Analytic approximation of the Tavis–Cummings ground state via projected states This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 Phys. Scr. 80 055401 (http://iopscience.iop.org/1402-4896/80/5/055401) Download details: IP Address: 131.170.6.51 The article was downloaded on 06/03/2013 at 21:36 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
Transcript
Analytic approximation of the Tavis–Cummings ground state via projected states
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2009 Phys. Scr. 80 055401
(http://iopscience.iop.org/1402-4896/80/5/055401)
Download details:
IP Address: 131.170.6.51
The article was downloaded on 06/03/2013 at 21:36
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Received 3 July 2009Accepted for publication 16 September 2009Published 27 October 2009Online at stacks.iop.org/PhysScr/80/055401
AbstractWe show that an excellent approximation to the exact quantum solution of the ground state ofthe Tavis–Cummings model is obtained by means of a semi-classical projected state. Thisstate has an analytical form in terms of model parameters and, in contrast to the exact quantumstate, allows for an analytical calculation of the expectation values of field and matterobservables, entanglement entropy between field and matter, squeezing parameter andpopulation probability distributions. The fidelity between this projected state and the exactquantum ground state is very close to 1, except for the region of classical phase transitions.We compare the analytical results with those of the exact solution obtained through directHamiltonian diagonalization as a function of atomic separation energy and matter–fieldcoupling.
PACS numbers: 42.50.Ct, 03.65.Fd, 64.70.Tg
(Some figures in this article are in colour only in the electronic version.)
1. Introduction
Progress in the technology of trapped-atom lasers and cavityquantum electrodynamics (QED) experiments has caught theinterest of researchers again in the basic models that describethe interaction between quantized radiation and atoms. TheDicke model (DM) [1] describes the interaction of a quantizedradiation field with a sample of N two-level atoms locatedwithin a distance smaller than the wavelength of the radiation.Dicke realized that under certain conditions a gas of radiatingmolecules shows collective behavior called superradiance.This phenomenon was observed experimentally in opticallypumped HF gas [2]. The simplest case N = 1 in the rotatingwave approximation is known as the Jaynes–Cummingsmodel (JCM) [3]. Many theoretical predictions of this lattermodel, such as the existence of collapse and revivals in Rabioscillations [4], the formation of macroscopic quantum statesor measures of entanglement associated with spin-squeezedstates, have been confirmed, and many experimental studiesof Rydberg atoms with very large principal quantum numbers
within single-mode cavities have been observed [5]. Itis well known that the JCM has served as a guide inunderstanding several quantum optics phenomena. Whilethis considers the rotating wave approximation in orderto discard the non-conserving energy terms, the model iswidely applicable. In 1991 it was proposed [6] that byusing circularly polarized light together with atomic selectionrules the non-conserving energy terms can be eliminated;additionally, the diamagnetic term can be taken into accountby properly changing the frequency of the field mode as afunction of the coupling interaction strength. Renormalizationof the field mode frequency can be understood by makinga Bogoliubov transformation on the field part of the DickeHamiltonian. This implies that, at least in some cases, theresults of the JCM can be valid beyond the rotating waveapproximation.
The JCM, which might well be called the standardmodel of quantum optics, has been studied intensively formore than 40 years. An experimental example is provided
by Aoki et al [7]. When there are many atoms interactingwith a single-mode quantized radiation field of one and thesame cavity, the exact solution can be obtained by meansof the so-called Tavis–Cummings model [8] (TCM) and itsrecent generalizations [9]. This model predicts the collectiveN -atom interaction strength to be γN =
√N gi , where gi
is the dipole coupling strength of each individual atom i .The TCM has also been considered to describe cavityQED with a Bose–Einstein condensate [10]. Exact solutionsof the generalized TCM have been found by means ofquantum inverse methods, an algebraic procedure based onfinding sets of solutions to the Bethe equations [11]. Otheralgebraic methods solve the eigenvalue problem of the TCMusing polynomially deformed algebras [12] where analyticalexpressions may be found up to third order in a specificperturbation theory.
Since the presence of superradiant phase transitionsin the DM was established [13], there have been severalcontributions simplifying the original computation and othersthat have also found the presence of phase transitionsin generalized DMs [14–16]. In particular, one of thecontributions showed the existence of phase transitionsin a Dicke Hamiltonian that includes the counter-rotatingatom–field interaction, although the necessary conditionbetween the coupling parameter and the field mode frequencywas modified [16].
In 1975 it was shown that the superradiant phasetransition was due to the absence of the diamagneticterm in the Hamiltonian describing the N two-level atomsinteracting with a one-mode electromagnetic field [17]. Itestablished, through the Thomas–Reiche–Kuhn sum rule,that the phase transition cannot be reached because itwould place contradictory bounds on the parameters ofthe model. Other authors [18–20] have established thatgauge invariance requires the presence of diamagnetic andcounter-rotating terms; moreover, the sum rule can bederived from charge-current conservation. For this reason, acriterion was sought to establish the validity of DMs anddetermine whether a superradiant phase transition can occur.It was shown that two-level atoms that make electric dipoletransitions cannot exhibit superradiant phase transitions,although their result does not rule out phase transitions basedon magnetic dipole interactions. In the last decade, the workof Crisp was extended to many two-level atoms, and itwas concluded that in the strong coupling regime, in thethermodynamical limit, it is not possible to have superradiantphase transitions [21].
It is however known that, for a one-electron atominteracting with a non-quantized electromagnetic field (that is,in the semiclassical approximation), a unitary transformationof the form U (Er , t)= exp( i e
h c Er · EA(0, t)) can be performedto eliminate terms depending on the electromagnetic vectorpotential, giving rise to an electric dipole–field interaction.The same procedure can be applied to the quantizedelectromagnetic field, the difference being that we aredealing with operators and there is an extra term, dueto the commutation relations, that represents dipole–dipoleinteraction [22]. In 2001, similar arguments on the interactionbetween radiation and molecular dipole moments weregiven [23], where a gauge-invariant formulation of molecular
electric dipole–photon interactions was used, which allowsfor superradiant phase transitions as in the Hepp–Liebformulation. It was shown that the dipole–field interaction isstrictly linear in the electric field, with an extra self-interactionterm. Thus, there are no quadratic terms, and therefore aphysical system displaying superradiant phase transitionsshould exist. The difference with previous gauge-invariantformulations is due to the time dependence of the unitarytransformation.
The considerations above render likely the physicalreality of systems presenting superradiant phase transitions,associated with a Hamiltonian of the type used to describelinear dipole–photon interactions. In particular, there arephysical systems where interactions between bosonic degreesof freedom with a collective spin [24, 25] are relevant. Then,the recently presented method [26], which allows a simpleand elegant determination of the stability properties and phasetransitions for a finite number of particles, is of importance.
In this work we obtain an excellent approximation to thesuperradiant states of the ground state of the Tavis–CummingsHamiltonian, which admits analytical expressions for bothfield and matter observables, including the entanglemententropy between field and matter, the squeezing parameterand the population probability distribution. The fitnessbetween this approximation and the exact quantumsolution is measured through the evaluation of the fidelityparameter [27].
Section 2 establishes the Tavis–Cummings Hamiltonianin terms of the constant of motion, and shows that thevariational tensorial product of coherent states has difficultiesin describing, for example, photon number fluctuations. Itis argued that restoring the symmetry by projecting thevariational state to a given value of the constant of motion,a much better (and analytical) approximation is obtained.This projected state is justified and built. In section 3, bymeans of the overlap of the projected state, the calculationof expectation values for field and matter observables isdone analytically for an arbitrary value of the constant ofmotion. The entanglement entropy between matter and field,the squeezing parameter of the state and the probabilitydistributions of photons and atoms are also calculated. Insection 4 the selection of the appropriate value of the constantof motion λ is discussed. Section 5 compares the results ofsection 3 with the corresponding ones for the exact quantumstate, and section 6 draws some conclusions.
2. Projected state
The TCM Hamiltonian for N identical two-level systems (e.g.atoms) immersed in an electromagnetic field is given by
HTCM = ωF a†a + ωA Jz +γN√
N
(a† J− + a J+
), (1)
where ωF is the field frequency, ωA the atomic energy-leveldifference and γN the dipole coupling. The operatorsa, a† denote the one-mode annihilation and creation photonoperators, Jz the atomic relative population operator and J±
the atomic transition operators.It is immediate that this Hamiltonian commutes with the
operator3= a†a + Jz . (2)
2
Phys. Scr. 80 (2009) 055401 O Castaños et al
γ γ
Figure 1. Squared fluctuation of photon number operator n, for the ground state, as a function of interaction strength γ , for a detuningparameter 1= 0.2 and N = 20 atoms. On the left, the continuous line shows the coherent state approximation to the exact solution (discreteline). The differences are large even when normalized by the number of atoms. On the right, the lower curve shows the approximation wecan obtain using a projected state, as explained in this paper. Note the different scales for the two plots.
It is then convenient to rewrite it by introducing a detuningparameter 1= ωF − ωA, and by dividing it by ωF (whichcan be thought of as the natural unit of frequency) and bythe total number of particles, having in this way an intensiveHamiltonian operator
H =1
N3−
1
NJz +
γ√
N N
(a† J− + a J+
), (3)
where 1= 1 −ωAωF
≡ 1 −ωA and γ =γN
ωF.
Following the variational procedure indicated in [26],it was found that the direct product of coherent states,Heisenberg–Weyl for the photon part |α〉 [5, 28] and SU (2)or spin for matter |ζ 〉 [29], i.e. |α, ζ 〉 = |α〉 ⊗ |ζ 〉, is agood approximation to the exact quantum solution of theTavis–Cummings Hamiltonian, in spite of the fact that theoverlap of the two states is small. This approximationdescribes very well the expectation values of the atomicobservables and most of the field observables. However,particular difficulties are found in the description of photonnumber fluctuations (1n)2, as shown in the left of figure 1,and for several observables when the detuning parameter 1takes values greater than 1.
The first of these problems arises because the variationalstate has contributions from all the eigenvalues λ of theoperator 3, violating its conservation symmetry. It is thennatural to propose as ground state, a state with the symmetryrestored by projecting the variational state to a given valueof λ, namely, the one that minimizes the classical energyobtained from the variational procedure and approximated tothe closest integer or half-integer according to whether N iseven or odd, respectively. The exact procedure is presented insection 4.
By projecting the state to one (as yet arbitrary) value of λ,we obtain
|ψ〉 =
|0〉 ⊗ | j, − j〉, ωA > γ2;
λ+ j∑ν=max[0, λ− j]
(2 j
j + λ− ν
)1/2
|ωA|6 γ 2;
×e−2 i ν φ ζ ν√ν!
|ν〉 ⊗ | j, λ− ν〉,
|0〉 ⊗ | j, j〉, ωA <−γ 2.
(4)
In this expression we have used j =N2 , and for |ωA|6 γ 2 the
state is not normalized, with
ζ = −
√N γ
2
(1 +
ωA
γ 2
). (5)
It has the advantage that it only contains one eigenvalue ofthe constant of motion, it is an analytic solution, it reproduceswell the matter and field observables (cf figure 1 (right) andnote the scale difference with figure 1 (left)), and the overlapwith the exact quantum solution is very close to one (seebelow).
Conditions on the parameters ωA and γ in equation (4)are associated with the existence of a separatrix in themodel, which defines the phase transitions in the semiclassicalsolution [26].
To build the projected state equation (4), we use thesemiclassical procedure, which consists of calculation ofthe expectation value of the Hamiltonian with respect tothe tensorial product of coherent states |α〉 ⊗ |χ〉,determination of the minima and use of the catastropheformalism [30] to find the stability properties [26]. We findthat the critical points for a minimum are given by
qc = −√
j γ sin θc cosφc,
pc =√
j γ sin θc sinφc,
where we have written α =1
√2(q + ip) in terms of the expe-
ctation values of the quadratures of the field, and χ =
eiφ tan ( θ2 ), with (θ, φ) denoting a point in the unit Blochsphere; at their critical values θc, φc, we have [26]
minima :
θc = 0, E0 = −N ωA
2, λc = − j, for ωA > γ
2,
θc = π, E0 =N ωA
2, λc = j, for ωA <−γ 2,
θc = arccos
(ωA
γ 2
), E0 = −
N (ω2A + γ 4)
4 γ 2, λc = j
−ωA (ωA + 2)+ γ 4
2 γ 2,
for |ωA|< γ 2.
(6)
This expression shows the minima critical points, the energyE0, the constant of motion λc and the conditions in parameter
3
Phys. Scr. 80 (2009) 055401 O Castaños et al
space to guarantee that they constitute an energy minimum.The energy surface of the TCM is φ-unstable, for whichreason φc can be taken arbitrarily.
The expression for the trial state |α〉 ⊗ |χ〉 that minimizesthe energy surface takes the following form [26]:
North Pole (ωA > γ2) : |ψnp〉 = |0〉 ⊗ | j, − j〉, (7)
South Pole (ωA <−γ 2) : |ψsp〉 = |0〉 ⊗ | j, j〉, (8)
Parallels (|ωA|< γ 2) : |ψpar〉 =
+ j∑m=− j
+∞∑ν=0
Am, ν |ν〉 ⊗ | j, m〉,
(9)
where we have defined the expansion coefficients
Am, ν =
(2 j
j + m
)1/2
exp
{−
j γ 2
4
(1 −
ω2A
γ 4
)+i ( j + m − ν) φ
}×
(−
√2 j γ
)ν√ν!
(1
2+ωA
2 γ 2
)( j−m+ν)/2(1
2−ωA
2 γ 2
)( j+m+ν)/2
.
One would usually perform this sum by selecting a valueof ν, making the sum over m and then proceeding withthe following value of ν, etc until we reach some type ofconvergence. In the TCM model λ= m + ν is a conservedquantity. By replacing m by λ− ν, we can write [26]
|ψpar〉 =
+∞∑λ=− j
λ+ j∑ν=max(0,λ− j)
Aλ−ν, ν |ν〉 ⊗ | j, λ− ν〉. (10)
The eigenstates of the North Pole (7), South Pole (8) andParallel region (10), for a given λ, justify the projected stateestablished in (4) with the unnormalized state for |ωA|6 γ 2
defined by
|ζ ; j, λ} ≡ |ζ } =
λ+ j∑ν=max[0, λ− j]
(2 j
j + λ− ν
)1/2
× e−2 i ν φ ζ ν√ν!
|ν〉 ⊗ | j, λ− ν〉, (11)
where we have retained only terms depending on the numberof photons ν from the coefficient Aλ−ν, ν .
3. Expectation values of field and matter observables
The effect of the transformation generated by the constantof motion exp[iφ3] in the plane of quadratures of theelectromagnetic field is a counterclockwise rotation by anangle φ, while for the matter observables Jx and Jy it isa clockwise rotation by an angle φ along the z-axis. Thistransformation then leaves invariant each term in the TCMHamiltonian. This gives our trial states their φ-invariance.Calculating for any value of φ allows one to recover the resultfor any other value of φ through the rotation above. Withoutloss of generality, then, we choose φ = 0 in what follows.
To determine the expectation values of the observables ofthe system, in the Parallels region, it is useful to calculate theoverlap
{ζ ′|ζ } =
λ+ j∑ν=max[0, λ− j]
(2 j
j + λ− ν
) (ζ ′ ζ
)νν!
. (12)
Considering this expression as a truncated expansion of thehypergeometric confluent function, we obtain
{ζ ′
|ζ}
=
L j−λ
j+λ (−ζ ζ′), |λ|6 j,
(2 j)!
( j + λ)!
(ζ ζ ′
)λ− jLλ− j
2 j (−ζ ζ′), λ> j,
(13)where Lαn (x) denotes the associated Laguerre polynomials,which are defined as [31]
Lαn (x)=x−α ex
n!
dn
dxn
(e−x xn+α
), n > 0,(
xd
dx
)Lαn (x)= n Lαn (x)− (n +α) Lαn−1(x), n > 1.
From expression (4) for the projected state, it isimmediate that the probability of finding ν photons dependson the relative values of ωA and γ , i.e.,
Pν =1{ζ |ζ
} ζ 2ν
ν!
(2 j
j + λ− ν
), (14)
for |ωA|6 γ 2, while Pν = δν, 0 outside that region. Theprobability of finding ne excited atoms can be obtained fromthe previous expression by replacing ν → λ+ j − ne.
Defining η ≡ ζ 2, and the overlap of the projected stateby Y ≡ {ζ |ζ }, we can calculate the expectation value ofthe photon number operator n = a†a and its correspondingfluctuations squared (1n)2:
〈n〉 =
λ+ j∑ν=max[0, λ− j]
ν Pν = ηd
dηln Y, (15)
(1n)2 =
(η
d
dη
)2
ln Y. (16)
Note that for λ= − j the overlap is Y = 1. On the other hand,for λ= j the overlap is Y = L0
2 j (−ζ2), which only equals 1
when ωA = −γ 2. At the North and South Poles, therefore,the expectation values for n and (1n)2 can be obtained byassuming Y = 1 there and the expressions (15) and (16) tobe valid for all values of the parameters γ and ωA. Analyticexpressions for 〈n〉 and (1n)2 in the Parallels region are givenin the appendix.
The expectation values of other matter and fieldobservables can be determined in terms of the equationsabove.
The quadrature components of the electromagnetic fieldare given by
q =1
√2
(a† + a
)p =
1√
2 i
(a†
− a);
4
Phys. Scr. 80 (2009) 055401 O Castaños et al
Figure 2. Energy surface of the ground state used to calculate the value λ of the constant of motion, for N = 20 atoms. The zoomon the right shows phase transitions when 1 is greater than 1 (see text for details).
as they change the value of λ for any state, their expectationvalues with respect to the projected state are equal to zero.Their corresponding fluctuations are therefore
〈q2〉 = 〈 p2
〉 = 〈n〉 + 12 . (17)
For the expectation values of matter observables, we have〈Jx 〉 = 〈Jy〉 = 0 for the same reason as above, and others takethe form
〈Jz〉 = λ− 〈n〉, (18)
(1Jz)2= (1n)2, (19)
〈J 2x 〉 = 〈J 2
y 〉 =12 j ( j + 1)− 1
2
(λ− 〈n〉
)2. (20)
Finally, the expectation values for the transition operatorsa† J− and a J+ are given by
〈a† J−〉 = 〈a J+〉 = ζ(
j + λ− 〈n〉). (21)
4. Determination of the constant of motion
The expectation value of the Hamiltonian with respect to theprojected state for an arbitrary value of λ is
〈H〉 ={ζ |H |ζ }
{ζ |ζ }=
λ
2 j(1 −1)+
2 γ
(2 j)3/2( j + λ) ζ
+1
2 j
[1−
2 γ√
2 jζ
]〈n〉. (22)
When λ= − j we have 〈n〉 = 0, and this expression simplifiesto 〈H〉 = −
12 (1 −1). By substituting the expression of the
expectation value of n given in equation (31) of the appendix,we obtain the energy surface
H(ζ )=λ+ j 1
2 j−
[1−
2 γ√
2 jζ
]
×
L j−λj+λ−1(−ζ
2)/
L j−λj+λ (−ζ
2), − j + 16 λ6 j,λ+ j2 j Lλ− j
2 j−1(−ζ2)/
Lλ− j2 j (−ζ
2), λ> j
(23)
and H(ζ )= 〈H〉.
We calculate this expression as a function of 1 and γ ,and for all possible values of λ with 1 ∈ [−2, 4] and γ ∈
[−5, 5], which allows us to choose the value of λ for which theenergy is minimum and onto which the coherent state is to beprojected. By noting that the classical value λc of the constantof motion, given in equation (6), when rounded to the nearestinteger or half-integer, never differs by more than one unitfrom its exact quantum counterpart, the minimizing proceduresimplifies to testing for |λ− λc|6 1, and approximating λ tothe nearest integer (if N is even) or half-integer (if N is odd).The minimum energy so obtained is plotted in figure 2, whilethe resulting constant of motion and its contour levels areshown in figure 3.
While similar, the behavior for 1> 1 appears to bedifferent than that for 16 1. The energy surface shows afictitious phase transition near γ ≈ 0.7 for 1> 1 (cf the rightof figure 2) that does not exist for 16 1. In fact, the regime1> 1 appears only when ωA < 0. One may visualize physicalsituations where this may occur: if the atoms are immersedin an external magnetic field and the energy levels appearas a Zeeman splitting of spectral lines, one may think ofcontinuously tuning the field intensity until a sign reversalis obtained, thus interchanging the excited and base levels.It will be seen, however, that the exact quantum solutiondoes not show this transition (cf figure 6). The case 1= 1(or equivalently ωA = 0) is, nevertheless, a special andboundary case: the Hamiltonian in equation (1) simplifiesconsiderably, the atomic levels are degenerate, the quantumenergy spectrum (see section 5) is also highly degenerate andthe poles (both North and South) contract to a single point. Infact (cf equation (6)), the North Pole is no longer a minimumcritical region, but a saddle point.
From the expressions derived for the observables insection 3, their expectation values can also be plotted asfunctions of interaction strength γ and detuning parameter1.It will be seen that their behavior will be inherited from thatof λmin itself. As an example, we show 〈Jz〉 in figure 4.
There is no field squeezing since, from the quadraturecomponents, 〈q〉 = 〈 p〉 = 0 and 〈q2
〉 = 〈n〉 + 12 = 〈 p2
〉.However, for the matter squeezing coefficient we have [32]
ξ =
√2(1J⊥)2
j, (24)
5
Phys. Scr. 80 (2009) 055401 O Castaños et al
Figure 3. Constant of motion λmin determined by minimizing the expectation value of the Hamiltonian with respect to the projected state,as a function of interaction strength γ and detuning parameter 1, for N = 20 atoms. Contour levels are shown on the right.
Figure 4. Expectation value of the population inversion operator Jz ,for the variational ground state, as a function of interactionstrength γ and detuning parameter 1, for N = 20 atoms.
where J⊥ is a component of EJ transverse to 〈 EJ 〉; as 〈EJ 〉 =
〈Jz〉ez , we take Jx to be this component, and show ξ in figure 5(left). ξ gives us information on how good the trial projectedstate is at approximating the exact solution of the eigenvalueproblem of the TCM Hamiltonian, and this comparison willbe made in section 5.
Finally, the entanglement entropy SE is zero for coherentstates since these are expressed as product states. In [26], inorder to calculate a useful expression for SE, we first tracedover the field and then again over one of the matter modes.When a projected state is used, it is no longer a product stateand we may trace over the matter (or equivalently over thefield) to obtain
SE = −
∑ν
Pν logPν, (25)
where Pν is already normalized, as given in equation (14).This is shown in figure 5 (right). Its behavior is very close tothat found in [26].
5. Projected coherent state versus exact solution
The exact solution for the ground state in the TCM waspresented in [26], where the natural basis |ν〉 ⊗ | j, λ− ν〉 wasused (natural because the Hamiltonian has 3 as a constant ofmotion):
|ψgs〉 =
λ+ j∑ν=max[0,λ− j]
cν |ν〉 ⊗ | j, λ− ν〉. (26)
Analytical solutions for values of λ up to λ= −N2 + 4
were also given, both at resonance (1= 0) [33] and awayfrom resonance (1 6= 0) [26]. For greater values, numericalsolutions were obtained.
When the exact solution is compared with that obtainedfrom the projected state, we find that the quantum phasetransitions are exactly reproduced, as shown in figure 6,even for a large number of atoms (N = 20) and when awayfrom resonance. The straight lines in the lower left figurecorrespond to the energy of the ground state for differentvalues of the constant of motion λ, starting from −10(0 photons) and up to −6 (4 photons). In each case theeigenstate is a mixture of 0 to λ+ N
2 photons. The projectedstate approximation is stunningly close; so much so, that bothgraphs (quantum and projected state) practically lie on top ofeach other and are indistinguishable. Whereas the projectedstate has a constant value of the ground state energy insidethe poles (North (|0〉 ⊗ | j, − j〉) and South (|0〉 ⊗ | j, j〉)), theexact quantum solution does not when 1> 1 (ωA < 0); thesolution therefore is very precise for a detuning of 1= 0.2,as shown on the left side of the figure, but fails slightly nearthe boundary of the poles for 1= 1.5, as shown on the lowerright.
6
Phys. Scr. 80 (2009) 055401 O Castaños et al
Figure 5. Squeezing parameter ξ for matter (left), and entanglement entropy SE between field and matter (right), for the variationalground state, as a function of interaction strength γ and detuning parameter 1, for N = 20 atoms.
4 2 2 4γ
6
5
4
3
2
1
E N
4 2 2 4γ
6
5
4
3
2
1
E N
0.86 0.88 0.90 0.92 0.94 0.96 0.98γ
0.415
0.410
0.405
0.400
E N
2 1 0 1 2γ
0.9
0.8
0.7
0.6
0.5
0.4
0.3
E N
Figure 6. Ground state energy per particle of the projected state solution compared with the exact quantum solution, as a function ofinteraction strength γ for N = 20 atoms. The plots on the left correspond to a detuning parameter 1= 0.2, while those on the rightcorrespond to 1= 1.5 (and therefore ωA < 0). The bottom plots are close-ups of those on top. Note that both solutions lie on top of eachother and are indistinguishable, except for the close-up for 1= 1.5.
The case 1= 1 is a particular boundary, as mentionedin the previous section, since it simplifies the Hamiltonian (3)making it independent of the inversion population operator Jz .In this case, if γ = 0 also, we have a dense degeneracy of theHamiltonian eigenvalues: matter does not see the radiationfield, so the degeneracy is that of the atoms themselves, i.e.2 j + 1. Any small deviation of γ away from zero unfolds theenergy levels, as shown in figure 7. In this case our methodfor choosing the λ-value onto which to project fails, as 2 j + 1values yield the same energy. We shall, in what follows,choose values for 1 above and below 1 to exemplify thebehavior of observables.
The constant of motion λmin is shown in figure 8 as afunction of interaction parameter γ for both the semiclassicalprojected and the quantum cases. Its discrete behavior isinherited by all the quantum observables of interest. The slopeobserved in the λ-steps is given precisely by the differencein energy between the absorbed photons and the atom’senergy level separation. This is a consequence of the systembeing away from resonance (1 6= 0). When in resonance,the steps are horizontal. The plots at the left show λ for adetuning of 1= 0.2, while those at the right show it for1= 1.5. Note (lower right plot) that even when ωA < 0, theprojected state solution tries to reproduce the non-constant
7
Phys. Scr. 80 (2009) 055401 O Castaños et al
Figure 7. Energy spectrum for 1= 1. When γ = 0 we have a (2 j + 1) fold degeneracy (the left plot shows the first two such levels), whichunfolds as soon as γ is perturbed away from zero. Spectra for γ = 0, 0.1, 0.2 and 0.3 are shown.
4 2 2 4γ
20
40
60
80
100
120
λ
4 2 2 4γ
20
40
60
80
100
120
λ
2 1 1 2γ
10
5
5
10
15
λ
2 1 0 1 2γ
10
15
20
λ
Figure 8. Constant of motion λmin of the projected state solution compared with the exact quantum solution, as a function of interactionstrength γ for N = 20 atoms. Note, once again, that both graphs practically lie on top of each other and are indistinguishable. The plots onthe left correspond to a detuning parameter 1= 0.2, while those on the right correspond to 1= 1.5 (and therefore ωA < 0). The bottomplots are close-ups of those on top.
4 2 2 4γ
0.5
0.4
0.3
0.2
0.1
⟨J ⟩z N
4 2 2 4γ
0.002
0.004
0.006
0.008
0.010
0.012
Jz2 N2
Figure 9. Expectation value for Jz and its dispersion (1Jz)2 of the projected state solution compared with the exact quantum solution, as
functions of interaction strength γ for N = 20 atoms. Both graphs practically lie on top of each other (cf the scale for (1Jz)2). The plots
correspond to a detuning parameter 1= 0.2.
behavior near the pole boundary. Otherwise the graphs areindistinguishable.
The photon number fluctuations (1n)2, as noted in theintroduction, very well resemble the projected state solution
to the TCM (cf figure 1 (right)). The differences are minute.The same trend is found for the dispersion in Jz , as is to beexpected, since 〈Jz〉 = λ− 〈n〉, 〈J 2
z 〉 = 〈n2〉 − 2λ〈n〉 + λ2 and
therefore (1Jz)2= (1n)2. Both the expectation value for Jz
8
Phys. Scr. 80 (2009) 055401 O Castaños et al
4 2 2 4γ
1.5
2.0
2.5
3.0
ξ
4 2 2 4γ
1.5
2.0
2.5
3.0
ξ
Figure 10. Squeezing parameter ξ of the projected state solution compared with the exact quantum solution, as functions of interactionstrength γ for N = 20 atoms. The plots correspond to detuning parameters 1= 0.2 (left) and 1= 1.5 (right). The round shape of the poleis badly approximated at its boundary, when 1> 1, while the approximation is exact for 1< 1.
4 2 2 4γ
0.5
1.0
1.5
2.0
SE
4 2 2 4γ
0.5
1.0
1.5
2.0
SE
Figure 11. Entanglement entropy SE of the projected state solution compared with the exact quantum solution, as functions of interactionstrength γ for N = 20 atoms. The plots correspond to detuning parameters 1= 0.2 (left) and 1= 1.5 (right). The round shape of the poleis badly approximated at its boundary, when 1> 1, while the approximation is excellent for 1< 1.
and its dispersion are plotted in figure 9 as functions of inter-action strength γ , slightly away from resonance (1= 0.2).Note the scale on the ordinate axis for (1Jz)
2.As a signature of the goodness of our trial projected
state in reproducing the exact quantum solution, one may usethe behavior of the squeezing spin coefficient ξ as given byequation (24). Even though it greatly exceeds the value of 1away from the poles (cf figure 10), thus suggesting a smalloverlap with the quantum state, both results lie exactly ontop of each other. Once again, for a large detuning makingωA < 0, the round shape of the pole is badly approximatedat its boundary. For 1< 1, however, the approximation isexact.
By taking the trace with respect to the field (matter)states, the reduced density matrix takes the form
%matter=
min{λ+ N2 , N }∑
n=0
∣∣cλ+ N2 −n
∣∣2|N − n, n〉〈N − n, n|, (27)
%field=
λ+ j∑ν=max{0,λ− j}
|cν |2|ν〉〈ν|, (28)
where cλ+ N2 −n (or cν) is determined from the Hamiltonian
diagonalization. As the reduced density matrix of matter isdiagonal, the matter–field entanglement entropy equals that
1
n 2 N2
0 1 2 3
0.0010.0020.0030.0040.0050.0060.007
Figure 12. (1n)2 as a function of 1 for the classical projected state(light curve), compared with the exact quantum solution (darkcurve), for N = 20 atoms and γ = 0.75.
between atoms occupying the hyperfine levels
SE = −
min{λ+ N2 , N }∑
n=0
∣∣cλ+ N2 −n
∣∣2 ln∣∣cλ+ N
2 −n
∣∣2 (29)
in both cases. This may be compared with that for theprojected state, equation (25), to give the result shown in
9
Phys. Scr. 80 (2009) 055401 O Castaños et al
4 2 2 4γ
0.994
0.996
0.998
1.000F
0.6 0.8 1.0 1.2 1.4 1.6 1.80.5
0.6
0.7
0.8
0.9
1.0
F
Figure 13. Fidelity F between projected state and exact quantum ground states, as a function of the interaction strength γ (left) anddetuning parameter 1 (right), for N = 20 atoms. The graph on the left corresponds to a detuning parameter 1= 0.2, and shows F = 1inside the North Pole dropping to F = 0.996 when crossing the separatrix into the Parallels region. The graph on the right corresponds toan interaction strength γ = 0.75, showing F ≈ 0.6 at the South Pole.
figure 11. The same behavior of excellent approximation for1< 1, and poor approximation at the south pole boundarywhen 1> 1, is obtained.
6. Discussion and conclusions
Our approximation is not exact. Several important observablessuch as ξ, SE, (1q)2, 〈n〉 and (1n)2 are symmetric undera reflection of about 1= 1 for the projected state, whilethis cannot be true for any observable coming from theHamiltonian (3). As an illustration of this we show (1n)2/N 2
as a function of 1, for both the projected state (light curve)and the quantum state (dark curve), in figure 12, for γ = 0.75.While the projected state has a constant value of 0 at bothpoles, the quantum state decays to 0 asymptotically as it entersthe South Pole.
A good measure of the distance between quantummechanical states is given by the fidelity parameter; for purequantum states it measures their distinguishability in the senseof statistical distance [27], but it is customary to use fidelityas a transition probability regardless of whether the states arepure or not. Measuring the fidelity between our projected stateand the exact quantum ground states,
F = |〈ψproj|ψgs〉|2, (30)
gives a result very close to 1, except in the region of classicalphase transitions. Figure 13 (left) shows the result as afunction of γ for a detuning parameter of 1= 0.2. F = 1inside the North Pole and drops to F = 0.996 when crossingthe separatrix into the Parallels region, only to approach itsvalue of 1 again as |γ | continues to increase. We have seenthat the South Pole is less well represented by the projectedstate for 1> 1; the behavior of F as a function of 1 nearthe South Pole is shown in figure 13 (right) for γ = 0.75, forwhich value the South Pole lies at 1> 1.56. It is seen thateven in this case F drops to approximately 0.6 and quicklyrecovers.
It has been emphasized that the DM presents quantumphase transitions when ωFγ =
√ωA/2. Although difficult to
satisfy for optical systems, some proposals to overcome the
experimental problems have been reported (see [34] andreferences therein).
More recently [35], the Tavis–Cummings Hamiltonianhas been physically realized for artificial atoms in the formof superconducting q-bits at fixed positions and coupled toa resonant cavity mode. The predicted
√N behavior for
the collective N -atom interaction strength of the model isobserved experimentally in good agreement. It is argued thatthe presented approach may enable novel investigations ofsuperradiant and subradiant states of articial atoms.
In this work, we have shown that an excellentapproximation to the exact quantum solution of the groundstate of the TCM is obtained by means of a semi-classicalprojected state. This state has an analytical form in termsof the model parameters and allows for the analyticalcalculation of the expectation values of field and matterobservables, entanglement entropy between field and matter,and squeezing parameter. In our discussions we have takenthe picture of two-level atoms interacting with a single-modeelectromagnetic field, but by re-interpreting the parametersthe results are useful to describe, for example, the superradiantproperties of condensed matter phases, or linear ion trapsproposed for quantum computation.
Acknowledgment
This work was partially supported by CONACyT-México andDGAPA-UNAM.
Appendix A. Expectation values of field operators
The expectation value of the photon number operator can befound analytically using overlap (13), with ζ ′
= ζ , to obtain
〈n〉 =( j + λ)−2 j L j−λ
j+λ−1(−η)/
L j−λj+λ (−η), − j + 16 λ6 j,
(λ+ j)− (λ+ j) Lλ− j2 j−1(−η)
/Lλ− j
2 j (−η), λ> j.
(31)
10
Phys. Scr. 80 (2009) 055401 O Castaños et al
In the same way we find that the squared fluctuations are(1n)2
=
2 j(
L j−λj+λ−1(−x)
/L j−λ
j+λ (−x)+ (2 j − 1)
×L j−λj+λ−2(−x)
/L j−λ
j+λ (−x)
−2 j[
L j−λj+λ−1(−x)
/L j−λ
j+λ (−x)]2 )
, − j + 16 λ6 j,
(λ+ j)(
Lλ− j2 j−1(−x)
/Lλ− j
2 j (−x)+ (λ+ j − 1)
×Lλ− j2 j−2(−x)
/Lλ− j
2 j (−x)
−(λ+ j)[
Lλ− j2 j−1(−x)
/Lλ− j
2 j (−x)]2 )
, λ> j.
(32)
References
[1] Dicke R H 1954 Phys. Rev. 93 99[2] Scrinabowitz N, Herman I P, MacGillivray J C and Feld M S
1973 Phys. Rev. Lett. 30 309[3] Jaynes E T and Cummings F W 1963 Proc. IEEE 51 89[4] Eberly J H, Narozhny N B and Sanchez-Mondragon J J 1980
Phys. Rev. Lett. 44 1323[5] Dodonov V V and Man’ko V I (ed) 2003 Theory of
Nonclassical States of Light (London: Taylor Francis)[6] Crisp M D 1991 Phys. Rev. A 43 2430
Crisp M D 1991 Phys. Rev. A 44 563[7] Aoki T, Dayan B, Wilcut E, Bowen W P, Parkins A S,
Kippenberg T J, Vahala K J and Kimble H J 2006 Nature443 671
[8] Tavis M and Cummings F W 1968 Phys. Rev. 170 379Tavis M and Cummings F W 1969 Phys. Rev. 188 692
[9] Dukelsky J, Dussel G G, Esebbag C and Pittel S 2004Phys. Rev. Lett. 93 050403
[10] Brennecke F, Donner T, Ritter S, Bourdel T, Koehl M andEsslinger T 2007 Nature 450 268
[11] Bogoliubov N M, Bullough R K and Timonen J 1996 J. Phys.A: Math. Gen. 29 6305
[12] Vadeiko I P and Miroshnichenko G P 2003 Phys. Rev. A 67053808
[13] Hepp K and Lieb E H 1973 Ann. Phys., NY 76 360[14] Wang Y K and Hioe F T 1973 Phys. Rev. A 7 831[15] Hioe F T 1973 Phys. Rev. A 8 1440[16] Comer D G 1974 Phys. Rev. A 9 418[17] Rzazewski K, Wodkiewicz K and Zakowicz W 1975 Phys.
Rev. Lett. 35 432[18] Knight J M, Aharonov Y and Hsieh G T C 1971 Phys. Rev. A
17 1454[19] Bialynicki-Birula I and Rzazewski K 1979 Phys. Rev. A 19 301[20] Gawedzki K and Rzazewski K 1981 Phys. Rev. A 23 2134[21] Liberti G and Zaffino R L 2004 Phys. Rev. A 70 033808[22] Milonni P W 1976 Phys. Rep. 25 1[23] Sivasubramanian S, Widom A and Srivastava Y N 2001
Physica A 301 241[24] Milburn G J and Alsing P 2001 Directions in Quantum Optics
(Lecture Notes in Physics vol 561) (Berlin: Springer) p 303[25] Brennecke F, Donner T, Ritter S, Bourdel T, Köhl M and
Esslinger T 2007 Nature 450 268[26] Castaños O, López-Peña R, Nahmad-Achar E, Hirsch J G,
López-Moreno E and Vitela J E 2009 Phys. Scr. 79 065405[27] Wooters W K 1981 Phys. Rev. D 23 357[28] Meystre P and Sargent M III 1991 Elements of Optics
(New York: Springer)[29] Arecchi F T, Courtens E, Gilmore R and Thomas H 1972
Phys. Rev. A 6 2211[30] Gilmore R and Narducci L 1978 Phys. Rev. A 17 1747
Gilmore R 1981 Catastrophe Theory for Scientists andEngineers (New York: Wiley)
[31] Andrews G E, Askey R and Roy R 2000 Special Functions(Cambridge: Cambridge University Press)
[32] Kitagawa M and Ueda M 1993 Phys. Rev. A 47 5138[33] Buzek V, Orszag M and Rosko M 2005 Phys. Rev. Lett.
94 163601[34] Goto H and Ichimura K 2008 Phys. Rev. A 77 053811[35] Fink J M, Bianchetti R, Baur M, Göppl M, Steffen L, Filipp S,
Leek P J, Blais A and Wallraff A 2009 Phys. Rev. Lett.103 083601
