Fluctuation theory for an equilibrium superradiant model

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Fluctuation theory for an equilibrium superradiant model

Maria Trache

To cite this version:

Maria Trache. Fluctuation theory for an equilibrium superradiant model. Journal de Physique I, EDP

Sciences, 1993, 3 (4), pp.957-967. �10.1051/jp1:1993177�. �jpa-00246776�

Classification Physics Abstracts

64.60

Fluctuation theory for

_an

equilibrium superradiant model

Mafia Trache

(*)

Laboratoire Aim£ Cotton, CNRS II, Bit. 505,

Campus d'orsay,

91405

Orsay

Cedex, France

(Received 19 August 1992, accepted in final

form

14 December 1992)

Abstract. Some critical

properties

for static and

dynamic equilibrium superradiant

models _are discussed, within _a Gaussian fluctuation

theory.

The temperature critical

region

and the A-critical

region

_are calculated and the

phase diagram

of the

superradiant phase

transition is obtained.

1. Introduction.

Large

fluctuations of the field variables _are

expected

in the

neighborhood

of _a

phase

transition.

They

influence the behavior of the

thermodynamic quantities

and lead _to deviations of the critical exponents ^{from their} mean field

values,

_as

predicted

for low dimensional models

II, 2].

The Gaussian

Theory

^of Fluctuations

(GTF)

is _an usual

approach

to obtain information

about the critical

region

: the interval of extemal parameters ^{of the} system ^{for which the} Mean Field

Theory (MFT)

fails and the fluctuations cannot be

neglected.

The

Ginsburg

criterion _or similar

analogous

statements _can be

applied

to calculate this critical

region [1-4].

The

superradiant

model

belongs

to a

large

class of models used in

Quantum Optics [5].

Its critical

properties

have been

previously

studied within _a Mean Field

Theory [6-1Ii. Beyond

the

mean field

behavior,

the model

presents

statistical

[12, 13]

and

quantum

fluctuations

[14, 15].

For _an

equilibrium model,

the quantum fluctuations of the field variables _are of thermal

origin.

The Method of Functional

Integration (MFI) [11, 12, 16]

is _a suitable formalism _to include the effect of both

types

of fluctuations.

The aim of this paper is to discuss

comparatively

the

validity

of the MFT for _a static and _a

dynamic superradiant

model and _to estimate the critical

regions. Thus,

in section

2,

the formal

interpretation

of the

superradiant phase

transition is

used,

in order to calculate the

specific heat,

for both

models,

within the Gaussian

approximation [6-10,

13,

15].

This section contains

a necessary review of results

previously

obtained.

In section 3 the

Ginsburg

criterion is

applied,

both for the static and the

dynamic

superradiant

model

[13, 17],

to estimate the T-critical

regions (where

T is the temperature of the

system).

(*) Permanent address Theoretical Physics Department,

Faculty

of

Physics, University

Bucharest,

Bucharest-Magurele,

PO Box MG-5211, Romania.

In section

4,

the formal

application

of the second order

phase

transition

theory

to a

superradiant

model is

replaced

with _{a new}

description supported by

the

experiments

on the

superradiant

effect in cavities

[18]. Thus,

the N-critical

region (where

N is the number of excited atoms in the

cavity)

_or

equivalently,

the A-critical

region (where

A is the

coupling

constant between atoms and

radiation)

_are

calculated,

_{at a} fixed

temperature

^{of the}

cavity.

Section 5 contains conclusions about the

phase diagram

of the

superradiant

model.

2. The

superradiant phase

transition.

Superradiance

is the collective effect of spontaneous emission exhibited

by

_a system ^of N two-level _atoms

prepared

in _a

fully

excited state.

We shall discuss _a

quasipunctiform

model whose characteristic

length

^{is less than} the

wavelength

of the spontaneous ^{emitted field and for which the}

propagation phenomena

_are

neglected.

Within the electric

dipole approximation

and

coupling

the _atoms with _a

single

resonant mode of the

electromagnetic field,

the Hamiltonian is

expressed

in the second

quantization

formalism _{as :}

Jf ~ ~+ _{~ ~} ^~

£ (~+

_~ ~+ _~

j

^l~

£ (~

~+

j(~+

_{~ ~} ~+ _~

j (jj

2

j

~~ ~~ ~~ ~~

fi,

^{~~ ~~ ~~ ~~}

where

a+

(a)

are the bosonic operators of the _resonant mode of energy w,

c),~(c~i,~)

are the fermionic

operators

associated with the

j-th

atom of

energies

±

e/2,

fi~ ^1/2

A

=

Di~

_w

~

^{is the}

^coupling

^parameter,

^Di~

^{is the electric}

^dipole

^matrix

^element,

So

so is the

permittivity

and V is the effective volume which contains N fixed emitters.

The statistical

properties

of the model _are obtained from the

partition

function written _{as a} functional

integral

_over the

complex

bosonic and the Grassmann fermionic variables

[16]

_:

iDa+

^{DaDc+ DC} exp S

Z/Zo

=

(2)

Da+ DaDc+ DC exp

So

The functional S is the _«

quasiclassical

_» action, defined

by

:

S

"

j~

^dT

_(a~ ^{(T ) 0~a(T )}

⁺

^{z cj} (T ) 0~cj~ (T ) H(T )) ⁽³¹

° Jo

where _r is the Matsubara

imaginary

time and

fl =1/(kT),

T is the temperature ^and

k is the Boltzmann constant. In

equation (3),

_a

=

1,

2 and

So

=

S(A

=

0).

Because

explicit

calculations of the effective actions in

MFI,

both for the static and the

dynamic

model _are

already

done

by

_many authors

[11-13, 15], only

_a few steps

leading

to the results will be

pointed

out further.

In order to discuss the

phase

transition

supposed

to occur as an ordered state _over the bosonic microstates associated with the

electromagnetic field,

the guess of the relevant variable for the critical behavior is

The

integration

over all the other variables

II, 12, 16]

leads _{to an} effective _« action _» of the critical model

S~ff

(A (r ))

=

N/2

i~

^dr

^{[w ~A~(r )}

⁺

^(a~A

^{)~ + N}

^Tr~

^In

^[I

^{A ~}

^{G~ AGI}

^A

⁽⁵⁾

o

where

Gi,~(r r')

_are the fermionic Green functions and the trace

Tr~

indicates the

integration

_over all the _«

quasiclassical

_»

trajectories

described

by

the continuous parameter

r

(r

e

[0, fl ).

The Fourier transforms b

(w

~

)

of the bosonic field variable A

(

_r

)

_are introduced

and the trace

Tr~

is estimated up to the second order of the

coupling

_parameter

A

[15].

If the contributions of the static mode

1~

=b(w~=0)

and of the

dynamic

_ones

b(w~

#

0)

_are

separated

and all the correlations between field variables _are

neglected,

the

approximate equation

of the effective action is obtained _:

fi~p ~2 ~2 S~fll'l, (b(W~)))=

~

+2N

Inch(flWd2)

N/2

£

[W^~ + W~ +

(2

A

ill ₎ K(W~)j b(W~) b(- W~)

+

~~~

n#0

where

w~ =

[(e/2

_{)~ +} A ^~

1~

~]l'~ (7)

and

K(w~ )

is calculated _{as a sum over} the fermionic

frequencies.

As such,

equation (6)

is valid for the

dynamic

model. It has the form of _a

Ginsburg

^{Landau functional} in the

dynwnic

modes, while the

perturbation

series in the static mode _are

exactly performed.

The static model

keeps only

the _w~

= 0 _term of the Fourier

expansion

of the field variables

[13]

and

therefore,

the

corresponding equation

for the action is exact

seffl'i )

"

~~ (

^~ _{+ 2 N in Ch}

_{(p WF/2) (8)}

The _mean field

equations

for the

dynamic

model _are

b(w~)=0

_1~ =0 for

T»T~

and

b(w~)

= 0 th

~~°~

=

~° ~°~

for T _<

T~, (9)

solutions which _assure the

stability

condition for the functional S~ff ^{in the}

specified

intervals of temperature. ^{The critical} temperature T~ ^{is defined} as the solution of the

equation

_:

as

usually

obtained in the

theory

of

phase

transitions.

Within a mean field

theory,

the

dynamic

and the static models

give

the _swne results for the

thermodynamic

functions and for their critical

behavior,

as the field variable involved is

only

the static _one.

Thus,

_a

typical

result for _a second order

phase

transition which is obtained is the

jump

of the

specific

heat at T

= T~ :

AC

=Ne~~32k~T(ch~z(thz-j)~ ⁽¹¹⁾

ch _z

fl~

_e

~~~~~ _~

4

The role of fluctuations in low dimensional models is

suggested by

the results of _a Gaussian

theory,

which is further discussed for the

superradiant

model.

For both sides of the critical

temperature,

^{the effective actions}

(Eq. (6)

and

Eq. (8))

_are

expanded

around the _mean field solutions of the field variables up ^to the second order. The effective actions will

get

^{the form of}

Ginsburg-Landau

functionals in all the field variables and the last

integrals

in the

partition

functions _can be

performed.

The

corresponding

ther-

modynamic potential

will

consequently

contain several contributions _:

a mean field _term

showing

_an extensive behavior

(~N

), ^the _same for both models

nonsingular

Gaussian contributions due to the

higher

modes

(w~

#

0),

which become

negligible

in the

thermodynamic

limit and _are present

only

in the

dynamic

model _;

singular

contributions of the lower modes

(w~

_m 0

)

which

proceed

from the

neighbor-

hood of the static _{one ;} this

vicinity

contains both the _w~

= 0 term

(n

=

0)

^{and small} terms w~ =

2 wnkT # 0, obtained from _n #

0,

when T

~ 0. This last contRbution would also be

negligible

in the

thermodynamic

limit if T _were far from the critical temperature ^{but becomes}

large

inside the critical

region,

_even for

large

values of the number of

particles

N.

The

singularities

_are

differently

introduced for the two

models,

in connection with their dimension. If the static critical model has the behavior of _a 0-dimensional

model,

the

dynwnic

model will be _a I-dimensional _one, due to the

supplementary

dimension introduced

by

the Matsubara

imaginary

time

[14, 15].

For both

models,

the

singular

contribution in the

specific

heat

becomes, respectively

_:

C)°~

₌ ^k/2

T([T- _{T~[~~ (12)}

as in reference

[13]

and

~~ 16fi~~~~~ ^~~

^{~~~ ~~~~}

as in reference

[15].

Therefore,

the critical exponent ^{of the}

specific

heat

changes

from

a[°~

= 2 in the static

model,

to _a[~~ = 3/2 in the

dynamic model,

both Gaussian results

satisfying

the

Josephson

rule

12j

_:

vd=2-a

(14)

where _v

=

1/2 is the critical exponent ^{of the} correlation

length.

3.

Ginsburg

criterion and the

temperature

critical

region.

The critical

region

is the interval of the parameter values where the MFT is _no

longer

valid. It

can be evaluated

by

the

Ginsburg

criterion

[1, 3]

_: if the

temperature

is chosen _{as a} control

parameter,

^the temperature ^critical

region

is

expressed

_as :

(AT)cr

₌

tT

_Tc

The

parameter (~

can be calculated _as the ratio between the

singular

term

(introduced by

fluctuations)

in the

specific

heat

C~

and the

jump

of the _same

quantity

AC at T

= T~. ^The

equation

for

C~

will include

only

the term of maximum

singularity

of the

specific heat,

_as

already

considered in

equation (12)

and

equation (13).

The _same critical exponents are

obtained for both sides of the critical

point.

For _a static

model,

the

Ginsburg

criterion states :

~~°~

~

~~~

~

AC T T~

/T~ (15)

and for _a

dynamic

model _:

c ^(I)

,jl)

3/2

j

=

(16)

~

IT Tc(/Tc

We intend _to

identify

the parwneter

(~

from the direct calculation of the

specific heat, already

done. In reference

[13], only

_a maximum value of the critical

region

for _a static model

was calculated and _some conclusions about the role of fluctuations _were drawn. In this paper, a

systematic

evaluation of fluctuations in the second order of the

perturbation theory

and

comparative

results for both models _are

expected.

Using equations (11, 12)

and

(13),

_one obtains

~~~

~

~ [~ ^{~ ~ 3/~}

~

x

(1

x~

)-1 ~ ^x~

~ ~ i~~

~

~~

~

(~~~

where

fl~

_e

No

x=th-=-.

(18)

We have defined

No

as a critical number of atoms whose minimum value _assures the

occurrence of the

phase

transition at the minimum

possible

temperature, T~ = 0 K _:

so eV

No

=

2

D(~

For the

dynwnic model,

the _same

quantity (~

is

given by

^the

equation

The parameters

(~ depend

not

only

on the adimensional

quantity

_x, but

explicitly

on the number of

particles

N.

If the

equation

of the critical

temperature expressed

with the _same parameter x is used _: T~ =

$ (In

^{~ ~} ,

(20)

2 k _x

the temperature ^critical

regions

_are obtained _:

~~~ ^~ i + ~ 5/2

~ i

~2

_~

~ l/2

~~~~~~

k

fi

~~ x1

^{~~ ~ ~} ₂ ^~~

x

~~~~

Approximate dependences

on the number of

particles

N are obtained for extreme values of x,

namely

:

in the static model

(AT

))$~ elk

#

for _{x ~}

0,

and

(AT

_))$~ elk

~

for _{x ~}

~~~~

N

No

and

in the

dynwnic

model _:

~~~~~~

~~~~~$)/

~~~ ~°~ _{~ ~}

~'

~"~

(24) (AT)))

elk ~

for _x

N

No

~

While the critical

temperature

_{T~ is}

only

_a function of _x, which is

already

a result of the

MET,

the temperature ^critical

region depends explicitly

_on the number of

particles

and _on the dimension of the model. This conclusion is

particularly important

^for finite systems, ^which have to show

large

fluctuations _{over a} wide

temperature

^{interval around the critical}

point.

The

mean field results for different

physical quantities

_{are no}

longer

dominant, _as soon as

(AT)

_<

(AT

_)~~.

For

macroscopic systems (N

_{~ oo}

), excepting

extreme values of _x, the critical

region

tends to zero and the MET is valid.

If _one compares the size of the

temperature

^critical

regions

with the critical temperature

itself,

one sees that in the

neighborhood

of _x

= I (T~ ~

0,

_or N

~

No)

both models show

large

critical

regions.

The fluctuations however _seem to decrease for the

dynamic model,

because

~lkT)jj~ fill/6

(AT )ll~ iln (I

_X)1~~~ ^{~~ ~ ~~~~}

Otherwise,

for _x

~ 0

(N

»

No,

_T~

~ oo

),

the critical

region

remains

large

for the static model, but is

strongly

^diminished for the

dynamic

_one. The ratio _:

(AT

_)~$~

N5/6

~~

= j ^»

(26)

(AT))~ No

The increase of the dimension of the model leads to the reduction of the effect of

fluctuations, excepting

the range of small critical temperatures, ^{when the number of}

particles

N tends also _to its minimum value

No. Figure1

shows both the

dependence

of the critical

temperature

^{and of the} temperature critical

regions

_{versus parameter x.}

Similar results

conceming

the

equations

of the temperature ^critical

regions

are obtained

by comparing

the _mean field value of the order

parameter (the

slowest field variable 1~) with its

corresponding dispersions [4],

calculated for each

model,

in the

neighborhood

of

the critical

temperature.

^The

dispersion

of the order

parameter being expressed

with the second order derivative of the effective action

(

(A~J

)~l

=

~~) ⁽²⁷⁾

~

a) I

1.

3

0.4 0.6 0.8

b) o 5

o.

o

0.

4

0.2 0.4 0.6 0.8 1

1.

Cl 1

0.

0.92 0.94 0.96 0.98

Fig.

I. a) The critical temperature kT~/s (curve I). The static critical

region

k(AT ))$~/s (curve 2). ^The

dynamic

critical region

k(AT)))~le

(curve 3) _{as a} function of the variable _x

=

No/N,

for N

=

2 _; b) Static critical

region

for N

=

2 (curve I) for N

= 200 (curve 2) Dynamic critical

region

for N

=

2 (curve 3) for N ₌ 200 (curve 4) c) A detail of figure 16 and of the critical temperature curve (the

crossing

curve) in the

neighborhood

of _x

=

1.

it shows

large singularities

_on both sides of the critical

point.

As _{soon as} the

dispersion

exceeds the _mean value of the order parameter, ^the temperature ^critical

region

is reached

(Fig. 2).

~

i(A~j2j

~

A

T~ T] T

Fig. 2. The temperature dependence of the order parameter ~ and of its dispersion

((A~ )~)

^for

different values of the

coupling

constant A.

41 The A -critical

region.

The theoretical

question

on the

validity

of the MFT and the role of fluctuations is

naturally

correlated with the

experimental

observation of the

phase

transition.

Only

this _one is able to

suggest ^how the critical behavior is reached and which _are the natural parameters to check.

The second order

phase transitions, typically

met in condensed matter

physics,

_use the temperature as a natural control parameter. ^{The behavior of} response

functions,

_as the

specific

heat _or

susceptibilities,

_announces the

instability

and the _occurrence of _a

phase

transition.

The _occurrence of the

superradiant

effect

[18, 19],

_as well _as that of other collective effects

studied in

Quantum Optics,

is

quite

different. A

macroscopic superradiant

state is maintained in _a

cavity

_{at a} fixed

temperature.

^{The ordered} state

(which

is observed _{as an}

oscillatory

dynamic

behavior of the spontaneous ^emitted

field)

is reached

only

if _a

sufficiently large

number of emitters _are

present

^{in the}

cavity

of _a

given

volume. This

implies

_{a necessary}

large coupling

constant A, between atoms and the emitted field.

Therefore,

_{a more} natural

«extemal» parameter seems to be the number of _atoms

N,

_or the

coupling

parwneter

A. This leads _{to an}

equivalent description

of the

superradiant phase

transition. For the purpose of this paper it is sufficient to formulate the

theory

in the A

representation,

that

is,

_{to express} both the order parameter ^{of the transition} and the critical

region using

A _{as a} control parameter.

For a

given

temperature ^T

(=

_T~~~), if A

~ A

~

the

expansion

of the nontrivial mean field

equation (Eq. (9))

around _1~

=

0 is obtained _:

~

3/2

p

^1/2

'~

I

A

/'~

~~ ~~

~~~

4

ch~(I

_em ^{~~ ~~~~~~ ~~~~}

The critical

coupling

is obtained from

equation 9,

for _1~

=

0 _:

~

2

thi~fl

em ~~~~

and has _a minimum value

Af~~

₌

(ew~/2)~'~,

_{if T~~~}

= 0 K. The critical number of

particles No

^introduced

by equation (18) corresponds

to the minimum critical value of the

coupling.

For finite

temperatures

^{of the}

cavity,

the critical

coupling increases,

_so that the condition

imposed

on tile number of

particles assuring

the _occurrence of the

phase

transition is _more restrictive.

The

dispersion

of the order parameter can also be

identified, using

the

Ginsburg-Landau expansion

of the effective action around the _mean field solution. It is easier _to write the second derivative

coming

from the disordered state (A _< A

~)

^the

dispersion

of the order

parameter

will show _a

singular

behavior when A _~

~.

°~~~~~ N

fl

_w ²

°'l~ n=o~ _A2 ^(A/-A~)

^and

⁽³⁰⁾

((A~I)~)[~~~

=

~c

~ ~

~~~°

^~ ~~

(31)

In order to obtain the A-critical

region,

_we

impose

the condition of the

validity

of

MFT,

that is _:

~J~» i(A~J

_)~>

(32)

One obtains _:

Al _p

~ ii~

~~ ~

/jw2 _e3/2

~~ ~~ ~"~

4

ch2(p

_em ^~~~~

and

identify

the A -critical

region,

_as the r.h.s. of the

inequality (33).

With the parameter z =

PEW already introduced,

_one obtains _:

A

~ z 1/2

~~~ ~"

4

$ ^{~~ ~~}

^~~~

ch~

z

~~~~

(the

connection between A~ and _z is

given by Eq. (29)).

In

figure

3 the curves of the order parameter ^{for different} temperatures ^{T and the} fluctuations in the

neighborhood

of the critical

coupling

value are shown. The

family

of _curves is limited

by

1(A~)2)

<

~

T

~~~ A

~

A

Fig. ^{3. The}

A-dependence

of the order parameter ~ and of its

dispersion (

(A~

)~),

for different values of the

cavity

temperature.

JOURNAL DE PHYSIQUE T 3, N'4, APRJL lW3 33

that of T~~~ =

0,

which starts at

Af~~.

^The curves

showing

the

A-dependence

of the order

parameter

are

going

to

infinity

and there is _a

region

of the parameter ^A _<

Af*

where _a

nontrivial solution _{can never} exist.

The A-critical

region (Eq. (34))

still

depends

_on the

temperature

^{of the}

cavity

and _on the number of

particles N,

_as the T-critical

region

_was

dependent

on the

coupling

constant A and the number of

particles

N.

Equations (10)

and

(29)

^which are

similar, represent equations

of _an

equilibrium

curve.

Using

_any of the two, a direct

equation

between the variations of the

temperature

T and

coupling

constant A could be

obtained,

_{as :}

~ ² AA

= ~ ~ ~

(AT(. (35)

4 kT _w ch

(fl

em

It

gives

the connection between the _two critical

regions.

The

equation (35)

has the form of _a

Clapeyron-Clausius equation.

5. Conclusions.

The above discussion

suggests

two

equivalent descriptions

of the

superradiant phase

transition.

The

temperature

parwneter ^T or the number of

particles

^N

(connected

with the A

coupling parwneter)

_are

possible

control

parameters

^{of the critical behavior. The}

experimental

conditions lead _to the choice of the _more natural

description

and _we

emphasize that,

for the

superradiant

effect in

cavities,

the

A-description

is _more suitable. A

N-description

^is

generally

convenient for systems ^with a variable and finite number of

particles.

The

phase diagrwn

of the

superradiant phase

transition is viewed in

figure

4 the

equilibrium

curve A

=

A

(T)

is

separating

the ordered and distordered _states. The _curve is extended _to

infinity,

but has _a critical end

point

_at

_Af~~

=

(w

^~ e/2 )~~ ^and _{Tm~~ =} 0. The infinite extension of

the _curve is _an argument ^for a second order

phase transition,

_as the

jump

of the

specific

heat

and the

continuity

of entropy ^{have also been} indicated. The

plane

of

parameters

^{is thus}

separated

and the transition between the ordered and disordered states is

possible only crossing

the

equilibrium

_curve.

A

O d

Fig.

4, The A-T

phase diagram

for the

superradiart phase

^transition : lo) ordered states ; id) disordered states.

AcknowledgJnent.

The

hospitality

of the

laboratory

^{AiJn£ Cotton}

(Orsay) during

^the final

stage

^{of this work is}

wamly acknowledged.

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