Coupled-harmonic-oscillator Hamiltonians, modelling different physical processes, are discussed along with their eigenfunctions, which are shown.
IL NUOVO CIMENTO
VOL. 111 B, N. 7
Luglio 1996
Coupled harmonic oscillators, generalized harmonic-oscillator eigenstates and coherent states G. DATTOLI(1), A. TORRE(1), S. LORENZUTTA(2) and G. MAINO(2) (1) ENEA, Dipartimento Innovazione, Settore Fisica Applicata Centro Ricerche Frascati - C.P. 65, 00045 Frascati, Roma, Italy (2) ENEA, Dipartimento Innovazione, Settore Fisica Applicata Centro Ricerche Bologna - Bologna, Italy (ricevuto il 16 Novembre 1995; approvato il 16 Febbraio 1996)
Summary. - - Coupled-harmonic-oscillator Hamiltonians, modelling different physical processes, are discussed along with their eigenfunctions, which are shown to be generalized harmonic-oscillator functions with many indices and variables. We introduce the relevant coherent states and analyse their peculiar properties. The physical and mathematical consequences of our study are f'mally considered.
PACS 0220 - Group theory. PACS 02.30.Gp - Special functions. PACS 02.90 - Other topics in mathematical methods in physics.
1. -
Introduction
Several physical problems in the field of quantum optics, including frequency converter, parametric amplifiers, Raman and Brillouin scattering etc., are modelled using coupled equations of motion which can be derived from coupled-oscillator Hamiltonians [1]. The evolution of quantum states ruled by coupled-harmonic-oscillator Hamiltonians (CHOH) deserves careful attention and the associated dynamical problems are significantly simplified if the proper eigenstates are found and the relevant diagonalization procedure is specified. This paper is addressed to this aspect of the problem. We show that CHOH can be diagonalized using multi-index multivariable generalized harmonic-oscillator eigenfunctions of the type introduced in ref. [2]. The main properties of these functions are summarized in these introductory remarks and their usefulness in applications will be discussed in the forthcoming sections. Limiting, for the moment, the analysis to two-variable and two-index functions, 811
812
G. DATTOLI~ A. TORRE~ S. LORENZUTTA and G. MAINO
we define a generalized harmonic-oscillator eigenfunction as (m and n integers) (see ref. [2])
f
(1)
~ Hm,~(x,y)
where z is a two-column vector specified by (2)
.T. denotes transpose.
and the superscript matrix given below:
Furthermore, 2~ is a symmetric 2 • 2
Finally, Hm, ~ (x, y) and G~, ~ (x, y) are two-variable two-index Hermite polynomials defined by H . . ~ (x, y) = ( - 1)~ § ~exp (4)
G.~(x,y)=(-1) W
The functions ~ ,
~x ~ ~Y~
,
exp[~w M-w]~--exp[-lwwM
Xw]
=Mz.
,~(x, y) and ~.;, ~ (x, y) form a biorthogonal system and thus q-co
--oo
+zc
--o~
The creation/annihilation operators associated to the generalized harmonic-oscillator eigenfunctions are
al, + =
(ax
+by)
~x
(6a)
1 c - - -
~x
~y
-F 2 x ,
COUPLEDHARMONICOSCILLATORS, ETC.
813
and 2, ~y
~,+= l(bx+cy) (6b)
~,
l{b a --~ ~
_ a~ ~
1 ~]+ ~Y,
which act on ~,~,n(x, y) according to
l al, +.~m,. -- V~'~
19g'm+ I,,~ ,
al, - ' ~ m , n = Vr-m.~m- l,n ,
(7)
The above creation/annihilation operators are not Hermitian conjugate to each other, thus justifying the biorthogonality of the ~ m , . and ~,~,. functions. We consider, therefore, the operators
2
(8a) ^
a'l,-
zl
~
'
a +1
~-~ ~ (ax -{--
and
i(b a oa / (8b)
+ l(bx+cy), which act on fire, ~ as the ~ on ~m, n. Denoting with a superscript ,~t,, the operation of Hermitian conjugation, it is easy to realize that (9)
~
•
= aj,~, =
j = l , 2.
The extension of the above formalism to more indices and variables will be discussed in the forthcoming sections. An idea of the behaviour of the functions ~a~m,~ and &~,~,~ is offered by fig. 1-3. In this section we have proved that more extended forms of harmonic-oscillator eigenfunctions exist; in the following sections we will prove that they are the natural tool to treat CHOH.
814
G. DATTOLI, A. TORRE, S. LORENZUTTA
and G. MAINO
Y 0.751
A. 2.987 x l O -6
-2
0
2 x
Fig. 1. - 3D view and contour plot of 9~o,o(X, y) = ~o,o(X, Y) (a = a, b = - 1, c = 1.5).
Y a)
/ -2
_J - 0 . 9 0 6 . . . . .
1
0
2 x
y b)
2
0.557
A -2 -0.557
K
-2
0
2
x
Fig. 2. - 3D view and contour plot of a) ~ 1 , o (x, y) and b) ~1, o (x, y) (same p a r a m e t e r s as fig. 1).
815
COUPLED HARMONIC OSCILLATORS, ETC.
y 2
a)
:iii
-2 T
-2
0
2 x
Y 2
b) 0.71
0
-4 -2 !
-2
0
-0.455
2x
Fig. 3. - 3D view and contour plot of a) ~2,z(x, Y) and b) ff2,2(x, Y) (same parameters as fig. 1).
2. - Quantum-mechanical treatment of coupled harmonic oscillators
In this section we analyse the quantum aspects of a system whose classical Hamiltonian is provided by
(10)
with p, and ~/being canonical conjugate variables. According to the usual quantization procedure, the quantum counterpart of (10) is
(11)
G. DATTOLI, A. TORRE, S. LORENZUTTA and G. MAINO
816
where az is the operator specified by the vector(1) (see ref. [3])
(12)
Sz =
We will prove that the eigenfunctions of the Hamiltonian (11) are just the functions : ~ , n and G~, ~ and it is worth noting that (11) is a clear two-variable generalization of the harmonic-oscillator Hamiltonian in one dimension. The kinetic-energy operator can be viewed as a generalized Laplacian, while the quadratic form on the r.h.s. generalizes the potential energy. It is now convenient to go back to the already introduced operators ~ and ~ and note that they satisfy a Weyl algebra in the sense that (13)
I [[~j' - ' , ~j'' 5j,, j , , ~j, ~_ ~j, , + ~ ]] = = O
j, j ' = l, 2 ,
[
and the same holds for the conjugate counterparts. Number operators are defined by the products A
(14)
N j - ~tj, + aj, _ ,
acting on ~ ,
A
A
j=1,2,
N j = ~j, § dj _ ,
~, for instance, as N 1 5 ~ , n = m.~Cm, n, N 2 "~'m, n = n.O~'m, n and, similarly,
on
Using the differential representations (6) and (8) we can easily prove that (15)
al,
+
al,
-[- a2, + ~2,
-
= al,
+
al,
-
-]- a2,
§
a2, - = - - ~ T / ~
l~z q- ! Z T / ~ z -- 1
4
'
thus implying that ~Vm,~ (z) and Win,n (z) are eigenfunctions of the Hamiltonian (11), the vector ~- being ~, = z / V h . The corresponding energy eigenvalues are provided by (16)
~m, n = h A ( m + n + 1).
The system energy is specified by the two quantum numbers m and n. The residual energy hA is due to the vacuum field fluctuation. The considerations and the formalism we have developed suggest the introduction of the states Im, n) and I m---~n) which are specified by the eigenvalue problems (17)
A
A
aj,+aj,
The v a c u u m
(18)
j = 1, 2, [~,n)
Njl~,n
).
is self-conjugated, in the sense that
Io, o) = Io, o).
(1) We have denoted by a T the transpose of (12), namely a T = ( ( 3 / a x ) ( 3 / 3 y ) ) .
COUPLED HARMONIC OSCILLATORS, ETC.
Ira, n)
According to eq. (5) the states
(19)
817
linen) satisfy
and
the condition
(m---7"~It, s) = ,~,,,~,,,,.
Let us now go back to the definition of the creation/annihilation operators, which can be more conveniently cast in the following vector form:
2
(20a)
1 ~-z
J_ =#-Is,+ and
(20b) A_
1^ a, + ~Mz,
where
(2Oc)
~•
~,
.
The above relations can be inverted, thus finding for the differential operators
{
z = # - I f t +
(21)
+~_,
a,= _ ! ~ + + ~#~i_. 2
2
We can now evaluate the r.m.s, values of positions and momenta using the properties of the .4• operators. Let us denote with (22)
(Y,~.i)(o.o)=(O,OlzizjlO, O),
(z~,i)(o,o)=(O,OIp,~p~jlO,O)
(i,j=l,2)
the entries of the position and momenta covariance matrices averaged on the vacuum state. In the above relations the following identifications Zl - x and z2 - y must be understood, and the state 10, 0) should be regarded as a function of ~. It is easy to prove that t(~ 2 ) ~(0, o) = (23) ~(0, 0) =
(=y)|\
= ~1,
(P~) (P'P~)/ = h_#. p2 (P~v,) (~) J(o,o) 4
G. DATTOLI,A. TORRE,S. LORENZUTTAand G. MAINO
818
We can, therefore, infer the following generalized uncertainty relation: (24)
Y.(0,0)~(0,0) = h21, 4
with 1 denoting the unit matrix. The vacuum I0, 0} seems to provide a minimum uncertainty state (m.u.s.), whose nature should be carefully understood. It should indeed be noted that 2
2
_
2
h2
2
ca
(p~)(o, o)(X )(o, o) - % )(o, o) (y )(o, o) = ~ ~ , (25)
h2 b 2
(xy)(o,o)(p~p~>(o,o)- ~ ~, according to which we can conclude that the x and y components reduce to m.u.s, for b = 0 only, that is for a decoupled system. More in general, as to the higher-order states Im, n} or I m--~n} the identity (24) does not hold, but one finds
4 The generalized number states cannot, therefore, be viewed as m.u.s., in the sense specified by the identity (24). The situation is reminiscent of the ordinary case. It is therefore natural to suggest the introduction of suitable coherent states, whose nature and role will be discussed in the next section.
3. -
Coupled-harmonic-oscillator coherent states
Generalized coherent states (g.c.s.) of CHO are defined in an almost natural way, i.e.
1 I-l,-~>:exp[-~(I.1 (27)
[
12+ I-~ 12)j.~,.~:o~ ] ~ a~a~ im,n},
+ 1fi212) ~, I/~l, ~2> ---- exp _ 1(1fll]2 2
o ~fiTfl~
I m--~--n}"
It is easily checked that
(28)
f al, Ic~1,c~2}= al lal, a2}, ~2, I-,, a2) = -2 I~,, .2), A
According to the biorthogonality of the states Im, n} and I vht,n}, we infer the
COUPLED HARMONIC OSCILLATORS, ETC.
819
further consequences
(~,nlal,a2)=exp[-~(lal f
(29)
]2
1
[ 1
(m, nlP~,p~)=exp -~(IP,
+
[2+
[a2 ]2)]~
a'~a'~
,
12]fl'~fl'2
IP~ ) V,-m-~.vn!
It is clear that, since we have two distinct.sets of number state, we also have two sets of g.c.s. The physical meaning of the last relations is that the probability for a [ f l ~ 2 ) or an [ al~-,-,ae) g.c.s, of being in the conjugate number state is a Poissonian. This conclusion does not hold for (m, n[al, a2) and (m---~n [ t i l d e ) . The above relation should be completed by the identities
(3Oa)
(~-P la, P) = 1
and
(30b)
(r.-51~,P)=exp
1 -~-[Irl~+
15 i~ + la is +
I P 1 2 - 2 ~ r * - 2 P 5 *] ] 9
Equation (30b) states that g.c.s, are not biorthogonal, but become approximately such when (1712 + 1512) + (lal 2 + Ifll 2 - 2 a t * - 2fl5" ) increases. This situation is reminiscent of the non-orthogonal properties of ordinary coherent states (see ref. [4]). A further important issue to be clarified is the coordinate representation of the g.c.s. (27). Using eqs. (27) and the differential realization of the creation operators we find
[ I0.T~ ]exp [[1-~(ax + by) - ~x
C~,(x, y) = (z, Ylal, a2) = exp - -~
a 1
"
"exp[az[ l (bx + cy) - ~y ]](x, y'O, O),
(3:)
Since (x, y[0, 0) = 5~0,0(x, y) and using the disentanglement theorem (see the appendix) we end up with (2)
(32)
co(x, y) =
[ - i2~~"*T
exp -
+
(2) The g.c.s. (33) can be expanded in terms of Hm, ~ polynomials as follows:
c a ( x , y) =
V-~ exp[_!~,M ~_ ~ * * 1
V~
i
4
o~
m,.:o(v~)m§
'
.
820
G. DATTOLI,A. TORRE,S. LORENZUTTAand G. MAINO In a similar way it is shown that
(33)
C~ (x, y) : 4V~~
exp [ - ~1 ~ T ~ 1exp [--l~/~-l~fl] exp
[--4zT~/lz+zT~],
A remarkable property of the states (32) and (33) is provided by the overlapping integral
I dy j dxC*(x,y) Ca(x,y)= 1.
(34)
Both Ca(x, y) and C~(x, y) are, therefore, not normalized to unity. Introducing two normalization constants JV'a and 4 , we define the normalized states
[
(35)
C~ (x, y) = ~ -
C~(x, y) -
1
Ca (x, y),
1
C/~(x,y)
and it is straightforward to prove that the covariance matrices Xc and ~c, whose entries have been calculated according to +~
+~c
-
+~
_ (36)
+~e
(hc)~,~ -
1
1
+~
! dy I dxC*(x,y)(zizj)C~(x,y),
+:e
! dy I dxC*(x,y)(p~p~j)C~(x,y)= _
1 I
! dy I dxC*(x'y)(Pz~PzJ)C~(x'Y)' .1-
-=
satisfy the identity (24a). The states (32), (33) or equivalently (27) are m.u.s, in the sense discussed in sect. 2. 4. - C o n c l u d i n g
remarks
The extension of the previous analysis to many variables and many indices ( > 2) is straightforward. It is indeed enough to replace the 2 • 2 matrix M given in (3) with an n • n matrix whose entries a~,j satisfy the condition (37)
ai, j > 0,
ai, k = ak, i;
correspondently z and az should be replaced by n-component vectors. From the conceptual point of view such extension does not produce any significant variation.
821
COUPLED HARMONIC OSCILLATORS, ETC.
We have stressed that the present formalism generalizes to the many-variable case most of the properties of the one-dimensional quantum oscillator. The prices to be paid are the introduction of number and coherent states, characterized by more general properties, than the usual case, as, e.g., biorthogonality rather than orthogonality. Generalized Hermite polynomials have already been exploited in the literature to define generalized coherent-squeezed states [5, 6]. These states are different from those introduced in this paper, in the sense that they are defined as eigenvalues of the b, b* operators provided by the Bogoliubov transformation (3)
(38)
(:W) = ( : *
~ ) (aat) ;
both a and b operators are n-component column vectors, the matrix on the r.h.s, of (38) is a symplectic 2N x 2N matrix consisting of N-dimensional complex square blocks. The operators ~ and ~* are ordinary annihilation/creation operators of a polymode field. The operators b and ~t are, therefore, Hermitian conjugate to each other in the ordinary sense and the different structure with respect to the operators (6) and (8) is evident. A well-known peculiarity of the ordinary harmonic-oscillator Hamiltonians and of their eigenfunctions is that they are left unchanged by a Fourier transform. This property holds time for CHOH. It can be indeed directly checked that the Fourier transforms of : ~ , ~ (s ~) and h~, ~(~, ~), namely 1
+~
+oo
[
[ cl~[ d~exp[+ 2kT.z]/9g''''~(s Y)/ (39)
satisfy the eigenvalue equation
In addition we also have [2]
(41)
- -
~exp
- lkT~I
(3) The transformation (42) is in general non-homogeneous.
J ~H~,,~(~, ~) "
822
G. DATTOLI, A. TORRE~ S. LORENZUTTA and G. MAINO
Before ending the paper let us add a few comments on the algebraic structure of the CHOH (11). It is well known that the ordinary HOH can be written in terms of SU(1, 1) generators. This is true for the coupled case, too. Defining indeed the operators
(42)
k+=-a
M-laz,
k =
zTMz, 4
ko=l+zTa,
it is directly checked that they satisfy the S U ( 1 , 1) commutation relations (43)
[fc+, k_ ] = - 2 k o ,
[k0, k_+ ] -- _+fc+ .
As a consequence, according to the above-developed considerations, ordering operational techniques of, e.g., the Magnus or Wei-Norman type [7] can be used to treat time-dependent evolution problems connected with coupled-oscillator dynamics. A final interesting point to be stressed is associated to the peculiar nature of the creation/annihilation operators (6) and (8). Using the Schwinger realization of angular-momentum operators, we can introduce the following generators: 1 ~ (44)
~3
=
~-[al,
+ a l , - -- a2, + 42, - ] ,
satisfying the SU(2) commutation rules
(as) We cannot, however,view the above realization as a standard SU(2) structure. The operators ~ + and ~ _ are indeed not Hermitian conjugate to each other. The same considerations can be extended to the bosonic realization of the S U ( 1, 1) group. These last considerations are a further example of the wealth of implications offered by the present approach to the theory of coupled oscillators. More general considerations will be developed in forthcoming investigations.
APPENDIX
According to eq. (32), the (x, y) representation of a CHO g.c.s, is provided by (A.1)
C . ( x , y) = e x p [ - ~ * W ~ l e x p
ai ~ ( a x + by) -
.
Using the disentanglement theorem [7], (A.2)
exp [A +/~] = exp [A] exp [/~] exp [ - c / 2 ] ,
[A, B] = c,
c = c - number,
823
COUPLED HARMONIC OSCILLATORS, ETC.
we can rewrite (A.1) as (A.3)
Ca(x, y) = exp[- l ~T ~]exp[lalby ]
e [lalo]e [ o1 ] [1 ][1][ a]
exp -2a2bx exp
"exp [ - l a ~ c ]
a2cY exp - a 2 ~
( ~~f ~ exp [ -
9
lzT~'Iz])"
Finally, using the displacement operator property: exp[;~(~/az)]f(x)=f(x end up with eq. (32), and in a similar fashion with eq. (33).
+ ;t) we
REFERENCES [1] See, e.g., MOLLOW]~. R. and GLAUBERR. J., Phys. Rev., 160 (1967) 1076; LOUISELLW. H., YARrVA. and SIEG~ta~NA., Phys. Rev., 124 (1961) 1646; Lu E. Y. C., Phys. Rev. A, 8 (1973) 1053. [2] DATrOLI G., LORENZUTTAS., MAINO G. and TORRE A., J. Math. Phys., 35 (1994) 4451; DATTOLI G. and TORRE A., submitted to Phys. Rev. A. [3] ROYERA., Phys. Rev. A, 43 (1991) 44. [4] LOUISELLW. H., Quantum Statistical Properties of Radiation (J. Wiley, New York, N.Y.) 1973, p. 104. [5] WOLFK. B., J. Math. Phys., 15 (1974) 1295; MALKINJ. A., MAN'KOV. I. and TRIFONOVD. A., J. Math.. Phys., 14 (1973) 576. [6] DODONOVV. V., MAN'KO O. V. and MAN'KOV. J., Phys. Rev. A, 50 (1994) 813. [7] DATTOLIG., GALLARDOJ. C. and TORRE A., Riv. Nuovo Cimento, 11 (1988).
