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Coherent and Squeezed States on Physical Basis PRAMANA © Printed in India Vol. 48, No. 3, __ journal of March 1997 physics pp. 787-797 Coherent and squeezed states on physical basis R R PURI Theoretical Physics Division, Central Complex, Bhabha Atomic Research Centre, Bombay 400 085, India MS received 6 April 1996; revised 24 December 1996 Abstract. A definition of coherent states is proposed as the minimum uncertainty states with equal variance in two hermitian non-commuting generators of the Lie algebra of the hamil- tonian. That approach classifies the coherent states into distinct classes. The coherent states of a harmonic oscillator, according to the proposed approach, are shown to fall in two classes. One is the familiar class of Glauber states whereas the other is a new class. The coherent states of spin constitute only one class. The squeezed states are similarly defined on the physical basis as the states that give better precision than the coherent states in a process of measurement of a force coupled to the given system. The condition of squeezing based on that criterion is derived for a system of spins. Keywords. Coherent state; squeezed state; Lie algebra; SU(m, n). PACS Nos 03.65; 02-90 1. Introduction The hamiltonian of a quantum system is generally a linear combination of a set of operators closed under the operation of commutation i.e. they constitute a Lie algebra. Since their introduction for a hamiltonian linear in harmonic oscillator (HO) oper- ators, the notion of coherent states [1] has been extended to other hamiltonians (see [2-9] and references therein). Physically, the HO coherent states are the minimum uncertainty states of the two quadratures of the bosonic operator with equal variance in the two. Mathematically, they are (i) the eigenstates of the bosonic annihilation operator or, equivalentally, (ii) the states generated from the vacuum by a unitary transformation linear in the bosonic operators. The extension of that notion to non-canonical systems is based on generalizing, not the physics aspect, but the two aforementioned mathematical ways of constructing the HO coherent states. The generalization based on (i) 14, 5] is limited to non-compact Lie algebras whereas that based on (ii) by Perelomov [6] has universal applicability. The coherent states in Perelomov sense are defined as the states obtained by a unitary transformation generated by the Lie algebra of the hamiltonian acting on a state in the Hilbert space of the associated group. The physical properties of the states so defined depend upon the choice of the fiducial state. Some choices of the fiducial state may lead to the physical property of equality of the variance in the minimum uncertainty relation for two observables in the Lie algebra of the hamiltonian but it is not a necessary require- ment for the construction of those states. For a discussion of the coherent states 787 R R Puri of a hamiltonian generating SU(m,n) transformations in the bosonic representa- tion, see [8]. The Perelomov construction, thus, does not carry the physical spirit of the HO coherent states. Here we present an approach to coherent states of non-canonical systems which is based on the generalization of the physics aspect of the HO coherent states. Following that approach, it turns out that the coherent states of a given non-canonical system fall into various classes. Our approach leads, besides the known Glauber coherent states, to a new class of coherent states of the harmonic oscillator. That new class consists of the states which are minimum uncertainty states of the number and position or momentum operator with equal variance. In the case of spins, it turns out that there is only one class of coherent states which consists of well-known spin coherent states. Like the notion of coherent states, the notion of squeezed states of a non-canonical system is based on the extension of the mathematical aspect of the canonical squeezed states. For, the notion of a canonical squeezed state was introduced to label those states of a HO prepared in which it gives better precision of measurement of a classical force coupled linearly to it than what is obtained if it is prepared in a coherent state [10, 11]. The canonical squeezed states so defined have smaller variance in a quadrature than that in a coherent state. It is the latter property that forms the basis of extension of the notion of squeezing to non-canonical systems. That approach, however, leads to ambiguities because it identifies some of the coherent states of a non-canonical system as squeezed whereas the coherent and squeezed states of a canonical system constitute mutually exclusive classes. The spirit of the notion of canonical squeezed states will be carried to non-canonical systems if it is based on the criterion of increased sensitivity as compared with that in a coherent state in an appropriate process of measurement. We derive a measure of the sensitivity of measurement of a force coupled linearly to a given system. The criterion of squeezing based on those considerations is derived for a system of spins. The paper is organized as follows. In § 2 we introduce the theory of coherent states based on the physical considerations. It is applied to a system of harmonic oscillator and of spins in § 3. In § 4 we discuss the criteria of squeezing in a system of spins. A measure of the sensitivity of measurement of a force interacting linearly with a dynamical system is derived in § 5. The main conclusions are summarized in § 6. 2. Coherent states In this section we introduce the notion of minimum uncertainty states and the coherent states of a hamiltonian. Recall that the variances AA:, AB 2 in the measurement of two observables represen- ted by hermitian operators A and B are related by the uncertainty relation AA 2 AB 2 >/¼[(F) 2 + (C)2], (1) where [A,B] -- iC; (F) = (AB + BA) - 2(A) (B). (2) Clearly, (F) is a measure of the correlation between A and B. IfA and B commute then (1) is the same as the relation between the variances and the correlation of two classical 788 Pramana - J. Phys., Voi. 48, No. 3, March 1997 Coherent and squeezed states on physical basis random variables A and B. The equality in (1) is obeyed by the states Iq~), called the minimum uncertainty states (MUS), which solve the eigenvalue equation [A + iAB]Iq~) = [(A) + i2(B)] [q~), (3) where 2 is a complex number. It can be shown from (3) that [9] AA 2-22AB 2 =i2(F), AA 2 + j~2AB2 = 2(C). (4) From (4) it follows that (F) = 0((C) = 0) for 2 real (2 imaginary). A coherent state of the given pair of observables is defined as that minimum uncertainty state for which (F) = 0 and AA 2 = AB 2 = 1(C)1/2. Those states are the solutions of(3) corresponding to2= _1. Consider now a quantum system described by a hamiltonian H given by H = ~ ~iXi, (5) i=1 where ~i-- (~i, ~2 ..... ~r) are real and {Xi} m (X1,X 2..... X,) are r independent her- mitian operators closed under the operation of commutation i.e. they constitute an r-dimensional Lie algebra. The elements of the algebra transform amongst each other under the group of unitary transformations U=exp[i ~ (6) where {a~} are real constants. If P1,P2 are two elements of the given Lie algebra and Q1 = UP1 U*, Q2 = UP2 Ut then the pairs (P1, P2) and (Q 1, Qz) are said to belong to the same pair-class. Any pair in the pair class is a generator of the other pairs in that class. In order to define the minimum uncertainty states of the given hamiltonian we first construct the distinct pair-classes generated by non-commuting independent hermitian generators of the Lie algebra of the hamiltonian. If the minimum uncertainty state for a pair in a pair-class is I~b) then it follows from (3) that the MUS for any other pair in that class for the same 2 is [~/'>uv = Ulq~>. (7) The state [~b> in (7) can be chosen to be the MUS for the pair of generators of the Lie algebra in the pair-class under consideration. The set of minimum uncertainty states for all distinct pair-classes determines completely the minimum uncertainty states of the given hamiltonian. We will see that the harmonic oscillator or h, algebra has two pair- classes whereas the spin or SU(2) algebra has one. The harmonic oscillator hamil- tonian, therefore, has two classes of minimum uncertainty states whereas spin has one. The generators of the SU(1, 1) algebra fall in three pair-classes [12, 13]. The minimum uncertainty states (7) corresponding to 2 -- _+ 1 are the coherent states IO)cs of the hamiltonian. The operators generating U in (7) may be divisible into two subalgebras: one the subalgebra h which consists of the generators of which I~b) is an eigenstate and the Pramana - J. Phys., VoL 48, No. 3, March 1997 789 R R Puri other, g, containing the rest of the generators. The group generated by h is called the stability or stationary group of Iq~) [3, 6]. The unitary transformation U, by virtue of the group property, can then be written as U = UgUh, where Ug and Uh are generated by the operators belonging to g and h respectively. The transformation Uh on j~b) evidently gives rise to only a phase factor. By absorbing the phase factor in the definition of [¢)Mv the expression (7) for the minimum uncertainty state can, therefore, be written as I¢ )Mu = U+l ~), (8) with the coherent states corresponding to it = +__ 1.
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