Generalized intelligent states and squeezing

Generalized intelligent states and squeezingD. A. Trifonov Citation: Journal of Mathematical Physics 35, 2297 (1994); doi: 10.1063/1.530553 View online: http://dx.doi.org/10.1063/1.530553 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/35/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Q-Deformed SU(1,1) and SU(2) squeezed and intelligent states and quantum entanglement AIP Conf. Proc. 1444, 241 (2012); 10.1063/1.4715427 Generalized squeezed states for the Jacobi group AIP Conf. Proc. 1079, 67 (2008); 10.1063/1.3043874 Dynamical evolution of the projector induced from the generalized squeezed state J. Math. Phys. 35, 560 (1994); 10.1063/1.530653 Quantum mechanical squeezed state Am. J. Phys. 61, 1005 (1993); 10.1119/1.17382 Parasqueezed states J. Math. Phys. 32, 783 (1991); 10.1063/1.529371

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Generalized intelligent states and squeezing D. A. Trifonov Institute for Nuclear Research and Nuclear Energy, Blv. Tzarigradsko chaussee, 72, I784 Sofia, Bulgaria

(Received 2 November 1993; accepted for publication 3 January 1994)

The Robertson-Schrodinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator XA +iB, X being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU( 1,l) and SU(2) groups are constructed and discussed. It is shown that SU( 1,l) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.

I. INTRODUCTION

The squeezed states (SS) of electromagnetic field have attracted (in view of their promising applications) due attention in the last decade (see the reviews*,2 and, for example, recent articles3-‘I). In SS of an electromagnetic field the fluctuations in one of the quadrature components Q and P (or in the amplitude) of the photon annihilation operator a = (Q + i P)lfi are smaller than those in the ground state IO). In the case of the quantum particle the quadratures Q and P are the canonical operators of the coordinate and momentum. In the recent years considerable interest has also been devoted to the non-Gaussian SS (Refs. 3-6, 8 and 9) and to the SS for other observables7.“-‘7 such as the spin components, 13*16 the number and phase operators,” the generators of the quasiunitary group SU(1,l) (Refs. 7 and 17), and the generators of the quantum group SU( 1,l )Q .I4 In Ref. 15 some general aspects of squeezing in classical and quantum systems are considered.

In the literature17 a criterion for SS was proposed according to which a state I@) is squeezed for the pair of dimensionless operators A and B if the variance of A or B in Ifi) is less than ( l/2) 1 (@I [A, B] I$) 1, where [A, B] is the commutator. This definition of SS has been applied in Ref. 7 to the quasispin operators [the generators of SU(l,l)] in the Perelomov SU(l,l) coherent states (CS) I [;k) (Ref. 18) and in the Barut and Girardello CS (BG CS) (Ref. 19) with the result that CS 1 l;k) exhibit strong squeezing. However one can show that the variances of the quasispin components in both types of SU(l,l) CS are always greater than their value of k/2 in the ground state IO;k). This fact reveals the relative character of the Eberly and Wodkiewicz criterion’7 and raises the problem to look for other quasispin states which could be more qualified as a SS. Let us also recall that the quasispin operators comprise a vector in the Minkowski space M,, therefore the squeezing of the quasispin components in a given state depends on the reference frame. Thus the CS 15; k) are (pseudo) rotationally equivalent to the ground state so that spacelike components orthogonal to the mean quasispin vector are with equal and minimal dispersions. Similarly the spin CS (Ref. 20) [the SU(2) CS or the Bloch CS (Ref. 21)] are squeezed after Eberly but are rotationally equivalent to the (nonsqueezed) spin ground state. In view of this coordinate dependence it is desirable to point out a natural state dependent frame to describe physical squeezing of the vector operators. In the case of the spin operators this is discussed, for example, in a recent articleI where such a natural frame was motivated to be with the third axis along the mean spin vector. Its analog for the quasispin case should be the frame with the timelike axis along the mean quasispin vector.

The aim of the present article is to consider some general properties of squeezing for an

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2298 D. A. Trifonov: Generalized intelligent states and squeezing

arbitrary pair of quantum observables A and B in states which minimize the Robertson- Schrodinger uncertainty relation (RS UR) (Ref. 22) and to construct explicitly such (squeezed) states for the spin operators and for the generators of the important in the squeezing phenomena SU(l,l).

In Ref. 16 the states which minimize the Heizenberg UR (which is a particular case of RS UR corresponding to the vanishing covariation of A and B) have been called intelligent states (IS). Keeping the analogy we call the states which minimize the Robertson-Schrodinger UR the generalized intelligent states (GIS). In the canonical case of A, B being the quadrature operators Q, P such minimal states (i.e., Q-P GIS) were constructed and discussed in Ref. 23 as correlated states. In fact they coincide24 with the Stoler states25 and the two photon CS (Ref. 26) (often called conventional or canonical SS or simply SS). The spin IS were constructed and studed in Ref. 16 and in a recent articleI the concepts of spin SS and the principles of their generation are discussed.

The article is organized as follows. In the next section we establish a sufficient condition [the eigenvalue Eq. (2)] for a quantum state to minimize the RS UR (1) which in the case of A (or B) with no discrete spectrum is also a necessary one. This means that in the latter case all A-B GIS are eigenstates of a linear in A and B operators, generalizing the eigenvalue property of the Glauber (canonical) CS and the conventional SS. We calculate in such GIS the three second momenta of the observables A and B and show that in the corresponding limits they can exhibit 100% squeezing in the sense of Eberly7’17 and in the sense of Eq. (7) as well. In the latter sense a state is an A-B SS if the variance of A or B is less than its value in the minimum uncertainties state (i.e., in the state with minimal equal variances of A and B). The existence of linear in A and B invariant operators is shown to be a sufficient condition for the evolution operator to keep all GIS stable.

In Sec. III we consider the case of A and B being the two generators K t and K, of SU( 1,l). We caI1 the K,-K, GIS the SU(1,l) GIS or quasispin GIS and construct them explicitly in the representation of the Barut and Girardello CS (the BG representation),” showing that they contain as two different particular cases all the Barut and Girardello CS and all the Perelomov SU(1,l) CS.18 The constructed GIS admit a squeezing operator and can exhibit arbitrarily strong squeezing. Among such GIS there is a family of SS in which the components orthogonal to the mean quasispin vector are strongly squeezed. We show that the evolution of all GIS is stable iff the Hamiltonian admits linear in the generators invariant and examine this condition for linear and quadratic Hamiltonians. In Sec. IV GIS for the spin components are considered. The spin CS are shown to minimize the RS UR. The spin GIS exhibit squeezing. Among spin GIS there is a family of states in which the components orthogonal to the mean spin vector are squeezed, as is required by the Kitagawa and Uedar3 definition of spin SS. At the end of this section it is shown that GIS exist for the quadrature operators of the Weyl generators of any semisimple Lie group and they take in particular the form of the Perelomov CS with maximal symmetry.2~19

II. GENERALIZED INTELLIGENT STATES A. The eigenvalue equation

For any two quantum observables A and B the corresponding second momenta in a given state obey the RS UR (Refs. 20 and 21)

u~u~ 2 ~(C)2+4a~B), CE-i[A, B], (1)

where o;, , (+s , and CAB are the dispersions and the covariation of A and B: cri=(A2)-(A)‘, uAB=~((AB + BA)) - (A)(B). The states that provide the equality in the RS UR (1) will be called here the generalized intelligent states (GIS). When the covariation oAB=O the RS UR coincides with the Heizenberg one.

The canonical SS (the two photon CS (Ref. 26) or the Stoler states25) are based on the equality in RS UR for Q and P (Refs. 23 and 24) and are eigenstates of the linearly transformed boson

J. Math. Phys., Vol. 35, No. 5, May 1994



D. A. Trifonov: Generalized intelligent states and squeezing 2299

operator ua + u a + (where u, v are complex numbers, I u I 2 - I u I 2 = 1). Our aim here is to generalize these properties for arbitrary observables A, B. For this purpose we have to prove first a similar sufficient condition for the equality in the UR (1). In Ref. 23 it was proven that if a pure state I+) with nonvanishing dispersion of the operator A minimizes the RS UR then it is an eigenstate of the operator XA + iB, where A is a complex number, related to (C) and to pi, i =A, B, AB. Here we prove that this is a sufficient condition for any state Ifi).

Proposition I: A state I (c) minimizes the RS UR (I) if it is an eigenstate of the operator L.(A) =AA +iB

L(h)lz,h)=zlz,Q, (2)

where the parameter X is an arbitrary complex number and z is the eigenvalue. Proofi We proceed in two steps. Let us first restrict the parameter A in the eigenvalue Eq. (2)

by the inequality Re A # 0. Then we express A and B in terms of L(A) and L+(A) and obtain

(C> (0 u;Cz,A)== 3 &zN=IA12 - 2ReA’ ~,&,A)= -(c> G 9 (3)

where (c)=(A,z~c~z,A). N ow we easily check that the obtained second momenta (3) obey the equality in RS UR (1).

Let now the eigenvalue equation (2) hold for Re X=0. This means that the state Iz, A) is now an eigenstate of the Hermitian operator rA + B with real r, r =Im A. We consider now the mean value of the non-negative operator Ft( r)F( r), where F(r) = rA + B - (r(A) + (B)) and r is any real number. Herefrom we easily get the uncertainty relation (valid for any state)

with the equality holding in the eigenstates of F(r) (and therefore of rA + B) only. One can consider the equality in Eq. (4) as the desired equality in the Robertson-Schrodinger UR if in these states the mean value of the operator C vanishes. And this is the case. Indeed, consider in Iz,ir) the mean values of the operator products A (rA + B) and (rA + B)A. We easily get the coincidence of the two mean values, wherefrom we obtain (ir,zlClz,ir)=O. Thus all eigenstates Iz, A) with any complex A are GIS.

One can prove that when the operator A has no discrete spectrum then for any I$) one has oA( #) # 0, thereby condition (2) is also necessary and all A-B GIS (for any B) are of the form Iz, A). Such are, for example, the cases of the canonical Q-P GIS and SU( 1,l) GIS, considered below. The above result stems from the following property of the dispersion of quantum observables: the dispersion oA vanishes in a state I+) iff it is an eigenstate of operator A

~~t@I)=0-+++)=&). (5)

Consider (for the sake of completeness) now the exceptional case of states which are eigenstates of A (and therefore with CA = 0). For such states to be GIS [i.e., to minimize the RS UR (l)] eigenvalue condition (2) is not necessary and they may have oB # 0. Note that if oA( rcI> = 0 then WlCbb~=O d an oAB( @) = 0. Thus apart from the eigenstates of A with nonvanishing dispersion of B all the other A-B GIS are eigenstates of the linear operator L(A).

The properties of the A-B GIS turned out to be sensitive about the spectral properties of the commutator C= - i[A, B] . Using the results of the second part of the proof of the Proposition 1 and the equivalence (5) we can prove

Proposition 2: If C is strictly positive (negative) then the Hermitian operators A, B and rA +B have no eigenstates in the Hilbert space.

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Then as consequences we have the following general properties of the GIS in the case of C>O (or C<O): (a) all GIS are solutions of the eigenvalue Eq. (2); (b) GIS ]z,A) exist for Re A # 0 only; and (c) the squeezing operator (defined below) belongs to the group SU(I,l).

A mixed state ~=Z~~l$,Jn)(ti~[ minimizes the RS UR if the orthonormalized pure states / &) are eigenstates of L(A) with a specific A determined in turn by means of the second momenta in accordance with Fq. (3).23 This is rather long procedure even for the pure states.

Let us consider the squeezing properties of the A-B GIS. All the noncorrelated GIS (i.e., states with vanishing covariation oAB) are IS and among them are the IS with equal variances of A and B, which all are eigenstates of the operator L =A + iB

Llz)=zlz) --+ ~~(z)=~zIcIz)=(T;(z). (6)

We denote by z. the label of the IS with minimal oA = ffB , i.e., (z~C~z)~(zo]~~zo) and adopt the definition of the A-B SS as states I$) satisfying the inequality

c~~(++)<$(zolClzo), i=A or B, (7)

i.e., It++) is SS if the variance of A or B is less than its value in the minimum uncertainty IS Izo). It is clear that if I+) is SS in the above sense it is an SS after Eberly and Wodkiewicz.17 Note that by construction the mean value (z/Clz) is always positive as can be seen from Eq. (3). In the canonical case and in the spin and quasispin states (see below) zo= 0, i.e., Izo) is the corresponding ground state. Now we observe [from Eqs. (3) and (2)] that for every fixed z the variance crB --P 0 when A -+ 0 and o,,, -t 0 when A + 00. The equal variance IS lz) are the analogs of the Glauber CS in the case of an arbitrary A-B pair. In the general case however the variances are equal but not constant, depending on the eigenvalue z.

B. The squeezing operator and stable evolution

In some cases the nonsqueezed (and noncorrelated) IS lz) are constructed. If we want to keep the analogy with the canonical GIS (i.e., with the conventional SS) we have to consider the linear transformation

L -+ L(A)=uL+uL+, (8)

where u=(X+ 1)/2, u=(X- 1)/2, Lt=A-iB. If this is a similarity transformation then GIS Iz,x) can be obtained by acting on the nonsqueezed IS lz) with the transforming operator S(A) (the generalized squeezing operator) as it was done by Stolerz3 in the canonical case

lz,~)=w4z). (9)

A necessary condition for the existence of S(A) is the operators L and L(X) to have the same spectra. This holds, for example, in the canonical case and in the case of SU( 1,l) considered below and does not occur in the SU(2) case. When in addition the squeezing operator is unitary, then one can show that the mean value (A ,z I Cl z, A) does not depend on A and all three second momenta are explicitly determined by Eq. (3) in terms of A and the real parameter (z] C/z) only.

It is worth noting that in the cases of positive commutator C= - i[A, B] (when GIS Iz,A) exist for Re A # 0 only) the linear transformation (8) is easily shown to belong to the SU(l,l) group and to preserve the commutation relation

[L”(A), L”+(A)]=[L, L+],

where L”(A) = L(A)Im. In the canonical case the operator L is the boson annihilation operator a. When the linear transformation of a and at preserves the commutator [a, at] it




belongs to SU( 1,l) and in addition it is a unitary similarity transformation according to the known von Neuman theorem. Now we see that the appearance of SU( 1 ,l) is due to the positivity of the commutator C.

Let us now consider the problem of time evolution of GIS. One can say that a given family of states can be realized for a certain quantum system if the evolved initial states remain in the same family of states, i.e., if the evolution of such initial states is stable. Then in cases when GIS admit a squeezing operator it is clear from Eq. (9) that the stable evolution occurs for a system with evolution operator U(r) of the form of the squeezing operator S(A(t)) with some time dependent A(t) . In the general case of GIS the condition is that the evolved state V(t) I z, A) = I t;z, A) should be an eigenstate of L(t)=A(t)A+iB with a time dependent eigenvalue z(t):L(t)lt;z,A) = z(t)1 t;z,A). Combining this with the Schrijdinger equation we get the necessary condition for the Hamiltonian H to admit the stable evolution of GIS (h= 1)

& L(r)--i[L(t), HI-ii(t)

Note that Es. (10) is necessary to be valid for the time evolved GIS only. If however it holds for the time evolution of any initial state then the operator in the bracket vanishes itself and this means that the operator L(r) -z(t) is an invariant operator. The same conclusion holds if the set (or some subset) of initial GIS jz,A) is overcomplete2t since in such a case an arbitrary state can be represented as a direct integral over such states. We shall show now that this is a sufficient condition for the stable evolution of GIS Iz, A). Indeed any invariant A(t) (( dldt)A(t) - i[A( t), H] = 0) can be written in the form of a similarity transformation by means of the evolution operator, A(t) = U( r)A (0) U- l(t) . We see now that if the initial state is an eigenstate of A(0) then the evolved state is an eigenstate of A(t) with the same eigenvalue. It is clear now that if the initial state is A-B GIS Iz, A) then the evolved state is again GIS if the invariant U(r)L(A) U- l(r) takes the form of L(r) up to a factor f(t) and a term g(t) (C numbers). Using Eq. (10) we can write this sufficient condition in the form

2 L’(t)-i[L’(t), H]=O,

where L’(t)=f(t)L(t)+g(t). Herefrom we see that the time evolved GIS V(t)Iz,A) is again a GIS (possibly up to a phase factor) of the form Iz(t), A( t)). If the set of Iz, A) is overcomplete then condition (11) is necessary and sufficient.

In the canonical case of A = Q and B= P such linear in the Q and P invariants and their eigenstates (now we could say Q-P GIS) for quadratic Hamiltonians were first constructed and studied in Ref. 27 (but with no reference to squeezing). The Q-P GIS are the conventional SS and they are overcomplete (see, for example Ref. 24 and the review2), so that the sufficient condition (11) is also a necessary one. Quadratic Hamiltonians are the most general ones which obey condition (11) and thus admit a stable evolution of Q-P GIS, preserving the equality in the RS UR ( 1).24 The general form of Hamiltonians which preserve the equality in the Heisenberg UR for Q and P was found in Ref. 28. The evolution of two photon CS for the quadratic Hamiltonians (the model Hamiltonians in quantum optics) was studied in detail by Yuen.26

III. SU(l ,l) GENERALIZED INTELLIGENT STATES

A. The eigenvalue equation and SU(1,l) CS

In this section we construct and discuss K,-K, GIS, where K, and K2 are the generators (the quasispin operators) of the discrete series D+(k) of representations of SU( 1,l) with Cazimir operator C2 = k(k - 1). From the commutation relation [K, , K2] = - iK, we see that one can apply the corresponding formulas of the previous section with A = K, , B = - K2, and C = K, . The




operator K, is positive with eigenvalues k+ m where m =0,1,2,..., and thus the mean value (K,)zk> l/4. Thereby GIS exist only if Re A>0 and one can safely use formulas (3) for the second momenta of K, ,2. In view of Re A>0 it is more convenient to consider the GIS Iz,A;k) as eigenstates of K(A): = (AK, - iK2)Im

K(A)~z,A;k)=z~z,A;k). (12)

As a consequence of Proposition 2 the operators K, ,2 have no discrete spectrum and so their variances are always greater than zero. Herefrom it follows that the eigenstates of K(A) will give us all the K,-K2 GIS.

The SU(l,l) equal variance IS Iz;k) (the eigenstates of K,--iK2=K-) have been constructed and studied by Barut and Girardello as groups.“”

“new coherent states associated with noncompact These states form an overcomplete family of states (resolving the unity by integration

with respect to an appropriate measure) and provide a representation of any state I#) by an entire analytic function (+lz;k) of z of order 1 and type 1 (exponential type).” In the Hilbert space of such entire analytic functions the generators of SU( 1,l) act as the following differential operators” (we shall call this the BG representation):

d d d2 K3=k+z z, K+=z, K-=2k-+zP. dz (13)

We use the BG representation to construct the SU( 1,1) GIS Iz ’ , A;k) (we denote for awhile the eigenvalue by z’). The eigenvalue equation (2) now reads

i 1 +vz @,~(z)=z’@~~(z), (14)

where the parameters u, v now are u = (A + 1)/2 JReh, u = (A - 1)/2 &?% By means of simple substitutions the above equation is reduced to the Kummer equation for the confluent hypergeometric function iF,(~,b;z),~~ so that we have the following solution of Eq. (14):

QZl(z)=exp(cz)iF,(a,b;-2cz), (15)

where a=k-z’/2uc, b=2k, c2=-v/u. Using the properties of the above hypergeometric function29 we arrive at the conclusion that this solution obeys the requirements of the BG representation iff

IcI=m<l@Re A>O, 06)

which is exactly the restriction on A imposed by the positivity of the commutator C= K3. Thus we obtain the SU( 1,l) GIS Iz’ , A;k) in the BG representation in the form (up to the normalization constant)

(k;A,z’Iz;k)=exp(c*z)iF,(u*,b;-2c*z), (17)

where the parameters a, 6, and c are defined in formula (15). Using the power series of ,F,(u,b;z) (Ref. 29) we get the coincidence of our solution (17) at A= 1 (u = 1, u = 0) with the solution of Barut and Girardello (Ref. 19): (k; X=1, z’lz; k)=oF,(2k;zz’*).

We note the twofold degeneracy of the eigenvalues of the operator L(A # 1) as it is seen from Eq. (3.4). We denote the two solutions as (+;k;A,z’lz;k). The degeneracy is removed at A= 1 when we get the BG solution. Thus A= 1 is a branching point for the operator K(A). It is worth noting that the degeneracy is also removed by the following constrain on the two complex parameters z’ and A in Eq. (17):




z’=2kG=kJ(l-A*)/Re A. (1%

With this relation both (+;k;A,z’lz;k) result in the same expression exp(zdm) which can be seen to be nothing but the BG representation of the Perelomov SU(1,l) CS l{;k) (Ref. 18) with 5 = d%(IO;k) z 15 = 0,k) =I k;k))

15;k)=(1-1512)k exp(lX+)lO;k). (19)

If we impose 2’ = -2k6 we get CS I-L;k). 0 ne can directly check [using the SU( 1,l) commutation relations only] that CS (19) are indeed eigenstates of K(A), Eq. (12), with eigenvalue (18) and with u, v (i.e., with A) determined by means of the relation 12= -v/u. In this way the Perelomov CS generalize the eigenvalue property of the Glauber canonical CS in a manner similar to that of the Barut and Girardello CS.

We calculate explicitly the first and second momenta of the generators Ki in CS 1c;k) (for crKj see also Ref. 7), (K-)=2k~(1-~[~2)-1, (K3)=k(l+]~~2)(1-~~~2)-1

Re 5 Im 5 2 -k 11+1212 2 -k IhY212 (TKIK2=-2k (l-1512)“’ oK,-2(1-]5/2)~’ Q2-Z(1-1512)2 (20)

and show that the equality in the RS UR (1) is satisfied. Thus all the Perelomov SU( 1,l) CS (and the BG CS as well) are a subset of the GIS.

We note that the above formulas for the first and second momenta of Ki in CS ( 5; k) hold also for the known Lipkin-Cohen representation with Bargman index k= l/4 (but not for k= 3/4). One can calculate the fluctuations of Q and P in CS I {;k= l/4,3/4) (Ref. 30) and show that they exhibit about 56% ordinary squeezing (Buiek3).

B. GE and quasispin squeezing

The squeezing of the variances of K,,, in CS 15; k) has been studied in Ref. 7: the CS can exhibit strong squeezing (after Eberly and Wodkiewicz) which tends to 100% when, for example, 5 tends to i. We note however that this is a relative squeezing since (using the property Id< 1) the variances (20) of K,,, can be estimated as follows:

c&;k)a; =&O;k), i=K,,K,,

i.e., no squeezing of oi in l[;k) in comparison with the ground state IO; k). Here the ground state is the minimum uncertainty state so that the quasispin CS are not SS according to the definition (7).

The IS Iz;k) (i.e., the BG CS) also are stretched states as the CS (19) are: the variances of Ki, i=1,2 are equal and given by a,,(z)=(k;zlK,lz;k)/2, where l9

(22)

oFl(u;z) being the confluent hypergeometric function.29 We again get agi(z)>k12. The new states Iz;A;k) however do exhibit strong squeezing. This takes place for example

when Re A--+a or A-+0 which can be easily derived even without explicit calculations of the variances. Indeed recall that Iz, A;k) are solutions of the eigenvalue equation (14) with the constrain I u/u] < 1 and note that the functions (15) are solutions of the eigenvalue equation for any u and u and these are entire analytic functions of u/u. When u - u tends to zero (i.e., Re A+m) they




tend continuously to the eigenfunctions of K, and when u + u tends to zero (i.e., X+0) they tend to the eigenfuctions of KZ. Then in view of the equivalence (5) we get strong squeezing in the above limits for the variances

(k;A,zlK,lz,kk), b12 a;,(z,A)=,,,, (k;A,zlK,lz,W). (23)

The SU( 1,l) squeezing operator S(A) exists and can be correctly defined by means of relation (9) since the spectra of L= K- and K(A) coincide and the set of BG CS Iz;k) is overcomplete.” It belongs again to group SU( 1,l) (see the discussion in Sec. II B) and its matrix elements (k;zlSlz;k) are explicitly given by the functions (17) with z ’ = z. These diagonal matrix elements determine S uniquely due to the analyticity property of the BG representation.” We recall that the same property of the diagonal matrix elements holds in the canonical (Glauber) CS representation (see, for example Refs. 2,21 and references therein).

The operators Ki , i = 1,2,3 are generators of the rotations in the Minkowski space M3 and they comprise a timelike vector in M, since the operator K, is strictly positive. The squeezing of the quasispin components clearly depends on the coordinate system. For every state I@) a natural state determined coordinate system for the quasispin is that with the time axis along the mean quasispin vector ($jKI&. In this system the mean spacelike components are vanishing: (IcIIK1,21@)=0, which f or our GIS Iz,A;k) results in z=O, i.e., in the states IO,A;k)=jA;k) [which are annihilated by the operator K(A)] where the mean quasispin vector is along the third (the timelike) axis. This means that the squeezing in these states could be considered as due to physical correlations between the quasispin components (following, for example, the ideas of Ref. 13 for the spin squeezing). It is worth noting that in the SS IA;k) the length of the mean quasispin vector is varying with A and is greater than its value k in the ground state IO;k) while in the Perelomov CS 15; k) this length is constant and is equal to k.

The obtained GIS Iz,X;k) could be realized in systems with Hamiltonians H which obey the stable evolution conditions (10) and (11). If H obey the sufficient condition (11) then any initial GIS is stable in time. If not then some particular states or subset of states still could have a stable evolution in the set of GIS. It is natural in our case here first to examine this condition for the Hamiltonians which are linear and quadratic in the SU( 1,l) generators (summation over repeated indexes)

H,=hj(t)Kj; HZ=hij(t)KiKj, j= 1,2,3. (24)

We get the validity of Eq. (11) only for H = h3( t)K, which in the case where the Bargman index k= l/4 is proportional to the stationary oscillator. The time evolution with this Hamiltonian is equivalent to a rotation about a third axis. An arbitrary initial GIS Iz,A; k) evolves into GIS Iz(t),A(t);k) with th e ime dependent parameters obeying the equations t’

$ A(t)+iA2(t)h,(t)-ih,(t)=O; z(t)=z exp[ -$ A(t)h,(l)dt]. (25)

We see that if initially A= 1 then A(t) = 1 which means that all the BG CS Iz; k) are stable under rotations in the spacelike plain “x,x2 .” For h, =const we have

A(t)= A cos hgt-i sin hgt Z

cos hjt-iX sin h3t ; z(t)= cos h3t-iA sin hjt * (26)




It is interesting to note that the ratios qi(t): = 2ai/(K,), i= 1,2 are pulsating (for real A) between l/A and A and between A and l/A exactly in the same manner as the dispersions 2~2, and 2a$ of the canonical P and Q ((K3) is replaced by 1) are pulsating with time in the evolution of an initial canonical SS of the harmonic oscillator3’

1 ql(t)=x cos2 h3t+A sin2 h,t. (27)

Note that the variances of K,,, are changing in time (unless A= 1) but the variances of the components orthogonal to the mean quasispin vector are constant. Different subsets of the GIS however could be stable for more general H which could generate squeezing from some initially nonsqueezed IS. For example, the subset of the Perelomov CS is stable for the general linear form Hi , Eq. (24) [more details about stable evolution of group related CS and in particular of SU( 1,l) CS can be found in Ref. 301. This is a good example of a stable subset for a Hamiltonian (here it is H,) which does not obey the sufficient condition (11). Inspite of the overcompleteness of SWlJ) CS l&k) y ou cannot get (11) from the necessary condition (10) since now the latter holds for all CS but not for a fixed operator: the parameter A in L(A(t))=L(t) in Eq. (10) varys with b in accordance with the constrain (18). Note that for k = l/4, 3/4 this is the most general quadratic in the Q and P Hamiltonian which models many systems in quantum optics.2 Other Hamiltonians should be considered elsewhere. Here we note that the evolution operators for such Hamiltonians should belong to the group of automorphisms of the Lie algebra of SU( 1,l). In the case of canonical SS the stable evolution Hamiltonians are quadratic in Q and P and they generate automorphisms of the Heizenberg algebra spanned by the operators 1, Q, and P.24

IV. GENERALIZED INTELLIGENT STATES FOR SU(2) AND SEMISIMPLE GROUP

A. Spin GIS

Let now A, B, and C be the generators J, , -J2, and -Js of the SU(2) group, i.e., the spin operators of spin j= l/2,1,... In this example the commutator C= -Js is not positive, the limit Re A=0 can be taken, and the operator A = J, has a discrete spectrum. Some of its eigenstates are examples of exceptional GIS which are not eigenstates of L(A). In Ref. 16 eigenstates (in their notations) I w,( r)) of the operator J(a) = J, - id2 were constructed, where N= 0,1,2.. . ,2 j, ?=(I-(U)/(lf ) cz , @ being an arbitrary complex number. These states are (for cy # 0) eigenstates also of L(A) = AJ, - iJ2, thereby they all are J,-J2 GIS, minimizing the RS UR (1). They can be represented in the general form IzA, ,A;j) with the eigenvalues zN = (j - N) $??. Among them (for N=O and N=2j) are the spin [the SU(2)] CS 17; -j) and I - 7; -j) (7 is any complex number)

]r;-j)=(l+lr]2)-j exp(rJ+)] j;-j). (28)

The mean values of Ji , i = 1,2,3 and J’ (and the dispersions oJ, and oJJ,) in spin CS are known.32 Here we also calculate the covariation oJ, ,J,

Re r Im r a~,,~~td=25’

j ll-PI2

(1+142)2’ o:P=2 (lfl# . (29)

We can directly check that in CS 17) the equality in the RS UR (1) holds for the spin operators J,,,. Thus the spin CS are a subset of the SU(2) GIS.

Let us discuss the properties of the SU(2) GIS. First of all for a given parameter A there are 2j + 1 independent GIS ]zN ,A; j). There are only two equal variance IS, namely, 1 j; If: j), the point A= 1 being again the branching point of L(A). From this fact it follows that the squeezing operator does not exist. Since the commutator C= i[J, , J2] = -J3 .is not positively defined the




limit Re A=0 in GIS is allowed and in the fluctuations formulas (3) as well since at this limit (C) = ( J3) = 0. The operator A = Jt has a discrete spectrum, therefore oA Z 0. From the explicit formula (29) for o;,(r) we see that this fluctuation vanishes at r2= 1. Therefore in virtue of property (5) the spin CS I T= t 1; - j) are eigenstates of J, which can also be checked directly, the eigenvalues being + j .

Unlike the SU( 1,l) GIS not all the SU(2) GIS are eigenstates of the operator AJi - iJ2. They are the eigenstates of Ji which are those exceptional states which minimize the RS UR (1) but are not of the form Iz, A). One can check this on the example of spin CS 1 T= + 1, -j).

Let us now briefly discuss the evolution of spin GIS for quantum systems. As in the case of SU(l,l) we first examine the Hamiltonians which are linear and quadratic in the spin operators: H, = hk( t)J, , H, = hik( t)JiJk . Here we get a result similar to the previous case, namely, all spin GIS ]z,A;j) are stable for H= h3(t)J3 only with the evolved parameters z(t), A(t) obeying Eqs. (25), (26). For constant h3 the parameter A(t) is either identically 1 or is given by the same formula (26) pulsating between the values A and l/A. For the ratios qi( t): = 20:i( t)l(J,), i = 1,2 we again obtain the pulsating formula (27).

As in the case of SU( 1,l) the subset of GIS in the form of spin CS 17; j) is stable for linear H, . Due to the nonpositivity of the commutator - i[J, , J2] =.I3 here the subset of GIS with Re A=0 is also stable for quadratic Hamiltonian H, provided a certain complicated relation between hii is maintained.

Among the spin GIS ]z, A) there is a subset of states, namely, [a(j) is a normalization constant and ?=(A- 1)/(X+ l)]

j j!(2j-I)! lA;j)-;lO.A;j)=a(j), l,(j-I), (-2rJ+)k-j)

in which the mean spin vector is along the third axis, and they are squeezed after Eberly for any A # 1 and squeezed after the definition (7) for Re Aa(j12)(J3) or Re A/]A]2>(j/2)(J3). For these states the definition (7) of spin SS coincides with the definition of the SS in Ref. 13 since K1,2 are orthogonal to the mean spin vector. In Ref. 13 squeezing of such orthogonal spin components is considered as physical squeezing due to the correlations between the elementary half spins. In the squeezing of other spin components the effect of coordinate rotation is present.

6. CS for semisimple Lie group as GIS

The above (generalized) intelligent properties of the SU( 1.1) and SU(2) CS can be partially extended to the group related CS for arbitrary semisimple Lie group~.~ Using the standard commutation relations of the generators of the group (written, for example, in Ref. 2) one can check that CS (Ref. 2)

I cJ=exp[v,E,- 772-J/0) (31)

are eigenstates of the linear combination uE_ n+ u E, with u and v determined by means of va and the algebraic structure

lJ - = -exp( - i+)tanh2] ~7~1 &Q. u (32)

Here E, is a Weyl generator related to the simple root a, va are complex parameters, and IO) is the extremal weight vector ]A&, annihilated by E -~ ; r$ is’the phase of va and urr is the value of the Killing form on the Cartan generator H, , (I~= (H, , H _ ,) . The obtained eigenvalue is




(33)

where Aa is the eigenvalue of H,:H,lA,A)=A,lA,A). Then in view of Proposition 1 these CS are GIS for the quadrature components A a, B, of

E-,, (E-,=A,+iB,) and they minimize the RS UR (1). In contrast to the SU(l,l) and SU(2) cases (where there are only two Weyl generators E +) however for larger semisimple groups the A,-B, GIS (31) form only a subset of all group related CS Iv}.2

V. CONCLUDING REMARKS

We have proven that the eigenstates /z, A) of the operator AA + iB with any complex A minimize the Robertson-Schrodinger uncertainty relation for the operators A and B and can exhibit strong squeezing. In the case of positive (negative) commutator C= - i[A, B] all states that minimize the UR (1) are of the form of ]z,A). Such minimal states [called here generalized intelligent states (GIS)] can be considered as a generalization of the canonical Q-P squeezed states. We have constructed and discussed here the SU( 1,l) and the SU(2) GIS. The SU( 1,l) GIS form a larger set of states which contain as two different subsets the Perelomov CS and the Barut and Girrardello CS. Thus the common feature of the two latter sets of the SU( 1,l) CS is that they both are eigenstates of non-Hermitian operators linear in the group generators and thereby they minimize the UR (1). The SU(2) GIS in turn contain the spin CS as a subset.

In the SU( 1,l) GIS the variances of the generators K 1,2 (the quasispin components) can be strongly reduced below their ground state value k/2 while in the SU(l,l) CS they are always greater than k/2 so that the GIS are more qualified as SS. Both in the case of spin and quasispin GIS there are states in which the components orthogonal to the mean spin (or quasispin) vector are strongly squeezed. In Ref. 13 such states in the spin case are defined as physical SS.

When the operators A and/or B are expressed in terms of the canonical pair Q, P one can look in the A-B GIS for the squeezing of Q and/or P as well. Such are, for example, the bosonic realizations of the generators of SU( 1,l) and SU(2).

If the A-B GIS can be obtained from the equal variance IS lz) by means of the invertible squeezing operator S(A) the latter belongs to SU( 1,l) as can be derived from Eq. (8). This fact shows that SU( 1,1) plays an important role in a wide class of squeezing phenomena (not only in the canonical Q-P case). For physical applications of GIS it is important to know the Hamilto- nians H which admit a stable evolution and could produce A-B squeezed states. For this purpose the necessary and sufficient conditions are derived and examined for H linear and quadratic in the generators of SU(l,l) and SU(2). In contrast to the canonical SS only particular GIS or subsets or GIS here could be stable for quadratic H. These problems should be considered in greater detail elsewhere.

The eigenvalue property (2) (and thus the GIS and corresponding squeezing properties) is established for the CS with maximal symmetry for any semisimple Lie group.

ACKNOWLEDGMENT

This work is partially supported by Bulgarian Science Foundation research Grant No. F-l 16.

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Generalized intelligent states and squeezing