J. Phys. B: At. Mol. Phys. 17 (1984) 2101-2128. Printed in Great Britain
Semiclassical many-mode Floquet theory: 11. Non-linear
multiphoton dynamics of a two-level system in a strong
bichromatic field
Tak-San Ho and Shih-I Chut
Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA
Received 16 November 1983
Abstract. The non-linear multiphoton excitation dynamics of two-level systems under the
influence of two strong linearly polarised monochromatic fields is studied exactly for the
first time using the semiclassical many-mode Floquet theory recently developed. The
present approach relaxes the restrictions of the conventional generalised rotating-wave
approximation (GRWA) and allows correct treatment of various non-linear features such
as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending
the nearly degenerate perturbative method of Salwen, approximate analytical expressions
for transition probabilities are obtained for multiphoton resonance processes for which
( n + 1) photons of one field are absorbed, whereas n photons of the other field are emitted,
where n is an arbitrary integer. Detailed comparison of the analytical, the G R W A and the
numerically exact results is given. Also presented is a case study of the time evolution
of spin f systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.
1. Introduction
Recently there has been considerable interest in the study of atomic and molecular
processes in the presence of two monochromatic radiation fields. Multiphoton doubleresonance experiments (Galbraith et al 1982, Jacques and Glorieux 1982), collisions
in two laser fields (de Vries et a1 1980), multiphoton dissociation (MPD) of polyatomic
molecules by two infrared lasers (Ambartzumian and Letakhov 1977, Alimpiev et a1
1983), and multiple quantum transitions in double frequemy pulsed NMR experiments
(Zur et al 1983), etc, have been studied extensively.
In this present paper we shall study the dynamics of a two-level system under the
influence of two linearly polarised monochromatic fields. Theoretical discussions on
this subject are abundant, but most of them are limited to the so-called ‘generalised
rotating-wave approximation’ (C;RwA)-an extension of the rotating-wave approximation (RWA) of the single monochromatic case. As early as 1955, Ramsey had already
discussed the effect of a second rotating wave (non-resonant) on the single quantum
resonance shift of a system exposed in a resonant rotating wave. Bucci et al (1970)
developed a second quantisation treatment to calculate the response, in NMR condition,
of a spin-i system irradiated by two nearly resonant waves of different frequency. The
modification of the energy levels of a spin-4 system by two rotating RF fields was
t Alfred P Sloan Foundation Fellow.
0022-3700/84/102101+28$02.25
@ 1984 The Institute of Physics
2101
2102
T-S Ho and S-I Chu
discussed by Tsukada (1974), while the possibility of population inversion for a
two-level system in a bichromatic field was investigated by Goreslavskii and Krainov
(1979). Most recently Zur et a1 (1983) studied the multi-quantum (MQ) effects on a
spin-; system in a double frequency pulsed NMR experiment. They employed the
ordinary single-mode Floquet theory (Shirley 1965) incorporating a special perturbative procedure designed for two nearly degenerate states (Salwen 1955) to describe
the evolution of the magnetisation vectors. The only work which had gone beyond
the generalised rotating-wave approximation on the subject was carried out by Guccione-Gush and Gush (1974). They postulated that the wavefunction of a two-level
system in a bichromatic field has the form of a double Fourier series, and obtained
the Green’s function of the system in terms of continued fractions, a method adopted
in an earlier paper for the case of a single monochromatic field (Autler and Townes
1955). In a previous communication (Ho et a1 1983) we have developed an exact
semiclassical multi-mode Floquet theory which enables us to treat the time-dependent
problem of any finite-level system exposed in polychromatic fields as an equivalent
time-independent infinite-dimensional eigenvalue problem. In this paper we shall
exploit in great detail the non-linear response of a two-level system illuminated by a
strong bichromatic field by using the many-mode Floquet theory along with an extended
version of Salwen’s (1955) perturbation theory. For simplicity we have neglected all
the relaxation processes. Atomic units will be used throughout the paper.
In 8 2 we give a brief account of the many-mode Floquet theory for a two-level
system. An extension of Salwen’s perturbation theory is developed in § 3.1. In § 3.2
we present simple analytic expressions, to some lower order of perturbation parameters,
for the transition probabilities of the resonance processes w o --- ( n + l ) w , - nuz with
w o the unperturbed energy spacing of the two-level system, w1 and w 2 the frequencies
of fields, and n an arbitrary integer. A comparison of the current (exact) approach
with the widely used generalised RWA approach is made in § 4. In § 5 we present and
discuss numerical results of the bichromatic quasi-energies, their connection with the
long time averaged transition probabilities at various field strengths, and time-dependent transition probabilities for various multiphoton resonance processes. A study of
the dynamics of an ensemble of the spin-: systems, subjected to a static magnetic field
in the 2 direction and two linearly polarised RF fields in the 2 direction is presented
in § 6. This is followed by conclusions in § 7.
2. Semiclassical two-mode Floquet theory
In this section we shall briefly describe the semiclassical many-mode Floquet theory
developed in a previous paper (Ho et a1 1983, to be referred to as I) for a two-level
system exposed in a strong bichromatic field. The two-level system will be treated
quantum mechanically, whereas the applied fields will be treated classically. The
evolution of the wavefunction is determined by the time-dependent Schrodinger
equation:
d
i - W ( t ) = HW( t )
dt
where
L
H = H o + 1 Vi(t),
i=l
Semiclassical many-mode Floquet theory: 11
2103
Ho is the unperturbed Hamiltonian of the two-level system (with eigenstates {la),IP)}
and eigenvalues {Eo,
E p } ) ,and the interaction Hamiltonian V,(t) in the electric dipole
approximation is given by
v,( t ) = -p
* E,
cos( w,t + +!)
i=1,2
(3)
with p the electric dipole moment operator; E , ( = I E , ~ ) , U , and +r, respectively, the
peak amplitude, the frequency and the initial phase of the ith monochromatic field.
In I we have shown that the time-dependent problem equation (1) can be transformed into an equivalent time-independent infinite-dimensional eigenvalue problem
(in Dirac notation):
where
HF:
A::
0
j
\I
B'
1
A
B
0
X*
0
r
0
1
0
X*
0
0
0
0
C-w,I
I
X"
x
j C-2W,I
0
0
0
r
r.
Figure 1. Linearly polarised bichromatic Floquet Hamiltonian for the two-level system,
constructed in a symmetric pattern. w 1 and w 2 are the two radiation frequencies and V:;
(i = 1 , 2 ) are the electric dipole coupling matrix elements for the ith field. Note that the
diagonal block A possesses an identical Floquet structure to that of the one-laser problem.
T-S Ho and S-I Chu
2104
with
and 'yl and y2 can be either a or /3. Without loss of generality, we shall assume the
matrix elements ( y I J p &,IyZ)are real. In figure 1 the structure of the matrix HF is
displayed by ordering components so that y runs over the unperturbed states, a and
p, before each change in n,, and nl, in turn, runs over before n2 changes. The
quasi-energy eigenvalues { AY,,,n2} and their corresponding eigenvectors { I A,, ,,,)} of HF
have the following useful periodicity forms, namely,
-
For simplicity we have chosen A, = Ayoo in equation (8).
The wavefunction * ( t ) at any instant of time t is evaluated via the relation
00;h ) I W t o ) )
the propagator 0(t;to) in its matrix form can be written
(10)
IWt)) =
where
Vy,,$;
tO)'(Yll0(C
3
tJIY2)
C
m
C
=
as
n , = - m n2=-m
( ~ l n ~ n 2 1 e x p [ - i E i , ( t - ~ ~ ~ ) l 1exp[i(nlw+
y200)
n 2 ~ ~ t 1 (11)
.
More explicitly, assuming the initial state of the system at to is la), the substitution of
equation (11) into equation (10) results in
n l n2
Equation (12) justifies the postulation that the wavefunction of a two-level system in
a bichromatic field has the form of a double Fourier series (Guccione-Gush and Gush
1974).
Using equation (11) (or (12)) and the periodic relations equations (8) and (9), the
probability of going from the initial state la) to the final state I@) under the influence
of the bichromatic fields has the form
P & + p ( t ;to) =
1 V,,(t; to)12
=
C 2
@klk21exp[-ifiF(t - to)]iaOO)exp[i(mlwl + m 2 ~ l ~ ) t 0 1
mi m2 k~ k2
x (amlmzi exp[i~F(t-to)llBk,kz).
(13)
Semiclassical many-mode Floquet theory: II
2105
Assuming the initial time to is not well defined, but determined by a random process,
the transition probability averaged over the initial time is then the quantity of experimental interest, and can be written as follows:
L
p =
cc
I(Pkl~21~rl,12)(~Yf,f2/~~0)!2.
(16)
klk2 Yllf2
Furthermore, it is easy to show that equation (16) can be cast into an alternative form,
with wo the unperturbed energy spacing of the two-level system,
which is in essence the extension of Shirley's (1965) expression for the case of a
two-level
system in a monochromatic field. Equation (17) shows that Fe+@S $, while
P&+pof equation (15) can be rewritten as
R - p =E + p
+
X
e' c c
mlfO
I(Pkl~ZI~yf,12)12
k I k 2 yf,l2
(amimzlAyi,i,)(A,i, i21a00)ex~[i(miwi
+ m2~z)foI
(18)
and, therefore, can be greater than
This suggests that the population inversion for
a two-level system is achievable only when one can define an initial phase seen by the
atom to a certain degree.
3. Almost degenerate perturbation treatment of the two-mode Floquet
Hamiltonian
3.1. General consideration
In this section we extend the perturbative technique of Salwan (1955) to approximate
the eigenvalues and their corresponding eigenvectors of the two-mode Floquet Hamiltonian HFwhen only two of its unperturbed Floquet eigenenergies are nearly degenerate, and are far away from all others in magnitude. We also assume that the off-diagonal
part HFlof HFis small (corresponding to weak fields case) compared with its diagonal
part HFo(here we have assumed HF= HFo+HFI)
in a sense to be made precise shortly.
The perturbation HF1includes two parts with each describing the coupling between
the unperturbed system and one of the two fields applied.
We shall only consider the resonant processes in which (n + 1) photons of one field
are absorbed and n photons of the other are emitted. Generalisation to other resonant
processes, e.g. ( n+ 3) photons absorbed and n photons emitted, is straightforward. In
T-S Ho and S-I Chu
2106
the following discussion we shall introduce the notations:
b2 = Rb exp(i4,)
b, = b exp(i&)
with
where V“d.2’ and V;i2’ are set to be zero, and E, = - t w o and Ep = t w o .
Now that E, = Eo - ( n f l ) w l + nw2, the two perturbed states corresponding to
~a)=~aOO
and
) ib)=l/3, -n--1, n ) will have large projections ( a l A ) and ( b l h ) on the
unperturbed states la) and 1 b), and approximately satisfy the eigenvalue equation
(
-two+
Va,(A)
Vab(A)*
vab(A)
i w o- ( n
+ 1)w , + nw2 + v b b ( A )
(19)
and the Green’s function operator Go(A) is defined as
G,(A) =
AI,
1
HFo‘
-
Here I , is a unity matrix of infinite dimension. The parameter A, the perturbed
eigenvalues, are to be determined self-consistently. The projections of these two
perturbed states on unperturbed states 1yl1l2)other than /aOO)and 1/3, - n - 1, n ) can
be expressed as
From equation (26), one can easily see the two necessary conditions for (y1,12/A)to
be small compared with ( a l A ) and (blA) are that
1
<<
(yll12/HFlla)
A - ( E , + 11 a i + 1 2 ~ 2 )
Semiclassical many-mode Floquet theory: II
2107
More explicitly, these two conditions imply that
l&l<<
and
l&l<<l
must be fulfilled for the perturbation approach adopted here.
To solve equation (19) to various order in the parameters b l / ( wo - w l ) and b2/ ( wo w 2 ) , we now introduce two additional small quantities q and A which are of the same
order of magnitude as b l / ( wo- wl) and b2/ ( w0 - w 2 ) , and are defined via the relations
where the unperturbed Floquet energies Ah"&, and AZ)n-l,n
A
Lo,;
are
-1
2wo
and
A p( 0, )- n - , , n ~ ~ W 0 - ( n + 1 ) W 1 + n W 2 .
Later one will see that the quantities q and A play important roles in describing the
effect of the external fields on the unperturbed two-level system. To a very good
approximation, equations (20) via (22) can be written as
O ( n )=
if n=O
otherwise.
(35)
From Floquet matrix HF,one immediately sees that
V,b(AEo))aRnb2n+1
exp[i(n+1)+,-inqh2]
(36)
and there are ( 2 n + l ) ! / ( n + l)!n! ways to connect the two nearly resonant states / a O O )
and Ip, -n - 1, n ) , i.e. (2n + l ) ! / ( n+ l)!n! non-vanishing summands under the summation in equation (31). Substituting equation (29) via (31) into equation (19), and
2108
T-S Ho and S-I Chu
replacing A by q through the relation (27), one can solve equation (19) to obtain
The eigenvalues and their corresponding eigenvectors of the two perturbed Floquet
states are thus written as
and
The normalisation constants Na and Nb are determined by relations
the prime on the summation indicates that ( yll ! 2 ) does not take the pair (aOO) and
( P , - n - 1, n ) .
Although we have only discussed how to evaluate approximately the perturbed
eigenvalues and eigenvectors corresponding to the unperturbed Floquet states I a00)
and I@, - n - 1, n ) when -$wo=$wo- ( n + l ) w l + nu2, there exist infinite duos of near
resonant states, i.e. {lamlmz),IP, ml - n - 1, m2+n ) } , whose corresponding perturbed
states can be computed similarly. The periodic relations (8) and ( 9 ) , furthermore,
suggest that these computations are unnecessary; the perturbed states IA,, ,,,J and
Ihp,ml-n-l,mz+n)
arid their corresponding eigenvalues can be directly obtained by using
equations (37) via (41) along with equations (8) and (9).
3.2, Transition probabilities-analytical expressioits
Equipped with the perturbation theory outlined in the previous section we are now
able to present the transition probability in simple analytic form for the near-resonant
processes. First we consider the n = 0 case (one photon), i.e. w g = w l . To the lowest
order (amounting to a two-state approximation), i.e. keeping only the dominant terms
in equations (13) or (14) we obtain the transition probability going from the state l a )
to the state IP) at any instant of time 1 :
where
2109
Semiclassical many-mode Floquet theory: I1
From equations (43) via (45) we found, besides the well known Bloch-Siegert shift
(Bloch and Siegert 1940), SBS= b2/w1, there exists an additional shift
caused by the second non-resonant field, on the resonant frequency. The width of the
resonance uo is equal to 4b as expected for the one-photon process.
In general, the transition probability at any instant of time t for the w o =
( n + l ) w l - nu2 resonant processes, when evaluated to the lowest order, can be written
as
(1
P,+p(t) =-[WO-
)2
2 ufl
( n + l ) w , +nw2+6,]2+(fu,)2
sin2 qnt
For n # 0, the resonance shift 6, is found via the relation
and is approximately proportional to b 2 / n (recall that ( w o - w l ) = n ( w l - w 2 ) and
(coo-- w 2 )= ( n + l ) ( w l - w 2 ) ) if one fixes w I and w2, but varies wo. The width of the
resonance can be computed from equation (31), e.g.
and
for the three- and five-photon resonances, respectively.
By retaining more terms in the sum of either equation (13), or equation (14), we
can obtain the transition probability to higher orders of accuracy. To the first order
in b l / ( o o - wl) and b 2 / ( w 0 - w 2 ) ,the transition probability, without averaging over the
2110
T-S Ho and S-I Chu
initial time, at any instant of time t on going from the state) . 1
to the state
I@)
is
= I up,(t)12
[sin( n u , t - no2t+
+-
Rb
WO
+W2
- n42)- sin( n41 - n42)]e(n )
{sin[(n+ 1)wlt--(n-- l ) w 2 t + ( n +l ) ~ $ - ( n - 1)42]
and, to the second order, the transition probability averaged over initial times to, while
keeping the elapsed time t’ = t - to constant, is
Semiclassical many-mode Floquet theory: II
x(
b2
~ 0 ~ [ ( n + 2 ) 0 , t ~ - n ~ , t ~ ] + e ( n ) ,cos(no,t’-nnw2t’)
b2
(WO+ W 1 )
+
R2b2
2111
(00-
01)
,C0S[(n+1)w,t’-(n-1)W,t’]
(WofW2)
+
+
R2b2
(WO - W 2 )
R2b2
(WO
+
+W2)
R2b2
(WO-
,C0S[(n+1)w,t‘-(n+1)W,t‘]
sin[(n+ l)w,t’-(n-- 1)u2t’]
,sin[(n+l)w,tf-(n+1)~,tf]
(53)
W2)
where
and Vb,, V;, and IV,bl are defined in equations (31), (32) and (33). We note that
the resonance is obtained when A = Vh, in equation (52) and (53). A comparison of
equations (52) and (53) shows that the initial phase informations of the two external
fields are contained explicitly in P & + p ( t )while
,
no phase information exists at all in
P,+p(t’) as one averages over the random initial times to. The long time averaged
transition probabilities, i.e. averaging over t in equation ( 5 2 ) , and over t’ in equation
(53), are
and
4;
At the resonance, i.e. A = U;,, both P&,, and Peep are equal to
the resonance
lineshape is a pure Lorentzian (with the width 41Vcrbl)for both quantities when we
2112
T-S Ho and S-I Chu
neglect the higher-order small terms in equations (54) and ( 5 5 ) . The effects of the
small terms on I? + B and pa+P,
respectively, are completely different, and exhibit very
interesting physics taking place. The
will show a dispersive feature (because the
factor ( A - V&,)changes sign) around the resonance centre, whereas the pa+@becomes
a fattened Lorentzian, but always stays below i.
4. Comparison with the generalised rotating-wave approximation
By expanding the total wavefunction 9(t ) in the basis of) . 1
eigenstates of H,, equation (1) becomes
and
(GRWA)
I@),
the unperturbed
2 b [ c o s ( w l t + ~ , ) + Rcos(~,t+&,)]
EP
When each cosine in the above equation is replaced by one exponential (in magnetic
resonance, when each of the two linearly oscillating fields is replaced by a rotating
field) we obtain
i d((.IW))
dt (PlWt))
=(
Ea
bl exp(+iwlt) + b2 exp(+iw2t)
b t exp(-iwlt)+ bT exp(-iw2t)
EP
This is the starting equation normally adopted for the problem of a two-level system
interacting with two classical monochromatic fields having frequencies w 1 and w 2 very
close to the transition resonance frequency wo (Ramsey 1955, Tsukada 1974, Goreslavskii and Krainov 1979, Zur et a1 1983). We shall call the replacement of equation
(56) by equation (57) the generalised rotating-wave approximation (GRWA). The usual
way of solving equation (57) is by transforming the problem to an appropriate rotating
frame so that the transformed Hamiltonian in the new frame becomes a periodic
function of time with a well defined frequency (or period). Thus the ordinary one-mode
Floquet theory can be evoked after the transformation (ZUIet al 1983).
With the newly developed multi-mode Floquet theory (I), one can directly construct
a timc-independent Floquet Hamiltonian (figure 2) equivalent to the equation (57)
without much effort. At this point, it is instructive to look into this GRWA Floquet
Hamiltonian from an intuitive angle: one understands that the time-independent
Schrodinger equation of a two-level system in the presence of a single rotating field
is equivalent to a 2 X 2 time-independent Floquet eigenvalue problem (Shirley 1965)
with the 2 X 2 Floquet Hamiltonian
Semiclassical many-mode Floquet theory: II
1
E ~ - Z W ~ + ~bq W Z 0
b; E p - 3 ~ + 2 ~ 2 0
0
b;
H,GRWA=
0
0
0
21 13
b2
0
E ~ - w < * w zbq
0
by E " - Z W I + W ~
0
0
6:
0
0
0
0
0
0
b2
0
Eo
0
61
b;
Ep-wq
0
0
- b;
0
0
0
0
0
0
0
0
E,+
W< - W ?
by
0
b;
bz
0
0
bi
0
b2
EO-WZ
0
0
0
E n + 2 w i - 2 ~ 2 bq
0
by
Ept~2w2
therefore only one photon can be absorbed or emitted at a time (conservation of the
total angular momentum). By introducing a second field (rotating in the same direction
as the first one) the corresponding time-independent (although infinite in
dimensionality) Floquet Hamiltonian can be obtained by juxtaposing the 2 X 2 singlefield Floquet Hamiltonians, similar to that of equation ( 5 8 ) , of different w2 blocks
(blocks labelled by n2w2with a common integer n2). These 2 X 2 single-field Floquet
Hamiltonians are coupled to one another via the second field in such a way that the
multiphoton processes under investigation can take place only by absorbing (emitting)
a photon of the one field, then emitting (absorbing) a photon of the other, i.e. the
ladder approximation (Guccione-Gush and Gush 1974). Other types of multiphoton
processes, e.g. the absorption of two photons of one field followed by an emission (or
absorption) of one photon of the other, are not permitted because the total angular
momentum of the system plus fields must again conserve. Moreover, the GRWA Floquet
Hamiltonian, figure 2, has a periodic structure with only the number of ( 0 1 - w 2 ) in
the diagonal elements varying from block to block.
The inclusion of both counter-rotating components of the two linear fields, i.e.
equation (S6), not only accounts for various multiphoton processes, besides GRWA
allowed transitions, but also provides the correct prediction on various non-linear
features, e.g. the resonant shift, the resonant line-width, etc, of the resonance transition.
In the GRWA limits, i.e. l q , - w , l < < wo and I w o - w 2 / < < wo, the perturbation approach
outlined in previous sections can be employed if the coupling strengths of the system
with the fields are extremely small (since we know the frequencies w 1 and w 2 of the
external fields are extremely close to the resonance frequency 0").All formulations
derived previously for the exact case can be equally applied in the GRWA case if one
neglects all anti-rotating terms, i.e. those with the denominator equal to w o + w , or
w o + w 2 . The resonance shift aiGRwA)
and the resonance linewidth u(,GRWA),
for n f 0 ,
can be expressed analytically in very simple forms:
and
in the GRWA limits. Comparisons of equations (59) and (60) with equations (47),
(48), ( S l a ) and (51b) show that the influence of the anti-rotating factors become
important when the detunings of the fields, w o - w l and w o - w 2 , are large, and,
21 14
T-S Ho and S-I Chu
therefore, cannot be ignored in the calculations. The dispersive feature of the long
time averaged transition probability, without averaging over the initial time, pk+pof
equation (54) remains in the GnwA limits, while this is not the case in the conventional
nwA limit. This indicates that the population inversion is still possible even without
anti-rotating components for the former case, but totally impossible for the latter case
(Goreslavskii and Krainov 1979).
5. Numerical results and discussions
5.1. Bichromatic quasi-energies and long time averaged transition probabilities
Shirley (1965) has discussed the significance of the monochromatic quasi-energy plot
in great detail. On the one hand, the quasi-energy levels represent physical energy
levels of the atom-radiation-field system in interaction and may be probed experimentally by exciting transitions between them with an additional weak field. On the
other hand, the quasi-energy plot gives information about the location and strength
of resonance. In this subsection we shall examine the bichromatic quasi-energy plots
along with the long time averaged transition probabilities ph+pand pm+pat various
field strengths b, and b2 for a two-level system. Two different cases will be discussed:
(i) the large detuning case where we have chosen w 1 = 1.0, w 2 = 0.799 and w o varying
from 0.35 to 1.45, and (ii) the extremely small detuning case, to simulate the GRWA
limits / w o - w l l << wo and Iwo- w21<< wo, where we have chosen w 1 = 1.0, w 2 = 0.99799
and wo varying from 0.9935 to 1.0045. Furthermore, we have assumed bl = b2 = b
and +1 = qb2 = 0 for simplicity. Multiphoton processes of (i) one photon absorbed, (ii)
two photons absorbed from one field, a third photon emitted to the other simultaneously, and (iii) three photons of one field absorbed, two photons of the other
emitted simultaneously, will be investigated as we proceed. To achieve numerical
convergences for all multiphoton processes just mentioned, it is sufficient to truncate
the bichromatic Floquet Hamiltonian, figure 1, to contain only 9-04 blocks (i.e.
nl = 0 , i l , 1 2 , i 3 and 1 4 in A of figure l ) , and 9-w2 blocks (i.e. n 2 = 0 , i l , * 2 , i 3
and *4). It is worthwhile to remark that the GRWA calculations, see figure 2, only
depend
on
three
parameters:
b l / ( w l- a 2 ) , b 2 / ( w 1- 4, and
x=
(2wo- w1 - w 2 ) / ( w 1 - w 2 ) (Goreslavskii and Krainov 1979), with the scaling factor
(wl - w 2 )characterising the periodic structure of the GnwA Floquet Hamiltonian, figure
2.
In figures 3, 4 and 5 we show the bichromatic quasi-energies, the time averaged
transition probabilities p&+pand pa+pas a function of x at fixed parameters w l , w 2 ,
bl = b2 = b and +1 = qb2 = 0 (or 7 = b / ( w , - w 2 ) ) . The two parameters x and 7 are
useful for discussing the deviations of the pure GnwA calculations from the exact ones.
Figure 3 ( a ) shows the bichromatic quasi-energy Ayl,:, as a function of x at w1 = 1.0,
w 2 = 0.799 and 7 - l = 8.04-a rather weak field case. Clearly one sees curves approach
each other closely, but they do not cross because of the like symmetry of coupled
Floquet states, at many locations (avoided crossings). Away from the crossings, curves
have well defined slope i (slopes of the unperturbed Floquet energies when considered
as a function of w o ) , and the Floquet states are not coupled to one another. At the
avoiding crossings, the Floquet states are strongly mixed, and resonance transitions
occur. There exists a unique relation, which is best depicted by equation (17), between
the time averaged transition probability po+,(wo), full curve in figure 3( b ) , and the
A,,,l, in figure 3 ( a ) . The curve p o + p ( w o )becomes significant only when the quantity
4
2115
Semiclassical many-mode Floquet theory: II
0
Xpo-I
.
2N
A
'
Aa3-3
-0.2
Xp-IO
ffl
s
A u 2-2
[si
-0.4
.-,
W
v1
0
2
xp-21
Xal-1
-0.6
xp-32
Xqoo
-0.8
0.61 ( b i
X
Figure 3. ( a ) Bichromatic quasi-energy ( A ) in au and ( b ) the time averaged transition
probabilities
(full curve) and Pb,,p (dotted curve) as functions of the dimensionless
at w , = l . O a u and w 2 = 0 . 7 9 9 a u . (c)
resonance parameter x - ( 2 w , - w , - w 2 ) / ( w , - w , )
The time averaged transition probabilities
and ph+p at w 1 = 1.0 au and w 2 =
0.997 99 au for the coupling strength v-' = 8.04. Note that lb,l = / b 2 =
/ b and
= +2 = 0.
+,
( d h / d w o ) starts to deviate from .ti, and reaches a peak value of 0.5 as ( d h / d w o ) is
equal to zero (maximum mixings). Furthermore, the width of each resonance peak is
determined by the minimum separation of curves at its associated avoided crossing.
Therefore, the five-photon peaks, near x = + 5 (in this case, positive sign indicates
three w 1 photons are absorbed, while two w 2 photons are emitted; and vice versa) are
extremely narrow, proportional to q 5 (in the x scale), while the one-photon ones,
near x = i 1, are very broad, proportional to q. Also presented in figure 3 ( b ) is the
time averaged (without averaging over the initial time to) transition probability
pL+p(wo) of equation (15), dotted curve. It shows a tiny dispersive profile around
each of the
peaks, and thus displays the sign of the population inversion, see
equations (18) and (54). In figure 3 ( c ) , the curves of pa+p(wo) and PL.+p(wo) are
plotted at w , = 1.00, w 2 = 0.997 99 and q - l = 8.04. The detunings w o - w 1 and wo- w 2
are very small, varying from 0.004 49 to 0.006 51 compared with wo which ranges
from 0.9935 to 1.0045, and the contribution of the anti-rotating components is thus
negligible in this case, i.e. a case in the GRWA limits. As discussed previously, the
2116
T-S HOsild S-I Chu
results should only depend on two parameters 7 and x (recall that bl = b2 = b
in this case), and, therefore, the curves pm+p(x)and P & + , ( x ) are completely synimetrical about x = 0 in figure 3( c). (We have noticed that the GRWA results at the
conditions of figure 3 ( b ) and the exact Floquet results at the conditions of figure 3(c)
are not distinguishable graphically when expressed in terms of dimensionless parameters
7 and x. The same observations are also found in figures 4 and 5 . ) Away from the
GRWA limits, i.e. figure 3( b ) , the counter-rotating components become important, and
the symmetry manifested in figure 3(c) is distorted: the resonance peaks at x > 0 are
broader, and more shifted from the unperturbed positions (i.e. x = * l , 1 3 and * 5 )
than their counterparts at x < 0. This, in turn, makes the p&+,(x)curve tilt counterclockwise, and the P,+,(x) tilted clockwise, when figure 3(b) is compared with figure
3(c). Approximately, all features encountered in figure 3 can be understood from
equations we derived by using the perturbation theory developed previously. Taking
into account only the rotating components, i.e. figure 3(c), the one-, three- and
five-photon resonances are shifted, respectively, toward x = 0 by amounts approximately equal to 4v2, 6 v 2 and y ~ (in
' the x scale) (see equation (59)), and have
widths, approximately equal to 87, 8v3 and 2v5, respectively (see equation (60)).
With anti-rotatirig components included, i.e. figure 3( b), the resonance shifts are
increased by amounts
GRWA
and the width by about zero, 8v2blW1and
respectively, for the one-, three- and five-photon resonance transitions at x > 0, while
the shifts (are) decreased by
and the widths decreased by about zero, 8v2blu2,and
respectively, at x < O (see equations (47), ( 5 0 ) , (51a) and (51b)). Nevertheless,
resonance peaks in figures 3( b) and 3( c) are well defined and can be fully explained
by the perturbation theory.
The results for v-'= 4.02 (coupling strength doubled), but otherwise the same
parameters as those used in figures 3, are given in figures 4 ( a ) , 4(b) and 4(c). First
we consider figures 4 ( a ) and 4( b), corresponding to o1= 1.0 and w 2 = 0.799. While
the relations between the avoided crossings of the bichromatic quasi-energy curves in
figure 4 ( a ) and the resonances of the transition probability pa+pin figure 4(b) remain,
the two one-photon resonances are less defined, although resolvable (the three- and
five-photon resonances are still well defined), because of the substantial mixing of the
Floquet states laOO), Ip, -1,0) and Ip, 0, -1) at an increasing coupling strength 7,
compared with the case in figure 3. Other apparent changes, besides the power
broadenings and the resonance shifts, in figures 4(b) and 4(c) (calculated at w1 = 1.0
2117
Semiclassical many-mode Floquet theory : II
Xpa-i
X e3-3
Xp-IO
ha?-2
Xp-21
XU1.1
X
Figure 4. Same as figure 3, except
v-'=
4.02.
and o 2= 0.997 99) are: (i) the dispersive feature of P&,,(x), dotted curves, is much
more enhanLed, and (ii) the asymmetries of both P&,,(x) and pm,p(x)
in figure 4(b),
when compared with figure 4(c) (where curves are symmetrical), become more visible.
In this case, it seems to indicate that the perturbation theory, where we have assumed
only two Floquet states were strongly mixed, outlined in § 3, tends to break down for
the one-photon resonance process, but remains a good approximation for the threeand five-photon ones.
Finally, we present the results for ? j - l = 2.01 (the same parameters as before
otherwise) in figures 5. Very strong mixing occurs for the Floquet states IaOO),
ip, -1, O), Ip,O, -l), Icy, -1, l), / a 91, -l), Ip, -2, 1) and ID, 1, -2) in the interval
-3 6 x s +3, and only the five-photon resonances stay nearly isolated at this very large
coupling strength. This is clearly indicated in the bichromatic quasi-energy plot figure
5 ( a ) where the once distinct avoided crossings representing four well separated resonances (two for one-photon and two others for three-photon, e.g. figure 3( b ) ) coalesce
into two broad ones, as revealed by the pm-p(x)curve in either figure 5 ( b ) or figure
5 ( c j . The most striking feature appearing in figures 5 ( b ) and 5 ( c ) is the existence of
a huge hump in p&,,(x) standing upright at the centre, and peaked at a height equal
to 0.7. Although the perturbation procedure is no longer useful in the case of immense
2118
T-S Ho and S-I Chu
0
hpo-:
ho3-3
N
F -0.2
A p-10
4
vi
Ad-1
p -0.4
ha-21
w
c
wI
3
Xn2-2
-0.6
XP-?Z
3
0
A000
-0.8
+
U
I
'
I
'
!
. /...
I
-......I
'
1
'
1
-
0.6-
0 1
-5
-
I
I
-3
I
1
,
-1
I
1
I
I
3
5
x
Figure S. Same as figure 3 except 7-l
= 2.01.
state mixing, the population inversion manifested in the quantity P&+p(
x ) has been
hinted at in equation (18), and can be interpreted as the interference effect arising from
all possible channels on going from Floquet states lam, m2)to Floquet states lpk, k 2 )
with m,, m2, k , and k2 arbitrary integers.
Table 1 displays the resonance frequencies in the x scale, in accordance with figures
3, 4 and 5 , for the one-, three- and five-photon transitions just discussed. Results of
both exact (or numerical), row a, and perturbative, equations (47) and ( 5 0 ) ,row b,
calculations are tabulated. At low field, i.e. 7-l = 8.04, the two calculations are in
very good agreement at all resonances, while at higher fields, q - l = 4.02, or 2.01, they
still agree fairly well with each other for the three- and five-photon cases. The
one-photon resonance becomes less and less defined as the coupling becomes larger:
the two results already show large disagreement at 7-l = 4.02, and, even more, they
do not give any sign at all on the existence of the isolated one-photon resonance at
7 - ' = 2.01. These have all been discussed in figures 3, 4 and 5.
We summarise, in this subsection, that the perturbative treatment of the two-level
bichromatic radiation system has provided a good understanding of all non-linear
features: the power broadening, the resonance shift, the dispersive profile and the
asymmetry, etc. The mystery of the population inversion is unveiled only when one
Semiclassical many-mode Floquet theory: 11
2119
Table 1. Resonance frequencies, measured in the dimensionless resonance parameter
x = (2w0- w1 - w 2 ) / ( w I - w 2 ) scale, for one-, three- and five-photon transitions shown in
figures 3, 4 and 5. Row a: exact calculations, row b: perturbation calculations. The
dimensionless coupling strength parameter q-l is defined in the text. The subscripts in x
indicate the field-free resonance positions.
8.04
4.02
2.01
b
-4.9484
-4.9486
-2.9065
-2.9073
-0.9403
-0.9382
0.9403
0.9380
2.9064
2.9071
4.9482
4.9483
0.799
a
b
-4.9675
-4.9677
-2.9224
-2.9238
-0.9503
-0.9528
0.9204
0.9250
2.8945
2.8953
4.9375
4.9376
0.99799
a
b
-4.7916
-4.7942
-2.6211
-2.6292
-0.5920
-0.7530
0.6020
-0.7520
2.6119
2.6282
4.7906
4.7932
0.799
a
b
-4.8674
-4.8710
-2.6756
-2.6955
-0.6915
-0.8113
0.5622
0.6999
2.5721
2.5812
4.7488
4.7504
0.99799
a
b
-4.1244
-4.1769
-1.5871
-1.5169
-
-
-0.0119
0.0079
1.5970
1.5129
4.1244
4.1730
a
b
-4.4179
-1.7861
-
-4.4838
-1.7819
-0.2451
1.4975
1.3250
3.9751
4.0017
0.99799
0.799
a
-0.2002
considers the whole problem from the viewpoint of the quasi-energy picture, or,
equivalently, the fully quantum mechanical scheme, i.e. the phenomenon manifested
by the interplay between the dressed states (Cohen-Tannoudji and Reynaud 1978).
Moreover, the two quantities P&,,, the long time averaged transition probability only
over the final time t while keeping the initial time to fixed at some instant, and pa+,,
the long time averaged transition probability over both the initial times to and the final
time t, should not be mixed up. The clarification of these two will prevent confusions
often made: e.g. Goreslavskii and Krainov (1979) have compared their calculations
of P&+,, which can be larger than 0.5 as discussed above, with the calculations of
Guccione-Gush and Gush (1974) made on Pa+,, which is always ~ 0 . 5 .
5.2. Time-dependent transition probabilities
In figures 6 and 7 we present the time-dependent transition probabilities P,+,(t), i.e.
equations (14) or ( 5 3 ) , as a function of the elapsed time t (note we have averaged
over the initial times to and replaced the elapsed time t - to by t ) at some specially
chosen one- and three-photon resonance cases (see below). Here we remark that the
time unit to used in figures 6 and 7 is chosen as Tq= 2 ~ / 2 q where
,
the quantity 29 is
defined as the minimum separation of the quasi-energy curves obtained from the exact
(numerical) analysis at each resonance case to be investigated. Furthermore, we have
= + 2 = 0, i.e. cases corresponding to
asumed w, = 1.0, w2 = 0.799, b, = b2 = b, and
figures 3 ( a ) , 3 ( b ) , 4 ( a ) and 4 ( b ) , for all calculations. Figures 6 ( a ) and 6 ( b ) depict
the time-dependent transition probabilities Pa+,( t ) for cases: (i) the one-photon
resonance at wo( = 0.8040) w2, i.e. the absorption of one w2 photon, and (ii) the
three-photon resonance at wo( = 0.6058) 2w2- w l , i.e. the absorption of two w2
photons and emission of one w1 photon at the coupling strength 7,' = 8.04 (figure
3( b ) ) . We have found that, at this field strength, the perturbative analysis almost gives
-
-
2120
T-S Ho and S-I Chu
0
0.2
0.6
0.8
1.0
Figure 6. Time-dependent transition probabilities for the case w 1 = 1.0 au, w 2 = 0.799 au,
b , = b2 = b = 0.025 au (or 7 / - ' = 8.04) and 41= = 0. ( a ) At one-photon resonance,
O J ~=
) 0.8040 au
w 2 and ( b ) at three-photon resonance, w g = 0.6058 au 2w2 - w l . Note
that in this case the analytical results, equation (531, and the exact ones, equation (14),
are indistinguishable graphically. The time unit is defined as Tq = 2 n / 2 q where 2q is the
minimum separation of the quasi-energy curves obtained from the exact calculations at
au.
each corresponding resonance. For ( a ) 2q -0.0492 au and for ( b ) 2q 5.93 X
-
-
the same results as the exact one does-they cannot be distinguished in figures 6 ( a )
and 6 ( b ) . At the lowest order of the perturbative procedure, equations (43) or (46),
the quantity Pa+/3( t ) is sinusoidal at the frequency 2q and the amplitude equal to unity
at the resonance; the predominant correction to it is an oscillation of frequency ( w , - w 2 )
and of amplitude order I b,/ ( w o- w l ) 1' or 1 b2/ ( wo - w 2 ).1' The small wiggly structure
appearing in figure 6 ( a ) can be attributed to the closeness of the two quantities 2q
( = 0.0492) and ( w 1 - w 2 ) = 0.201, while very fast oscillations, forming a heavy band
(wider at the two lower ends). appearing in figure 6 ( b ) is caused by the very small
ratio of 2q (-5.93 X lo-') and ( U , -az)(= 0.201). Physically the sinusoidal profile,
corresponding to the lowest order of the perturbation, is due to the equal mixing
(because of the field couplings) of the two nearly degenerate unperturbed Floquet
states (or true quantum states) at the resonance condition. while the first-order
correction is attributed to the most nearby unperturbed Floquet states, being at a
2121
Semiclassical man y-mode Floquet theory: 11
1.0
I
'
I
'
I
'
I
'
la)
+
c
U
W
2n
0.8
U
I
W
E
b-
0.6
0.4
0.2
0.2
0
0.4
0.6
0.8
1.o
'Y
Figure 7. Time-dependent transition probabilities for the case w , = 1.0 au, w 2 = 0.795 au,
b, = b, = b = 0.05 au (or 7 -'= 4.02) and
= $* = 0. ( U ) At one-phbton resonance, w,, =
0.8300 au w , and ( b ) at three-photon resonance, wo = 0.6306 au 2 w 2 - w l . Full curve:
exact results; dotted curve: analytical results. The time unit Tq is defined in the same way
as in figure 6. For ( a ) 2q = 0.0924 au and for ( b ) 2q = 0.0049 au.
-
-
distance ( w l - w 2 ) away in this case, see figure 3 ( a ) , from these two degenerate states.
The corrections due to other even farther away states are negligible.
In figures 7 (a ) and 7 ( b ) , we present the results for the same processes as in figures
6 ( a ) and 6 ( b ) , but at 7,'=4.02, cf figures 4 ( a ) and 4(b). The deviations of the
perturbative calculations, equation ( 5 3 ) , from the exact (numerical) ones, equation
(14), become evident, although small. At the one-photon resonance (oa= 0.8300),
figure 7 ( a ) ,the corresponding minimum separation 2q of the quasi-energy curve is
about equal to 0.0924, compared with ( w l - w 2 ) = 0.201 for the dominant correction;
the sinusoidal profile is lifted upwards on the sides, but lowered down, from one, at
the centre. At the three-photon resonance (w,=0.6306), figure 7 ( b ) ,2q is equal to
- w 2 ) ; the large
approximately 0.0049, which is about 40 times smaller than (ol
oscillations due to the correction terms are carried through most parts of the curve,
this, in turn, causes the substantial decrease of the lower-order amplitude. 'The
perturbative calculations in this second case, figure 7 (b ) , tend to exaggerate the
higher-order oscillations, and, therefore, further push down the whole profile.
2122
T-S Ho and S-I Chu
6. Multiphoton
NMR
spectroscopy of a spin-; system
The equivalence of a two-level system to a spin-$ system has been well known (Feyman
et al 1957). The formulations developed in p;evious sections can be applied as well
to the case where a spin-; system is placed in a static magnetic field Bo in the 2 direction
and two linearly polarised radiofrequency fields Bl(t) and B 2 ( t )in the x^ direction.
For the spin-; case, we replace wo and p E , ( t ) , i = 1 , 2 , for the two-level system by
yBo and i y B i ( t ) ,respectively. The parameter y designates the gyromagnetic ratio of
the spin-; under discussion and is assumed to be positive. In the following we shall
study the time evolution of the average magnetisation of an ensemble of these systems.
We assume that the ensemble initially is in thermodynamic equilibrium at temperature
p, and the magnetisation vector ( M O )points in the 2 direction. Furthermore, the
relaxation processes are ignored for simplicity. The normalised magnetisation vector
( m ) ( t )of the systems, during the illumination of the radiofrequency (RF) fields, can
be expressed in terms of the density matrix, namely,
-
where g ( t )is the evolution propa4ator, i.e. equation ( l l ) ,
is the spin angular
momentum, 3z the 2 component of S, and the density matrix operator of the systems
at the initial moment to = 0 is
with the partition function 2, = exp($pwo)+ exp( - $ U , ) . The components of ( m ) (t )
in x , y and z direction can be written in simple forms, after substituting equation (62)
in equation (61),
where a and p label the spin-up and spin-down states, respectively. Equations (63)
via (65) can be computed directly by numerically diagonalising the truncated Floquet
Hamiltonian (figure 1). When near the resonance, e.g. wo ( n + l ) w l - no2, and RF
fields weak, ( ~ Z ) ~ ((Zm) ),, ( t ) and ( m ) , ( t ) can be approximated by simple analytical
expressions:
-
Semiclassical many-mode Floquet theory: 11
Rb
+-
WO
+2
+W2
cos(2At-42)
+-
Rb
WO
( A - VLa) sin 2qt
[(A- Vh,)2+IV,b/2]1'2
-0
(
2123
cos(2At + &))
2
sin(2At- &)
Wo+Wl
b
+ e ( n )___
~ 0 ~ [ 2 A t + ( 2 n1)41-2n$~2]
+
w2-
+-
Rb
U 1
COS[ 2At
+ (2n + 2) $1
- (2n - 1)421
WO+%
+-
Rb
wo-wz
+
~ 0 ~ [ 2 A t + ( +2)412n
(2n + 1)44
'
Vobl(A(1- cos2qt)sin[2At + ( n l ) w 1 - n4J
(A - vLa + I vab 1
+
b
+ e ( n )sin(wo-wl-c$l)
W1
WO-
+e(n)
b
~
"O-WZ
sin(2At+ 41)
2124
T-S Ho and S-I Chu
Rb
sin(2AtWO + W 2
+-
Rb
42)+-
WO - W 2
sin(2At+ 42))
(
(A - Via) sin 2qt
cos(2At- &)
[(A- Vha)2+)Vab/2]1’2 W o + W l
-2
+O(n)-----
b
cos(2At+ &)
WO-Wl
Rb
cos(2At - &.)
WO + W 2
+-
+-
Rb
WO - W 2
+ I Va,I2( 1-cos2qt)
( A - vhu)2+IVab12
cos(2At+ 42))
sin[2At + (2n + 3)41- 2n4J
WO+Wl
b
sin[2At+(2n + 1 ) ~- 2$n~4 , ]
+ O ( n )WO-Wl
+-
Rb
+-
Rb
wo+w2
WO -’ W 2
and
sin[2At + (2n + 2)41- (2n - 1)4,]
sin[2At + ( 2 n + 2)4, - (2n + 1)4,]
Semiclassical man y-mode Floquet theory: 11
Rb
+-wo-w2
+-
{cos[(n+ l ) o l t - ( n
Rb
WO
+ W2
- cos[( n
2125
+ 1)w2t+(n+ 1 ) # ~(n~ +- 11421
{cos[(n+ 1 ) q t - ( n - l ) 0 2 t + ( n + 1)&- ( n - 11421
+ 1)41- ( n - 1)s2]}) + O (b2).
(681
Here we remark that equations (66) and (67) were obtained in the rotating frame of
the natural spectra line, i.e. the frame rotating with frequency o0 with respect to the
laboratory frame, while equation (68) is independent of the reference frame of the
observation.
Zur et a1 (1983) have given a detailed study on the dynamics of spin-4 systems in
a double frequency RF field, but within the limit of the GRWA. In figure 8 we show
the results of our exact calculations, along with those of the GRWA, on the magnetisation
0.8
0.4
-*
C
O
h
e -0.4
v
-0.8
0
0.2
0.4
0.8
0.6
1.o
T,
Figure 8. ( a ) 2 and ( b ) 2 components of the magnetisation vector of spin-; systems as a
function of time t at the three-photon resonance, w o 2 w z - w1 with w1 = 1.0 au, w 2 =
0.799 au, b, = b, = b = 0.05 (or t7-I = 4.02) and &J,
= &J2 = 0. The resonance shift 2A and
the minimum separation of the resonant quasi-energy curves 2q are approximately
0.0326 au and 0.0049 au, respectively, for the exact calculations (full curve), while they
are approximately 0.0388 au and 0.0066 au, respectively, for the GRWA calculations
(dotted curves). The time unit Tq is defined in the same way as in figure 7 ( b ) .
-
2126
T-S Ho and S-I Chu
-
vector (m)(
t ) for the three-photon resonance process wo 2w2- w1 with w1 = 1.0 and
w 2 = 0.799 at the coupling strength ~ - l =
4.02 (or b1 = b2 = b = 0.05) and the phases
rbl = rb2 = 0. The resonance frequency, in this case, obtained from the exact analysis
is
= 0.6306, while that from the GRWA is w r = 0.6368. Furthermore, the minimum
separation of the corresponding quasi-energy curves 2q is equal to approximately
0.0049 for the former, and equal to about 0.0066 for the latter. These differences in
the resonance characteristics imply very diverse observations on the dynamics of the
spin-4 systems under study. With the same time unit as that used for figure 7( b ) (the
two are under the same conditions), we depict the time evolution of the z component
of the magnetisation vector ( m ) , ( t )in figure 8 ( a ) . If we neglect the fast oscillations
(of frequency w1 - w 2 ) for a moment, the exact result (full curve) is seen to be oscillating
(with the frequency 2q) slower than the GRWA (dotted curve) because 2qeXact=
0.0049<2q,,,,=0.0066.
In figure 8 ( b ) we show the x component, i.e. the precession, of the magnetisation vector ( m ) , ( t ) in the w o rotating frame. The time
evolution of the ( m ) , ( t ) is defined by the resonance shift 2A, e.g. see equation (66).
In the current case, the shift predicted by the exact and the GRWA analysis are,
ors
T,
Figure 9. ( a ) 2 and ( b ) x^ components of the magnetisation vector of spin-4 systems as a
function of time t at the three-photon resonance, wo- 2 w 2 - w 1 with w 1 = 1.0 au, w 2 =
0.799 au, b, = b, = b = 0.05 au (or 7-l = 4.02) and
=
= 0. The analytic calculations
(dotted curves) have been normalised in such a way that the total normalised magnetisation
vector has the magnitude of unity. The frequency w o is set equal to the exact resonance
frequency, see figure 8, and the time unit Tq is defined in the same way as in figure 7 ( b ) .
The exact calculations are presented as full curves.
# J ~
Semiclassical many-mode Floquet theory: II
2127
respectively, equal to approximately 0.0326 and 0.0388; the periods which define the
precession of the magnetisation vector are, therefore, different between the exact
analysis (full curve) and the GRWA (dotted curve) which overestimates the resonance
shift in this case, i.e. wo 2w2- w1 and w 2 < wl. We note that the latter would have
underestimated the resonance shift, had we chosen the case wo 2w1- w 2 and w1 > w 2 ,
see figures 4(b) and 4 ( c ) . Figures 8 ( a ) and 8 ( b ) have demonstrated the importance
of including the anti-rotating components in the full analysis when the detunings of
external fields are no longer small compared with the natural frequency wo.
In figure 9 we present the analytic results obtained from equations (66) and ( 6 8 ) ,
dotted curves, at the same conditions as those used in figures 8 ( a ) and 8 ( b ) . Also
shown in figure 9 are the exact results (full curves) for easier comparisons. We note
that our analytic calculations have been normalised in such a way that the total
normalised magnetisation vector ( m ) ( t )has the magnitude of unity at any instant of
time t. It is found that our analytic results in general agree pretty well with the exact
predictions except in some really minor details which are not important: e.g. the 2
component ( m ) , ( t ) at each turning point, i.e. at B = 0" or 180", tends to downgrade
the performance of the fast oscillations, but the x^ component follows its exact counterpart almost step-by-step everywhere.
-
-
7. Conclusions
The detailed non-linear responses of two-level systems under the influence of linearly
polarised bichromatic fields are explored by employing the semiclassical many-mode
Floquet theory. The present (numerically) exact approach not only establishes the
validity of the GRWA method in the limits of weak coupling and small detunings but
provides correct treatments of many non-linear features and new multiphoton transitions beyond the GRWA limits. In addition, the controversial problem of population
inversion of two-level systems in bichromatic fields is resolved. Useful analytical
expressions for multiphoton transition probabilities, resonance shifts and widths, and
spin magnetisations, etc, are also obtained by appropriate extension of the almost
degenerate perturbation method of Salwen.
We are currently extending the method to incorporate the effects of relaxation
mechanisms and spin-spin interactions. Results will be published elsewhere.
Acknowledgments
This work was supported in part by the Department of Energy, Division of Chemical
Sciences, by the University of Kansas General Research Fund, and by the Alfred P
Sloan Foundation. The authors are also grateful to the United Telecom Computing
Group for generous support of the CRAY computer time.
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