Lasing without inversion in three-level systems: Self-pulsing in the cascade schemes
J. Mompart,
*
C. Peters, and R. Corbala
´
n
Departament de Fı
´
sica, Universitat Auto
`
noma de Barcelona, E-08193 Bellaterra, Spain
~Received 28 January 1997; revised manuscript received 27 October 1997!
Lasing without inversion ~LWI! in specific models of closed three-level systems is analyzed in terms of
nonlinear dynamics. From a linear stability analysis of the trivial nonlasing solution of the homogeneously
broadened systems with on-resonance driving and laser fields, we find that, near lasing threshold, resonant
closed L and V schemes yield continuous-wave LWI while resonant cascade schemes can give rise to self-
pulsing LWI. The origin of this different behavior is discussed. For parameters of a real cascade system in
atomic
138
Ba we check numerically that the self-pulsing solution is stable in a broad range of nonzero
detunings. It is shown that the self-pulsing emission can still be observed when the typical residual Doppler
broadening of an atomic beam is taken into account. @S1050-2947~98!06803-6#
PACS number~s!: 42.50.Gy, 32.80.Qk, 42.65.Vh
I. INTRODUCTION
In recent years, lasing ~LWI! and amplification ~AWI!
without inversion have been discussed in many of the pos-
sible three- and four-level schemes. Recently, Sanchez-
Morcillo et al. applied the techniques of nonlinear dynamics
to the analysis of LWI considering explicitly the generation
of the coherent laser field in a cavity @1#. They found that
LWI based on a three-level system with a continuous-wave
~cw! driving field can, in principle, show self-pulsing emis-
sion. Nevertheless, with their very general treatment of the
necessarily involved incoherent processes it was impossible
to predict the behavior of the different three-level systems of
Fig. 1 as a function of the parameters which can be con-
trolled experimentally.
We adopt the theoretical treatment of @1#, but introduce
the incoherent processes in the usual explicit manner, which
allows us to demonstrate a fundamental difference between
the closed three-level systems shown in Fig. 1 with on-
resonance driving and laser fields: near lasing threshold, L
and V systems generate cw laser light, whereas the cascade
systems can lead to self-pulsing emission. The origin of this
different behavior of the resonantly driven folded and cas-
cade schemes is to be traced back to the fact, shown by
Mandel and Kocharovskaya @2#, that, with the incoherent
pumping and relaxation schemes assumed here ~see Fig. 1!,
AWI arises either at line center in the closed folded schemes,
or at two symmetrical sidebands placed outside the absorb-
ing Rabi sidebands in the closed cascade schemes @2#.We
show that the simultaneous amplification of these two sym-
metrical sidebands gives rise to a periodically modulated la-
ser output at the probe transition frequency. It is well known
@3# that lasers in which two fields can be simultaneously
amplified can exhibit also asymmetric emission regimes for
which the two generated fields have different intensities,
even in the fully symmetric case. We focus here on the self-
pulsing LWI regime and have identified numerically a do-
main of parameters for a real cascade system in atomic
138
Ba
where the self-pulsing solution is stable. Including the Dop-
pler broadening for the cascade system in the barium atom,
we find that the self-pulsing emission should be observable
in an atomic beam experiment.
After submission of this paper we became aware of a
systematic study of nonlinear dynamics in homogeneously
broadened single mode three-level lasers without inversion,
where the general case with cavity and driving field detun-
ings is considered @4#. The analysis of the resonant case in
that study confirms our main conclusion that, with incoherent
relaxation and pumping processes as in Fig. 1, only the cas-
cade schemes can exhibit stable self-pulsing LWI near lasing
threshold.
II. THREE-LEVEL MODELS
We consider the four closed three-level schemes shown in
Fig. 1. In all schemes, a coherent driving field E
b
with Rabi
frequency 2
b
interacts with a transition labeled 3-2 and pre-
pares the atoms in order to generate a laser field E
a
with
Rabi frequency 2
a
in a ring laser cavity. In addition, the
upper level of the lasing transition is populated by an inco-
*
Fax number: ~34!-3-581 2155. Electronic address:
ifop0@cc.uab.es
FIG. 1. The considered level schemes: ~a! cascade scheme with
the driving field in the lower transition, ~b! cascade scheme with the
driving field in the upper transition, ~c! V scheme, ~d!Lscheme.
PHYSICAL REVIEW A MARCH 1998VOLUME 57, NUMBER 3
57
1050-2947/98/57~3!/2163~6!/$15.00 2163 © 1998 The American Physical Society
herent pump process interacting with this transition and rep-
resented by a rate L.
From the nonlinear dynamics point of view, a lasing so-
lution corresponds to the destabilization of the trivial solu-
tion of the Maxwell-Schro
¨
dinger equations with the electric
field amplitude
a
equal to zero. Thus we calculate this solu-
tion and perform a linear stability analysis.
A. Cascade schemes
In the following, we will discuss explicitly the cascade
configuration of Fig. 1~a! with both coherent fields and the
laser cavity on resonance with the corresponding transition.
In the framework of the semiclassical theory and using the
standard density matrix formalism with the rotating wave
and slowly varying envelope approximation, the Maxwell-
Schro
¨
dinger equations of this scheme can be written as
r
˙
11
52
g
12
r
11
1 L
~
r
22
2
r
11
!
1 i
@
a
r
12
*
2 c.c.
#
, ~1a!
r
˙
22
52
g
23
r
22
1
g
12
r
11
1 L
~
r
11
2
r
22
!
1 i
@
b
r
23
*
1
a
*
r
12
2 c.c.
#
, ~1b!
r
˙
33
5
g
23
r
22
2 i
@
b
r
23
*
2 c.c.
#
, ~1c!
r
˙
12
52G
12
r
12
1 i
@
a
~
r
22
2
r
11
!
2
b
*
r
13
#
, ~1d!
r
˙
23
52G
23
r
23
1 i
@
b
~
r
33
2
r
22
!
1
a
*
r
13
#
, ~1e!
r
˙
13
52G
13
r
13
1 i
@
a
r
23
2
b
r
12
#
, ~1f!
a
˙
52
ka
1ig
r
12
, ~1g!
with
r
11
1
r
22
1
r
33
5 1.
k
designates the damping rate of the
lasing field E
a
due to cavity losses and g5
pn
a
N
m
a
2
/(\«
0
)
the unsaturated gain of the lasing transition.
n
a
is the corre-
sponding transition frequency,
m
a
the dipole matrix element,
N the density of atoms, \ Planck’s constant, and «
0
the
dielectric permittivity. The decay rates
g
12
and
g
23
describe
phenomenologically the spontaneous relaxation of driving
and lasing transitions. Depletion of the driving field is ne-
glected. In the radiative limit, the decay rates of the coher-
ences are given by
G
12
5
1
2
~
g
12
1
g
23
1 2L
!
, ~2a!
G
23
5
1
2
~
g
23
1 L
!
, ~2b!
G
13
5
1
2
~
g
12
1 L
!
. ~2c!
In resonance, it is possible to take
a
5
a
*
,
b
5
b
*
and the
coherences can be expressed as
r
12
[iy
12
,
r
23
[iy
23
, and
r
13
[x
13
with the real variables y
12
, y
23
, and x
13
@5#.Inour
notation,
a
y
12
. 0(,0) and
b
y
23
. 0(,0) lead to absorp-
tion ~amplification! of the corresponding field.
Taking
a
5 0 ~and thus
r
12
5
r
13
5 0) and all time deriva-
tives in Eqs. ~1! equal to zero, the nonlasing solution is
r
11
5
4L
b
2
A
, ~3a!
r
22
5
4
~
g
12
1 L
!
b
2
A
, ~3b!
r
33
5
~
g
12
1 L
!
@
g
23
~
g
23
1 L
!
1 4
b
2
#
A
, ~3c!
y
23
5
2
g
23
~
g
12
1 L
!
b
A
, ~3d!
with
A5
g
23
~
g
12
1 L
!
~
g
23
1 L
!
1 4
~
2
g
12
1 3L
!
b
2
. ~3e!
Clearly, neither of both transitions is inverted, and, since
b
y
23
. 0, the driving field is absorbed.
The stability of the nonlasing solution is governed by a
73 7 matrix which splits into two independent submatrices.
One of them governs the stability of the variables
a
, y
12
,
and x
13
and therefore the generation of the lasing field. The
characteristic polynomial of this submatrix is
l
3
1 c
1
l
2
1 c
2
l1 c
3
5 0, ~4!
with the coefficients
c
1
5 G
12
1 G
13
1
k
, ~5a!
c
2
5
k
~
G
12
1 G
13
!
1 G
12
G
13
1
b
2
1 gn
21
, ~5b!
c
3
5
k
~
G
12
G
13
1
b
2
!
1 g
~
G
13
n
21
1
b
y
23
!
, ~5c!
where n
21
[
r
22
2
r
11
. We apply the Hurwitz criteria for de-
termining the instabilities associated with the above polyno-
mial: c
1
, c
2
, c
3
. 0 and H
2
[c
1
c
2
2 c
3
. 0 signify negative
real parts of all eigenvalues which means stability of the
nonlasing solution. The destabilization of the trivial solution
occurs through a pitchfork bifurcation ~static instability! if
c
3
, 0 or, alternatively, through a Hopf bifurcation ~self-
pulsing instability! if H
2
, 0. In this case,
A
c
2
gives the an-
gular pulsation frequency of E
a
at the destabilization point.
Here, H
2
is
H
2
5
~
G
12
1 G
13
!
@
k
~
k
1 G
12
1 G
13
!
1 G
12
G
13
1
b
2
#
1 g
@~
G
12
1
k
!
n
21
2
b
y
23
#
. ~5d!
As n
21
. 0, the only term that can contribute to the destabi-
lization of the nonlasing solution is
b
y
23
. It follows from
Eq. ~3! that the driving field is absorbed (
b
y
23
. 0) and,
consequently, the destabilization of the nonlasing solution
can occur only via a Hopf bifurcation which gives rise to
self-pulsing laser emission. For
b
5 0 as well as for very
large values of
b
2
, H
2
is positive and the nonlasing solution
is stable. Varying the driving field intensity
b
2
, there are
consequently two ways to obtain the destabilization of the
nonlasing solution: increasing
b
2
starting with a very small
value or, alternatively, decreasing
b
2
from a very large
value. It should be emphasized that a direct calculation of the
probe field amplification without cavity as it has been carried
2164 57J. MOMPART, C. PETERS, AND R. CORBALA
´
N
out in @6–8# leads to steady-state probe field absorption when
probe and driving fields are taken on resonance. This dem-
onstrates the difference between AWI and LWI in the cas-
cade schemes.
Substituting the nonlasing stationary solution ~3! into Eq.
~5d!, it is easily seen that the inequality
g
23
. 2
g
12
~6!
is a necessary condition for LWI. This means that the spon-
taneous decay rate of the driving field transition has to be at
least twice that of the lasing transition. The threshold values
of all parameters can be obtained analytically from condition
H
2
, 0. For example, the threshold value of the incoherent
pump rate L is given by
L.
g
12
g
12
1 2
k
g
23
2 2
g
12
1 o
S
1
g
D
. ~7!
The threshold values for all schemes of Fig. 1 will be dis-
cussed elsewhere.
Figure 2 shows different curves H
2
5 0 as a function of
the parameters
b
and L for different values of the gain pa-
rameter g @9#. For a given gain, LWI is obtained within a
closed curve in the
b
-L plane (H
2
, 0). Outside this curve
the nonlasing solution is stable. The cross marks the values
of
b
and L which are used in Figs. 3 and 4. We choose a L
value just above threshold since the realization of an efficient
incoherent pump represents the main difficulty in AWI and
LWI experiments.
Figures 3~a! and 3~b! represent the results of a numerical
integration of Eqs. ~1! using a seventh- to eighth-order
Runge-Kutta-Fehlberg routine. The parameters are
b
5 10
MHz, L52 MHz, and g515 000 MHz
2
with
g
12
,
g
23
, and
k
as in Fig. 2. After a transient which is shown in the insets,
the laser field amplitude E
a
oscillates symmetrically around
zero with an angular frequency 2
p
(8.75)5 55 MHz, while
the populations oscillate with very small amplitudes, and nei-
ther of the transitions is inverted. A numerical study of AWI
for the parameters of Fig. 3 shows that the maxima in am-
plification of the additional sidebands appear for probe field
detunings D
a
5655 MHz. This shows that the self-pulsing
emission at line center is due to the simultaneous amplifica-
tion of these two additional sidebands. Since we have con-
sidered thus far a resonant laser field, Eqs. ~1! do not admit
solutions with time-independent intensity, corresponding to
the amplification of only one of the ~detuned! sidebands.
FIG. 2. LWI regions in the plane of parameters
b
and L for
various values of the gain parameter g. The values of g are given in
MHz
2
. The other parameters are
g
12
5 3.5,
g
23
5 19, and
k
5 0.5 ~all
in MHz!.
FIG. 3. ~a! Evolution of the laser field amplitude, ~b! corre-
sponding evolution of the atomic populations. The parameters are
given in the text.
57
2165LASING WITHOUT INVERSION IN THREE-LEVEL . . .
Therefore these equations do not allow us to investigate
whether the self-pulsing state emerging from the Hopf bifur-
cation is stable or unstable. This issue is studied analytically
in Ref. @4#. Instead, we have checked numerically that, for
the conditions of Fig. 1~a!, this self-pulsing state is stable.
For this purpose we considered a full set of equations analo-
gous to Eqs. ~1!, but including both cavity and driving field
detunings D
a
and D
b
, respectively @see Eq. ~14! below#.We
studied first the laser behavior as a function of D
a
, with
D
b
5 0.1 MHz and other parameters as in Fig. 3. For various
cavity detunings up to D
a
.15 MHz we obtained stable self-
pulsing emission, with both the maximum generated field
intensity and the modulation depth decreasing with increas-
ing cavity detuning. In all cases the intensity pulsing fre-
quency was, as in resonance, 110 MHz. For D
a
. 15 MHz
we obtained cw emission. A similar behavior was observed
when the cavity detuning was kept fixed at D
a
5 0.1 MHz
and the driving field detuning D
b
was varied, except that we
obtained self-pulsing emission up to D
b
.0.7 MHz and a cw
regime for D
b
. 0.7 MHz. The system is therefore more sen-
sitive to driving field detuning, which breaks the symmetry
between the two amplifying sidebands, than to cavity detun-
ing @4#. We have checked numerically that it is possible to
increase the cavity and driving detuning domain for which
the self-pulsing regime is stable by increasing the unsatur-
ated gain parameter or, alternatively, by decreasing the cav-
ity losses.
For the cascade scheme of Fig. 1~b! the results are similar.
Again, the only way to destabilize the nonlasing solution is
through a Hopf bifurcation giving rise to self-pulsing LWI
emission with the necessary condition
g
23
. 2
g
31
. In the
same way as for the other cascade scheme, we have checked
numerically that there is also a broad domain of parameters
where the self-pulsing solution is stable.
It is well known that a conventional incoherently pumped
laser without driving field can show self-pulsing and even
chaotic emission if the gain of the lasing transition and the
cavity losses are sufficiently large @10#. This dynamical be-
havior corresponds to a destabilization of a cw lasing solu-
tion while the destabilization of the nonlasing solution oc-
curs always through a pitchfork bifurcation leading primarily
to cw output. In contrast, the self-pulsing output of the cas-
cade schemes is obtained directly through a destabilization
of the nonlasing solution. Furthermore, the self-pulsing ap-
pears even without cavity losses.
It should be mentioned that in Ref. @11# cw laser emission
has been observed experimentally in a closed cascade sys-
tem. However, a comparison with our results is not possible
since in this experiment the incoherent pump process was
substituted by a second coherent driving field with the same
frequency as the laser field.
B. Folded schemes
For the folded schemes the procedure is analogous, and
the destabilization of the trivial solution is again governed by
a333 matrix. For the V-type system of Fig. 1~c!, the coef-
ficients of the characteristic polynomial are
c
1
5
k
1 G
12
1 G
13
, ~8a!
c
2
5
k
~
G
12
1 G
13
!
1 G
12
G
13
1
b
2
1 gn
31
, ~8b!
c
3
5
k
~
G
12
G
13
1
b
2
!
1 g
~
G
12
n
31
2
b
y
23
!
, ~8c!
H
2
5
~
G
12
1 G
13
!
@
k
~
k
1 G
12
1 G
13
!
1 G
12
G
13
1
b
2
#
1 g
@~
G
13
1
k
!
n
31
1
b
y
23
#
, ~8d!
with n
31
5
r
33
2
r
11
, and for the L-type system of Fig. 1~d!
c
1
5
k
1 G
12
1 G
13
, ~9a!
c
2
5
k
~
G
12
1 G
13
!
1 G
12
G
13
1
b
2
1 gn
12
, ~9b!
c
3
5
k
~
G
12
G
13
1
b
2
!
1 g
~
G
12
n
12
2
b
y
23
!
, ~9c!
H
2
5
~
G
12
1 G
13
!
@
k
~
k
1 G
12
1 G
13
!
1 G
12
G
13
1
b
2
#
1 g
@~
G
12
1
k
!
n
12
1
b
y
23
#
, ~9d!
with n
12
5
r
11
2
r
22
. As expected, in the limit
b
→0 the sta-
bility condition is fulfilled. From the Maxwell-Schro
¨
dinger
equations it is easy to verify that in both cases the lasing
transition is not inverted, i.e., n
31
. 0 for the V-type system
and n
12
. 0 for the L-type system. Furthermore, the driving
field is again absorbed, i.e.,
b
y
23
. 0. Consequently, the only
way to destabilize the nonlasing solution is now via a pitch-
fork bifurcation which gives rise to continuous wave LWI.
This is consistent with the result @2# that closed folded
schemes driven in resonance show AWI at line center. Nec-
essary conditions for c
3
, 0 are now
g
23
.
g
13
~
V scheme
!
and
g
23
.
g
12
~
L scheme
!
.
~10!
FIG. 4. Maximal laser field strength
A
e
2
1 f
2
of the self-pulsing
after the transient and pulsation frequency
v
of this quantity as a
function of the Doppler broadening D
v
D
a
. The parameters are the
same as in Fig. 3.
2166 57
J. MOMPART, C. PETERS, AND R. CORBALA
´
N
In the high gain limit, this bifurcation occurs in the V-type
system for
b
y
23
G
12
. n
31
. ~11!
The usual discussion of continuous-wave amplification
without inversion ~AWI! in the V scheme leads to the am-
plification condition @12#
2 y
13
. 2
a
n
13
1
b
x
12
g
13
1 2L
, ~12!
which involves directly the real part of the two-photon co-
herence x
12
, and AWI is explained as due to the contribution
of this coherence. In contrast, the lasing condition ~11! for
LWI does not involve x
12
. At the destabilization point of the
nonlasing solution we have x
12
'0. Consequently, following
Eq. ~12! one can think naı
¨
vely that, at least at the destabili-
zation point, population inversion is required to achieve am-
plification. It is important to remark that at the bifurcation
point Eq. ~12! does not hold and should be substituted by Eq.
~11!, which has been obtained through a nonlinear dynamical
analysis that allows us to characterize the bifurcation. On the
other hand, in the limit G
12
→`, where the two-photon co-
herence could not be generated, Eq. ~11! indicates that
LWI is not possible. This shows that this coherence is also
essential for LWI.
In order to realize frequency up conversion with the
closed three-level schemes, the frequency of the lasing tran-
sition must be larger than the frequency of the driving tran-
sition 3-2. Due to the scaling of the spontaneous emission
probability with the cube of the transition frequency, it is
difficult to find three-level systems in real atoms which fulfill
the conditions ~6! or ~10! and permit at the same time fre-
quency up conversion. A possibility could be to use dipole-
forbidden laser transitions ~i.e., metastable levels!.
III. DOPPLER BROADENING
In order to investigate if the self-pulsing emission of the
cascade schemes will be observable experimentally, it has to
be taken into account that experiments are made with free
atoms. Atomic motion leads unavoidably to Doppler broad-
ening and can change significantly the atom-light interaction.
Typical values of the Doppler broadening at optical frequen-
cies are 1 GHz for a vapor cell, a few MHz for a collimated
atomic beam, and a few hundred kHz for an atomic trap. The
influence of the Doppler broadening on cw LWI has been
discussed in @13# for L and V schemes and in @8# for the
cascade schemes.
The Doppler effect leads to a velocity-dependent shift
D
a
,
b
of the cavity and driving field frequencies with respect
to the atomic transition frequencies:
v
a
,
b
(
v
z
)5
v
a
,
b
(
v
z
5 0)1D
a
,
b
(
v
z
)5
v
a
,
b
(
v
z
50)(16
v
z
/c).
v
z
designates the
atom velocity component parallel to the direction of the
light, c the velocity of the light. For the atomic velocities, we
assume a Maxwell-Boltzmann distribution
r
~v
z
!
5
1
v
zp
A
p
exp
~
2
v
z
2
/
v
zp
2
!
, ~13!
with the most probable velocity
v
zp
. The Doppler broaden-
ings ~full width at half maximum! are then given by
D
v
D
a
,
b
[2D
a
,
b
(
v
zp
)ln(2) with D
v
D
b
5 (l
a
/l
b
)D
v
D
a
.
The laser field E
a
is generated by all atoms. If the cavity
is resonant with the frequency
v
a
(
v
z
5 0), E
a
will also be in
resonance. To simulate the generation of E
a
, we consider n
v
velocity classes, each of them with the corresponding transi-
tion frequencies
v
a
(
v
z
) and
v
b
(
v
z
) and with a set of equa-
tions such as Eqs. ~1a!–~1f!, but with the equations of the
coherences being now
r
˙
12
j
52
~
G
12
2 iD
a
j
!
r
12
j
1 i
@
a
~
r
22
j
2
r
11
j
!
2
b
*
r
13
j
#
,
r
˙
23
j
52
~
G
23
2 iD
b
j
!
r
23
j
1 i
@
b
~
r
33
j
2
r
22
j
!
1
a
*
r
13
j
#
, ~14!
r
˙
13
j
52
~
G
13
2 i
@
D
a
j
1 D
b
j
#
!
r
13
j
1 i
@
a
r
23
j
2
b
r
12
j
#
.
The index j numbers the atomic velocity classes. The differ-
ential equation for E
a
, Eq. ~1g!, becomes
a
˙
52
ka
1ig
(
v
z
r
12
~v
z
!
. ~15!
Since
a
is now a complex number,
a
[e1 if, the total num-
ber of real equations becomes 9n
v
1 2. The Maxwell-
Boltzmann distribution is included in the initial conditions of
the atomic populations. In the calculations presented in Fig.
4, we have used n
v
5 40. Different initial values for e and f
yielded the same final results. It is assured that further in-
creasing of n
v
does not change the results.
For the simulation of the Doppler broadening, we choose
the parameters of a real cascade system in atomic
138
Ba
which has already been used experimentally by Sellin et al.
@11#. The wavelengths of the atomic transitions are l
a
5 821 nm and l
b
5 554 nm, the corresponding decay rates
g
12
5 3.5 MHz and
g
23
5 19 MHz. Thus the system works
with frequency down conversion, nevertheless it allows the
observation of the self-pulsing emission. Figure 4 shows the
results of our calculations with a Runge-Kutta-Fehlberg rou-
tine of order 7 to 8, the other parameters have the same
values as in Fig. 3. We represent the maxima of the laser
field strength
u
a
u
5
A
e
2
1 f
2
within the cavity and the self-
pulsing frequency
v
of the laser field intensity as a function
of the Doppler broadening D
v
D
a
. As has already been dis-
cussed for AWI, counterpropagating coherent fields are more
favorable for the cascade systems while copropagating fields
are more favorable for L and V systems @13,8#. With coun-
terpropagating fields, laser output can be observed up to
D
v
D
a
5 20
p
MHz which is much more than the typical
value for a collimated atomic beam. Thus the self-pulsing
should be observable in laser-cooled or atomic beam experi-
ments, whereas the use of gas or vapor cells would hinder
laser oscillations.
IV. CONCLUSIONS
We have analyzed the inversionless generation of a laser
field in a resonant cavity, the active medium consisting of
specific models of closed three-level atoms in different con-
figurations driven in resonance with a continuous wave.
While the closed L and V systems lead to cw laser emission
57 2167LASING WITHOUT INVERSION IN THREE-LEVEL . . .
