Deviation from the sech2 superradiant emission law in a two-level atomic system

PHYSICAL REVIE%'A VOLUME 42,NUMBER 5 1SEPTEMBER 1990

Deviation

from

the

sech snperradiant

emission

law in

a

two-level

atomic

system

A.

Edson Goncalves

Departamento deFisica, Fundagdo Universidade Estadual de Londrina, Londrina, 86051 PR, Brazil Salomon

S.

Mizrahi

Departamento deFI'sica, Universidade Federal deSaoCarlos, Via 8'ashington Luiz, Em 235, SaoCarlos, 13560SP,Brazil

B.

M.Pimentel*

Instituto deFl'sica Teorica,

RESP,

Rua Pamplona, 145,SaoPaulo, 01405 SP,Brazil (Received 29 May 1990)

The atomic superradiant emission istreated in the single-particle mean-field approximation. A single-particle Hamiltonian, which represents a dressed two-level atom in a radiation field, can be obtained and it is verified that it describes the transient regime ofthe emission process. While the line-shape emission for a bare atom follows the sech' law, for the dressed atom the line shape devi-ates appreciably from this law and it is verified that the deviation depends crucially on the ratio of

the dynamic frequency shift to the transition frequency. This kind ofdeviation isobserved in exper-imental results.

Since the theoretical prediction

of

the superradiance (SR)phenomenon by Dicke in 1954 (Ref. 1)and the first experimental observation _performed by Skribanowitz et a/. in 1973 (Ref. 2) a lot

of

work has been devoted to the subject, ' in _{the experimental} as in the theoretical areas. The superradiant emission consists essentially in the spontaneous collective emission

of

an electromagnetic (EM) pulse that takes place in a moderately dense system composed by

E

two-level atoms (or molecules). The atoms are prepared in an initial population inversion state which reverts, spontaneously, tothe ground state in

a time inversely proportional to the number

of

atoms,

'T

~

_N

',

such that the emission intensity turns out to be proportional to N

.

The

SR

pulse occurs due to the in-duced correlations that develop among the transition di-pole moments

of

the atoms as they interact with each other, not directly, but through the coherent radiation they emit.

The

SR

phenomenon turns out to be simpler than the process

of

the laser emission because one can ignore the pumping and relaxation mechanism

of

the walls, it is

suScient to consider the evolution

of

the atoms coupled exclusively to their own radiation field. Several ap-proaches have been advanced toward a theoretical description

of

the

SR,

all

of

them arrive either to a mas-ter equation for the atomic system (Schrodinger picture) or to aset

of

coupled first-order differential equations, the Maxwell-Bloch equations (Heisenberg picture). Further-more several approximations are introduced in order to obtain physically transparent analytical expressions. Among these one can cite the Born-Markov approxima-tion, the rotating-wave approximation (RWA), the small sample size assumption (the sample size is smaller than the wavelength

of

the emitted radiation) or the large cy-lindrical pencil-shaped sample model (the linear

dimen-sion

L,

is much larger than both the wavelength A, and

the transverse dimension d satisfying the relation

I.

»A,

»d).

The full Hamiltonian

of

the radiation-matter system is assumed tobe composed by three terms,

H

=Hs+Hz

+

V

with

Hs

=no

_y.

so(i»

(2a)

H„=

_g

co„b„b„,

(2b)

V=

_ga„[b„s

_{(i)+b„s+(i)],}

n,i

(2c)

p

=

i

[A(p),

p]—

2

[n(s s+p

—

_2s+ps

_+ps

_s+

₎

+(n

+

1)(s+s

p

—

2s

ps+

+ps+

s

)],

(3) where _Hs isthe Hamiltonian

of

the emitters (bare atoms),

cooisthe transition frequency

of

the two-level atoms, H~

is the Hamiltonian

of

the radiation field, and V is the matter-radiation interaction considered in the RWA, where

~„stands

forthe radiation-matter coupling param-eter.

Considering that the radiation field acts like a heat bath that surrounds the atomic system, this one evolves irreversibly from an initial to a final state with the evolu-tion governed by a master equation for the reduced sta-tistical operator. In the one-body mean-field approxirna-tion the atomic master equation is

3130

BRIEF

REPORTS 42

where

%(p) =coso+(N

—

1)bco(s

(s+

)

+s+

(s

)

)

+i

(N

—

1)~(s

(s

)

—

s

(s

)

),

(4a)

3000

2500

=Vs,

+(N

—

1)[(2bco(s

)

—

y(s

))s +(2bco(s

)

+y(.

„)

)s,

]

(4b)

is the single-particle mean-field Hamiltonian, which de-scribes a dressed atom, and

( )

stands for mean value. The parameters that enter the master equation (3) are

&-

₂₀₀₀

~

1500

O

l000

6 —

cop+Aco,

where

(5)

500

ln y

277 COp

CO

—

1

Np

(6)

isthe dynamical frequency shift, _y isthe spontaneous de-cay constant, and

_~,

isacutoff frequency, while

0

0.

00

0.08

TlME

0.

15

+2(N

—

1)bco(S

)(S,

),

(7a)

(s,

)

=co(s„)

—

y(n+-,

')(s,

)+(N

—

1)y(s,

)

(s,

)

—

_2(N

—

_1)bee(s„)

_(s,

_),

&s,)

=

—

—,

'y

—

2y(n+

—,

')(s,

)

(7b) l

exp(irido/kz T)

1—

isthe mean photon number.

The master equation permits one to write the time derivative

of

the spin components' mean values,

(,

)

=

—

(

)

—

y(

+-,

')(

„)+(N

—

1)y(

„)

(,

)

FIG.1. The plot isof

2

vs t, for

n=0.

The dashed curve cor-responds to Ace=0 while the solid curves labeled by a, b,and c are for the set of values b,co/co=2X 10

',

4X10

',

and

6X10,

respectively. The time is given in units ofthe decay

constant y while the intensity

2

in units ofAce.

The intensity J" and time t are plotted for several values

of

b,co/co and for two values

of

n. One has assumed

N=100

and attributed values for he@/co such that the condition

(Ace/co)N(1

be valid. ' In Fig. 1,

n=0;

the dashed curve corresponds to

Ace=0;

it isthe case

of

con-sidering the bare atom Hamiltonian

&=cosp'

it preserves the symmetry

of

the sech emission law about to (the de-lay time or time

of

maximuin intensity emission). The

—

_y(N

—

_1)(&

_„)'+&,)'),

(7c)

and one verifies that this system is not integrable because it has not any constant

of

motion, however for the mean-field Hamiltonian there is an invariant, the Bloch vector

(s„)s„+

(s~

)s~+

(s,

)s,

which permits an exact solution

of

Schrodinger equation,

iB,

Q=&(p)f

'As

we h.ave

in-troduced the conjecture that the Hamiltonian (4) represents a dressed atom, then one defines the intensity

of

the radiation field (in units

of

fico)as

3000

2500—

W

2000—

V)

LU

I- 1500—

=

—

+2(n+

—,

')(S,

)

N Cl

1000—

+

1+4

(~+-,

')

—

4

"(s,

)

(&s.

)'+(s,

)')

instead

of

the usual definition which makes use

of

the bare Hamiltonian.

"

Here one has

(

S

)

=

N(

s„),

(S

)

=N(s

),

(S,

)

=N(s,

)

and the set

of

equations (7) were used in order to introduce the dissipative terms in

Eq.

(8).

0.

00

0.08

TlME'

0.

15

42

BRiEF

REPORTS 3131

other curves labeled by

a,

b, and ccorrespond

respective-ly to b,

co/oi=2X10,

4X10,

and

6X10,

and their

main features are the following.

(i) With deviation from the symmetry about the delay time, the slope

of

the curve issmoother on the right than on the left. On the other hand, the inverse behavior occurs ifone considers b,co

(0

(see Figs. 1 and 2

of Ref.

2).

(ii)The peak intensity ishigher the greater the value

of

the ratio hco/to. This behavior occurs because the emis-sion line becomes narrower whereas the area under the curve isconserved.

Identical conclusions are obtained for

n%0

(see

Fig.

2) and for the same set

of

values

of

b,co/to; all the same here, the peak intensity isless, due tothe effects

of

temperature

that enhance the contribution

of

the dissipative terms over the ones responsible for loss

of

coherence, the com-mutator in

Eq.

(3).

In summary, using the dressed atom conjecture Eqs. (4) and (8), the line shapes deviates appreciably from the sech law and this feature is observed in experimental re-sults,

i.e.

, pulse signals, oscilloscope traces,

etc.

So the mean-field Hamiltonian, Eqs. (4a)or (4b), can explain a further aspect

of

the

SR

emission which manifests through the presence

of

the dynamic frequency shift in the emission intensity expression,

Eq.

(8).

S.S.

M. and

B.

M.

P.

received partial financial support from the CNPq, Brazil.

Electronic address: uesp@brfapesp. bitnet.

'R.

H. Dicke, Phys. Rev. 93, 99 (1954).

N.Skribanowitz et al.,Phys. Rev. Lett. 30, 309 (1973).

M.Gross et al.,Phys. Rev. Lett. 36,1035(1976).

4H.M.Gibbs et al.,Phys. Rev. Lett. 39,547(1977).

5M. Gross et al.,Phys. Rev.Lett.40, 1711 (1978). A.Crubellier et al.,Phys. Rev.Lett. 41,1237(1978)~

~M. Gross et al.,Phys. Rev. Lett. 43,343(1979).

8A. V. Andreev et al., Usp. Fiz. Nauk 131,653 (1980) [Sov.

Phys.

—

Usp.23,493 (1980)].

M.Gross and S.Haroche, Phys. Rep.93,301 (1982)~

' _{C.Leonardi et al.}

,Riv. Nuovo Cimento 9,1(1986).

"G.

S.Agarwal, Quantum Statistica! Theories

of

Spontaneous Emission, Vol. 70 of Springer Tracts in Modern Physics (Springer-Verlag, Berlin, 1984), and references therein. ' _{S.S.Mizrahi, Phys. Lett.A 144,282(1990).}