Dicke superradiance on Landau levels
It is shown that in the inverted system of nonrelativistic electrons in rarefied magnetized plasma, when electron density on high Landau levels exceed some critical value defined by its transversal energy, magnetic field and temperature, the nonequilibrium phase transition occures with domain orderi...
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2001
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| Cite this: | Dicke superradiance on Landau levels / P.I. Fomin, A.P. Fomina // Вопросы атомной науки и техники. — 2001. — № 6. — С. 45-48. — Бібліогр.: 8 назв. — англ. |
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| citation_txt | Dicke superradiance on Landau levels / P.I. Fomin, A.P. Fomina // Вопросы атомной науки и техники. — 2001. — № 6. — С. 45-48. — Бібліогр.: 8 назв. — англ. |
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| description | It is shown that in the inverted system of nonrelativistic electrons in rarefied magnetized plasma, when electron density on high Landau levels exceed some critical value defined by its transversal energy, magnetic field and temperature, the nonequilibrium phase transition occures with domain ordering of mutual orientations of interacting rotated dipoles. The intensity of cyclotron radiation of each domain in ordered phase becomes proportional to square of electron number in it.
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Q U A N T U M F I E L D T H E O R Y
DICKE SUPERRADIANCE ON LANDAU LEVELS
P.I. Fomin, A.P. Fomina
Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
e-mail: pfomin@bitp.kiev.ua
It is shown that in the inverted system of nonrelativistic electrons in rarefied magnetized plasma, when electron
density on high Landau levels exceed some critical value defined by its transversal energy, magnetic field and tem-
perature, the nonequilibrium phase transition occures with domain ordering of mutual orientations of interacting ro-
tated dipoles. The intensity of cyclotron radiation of each domain in ordered phase becomes proportional to square
of electron number in it.
PACS: 41.60.-m; 42.50.Fx
1. INTRODUCTION
The phenomenon of superradiance (SR) was consid-
ered first in the famous work by Dicke [1] on the exam-
ple of two-level model. Now rather significant literature
(see for example, reviews [2,3,4]) is devoted to it, but,
as it is marked by many authors, a theme is far from be-
ing exhausted, many interesting and physically impor-
tant questions and situations remain not investigated.
In the present work the question of possbility and
conditions of SR formation in the inverted system of
electrons on high Landau levels (1) with 1> >n are in-
vestigated
,...2,1,0, ==⊥ nnE Hω (1)
mceHH /=ω . (2)
As it is known, for generation of the induced coher-
ent radiation in systems like masers the equidistancy of
levels is an obstacle because of specific competition of
radiation and absorption processes in this case. In a case
of SR we deal not with induced, but with spontaneous
radiation and here, as we shall see, equidistancy appears
of advantage. This is, first, because the SR regime is
usually realized in open finite systems without mirrors
when radiation leaves active volume of generation
quickly enough, practically having no time to get in ab-
sorption regime [3]. Second, in this case all inverted
electrons, as a rule, occupying not one level, but some
significant interval of high levels )1( > >∆> >∆ nnn , be-
cause of equidistancy, radiate the same mode on fre-
quency (2), and also, as we shall see, in the same rate,
do not depend on initial energy.
The phenomenon of SR arises when in "coherence
domains", with the sizes 0R smaller than a wave length
λ all 0N radiating dipoles gradually, during radiation,
are aligned in one direction by a dipole - dipole interac-
tion between them in "a near zone" λ< <0R , - so that in
result the total dipole of the domain D
appears in 0N
times larger than an elementary dipole d
. Therefore the
intensity of collective dipole radiation becomes proportional
to 2
0N - as opposite to 0N in the case of radiation of uncor-
related dipoles.
Transition in such a correlated polarized state is sim-
ilar to phase transition in magnetics or ferroelectrics,
and for its description it is convenient to use the Weiss
method of mean self-consistent field [6]. Let's note, that
the considered phase transition is nonequilibrium and
has all features of the self-organizing phenomenon in
dissipative systems.
2. CYCLOTRON SUPERRADIATION ON
LANDAU LEVELS
Levels (1) at high n correspond to quantum states
which wave functions are located near the classical Lar-
mor orbits with radiuses HL Vr ω/⊥= .
In this connection we can proceed to (quasi-) classi-
cal description of such states and transitions between
them. In a classical limit in сoordinate system, in which
longitudinal movement is excluded, these orbits are giv-
en as
}0),sin(),{cos()( αωαω ++=⊥ ttrtr L
, (3)
where in braces the cartesian components of the vector
⊥r
are written out. In other words, quantum states are set
by narrow enough wave packages at "Landau orbits"
which represent the certain superpositions of the Landau
wave functions [5].
Being inverted at the beginning on high levels, elec-
trons start to fall downwards on stear steps (1), radiating
quants with frequency (2). Differential (on corners) and
integral intensities of dipole radiation of one electron in
a classical limit are described by the known formulas [7]
kkncndddI kk /,4/][/ 3
≡×=Ω π , (4)
⊥
⊥ == E
mc
e
c
Ve
I HH
3
22
3
222
3
4
3
2 ωω
, (5)
From (5) the following evolution the law of electron
energy follows
22
3
4
3,
)(
He
mcE
I
dt
tdE
ω
τ
τ
=−=−= ⊥⊥ , (6)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 45-48. 45
)/exp()0()( τtEtE −= ⊥⊥ . (7)
We see that time of radiation τ does not depend on
⊥E , i.e. electrons with various (within the limits of dis-
persion )0()0( ⊥⊥ < <∆ EE ) initial energies will fall
downwards at the same rate. It is easy to see, that at the
same rate the dispersion of energy will decrease also
)/exp()0()( τtEtE −∆=∆ ⊥⊥ , (8)
so that the relation is true
1)(/)( < <=∆ ⊥⊥ ConsttEtE . (9)
This allows judging about evolution of the whole
collective of electrons by evolution of their mean energy
and other mean quantities. Therefore, for simplicity, we
shall understand further under the symbols
LerdVE =⊥⊥ 0,, and etc., their average values over en-
semble of inverted electrons.
It is possible to divide the total volumeV , occupied
by N inverted electrons on subvolumes cohV , or "coher-
ence areas", which sizes 0R are smaller in comparison
with radiated wave length λ but are greater in compari-
son with radiuses of Larmor orbits, describing the sizes
of elementary dipoles
λ< << < 0RrL . (10)
Thus, a big enough number of 0N radiating dipoles
will be in volume cohV
10 > >=> >= cohee VnNNVn (11)
Let us consider a total dipole of such subsystem
∑∑
=
⊥
=
==
00
11
)()()(
N
j
j
N
j
j tretdtD
. (12)
Similarly to (4) for one electron, collective dipole radia-
tion of our subsystem will be described by the formula
[ ]
π
ω
π 4
)(
4
][ 22
3
4
3
2 Ω⋅−=Ω×
= dnDD
c
d
c
nD
dI k
Hk
. (13)
By substituting here (12) and (3) and averaging over
the period HT ωπ /2= , we have
+> =< ⊥
03
222
[N
c
VedI Hω
π
αα
42
1)]cos(
2)1( 00 Ω+−+ ∑
−
≠
dnz
NN
ji
ji . (14)
The multiplier 2/)1( 2
zn+ reflects the anisotropy
properties of dipole radiation of nonrelativistic elec-
trons. The second member in square brackets describes
the correlation effects connected with mutual aligning of
dipoles. If correlations are not present, i.e. contributions
of all )cos( ji αα − are mutually compensated and in the
sum give zero, then in (14) works the first member only
that corresponds to the total radiation of 0N indepen-
dent elementary dipoles.
In the case of full correlation, when all
1)cos( =− ji αα , the formula (14) gives
π
ω
42
1 2
3
222
2
0
Ω+=>< ⊥ dn
c
VeNdI zH
corr , (15)
i.e. in comparison with radiation of 0N noncorrelative
dipoles the intensity grows in 0N times. This is just SR
[1].
Radiation time of such correlated dipoles radiating
coherently will decrease in 0N times in comparison
with time (6)
0/ Ncoh ττ = (16)
Really, it is possible to expect only partial positive
correlation of phases, i.e. partial aligning of all dipoles,
when the average over ensemble value of phase differ-
ence cos is positive. Having replaced in (14) all
)cos( ji αα − by their average value, we receive
+> =< ⊥
03
222
2
0 [N
c
VeNdI Hω
π
α
42
1]cos)1(
2
00
Ω+><−+ dnNN z (17)
In this case the intensity of coherent radiation is pro-
portional to >< αcos2
0N .
We proceed now to consideration of the mechanism
of spontaneous aligning of the dipoles giving rise to SR
regime.
3. POLARIZATION PHASE TRANSITION IN
THE “COHERENCE DOMAINS”
To solve the problem of the phase transition we ap-
ply here the Weiss method of self-consistent mean field
[6] approved in the theory of spontaneous magnetiza-
tion. Let us consider the potential energy of trial dipole
),( 00 trd
with the electric field ),( 0 trE
, created at the
point 0r
by the other )1( 0 −N dipoles
),(),()( 0000 trEtrddU
⋅−= , (18)
where
∑
−
−
−⋅
=
1
3
0
0
0 )())((3
),(
N
j j
jjjj
rr
tdtdnn
trE
,
jjj rrrrn
−−= 00 /)( . (19)
All dipoles rotate under the law (3) and radiate and
their electric field is not static, but it also rotates with
frequency Hω . Therefore, the use of expression (19) for
a field )(tE
demands explanation. The matter is that the
conditions (10), determining the coherence volume,
mean that various dipoles of the domain are in the so-
called “near zone” )( 0 λ< <R relativly to one another,
where the main member in decomposition of retarded
potentials and fields over degrees of small parameters
)/( 0RrL and )/( λLr appears just in expression (19)
(see on this subject, for example, [7]).
Averaging (18) over the rotation period, we notice,
that points 0r
and jr
characterize not instant positions
of rotating electrons, but positions of the motionless ro-
tation centres and consequently do not depend on time.
Having substituted (19) to (18) and averaging over the
period, we receive
46
)cos(
)(31
2
)( 0
1
1
3
0
22
0
0
0
j
N
j j
jz
rr
nd
rU αα −
−
−
−> =< ∑
−
=
, (20)
where jjjzL rrzznerd
−−== 000 /)(, . It is possible
to replace the sum (20) approximately by the integral
over coordinates in limits of "coherence volume" with
an obvious measure jedVn , representing an average
number of dipoles in the element of volume
)( jj rddV
≡ in a vicinity of a point jr
. But before writ-
ing out this integral, we shall notice, that because the po-
sition of our trial dipole should not be allocated, it is
necessary to average (20) over this parameter, i.e. to en-
ter additional integration cohVrd /)( 0
. Besides, likewise
an average field method, we replace )cos( 0 jαα − in
(20) by its value averaged over ensemble. After all these
averagings we get the expression
3
0
2
0
2
0 )(31)(
cos
2
j
jz
j
coh
e
rr
n
rd
V
rdnd
U
−
−
><−> =< ∫ ∫α . (21)
By the replacement of variables
)}(2/1,{},{ 000 jjj rrRrrrrr
+=−=→ and integrat-
ing over dR, we get
)(3cos
2 5
222
0 rd
r
zrnd
U
cohV
e −><−> =< ∫α . (22)
Aligning of dipoles over directions is energetically
favourable by virtue of increasing, thus, of the negative
contribution to potential energy < U >. Therefore, the
correlations necessary for that will occur only in that
part of coherence volume where the area of integration
over relative coordinates satisfies the condition
03 22 >− zr , (23)
i.e. in the area like a flattened out circular cylinder. We
will name this part of coherence area a "coherence do-
main" or otherwise a "domain of self-polarization". In
the similar next domain the direction of an average vec-
tor of polarization should be close to opposite to mini-
mize in system as a whole the positive energy of polar-
ization electric field. It is known that by virtue of the
similar reason the macroscopical volumes of magnetics
and ferroelectrics are broken into domains also. Do-
mains are divided by transition regions ("domain walls")
within which limits the turn of polarization vector occur.
We return now to an estimation of integral in (22)
with restriction (23). It is convenient, obviously, to cal-
culate it in cylindrical coordinates, in which
023, 2222222 >−=−+= zzrzr ρρ . (24)
We limit the integration over ρ by ),( 21 ρρ , so it is
easy to show that
><
−> =< α
ρ
ρπ cosln
33
2 2
0
2
1
2
endU . (25)
We will consider now a question about the minimal
and maximal limits ),( 21 ρρ over relative coordinate
2
21
2
21 )()( yyxx −+−=ρ in a plane between two
dipoles in the coherence domain. We remind that the
characteristic sizes of initial "coherence volume" were
determined by conditions (11): λ< << < 0RrL . The in-
equality (24) means that in relative coordinates "the co-
herence domain" has the form of flattened out circular
cylinder with radius 02R . Hence, the maximal value of
ρ is 02 2R=ρ , the minimal value should be taken
about double Larmor radius Lr2~1ρ , because at small-
er distances between centres of dipoles the interaction
between pair of electrons does not carry dipole character
any more and it is impossible to use dipole formulas.
Thus, we can write )/ln()/ln( 012 LrR≈ρρ . As the re-
lation LrR /0 enters under a mark of the logarithm the
result is poorly sensitive to exact value of this relation.
Thus, taking into account a condition λ< <0R we can
substitute here 0R by the quantity of the order of 10/λ .
Taking into account, that HL Vr ω/⊥= and cH πλ ω 2= ,
we can write
≈
≈
⊥⊥ E
mc
V
H
5
ln
10
lnln
222
1
2 λ ω
ρ
ρ
. (26)
With the account of (26) the expression (25) takes the
following form
><
−> =<
⊥
απ cos
5
ln
33
2 2
0
22
end
E
mcU . (27)
To find the >< αcos , we address now to Weiss
method [6]. For this purpose at the beginning it is neces-
sary to consider the response of our system of rotating
dipoles in plane (x,y)
}0),sin(),{cos(0 jjj ttdd αωαω ++=
(28)
to the external homogeneous electric field
}0),sin(),{cos( 00 αωαω ++= ttEE ee
(29)
rotating synchronously with dipoles. The potential ener-
gy of a dipole in this field, averaged on the period of ro-
tation, obviously, is
)cos()()( 00 αα −−=⋅−= jeej EdtEtdU
, (30)
thus it is seen, that the aligning of dipoles along the field
with radiation of released energy is energetically
favourable. The thermal fluctuations resist this tendency.
They are realized in rarefied magnetized plasma mainly
in the form of plasma fluctuations and Alfven waves.
The distribution over the phase differences is given by
Bolzman formula
=
−= αααρ cosexp)(exp)( 0
kT
Ed
C
kT
UC e (31)
with normalizing factor
=
= ∫−
kT
Ed
I
kT
Ed
dC ee 0
0
0
2
0
1 2cosexp αα
π
, (32)
where )(0 xI is a modified Bessel function of a zero or-
der [8]. Average value of >< αcos is determined by
the integral
)(
)(
)(ln)(coscos
0
1
0
2
0
xI
xIxI
dx
dd ==> =< ∫ ααραα
π
, (33)
where
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
47
kTEdx e /0= (34)
So, the external field induces the nonzero correlator
of a phase difference (33), i.e., otherwise, polarizes the
system. A polarization measure is an average dipole mo-
ment of unit volume
><= αcos0dnP e . (35)
Polarization generates an additional internal electric
field
PE p ⋅= ν , (36)
where ν is some dimensionless parameter which we
shall define later. The field pE , in turn, strengthens the
polarization. This feedback effect will be taken into ac-
count, if in (34) we replace eE by the sum
PEEE epe ν+=+ that corresponds to ideology of
self-consistent mean field by Weiss. As a result, we re-
ceive the nonlinear equation for polarization P
+
> =<=
kT
PEd
FdndnP e
ee
)(
cos 0
00
ν
α , (37)
)(/)()( 01 xIxIxF = . (38)
Excluding an external field, we receive "the equation
of the self- consistency"
⋅= P
kT
d
FdnP e
ν0
0 . (39)
We consider now a condition of existence of its non-
trivial solution. Having entered a variable
kT
Pd
z
ν0= ,
we rewrite the equation (39) in the following form
)(
2
0 zF
kT
nd
z e ⋅
⋅
=
ν . (40)
Function F (z) has asymptics [8]
)1(...,
8
1
2
11)( 2 > >−−−= z
zz
zF (41)
)1(,...
8
1
2
)( < <
+−= zzzzF . (42)
From (40) and (41) it follows that at large z the solution
exists and corresponds to the maximal polarization
1
2
0 > >
⋅
=
kT
nd
z eν , (43)
00
0max d
kTz
d
kTdnP e νν
> >⋅== . (44)
To determine threshold value of electron density,
higher of which there is a nontrivial solution of the
equation (39), it is necessary to consider the asymptotic
(42). Being limited by the first member, we receive from
(40)
)1(,
2
2
0 < <⋅
⋅
= zz
kT
nd
z eν . (45)
It follows from here, that the density en has a critical
(threshold) value, if it satisfies the condition
1
2
2
0 =
⋅
c
e
kT
ndν
, (46)
at ece nn ≥ a nontrivial domain self-polarization ap-
pears.
To define the factor ν , it is necessary to compare
the expression for potential energy following from (35),
(36) and (30)
><−> =< αν cos2
0dnU e , (47)
with the expression (27) received earlier. Demanding
equality between them, we receive
)5/(,
5
ln
33
2 2
2
mcE
E
mc < <
= ⊥
⊥
πν . (48)
It is useful to express 2
0d by energy ⊥E and mag-
netic field H
2
2
22
0
2
)(
H
Emcerd L
⊥== . (49)
As a result, the criterion of occurrence of domain
self-polarization of the inverted electron system on high
Landau levels, leading to SR, takes the form
1
5
ln
33
2
2
22
≥
⊥
⊥ kT
En
H
mc
E
mc eπ
. (50)
ACKNOWLEDGMENT
The authors would like to thank V.N. Mal`nev for
helpful discussion.
REFERENCES
1. T.L. Dicke R.H. Coherence in Spontaneous Radia-
tion Processes // Phys. Rev. 1954, v. 93, № 1-2, p. 99-
110.
2. V.V. Zheleznyakov, V.V. Kocharovskiy, and
Vl.V. Kocharovskiy. Polarization waves and super-
radiance in active media // Uspekhi Fizicheskikh
Nauk. 1989, v. 159, № 2, p. 193-260 (in Russian).
3. A.V. Andreev. Optical superradiation: new ideas and
new experiments // Uspekhi Fizicheskikh Nauk.
1990, v. 160, № 12, p. 1-46 (in Russian).
4. L.I. Men`shikov. Superradiance and related phenom-
ena // Uspekhi Fizicheskikh Nauk. 1999, v. 169,
№ 2, p. 113-154 (in Russian).
5. L.D. Landau, E.M. Lifshitz. Quantum mechanics,
M.: “Nauka”, 1974, 752 p.
6. J.S. Smart. Effective field theoris of magnetism.
1966, W. Saunders Company, Philadelphia-London.
7. L.D. Landau, E.M. Lifshitz. Field theory. M.: ”Nau-
ka”, 1973, 504 p.
8. Handbook of Mathematical Functions, Ed-
s.M. Abramovitz and I. Stegun, Nat. Bureau of
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циям, 1979, M.: “Наука”).
48
P.I. Fomin, A.P. Fomina
Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
|
| id | nasplib_isofts_kiev_ua-123456789-79421 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:34:18Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Fomin, P.I. Fomina, A.P. 2015-04-01T19:20:29Z 2015-04-01T19:20:29Z 2001 Dicke superradiance on Landau levels / P.I. Fomin, A.P. Fomina // Вопросы атомной науки и техники. — 2001. — № 6. — С. 45-48. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 41.60.-m; 42.50.Fx https://nasplib.isofts.kiev.ua/handle/123456789/79421 It is shown that in the inverted system of nonrelativistic electrons in rarefied magnetized plasma, when electron density on high Landau levels exceed some critical value defined by its transversal energy, magnetic field and temperature, the nonequilibrium phase transition occures with domain ordering of mutual orientations of interacting rotated dipoles. The intensity of cyclotron radiation of each domain in ordered phase becomes proportional to square of electron number in it. The authors would like to thank V.N. Mal`nev for helpful discussion. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Quantum field theory Dicke superradiance on Landau levels Сверхизлучение Дикке на уровнях Ландау Article published earlier |
| spellingShingle | Dicke superradiance on Landau levels Fomin, P.I. Fomina, A.P. Quantum field theory |
| title | Dicke superradiance on Landau levels |
| title_alt | Сверхизлучение Дикке на уровнях Ландау |
| title_full | Dicke superradiance on Landau levels |
| title_fullStr | Dicke superradiance on Landau levels |
| title_full_unstemmed | Dicke superradiance on Landau levels |
| title_short | Dicke superradiance on Landau levels |
| title_sort | dicke superradiance on landau levels |
| topic | Quantum field theory |
| topic_facet | Quantum field theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79421 |
| work_keys_str_mv | AT fominpi dickesuperradianceonlandaulevels AT fominaap dickesuperradianceonlandaulevels AT fominpi sverhizlučeniedikkenaurovnâhlandau AT fominaap sverhizlučeniedikkenaurovnâhlandau |