Quantum phase transitions in an effective Hamiltonian: fast and slow systems

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by  Isabel Sainz

Quantum phase transitions in an effective Hamiltonian: fast and slow systems
Isabel Sainz,1 A. B. Klimov,2 and Luis Roa3
1School of Information and Communication Technology,
Royal Institute of Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden
2Departamento de F´ısica, Universidad de Guadalajara,
Revoluci´on 1500, 44420 Guadalajara, Jalisco, Mexico.
3Center for Quantum Optics and Quantum Information, Departamento de F´ısica,
Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.
(Dated: February 3, 2008)
An effective Hamiltonian describing interaction between generic fast and a slow systems is ob-
tainedinthestronginteractionlimit. Theresultisappliedforstudyingtheeffectofquantumphase
8 transition as a bifurcation of the ground state of the slow subsystem in the thermodynamic limit.
0 Examples as atom-field and atom-atom interactions are analyzed in detail.
0
2 PACSnumbers: 42.50.Ct,42.50.Hz,42.50.Fx

n
a
J I. INTRODUCTION systems can be different: either continuous or discrete.

0 The b) case correspondsto a situationwhere the reso-
3 Frequently, in the process of interaction between two nance expansionis applicable. This particularcase leads
quantum systems, only one of them can be detected ex- todispersive-likeinteractions[4]. AsitwasshowninRef.
h] perimentally. In this case, a variety of physical effects [3],theevolutionisgovernedbyaneffectiveHamiltonian
p appearintheprocessofsuchinteractionwhichcanbede- describing a certain resonant interaction and the repre-
- scribed in terms of an effective Hamiltonian correspond- sentationspaceofthetotalsystemcanbealwaysdivided
t
n ingtotheobservedsystem. Thesimplestexampleofsuch into (almost) invariant subspaces.
a asituationariseswhenafastsystem interactswithaslow The last case c) possesses a peculiar property: besides
u system. Then, the fast system canbe adiabaticallyelim- of finding a corresponding effective Hamiltonian, we can
q
inated and the slow system is described by an effective also project it out to the lower energy state of the fast
[
Hamiltonian. These considerations were assumed in the system, which would never get excited under given re-

1 famousBorn-Oppenheimerapproximation. Aregularap- lations between the system’s parameters, and thus, de-
v proachto the quantum dynamics ofthe observedsystem scribe an effective dynamics of the slow system in the
9
isprovidedbytheLietransformationmethod[1,2]. The limit of strong interaction. It is well known that in this
8
advantage of this method consists in the possibility of regimesuchaninterestingeffectasQuantumPhaseTran-
6
4 varying the system’s parameters, changing relations be- sitions may occur.
. tweenthem,whichallowsustodescribedifferentphysical The quantum phase transitions (QPT) are a common
1
regimes using the same mathematical tool. In particu- feature of non-linear quantum systems. Such transitions
0
lar, such an important example as expansion on the res- occur at zero temperature and are associated with an
8
0 onances in quantum systems not preserving the number abrupt change in the ground state structure. QPT are
: of excitations can be obtained [3]. In this case a generic related to singularities in the energy spectrum and, at
v
Hamiltoniangoverninginteractionoftwosubsystemsbe- the critical points defining QPT, the ground state en-
i
X yond the Rotating Wave Approximation (RWA) can be ergy is a non-analytic function of the system’s parame-
r representedasaseriesinoperatorsdescribingallpossible ters [6]. Qualitatively, for a wide class of quantum sys-
a transitions in the system. tems, several important properties of QPT can be stud-
Severalinterestingfeaturesappearingintheprocessof ied in the so-called thermodynamic (semiclassical) limit
interactionofquantumsystemscanberealizedbystudy- [7,8]. Then,QPTcanbeanalyzedintermsofaclassical
ing evolution of only two generic quantum system with effective potential energy surface [9]. In this language
onequantumchannel. Eveninsuchasimplecasewemay QPT are related to the appearance of a new classical
discriminateatleastthreeinterestinglimits: a)whenthe separatrixwhen the coupling parameters acquire certain
interactionconstantg ismuchhigherthanthecharacter- values. Accordingtothestandardsemiclassicalquantiza-
isticfrequenciesofbothinteractingsystems;b)wheng is tionschemeandthecorrespondenceprinciple,theenergy
smaller than the frequencies of the systems and c) when density is proportional to the classical period of motion,
g is higher than the frequency of one system but smaller divergingontheseparatrix,whichexplainsahighdensity
that the frequency of the other one. of quantum states at the critical points.
The a) case of very strong coupling should be studied Inthis articlewestudyeffectiveHamiltoniansdescrib-
carefully, because using the expansion parameter like an ing evolution of a generic quantum system X interact-
interactionconstantoveracharacteristicfrequencycould ing with a quantum system Y in the case where the
be quite tricky. For instance, the type of the spectrum characteristic frequency of the system X is essentially
correspondingtothenon-perturbedandtotheperturbed lower than the corresponding frequency of the system
2
Y, ω ≪ ω , and the interaction constant g satisfies Now, we will be interested in the limit where the slow
X Y
the strong coupling condition: ω . g ≪ ω . We system frequency is less than/or of the order of the cou-
X Y
show that, depending on the type of interaction and the pling constant, ω . g ≪ ω . Following the method
1 2
nature of quantum systems different physical situations described in Ref. [2] we can adiabatically remove all
take place, but generically such effective Hamiltonians the terms that contain the fast system’s transition op-
describe QuantumPhase Transitions in the slow system. erators, Y . In particular, the counter-rotating term
±
X Y +X Y andtherotatingtermX Y +X Y can
+ + − − + − − +
be eliminated from the Hamiltonian (1) by a subsequent
II. EFFECTIVE HAMILTONIAN application of the following Lie-type transformations:
Let us consider the following generic Hamiltonian de-
U = exp[ε(X Y −X Y )], (4a)
scribing an interaction between two quantum systems: 1 + + − −
U = exp[ǫ(X Y −X Y )], (4b)
2 + − + −
H =ω X +ω Y +g(X +X )(Y +Y ), (1)
1 0 2 0 + − + −
where the small parameters, ε and ǫ, are defined by
whereX andY arethe freeHamiltonians oftheX and
0 0
Y systems respectively, and such that ω ≪ ω . The
1 2
above Hamiltonian does not preserve the total excita- g g
ε= ≪1 and ǫ= ≪1. (5)
tion number operator N = X0 + Y0 and, in the limit ω2+ω1 ω2−ω1
ω ,ω ≫ g, leads to the appearance of multiphoton- The transformations (4a) and (4b) generate different
1 2
type interactions of the form XnYm which, under cer- kinds of terms: such as XnYk + h.c., XnYk + h.c.,
+ − ± ± ± ∓
tain physicalconditions onthe frequencies ω , describe Yn + h.c., and Xn + h.c. with coefficients depending
1,2 ± ±
resonant transitions between energy levels of the whole on X and Y . Under the condition ω ,g ≪ ω all the
0 0 1 2
system (see [3] and references therein). rapidly oscillating terms, i.e. those containing powers of
The raising-lowering operators X±, Y± describe tran- Y±, can be removed by applying transformations similar
sitions between energy levels of the systems X and Y to (4), with properly chosen parameters. Then, the ef-
respectively and consequently obey the following com- fective Hamiltonian is diagonal for the operators of the
mutation relations: Y system. The result can be expressedas a power series
of the single parameter δ =g/ω ≪1.
2
[X ,X ]=±X , [Y ,Y ]=±Y . (2)
0 ± ± 0 ± ± It is worth noting that it is not enough that δ be a
smallparameterforthe formalexpansionin(4)(andthe
We do not impose any condition on the commutators
subsequent transformations). A balance is necessary be-
between transition operators, which are generally some
tween the effective dimensions of the subsystems and δ.
functions of diagonal operators and of some integrals of
The effective dimensions of the system depend on the
motion [N ,X ]=[N ,Y ]=0:
1 0 2 0
order of the polynomials φ , and on the powers of the
1,2
elements X andY involvedin eachtransformation.
[X+,X−] = ∇X0φ1(X0,N1), (3a) ±,0 ±,0
It was shown before [3], that the powers of the small
[Y ,Y ] = ∇ φ (Y ,N ), (3b)
+ − Y0 2 0 2 parametersareincreasingfasterthanthe powersofX
±,0
andY ,whichimpliesthatwecanfocusontheeffective
where φ (X ,N ) = X X and φ (Y ,N ) = Y Y ±,0
1 0 1 + − 2 0 2 + − dimensions introduced with (4).
are some polynomials of X and Y respectively (from
0 0
now on we omit the dependence on integrals N in Taking into account the above mentioned considera-
1,2
the arguments) and ∇ φ(z) = φ(z) − φ(z + 1). The tions, keeping only terms up to third order in δ and dis-
z
objects (X ,X ) and (Y ,Y ) are known as polynomial regardingsmallcorrectionstotheeffectivetransitionfre-
0 ± 0 ±
deformed algebras sl (2,R) [11]. quencies,wearriveatthefollowingeffectiveHamiltonian:
pd
H = ω X +ω Y −2ω δ2∇ Φ(X ,Y +1)+gδ∇ φ (Y )(X +X )2
eff 1 0 2 0 1 x,−y 0 0 y y 0 + −
1
+2gδ3∇y(cid:0)φy(Y0)∇2yφy(Y0−1)(cid:1)(X++X−)4, (6)
where as
Φ(X ,Y +1)=∇ [φ (X )φ (Y )], (7) ∇ f(X ,Y )=f(X ,Y )−f(X +m,Y +n),
0 0 X0,Y0 1 0 2 0 mX0,nY0 0 0 0 0 0 0
and the generalized displacement operators are defined for m and n integers.
3
Because the effective Hamiltonian (6) is diagonal for we immediately detect that QPT in this case is re-
theoperatorsoftheY fast system,wemayprojectitout lated to the bifurcation of the effective potential U(x)=
ontoaminimalenergyeigenstateoftheY system,|ψ i , (ω˜ /2−2Agδ)x2 +4gAδ3x4 from a single minimum at
0 Y 1
substituting Y by its eigenvalue y : Y |ψ i =y |ψ i . A<ω˜ /(4gδ)toadoublewellstructureatA>ω˜ /(4gδ).
0 0 0 0 Y 0 0 Y 1 1
The first order effect then comes from the term ∼ The physical effect associated with this QPT consists of
(X +X )2, while the term ∼ (X +X )4 defines a a spontaneous generation of photons in the field mode.
+ − + −
fine structure of the effective potential, obtained after In some sense, the virtual photons, always presented in
projecting the effective Hamiltonian (6) onto the state the Dickemodel(8),arecondensed intothe realphotons
|ψ i . aftercrossingthecriticalpointA=ω˜ /(4gδ). Itisworth
0 Y 1
It is important to stress that, although δ is a small noting that this does not happen if the RWA is applied
parameter, the effect of the terms ∼ δn, n ≥ 1, could to (8).
be in principle comparable with the main diagonal term
ω X , especially if the algebraof X operatorsdescribe a
1 0
big subsystem, i.e., large spin or big photon number. In 2. Effective atomic dynamics
this case non-trivial effects such as QPT may occur.
Now, we may proceed with analysis of the effective In the opposite case, when the atoms form a slow sub-
Hamiltonian (6) in the thermodynamic limit, focusing system we have
on the possible bifurcation of the ground state.
X =S , X =S , Y =nˆ, Y =a†, Y =a.
0 z ± ± 0 + −
Projecting the effective Hamiltonian onto the minimum
III. EXAMPLES
energy state of the field mode |0i , so that y = 0, the
f 0
effective Hamiltonian acquires the form
A. Atom-field interaction (Dicke model)
H =ω˜ S −4gδS2+2ω δ2S2+16gδ3{S2,S }, (10)
eff 1 z x 2 z x z
The Hamiltonian governing the evolution of A sym-
metrically prepared two-level atoms interacting with a where ω˜1 =ω1−2ω2δ2−16gδ3.
single mode of quantized field has the form Forouranalysisitisconvenienttoperformaπ/2rota-
tion in (10) around axis y, transforming the (10) Hamil-
H =ω1nˆ+ω2Sz+g(S++S−)(cid:0)a†+a(cid:1), (8) tonian into
H˜ =−ω˜ S −4gδS2+2ω δ2S2−16gδ3{S2,S }. (11)
where nˆ = a†a and S are generators of the (A+1)- eff 1 x z 2 x z x
z,±
dimensional representation of the su(2) algebra.
In the thermodynamical limit we may replace the
atomic operators by the corresponding classical vectors
over the two-dimensionalsphere, i.e.,
1. Effective field dynamics
A A A
S → cosθ, S → sinθcosφ, S → sinφsinθ,
z x y
Firstletussuppose thatthe atomsformafast subsys- 2 2 2
tem so that,
andthusrewritetheeffectiveHamiltonian(11)asaclas-
sical Hamiltonian function,
X =nˆ, X =a†, X =a, Y =S , Y =S ,
0 + − 0 z ± ±
A
andthus, φy(Y0)=C2−Sz2+Sz andφx(X0)=nˆ, where Hcl = −2(ω˜1cosφsinθ+2Agδcos2θ
C =A/2(A/2+1)istheeigenvalueoftheCasimiropera-
2 −Aω δ2cos2φsin2θ
torofthesu(2)algebra(integralofmotioncorresponding 2
to the atomic subsystem). +4A2gδ3cosθcosφsin2θ). (12)
Projecting the effective Hamiltonian onto the mini-
The first two terms in the above expression describe the
mum energy state of the atomic system |0i , so that
at thermodynamical limit of the Lipkin-Meshkov model [8]
y = −A/2, we obtain the following effective Hamilto-
0 anddeterminethecriticalpointofQPT,ξ =4Agδ/ω˜ =
nian for the field mode: 1
1,whichagainis relatedto the bifurcationofthe ground
Heff =ω˜1nˆ−Agδ(cid:0)a+a†(cid:1)2+gAδ3(cid:0)a+a†(cid:1)4, (9) state: a single minimum at sinφ = 0, cosθ∗ = 0 splits
into two minima at sinφ = 0, cosθ = ±p1−ξ−2 for
∗∗
where ω˜ =ω (1−2Aδ2). ξ > 1. It is worth noting that the global minimum of
1 1
H at ξ < 1 converts into a local maximum for ξ > 1,
Rewriting (9) in terms of position and momentum op- cl
so that H (θ ) < H (θ ). This means that the atoms,
erators, cl ∗∗ cl ∗
initially prepared at the minimum of the Hamiltonian
ω˜ function,spontaneouslychangetheirgroundstateenergy
Heff = 1(p2+x2)−2Agδx2+4gAδ3x4, at some value of the system’s parameters. Classically,
2
4
this implies appearance of a separatrix, which leads to ing system when the total excitation number is not pre-
the discontinuity on the energy density spectrum in the served. Analyzing those effective Hamiltonians in the
thermodynamiclimit. Itisalsoworthnotingthatthereis thermodynamic limit we have observed a bifurcation of
alossoftherotationalsymmetryinthisprocess: thenew the ground state leading to the effect of the Quantum
ground state is obviously not invariant under rotations Phase Transitions.
around axis x, while the initial ground state is clearly Itisinterestingtonotethat, formultidimensionalsys-
invariant under x-rotations. tems,whenalgebraicallythe X systemisadirectsumof
It is easy to see that the last two terms in (12) are of several non-interacting subsystems, an interesting effect
lowerorderin the parameter Aδ andcan be neglectedin of generation of entangled states (in the non-preserving
the first approximation for description of QPT at ξ =1. excitation case) can be observed. Really, let us suppose
that in (1) X = X +X , [X ,X ] = 0,j =
0,± 0,±1 0,±2 j,1 j,2
0,±;thenthecorrespondingeffectiveHamiltonian(upto
B. Spin-spin interaction a first non-trivial order in δ) takes the form
Asasecondexampleletusconsideradipole-dipolelike H ≈ ω (X +X )+ω Y +
eff 1 0,1 0,2 2 0
interaction, that is,
gδ[(X +X )2+(X +X )2
+,1 −,1 +,2 −,2
X0 =Sz1, X± =S±1, Y0 =Sz2, Y± =S±2. +2(X+,1+X−,1)(X+,2+X−,2)]∇yφy(Y0),
TheeffectiveHamiltonianfortheslow spin system (after where we can clearly see that the last term contains
projecting onto the lowest state of the fast spin system the operator product ∼ X X which, together with
+,1 +,2
with eigenvalue −A2/2) takes the form similar to (10), quadratic terms in X±,1(2), implies a spontaneous gen-
with eration of entangled states of X and X starting from
1 2
the minimum energy state. This can be corroboratedby
H = ω˜ S −2A gδS2 +2A ω δ2S2
eff 1 z1 2 x1 2 1 z1 the entangling power measure by considering a uniform
+16gA δ3S4 +24gδ3A2{S2,S }, (13) distribution of the initial factorized states [12]. Thus we
2 x1 2 x z
can say that, in the regime studied here, entanglement
where ω˜1 = ω1 − 2A1ω1δ2 − 20gδ3A21. The first two can be generated in a bipartite system the vicinity of a
terms are dominant for δ ≪ 1 and describe the Lipkin- Phase Transition.
Meshkov model, so that the critical point is reached at
ξ =4A A2gδ/ω˜ =1inthe thermodynamicallimit. The
2 1 1
effect of the rest of the terms in (13) is negligible in the
vicinity of ξ =1. Acknowledgments
One of the authors (I. S.) thanks STINT (Swedish
IV. CONCLUSIONS Foundation for International Cooperation in Research
and Higher Education) for support. This work was sup-
We deduce the effective Hamiltonian of a generic slow ported by Grants: CONACyT No45704, Milenio ICM
quantum system interacting with another fast oscillat- P06-067-Fand FONDECyT No 1080535.
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