Eur. Phys. J. D (2012) 66: 201DOI: 10.1140/epjd/e2012-30197-x

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Quantum Fisher information and chaos in the Dicke model

L.J. Song1, J. Ma2, D. Yan1,3,a, and X.G. Wang2

1 Department of Physics, School of Science, Changchun University, Changchun 130022, P.R. China2 Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China3 College of Physics, Jilin University, Changchun 130012, P.R. China

Received 25 March 2012 / Received in final form 6 May 2012Published online 16 August 2012 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2012

Abstract. We used quantum Fisher information (QFI) to distinguish quantum chaotic and regular dy-namics in Dicke model. The QFI, which reflects the geometric properties of a state in parameter space,characterizes the sensitivity of the state with respect to SU(2) rotations. With numerical computation wefound that, quantum dynamics in chaotic and regular cases are well distinguished by QFI, especially inthe long-time evolution, where the QFI in the chaotic situation is evidently larger than that in regularone, i.e. the chaotic nature enhances the sensitivity of the system, as expected usually. We also plotted thetime-averaged QFI in phase space, and found a good quantum-classical correspondence.

1 Introduction

Chaos plays an important role in many fields of physics. Inclassical regime, one of the most distinct features of chaosis the extreme sensitivity of the trajectories with respectsto perturbation, i.e. the distance between two initially veryclose trajectories may deviate far from each other duringthe time evolution. However, in the quantum world, fora close system, due to the unitarity of quantum evolu-tions, the overlap (or fidelity) between two initially sepa-rated states is invariant during the evolution, thus thereis no well-accepted definition of quantum chaos. To inves-tigate quantum chaos, one of the best ways is to studyintrinsic quantum properties of the physical system [1],such as the spectral properties of the generating Hamilto-nian [2], phase space scarring [3], hypersensitivity to per-turbation [4], phase transition [5], and fidelity decay [6–9].Moreover, quantum chaos reflected by the states duringthe evolution has been identified by entanglement [10–19]and spin squeezing [18–20], and these two quantities haveclose relations [21,22]. All the above studies indicate thatthere is an underlying chaotic presence in the correspond-ing quantum dynamics.

The Fisher information determines the best estima-tion precision of a parameter in a probability distribution,such as P (x|λ), with x the random variable, and is char-acterized by a parameter λ. In this paper, we study quan-tum chaos by using quantum Fisher information (QFI),which is the extension of Fisher information in quantumregime [23,24]. Consider a density matrix ρ (λ), the QFIfor the parameter λ is give by Fλ = Tr

[ρ (λ)L2

λ

]with Lλ

the so-called symmetry logarithm derivative determined

a e-mail: ydbest@126.com

by the following equation

∂ρ (λ)∂λ

=12

[ρ (λ)Lλ + Lλρ (λ)] . (1)

In a more intuitive view, Fλ characterizes the sensitiv-ity of states with respect to the change of λ. In this pa-per, we consider Fθ, and θ is generated by an SU(2) rota-tion, thus Fθ characterizes the rotational sensitivity of astate ρ(θ), more precisely, Fθ is proportional to the met-ric of ρ(θ) in parameter space. This geometric interpreta-tion of QFI is an important motivation of this paper. Asis known that, spin squeezing parameters can also quan-tify the rotational sensitivity of a state, whereas this sen-sitivity is restricted by specific measurements: only col-lective spin operators are measured. Compared with spinsqueezing parameters, QFI gives the ultimate bound of thesensitivity mathematically. For example, the spin squeez-ing parameter ξ2R [25] of the Greenberger-Horne-Zeilinger(GHZ) state (a maximally entangled state) is divergent,that means a vanishing rotational sensitivity; however, ac-cording to the QFI, the GHZ state has such a high ro-tational sensitivity that it attains the Heisenberg limit[26], and this ultimate sensitivity is attained by measur-ing the parity operator [26]. To study quantum chaos, wechoose the Dicke model [27], also known as the N -atomJaynes-Cummings model, which describes a collection oftwo-level atoms interacting with a single-mode field. Itsclassical counterpart displays chaotic behaviors, and itsquantum chaos was studied by using linear entropy [11]and spin squeezing [19]. The former one quantifies theentanglement between atoms and field, while the latterone reflects pairwise entanglement in atoms. These stud-ies found that, quantum chaos strongly impacts the initialshort time behaviors of the entanglement and spin squeez-ing. For initial states in the chaotic region, entanglement

Page 2 of 5 Eur. Phys. J. D (2012) 66: 201

decays and spin squeezing vanishes very fast, and thesephenomena can be interpreted as an enhancement of de-coherence by quantum chaos [11]. However, for long timeevolution the linear entropy saturates to a plateau [11]and the spin squeezing vanishes [19], in spite of the ini-tial conditions. This implies that, linear entropy and spinsqueezing are not suitable for detecting chaotic dynamicsfor long-time evolutions. Nevertheless, in our paper, wefound that the long-time behaviors of the QFI can welldistinguish the chaotic and regular dynamics. With nu-merical computation, we found that the system in chaoticcase is evidently more sensitive than in regular one, whichsatisfies our usual expectation. Thus the behaviors of QFIare strongly connected with the underlying chaotic dy-namics, and a geometric description of the quantum chaosis revealed. Besides, a good quantum-classical correspon-dence is given by plotting the time-averaged QFI in thephase space.

2 Dicke model

We consider the Dicke model that describes a collectionof N two-level atoms interacting with a single-mode radi-ation field. The Hamiltonian is written as (� ≡ 1)

H = ωJz + ω0a†a+

R√2j

(J+a+ J−a†

)

+R′√

2j(J+a

† + J−a), (2)

where ω and ω0 are frequencies of free Hamiltoniansfor atoms and field, respectively, and Jz =

∑Ni=1 σiz/2,

J± =∑N

i=1(σx ± iσy)/2 are collective angular momentumoperators, with σα(α = x, y, z) the Pauli matrices. Param-eters R, R′ are coupling strengthes. The usual rotating-wave approximation is recovered by setting R′ = 0. Wefirst present the classical counterpart of the Dicke modelexpressed in coherent state (CS) representation. The CSsfor atoms and field are defined as

|μ〉 = (1 + μμ∗)−jeμJ+ |j,−j〉, (3)

|ν〉 = e−νν∗/2eνa† |0〉, (4)

where j = N/2 and the variables μ and ν can be written asfunctions of the classical variables in corresponding phasespaces, (q1, p1) for the atomic degree of freedom, and(q2, p2) for the field

μ =p1 + iq1√

4j − (p21 + q21)

, (5)

ν =1√2(p2 + iq2), (6)

where |0〉 is the bosonic field ground state, and q1, p1, q2,p2 describe the phase space of the system under consid-eration, indices 1 and 2 denote the atomic and field sub-system, respectively. The CS, also known as the minimum

Fig. 1. Poincare sections with coupling parameters R = 0.5and R′ = 0, 0.1, 0.2, 0.3 for the resonant case (ω = ω0 = 1).The energy E = 8.5 and the total angular momentum j =9/2. In the case the non-integrable parameter R′ = 0 (a), thePoincare section consists of two fixed points that are enclosedby KAM tori. From (b) to (d), with the increase of R′, chaoticregions appear and enlarge, while the regular regions shrink.

uncertainty state, or the classical state, enables us to studythe transition between classical and quantum worlds. Theclassical Hamiltonian corresponding to equation (2) canbe obtain by a standard procedure as [28]

Hcl(ν, ν∗, μ, μ∗) ≡ 〈μν|H |μν〉 , (7)

and it can be rewritten in terms of the phase-space vari-ables as

Hcl(q1, p1, q2, p2) =ω

2(p2

1 + q21 − 2j) +ω0

2(p2

2 + q22)

+

√4j−(p2

1+q21)4j

(R+p1p2+R−q1q2),

(8)

where R± = R+ ± R−, and the equations of motion canbe derived readily.

To reveal the classical chaotic behaviors, in Figure 1,we show Poincare sections of the spin degrees of freedom(q2 = 0 and p2 > 0) for different interaction strengthes R′.The rotation-wave approximation, R′ = 0, corresponds tothe integrable case, and R′ �= 0 to the non-integrable case.From Figure 1a to 1d, with the increase of R, the regu-lar regions, including fixed points and the KAM tori, tendto disappear, and are replaced by chaotic regions. We seethat (Fig. 1d) for R′ = 0.3, most areas of the Poincaresection display chaos. When we turn to study quantumchaos, we use Schrodinger equation instead of the equa-tions of motion derived from the classical Hamiltonian (8),whereas the initial states are also CSs

|ψ(0)〉 = |μ〉 ⊗ |ν〉 ≡ |μν〉. (9)

The parameters determining the initial CSs, as shownin equations (3) and (4), are chosen according to these

Eur. Phys. J. D (2012) 66: 201 Page 3 of 5

Poincare sections shown in Figure 1 in order to makequantum-classical connections.

3 Maximal mean QFI

Now, we consider the rotational sensitivity of a collectionof N two-level atoms. The state of the system is ρ(θ), andθ is generated via ρ (θ) = exp (−iJnθ) ρ exp (iJnθ), whereJn = J ·n, and n = (nx, ny, nz) is a normalized directionvector. From equation (1), the QFI of θ is derived as

F [ρ (θ) , Jn] =∑

α,β=x,y,z nαCα,βnβ = nCnT , (10)

where the matrix elements

Cα,β =∑

i�=j

(pi − pj)2

pi + pj[〈ϕi |Jα|ϕj〉 〈ϕj |Jβ |ϕi〉 + h.c.] ,

(11)where pi and |ϕi〉 are the ith eigenvalue and eigenvector forρ (θ), therefore, they are all functions of θ. The maximalQFI is equal to the largest eigenvalue of C, denoted asλmax. Next, we introduce the maximal mean QFI

Fmax =λmax

N, (12)

which is generally not equal to the maximal QFI of a sin-gle atom due to correlations, and this definition is intro-duced for convenience of the following discussion. It wasproven that [29], Fmax > 1 implies entanglement, other-wise, Fmax ≥ 1/ξ2R, where ξ2R is the spin squeezing param-eter given in reference [25],

ξ2R =N min

(ΔJ2

n⊥

)

|〈J〉|2 , (13)

where n⊥ refers to the direction perpendicular to themean spin direction n = 〈J〉/|〈J〉|. Parameter ξ2R < 1means that the sensitivity of ρ (θ) is higher than that ofthe coherent spin state, for which Fmax = ξ2R = 1, oth-erwise, it also implies multipartite entanglement [21]. Indetecting entanglement, there are merits to both Fmax andξ2R. The entanglement criterion Fmax > 1 is stronger thanξ2R < 1, while the squeezing parameter ξ2R is easier to mea-sure in experiments.

4 QFI and chaos

Figure 2 shows comparisons between the dynamics of themaximal mean QFI Fmax and the linear entropy E, forboth chaotic and regular situations. In the early time, thechaotic nature induces strong decoherence effects, revealedby the fast increase of E and the drastic decrease of Fmax,which drops almost to zero, implying that the atomic stateis close to a maximally mixed state. For regular situation,E increases with short periodic oscillation. After the earlyoscillations, irrespective of the initial conditions, the lin-ear entropy E saturates to a plateau due to the finiteness

0 20 40 60 800

0.5

1

1.5

2

t

Fm

ax

(a)

0 20 40 60 800

0.2

0.4

0.6

0.8

1

tE

(b)

chaoticregular

chaoticregular

Fig. 2. (Color online) Comparisons between the maximalmean QFI Fmax and the linear entropy E as functions of tin both chaotic and regular regions, with parameters chosen asFigure 1c: R = 0.5 and R′ = 0.2. Green (lighter) lines repre-sent chaotic evolution, the initial state parameters are p1 = 0,q1 = 0. Blue (darker) lines denote the regular case, with ini-tial state parameters p1 = −3.5, q1 = 0. In the early period,the linear entropy E increases faster in chaotic case than inregular one, while the Fmax undergoes dramatic oscillations inboth cases. In the long-time period, the linear entropy enters aplateau for both chaotic and regular case, however, Fmax in thechaotic case is evidently larger than that in the regular one.

of the Hilbert space of the spin degrees of freedom, andundergoes comparative weak oscillations, thus the chaoticand regular dynamics are nearly indistinguishable. Thebehaviors of Fmax are very different, we observe that thegeometric properties of the states are affected stronglyby the dynamics of the system. Firstly, for both chaoticand regular situations, Fmax oscillates around stable meanvalues, while the chaotic dynamics induces larger fluctu-ation. Remarkably, the mean value of Fmax is also largerfor chaotic case, and we check this for t = 1000. This re-sult gives an intuitive description of quantum chaos, forwhich the states are shaped to be more sensitive, whichcannot be reflected by using linear entropy, which onlyinvolves summarizing the eigenvalues, without concern-ing the properties of eigenstates. Moreover, we observethat Fmax happens to be larger than 1 frequently in thechaotic situation, at which time the system is entangledand is more sensitive than that of the coherent spin state.This is interesting as compared with the results of thespin squeezing parameters [19], which become larger than1 (parameter ξ2R becomes even divergent) after the veryinitial evolution, thus cannot reflect the entanglement andintrinsic sensitivity of the states. The time-averaged Fmax

Page 4 of 5 Eur. Phys. J. D (2012) 66: 201p 1 0

5

0.6

1.6

0.6

1

q1

p 1

−5 0 5−5

0

5

0.6

1

q1

0 5

0.6

0.95

(a) R’=0 (b) R’=0.1

(c) R’=0.2 (d) R’=0.3

Fig. 3. (Color online) Contourf plot of maximal mean QFIFmax in phase space, we average Fmax from t = 0 to t = 50,for R = 0.5 and R′ = 0, 0.1, 0.2, 0.3, ω = ω0 = 1, the energyE = 8.5 and j = 9/2. The initial quantum states are CSs withcorresponding (p1, q1).

in phase space is plotted in Figure 3, as compared withthe Poincare sections shown in Figure 1, we observe a verygood quantum-classical correspondence, especially whenR′ is small. The time-averaged E is also shown in Fig-ure 4, and the quantum-classical correspondence cannotbe reflected well. Comparing Figures 3 and 4, we find thestructure of the phase space is characterized better byFmax, and we believe this is partially because the linearentropy only involves summarizing the eigenvalues of thestates, whereas Fmax describes states in a different way,therefore, it reflects the intrinsic geometric structures ofthe states in parameter space, naturally, it distinguishesquantum chaos better.

5 Conclusion

We found that, the QFI distinguishes the chaotic and reg-ular dynamics of the Dicke model. It is better than thelinear entropy and spin squeezing parameters in some as-pects to indicate underlying chaos. This helps to solve thelong-standing question for searching quantum signatureof classical chaos. Remarkably, by means of the geomet-ric description of the QFI, we found that states under-going chaotic evolution have comparatively higher sen-sitivity, which satisfies our usual expectation of chaos.We also found a good correspondence between the quan-tum dynamics and the classical phase space structure,which cannot be given by using linear entropy. It is alsopromising to experimentally test our proposition to useQFI for detecting quantum chaos, such as the quantumkicked top [2], for which quantum chaos has been observedexperimentally [30].

p 1 0

5

0.74

0.86

0.78

0.86

p 1

−5 0 5−5

0

5

0.8

0.86

q1

q1

0 5

0.83

0.865

(a) R’=0 (b) R’=0.1

(c) R’=0.2 (d) R’=0.3

Fig. 4. (Color online) Contourf plot of linear entropy E(t)in phase space from t = 0 to t = 50, for R = 0.5 and R′ =0, 0.1, 0.2, 0.3, ω = ω0 = 1, the energy E = 8.5 and j = 9/2.The initial quantum states are CSs with corresponding (p1, q1).

This work is supported by NFRPC with Grant No.2012CB921602, National Natural Science Foundation of China(NSFC) with Grants Nos. 11025527, 10935010, 10947019, theKey Project of Chinese Ministry of Education with Grant No.211040, the Scientific Foundation of the Education Departmentof Jilin Province with Grant No. 2012245, and the Youth Sci-entific Foundation of Jilin Province with Grant No. 201201140.

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