Schrodinger-equation formalism for a dissipative quantum system

0.44 MB·English

by  E. Anisimovas

S
hrödinger-equation formalism for a dissipative quantum system
∗
E. Anisimovas
Department of Theoreti
al Physi
s, Vilnius University, Saule(cid:10)tekio al. 9, LT-10222 Vilnius, Lithuania
†
A. Matulis
Semi
ondu
tor Physi
s Institute, Go²tauto 11, LT-01108 Vilnius, Lithuania
7 (Dated: February 6, 2008)
0
0 We
onsider a model dissipative quantum-me
hani
al system realized by
oupling a quantum
2 os
illator to a semi-in(cid:28)nite
lassi
al string whi
h serves as a means of energy transfer from the
os
illator to thein(cid:28)nityand thus playstherole of a dissipative element. The
oupling betweenthe
n
two(cid:22)quantumand
lassi
al (cid:22)partsofthe
ompoundsystemistreatedinthespiritofthemean-
a
(cid:28)eld approximation and justi(cid:28)
ation of the validity of su
h an approa
h is given. The equations
J
of motion of the
lassi
al subsystem are solved expli
itly and an e(cid:27)e
tive dissipative S
hrödinger
9
equation for the quantum subsystem is obtained. The proposed formalism is illustrated by its

appli
ationtotwobasi
problems: thede
ayofthequasi-stationarystateandthe
al
ulationofthe
]
r nonlinear resonan
e line shape.
e
h
PACSnumbers: 03.65.Sq,03.65.Yz,73.21.La
t
o
.
t
a I. INTRODUCTION dissipative nonlinear S
hrödinger equation [10℄, and Al-
m bre
ht[11℄ workedoutthegeneralprin
ipleslimitingthe
a

eptable form of the nonlinear terms.
- The rapid expansion of resear
h frontiers into the
d
nanometer and femtose
ond regime demands a
areful The problem of dissipation in quantum me
hani
s is
n
o
onsideration of dissipation as a way to
ounterbalan
e
losely
onne
ted to the use of mixed quantum-
lassi
al
c theenergyin(cid:29)uxfromtheexternal(cid:28)eldsinquantumme- treatments[12,13℄wherebyonlyafewdegreesoffreedom
[
hani
al systems. of a
omplex system are treated quantum-me
hani
ally

while the remaining ones are des
ribed
lassi
ally. Al-
1 The subje
t takes roots in the seminal work by Feyn-
though Egorov and
oworkers [14℄ point out that not
v man and Vernon [1℄ who treated a quantum obje
t
ou-
3 pled to an in(cid:28)nite
olle
tion of os
illators as a model always an a

urate des
ription is attained and so this
7 of a linear dissipative environment. By employing the approa
h must be used with
are, it is re
ognized as a
1 method of
hoi
e when a signi(cid:28)
ant redu
tion of
om-
path-integral te
hniques they were able to eliminate the
1 plexity is desired.
environment variables and arrived at a dissipative mod-
0
i(cid:28)
ation of the Green's fun
tion of the quantum obje
t. Inthispaperwepresentaderivationofarathersimple
7
0 This approa
hwasfurther used by a number of authors, nonlinear S
hrödinger equation that in
ludes a dissipa-
/ most notably by Caldeira and Leggett [2℄, who applied tiveterm. Thisequationisobtainedfroma
onsideration
t
a it to a spe
i(cid:28)
problem ofdissipative e(cid:27)e
ts in tunneling of a physi
al model: a quantum os
illator
onne
ted to
m on a ma
ros
opi
s
ale. A detailed a

ount on the prob- an in(cid:28)nite
lassi
al string that a
ts as a means to trans-
- lem of dissipation in quantum-me
hani
al systems
an ferenergyawayfromthequantumsystem(seealso[15℄).
d befoundinamonographbyWeiss[3℄whilesemi
lassi
al The obtained S
hrödinger-like equation is illustrated by
n
approa
hesare also reviewed in [4℄.
onsidering two spe
i(cid:28)
examples. The (cid:28)rst one, the de-
o

ay of a quasi-stationary state, is an auxiliary one and
c Generally, the main mode of atta
k uses the density-
: matrixformalism[5℄andthepopularLindbladte
hnique shows that the results obtained by using our analysis do
v
not
ontradi
t the ones derived by the usual and more
[6℄. Thisapproa
hisrelevantforopen quantumsystems.
i
X In
ontrast, the
onsiderably less arduous S
hrödinger-
omplexmethodsusingthedensity-matrixformalismand
a double set of variables. The se
ond example, the non-
r equationformalismisregardedtobeappropriateforiso-
a linear resonan
e, is interesting on its own and demon-
lated systems[7℄thataredes
ribedbyahermitianHamil-
stratesthatthequantum-me
hani
alnonlinearresonan
e
tonian.
is physi
ally ri
her than its
lassi
al analog.
Despite this fa
t, thereweresu

essfulattempts to in-

lude dissipation dire
tly in the S
hrödinger equation. Our paper is organized in the following way. In
Most often they based on the de Broglie-Bohm pi
ture Se
t. II, the model system is introdu
ed and the dissi-
[8℄ and employed modi(cid:28)
ations to the quantum poten- pativeS
hrödinger-likeequationforthequantumsystem
tial [9℄. These ideas were used by Kostin to derive a is derived. In Se
t. III, the obtained equation is simpli-
(cid:28)ed employing the rotating-wave approximation. In the
next two Se
tions, some illustrations of the developed
te
hnique are presented: in Se
t. IV we
onsider the de-
∗
†Ele
troni
address: egidijus.anisimovas(cid:27).vu.lt
ay of the quasi-stationary state, and Se
t. V is devoted
Ele
troni
address: amatulistakas.lt to the non-linear resonan
e. Finally, our
on
lusions are
2
formulated in Se
t. VI. obtains a
onsistent des
ription. This averaging
onsti-
tutes the main assumption of the quasi
lassi
alapproxi-
mationandwasusedearliertotreatthe
ouplingof
las-
II. INTERACTION OF THE QUANTUM si
alandquantumdegreesoffreedom[16℄. Wegivesome
OSCILLATOR WITH A CLASSICAL CHAIN further justi(cid:28)
ation of this
ru
ial step below.
Eqs.(3)leadtothestandard
lassi
alequationsforthe
Let us begin by introdu
ing our model of weakly
ou-
hain
pled quantum and
lassi
al subsystems. The quantum Mx¨ +K(x x )+κx = κ x ,
m 0 0− 1 0 h i (5a)
systemismodelekdasaharmoni
os
illatorofmass and Mx¨n K(xn−1 2xn+xn+1) = 0, n>1.
spring
onstant , des
ribedby the quantum-me
hani
al − − (5b)
xˆ pˆ
position and momentum operators and . The role of
The se
ond of these equations (5b) is readily solved by
the
lassi
al system is played by a semi-in(cid:28)nite
lassi
al
the Fourier transform
string whi
h, for the sake of
onvenien
e, is dis
retizMed 1 ∞
and represented as a
hain of identi
al balls of maass xn = 2π Z dqeß(qan−ωt)f(q), (6)
inter
onne
ted by springs of equilibrium length and −∞
K
spring
onstant . The two subsystems are linked by a
κ
produ
ing the standard dispersion law
di(cid:27)erentspring(spring
onstant )
onne
tingthequan-
4K
tum os
illator to the ball sitting at the end of the
hain ω2 = sin2(qa/2).
M (7)
as depi
ted in Fig. 1. The
orresponding Hamiltonian
reads
1 k κ From here on we restri
t ouqra
ons1ideration to the long-
H = pˆ2+ xˆ2+ (xˆ x0)2 wavelength approximation ( ≪ ) thus negle
ting the
2m 2 2 −
∞ dis
retization and e(cid:27)e
tively returning to a
ontinuous-
+ 1 p2 + K(x x )2 , (1) string model for the
lassi
al subsystem. In this limit,
nX=0(cid:20)2M n 2 n− n+1 (cid:21) we have a linear dispersion law
ω =vq
x n (8)
n
with denoting the
oordinate of thep -th ball mea- v = Ωa Ω =
suredfromitsequilibriumposition,and n therespe
tive witKh/Mthe wave-propagation velo
ity and
momentum. .
p
Inserting the Fourier transform (6) into the boundary

ondition(5a)andusingthelong-wavelengthapproxima-
tion we are able to simplify its left-hand side to
1 ∞
dqe−ßωt K 1 eßqa Mv2q2+κ f(q)
2π Z − −
−∞ (cid:8) (cid:0) (cid:1) (cid:9)
FIG. 1: The layout of themodel system. 1 ∞
dqeßvqt( iKqa+κ)f(q)
≈ 2π Z − (9)
−∞
Ka
Following [16℄, we assume that theΨq(uxa,ntt)um os
illator = v x˙ +κx0.
isdes
ribedbyitsownwavefun
tion whi
hsolves
the S
hrödinger equation
The resulting equation of motion for the (cid:28)rst ball
oor-
x
∂ 0
ß~ Ψ(x,t)=HΨ(x,t) dinate
∂t (2) Ka
x˙ +κx =κ x
0 0
v h i (10)
withtheaboveHamiltonian(1)whilethedynami
alvari-
ablesoftheballsinthe
hainobeythe
lassi
alHamilton may be interpreted as the equation of motion for a
equations dissipative
lassi
al mode. Eq. (10) together with the
∂ ∂ S
hrödingerequation (2) for the quantum variables
on-
x˙ = H , p˙ = H ,
n ∂p h i n −∂x h i (3) stitutethe
ompleteequationsetdes
ribingthebehavior
n n
of the dissipative quantum system.
Now we turn to the examination of the validity of the
with the Hamiltonian operatorrepla
edby its quantum-
employed quasi
lassi
al approximation. A smooth way
me
hani
al average over the state of the quantum os
il-
to pro
eedis to introdu
ethe dimensionlessvariables by
lator
∞ means of s
aling
H = dxΨ∗(x,t)HΨ(x,t). ~
h i Z−∞ (4) xˆ→lxˆ, pˆ→ lpˆ, x0 →l0x0, t→ω0−1t,
k ~ ~ (11)
Inthisway,quantumvariablesandoperatorsdonotenter ω = , l= , l = .
0 rm rmω 0 rMΩ
theequationsforthe
lassi
aldegreesoffreedomandone 0
3
l l
0
Here and are, respe
tively, the
hara
teristi
length for the des
ription of the intera
tion between the quan-
ω
0
s
ales of the quantum and
lassi
al subsystems, and tum and
lassi
al subsystems. A similar argument was
stands for the frequen
y of the quantum os
illator. The alsopresentedinthedevelopmentofthemixedquantum-
s
alingenablesustorewritetheobtainedsetofequations
lassi
al dynami
s [12℄.
as Alternatively,onemayarguethatduethesmallnessof
∂ the adiabati
parameter the
hara
teristi
length of the
ß Ψ(x,t)=HΨ(x,t), l0
∂t (12a) wavefun
tions ofheavlyballs ismu
hsmallerthanthe
1 1
hara
teristi
length of the wave fun
tion of the os
il-
H = pˆ2+ xˆ2 λxˆx ,
2 2 − 0 (12b) lator. Consequently, the balls may be
onsidered to be
x˙ +λξx =λ x , following their well-de(cid:28)ned
lassi
al traje
tories whereas
0 0
h i (12
) theos
illatorhasto bedes
ribedquantum-me
hani
ally.
with
κ ω m l III. ROTATING-WAVE APPROXIMATION
0 0
λ= , ξ = .
k rΩM l (13)
0
Let us now pro
eed to some illustrations of the devel-
oped formalism based on a weakly nonlinear os
illator.
Note that we ex
luded the terms whi
h do not depend
xˆ pˆ
Namely,weassumethattheHamiltonianofthequantum
on the operators and from the quantum Hamilto-
subsystem
onsists of a sum of Eq. (12b) and Eq. (14).
nian(12b)asthey
analwaysbeabsorbedintothewave-
Besides, we restri
t the treatment to the
ase of a weak
fun
tion phase. The obtained equations des
ribe the in-
nonlinearity, that is, we assume that the anharmoni
ity
tera
tion of a quantum-me
hani
al system with a single α f λ
x
0 , the for
e amplitude , and the
oupling
onstants
dissipative
lassi
almode . Infa
t,thederivationdoes ξ
and aresmallquantities
omparedtotheunity. Wealso
not depend on the assumption that the quantum system
restri
t the frequen
y of the driving (cid:28)eld to the immedi-
is a harmoni
os
illator. Thus, the equationsare general
ate vi
inity of the resonant frequen
y of the harmoni
and
anbe applied to anyquantum system withdissipa-
os
illatorwhi
hisunityinthedimensionlessunits. Thus
tion. For instan
e,
onsidering an anharmoni
os
illator
we are able to treat the frequen
y deviation from the
driven by an external time-periodi
for
e (we treat this η =ω 1
resonan
e as yet another small parameter − .

ase as an illustration) the aboveHamiltonian is supple-
We expand the sought wavefun
tion into the seriesof
mented with the following additional terms
harmoni
-os
illatoreigenfun
tions
∆H = xˆfcos(ωt)+αxˆ4,
− (14) ∞
Ψ(x,t)= a (t)e−ßωEntψ (x).
n n
f α (15)
with expressing the for
e amplitude and being the nX=0
anharmoni
ity
oe(cid:30)
ient.
In order to have an appre
iable (that is, not negligi- Here, the dimensionless energies and the
orresponding
blysmall)intera
tionbetweenthequantumand
lassi
al eigenfun
tions read
subsωys0temsΩonehastoassumethe
hara
teristi
frequen- E =n+ 1, ψ (x)= 1 e−x2/2H (x),

ies and to beofthesameorderofmagnitude. Un- n n n
2 2nn!√π (16)
der this assumption the quantum os
illator will be able
p
to emit (cid:16)phonons(cid:17) into the string. Furthermore,the
ou-
κ H (x) n
n
pling
onstant representingtheintera
tionofquantum and the symbol standsforthe -th Hermite poly-
kos
illator with the
hain should not ex
eed the
onstant noem−ißaEl.ntNote that instead of the usual time dependen
e
des
ribing the os
illator potential itself. Otherwise, ∼ of a harmoni
-os
illatoreigenfun
tion Ee−q.ßω(E15n)t
the os
illator
annot be
onsidered as a separate entity. features a slightly modi(cid:28)ed exponential fa
tor
In view of the above
onstraints, the dimensionless
ou- thatabsorbsapartofthetimedependen
eoftheexpan-
λ ξ a (t)
n
pling
onstants and depend essentially only on the sion
oe(cid:30)
ient .
m/M
adiabati
parameter . Using the above expansion of the wave fun
tion (15)
Thus, in the
ase of a light quantum os
illator inter- the
oordinate average
an be presented as
a
ting with a
hain of heavy balls the above adiabati
∞
parameter is small and so are the
oupling
onstants. hxi= eßω(En−Em)ta∗namhn|x|mi
The smallness of the
oupling
onstant ensures a weak n,Xm=0
(17)
perturbation of the
hain. The
hain under
onsider- =√2Re eßωta
ation may be treated as a
olle
tion of nonintera
ting
(cid:0) (cid:1)
harmoni
modes. Meanwhile, the quantum-me
hani
al
des
ription of a harmoni
mode gives the same result as where
∞
the
lassi
al one. Therefore, one may
on
lude that the
m/M a= a a∗ √n+1
smallness of the adiabati
parameter guarantees n n+1 (18)
the validity of the applied quasi
lassi
al approximation nX=0
4
is the usual polarization of the quantum os
illator.
Inserting the
oordinate average (17) into Eq. (12
),
and solving it to the leading term in the small
oupling

onstants we obtain the following expression for the
o-
ordinate of the (cid:28)rst ball
x =√2λ dtRe eßωta =√2λIm eßωta .
0
Z (19)
(cid:0) (cid:1) (cid:0) (cid:1)
Next, inserting the obtained expression (19) and the
wave fun
tion (15) into Eq. (12a), proje
ting this equa-
n
tion onto the -th state, and negle
ting all os
illating
terms (this amounts to the usual rotating-waveapproxi-
mation)wearriveatthe(cid:28)nalsetofequationsforthetime
evolution of the wave fun
tion expansion
oe(cid:30)
ients
ßa˙n = ηEn α nx4 n an |a1|2
√n(F−+(cid:8)ßγa∗−)an−h1| √| ni(cid:9)+1(F ßγa)an+1.
FuIrGv.e,2:|a0D|2e
(cid:21)aydaosfhqedua
sui-rsvtea,tiaon(cid:21)ardyotstteadte
:urve, an(cid:21)dthxi0
k(cid:21)stohliidn
− − − (20)
solid
urve.
Here, for the sake of simpler notation we denoted
F =f/2√2, γ =λ2/2.
(21)
indu
edduringthequantumtransition,andthethinsolid
The obtained set of equations (20) was solved in the xlin(et)showsthe
lassi
al
oordinateof(cid:28)rstballinthe
hain
0
two- and three-level approximations, and the results are whi
his
al
ulatedbymeansofexpression(19). As
presented in the following se
tions. a matter of fa
t, this
urve represents the
oordinate of
any ball in the
hain be
ause a

ording to Eq. (6) the
n

oordinate of the -th ball
an be expressed as
IV. DECAY OF THE QUASI-STATIONARY
x (t)=x (t an/v).
n 0
STATE − (26)
Therefore,duringthede
ayofthequantumstateastring
In order to illustrate the validity of the proposed
ex
itation of the shape shown by thin solid line is emit-
method we
onsider a well-known problem of the de
ay
ted and travelsawayfrom the quantum os
illatorwith a
of a quasi-stationary state. v

onstantvelo
ity . It
anbeinterpreted asthe
lassi
al
Forthis purpose, weassumethat the quantum system
analogofanemittedphonon. Naturally,thetotalenergy
has only two states and there is no anharmoni
ity and
F = η = α = 0

arriedbythestringex
itationisequaltotheenergydif-
no external for
e ( ). At the beginning
feren
e between the two quantum levels. Consequently,
the system is prepared in the upper level, that is, the
we may
on
lude that the quantum subsystem provides
boundary
ondition reads
the energy quantization in the
lassi
al one.
t= , a =0, a =1.
0 1
−∞ (22) It is worth mentioanin2g+thaat2ta=kin1g into a

ount the
0 1
norm
onservation (| | | | ) Eqs. (20)
an be
A

ordingto Eq. (20), the behaviorof this simplesys- transformed into another set of equations govIer=ninag t2he
tem is des
ribed by the following two equations baeh2aviorofthreerealvariables: theinvaersion | 1| −
a˙0 =γaa1, a˙1 =−γa∗a0, with a=a0a∗1. (23) | 0| and the (
omplex) polarization . The linearized
versionoftheseequationsinthevi
inityofthestationary
n=0 I = 1
These equations are readily solved analyti
ally, and the state ( − ) reads
solution reads I˙= 2γI, a˙ = γa.
1 1 − − (27)
a 2 = , a 2 = .
| 0| e−2γt+1 | 1| e2γt+1 (24)
Itisremarkablethat theseequations
oin
idewith those

onsidered in the density matrix formalism [17℄, more-
These(cid:28)llingfa
torsareshowninFig.2bythethi
ksolid
over, from our Eq. (27) we are able to extra
t the relax-
(ex
ited level) and dashed (ground level) lines. We see T =1/2γ
1
ationtimes: thelongitudinalrelaxationtimeis
that the quantum transition takγe−s1pla
e during a (cid:28)nite T2 =1/γ
andtheperpendi
ularoneis . Weseethatthese
period of time proportional to , the re
ipro
al fri
- T = 2T
2 1
relaxation times obey the relation inherent to
tion
oe(cid:30)
ient. The dotted
urve shows the os
illator
the pure state when no ensemble average is
arried out
polarization
andthisfa
t
on(cid:28)rmstheadequa
yofourattempttode-
1
a= s
ribe a dissipative quantum system in the S
hrödinger-
2cosh(γt) (25)
equation formalism.
5
Two more
omments are in order. First, a

ording to We observe the following typi
al behavior: For a weak
t = F < 0.5γ
0
Eqs. (22) we used the boundary
ondition at −∞. for
e ( ) the shape of the resonan
e resembles
Thisisrelatedtothe
lassi
aldes
riptionofthe
hainex- a Lorentzian, and its peak value is in
reasing with in-

itations when there is no spontaneous emission. When
reasing for
e. When the for
e ex
eeds the
riti
al value
F =0.5γ
0
solving the equations numeri
ally one has to add some the peak stops growingand is (cid:29)attened at the
initial vibration (cid:28)eld (cid:29)u
tuation in order to for
e the topduetothesaturationatthe
riti
alpowerabsorption
P =0.5γ
0
quantum transition. value indi
ated by the thin dotted line.
Another point is related to the Fourier transform of
the
oordinate of the (cid:28)rst ball 0.5
∞ π/2γ
x (ω)= dωeßωtx (t) F/g =5
0 0
Z ∼ cosh(ηπ/2γ) (28) g
−∞ P/
2
whi
h gives us the shape of the emission line. We see
η 1
0.25
that
lose to the resonant frequen
y, when | | ≪ , the
line shape is Lorentzian
oin
iding with the result fol- 0.3 1
lowing from a simple
al
ulation based on the Fermi's
0.5
golden rule. However, further away from the resonan
e, 0.2
the line shape deviates from the Lorentzianand features
exponential tails. 0 0.1
-3 -2 -1 0 1 2 3
V. RESONANT POWER ABSORPTION h/g
The following problem is that of the resonant power FIG. 3: Power absorption in the two-level approximation.
absorbtion whi
h we
al
ulated solving the equation set The numbers on
urves indi
ate the values of the external
F/γ
(20) using two-level for
e amplitude .
η
a˙ = ß a +ß(F ßγa)a ,
0 0 1
2 − (29a)
3η In the next two Figures the results for power ab-
a˙ = ß a +ß(F +ßγa∗)a ,
1 2 1 0 (29b) sorption
al
ulated in the three-level approximation are
a = a a∗. shown. InFig.4weplotthepowerabsorptioninalinear
0 1 (29
) α=0
os
illator ( ).
and three-level
a˙ = ß(η/2 α)a +ß(F ßγa)a ,
0 0 1
− − (30a)
a˙ = ß(3η/2 5α)a
1 1
−
+ß(F +ßγa∗)a +√2ß(F ßγa)a ,
0 2
− (30b)
a˙ = ß(5η/2 13α)a +√2ß(F +ßγa∗)a ,
2 1 1
− (30
)
a = a a∗+√2a a∗.
0 1 1 2 (30d)
approximations. Notethatweex
ludedanonlinearterm
from the equations of the two-level approximation be-

ause in this
ase it leads only to an inessential shift of
the resonant frequen
y. The power absorption was
al-

ulated by averaging the instantaneous power over the
period of the external for
e, that is,
P = 2FIma.
− (31)
FIG.4: Powerabsorptioninthethree-levelapproximationin
The power absorption in the two-level approximation the
ase of the linear os
illator. The numbers on the
urves
η F/γ
as a fun
tion of the deviation is shown in Fig. 3 for indi
ate thevalues of theexternal for
e amplitude .
various for
e amplitudes. Note that Eqs. (29) and (30)
depend linearly on their parameters, therefore, by us-
ing an appropriate s
aling of the time we may redu
e The general shape of the absorption
urves
losely
the number of parameters by one. Therefore, we
hoose resembles those obtained in the two-level approxima-
F P η α γ
to express the quantities , , and in units. tion, ex
ept that now the saturation sets in at a larger
6
F = γ
0
for
e value , and the saturated power absorption of the quantum anharmoni
os
illator in the absorption
P =1.5γ
0
is larger as well. gap more
arefully. In Fig. 6 the right-hand side of this
F = 0.5γ
A more essential di(cid:27)eren
e is seen in the uppermost absorption gap for the parameter values and
F = α= 0.25γ

urve
orresponding to the for
e whose amplitude ( is shown in more detail. The simplest way to
5γ
) strongly ex
eeds the
riti
al value. Here we see a distinguish various types of dynami
behavior is to plot
narrowgap in the absorptionwhi
h appears
lose to the thephaseportraitofthesystem. Astherearethree
om-
resonant frequen
y. It resembles an analogous gap that plexvariables
onstrainedby asinglenormalization
on-
appears in the power absorption
al
ulated by means of dition the phase spa
eis(cid:28)ve-dimensional. Therefore, we
the density matrix equation te
hnique [18℄ and is
aused 0.5
by the intera
tion of
oherent light with an inhomoge-
1
neously broadened resonan
e line. 2
0.4 3
0.5
a =0 0.3
0.4 P/g
0.2
P/g 0.3
0.1
0.2
0.075
4
0.1 0.05 0.0
0.525 0.550 0.575 0.600
0.025 h/g
0.0
0.5 0.0 0.5 1.0 1.5 F =
h/g F0.I5Gγ. 6: Pαow=er0.a2b5sγorption gap for the parameter values
and .
FIG.5: Powerabsorptioninthethree-levelapproximationin
the
Fase=o0f.5aγn anharmoni
os
illator obtained for theparam- restri
tourselvesto a pro(jRe
etiao0n,Rofetah1e)phase spa
e onto
eter . The
urvesarenumberedbythevaluesof the a two-dimensional plane shown in Fig. 7.
α/γ
anharmoni
ity
oe(cid:30)
ient . The panel numbers
orrespond to the frequen
ies indi-

atedbydotsinFig.6. Weseethatontheregular(cid:29)ankof
In Fig. 5 the results for the nonlinear os
illator are
presented for a set of values of the anharmoni
ity
oe(cid:30)-

ient and the for
e amplitude
orresponding to the
riti-
F = 0.5γ

al value of the two-level approximation, i. e. .
α = 0
We see that for the linear-os
illator
ase ( ) the
absorptiondemonstratestheexpe
tedLorentzianbehav-
ior. When the anharmoni
ity is in
remented the reso-
nant peak moves to the right, be
omes asymmetri
and
the saturation manifests itself as a widening absorption
gap
lose to the shifted resonan
e frequen
y.
The absorptiongapresemblesthe gapin nonlinear
y-

lotronresonan
e[19℄whi
hwasdes
ribedusingthebal-
an
e equations. In that work the gap was related to
the dynami
al
haos appearing in the
lassi
al nonlin-
ear equations of motion. Quantum
haos is a subje
t
of great interest as well (see [20℄ for referen
es), and is
largely
on
erned with the quantum signatures of
lassi-

ally
haoti
systems. Su
h signatures are often sought
by
omparing the wave-fun
tion patterns
orresponding
to various eigenstates with the
lassi
al traje
tories [21℄.
It follows from our quasi
lassi
al
onsideration,
FIG. 7: Two-dimensional proje
tion of the phase spa
e. The
Eqs. (20), that the behavior of the quantum system
an
numbersonthepanels
orrespondtothefrequen
iesindi
ated
alsobevisualizedandinterpretedintermsofphase-spa
e
bydots in Fig. 6.
traje
tories obtained by treating the wave fun
tion ex-
a
n
pansion
oe(cid:30)
ients as generalized
oordinates. In or-
dertoillustratethispossibility,welookedatthebehavior theresonan
e
urve(point1)theos
illatordemonstrates
7
a rather simple behavior (cid:22) the limit
y
le. As a mat- system. This equation is derived on the basis of a qua-
a
n
ter of fa
t, in this
ase all three
oe(cid:30)
ients os
illate si
lassi
al treatment of
oupled quantum and
lassi
al
a (t) = g exp(ßζt)
n n
with the same frequen
y, namely, . subsystems and justi(cid:28)ed by the di(cid:27)eren
e of masses of
Therefore,inthis
asethepowerabsorptionproblem
an the two subsystems whi
h leads to the smallness of adi-
be simpli(cid:28)ed essentially and transformed into a nonlin- abati
parameters. The
onsidered de
ay of a quasi-
ζ
eareigenvalueproblemde(cid:28)ningthe frequen
y , and the stationarystatedemonstratesthattheproposeddes
rip-
g
n
stationary
oe(cid:30)
ients aresimply relatedto the (cid:28)lling tion gives identi
al results to those obtained from the
fa
tors of the harmoni
os
illator levels. density-matrixapproa
hwhen only homogeneousbroad-
The behavior of the anharmoni
os
illator in the ab- eningistakenintoa

ount. Themodel
al
ulationofthe
sorption gap (points 2, 3, 4) is quite di(cid:27)erent. Here, the nonlinear absorption indi
ates that many
lassi
al phe-
a
n

oe(cid:30)
ients os
illate with multiple frequen
ies, more- nomena(cid:22)su
hastheasymmetryoftheresonan
eshape
over, the os
illator
an demonstrate both periodi
be- and absorption dips (cid:22) brought about by the loss of
o-
havior(point 2) and non-periodi
behavior(points 3, 4). heren
e between the driving for
e and the response may
This non-periodi
ity is in fa
t responsible for the gap in be featured by a quantum system as well. We believe
the power absorption as it
auses the loss of
oheren
e that the proposed S
hrödinger-equation formalism with
betweenthe drivingfor
e and the system response. This dissipation may (cid:28)nd appli
ation to quantum nanostru
-
is the so-
alled quasiperiodi
behavior but the appear- tures and quantum
haos.
an
e of a strange attra
tor is also expe
table at larger
α
values of the anharmoni
ity
oe(cid:30)
ient .
A
knowledgments
VI. CONCLUSIONS We thank Prof. K. Pyragas and J. Ruse
kas for help-
ful dis
ussions. This work was supported by the Lithua-
In
on
lusion, we propose a S
hrödinger-like equation nianS
ien
eand Studies Foundationunder grantNo.T-
for the des
ription of a dissipative quantum-me
hani
al 05/18.
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