Volume 134, number 5
PHYSICS LETTERS A
9 JanuarY 1989
THERMAL COHERENT STATES A. MANN and M. REVZEN Department of Physics, Technion — Israel institute of Technology, Haifa 32000, Israel Received 26 May 1988; accepted for publication 7 November 1988 Communicated by J.P. Vigier
The formalism of thermo field dynamics is used to define a thermal coherent state, and thus calculate some correlation functions, The relation of the thermal coherent state so defined to earlier definitions is briefly discussed.
Thermo field dynamics (TFD) [1,21 was developed as an extension of zero temperature quantum field theory to finite temperature problems. The price for this convenience is that in TFD one deals with a space that is a direct product [3] of the ordinary zero temperature Hubert space (~R)and a socalled tilde space (~). Thus every state I n> in ¶R has a corresponding state ñ> in A similar rule holds for operators: every operator A acting on 9~has an image (tildian) operatorAactingon~.The association of an operator A with A is called tilde conjugation. The basic relation between the operators is given in ref. [2]. The equivalence of the equilibrium TFD to the C~algebra approach was proven by Ojima [31. We give a brief summary of the basic TFD formulae that we shall need. Details of the derivation and discussion are given in the cited references [1,21: ~.
(akak)
=aka/~ ,
(clak+c2ak) 
=clak+c2ak,
— —
where H is the hamiltonian. The transformation from the zero temperature (/1= x) free bosons to the finite temperature case is generated by [21 Gr—i
~
Ok(P)(a~a)~—akak).
(4)
The transformation for the annihilation operators is [21 ak(P)=e”ake’~
(5)
UPakU~.
Thus ak—~ak(/J)=akcosh 0k —ã~sinh 0k,
(6a)
ak—*ak(P)=akcoshOka~sinhOk.
(6b)
(la)
c,eC,
(lb)
(a) =(at) . (lc) Both the “normal” operators (a, at) and the tildian operators (a, at) obey the Bose commutation rules and they commute with each other, [a, a] = [a, at] = 0.
vacuum is defined so that, for any “normal” operator A, (3 <0(P) A 0(fl)> Tr ( e~11A) /Tr e
(2)
At the heart of equilibrium TFD we have the thermal vacuum, 10(P)>, where P is the inverse ternperature. Physical quantities are calculated as thermal vacuum expectation values. (The chemical potential term is included in the hamiltonian.) The thermal
In the above [2] cosh Ok(P)= [1 —exp( —Ikk)l l/2~ Uk(P) sinh Ok(fl) = [exp(/kk) 1]_h/’2 = vk(IJ) . —
—
(7a) (7b)
Henceforth we shall delete the index k. The thermal vacuum is related to the zero temperature one, 0, 0>, via [2] I0(P)>=exp(—iG)I0,O>~U~I0,O>.
(8)
In this Letter we use TFD formalism to extend to finite temperatures the well known coherent states
03759601/89/s 03.50 © Elsevier Science Publishers B.V. (NorthHolland Physics Publishing Division)
273
Volume 134, number 5
PHYSICS LETTERS A
[4—6].These states are defined to be the eigenstates of the annihilation operator, a, ala>=ala>,
ja>=exp(aat_a*aflO>
—
/3>
(17b)
_i.~f
P=
[u(/3) (a_a*) +v(/3) (~_?K) ] =
(10)
.
The formalism of TFD allows a natural extension of the definition to thermal coherent states (TCS): ~‘;
~=
(9)
i.e., the normalized coherent state is
a,
9 January 1989
exp [aat (/3) + ?~at (/3)
(17c)
P=isf~[u(p)(7*_y)+v(P)(a*_a)]
=
(1 7d) TCS were discussed by Emch and Hegerfeldt [7].
a*a (/3) 5~ä(/3)] 0(P)> .
(11)
—
These states are, obviously, eigenstates ofa (/3) and a(fl)la,5i;P>=ala,5;P>,
(12a)
a(/3)Ia,5~J3>=5*Ia,5i;p>.
(l2b)
Since the algebra of the thermal operators is identical to that ofthe zero temperature operators, all the formal properties of the “usual” (eq. (10)) coherent states have a natural correspondence in the TCS. We now evaluate the characteristic function (CF) for bosons of mass m = 1 and frequency w such that their operators (a, at, a, at) are related to the Canonical variables (Q, F, P) by (h= 1)
The two approaches are essentially the same (see below), if we put p=4=O. Setting these values in eq. (15) we recover the definition ofref. [7]. In our notation the TCS as defined in ref. [7] is IOD(/3)>=exp(—iGD)IOD,OD;oo>.
(18)
In this equation I 0~,OD; cx> is the displaced vacuum of TFD; it satisfies (note: we display only the relevant mode) bIOD, OD; ~> =bIOD, OD; cx> =0, with b=a—a,
(19)
6=ä_a*.
~,
Q=
(a+at),
P=
~
~
(a_at),
(13a)
~=__‘_(a+at), P=_1~/~(a_at).(13b) The characteristic function is evaluated in the TCS: CF=
.
(14)
Here the lower case letters (q, p, ~) are real c numbers. The calculations are straightforward. The result is CF=exp{— ~ø[w(p2+~2)+ (q2+~2)/w] ~,
—A ( wp~+ q~/w)} xexp[—i(qQ+pP+~+~P)].
Its explicit form is IOD, 0D; ~> ~D(a, à)l0,
~exp(aat_a*a+a*at_aa)I0,O;oo>, while GD is given by
(20)
GD=_iO(/3)(btbt_bb).
(21)
This definition is related to the displaced harmonic oscillator hamiltonian and with it one has y= a* (i.e. the eigenvalues of a and a are related as both appear in b, 5 of eq. (21 )). Thus the formal relation between the two definitions (G refers to eq. (4), GD to eq. (21)) is I0~(P)>=e”~D(a,à)I0, O; cx>
(15) =D(a, ã)e~GI0,0; ~>
Here (h=l) t9=coth(/3w/2). A=1 cosech(/Jw/2),
while our definition is
7; p>=e~D(a,7)10, 0;
x>
,
(23)
[u(p)(a+a*)+v(/J)(7+fl]= (1 7a)
274
(22)
,
(16) a,
—~—
~ cx>
i.e. we allow unequal a and 7, which leads to the gen
Volume 134, numberS
PHYSICS LETTERS A
eral characteristic function, eq. (15). Eq. (15) also implies
(QQ> =A / w,
(24a)

=Aw.
(24b)
with P=p—P~etc. These results, eqs. (24), are independent ofa and 7 and were obtained because in TFD the thermal degrees of freedom are explicitly displayed via the tilde degrees of freedom. For fl—.oo, A—~0and we recover the zero temperature results. For /3—~0we obtain (kT is the temperature) (25) <~~> =A2—~(kT)2/w2, i.e. the thermal degrees of freedom (the tildians) are strongly tied to the “normal” operators. It should be noted that the concept of a squeezed state (SS) [9] can be extended to finite temperatures in an analogous way. Thus we define thermal squeezed states (TSS) by ITSS> =exp( —iG) ISS>
,
(26)
where ISS> is the ordinary (zero temperature) squeezed state. In this connection it is interesting to note that formally the TFD transformation (eqs. (8) and (4)) is equivalent to a twomode squeezed state ~suJ. To summarize: In this Letter thermal coherent states (TCS) were defined as a natural extension of
9 January 1989
their zero temperature definition. The extension utilizes the formalism of thermo field dynamics (TFD) which is an extension to finite temperature of zero temperature field theory. For the thermal coherent states the correlation between the socalled thermal degrees offreedom and the normal (i.e. usual) degrees of freedom was derived. The research was supported by the Fund for the Promotion of Research at the Technion.
References [1] H. Umezawa, H. Matsumoto and M. Tachiki, Thermo field dynamics and condensed states (NorthHolland, Amsterdam, 1982). [2] Y. Takahashi and H. Umezawa, Collect. Phenom. 2 (1975) [3]I. Ojima, Ann. Phys. (NY) 139(1981)1. [41 I.R. Senitzky, Phys. Rev. 95 (1954) 904. [5] R. Glauber, Phys. Rev. 131(1963) 2766. (6]J.R. Klauder and B.S. Skagerstam, Coherent states— application in physics and mathematical physics (World Scientific, Singapore, 1985). [7] GO. Emch and G.C. Hegerfeldt, J. Math. Phys. 27 (1986) 2731. [8] Y. Aharonov, D. Falkoff, E. Lerner and H. Pendleton, Ann. Phys. (NY) 39 (1966) 498. [9] J. Opt.Caves Soc. Band 4, No. (1987), and references [10]C.M. B.L. 10 Schumaker, Phys. Rev. Atherein. 31(1985) 3068; B.L. Schumaker and C.M. Caves, Phys. Rev. A 31(1985) 3093.
275