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Squeezing operations in Fock space and beyond
PHYSiCA ELSEVIER
Physica A 242 (1997) 423-438
Squeezing operations in Fock space and beyond Reinhard Honegger, Alfred Rieckers* Institut fiir Theoretische Physik, Universitdt Tiibingen D-72076 Tiibingen, German), Received 25 October 1996
Abstract Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.
PACS: 03.65.Fd; 05.30.Jp; 42.50.Dv; 02.30.Tb Keywords: Weak and strong squeezing operations; Existence of quadratic Boson Hamiltonians; Inequivalence of squeezed vacua
1. Introduction In the theoretical descriptions o f squeezing processes the Hamiltonians mostly are approximated by quadratic expressions o f the photon field (interaction picture). Let us start our discussion with these formal expressions and deal later on with their deeper meaning and mathematical definition. The Hamiltonians are meant to describe essential features o f the dynamics which takes place in the non-linear optical medium. There are (at least) two types o f photons, belonging to the signal resp. to the idler beam. Thus, usually two types o f quadratic Hamiltonians are distinguished: • The degenerate case [1]: N 1 Hq, d : ~ ~ ( ~ n a * a * n +-~nanan), n-I * Corresponding author. Fax: 07071 29 5850; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved. PII S0378-4371 (97)001 81-7
(1)
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where the squeezing parameters (n E C (and their complex conjugates (n) arise from a classical (macroscopic) pumping field. Here the a n* =-a*(en) are the creation, and the a n - - a ( e n ) are the annihilation operators of the orthonormalized modes {e~ ..... eN}, which fulfill the canonical commutation relations (CCR) [a(en),a*(em)]=rn, m~,
[a(en),a(em)]=O,
l ~m,n<<,N.
(2)
• The nondegenerate case [2]: M
nq, nd = Z (rlkas, kai, k q- qk as, kai, k ) ,
(3)
k=l
where the index s stands for the signal modes Us,k and i for the idler modes ui, k (which all are supposed to be mutually orthogonal and normalized), with the annihilation operators as.k =--a(us, k) for the signal modes Us,k, and ai.k =a(Ui,k) for the idler modes ui, k. The r/k E C are classical pumping parameters, too. Usually only finitely many modes are considered, but there also are some investigations with infinitely many modes [3]. By not specifying N and M we cover all the cases of finitely, and infinitely, respectively, continuously many photon modes. The above two cases of squeezing quadratic Hamiltonians are not so different as they seem to be: By superposing the idler and the signal modes in the smeared field formalism, we transform the nondegenerate squeezing Hamiltonians Hq, nd into the form (1). This reveals the degenerate form of a squeezing Hamiltonian to be the more general one. Then we investigate the Bogoliubov transformations arising from the dynamics associated with the Hamiltonians Hq,d and Hq, nd. We show that for infinitely many modes the squeezing operations associated with the classical pumping parameters ~n, resp. r/k may become so strong that the quadratic Hamiltonians have no meaning as operators in Fock space, and the squeezing Bogoliubov transformations lead beyond the Fock representation. In this case of a strong squeezing, the squeezing procedure leads into a representation of the CCR which is inequivalent to the Fock representation. Nevertheless, it is a well-defined squeezing transformation in the smeared field formulation, and the squeezed vacuum state (characterized by some reduced fluctuations) has a well-defined vector in these non-Fock representation spaces. For strong squeezing one has, however, no (vacuum) vector which is annihilated by the Bogoliubov transformed annihilation operators. These and other facts are derived here in the smeared field formalism in an almost self-contained fashion. Moreover, we try hard to explain the basic notions of this formalism in such a way, that also those readers, who had worked hitherto with the Fock space formalism only, should be able to realize the implications of our statements. This much wider frame of the quantized field formalism is unavoidable, if one deals with the optical properties of photon states. This has been pointed out in the optical coherence theory [4] (which is based on Glauber's original works [5]) and the macroscopic
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radiation theory [6], and will be further substantiated in terms of special squeezing transformations.
2. The smeared field formalism
In the smeared field formalism the creation and annihilation operators are "smeared" by testfunctions f , which are taken from the so-called one-photon testfunction space E. What about the physical meaning of E? E is spanned by the photon modes taken into account, e.g., {el ,eu} or {u.,.l . . . . . Igs,M,Ui, I,...,Ui, M}. Since each mode is a normalized function we have that E is a complex subspace of a complex Hilbert space Y~ of square integrable functions with scalar product (. ] . ) (which is complexlinear in the right factor and complex-anti-linear in the left one). For example, when quantizing in the Coulomb gauge in position space with quantization volume A C R 3 and taking all directions of polarization into account we have that ~ consists of all square integrable divergence-free (i.e. 27. f = 0) functions f on A with f ( x ) E C 3 for all x E A. ~J has the scalar product ( f l g) = JA f ( x ) . g(x) d3x for f , g E J~, where a. b---albj +a2b2 +a3b3 for a=(al,az,a3)cC 3 and b=(bj,b2,b3)cC 3, [7-9]. With an orthonormal basis {el eu} of the complex one-photon space E the smeared annihilation and creation operators are given by ([10], cf. also [l l] Section 13.6) ....
.....
N
a ( f ) = ~ (fle,,)a(e,,),
N
a*(f)=~(e,,If)a*(e,,),
n=l
N
f -- ~ (e,,I.f)e,,EE,
n--I
n--I
(4) from which it immediately follows that a ( f ) and a * ( f ) are adjoint to each other, and an important point that the operator-valued mapping f E E~-+a*(f) is complexlinear, and f E E H a ( f ) is complex-anti-linear, i.e., for all f,.q E E and ~,fl E C, one has
a*(:~f +flg)=c~a*(f)+fla*(g),
a(:~f + f i g ) = ~ a ( f ) + f i a ( g ) .
(5)
One also easily checks from (2) that in the smeared formalism the canonical commutation relations (CCR) with the commutator [A,B] = A B - BA are expressed as
[a(f),a(g)]=[a*(f),a*(9)]=O,
[a(f),a*(g)]=(flg)n,
Vf, g E E .
(6)
The advantage of the smeared formalism is the independence from the orthonormal basis of E, which makes many manipulations more transparent, and is indispensable for a mathematically consistent formulation of non-relativistic QED. Let us turn to the smeared selfadjoint field operators ~ ( f ) : = 2 - W 2 ( a ( f ) + a*(f)), f E E. Here the mapping f E E ~-~ t/,(f) is only real-linear, i.e,
cb(af + f l O ) = a ~ ( f ) + f l ~ ( g ) Vet,tiER,
Vf, g E E .
(7)
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From the quantization procedure in the Coulomb gauge it is well known, that for realvalued testfunctions f the ~ ( f ) are the observables of the smeared vector potential, and for proper imaginary f (i.e. if(x)C ~3 VxEA) the ~ ( f ) are proportional to the observables of the smeared transversal electric field, [7-9, 12]. Thus, the selfadjoint field operators ~ ( f ) , f E E , are the fundamental observables for the quantum description of the photon field, since they generate all other observables. In the smeared formalism the field operators fulfill the CCR in the following form (with the scaling h=l)
[~(f),¢(g)]=ilm(fl9)~,
Vf, oEE,
(8)
which give rise to the Heisenberg uncertainty relations for the expectation values of conjugate field observables ~ ( f ) and cP(if), f cE. Obviously, the smeared annihilation and creation operators, a(f) resp. a*(f), are obtained back from the field operators by 1
a(f)=2-1/2(q~(f)+i~(if)),
a*(f)=2-1/e(~(f)-i~(if)).
(9)
Let us now consider the nondegenerate squeezing Hamiltonian Hq,.d. The considered modes are the signal and idler modes {Us,1,...,u~,M, ui,1..... ui,M}, which generate an orthonormal basis of the testfunction space E. Mixing these modes we obtain a new orthonormal basis {e~,...,e2M} of E according to
1
e2k-a:='-~(Us, k+ui, k),
1
e2k:=--~(Us, k--Ui, k),
l<~k<~M.
From the complex-linearity resp. -anti-linearity of f ~ a # ( f ) we obtain the simple transformations a#(Us,k)a#(ui,k)= ~(a I # (e2k-1)2 -a#(e2k) 2) which are valid for both the smeared creation operators (i.e., a#(h)=_a*(h)) and annihilation operators (i.e., a#(h)=a(h)). Inserting this into Eq. (3) yields Hq,nd=½ ~-~n= N 1 (~na*(en)2 + ~n"--a(en)2 ) with ff2k-I =glk,
~2k = -- r/k,
1 <~k<~M,
N=2M,
(10)
which agrees with the form of a degenerate quadratic Hamiltonian. The restrictions (10) demonstrate that the degenerate form (1) of a quadratic Hamiltonian is more general than the nondegenerate form (3). Thus in the following we only treat quadratic Hamiltonians in the degenerate form (1) and write Hq instead of Hq,d.
3. Symplectic- and Bogoliubov-transformations for squeezing Let us start from the degenerate quadratic Hamiltonian Hq. By purely algebraic (representation independent) calculations using the power series for the exponentials I For an up-to-date description of these basic structures cf. [10,4].
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and the CCR one obtains that the squeezing operations associated with Hq are given in terms of Bogoliubov transformations (cf. also [1,13]),
exp{itHq}CI)(f)exp{-itHq} ---=qb(Ttf) =: ~T,(U) V f E E,
t E ~,
(11 )
which are associated with a group of real-linear transformations Tt on the one-photon testfunction space E. More precisely one has
Tt = exp{tJS} = cosh(tS) + J sinh(tS),
t E ~,
(12)
with the complex-linear positive self-adjoint operator S on E and the complex-antilinear involution J on E (i.e., J = J * = j - l ) , where we put (n/](,I := 1 for ~, = 0 , N
S=Zfnlle,,)(enl ,
~n
Jen=l-~nle n VnE{1 ..... N}.
(13)
n=l
Especially for the transformed annihilation operators we obtain by means of Eq. (9) for each f E E exp{itHq } a ( f ) e x p ( - i t H q } = a(cosh(tS)f) + a* (J sinh(tS)f) = at, ( f ) , (14) from which with the en as testfunctions one gets the more familiar formulas [l,2]
ar,(e,) = cosh(tl~n])a(e,,) + ~~n sinh(ti~nl)a*(en) Observe that the complex-linear part of the real-linear Tt is given by (Tt)l = cosh(tS) and the complex-anti-linear part is (Tt)a =Jsinh(tS), which occur in Eq. (14). The ar,(f) are still annihilation operators: they firstly satisfy the CCR (6) with the transformed creation operators a~,(f) = exp{itHq}a*(f)exp{-itHq}, secondly are complexanti-linear in the tesffunction argument, and thirdly are given with the transformed field operators from Eq. (11) by ar,(f)=2-1/2(ebr,(f)+iq)v,(if)). This is due to the fact that the real-linear transformations Tt leave the imaginary part of the scalar product invariant (Eq. (15) below). More generally one defines: A mapping T acting on the one-photon testfunction space E is called a symplectic transformation on E, if it is real-linear, with range T(E)= E, and fulfills
Im(TflTy)=Im(flg)
Vf, g E E .
(15)
For each symplectic T on E the transformation, which maps every field operator ~b(f) onto the transformed field operator ~ r ( f ) - = eb(Tf) is called the associated Bogoliubov transformation. Because of Eq. (15) the transformed q~r(f), f E E , also fulfill the CCR (8). The Bogoliubov transformed annihilation and creation operators, a v ( f ) and a*(f), f EE, are given in terms of the complex-linear part T/ and the complexanti-linear part Tu for our (real-linear) symplectic T - - that is: T = TI + Tu with T I = ~1 ( T - iTi) and Ta = ½(T + iTi) - by aT( f ) = 2 -1/2 . ( ~ r ( f ) + iq~r(if)) = a(TJ) +a*(T~f), and a~(f) as the adjoint [13]. They obviously fulfill the CCR (6).
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4. The Fock space
The field operators q~(f) and the transformed ones q~r,(f) - respectively, the by (9) associated annihilation and creation operators and their transforms - from the previous sections are formal expressions as long as we do not specify the Hilbert space (and therein their domains) on which they are defined to act. Commonly in the quantum optic literature these operators are adopted to act on Fock space. In the present section we will do so, too. Let us, for later reference, recall the basic ingredients of the Fock space ~ + ( E ) for our complex one-photon testfunction space E with orthonormal basis {el . . . . . eN}. The Fock space ~ ( E ) is constructed from the normalized vacuum vector 1 0 ) E ~ + ( E ) satisfying a(f)lO) = 0
VfEE.
(16)
The ruth particle subspace ~ + is generated by the orthonormalized Fock state vectors 2
) [ml;m2;...;mN):= H
N
1 a*(ej) m' I0>,
j=l
= m,
mj E ~o .
j=l
(17) Then , ~ + ( E ) - (~m=0 m is the direct sum of all the m-photon spaces, where .~0+ := C[(9). It is well known that
a*(eD [ . . . ; m k ; . . . )
= V/~-k + I I ...;mk + l ; . . . ) ,
a(ek)l . . . ; i n k ; . . . ) = v/-~-I . . . ; m k - 1 ; . . . ) .
(18) (19)
With Eq. (14) it is immediate that the vacuum vector IO)r for the Bogoliubov transformed annihilation operators at,(f), f E E, is given by 16~)r, = exp{itHq}lO),
(20)
that is, we have the relations ar,(f)lO)r, = 0
VfEE.
(21)
It is obvious that our above reasoning is valid, if Hq is a well defined self-adjoint operator on the Fock space Y+(E). In the following sections we investigate under which conditions - concerning the macroscopic pumping parameters ~n E C - Hq has a meaning as an operator on Fock space, and when not. Since for arbitrary pumping parameters ~n E C for each t E ~ the real-linear mapping Tt from Eqs. (12) and (13) is a well-defined symplectic transformation on E, the associated Bogoliubov transformed field respectively annihilation operators, ~r, ( f ) = ~ ( T t f ) and ar,(f)=a((Tt)lf)+a*((Tt)af), f E E, are well defined on Fock space. Thus, for 2 [~0 are the natural numbers including zero.
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the case where Hq does not exist, two questions arise: First, are there nevertheless some unitaries Ut on Fock space such that q ~ r , ( f ) - - U r n ( f )U* similarly to the equations (11) and (14)? Secondly, are there some vectors [O)r, in Fock space fulfilling the relations (21)? For a clarification o f our observations and results we have to deal with the representation theory of the CCR, respectively, o f the photon Weyl algebra.
5. Representations o f the C C R
By construction the quantization procedure leads to the CCR (8) and the real-linearity (7) for the fundamental field observables o f the quantum describtion of the photon field - the observables of the magnetic and electric field summarized in the smeared field expressions ( b ( f ) with complex testfunctions f E E . These purely algebraic relations have to be realized on some complex Hilbert spaces 9¢f by complex-linear self-adjoint field operators ~ ( f ) , f E E, acting on ~ , [7,8]. Every such a realization usually is called a realization of the CCR on the realization Hilbert space ~vg~. It depends on the physical problem and the physical states under consideration to choose the relevant realization(s) o f the CCR. How to obtain the physically relevant realizations of the CCR? In (non-relativistic) algebraic QED the Weyl algebra ~#/'(E) over the one-photon testfunction space E is regarded as the C*-algebra o f fundamental observables for our photon field [8]. ~ ( E ) is uniquely generated by unitary elements W ( f ) , f E E, satisfying the Weyl form of the CCR (often also denoted the Weyl relations), [10, Theorem 5.2.8],
W ( f ) W ( g ) = e x p { - ~ I m ( f l g ) } W ( f +g),
W(f)* = W(-f)
Vf, g E E . (22)
For obtaining realizations o f the CCR one considers (Hilbert space) representations 3 of the photonic C*-algebra ~V'(E). Indeed, for each regular 4 representation /7 of ~ ( E ) one obtains a realization of the CCR on the representation Hilbert space ~ ~- -~)7 with the field operators ~ ( f ) , f E E , as the generators of the represented unitary groups { F l ( W ( t f ) ) l t E ~} [10], d (if
cb(f)=-i-;7II(W(tf))lt=o
for every f E E .
(23)
3 The W(f), f C E, only are elements of the (abstract) C*-algebra ~¢/'(E), especially they do not act on a Hilbert space. Nevertheless they often are called Weyl operators. In a representation 17 of ~-(E) on the (complex) representation Hilbert space ~ ------~rt every element A E ~(E) is mapped onto a complex-linear bounded operator ll(A) acting on ~ Especially, the represented Weyl operators ll(W(f)) now really are operators, which act on .~ 4 Regularity of a representation H of "#/'(E) is the smoothness condition that for each f E E the represented unitary group {H(W(tf)) I t E ~} be continuous with respect to the real parameter t. Other representations of ~#"(E) than the regular ones are void of physical interest.
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The self-adjoint field operators ~ ( f ) , f E E, fulfill the CCR (8) and the real-linearity in the testfunction argument (7). Further, as in Eq. (9) one immediately reconstructs the annihilation and creation operators, a(f) resp. a*(f), which satisfy the relations (6) and (5). By construction the field, annihilation and creation operators depend on the considered regular representation/7 of ~¢/'(E) and act on the representation Hilbert space jig. Different regular representations of ~ ( E ) give different realizations of the CCR. 5 For an infinite-dimensional one-photon testfunction space E one has a continuum of inequivalent (regular) representations of ~#/'(E), which describe physically different aspects of the photon system. For finite dimensional E, however, Stone-von Neumann's uniqueness result (cf. e.g. [10,16,17]) ensures that each regular representation of ~¢'(E) is a direct sum of Fock representations, or equivalently, every (regular) field state on ~¢/'(E) is given by a density operator on Fock space. Thus, from the algebraic point of view, the Fock representation/7~ of ~W(E) with the Fock space ~+(E) as representation Hilbert space is only one special representation of ~¢/~(E), respectively, gives only one special realization of the CCR. It is selected by the demand of the existence of a vacuum vector - defined to fulfill (16) - as an element of the representation Hilbert space. For distinguishing the Weyl operators W(f) as element of the C*-algebra ~¢/(E) from those represented and acting on Fock space in the following we write W~-(f) - / 7 ~ ( W ( f ) ) . 6
6. Bogoliubov transformations on the photon C*-Weyl algebra In the algebraic formalism the Bogoliubov transformation associated with the symplectic transformation T on E is the (unique) .-automorphism Ctr on ~¢/'(E) satisfying 7
~r(W(f))~- W(Tf)
Vf EE.
(24)
By (23) the relation (24) leads for each regular representation of ~¢F(E) to the above notion of a Bogoliubov transformation, where every field operator ~ ( f ) is mapped 5 However, there are realizations of the CCR, which do not arise from a regular representation of W ( E ) (cf. e.g. [14-16, Section VIII.5, 17, Ch. 6]). But there are physical reasons (cf. [11, Section 2.12] and references therein), which imply the Weyl relations (22) to be the more fundamental relations than the CCR (8). And the examples of realizations of the CCR, which do not arise from a regular representation of "W(E), violate the isotropy and homogenity of the testfunction space E. Therefore in QED only those realizations of the CCR are of interest, which arise from the regular representations of W'(E). 6 In the quantum optic literature the Fock represented Weyl operators l ( ~ ( f ) , f C E, sometimes are called displacement operators, a naming which stems from Eq. (28) below. With the orthonormal basis {et . . . . . em} of E using the Eqs. (5) and (4) with 7~ :=2-1/2(e, l i f ) one easily shows that N
/J(~-(f) = e x p { i ~ ( f ) } =exp{2-1/Z(a*(if) -- a ( i f ) ) } = I J exp{~Ta*(e-) -- 7f~a(e,)}. n
I
7 A *-automorphism c~: d --~ ~¢ on a C*-algebra d is a bijective complex-linear mapping, which preserves the algebraic stucture, that is, ~(AB)=~(A)c~(B) and c~(A*)=~(A)* for all elements A and B of d . This concept is a more precise version of the notion of a canonical transformation.
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onto the transformed field operator ~ r ( f ) - ~ ( T f ) acting on the same representation Hilbert space. The Bogoliubov transformation ~r on ~/¢/'(E) is called unitarily implementable with respect to the representation 11 on the representation Hilbert space fig, if there exists a unitary operator U (depending on the symplectic T) acting on fig so that 11(~T(A)) = U H(A) U* for all A E ~/'(E). U in general may be non-unique. It is known that ~T is unitarily implementable with respect to the Fock representation, if and only if the complex-anti-linear part T, of the real-linear symplectic T is a HilbertSchmidt operator on the one-photon testfunction space E (see [13] and references therein). Hence, for infinite-dimensional E there exist many Bogoliubov transformations which are not describable by some unitaries on Fock space. For the symplectic Tt from (12) the unitary implementability of ~r, in Fock space is treated by Theorem 2 below.
7. Squeezed vacua and general states of the photon-field Consider any (regular) representation H of the algebra ~¢/~(E) of photon observables with the representation Hilbert space fig. Then every density operator ~ on fig gives a (possibly mixed) state co of the photon field with expectation values for the observables (A),,~ = (oo; A) = t r ( 0 H ( A ) ) ,
A E ~F(E),
where tr(.) denotes the usual trace on the representation Hilbert space fig. Considering the density operators for all representations of the photonic C*-algebra #"(E), one can show that the set of all states of the photon field is just given by the convex set of all positive, normalized (i.e., (~o; l) = 1 ) complex-linear functionals ~o on ~/¢~(E). This agrees with the C*-algebraic definition of the state space of a C*-algebra (e.g. [10, Subsection 2.3.2]). The expectation mapping C,,, for the Weyl operators E ~ f~--~(W(f)),~,= (o9; W ( f ) ) = C o , ( f )
(25)
commonly is denoted as the characteristic function of the field state vJ (see [18, p. 58 and p. 62 f]). The characteristic function C,,, of the photon field state ~o (resp. 0) 8 s For finite-dimensional testfunction space E let us exhibit the connection to the usually treated phase space formalism. Taking an orthonormal basis of modes {ej . . . . . eN} for E (where N C 1~ is finite) each ./'E E decomposes f = ~ N 1 fl.e. with //. = (e. I f ) , defining a unitary representation of the N-dimensional testfunction space E as the phase space C N ~ ~2N with phase space points fl = ([fi . . . . . /~N) C C u. Decomposing the characteristic function C~ from Eq. (25) as
C,,, -iv~
[~,e, = C~,,(fl) s =
e x p { - 2lfll ] 2 }c,,,(fl)=exp{ x l 2lfilZ}c~(fl) ,
n=l
where I/~12- ~ N 1 I/~. 12, we obtain the well-known characteristic functions C~, C N, and cA in symmetric (or Weyl), normal, and antinormal ordering, which by Fourier transformation give the W- (or Wigner-), P-, and Q-representation of our photon state 09, respectively, [7,18].
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contains all statistical informations about the distribution of the field expressions q~(f), f E E: the expectation values of the moments of the field observable ~ ( f ) are obtained by differentiating Co~(tf) = (09 ; e x p { i t ~ ( f ) } ) with respect to the real parameter t in a similar way as in classical probability theory. For example, one calculates the variances as
Var(09;q~(f)) = (09; q ~ ( f ) 2 ) _ (09; qb(f))2
- (d(09; W(tf)) t=o)2_ d2(09; W(tf)) t=0 dt
dt 2
"
Let us now describe the above squeezing operations with the introduced state concept using characteristic functions. The characteristic function of the vacuum state 09o is
C~oo(f) =- (09o; m ( f ) ) = e x p { - l l l f H 2) V f E E , which obviously in the Fock representation is given with the vacuum vector ]O) by C~0(f) = (OiW~(f)[O) Vf rE. For each t C ~ the squeezed vacuum state 09T, is given in terms of the Bogoliubov transformation ~r, associated with the symplectic Tt by the relation 09r, -- 09o o ~r,, which leads to the characteristic function
Coor,(f)=(09r,; W ( f ) ) = ( 0 9 0 ; W(Ttf))---exp{-¼llTtfl[ 2} V f CE.
(26)
For every testfunction f 6 E with J f = - f we have Ttf = e x p { - t S } f , and hence for t > 0 the variances Va r(09r,; 4 ( f ) ) = ½[ITtf][ 2 are smaller than the vacuum fluctuations Var(090; q~(f))-- ½1lUll 2 [19]. This illustrates, in which way the squeezed vacuum states can be dealt with in a representation independent manner. Nevertheless it is important to know, under which conditions - concerning the macroscopic pumping parameters ~n - the squeezed vacuum states 09r, are given by vectors qT, from Fock space, i.e., C o ~ , ( f ) = (qr,[Wj(f)[ ~lr,) Vf E E. Obviously, if the squeezing Hamiltonian Hq is a well-defined self-adjoint operator on Fock space, then the r/r, are given by (see also Theorem 3 below)
qr, = exp{-itHq}lO) .
(27)
Also if qT, is not a vector in Fock space, the characteristic function C~T, is well defined in (26) and belongs to the well defined algebraic state 09r, of the photon field (since the symplectic transformations Tt, t E R, are well defined for arbitrary pumping parameters fin). By means of the GNS-representation theorem (from the mathematicians Gelfand, Naimark and Segal, cf. [10, Theorem 2.3.16]), one has also in this case a representation Hilbert space ovgr, with associated representation Hr,, and a normalized vector qr, E ~¢~r, such that (a) C,or,(f)=(qzlHr,(W(f))lqr,) for all f E E ; (b) qr, is cyclic for the represented Weyl algebra 1-IT,(~f(E)). Furtheron the GNS-representation is uniquely determined by (a) and (b) up to unitary equivalence.
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8. Weak and strong squeezing Here we take into account infinitely many orthonormalized modes {e, I nE ~} of the photon field, that is, we suppose N = oc. In the present section we treat the Fock representation H ~ , i.e., the Fock space ~+(E) is the representation Hilbert space. Especially the field, annihilation and creation operators as well as their Bogoliubov transforms here are considered in the Fock representation. We first investigate under which conditions the squeezing quadratic Hamiltonian 1
OC
Hq = ~ ~
(~a*(e~) 2 +-~a(e,,) 2) ,
n=l
associated with the macroscopic pumping parameters (, E C, n E N, is a well-defined self-adjoint operator acting on Fock space. By definition Hq is approximated with the sequence of Hamiltonians H~ := i1 ~ =Ll ((,a*(en) 2 + ~a(e~)2), where L E ~. For a first impression for the condition under which Hq is a well defined operator on Fock space, let us determine the norm with the (Fock-) vacuum vector 10). With the Eqs. (16) and (18) it is immediate that IlnqlO)ll2-- E ~ I~12. We conclude that the vacuum vector 169) is an element of the domain of definition for/-/q, if and only if the sequence ( ( , In E ~ } of pumping parameters is square summable.
Theorem 1. For Hq the following two assertions are valid: (a) If ~-~n~_l [~n[2 m. The displacement relations i a(g)W~(h ) = W~(h )(a(g) + ~(glh) a*(g)VF¢(h) = W:~(h)(a*(g)- i
l),
(28)
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then imply W~(h)q E ~ for all h C E which are a finite linear combination of the basis elements {en In E N}. Now [10] Proposition 5.2.4 shows that .~ is a norm-dense complex subspace of ~+(E), which gives the existence of a ff E-~ with 4-particle component ~k4E ~4+ satisfying II~O41l 2 < 6 -1 and the 0-particle component to be exactely the vacuum vector [O). For the Fock vectors ]ml;m2;...) from Eq. (17) in the following we write only those numbers which are occupied, e.g., [ mk = 2) _= [ 0;... ; 0; mk = 2; 0;... ) means that the kth mode is two times occupied and all other modes are not. From now on we assume ~, C E (turn over to the phase-shifted modes exP{½arg(~n)}en). For 4 = ~-~k~_l ~k [ m k = 2 ) E ~ + we obtain with (18) and (19) for all LC L
~n((a*(en)2~t~t4) + (a(e")2~[O))
2(¢ ] H~k) = 2(H~¢ [ ~9) = Z n=l
= v/2 Z
~n
~(mn = 4 ] ~k4)+
~-(m, = 2; mk = 21 ~k4) + ~ n n~k=l
n=l
Za°, n=l
k=l
where {. I • )~ is the scalar product of the complex Hilbert space o ~ - 12(N) of complex square summable sequences fl = (fin c C I n c N), and ~ := (e~ I n c N), ~L := (~1, .... ~L,0,0 .... ), and A is the matrix operator on ~ defined by (Afl)~ := }-~--i a,,kflk with a~,~:=v'~(O41m~=4) and a~,k:=(O41mn=2; ink=2) for k#n. Since the Im~ = 2; mk =2) and Im~ =4) form an orthonormal system for the 4-particle space o~ + , A is a Hilbert-Schmidt operator on ~ (see e.g. [20] Theorem 6.22). The estimation
IIAII2~IIAII~s= ~
la.,k12~<611~'4112<1
n,k=l
(flAIIHs denotes the Hilbert-Schmidt norm of A) implies that ±1 are contained in the resolvent set of the operator A, and hence the range of A + I is all of ~g'. Since the above conclusion holds for all ¢ E ~ + of the form ¢ = ~-']~ ~ I m~ = 2) ~ + the sequences ~ range over all of Of. Hence, because ~9~ .~, we conclude that the weak limit w - l i m c ~ ff~ exists in ~(. But, since each weakly convergent sequence has to be norm-bounded (e.g. [20, Theorem 4.24]), we have a contradiction to 2 ~ lim IIffL I1.*=
L---+~
I~n12= ~ 0 "
n=l
Consequently, our assumption that there exists a 0 # r/~ ~ is wrong.
[]
Let us mention that every Glauber vector and every finite particle vector is an element of the domain ~N of the number operator. Thus, by part (a) of the above
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theorem, the Glauber vectors and the finite particle vectors are contained in the domain of definition for Hq, whenever the pumping parameters are square summable. For the following it is important to note that for arbitrary parameters (, (i.e., also for ~nC~l ](nl 2 =OO) the symplectic transformations Tt on E from the Eqs. (12) and (13) are well defined. Thus in any case of the (n the (algebraic) squeezed vacuum states tot, = ~o0 o ~r, exist and are given by the characteristic functions (26).
Theorem 2. Let Tt be the group of symplectic transformations given by the ~ according to (12) and (13). Then for each real t ~ 0 we have the following equivalences: (ii) the Bogoliubov transformation ~r associated with Tt is unitarily implementable in Fock space, or equivalently, there exists a unitary U, on Fock space such that
W~(TtJ')= Ut W.~(f)U7
V f EE.
(29)
Especially, (i) implies (ii) to be valid for all t E ~. The implementing unitaries Ut, t E ~, are uniquely (up to an arbitrary phase) given by Ut = exp{itHq}. Further, (29) implies q~r,(f) = ¢P(Ttf) = U, ~ ( f ) [Jr* (cf. the Eqs. (11 ) and (14)).
Proof ( i ) ~ (ii): By Theorem 1 Hq is well defined on Fock space. The above algebraic calculations lead to (11) valid in Fock space, which is equivalent to (29) with U, := exp{itHq }. (ii)=~(i): As mentioned above, the complex-anti-linear part (Tt)a =Jsinh(tS) from Tt has to be a Hilbert-Schmidt operator on E, that is,
I~nl2~0
~ > ~-~sinh(tl~nl)2 = ~1 ~-~(cosh(2tl~.l)- 1) ~t 2 n--1
n--I
n--I
which gives (i), since t ¢ 0. But then Hq is well defined and fulfills
exp{itHq}W~(f)exp{-itHq} = W#(Ttf) = Ut v/~(f)U*,
f E E,
which implies Ut*exp{itHq}W~(f ) = W ~ ( f ) Ut*exp{itHq}. But { W ~ ( f ) I f c E} is an irreducible set of operators on Fock space, and thus Ut*exp{itHq}=C,~ with some
ct E C. [] Theorem 3. Let Tt again be the group of symplectic transformations given with the ~n according to (12) and (13). Then for each real t ¢ 0 we have the following equivalences:
(i) En% I~.12<~, (ii) the (algebraic) squeezed vacuum state O9y, is given by a normalized vector qv, from Fock space, i.e., C,,< ( f ) = (ttv, IW.~(f) I qv,) Vf E E, (iii) in Fock space there exists a vector 16))y, fulfilling av,(f)[tg)Z = 0 y f ~E, that is,
Eq. (21).
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Especially, (i) implies (ii) and (iii) to be valid for all t C ~. The vectors t/T' and IO)7;, from (ii) resp. (iii) are uniquely (up to an arbitrary phase) given by the Eqs. (27) resp. (20), that is, t/T, =exp{-itHq}lO) and IO)7;, =exp{itHq}lO). If ~ = l I~.12= ~ , then all squeezed vacua coT' may be represented also by vectors t/T, but from the GNS-Hilbert spaces YET,. The YET, are pairwise non-equivalent representation Hilbert spaces for every pair t ~ t ~. Moreover, for t ~ 0 neither in Fock space nor in YET' there exists a non-zero vector IO)T, fulfilling (21).
Proof. ( i ) ~ (ii) and (i)=~> (iii) follow from the above theorems with t/T, :=exp{--itnq} IO) and IO)T, :=exp{itHq}lO). Let us prove (ii) ~ (i): From toT, = too o aT, it follows (t/rlH~(A) I ~T') = (OII!,~(~r,(A))lO) VA E ~t¢/~(E), which implies for all B E ~/¢r(E) II//.~(B)t/T, II2 = (t/T, IIIF( B* B ) I t/T' ) = ( OlI75( ~T'( B* B ) )I O) = III1F(aT'( B ) )IO) II2.
Because of this, and since the Fock representation/7.~ of ~q/'(E) is irreducible, the operator Ut defined by UtFI~(B)t/T, :=//,~(~T'(B))IO) for all B c ~qF(E) uniquely extends to a unitary acting on Fock space. Further, we have
U,17 ~(A )s-I.~( B )t/T' = U, II.~(AB )t/T' = II.~( ~T'(AB ) )IO ) = 17.~(~T,(A))II.~(~T'(B))IO) =
H~(OCT,(A))Ufl7~(B)n z
VA,B E ~g/~(E),
which yields UtlI ~(A ) = FI ~( CtT,(A) )Ut, respectively, FI~( ctT'(A) ) = UtlI ~(A )U*VA E ~g/'(E), the unitary implementability of the Bogoliubov transformation aT, in Fock space. From Theorem 2 it follows (i) and that Ut is uniquely (up to a phase) given by Ut =exp{itHq }. Let us prove ( i i i ) ~ (i): As in Eq. (17) we may construct an orthonormalized system of vectors in Fock space oo
Iml;m2;...)T,:=H t~aT" j=l
(ej) ) IO) T,, mj E No,
y~. mj < c~3, j=l
for which the Eqs. (18) and (19) are analoguously valid. Let ~ be the sub-Hilbert space of the Fock space ~-+(E), which is spanned by the Iml;mz;...)T,. Then the transformed field operators ~ T ' ( f ) = ~(Ttf) leave 3¢" invariant, and so do the exponentials W~(g) =exp{i~(g)} =exp{i~T'(Tt-lg) } for every g E E. But { W~(g) 19 C E} is an irreducible set of operators on Fock space, and consequently ~ = ~+(E). Now define the unitary Ut on Fock space by U t l m l ; m 2 ; . . . ) : = Iml;m2;...)T'. It follows Ut a#(f)Ut* = a ~ ( f ) , which yields (29). From Theorem 2 it follows (i) and that Ut is uniquely (up to a phase) given by Ut =exp{itHq}. That all squeezed vacua toT, may be represented by vectors t/r, from Hilbert spaces YET' follows from the GNS-representation theorem as mentioned at the end of the previous section. By Theorem 2 for each t ¢ 0 the GNS representation of the squeezed
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vacuum state ~or, on the representation Hilbert space ~f~r~ is non-equivalent to the Fock representation, if and only if ~--],~1 [~,[2 = ~ . Since :~v,, t E E, is a group of *automorphisms on ~W(E) shifting the unitary equivalence, this implies that the ~ r , are pairwise non-equivalent representation Hilbert spaces for every pair t ¢ / . If for t ¢ 0 in o'¢gv,there is a vector 0 ¢ IO)r, E ~ r , fulfilling (21), then as above one may construct a unitary Ut from ~T, onto the Fock space. Thus, the GNS representation of ~or, and the Fock representation are equivalent, which by Theorem 2 is equivalent to (i). [] We see that for the squeezing transformations associated with the classical pumping parameters ~, there are two different cases: • Weak squeezing, that is, ~n~l ]~,[2< cx~. The squeezing transformations are unitarily implementable in the Fock representation by the unitaries exp{itHq}, and the squeezed vacuum states are given by vectors in Fock space (with infinitely many m-particle components). • Strong squeezing, that is, ~,~1 ]~.]2 = ~ . The squeezed vacua are non-Fock and mutually inequivalent. For the notion of a squeezed vacuum one should notice, that for both weak and strong squeezing there exists the cyclic GNS-vector qr~ for the algebraic state ~or,. But only for weak squeezing r/r, may be realized in Fock space. And also only for weak squeezing there exists a non-zero vector IO)r, which is annihilated by the Bogoliubov transformed annihilation operators ar,(f) as in Eq. (21), and this vector is always in Foek space. The fluctuations of the Bogoliubov transformed field operators CbT,(f) are reduced in qr,, but are not squeezed in ]O)v,. ~br,(f) and ]O)v, are simultaneously obtained from • ( f ) and ]O) by the unitary transformation exp{itHq } in Fock space, and thus are only a different realization of the original Fock structure. In this sense the Fock structure is unique. 9
Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft.
References [1] H.P. Yuen, Generalized coherent states and the statistics of two-photon-lasers, Phys. Lett. A 51 ( I + 2 ) (1975); Two-photon coherent states of the radiation field, Phys. Rev. A 13 (1976) 2226-2243; H.P. Yuen, J.H. Shapiro, Optical communication with two-photon coherent states, Part 1 and IlL IEEE Trans. Inf. Theory IT-24 (1978) 657-668, and, IT-26 (1980) 78 93; Part II Opt. Lett. 4 (1979) 334; D.F. Walls, Squeezed states of light, Nature 306 (1983) 141 146; R. Loudon, P.L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709-759. 9 This uniqueness is however lost, if one employs for the transition from classical electrodynamic to QED (quantization) a different complexification of the original real, sympleetic testfunction space.
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[2] J. Huang, P. Kumar, Photon-counting statistics of multimode squeezed light, Phys. Rev. A 40 (1989) 1670-1673; X. Ma, W. Rhodes, Multimode squeeze operators and squeezed states, Phys. Rev. A 41 (1990) 4625-4631; A.K. Ekert, P.L. Knight, Relationship between semiclassical and quantummechanical input-output theories of optical response, Phys. Rev. A 43 (1991) 3934-3938. [3] C.M. Caves, Quantum limits on noise in linear amplifiers, Phys. Rev. D 26 (1982) 1817-1839; C.M. Caves, B.L. Schumaker, Broadband squeezing, Vol. 12, Springer Proc. Phys., Springer, Berlin, 1986, pp. 20-30; C.M. Caves, D.D. Crouch, Quantum wideband traveling-wave analysis of a degenerate parametric amplifier, J. Opt. Soc. Am. B 4 (1987) 1535-1545. [4] R. Honegger, A. Rieckers, The general form of non-Fock coherent Boson states, Publ. RIMS Kyoto Univ. 26 (1990) 397-417; First order coherent Boson states, Helv. Phys. Acta 65 (1992) 965-984; On higher order coherent states on the Weyl algebra, Lett. Math. Phys. 24 (1992) 221 225; R. Honegger, A. Rapp, General Glauber coherent states on the Weyl algebra and their phase integrals, Physica A 167 (1990) 945-961. [5] R.J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130 (1963) 2529-2539; Coherent and incoherent states of the radiation field, Phys. Rev. 131 (1963) 2766-2788; In: C. de Wit-t, A. Blandin, C. Cohen-Tannoudji (Eds.), Quantum Optics and Electronics, Proc. Les Houches 1964, Gordon and Breach, New York, 1965. [6] R. Honegger, A. Rieckers, Quantized radiation states from the infinite Dicke model, Publ. RIMS Kyoto Univ. 30 (1994) 111-138. R. Honegger, The dynamical generation of macroscopic coherent light, in: W. Gans, A. Blumen, A. Amann (Eds.), Large-Scale Molecular Systems: Quantum and Stochastic Aspects. Beyond the Simple Molecular Picture, Proc. NATO ASI Maratea, Italy, Plenum Press, 1991; Quantized radiation from collectively ordered atoms, in: Classical and Quantum Systems Foundation and Symmetries, Proc. II. Intemat. Wigner Symp. Goslar, Germany, July 1991, World Scientific, 1992; The weakly coupled infinite dicke model, Physica A 225 (1996) 391-411; Th. Gerisch, R. Honegger, A. Rieckers, The Josephson Microwave Radiation States from Tunneling Cooper Pairs, T/ibingen, 1997, preprint. [7] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons & Atoms, Introduction to QED, Wiley, New York, 1989. [8] R. Honegger, Globale Quantentheorie der Strahlung, Thesis, Tfibingen 1991. [9] H. Haken, Light 1, North-Holland, Amsterdam, 1981. [10] O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics I, II, Springer, Berlin, 1979, 1981. [11] A. Galindo, P. Pascual, Quantum Mechanics I, II, Springer, Berlin, 1989, 1991. [12] R. Loudon, The Quantum Theory of Light, Clarendon Press, Oxford, 1979. [13] R. Honegger, A. Rieckers, Squeezing Bogoliubov transformations on the infinite mode CCR-algebra, J. Math. Phys. 37 (1996) 4292-4309. [14] C.R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, Berlin, 1967. [ 15] R. Honegger, On Heisenberg's uncertainty principle and the CCR, Z. Naturforsch. A 48 (1993) 447-451. [16] M. Reed, B. Simon, Methods of Modem Mathematical Physics I, Academic Press, New York, 1980. [17] E. Prugoverki, Quantum Mechanics in Hilbert Spaces, Academic Press, New York, 1971. [18] H.M. Nussenzveig, Introduction to Quantum Optics, Gordon and Breach, London, 1973. [19] R. Honegger, A. Rieckers, Squeezing of Optical States on the CCR-Algebra, Publ. RIMS Kyoto Univ., to appear; Squeezed Variances of Smeared Boson Fields, Helv. Phys. Acta 70 (1997) 507-541. [20] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, Berlin, 1980.
Physica A 242 (1997) 423-438
Squeezing operations in Fock space and beyond Reinhard Honegger, Alfred Rieckers* Institut fiir Theoretische Physik, Universitdt Tiibingen D-72076 Tiibingen, German), Received 25 October 1996
Abstract Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.
PACS: 03.65.Fd; 05.30.Jp; 42.50.Dv; 02.30.Tb Keywords: Weak and strong squeezing operations; Existence of quadratic Boson Hamiltonians; Inequivalence of squeezed vacua
1. Introduction In the theoretical descriptions o f squeezing processes the Hamiltonians mostly are approximated by quadratic expressions o f the photon field (interaction picture). Let us start our discussion with these formal expressions and deal later on with their deeper meaning and mathematical definition. The Hamiltonians are meant to describe essential features o f the dynamics which takes place in the non-linear optical medium. There are (at least) two types o f photons, belonging to the signal resp. to the idler beam. Thus, usually two types o f quadratic Hamiltonians are distinguished: • The degenerate case [1]: N 1 Hq, d : ~ ~ ( ~ n a * a * n +-~nanan), n-I * Corresponding author. Fax: 07071 29 5850; e-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved. PII S0378-4371 (97)001 81-7
(1)
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where the squeezing parameters (n E C (and their complex conjugates (n) arise from a classical (macroscopic) pumping field. Here the a n* =-a*(en) are the creation, and the a n - - a ( e n ) are the annihilation operators of the orthonormalized modes {e~ ..... eN}, which fulfill the canonical commutation relations (CCR) [a(en),a*(em)]=rn, m~,
[a(en),a(em)]=O,
l ~m,n<<,N.
(2)
• The nondegenerate case [2]: M
nq, nd = Z (rlkas, kai, k q- qk as, kai, k ) ,
(3)
k=l
where the index s stands for the signal modes Us,k and i for the idler modes ui, k (which all are supposed to be mutually orthogonal and normalized), with the annihilation operators as.k =--a(us, k) for the signal modes Us,k, and ai.k =a(Ui,k) for the idler modes ui, k. The r/k E C are classical pumping parameters, too. Usually only finitely many modes are considered, but there also are some investigations with infinitely many modes [3]. By not specifying N and M we cover all the cases of finitely, and infinitely, respectively, continuously many photon modes. The above two cases of squeezing quadratic Hamiltonians are not so different as they seem to be: By superposing the idler and the signal modes in the smeared field formalism, we transform the nondegenerate squeezing Hamiltonians Hq, nd into the form (1). This reveals the degenerate form of a squeezing Hamiltonian to be the more general one. Then we investigate the Bogoliubov transformations arising from the dynamics associated with the Hamiltonians Hq,d and Hq, nd. We show that for infinitely many modes the squeezing operations associated with the classical pumping parameters ~n, resp. r/k may become so strong that the quadratic Hamiltonians have no meaning as operators in Fock space, and the squeezing Bogoliubov transformations lead beyond the Fock representation. In this case of a strong squeezing, the squeezing procedure leads into a representation of the CCR which is inequivalent to the Fock representation. Nevertheless, it is a well-defined squeezing transformation in the smeared field formulation, and the squeezed vacuum state (characterized by some reduced fluctuations) has a well-defined vector in these non-Fock representation spaces. For strong squeezing one has, however, no (vacuum) vector which is annihilated by the Bogoliubov transformed annihilation operators. These and other facts are derived here in the smeared field formalism in an almost self-contained fashion. Moreover, we try hard to explain the basic notions of this formalism in such a way, that also those readers, who had worked hitherto with the Fock space formalism only, should be able to realize the implications of our statements. This much wider frame of the quantized field formalism is unavoidable, if one deals with the optical properties of photon states. This has been pointed out in the optical coherence theory [4] (which is based on Glauber's original works [5]) and the macroscopic
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radiation theory [6], and will be further substantiated in terms of special squeezing transformations.
2. The smeared field formalism
In the smeared field formalism the creation and annihilation operators are "smeared" by testfunctions f , which are taken from the so-called one-photon testfunction space E. What about the physical meaning of E? E is spanned by the photon modes taken into account, e.g., {el ,eu} or {u.,.l . . . . . Igs,M,Ui, I,...,Ui, M}. Since each mode is a normalized function we have that E is a complex subspace of a complex Hilbert space Y~ of square integrable functions with scalar product (. ] . ) (which is complexlinear in the right factor and complex-anti-linear in the left one). For example, when quantizing in the Coulomb gauge in position space with quantization volume A C R 3 and taking all directions of polarization into account we have that ~ consists of all square integrable divergence-free (i.e. 27. f = 0) functions f on A with f ( x ) E C 3 for all x E A. ~J has the scalar product ( f l g) = JA f ( x ) . g(x) d3x for f , g E J~, where a. b---albj +a2b2 +a3b3 for a=(al,az,a3)cC 3 and b=(bj,b2,b3)cC 3, [7-9]. With an orthonormal basis {el eu} of the complex one-photon space E the smeared annihilation and creation operators are given by ([10], cf. also [l l] Section 13.6) ....
.....
N
a ( f ) = ~ (fle,,)a(e,,),
N
a*(f)=~(e,,If)a*(e,,),
n=l
N
f -- ~ (e,,I.f)e,,EE,
n--I
n--I
(4) from which it immediately follows that a ( f ) and a * ( f ) are adjoint to each other, and an important point that the operator-valued mapping f E E~-+a*(f) is complexlinear, and f E E H a ( f ) is complex-anti-linear, i.e., for all f,.q E E and ~,fl E C, one has
a*(:~f +flg)=c~a*(f)+fla*(g),
a(:~f + f i g ) = ~ a ( f ) + f i a ( g ) .
(5)
One also easily checks from (2) that in the smeared formalism the canonical commutation relations (CCR) with the commutator [A,B] = A B - BA are expressed as
[a(f),a(g)]=[a*(f),a*(9)]=O,
[a(f),a*(g)]=(flg)n,
Vf, g E E .
(6)
The advantage of the smeared formalism is the independence from the orthonormal basis of E, which makes many manipulations more transparent, and is indispensable for a mathematically consistent formulation of non-relativistic QED. Let us turn to the smeared selfadjoint field operators ~ ( f ) : = 2 - W 2 ( a ( f ) + a*(f)), f E E. Here the mapping f E E ~-~ t/,(f) is only real-linear, i.e,
cb(af + f l O ) = a ~ ( f ) + f l ~ ( g ) Vet,tiER,
Vf, g E E .
(7)
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From the quantization procedure in the Coulomb gauge it is well known, that for realvalued testfunctions f the ~ ( f ) are the observables of the smeared vector potential, and for proper imaginary f (i.e. if(x)C ~3 VxEA) the ~ ( f ) are proportional to the observables of the smeared transversal electric field, [7-9, 12]. Thus, the selfadjoint field operators ~ ( f ) , f E E , are the fundamental observables for the quantum description of the photon field, since they generate all other observables. In the smeared formalism the field operators fulfill the CCR in the following form (with the scaling h=l)
[~(f),¢(g)]=ilm(fl9)~,
Vf, oEE,
(8)
which give rise to the Heisenberg uncertainty relations for the expectation values of conjugate field observables ~ ( f ) and cP(if), f cE. Obviously, the smeared annihilation and creation operators, a(f) resp. a*(f), are obtained back from the field operators by 1
a(f)=2-1/2(q~(f)+i~(if)),
a*(f)=2-1/e(~(f)-i~(if)).
(9)
Let us now consider the nondegenerate squeezing Hamiltonian Hq,.d. The considered modes are the signal and idler modes {Us,1,...,u~,M, ui,1..... ui,M}, which generate an orthonormal basis of the testfunction space E. Mixing these modes we obtain a new orthonormal basis {e~,...,e2M} of E according to
1
e2k-a:='-~(Us, k+ui, k),
1
e2k:=--~(Us, k--Ui, k),
l<~k<~M.
From the complex-linearity resp. -anti-linearity of f ~ a # ( f ) we obtain the simple transformations a#(Us,k)a#(ui,k)= ~(a I # (e2k-1)2 -a#(e2k) 2) which are valid for both the smeared creation operators (i.e., a#(h)=_a*(h)) and annihilation operators (i.e., a#(h)=a(h)). Inserting this into Eq. (3) yields Hq,nd=½ ~-~n= N 1 (~na*(en)2 + ~n"--a(en)2 ) with ff2k-I =glk,
~2k = -- r/k,
1 <~k<~M,
N=2M,
(10)
which agrees with the form of a degenerate quadratic Hamiltonian. The restrictions (10) demonstrate that the degenerate form (1) of a quadratic Hamiltonian is more general than the nondegenerate form (3). Thus in the following we only treat quadratic Hamiltonians in the degenerate form (1) and write Hq instead of Hq,d.
3. Symplectic- and Bogoliubov-transformations for squeezing Let us start from the degenerate quadratic Hamiltonian Hq. By purely algebraic (representation independent) calculations using the power series for the exponentials I For an up-to-date description of these basic structures cf. [10,4].
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and the CCR one obtains that the squeezing operations associated with Hq are given in terms of Bogoliubov transformations (cf. also [1,13]),
exp{itHq}CI)(f)exp{-itHq} ---=qb(Ttf) =: ~T,(U) V f E E,
t E ~,
(11 )
which are associated with a group of real-linear transformations Tt on the one-photon testfunction space E. More precisely one has
Tt = exp{tJS} = cosh(tS) + J sinh(tS),
t E ~,
(12)
with the complex-linear positive self-adjoint operator S on E and the complex-antilinear involution J on E (i.e., J = J * = j - l ) , where we put (n/](,I := 1 for ~, = 0 , N
S=Zfnlle,,)(enl ,
~n
Jen=l-~nle n VnE{1 ..... N}.
(13)
n=l
Especially for the transformed annihilation operators we obtain by means of Eq. (9) for each f E E exp{itHq } a ( f ) e x p ( - i t H q } = a(cosh(tS)f) + a* (J sinh(tS)f) = at, ( f ) , (14) from which with the en as testfunctions one gets the more familiar formulas [l,2]
ar,(e,) = cosh(tl~n])a(e,,) + ~~n sinh(ti~nl)a*(en) Observe that the complex-linear part of the real-linear Tt is given by (Tt)l = cosh(tS) and the complex-anti-linear part is (Tt)a =Jsinh(tS), which occur in Eq. (14). The ar,(f) are still annihilation operators: they firstly satisfy the CCR (6) with the transformed creation operators a~,(f) = exp{itHq}a*(f)exp{-itHq}, secondly are complexanti-linear in the tesffunction argument, and thirdly are given with the transformed field operators from Eq. (11) by ar,(f)=2-1/2(ebr,(f)+iq)v,(if)). This is due to the fact that the real-linear transformations Tt leave the imaginary part of the scalar product invariant (Eq. (15) below). More generally one defines: A mapping T acting on the one-photon testfunction space E is called a symplectic transformation on E, if it is real-linear, with range T(E)= E, and fulfills
Im(TflTy)=Im(flg)
Vf, g E E .
(15)
For each symplectic T on E the transformation, which maps every field operator ~b(f) onto the transformed field operator ~ r ( f ) - = eb(Tf) is called the associated Bogoliubov transformation. Because of Eq. (15) the transformed q~r(f), f E E , also fulfill the CCR (8). The Bogoliubov transformed annihilation and creation operators, a v ( f ) and a*(f), f EE, are given in terms of the complex-linear part T/ and the complexanti-linear part Tu for our (real-linear) symplectic T - - that is: T = TI + Tu with T I = ~1 ( T - iTi) and Ta = ½(T + iTi) - by aT( f ) = 2 -1/2 . ( ~ r ( f ) + iq~r(if)) = a(TJ) +a*(T~f), and a~(f) as the adjoint [13]. They obviously fulfill the CCR (6).
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4. The Fock space
The field operators q~(f) and the transformed ones q~r,(f) - respectively, the by (9) associated annihilation and creation operators and their transforms - from the previous sections are formal expressions as long as we do not specify the Hilbert space (and therein their domains) on which they are defined to act. Commonly in the quantum optic literature these operators are adopted to act on Fock space. In the present section we will do so, too. Let us, for later reference, recall the basic ingredients of the Fock space ~ + ( E ) for our complex one-photon testfunction space E with orthonormal basis {el . . . . . eN}. The Fock space ~ ( E ) is constructed from the normalized vacuum vector 1 0 ) E ~ + ( E ) satisfying a(f)lO) = 0
VfEE.
(16)
The ruth particle subspace ~ + is generated by the orthonormalized Fock state vectors 2
) [ml;m2;...;mN):= H
N
1 a*(ej) m' I0>,
j=l
= m,
mj E ~o .
j=l
(17) Then , ~ + ( E ) - (~m=0 m is the direct sum of all the m-photon spaces, where .~0+ := C[(9). It is well known that
a*(eD [ . . . ; m k ; . . . )
= V/~-k + I I ...;mk + l ; . . . ) ,
a(ek)l . . . ; i n k ; . . . ) = v/-~-I . . . ; m k - 1 ; . . . ) .
(18) (19)
With Eq. (14) it is immediate that the vacuum vector IO)r for the Bogoliubov transformed annihilation operators at,(f), f E E, is given by 16~)r, = exp{itHq}lO),
(20)
that is, we have the relations ar,(f)lO)r, = 0
VfEE.
(21)
It is obvious that our above reasoning is valid, if Hq is a well defined self-adjoint operator on the Fock space Y+(E). In the following sections we investigate under which conditions - concerning the macroscopic pumping parameters ~n E C - Hq has a meaning as an operator on Fock space, and when not. Since for arbitrary pumping parameters ~n E C for each t E ~ the real-linear mapping Tt from Eqs. (12) and (13) is a well-defined symplectic transformation on E, the associated Bogoliubov transformed field respectively annihilation operators, ~r, ( f ) = ~ ( T t f ) and ar,(f)=a((Tt)lf)+a*((Tt)af), f E E, are well defined on Fock space. Thus, for 2 [~0 are the natural numbers including zero.
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the case where Hq does not exist, two questions arise: First, are there nevertheless some unitaries Ut on Fock space such that q ~ r , ( f ) - - U r n ( f )U* similarly to the equations (11) and (14)? Secondly, are there some vectors [O)r, in Fock space fulfilling the relations (21)? For a clarification o f our observations and results we have to deal with the representation theory of the CCR, respectively, o f the photon Weyl algebra.
5. Representations o f the C C R
By construction the quantization procedure leads to the CCR (8) and the real-linearity (7) for the fundamental field observables o f the quantum describtion of the photon field - the observables of the magnetic and electric field summarized in the smeared field expressions ( b ( f ) with complex testfunctions f E E . These purely algebraic relations have to be realized on some complex Hilbert spaces 9¢f by complex-linear self-adjoint field operators ~ ( f ) , f E E, acting on ~ , [7,8]. Every such a realization usually is called a realization of the CCR on the realization Hilbert space ~vg~. It depends on the physical problem and the physical states under consideration to choose the relevant realization(s) o f the CCR. How to obtain the physically relevant realizations of the CCR? In (non-relativistic) algebraic QED the Weyl algebra ~#/'(E) over the one-photon testfunction space E is regarded as the C*-algebra o f fundamental observables for our photon field [8]. ~ ( E ) is uniquely generated by unitary elements W ( f ) , f E E, satisfying the Weyl form of the CCR (often also denoted the Weyl relations), [10, Theorem 5.2.8],
W ( f ) W ( g ) = e x p { - ~ I m ( f l g ) } W ( f +g),
W(f)* = W(-f)
Vf, g E E . (22)
For obtaining realizations o f the CCR one considers (Hilbert space) representations 3 of the photonic C*-algebra ~V'(E). Indeed, for each regular 4 representation /7 of ~ ( E ) one obtains a realization of the CCR on the representation Hilbert space ~ ~- -~)7 with the field operators ~ ( f ) , f E E , as the generators of the represented unitary groups { F l ( W ( t f ) ) l t E ~} [10], d (if
cb(f)=-i-;7II(W(tf))lt=o
for every f E E .
(23)
3 The W(f), f C E, only are elements of the (abstract) C*-algebra ~¢/'(E), especially they do not act on a Hilbert space. Nevertheless they often are called Weyl operators. In a representation 17 of ~-(E) on the (complex) representation Hilbert space ~ ------~rt every element A E ~(E) is mapped onto a complex-linear bounded operator ll(A) acting on ~ Especially, the represented Weyl operators ll(W(f)) now really are operators, which act on .~ 4 Regularity of a representation H of "#/'(E) is the smoothness condition that for each f E E the represented unitary group {H(W(tf)) I t E ~} be continuous with respect to the real parameter t. Other representations of ~#"(E) than the regular ones are void of physical interest.
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The self-adjoint field operators ~ ( f ) , f E E, fulfill the CCR (8) and the real-linearity in the testfunction argument (7). Further, as in Eq. (9) one immediately reconstructs the annihilation and creation operators, a(f) resp. a*(f), which satisfy the relations (6) and (5). By construction the field, annihilation and creation operators depend on the considered regular representation/7 of ~¢/'(E) and act on the representation Hilbert space jig. Different regular representations of ~ ( E ) give different realizations of the CCR. 5 For an infinite-dimensional one-photon testfunction space E one has a continuum of inequivalent (regular) representations of ~#/'(E), which describe physically different aspects of the photon system. For finite dimensional E, however, Stone-von Neumann's uniqueness result (cf. e.g. [10,16,17]) ensures that each regular representation of ~¢'(E) is a direct sum of Fock representations, or equivalently, every (regular) field state on ~¢/'(E) is given by a density operator on Fock space. Thus, from the algebraic point of view, the Fock representation/7~ of ~W(E) with the Fock space ~+(E) as representation Hilbert space is only one special representation of ~¢/~(E), respectively, gives only one special realization of the CCR. It is selected by the demand of the existence of a vacuum vector - defined to fulfill (16) - as an element of the representation Hilbert space. For distinguishing the Weyl operators W(f) as element of the C*-algebra ~¢/(E) from those represented and acting on Fock space in the following we write W~-(f) - / 7 ~ ( W ( f ) ) . 6
6. Bogoliubov transformations on the photon C*-Weyl algebra In the algebraic formalism the Bogoliubov transformation associated with the symplectic transformation T on E is the (unique) .-automorphism Ctr on ~¢/'(E) satisfying 7
~r(W(f))~- W(Tf)
Vf EE.
(24)
By (23) the relation (24) leads for each regular representation of ~¢F(E) to the above notion of a Bogoliubov transformation, where every field operator ~ ( f ) is mapped 5 However, there are realizations of the CCR, which do not arise from a regular representation of W ( E ) (cf. e.g. [14-16, Section VIII.5, 17, Ch. 6]). But there are physical reasons (cf. [11, Section 2.12] and references therein), which imply the Weyl relations (22) to be the more fundamental relations than the CCR (8). And the examples of realizations of the CCR, which do not arise from a regular representation of "W(E), violate the isotropy and homogenity of the testfunction space E. Therefore in QED only those realizations of the CCR are of interest, which arise from the regular representations of W'(E). 6 In the quantum optic literature the Fock represented Weyl operators l ( ~ ( f ) , f C E, sometimes are called displacement operators, a naming which stems from Eq. (28) below. With the orthonormal basis {et . . . . . em} of E using the Eqs. (5) and (4) with 7~ :=2-1/2(e, l i f ) one easily shows that N
/J(~-(f) = e x p { i ~ ( f ) } =exp{2-1/Z(a*(if) -- a ( i f ) ) } = I J exp{~Ta*(e-) -- 7f~a(e,)}. n
I
7 A *-automorphism c~: d --~ ~¢ on a C*-algebra d is a bijective complex-linear mapping, which preserves the algebraic stucture, that is, ~(AB)=~(A)c~(B) and c~(A*)=~(A)* for all elements A and B of d . This concept is a more precise version of the notion of a canonical transformation.
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onto the transformed field operator ~ r ( f ) - ~ ( T f ) acting on the same representation Hilbert space. The Bogoliubov transformation ~r on ~/¢/'(E) is called unitarily implementable with respect to the representation 11 on the representation Hilbert space fig, if there exists a unitary operator U (depending on the symplectic T) acting on fig so that 11(~T(A)) = U H(A) U* for all A E ~/'(E). U in general may be non-unique. It is known that ~T is unitarily implementable with respect to the Fock representation, if and only if the complex-anti-linear part T, of the real-linear symplectic T is a HilbertSchmidt operator on the one-photon testfunction space E (see [13] and references therein). Hence, for infinite-dimensional E there exist many Bogoliubov transformations which are not describable by some unitaries on Fock space. For the symplectic Tt from (12) the unitary implementability of ~r, in Fock space is treated by Theorem 2 below.
7. Squeezed vacua and general states of the photon-field Consider any (regular) representation H of the algebra ~¢/~(E) of photon observables with the representation Hilbert space fig. Then every density operator ~ on fig gives a (possibly mixed) state co of the photon field with expectation values for the observables (A),,~ = (oo; A) = t r ( 0 H ( A ) ) ,
A E ~F(E),
where tr(.) denotes the usual trace on the representation Hilbert space fig. Considering the density operators for all representations of the photonic C*-algebra #"(E), one can show that the set of all states of the photon field is just given by the convex set of all positive, normalized (i.e., (~o; l) = 1 ) complex-linear functionals ~o on ~/¢~(E). This agrees with the C*-algebraic definition of the state space of a C*-algebra (e.g. [10, Subsection 2.3.2]). The expectation mapping C,,, for the Weyl operators E ~ f~--~(W(f)),~,= (o9; W ( f ) ) = C o , ( f )
(25)
commonly is denoted as the characteristic function of the field state vJ (see [18, p. 58 and p. 62 f]). The characteristic function C,,, of the photon field state ~o (resp. 0) 8 s For finite-dimensional testfunction space E let us exhibit the connection to the usually treated phase space formalism. Taking an orthonormal basis of modes {ej . . . . . eN} for E (where N C 1~ is finite) each ./'E E decomposes f = ~ N 1 fl.e. with //. = (e. I f ) , defining a unitary representation of the N-dimensional testfunction space E as the phase space C N ~ ~2N with phase space points fl = ([fi . . . . . /~N) C C u. Decomposing the characteristic function C~ from Eq. (25) as
C,,, -iv~
[~,e, = C~,,(fl) s =
e x p { - 2lfll ] 2 }c,,,(fl)=exp{ x l 2lfilZ}c~(fl) ,
n=l
where I/~12- ~ N 1 I/~. 12, we obtain the well-known characteristic functions C~, C N, and cA in symmetric (or Weyl), normal, and antinormal ordering, which by Fourier transformation give the W- (or Wigner-), P-, and Q-representation of our photon state 09, respectively, [7,18].
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contains all statistical informations about the distribution of the field expressions q~(f), f E E: the expectation values of the moments of the field observable ~ ( f ) are obtained by differentiating Co~(tf) = (09 ; e x p { i t ~ ( f ) } ) with respect to the real parameter t in a similar way as in classical probability theory. For example, one calculates the variances as
Var(09;q~(f)) = (09; q ~ ( f ) 2 ) _ (09; qb(f))2
- (d(09; W(tf)) t=o)2_ d2(09; W(tf)) t=0 dt
dt 2
"
Let us now describe the above squeezing operations with the introduced state concept using characteristic functions. The characteristic function of the vacuum state 09o is
C~oo(f) =- (09o; m ( f ) ) = e x p { - l l l f H 2) V f E E , which obviously in the Fock representation is given with the vacuum vector ]O) by C~0(f) = (OiW~(f)[O) Vf rE. For each t C ~ the squeezed vacuum state 09T, is given in terms of the Bogoliubov transformation ~r, associated with the symplectic Tt by the relation 09r, -- 09o o ~r,, which leads to the characteristic function
Coor,(f)=(09r,; W ( f ) ) = ( 0 9 0 ; W(Ttf))---exp{-¼llTtfl[ 2} V f CE.
(26)
For every testfunction f 6 E with J f = - f we have Ttf = e x p { - t S } f , and hence for t > 0 the variances Va r(09r,; 4 ( f ) ) = ½[ITtf][ 2 are smaller than the vacuum fluctuations Var(090; q~(f))-- ½1lUll 2 [19]. This illustrates, in which way the squeezed vacuum states can be dealt with in a representation independent manner. Nevertheless it is important to know, under which conditions - concerning the macroscopic pumping parameters ~n - the squeezed vacuum states 09r, are given by vectors qT, from Fock space, i.e., C o ~ , ( f ) = (qr,[Wj(f)[ ~lr,) Vf E E. Obviously, if the squeezing Hamiltonian Hq is a well-defined self-adjoint operator on Fock space, then the r/r, are given by (see also Theorem 3 below)
qr, = exp{-itHq}lO) .
(27)
Also if qT, is not a vector in Fock space, the characteristic function C~T, is well defined in (26) and belongs to the well defined algebraic state 09r, of the photon field (since the symplectic transformations Tt, t E R, are well defined for arbitrary pumping parameters fin). By means of the GNS-representation theorem (from the mathematicians Gelfand, Naimark and Segal, cf. [10, Theorem 2.3.16]), one has also in this case a representation Hilbert space ovgr, with associated representation Hr,, and a normalized vector qr, E ~¢~r, such that (a) C,or,(f)=(qzlHr,(W(f))lqr,) for all f E E ; (b) qr, is cyclic for the represented Weyl algebra 1-IT,(~f(E)). Furtheron the GNS-representation is uniquely determined by (a) and (b) up to unitary equivalence.
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8. Weak and strong squeezing Here we take into account infinitely many orthonormalized modes {e, I nE ~} of the photon field, that is, we suppose N = oc. In the present section we treat the Fock representation H ~ , i.e., the Fock space ~+(E) is the representation Hilbert space. Especially the field, annihilation and creation operators as well as their Bogoliubov transforms here are considered in the Fock representation. We first investigate under which conditions the squeezing quadratic Hamiltonian 1
OC
Hq = ~ ~
(~a*(e~) 2 +-~a(e,,) 2) ,
n=l
associated with the macroscopic pumping parameters (, E C, n E N, is a well-defined self-adjoint operator acting on Fock space. By definition Hq is approximated with the sequence of Hamiltonians H~ := i1 ~ =Ll ((,a*(en) 2 + ~a(e~)2), where L E ~. For a first impression for the condition under which Hq is a well defined operator on Fock space, let us determine the norm with the (Fock-) vacuum vector 10). With the Eqs. (16) and (18) it is immediate that IlnqlO)ll2-- E ~ I~12. We conclude that the vacuum vector 169) is an element of the domain of definition for/-/q, if and only if the sequence ( ( , In E ~ } of pumping parameters is square summable.
Theorem 1. For Hq the following two assertions are valid: (a) If ~-~n~_l [~n[2
l),
(28)
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then imply W~(h)q E ~ for all h C E which are a finite linear combination of the basis elements {en In E N}. Now [10] Proposition 5.2.4 shows that .~ is a norm-dense complex subspace of ~+(E), which gives the existence of a ff E-~ with 4-particle component ~k4E ~4+ satisfying II~O41l 2 < 6 -1 and the 0-particle component to be exactely the vacuum vector [O). For the Fock vectors ]ml;m2;...) from Eq. (17) in the following we write only those numbers which are occupied, e.g., [ mk = 2) _= [ 0;... ; 0; mk = 2; 0;... ) means that the kth mode is two times occupied and all other modes are not. From now on we assume ~, C E (turn over to the phase-shifted modes exP{½arg(~n)}en). For 4 = ~-~k~_l ~k [ m k = 2 ) E ~ + we obtain with (18) and (19) for all LC L
~n((a*(en)2~t~t4) + (a(e")2~[O))
2(¢ ] H~k) = 2(H~¢ [ ~9) = Z n=l
= v/2 Z
~n
~(mn = 4 ] ~k4)+
~-(m, = 2; mk = 21 ~k4) + ~ n n~k=l
n=l
Za°, n=l
k=l
where {. I • )~ is the scalar product of the complex Hilbert space o ~ - 12(N) of complex square summable sequences fl = (fin c C I n c N), and ~ := (e~ I n c N), ~L := (~1, .... ~L,0,0 .... ), and A is the matrix operator on ~ defined by (Afl)~ := }-~--i a,,kflk with a~,~:=v'~(O41m~=4) and a~,k:=(O41mn=2; ink=2) for k#n. Since the Im~ = 2; mk =2) and Im~ =4) form an orthonormal system for the 4-particle space o~ + , A is a Hilbert-Schmidt operator on ~ (see e.g. [20] Theorem 6.22). The estimation
IIAII2~IIAII~s= ~
la.,k12~<611~'4112<1
n,k=l
(flAIIHs denotes the Hilbert-Schmidt norm of A) implies that ±1 are contained in the resolvent set of the operator A, and hence the range of A + I is all of ~g'. Since the above conclusion holds for all ¢ E ~ + of the form ¢ = ~-']~ ~ I m~ = 2) ~ + the sequences ~ range over all of Of. Hence, because ~9~ .~, we conclude that the weak limit w - l i m c ~ ff~ exists in ~(. But, since each weakly convergent sequence has to be norm-bounded (e.g. [20, Theorem 4.24]), we have a contradiction to 2 ~ lim IIffL I1.*=
L---+~
I~n12= ~ 0 "
n=l
Consequently, our assumption that there exists a 0 # r/~ ~ is wrong.
[]
Let us mention that every Glauber vector and every finite particle vector is an element of the domain ~N of the number operator. Thus, by part (a) of the above
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theorem, the Glauber vectors and the finite particle vectors are contained in the domain of definition for Hq, whenever the pumping parameters are square summable. For the following it is important to note that for arbitrary parameters (, (i.e., also for ~nC~l ](nl 2 =OO) the symplectic transformations Tt on E from the Eqs. (12) and (13) are well defined. Thus in any case of the (n the (algebraic) squeezed vacuum states tot, = ~o0 o ~r, exist and are given by the characteristic functions (26).
Theorem 2. Let Tt be the group of symplectic transformations given by the ~ according to (12) and (13). Then for each real t ~ 0 we have the following equivalences: (ii) the Bogoliubov transformation ~r associated with Tt is unitarily implementable in Fock space, or equivalently, there exists a unitary U, on Fock space such that
W~(TtJ')= Ut W.~(f)U7
V f EE.
(29)
Especially, (i) implies (ii) to be valid for all t E ~. The implementing unitaries Ut, t E ~, are uniquely (up to an arbitrary phase) given by Ut = exp{itHq}. Further, (29) implies q~r,(f) = ¢P(Ttf) = U, ~ ( f ) [Jr* (cf. the Eqs. (11 ) and (14)).
Proof ( i ) ~ (ii): By Theorem 1 Hq is well defined on Fock space. The above algebraic calculations lead to (11) valid in Fock space, which is equivalent to (29) with U, := exp{itHq }. (ii)=~(i): As mentioned above, the complex-anti-linear part (Tt)a =Jsinh(tS) from Tt has to be a Hilbert-Schmidt operator on E, that is,
I~nl2~0
~ > ~-~sinh(tl~nl)2 = ~1 ~-~(cosh(2tl~.l)- 1) ~t 2 n--1
n--I
n--I
which gives (i), since t ¢ 0. But then Hq is well defined and fulfills
exp{itHq}W~(f)exp{-itHq} = W#(Ttf) = Ut v/~(f)U*,
f E E,
which implies Ut*exp{itHq}W~(f ) = W ~ ( f ) Ut*exp{itHq}. But { W ~ ( f ) I f c E} is an irreducible set of operators on Fock space, and thus Ut*exp{itHq}=C,~ with some
ct E C. [] Theorem 3. Let Tt again be the group of symplectic transformations given with the ~n according to (12) and (13). Then for each real t ¢ 0 we have the following equivalences:
(i) En% I~.12<~, (ii) the (algebraic) squeezed vacuum state O9y, is given by a normalized vector qv, from Fock space, i.e., C,,< ( f ) = (ttv, IW.~(f) I qv,) Vf E E, (iii) in Fock space there exists a vector 16))y, fulfilling av,(f)[tg)Z = 0 y f ~E, that is,
Eq. (21).
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Especially, (i) implies (ii) and (iii) to be valid for all t C ~. The vectors t/T' and IO)7;, from (ii) resp. (iii) are uniquely (up to an arbitrary phase) given by the Eqs. (27) resp. (20), that is, t/T, =exp{-itHq}lO) and IO)7;, =exp{itHq}lO). If ~ = l I~.12= ~ , then all squeezed vacua coT' may be represented also by vectors t/T, but from the GNS-Hilbert spaces YET,. The YET, are pairwise non-equivalent representation Hilbert spaces for every pair t ~ t ~. Moreover, for t ~ 0 neither in Fock space nor in YET' there exists a non-zero vector IO)T, fulfilling (21).
Proof. ( i ) ~ (ii) and (i)=~> (iii) follow from the above theorems with t/T, :=exp{--itnq} IO) and IO)T, :=exp{itHq}lO). Let us prove (ii) ~ (i): From toT, = too o aT, it follows (t/rlH~(A) I ~T') = (OII!,~(~r,(A))lO) VA E ~t¢/~(E), which implies for all B E ~/¢r(E) II//.~(B)t/T, II2 = (t/T, IIIF( B* B ) I t/T' ) = ( OlI75( ~T'( B* B ) )I O) = III1F(aT'( B ) )IO) II2.
Because of this, and since the Fock representation/7.~ of ~q/'(E) is irreducible, the operator Ut defined by UtFI~(B)t/T, :=//,~(~T'(B))IO) for all B c ~qF(E) uniquely extends to a unitary acting on Fock space. Further, we have
U,17 ~(A )s-I.~( B )t/T' = U, II.~(AB )t/T' = II.~( ~T'(AB ) )IO ) = 17.~(~T,(A))II.~(~T'(B))IO) =
H~(OCT,(A))Ufl7~(B)n z
VA,B E ~g/~(E),
which yields UtlI ~(A ) = FI ~( CtT,(A) )Ut, respectively, FI~( ctT'(A) ) = UtlI ~(A )U*VA E ~g/'(E), the unitary implementability of the Bogoliubov transformation aT, in Fock space. From Theorem 2 it follows (i) and that Ut is uniquely (up to a phase) given by Ut =exp{itHq }. Let us prove ( i i i ) ~ (i): As in Eq. (17) we may construct an orthonormalized system of vectors in Fock space oo
Iml;m2;...)T,:=H t~aT" j=l
(ej) ) IO) T,, mj E No,
y~. mj < c~3, j=l
for which the Eqs. (18) and (19) are analoguously valid. Let ~ be the sub-Hilbert space of the Fock space ~-+(E), which is spanned by the Iml;mz;...)T,. Then the transformed field operators ~ T ' ( f ) = ~(Ttf) leave 3¢" invariant, and so do the exponentials W~(g) =exp{i~(g)} =exp{i~T'(Tt-lg) } for every g E E. But { W~(g) 19 C E} is an irreducible set of operators on Fock space, and consequently ~ = ~+(E). Now define the unitary Ut on Fock space by U t l m l ; m 2 ; . . . ) : = Iml;m2;...)T'. It follows Ut a#(f)Ut* = a ~ ( f ) , which yields (29). From Theorem 2 it follows (i) and that Ut is uniquely (up to a phase) given by Ut =exp{itHq}. That all squeezed vacua toT, may be represented by vectors t/r, from Hilbert spaces YET' follows from the GNS-representation theorem as mentioned at the end of the previous section. By Theorem 2 for each t ¢ 0 the GNS representation of the squeezed
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vacuum state ~or, on the representation Hilbert space ~f~r~ is non-equivalent to the Fock representation, if and only if ~--],~1 [~,[2 = ~ . Since :~v,, t E E, is a group of *automorphisms on ~W(E) shifting the unitary equivalence, this implies that the ~ r , are pairwise non-equivalent representation Hilbert spaces for every pair t ¢ / . If for t ¢ 0 in o'¢gv,there is a vector 0 ¢ IO)r, E ~ r , fulfilling (21), then as above one may construct a unitary Ut from ~T, onto the Fock space. Thus, the GNS representation of ~or, and the Fock representation are equivalent, which by Theorem 2 is equivalent to (i). [] We see that for the squeezing transformations associated with the classical pumping parameters ~, there are two different cases: • Weak squeezing, that is, ~n~l ]~,[2< cx~. The squeezing transformations are unitarily implementable in the Fock representation by the unitaries exp{itHq}, and the squeezed vacuum states are given by vectors in Fock space (with infinitely many m-particle components). • Strong squeezing, that is, ~,~1 ]~.]2 = ~ . The squeezed vacua are non-Fock and mutually inequivalent. For the notion of a squeezed vacuum one should notice, that for both weak and strong squeezing there exists the cyclic GNS-vector qr~ for the algebraic state ~or,. But only for weak squeezing r/r, may be realized in Fock space. And also only for weak squeezing there exists a non-zero vector IO)r, which is annihilated by the Bogoliubov transformed annihilation operators ar,(f) as in Eq. (21), and this vector is always in Foek space. The fluctuations of the Bogoliubov transformed field operators CbT,(f) are reduced in qr,, but are not squeezed in ]O)v,. ~br,(f) and ]O)v, are simultaneously obtained from • ( f ) and ]O) by the unitary transformation exp{itHq } in Fock space, and thus are only a different realization of the original Fock structure. In this sense the Fock structure is unique. 9
Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft.
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