenNandr = 1 foroddN)andthelabelm∈Ziscon-
2
stricted by |m|≤j. These vectors do not form a basis of
When considering a system of N two-level atoms, we
use the collective spin operators J , J , J defined as (cid:0)C2(cid:1)⊗N for N >2, as the dimension of their linear span
z − +
is
J =S ⊗I⊗(N−1)+I ⊗S ⊗I⊗(N−2)
(cid:5) (cid:5) 2 2 (cid:5) 2 N
+···+I⊗(N−2)⊗S ⊗I +I⊗(N−1)⊗S (5) dim(cid:110)span(cid:110){|j,m(cid:105)}j=r,...,N2(cid:111)(cid:111)=(cid:88)2 (2j+1)≤2N.
2 (cid:5) 2 2 (cid:5) |m|≤j
j=r
whereI istheidentityoperatoronC2and(cid:5)∈{z,−,+}.
2
These collective spin operators satisfy the commutation We will denote by H the subspace of (cid:0)C2(cid:1)⊗N gener-
A
relations j=r,...,N
ated by the states {|j,m(cid:105)} 2.
|m|≤j
There are two main results concerning the states
[J+,J−]=2Jz, [Jz,J±]=±J± (6) {|j,m(cid:105)}j|m=|r≤,.j..,N2 and the space HA: the first comes from
noticing that (cid:2)H,J2(cid:3) = 0, which means that the la-
and act, in principle, on the complex Hilbert space
(cid:0)C2(cid:1)⊗N; however, working with this space is physically bel j of the eigenvalues of J2 remains constant during
the system’s evolution; the second is the decomposition
equivalenttostudyingasystemofNfullydistinguishable
N
atoms, which we don’t usually have in the experimental (cid:76)2
H = H ,whereeachH isthesubspaceofdimension
A j j
setups used in the study of the QPT in the Dicke model.
j=r
3
(cid:60)Jz(cid:62)j(cid:61)1 (cid:60)Jz(cid:62)j(cid:61)5 (cid:60)Jz(cid:62)j(cid:61)9
(cid:45)4
(cid:45)0.3
(cid:45)2.5
(cid:45)6
(cid:45)0.6 (cid:45)3.5
(cid:45)0.9 (cid:45)4.5 (cid:45)8
Γ u Γ u Γ u
(cid:45)5 (cid:45)2 2 5 2 e (cid:45)2 (cid:45)1 1 2 2 e (cid:45)1 1 2 e
Figure3. ExpectationvalueofJ asafunctionofγ obtainedusingCS(darkgraydashed/redonline),SASwithCS’sminima
z 2
(light gray dashed / orange online), SAS minimized numerically (dark gray / blue online) and quantum solution (light gray /
cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light gray / cyan
online), SAS minimized numerically(cid:72)(d(cid:76)ark gray / blue online) and SAS with C(cid:72)S’s(cid:76)minima (light gray dashed / orange onli(cid:72)ne)(cid:76).
Left: j=1, center: j=5, right: j=9. Assuming k =2 and using N =18, ω =Ω =Ω =2u , γ = 1u , where u stands for
A 1 2 e 1 2 e e
any energy unit ((cid:126)=1).
(cid:60)Jz(cid:62)j(cid:61)10.2 0.4 1 ΩA ue (cid:60)Jz(cid:62)j(cid:61)5 0.5 1.5 ΩA ue (cid:60)Jz(cid:62)j(cid:61)9 1.5 3 ΩA ue
(cid:45)0.2 (cid:45)1 (cid:45)2
(cid:45)0.6 (cid:45)3 (cid:45)5
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
(cid:45)8
(cid:45)1 (cid:45)5
Figure 4. Expectation value of J as a function of ω obtained using CS (dark gray dashed / red online), SAS with CS’s
z A
minima(lightgraydashed/orangeonline),SASminimizednumerically(darkgray/blueonline)andquantumsolution(light
gray / cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light gray
/ cyan online), SAS minimized numerically (dark gray / blue online) and SAS with CS’s minima (light gray dashed / orange
online). Left: j=1, center: j=5, right: j=9. Assuming k=2 and using N =18, Ω =Ω =2u , γ = 1u , γ =1u , where u
1 2 e 1 2 e 2 e e
stands for any energy unit ((cid:126)=1).
dim{H }=2j+1 generated by the states {|j,m(cid:105)} In this paper we restrict our analysis to the space H ,
j |m|≤j A
with a fixed j. In this treatment, in order to study in- as it allows us to choose j as an initial condition (which
distinguishable atoms, we are ignoring the multiplicities will remain constant) and work in Hj, where the atoms
g(j) of the irreducible representations of SU(2), i.e. the are indistinguishable.
number of timesthat each H appears inthe full decom-
j
N
position (cid:0)C2(cid:1)⊗N = (cid:76)2 g(j)Hj. Methodology
j=r
TomakeitclearthatthespaceH istheonewemust
A
Therehavebeenvariouscontributionstothestudyofthe
work with when indistinguishable atoms are considered
phase transition in the Dicke model (and other two-level
we should inquire into the physical interpretation of the
models) [14–18] and different approaches such as Husimi
labels j and m. In order to give a physical interpreta-
function analysis [19], entropic uncertainty relations [20]
tion to the label j we must notice that the energy of the
andenergysurfaceminimization[21–26], havebeenused
atomic system is bounded by ±jω independently of the
number of atoms N (but with the restriction j ≤ N), for its investigation.
2
this leads us to interpret the quantity 2j as the effec- In this work we use the energy surface minimization
tive number of atoms in the system and define it as the method,whichconsistsonminimizingthesurfacethatis
cooperation number. To make the notion of the coopera- obtainedbytakingtheexpectationvalueofthemodeling
tion number more intuitive, Dicke, in his original paper Hamiltonian with respect to some trial variational state.
[1], compares a state with j = 0, which exists only for Thestrengthofthismethodliesonthechoiceofthetrial
an even number of atoms, with a classical system of an state, as it is the latter, after minimization, the one that
even number of oscillators swinging in pairs oppositely will be modeling the ground state of the system.
phased. The interpretation of the label m is clear from Here we take a variational approach for both matter
the definition of J : m = 1(n −n ), where n and n and radiation fields, and show how to calculate the QPT
z 2 e g e g
arethenumberofatomsintheexcitedandgroundstates, of the system modeled by the Hamiltonian H given in
respectively. eq. (2) via four means:
4
(cid:60)Ν (cid:62) (cid:60)Ν (cid:62) (cid:60)Ν (cid:62)
2 j(cid:61)1 2 j(cid:61)5 2 j(cid:61)9
0.5 4
3
0.3 2 2.5
1
1
0.1
Γ u Γ u Γ u
(cid:45)4 (cid:45)2 2 4 2 e (cid:45)1.5 (cid:45)0.5 0.5 1.5 2 e (cid:45)1 (cid:45)0.5 0.5 1 2 e
Figure5. Expectationvalueofν =a†a asafunctionofγ obtainedusingCS(darkgraydashed/redonline),SASwithCS’s
2 2 2 2
minima(lightgraydashed/orangeonline),SASminimizednumerically(darkgray/blueonline)andquantumsolution(light
gray / cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light gray
/ cyan online), SAS minimized num(cid:72)eri(cid:76)cally (dark gray / blue online) and SAS(cid:72)wi(cid:76)th CS’s minima (light gray dashed / ora(cid:72)ng(cid:76)e
online). Left: j=1, center: j=5, right: j=9. Assuming k = 2 and using N = 18, ω = Ω = Ω = 2u , γ = 1u , where u
A 1 2 e 1 2 e e
stands for any energy unit ((cid:126)=1).
(cid:60)Ν (cid:62) (cid:60)Ν (cid:62) (cid:60)Ν (cid:62)
2 j(cid:61)1 2 j(cid:61)5 2 j(cid:61)9
1.4
0.05 4
0.03 0.8
2
0.01 0.2
Ω u Ω u Ω u
0.2 0.6 1 A e 0.5 1.5 A e 1 2 3 A e
Figure 6. Expectation value of ν = a†a as a function of ω obtained using CS (dark gray dashed / red online), SAS with
2 2 2 A
CS’s minima (light gray dashed / orange online), SAS minimized numerically (dark gray / blue online) and quantum solution
(light gray / cyan online). Vertical lines show the transition according to the quantum solution via fidelity’s minimum (light
gray / cyan online), SAS minimize(cid:72)d n(cid:76)umerically (dark gray / blue online) an(cid:72)d S(cid:76)AS with CS’s minima (light gray dashe(cid:72)d /(cid:76)
orange online). Left: j=1, center: j=5, right: j=9. Assuming k = 2 and using N = 18, Ω = Ω = 2u , γ = 1u , γ = 1u ,
1 2 e 1 2 e 2 e
where u stands for any energy unit ((cid:126)=1).
e
1. Using a tensor product of Heisenberg-Weyl HW(1) Coherent states (CS)
coherent states for each mode of the electromag-
neticfieldandSU(2)coherentstatesfortheatomic For each mode of the electromagnetic field the annihila-
fieldastrialstates,andanalyticallyminimizingthe tion and creation operators a and a†, appearing in the
ı ı
obtained energy surface with respect to its param- modeling Hamiltonian H, satisfy the commutation rela-
eters. tions(3)oftheLiealgebrageneratorsoftheHeisenberg-
WeylgroupHW(1);hence,anaturalchoiceofatrialstate
2. Using a projection operator on HW(1) coherent for the radiation field is a tensor product of k (number
states and SU(2) coherent states to obtain trial of modes) coherent states of HW(1)
states that preserve the parity symmetry of the
Hamiltonian with respect to the total excitation
number of the system (symmetry adapted states), |α¯(cid:105):=|α (cid:105)⊗···⊗|α (cid:105), (7)
1 k
andnumericallyminimizetheobtainedenergysur-
face with respect to its parameters. where each |α (cid:105) is defined as
ı
3. Usingsymmetryadaptedstates,asin(2)above,to
othbetaminintihmeizeinnegrgpyarsaumrfeatceersaonbdta“imneindiminiz(e1”)itabwoivteh, |αı(cid:105):=eαıa†ı−α∗ıaı|0ı(cid:105)=e−|α2ı|2 (cid:88)∞ √ανıνı!|νı(cid:105). (8)
thus allowing us to have analytic expressions for νı=0 ı
the ground state.
Furthermore, the commutation relations of the collec-
tive spin operators J , J and J (6) are the same as
− + z
4. Numerically diagonalizing the Hamiltonian, which the ones of the Lie algebra generators of the special uni-
gives us the exact quantum solution. tarygroupSU(2). Thus,analogouslyasfortheradiation
field, we use the coherent states of SU(2)
5
(cid:60)Ν(cid:62) (cid:60)Ν(cid:62) (cid:60)Ν(cid:62)
1 j(cid:61)1 1 j(cid:61)5 1 j(cid:61)9
0.014 0.35
1
0.009
0.2
0.6
0.004
0.05 0.2
Γ u Γ u Γ u
(cid:45)10 (cid:45)5 5 10 2 e (cid:45)3 (cid:45)1 1 3 2 e (cid:45)2 (cid:45)1 1 2 2 e
Figure 7. Expectation value of ν = a†a as a function of γ obtained using CS (dark gray dashed / red online), SAS with
1 1 1 2
CS’s minima (light gray dashed / orange online), SAS minimized numerically (dark gray / blue online) and quantum solution
4j2γ2
(light gray / cyan online). Horizon(cid:72)tal(cid:76)dashed line shows the asymptote of (cid:104)ν1(cid:72)(cid:105) a(cid:76)t NΩ21. Vertical lines show the transi(cid:72)tion(cid:76)
1
according to the quantum solution via fidelity’s minimum (light gray / cyan online), SAS minimized numerically (dark gray
/ blue online) and SAS with CS’s minima (light gray dashed / orange online). Left: j=1, center: j=5, right: j=9. Assuming
k=2 and using N =18, ω =Ω =Ω =2u , γ = 1u , where u stands for any energy unit ((cid:126)=1).
A 1 2 e 1 2 e e
Entropy of entanglement (S )
ε
(cid:12) (cid:29)
|ξ(cid:105) :=(cid:12)(cid:12)υtan|υ| :=eυJ+−υ∗J−|j,0(cid:105) Entropy of entanglement is defined for a bipartite sys-
j (cid:12) |υ| j temastheVonNeumannentropyofeitherofitsreduced
2j (cid:18) (cid:19)1 states, that is, if ρ is the density matrix of a system in a
1 (cid:88) 2j 2
= (cid:16) (cid:17)j m ξm|j,m−j(cid:105). (9) Hilbert space H=H1⊗H2, its entropy of entanglement
1+|ξ|2 m=0 is defined as
as trial states for the matter field.
S :=−Tr{ρ logρ }=−Tr{ρ logρ }, (11)
ε 1 1 2 2
Symmetry adapted states (SAS)
where ρ =Tr {ρ} and ρ =Tr {ρ}.
1 2 2 1
Our Hamiltonian H models a bipartite system formed
The modeling Hamiltonian we are considering has a
parity symmetry given by (cid:2)eiπΛ,H(cid:3) = 0, where Λ = by matter and radiation subsystems, which means that
their entropy of entanglement can be used to see the in-
(cid:113) (cid:88)k fluence of the QPT on its behavior; this we do below.
J2+ 1 − 1 + J + a†a is the excitation number
4 2 z ı ı
ı=1
k Fidelity between neighboring states (F)
(cid:88)
operator with eigenvalues λ=j+m+ ν . This sym-
ı
ı=1 Fidelityisameasureofthe"distance"betweentwoquan-
metry allows us to classify the eigenstates of H in terms
tum states; given |φ(cid:105) and |ϕ(cid:105) it is defined as
oftheparityoftheeigenvaluesλ;however,asstateswith
oppositesymmetryarestronglymixedbytheCSdefined
in the previous section, we should then adapt this sym- F(φ,ϕ):=|(cid:104)φ|ϕ(cid:105)|2. (12)
metry to the CS by projecting them with the operator
P± = 21(cid:0)I±eiπΛ(cid:1), i.e. Across a QPT the ground state of a system suffers a
sudden, drastic change, thus it is natural to expect a
drop in the fidelity between neighboring states near the
|α¯,ξ (cid:105) :=N P |α¯(cid:105)⊗|ξ(cid:105)
j ± ± ± j transition. Thisdrophasbeen, infact, alreadyshownto
(cid:16) (cid:17) happen [15, 25] for the case 2j = N. We study it here
=N |α¯(cid:105)⊗|ξ(cid:105) ±|−α¯(cid:105)⊗|−ξ(cid:105) , (10)
± j j also, and its behavior with the cooperation number.
(cid:16) (cid:17)−1
with N = 2±2E(−cosθ)2j 2 the normalization
±
Results
factors for the even (+) and odd (-) states (where
(cid:40) k (cid:41)
E =exp −2(cid:88)(cid:12)(cid:12)α2(cid:12)(cid:12) ). Writing the complex labels αı and ξ as αı =qı+ipι and
ı ξ =tan(cid:0)θ(cid:1)eiφ,withq ,p ∈R,θ ∈[0,π),φ∈[0,2π),the
ı=1 2 ı ı
Asweareinterestedinthegroundstateofthesystem, CS’senergysurfaceisobtainedbytakingtheexpectation
which has an even parity, we only focus on the state valueofthemodelingHamiltonianH withrespecttothe
|α¯,ξ (cid:105) . state |α¯(cid:105)⊗|ξ(cid:105) , and has the form
j + j
6
S(cid:182)j(cid:61)1 S(cid:182)j(cid:61)5 S(cid:182)j(cid:61)9
1.4
1 1.2
0.6 0.7 0.8
0.2 0.2 0.2
Γ u Γ u Γ u
(cid:45)8 (cid:45)4 4 8 2 e (cid:45)2 (cid:45)1 1 2 2 e (cid:45)1.2 (cid:45)0.6 0.6 1.2 2 e
Figure 8. Entropy of entanglement as a function of γ obtained SAS with CS’s minima (middle gray / magenta online), SAS
2
minimized numerically (dark gray / blue online) and quantum solution (light gray / cyan online). Vertical lines show the
transition according to the quantum solution via fidelity’s minimum (light gray / cyan online), SAS minimized numerically
(dark gray / blue online) and SAS(cid:72)wi(cid:76)th CS’s minima (middle gray dashed / m(cid:72) ag(cid:76)enta online). Left: j=1, center: j=5, ri(cid:72)ght(cid:76):
j=9. Assuming k=2 and using N =18, ω =Ω =Ω =2u , γ = 1u , where u stands for any energy unit ((cid:126)=1).
A 1 2 e 1 2 e e
S(cid:182)j(cid:61)1 S(cid:182)j(cid:61)5 S(cid:182)j(cid:61)9
0.35 1.2 1.4
0.8
0.2 0.8
0.4
0.05 0.2
Ω u Ω u Ω u
0.2 0.6 1 A e 0.5 1.5 A e 1 2 3 A e
Figure 9. Entropy of entanglement as a function of ω obtained SAS with CS’s minima (middle gray / magenta online), SAS
A
minimized numerically (dark gray / blue online) and quantum solution (light gray / cyan online). Vertical lines show the
transition according to the quantum solution via fidelity’s minimum (light gray / cyan online), SAS minimized numerically
(dark gray / blue online) and SAS(cid:72)wi(cid:76)th CS’s minima (middle gray dashed / m(cid:72) ag(cid:76)enta online). Left: j=1, center: j=5, ri(cid:72)ght(cid:76):
j=9. Assuming k=2 and using N =18, Ω =Ω =2u , γ = 1u , γ =1u , where u stands for any energy unit ((cid:126)=1).
1 2 e 1 2 e 2 e e
Substituting these values into (13) we obtain the en-
ergy of the coherent ground state as a function of the
H (q ,p ,θ,φ):=(cid:104)α¯|⊗(cid:104)ξ| H|α¯(cid:105)⊗|ξ(cid:105) Hamiltonian parameters,
j,CS ı ı j j
k k
=−jωAcosθ+(cid:88)Ωı(cid:0)qı2+p2ı(cid:1)−√4Nj sinθcosφ(cid:88)γıqı.  −(cid:18)jωA (cid:19) , forδ ≥1
ı=1 ı=1 (13) ECS(ωA,γı)=−jω2A 1δ +δ , forδ <1, (14)
The critical points which minimize it are then found
to be where we have defined δ = NωA with ς =(cid:88)k γı2. Us-
8jς Ω
ı
ı=1
ingtheinformationofthiscoherentgroundstatewealso
8j(cid:88)k γ2 obtaintheexpectationvaluesoftheatomicrelativepopu-
θ =q =p =0, forω ≥ ı ,
c ıc ıc A N Ωı lationoperatorJz andofthenumberofphotonsofmode
ı=1 ı operator ν :=a†a :
ı ı ı
coqsıcθc==Ω2φıNj√c8γωjNı=Ac(cid:32)0o,s(cid:88)ıπ=kφ,1cΩγsı2iın(cid:33)θ−c1, , forωA < 8Nj(cid:88)ı=k1Ωγı2ı. (cid:104)νı(cid:105)CS(ω(cid:104)JAz,(cid:105)γCı,Sγ(ω)A=,γı)Ωγ=ı22ı (cid:26)j2ω−ς−Ajjδ0(cid:18)1δ,, −ffoorδr(cid:19)δδ<≥,, 11ff,oorrδδ<≥((1111.65))
pıc =0 Analogously as for the CS’s energy surface, the SAS’s
energy surface is obtained by taking the expectation
7
F Q,Q(cid:43)∆ j(cid:61)1 F Q,Q(cid:43)∆ j(cid:61)5 F Q,Q(cid:43)∆ j(cid:61)9
1 1 1
0.9999(cid:72)998 (cid:76) 0.999(cid:72)985 (cid:76) (cid:72) (cid:76)
0.99993
0.9999996
0.999970
0.99986
(cid:45)8 (cid:45)4 4 8 Γ2 ue (cid:45)2 (cid:45)1 1 2 Γ2 ue (cid:45)1.2 (cid:45)0.6 0.6 1.2 Γ2 ue
Figure 10. Fidelity between neighboring quantum states as a function of γ . Left: j=1, center: j=5, right: j=9. Assuming
2
k = 2 and using N = 18, ω = Ω = Ω = 2u , γ = 1u , where u stands for any energy unit ((cid:126) = 1). Vertical black line
A 1 2 e 1 2 e e
shows the fidelity’s minimum (i.e. the quantum phase transition) for j =5 and j =9.
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
F Q,Q1(cid:43)∆ j(cid:61)1 F Q,Q(cid:43)∆ j(cid:61)5 F Q,Q(cid:43)∆ j(cid:61)9
1 1
0.999994 0.999995
0.9(cid:72)999 (cid:76)
(cid:72) (cid:76) (cid:72) (cid:76)
0.999988 0.99999
0.9998 0.1 0.2 0.3ΩA ue 1 2 3 ΩA ue 1 2 4 ΩA ue
Figure 11. Fidelity between neighboring quantum states as a function of ω . Left: j=1, center: j=5, right: j=9. Assuming
A
k=2 and using N =18, Ω =Ω =2u , γ = 1u , γ =1u , where u stands for any energy unit ((cid:126)=1). Vertical black line
1 2 e 1 2 e 2 e e
shows the fidelity’s minimum (i.e. the quantum phase transition) for j =5 and j =9.
(cid:72) (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
value of the modeling Hamiltonian H with respect to
the state |α¯,ξ (cid:105) , and has the more complicated form
j +
E (ω ,γ )
SAS A ı
 −jω , forδ ≥1
A
 (cid:34) (cid:32)1+ε(−δ)2j−2(cid:33)
= −jω δ (18)
Hj,SAS(qı,pı,θ,φ):=(cid:104)α¯,ξj|+H|α¯,ξj(cid:105)+  A +12(cid:18)11δ+−εδ((cid:19)−(cid:21)δ)2j , forδ <1,
(cid:32) (cid:33)
1+E(−cosθ)2j−2
= (−jω cosθ)
1+E(−cosθ)2j A (cid:104)J (cid:105) (ω ,γ )
z SAS A ı
+(cid:32)1−E(−cosθ)2j(cid:33)(cid:88)k Ω (cid:0)q2+p2(cid:1)  (cid:32) −j (cid:33) , forδ ≥1
1+E(−cosθ)2j (cid:96) (cid:96) (cid:96) = 1+ε(−δ)2j−2 (19)
(cid:96)=1 −jδ , forδ <1,
−√4j sinθ(cid:88)k (cid:40)cosφγ(cid:96)q(cid:96)+E(−cosθ)2j−1sinφγ(cid:96)p(cid:96)(cid:41).  1+ε(−δ)2j
N 1+E(−cosθ)2j
(cid:96)=1
(cid:104)ν (cid:105) (ω ,γ ,γ )
(17) ı SAS A ı 

0 , forδ ≥1
=γı2 jωA (cid:18)1 −δ(cid:19)(cid:32)1−ε(−δ)2j(cid:33) , forδ <1,
Ω2 2ς δ 1+ε(−δ)2j
ı
(20)
As a first approximation, we may substitute the criti-
cal values obtained for the CS’s energy surface into (17), where ε=exp(cid:26)−jωAσ(cid:27) with σ =(cid:88)k γı2.
we obtain the trial state which approximates the low- ς ı=1Ω2ı
est symmetry-adapted energy state, and with respect to Of course, we can minimize eq. (17) numerically for
which we evaluate the expectation values of the observ- theSASandobtaintheexpectationvalueoftherelevant
ables H, J and ν : matter and field observables.
z ı
8
In our numerical analysis we study the case with two quantumsolution: thecomputationaltime. SASnareob-
modes of the radiation field, as it is the maximum num- tained by numerically minimizing a real function, which
ber of orthogonal modes that can be present in a 3D is far easier to do (computationally speaking) than nu-
cavity with the restrictions that the modes interact with merically diagonalizing the Hamiltonian matrix.
the electric dipole moment of the atoms and to be in
resonance with the frequency associated with the energy
FSASc,Qj(cid:61)1 FSASc,Qj(cid:61)5
difference between the two levels of the atoms. This lat- 1 1
ter restriction is just considered to have the maximum
transition probability between states. (cid:72)0.95 (cid:76) (cid:72)0.91 (cid:76)
For the exact quantum solution we must resort to nu-
merical diagonalization of the Hamiltonian and use the 0.9 0.82
lowest eigenstate to compute the expectation values of (cid:45)8 (cid:45)4 4 8 Γ2ue (cid:45)2 (cid:45)1 1 2 Γ2ue
the relevant observables. FSASc,Qj(cid:61)9
1
The results, properties of the ground state related to
theCS,thoserelatedtotheSASusingthecriticalpoints (cid:72) (cid:76) (cid:72) (cid:76)
(cid:72)0.89 (cid:76)
of the CS (which have the advantage of also providing
analyticalsolutions),thoseoftheSASminimizednumer-
0.78
icallyandthoseofthequantumsolutionthroughnumer- (cid:45)1.2 (cid:45)0.6 0.6 1.2Γ2ue
ical diagonalization, are shown in figures 1 - 15 and are
discussed below. Figure12. FidelitybetweenSASwithCS’sminimaandquan-
One advantage of having analytical solutions is, of tum solution as a function of γ . Up (left): j=1, up (right):
2
course, that the order of the transition may be easily j=5, down: j=9. Assuming k = 2 an(cid:72)d(cid:76)using N = 18,
found. Equations (14) and (18) show a second-order ωA = Ω1 = Ω2 = 2ue, γ1 = 12ue, where ue stands for any
QPT at δ = NωA = 1 with the CS and SAS using energy unit ((cid:126)=1).
8jς
CS’s minima (SASc) approximations. In figure 1 it can
be seen that the data of the SAS using numerical min-
imization (SASn) has a small discontinuity (the QPT) FSASc,Qj(cid:61)1 FSASc,Qj(cid:61)5
1 1
at γ ≈ 1.485 for j = 5 and γ ≈ 1.015 for j = 9,
2 2
while in figure 2 this discontinuity is at ωA ≈ 0.975 for (cid:72)0.98 (cid:76) (cid:72) 0.9 (cid:76)
j = 5 and ω ≈ 1.965 for j = 9. Note that the SASn
A
solution always approximate better the exact quantum 0.96
result, as the cooperation number increases this approx- 0.3 0.6 0.9 ΩAue 0.8 1 2 3ΩAue
imation gets better, in fact, for 2j = 18 = N the loci of FSASc,Qj(cid:61)9
1
the separatrix between the normal and collective regions
for the quantum and SASn solutions are indistinguish- (cid:72)0.88 (cid:76) (cid:72) (cid:76) (cid:72) (cid:76)
able (except in the zoomed inset). The true loci of the
0.76
QPT may be found through the fidelity: figures 10 and 1 2 4ΩAue
11 show the fidelity between neighboring states of the
quantum solution, where the exact QPT is characterized Figure 13. Fidelity between SAS with CS’s minima and
by the minimum, which is localized at γ2 ≈ 1.550 for quantum solution as a function of ωA. Up (left): j=1, up
j =5, γ ≈1.031 for j =9 in figure 10; and ω ≈0.817 (right): j=5, down: j=9. Assuming k=2(cid:72)an(cid:76)d using N =18,
2 A
for j =5, ωA ≈1.870 for j =9 in figure 11. Ω1 =Ω2 =2ue, γ1 = 21ue, γ2 =1ue, where ue stands for any
The discrepancies between the transition values of energy unit ((cid:126)=1).
the SASc approximation and the exact quantum solu-
tion become obvious when looking at figures 12 and 13, Figures1-7showthecomparisonbetweenthedifferent
where the fidelity between SASc and the quantum so- approximationstothegroundstate: CS,SASc,SASnand
lution drops (and oscillates) in a vicinity of the sepa- quantum solution. We show the behavior of E := (cid:104)H(cid:105),
ratrix. Therefore, we conclude that SASc offer a good (cid:104)J (cid:105) and (cid:104)ν (cid:105) as functions of the atomic frequency ω
z ı A
approximation(withananalyticexpression)totheexact and one of the coupling constants γ , for different coop-
2
quantum solution far from the QPT for low cooperation erationnumbers. Itcanbenoticedthatthediscontinuity
numbers,butasj →∞,theintervalwheretheSAScfail in the second derivative of the energy (as modeled with
to reproduce the correct behavior, becomes smaller. CS and SASc) translates into a discontinuity in the first
Figures14and15showthefidelitydropatthesepara- derivative of (cid:104)J (cid:105) and (cid:104)ν (cid:105), thus characterizing the QPT
z ı
trix for the SASn. The resemblance to figures 10 and 11 by means of an abrupt change in the expectation values
is uncanny, showing the benefits of restoring the Hamil- oftheobservables. Ingeneral,itcanbeobservedthatthe
tonian symmetry into the trial variational states. This four methods (CS, SASc, SASn and quantum solution)
improvement comes with the disadvantage of losing the converge in the limit δ → 0, where the case j → ∞ is
analytic expression, but still has an advantage over the particularly interesting as the interval around the QPT,
9
wherealltheapproximationsfailtoreproducethecorrect FSASn,Q j(cid:61)1
behavior, becomes smaller. 1
0.998
Itisworthmentioningthesignificanceandimportance (cid:72) (cid:76)
offigures7and16astheyshowaspectsofthemulti-mode 0.996
Dicke model which are not present in the single-mode
case. In figure 7 it is shown how the different modes of 0.5 1 2ΩA ue
radiation (orthogonal in principle) interact through the FSASn,Q j(cid:61)5
1
matterfield,analogouslyasitoccurswithdifferentatoms
interacting through the radiation field. On the other
(cid:72) (cid:76)
hand, figure 16 shows (pictorically) the phase diagrams (cid:72)0.94 (cid:76)
ofthetwo-modesystem,inwhichitcanbeobservedthat
any two points in the super-radiant region can be joined 0.88 0.5 1 2ΩA ue
by a trajectory that does not cross the normal region, a
characteristicthatthesingle-modesystemdoesnothave. FSASn,Q j(cid:61)9
1
Figures 8 and 9 show the comparison between SASc, 0.9 (cid:72) (cid:76)
(cid:72) (cid:76)
SASn and the quantum solution for the entropy of en-
tanglement S as a function of the atomic frequency ω
ε A 0.8
and one of the coupling constants γ2, using different co- 1 2 4ΩA ue
operation numbers. A characterization of the QPT can
be made by observing that the entropy of entanglement Figure 15. Fidelity between SAS minimized numerically and
obtained using the quantum solution shows a maximum quantum solution as a function of ωA. Up: j=1, center: j=5,
at the transition, an attribute that SASc and SASn ap- down: j=9. Assuming k = 2 and using N =(cid:72) 1(cid:76)8, Ω1 = Ω2 =
2u ,γ = 1u ,γ =1u ,whereu standsforanyenergyunit
proximations fail to reproduce. e 1 2 e 2 e e
((cid:126)=1).
FSASn,Q j(cid:61)1 ΩAue
1
(cid:72)0.99 (cid:76) (cid:72) (cid:76)
normal
0.98
(cid:45)8 (cid:45)4 4 8 Γ2 ue
FSASn,Q j(cid:61)5
1
super(cid:45)radiant
(cid:72) (cid:76) Γ2ue
(cid:72)0.93 (cid:76) Γ2ue
0.86 (cid:72) (cid:76)
(cid:45)2 (cid:45)1 1 2 Γ2 ue (cid:72) (cid:76)
FSASn,Q j(cid:61)9 normal
1
Γ1ue
(cid:72) (cid:76)
(cid:72) 0.9 (cid:76)
super(cid:45)radiant
(cid:72) (cid:76)
0.8
(cid:45)1.2 (cid:45)0.6 0.6 1.2 Γ2 ue
Figure16. Pictographicrepresentationofthephasediagrams
Figure 14. Fidelity between SAS minimized numerically and intheplane(γ ,ω )(up)and(γ ,γ )(down)obtainedusing
2 A 1 2
quantum solution as a function of γ . Up: j=1, center: j=5, CS.Normalregion(white)isdefinedastheregionwhereδ≥1
2
down: j=9. Assuming k = 2 and using N = 18, ω = Ω = and super-radiant region (gray / light blue online) is defined
(cid:72) (cid:76) A 1
Ω = 2u , γ = 1u , where u stands for any energy unit as the region where δ < 1. u stands for any energy unit
2 e 1 2 e e e
((cid:126)=1). ((cid:126)=1).
10
Discussion and Conclusions observables. In general, it can be observed that the four
methods (CS, SASc, SASn and quantum solution) con-
From figures presented we conclude that SASc offer a vergeinthelimitδ →0,wherethecasej →∞ispartic-
good approximation (with an analytic expression) to the ularly interesting as the interval around the QPT, where
exact quantum solution far from the QPT for low co- all the approximations are weaker, becomes smaller.
operation numbers, but as j → ∞, the interval where Inconclusion,wehaveshownhowtheuseofvariational
theSAScfailtoreproducethecorrectbehavior,becomes states to approximate the ground state of a system can
smaller. beusefultocharacterizetheQPTinamulti-modeDicke
model using the expectation value of the observables rel-
On the other hand, the SASn provide a better ap-
evanttothesystemandtheentropyofentanglementbe-
proximationtothequantumsolution. Thisimprovement
tween matter and radiation. We have also introduced
comes with the disadvantage of losing the analytic ex-
a not very commonly used dependence: the cooperation
pression, but still has the advantage over the quantum
number, showing its influence over the behavior of the
solution of the computational time. SASn are obtained
system, paying particular attention to the QPT and the
by numerically minimizing a real function, which is far
accuracy of the used approximations. Some aspects of
easiertodo(computationallyspeaking)thannumerically
the multi-mode Dicke model which are not present in
diagonalizing the Hamiltonian matrix.
the single-mode case were also briefly discussed.
AcharacterizationoftheQPTcanbemadebylooking
at the entropy of entanglement; that obtained using the
quantum solution shows a maximum at the transition,
Acknowledgments
an attribute that SASc and SASn approximations fail to
reproduce. WethankR.López-Peña,O.CastañosandS.Corderofor
The behavior of the expectation values of the relevant their comments and discussion. This work was partially
observables of the system (cid:104)H(cid:105), (cid:104)J (cid:105) and (cid:104)ν (cid:105), is also af- supported by DGAPA-UNAM under project IN101217.
z ı
fected by the QPT (figures 1 - 7), thus allowing us to L. F. Q. thanks CONACyT-México for financial support
characterize the QPT by means of its influence over the (Grant #379975).
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