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The dicke maser model
Volume 48A, number 6
PHYSICS LETTERS
29 July 1974
THE DICKE MASER MODEL G. VERTOGEN and A.S. De VRIES Institute for Theoretical Physics, University of Groningen, the Netherlands Received 10 October 1973 A simple method is presented to calculate the thermodynamic properties of the Dicke maser modeL
Recently Hepp and Lieb [1] calculated the thermodynamic properties of the Dicke maser model [2] a rigorous and very sophisticated way. In the conclusion they remark that their findings are typical of socalled van der Waals limits in systems with a longrange interaction. There is one important difference, however, their “mean field” is entirely off-diagonal. It is the purpose of this letter to present a calculation of the model, leading to the same results, based on the “Bogoliubov trick” [3], which is nothing but a mean field approach for bosons. Owing to difficulties inherent to a boson system (i.a. the unboundedness of the creation and annihilation operators) we cannot prove that “Bogoliubov’s trick” is allowed, although the analogy with the fermion case is striking, The Hamiltonian of the Dicke maser model reads N
HN
=
a~a+
E n1
N
E (as; +a~S~),A
5Z
~‘
‘~
N
Z~s~,
(2a)
N
[HN,a] =
—
a
——-a— E s
~
(2b)
,
=
—
~ (2e)
eS~+—~aSk .
Consider lmlN..+ (l /IV)~=1S~hmN~(1 /A~1S and lIrnN (1IN)Y~[~1S~. These operators are spaceaverages. If our a!gebra is a C.A.R.-algebra, which is the case without bosons, and assuming the state to be translationally invariant, which is quite reasonable from the physical point of view, we know the limits to exist and to belong to the center of the induced representation [4]. In this case we have a tensor product of a boson algebra and a C.A.R.-algebra. We conjecture the theorem to be also valid in this situation. Besides we assume the state to be such that the spaceaverages are c-numbers denoted by 5~,5+and S Consider a/,./~and a+/,.JY~The matrix elements of these operators are clearly non-diagonal, because they connect states with N0 and N0 1 photons. We have (N0 1 Ia/~WI N0) = ~jAç7Neiatand ~ —
0. (1)
The units have been chosen such that the photon mode has unit energy. Using the well-known commutation ielations for bosom of andmotion spin operators we obtain the following equations [HN, a~’]=a’+—~—
2X [HN, Ski
(2c)
—
N 00 such that N0/N = n0~where n0 denotes the density of the photons, one2eia, getslikewise ‘~~~N’°~ (n0 IaIv~NIn0> = n0no1/h/2e—1a. In this limit 11mN_~=(nOIa~/\~~InO)= the operators lin ~a/~~and limN~a”/~possess finite diagonal elements. They incommute withWe each other and with all other operators the algebra. therefore replace these operators b~c-numbers. This is known as the “Bogoliubov trick” [5]. As far as the spin operators are concerned time translation is induced by o°and
-~
2ei~ES+AnohI2e_~ES~, fl (3) In the thermodynamic limit, provided the following “gap-equations” hold [6]
H
5 =e ES~+Xn0l/
EHN, S] eS~—:
~
(2d)
451
Volume 48A, number 6
PHYSICS LETTERS
~—1 ,~0l/2e_ia + AS~=0,
[HN, ~j N-+~oL
(4a)
=
29 July 1974
N T~=limb~T—~-tanh~flC N-’
(10 urn 1HN,—~-—1=
n 1/~e~ —AS 0
—
N-~~~L
=
0,
(4b)
eSz_2A2((Sx)2 +(SY)2)
= =
aSz +,3SX +7SY
c
~
lirn
The solutions of the “gap-equations” are: N
EHN,b~S~=
—
An
2einS~
(1)
0 l/
0=O;
(4 c)
1/2e4’~S=O, +An0
(2)
S~—--~-—, 2 A2
~
~rn[HN~
sx =Sy=o, i.e. n
N
~s;]
=
eS~ 2Xn
2einSz
—
=
0 lt
0, (4d)
this implies ~ A2 in the case of spinS = Solution (1):
~-
operators.
N lirn[HN,~ES,]=_eS_+2Ano1I2e_1’~Sz=0.
(4e) Using the “gap-equations”, the internal energy per atom is given by
(HN> eSzA2S+S
(5)
,
u(13)=lim —j~--=
sz=_~~tanii4~e, u(j3)=—~etanh~13e, p f(13)=~fuO3’)d~’=—~ln[2cosh(~-13e)] 0
and
Hs=C~(aS~+t3S~+iS~), with
Sx=Sy=0_~T~Sz
(6)
n
Solution (2): 2A2
_2A2Sx
c
‘
~=
—2A25Y
c
C=Atanh(413XC), —
and
4A2
C= [e2+4X4((Sx)2+(SY)2)]l/2 (7) Changing to a new spin representation defined by
c2 ~‘
f(~)= ~ln [2cosh (~3AC)]+ —
c~
~.
‘~
(8 a)
aS~+ (iS~+ = v’a2+P2 (~5~aS~), =
(8b)
—
=
v’a2 +j32 [ ~
+ 137 k a2 +132
5Z
~s)~1 k Id
5X
a2 +132 the time translation of Tk is described by =
C ~ T~.
(8c)
(9)
Now we assume to deal with a spin S = ~ system. By means of the K.M.S. boundary condition [71 we arrive immediately at 452
4X~
The critical temperature T~is determined by the relation I3~= ~tanh’ (-i). (11)
3c solution 11$ <13~, then we have solution (1), for j3 > ‘ (2) has the lowest.free energy. The system exhibits a second order phase transition from the normal to the superradiant state. Finally it should be remarked that, although all proofs of the correctness of our calculations are lacking, we feel confident about the exactness of our approach in view of the fact that for fermion systems this kind of substitutions are allowed [6]. Over against
Volume 48A, number 6
PHYSICS LETTERS
that the method has the advantage of being very simple to carry out and can be easily generalized to a maser model with a finite number of photon modes. This work is part of the research of the “Stichting voor Fundamenteel Onderzoek der Materie” (Foundation for Fundamental Research on Matter F.O.M.) and was made possible by the financial support from the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek” (Netherlands Organization for the Advancement of Pure Research Z.W.O.). —
29 July 1974
References [11 K. Hepp and E.H. Lieb, Ann. of Phys. 76 (1973) 360. [2] R.H. Dicke, Phys. Rev. 93 (1954) 99. [31 N.N. Bogoliubov, J. Phys. Moscow 11(1947)23. [4] D. Kastler and D.W. Robinson, Comm. Math. Phys. 3 (1966) 151. [51 N.M. Hugenholtz, Rep. Progr. Phys. 28 (1965) 201. [6) A.S. de Vries, thesis (Groningen) 1972. [7] R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Ph’s. 5 (1967) 215.
—
453
PHYSICS LETTERS
29 July 1974
THE DICKE MASER MODEL G. VERTOGEN and A.S. De VRIES Institute for Theoretical Physics, University of Groningen, the Netherlands Received 10 October 1973 A simple method is presented to calculate the thermodynamic properties of the Dicke maser modeL
Recently Hepp and Lieb [1] calculated the thermodynamic properties of the Dicke maser model [2] a rigorous and very sophisticated way. In the conclusion they remark that their findings are typical of socalled van der Waals limits in systems with a longrange interaction. There is one important difference, however, their “mean field” is entirely off-diagonal. It is the purpose of this letter to present a calculation of the model, leading to the same results, based on the “Bogoliubov trick” [3], which is nothing but a mean field approach for bosons. Owing to difficulties inherent to a boson system (i.a. the unboundedness of the creation and annihilation operators) we cannot prove that “Bogoliubov’s trick” is allowed, although the analogy with the fermion case is striking, The Hamiltonian of the Dicke maser model reads N
HN
=
a~a+
E n1
N
E (as; +a~S~),A
5Z
~‘
‘~
N
Z~s~,
(2a)
N
[HN,a] =
—
a
——-a— E s
~
(2b)
,
=
—
~ (2e)
eS~+—~aSk .
Consider lmlN..+ (l /IV)~=1S~hmN~(1 /A~1S and lIrnN (1IN)Y~[~1S~. These operators are spaceaverages. If our a!gebra is a C.A.R.-algebra, which is the case without bosons, and assuming the state to be translationally invariant, which is quite reasonable from the physical point of view, we know the limits to exist and to belong to the center of the induced representation [4]. In this case we have a tensor product of a boson algebra and a C.A.R.-algebra. We conjecture the theorem to be also valid in this situation. Besides we assume the state to be such that the spaceaverages are c-numbers denoted by 5~,5+and S Consider a/,./~and a+/,.JY~The matrix elements of these operators are clearly non-diagonal, because they connect states with N0 and N0 1 photons. We have (N0 1 Ia/~WI N0) = ~jAç7Neiatand ~ —
0. (1)
The units have been chosen such that the photon mode has unit energy. Using the well-known commutation ielations for bosom of andmotion spin operators we obtain the following equations [HN, a~’]=a’+—~—
2X [HN, Ski
(2c)
—
N 00 such that N0/N = n0~where n0 denotes the density of the photons, one2eia, getslikewise ‘~~~N’°~ (n0 IaIv~NIn0> = n0no1/h/2e—1a. In this limit 11mN_~=(nOIa~/\~~InO)= the operators lin ~a/~~and limN~a”/~possess finite diagonal elements. They incommute withWe each other and with all other operators the algebra. therefore replace these operators b~c-numbers. This is known as the “Bogoliubov trick” [5]. As far as the spin operators are concerned time translation is induced by o°and
-~
2ei~ES+AnohI2e_~ES~, fl (3) In the thermodynamic limit, provided the following “gap-equations” hold [6]
H
5 =e ES~+Xn0l/
EHN, S] eS~—:
~
(2d)
451
Volume 48A, number 6
PHYSICS LETTERS
~—1 ,~0l/2e_ia + AS~=0,
[HN, ~j N-+~oL
(4a)
=
29 July 1974
N T~=limb~T—~-tanh~flC N-’
(10 urn 1HN,—~-—1=
n 1/~e~ —AS 0
—
N-~~~L
=
0,
(4b)
eSz_2A2((Sx)2 +(SY)2)
= =
aSz +,3SX +7SY
c
~
lirn
The solutions of the “gap-equations” are: N
EHN,b~S~=
—
An
2einS~
(1)
0 l/
0=O;
(4 c)
1/2e4’~S=O, +An0
(2)
S~—--~-—, 2 A2
~
~rn[HN~
sx =Sy=o, i.e. n
N
~s;]
=
eS~ 2Xn
2einSz
—
=
0 lt
0, (4d)
this implies ~ A2 in the case of spinS = Solution (1):
~-
operators.
N lirn[HN,~ES,]=_eS_+2Ano1I2e_1’~Sz=0.
(4e) Using the “gap-equations”, the internal energy per atom is given by
(HN> eSzA2S+S
(5)
,
u(13)=lim —j~--=
sz=_~~tanii4~e, u(j3)=—~etanh~13e, p f(13)=~fuO3’)d~’=—~ln[2cosh(~-13e)] 0
and
Hs=C~(aS~+t3S~+iS~), with
Sx=Sy=0_~T~Sz
(6)
n
Solution (2): 2A2
_2A2Sx
c
‘
~=
—2A25Y
c
C=Atanh(413XC), —
and
4A2
C= [e2+4X4((Sx)2+(SY)2)]l/2 (7) Changing to a new spin representation defined by
c2 ~‘
f(~)= ~ln [2cosh (~3AC)]+ —
c~
~.
‘~
(8 a)
aS~+ (iS~+ = v’a2+P2 (~5~aS~), =
(8b)
—
=
v’a2 +j32 [ ~
+ 137 k a2 +132
5Z
~s)~1 k Id
5X
a2 +132 the time translation of Tk is described by =
C ~ T~.
(8c)
(9)
Now we assume to deal with a spin S = ~ system. By means of the K.M.S. boundary condition [71 we arrive immediately at 452
4X~
The critical temperature T~is determined by the relation I3~= ~tanh’ (-i). (11)
3c solution 11$ <13~, then we have solution (1), for j3 > ‘ (2) has the lowest.free energy. The system exhibits a second order phase transition from the normal to the superradiant state. Finally it should be remarked that, although all proofs of the correctness of our calculations are lacking, we feel confident about the exactness of our approach in view of the fact that for fermion systems this kind of substitutions are allowed [6]. Over against
Volume 48A, number 6
PHYSICS LETTERS
that the method has the advantage of being very simple to carry out and can be easily generalized to a maser model with a finite number of photon modes. This work is part of the research of the “Stichting voor Fundamenteel Onderzoek der Materie” (Foundation for Fundamental Research on Matter F.O.M.) and was made possible by the financial support from the “Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek” (Netherlands Organization for the Advancement of Pure Research Z.W.O.). —
29 July 1974
References [11 K. Hepp and E.H. Lieb, Ann. of Phys. 76 (1973) 360. [2] R.H. Dicke, Phys. Rev. 93 (1954) 99. [31 N.N. Bogoliubov, J. Phys. Moscow 11(1947)23. [4] D. Kastler and D.W. Robinson, Comm. Math. Phys. 3 (1966) 151. [51 N.M. Hugenholtz, Rep. Progr. Phys. 28 (1965) 201. [6) A.S. de Vries, thesis (Groningen) 1972. [7] R. Haag, N.M. Hugenholtz and M. Winnink, Comm. Math. Ph’s. 5 (1967) 215.
—
453