Coherent and squeezed states on physical basis

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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 P U R I

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 hamiltonian. 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 q u a n t u m system is generally a linear c o m b i n a t i o n of a set of operators closed u n d e r the operation of c o m m u t a t i o n i.e. they constitute a Lie algebra.

Since their i n t r o d u c t i o n for a hamiltonian linear in h a r m o n i c oscillator (HO) operators, the n o t i o n of c o h e r e n t states [1] has been extended to other hamiltonians (see [ 2 - 9 ] and references therein). Physically, the H O c o h e r e n t states are the m i n i m u m uncertainty states of the two quadratures of the bosonic o p e r a t o r with equal variance in the two. Mathematically, they are (i) the eigenstates of the bosonic annihilation o p e r a t o r or, equivalentally, (ii) the states generated from the vacuum by a unitary transformation linear in the bosonic operators. T h e extension of that n o t i o n to non-canonical systems is based on generalizing, n o t the physics aspect, but the two aforementioned mathematical ways of constructing the H O coherent states. T h e generalization b a s e d on (i) 14, 5] is limited to n o n - c o m p a c t Lie algebras whereas that based on (ii) by P e r e l o m o v [6] has universal applicability. T h e coherent states in P e r e l o m o v sense are defined as the states obtained by a unitary t r a n s f o r m a t i o n generated by the Lie algebra of the hamiltonian acting o n a state in the Hilbert space of the associated group. T h e physical properties of the states so defined depend u p o n the choice of the fiducial state. Some choices of the fiducial state m a y lead to the physical p r o p e r t y of equality of the variance in the m i n i m u m uncertainty relation for two observables in the Lie algebra of the hamiltonian but it is not a necessary requirement for the construction of those states. F o r a discussion of the coherent states 787

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of a hamiltonian generating S U ( m , n ) t r a n s f o r m a t i o n s in the bosonic representa- tion, see [8].

T h e P e r e l o m o v construction, thus, does not c a r r y the physical spirit of the H O coherent states. Here we present an a p p r o a c h to c o h e r e n t states of non-canonical systems which is based on the generalization of the physics aspect of the H O coherent states. Following that approach, it turns out that the c o h e r e n t states of a given non-canonical system fall into various classes. O u r a p p r o a c h leads, besides the k n o w n G l a u b e r coherent states, to a new class of coherent states of the harmonic oscillator.

T h a t new class consists of the states which are m i n i m u m uncertainty states of the n u m b e r and position or m o m e n t u m o p e r a t o r 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 n o t i o n 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 n o t i o n of a canonical squeezed state was i n t r o d u c e d to label those states of a H O p r e p a r e d in which it gives better precision of m e a s u r e m e n t of a classical force coupled linearly to it than what is obtained if it is p r e p a r e d in a coherent state [10, 11].

T h e canonical squeezed states so defined have smaller variance in a q u a d r a t u r e than that in a c o h e r e n t state. It is the latter property that forms the basis of extension of the notion of squeezing to non-canonical systems. T h a t a p p r o a c h , 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. T h e 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 c o m p a r e d with that in a coherent state in an a p p r o p r i a t e 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 p a p e r is organized as follows. In § 2 we introduce the t h e o r y 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 u n c e r t a i n t y states and the coherent states of a hamiltonian.

Recall that the variances AA:, AB 2 in the m e a s u r e m e n t of two observables represen- ted by hermitian operators A and B are related by the u n c e r t a i n t y relation

AA 2 AB 2 > / ¼ [ ( F ) 2 + ( C ) 2 ] , (1)

where

[ A , B ] -- iC; ( F ) = ( A B + B A ) - 2 ( A ) ( B ) . (2) Clearly, ( F ) is a measure of the correlation between A and B. I f A and B c o m m u t e 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

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Coherent and squeezed states on physical basis

r a n d o m variables A and B. The equality in (1) is obeyed by the states I q~), called the minimum uncertainty states (MUS), which solve the eigenvalue equation

[A + iAB]Iq~) = [ ( A ) + i 2 ( B ) ] [q~), (3) where 2 is a complex number. It can be shown from (3) that [9]

AA 2 - 2 2 A B

2 =i2(F),

A A 2 + j~2AB2 = 2 ( C ) . (4)

F r o m (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 m i n i m u m uncertainty state for which ( F ) = 0 and AA 2 = AB 2 =

1(C)1/2.

Those states are the solutions of(3) corresponding t o 2 = _ 1 .

Consider now a q u a n t u m system described by a hamiltonian H given by

H = ~ ~iXi,

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i=1

where ~i-- (~i, ~2 . . . ~r) are real and {Xi} m (X1,X 2 .. . . . X,) are r independent hermitian operators closed under the operation of c o m m u t a t i o n 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,

Q z)

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 m i n i m u m uncertainty state for a pair in a pair-class is I~b) then it follows from (3) that the M U S 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 M U S for the pair of generators of the Lie algebra in the pair-class under consideration. The set of m i n i m u m uncertainty states for all distinct pair-classes determines completely the m i n i m u m 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 hamiltonian, 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 m i n i m u m uncertainty states (7) corresponding to 2 -- _+ 1 are the coherent states IO)cs of the hamiltonian.

The operators generating U in (7) m a y be divisible into two subalgebras: one the subalgebra h which consists of the generators of which I~b) is an eigenstate and the P r a m a n a - J. Phys., VoL 48, No. 3, March 1997 7 8 9

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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 ~),

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with the coherent states corresponding to it = +__ 1.

The approach developed here has been applied to construct the coherent and minimum uncertainty states of SU(1, 1) [12, 13]. In that case there are three pair- classes, two of which are discussed extensively in refs [12] a n d [13]. Recall that according to usual approaches, as has been mentioned in § 1, the coherent states of a non-compact group like SU(1, 1) can be constructed in two different ways: (i) as the eigenstates of a lowering operator [4, 5] and (ii) by following the approach of Perelomov [6]. However, by following the approach presented here, those two coherent states emerge as special cases of the coherent states of a pair-class whereas the coherent states belonging to the other pair class are new [12, 13]. Thus the proposed approach is able to present a unified view of the coherent states of SU(1, 1). A different approach to the unification of the SU(1, 1) coherent states is presented in ref. [14].

In the following we construct all the minimum uncertainty states of a system of H O a n d of spins and show that a new class of coherent states emerges even for a system of harmonic oscillator.

3. M i n i m u m uncertainty states o f H O and spins

In this section we apply the considerations of the last section to obtain complete set of minimum uncertainty states for a harmonic oscillator a n d a system of spins.

3.1 Minimum uncertainty states of harmonic oscillator

Consider a hamiltonian linear in the harmonic oscillator operators {a t a, q - (a + at)Ix~r2, p - i (a t - a)/x/~, I} where a, a t are the bosonic operators. T h a t set of operators spans the Lie algebra h 4 [3]. Its subset of canonical operators q - (a + at)/v/2, p - (a t - a)/x/~, I spans an invariant subalgebra denoted by h a [3]. Clearly there are two pair-classes of h 4. One is constituted by the operators (q, p) a n d another by (a t a, q) or equivalently by (a t a, q). The minimum uncertainty states of the class (q, p) are given by (8) where [~b) is the solution of the equation

1 [(1 + ;~)a + (1 - , ~ ) a t ] ]~/,) = z l ¢ ) . (9)

Here z = (1/2)((1 + 2)a + (1 - i t ) a t ] ) is a complex number. It can be shown that the normalizable solutions of (9) exist only if Re(it) > 0. Hence the coherent states, which correspond to the solution of(9) for it = _+ 1 are defined only for it = 1. That solution is evidently the coherent state [z):

Iz) = exp [za t - z* a] 10). (10)

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Coherent and squeezed states on physical basis

Note that a H O group element acting on a coherent state again gives a coherent state.

For 2 ¢ 1, (9) can be solved by rewriting it as

e x p ( - Oata) l,a + a t] exp(Oata)l~b) = zl~b), (11) where exp(0) = x/(1 + 2)/(1 - 2). The solution of(11) m a y be written as

I~b) = e x p ( - Oat a)lx), (12)

where ix) is the eigenstate of q with eigenvalue x. In terms of the zero eigenvalue state 10) of q, (12) reads

J~b) = exp(-- Oat a)exp { -- x(a -- at)/w/2 } 10). (13) The state Ith) is thus generated by a linear non-unitary transformation on an eigenstate of the position operator. In terms of the eigenstates Ira) of at a it reads

[ ~ ]1/2

Iq~) = exp(-- x2/2) / _ _ , H m ( x ) e x p ( - - O m ) l m ) ,

(14) L2",,/'~ m~ J

m=O

where H,,,(x) are the Hermite polynomials.

The expression (12) is not the familiar form 1-15] of the solution of(9). It is introduced here in order to compare it with the expression for the minimum uncertainty state for spins. The familiar form 1-15] is obtained by writing (9) as

exp l' -- ½ (~ at2 -- ~ *a2)] a exp l'½ (( at2 -- ~* a2)] [~b ) = ct[ t~), (15) where tanh(l~J) = (1 - 2)/(1 + 2). Its solution is given by

[q6) = [exp( - ½(~a tz - ¢*a2)]lct), (16)

which is the familiar form of the minimum uncertainty states of canonical operators.

The state I~b) is thus obtained as a result of a non-linear unitary transformation on a coherent state. It is straightforward to verify that (16) leads to (14).

Next we construct the minimum uncertainty states for the second pair class which is generated by (a t a, p) or equivalently by (a t a, q). By choosing (a t a, p) as the generating pair, the desired states are given by (8) where J~) is the solution of the equation

l,a t a + i 2p]

I ~)

-- exp (2q) I-at a +/~2] exp ( -- 2q) l ~b) = z J qS), (17) which is evidently solved by z = m + 2 2 with the corresponding solution given by

Iq~,,) = exp().q)lm), m = 0 , 1 , 2 . . . (18) The m i n i m u m uncertainty states are given by substituting (18) in (8) where Ug is now the full group element. The MUS in this case is thus obtained by the action of the unitary transformation generated by h 4 on a state generated from a basis state by a non-unitary transformation. F o r m = 0, that non-unitary transformation is evidently equivalent to a unitary one. In that case IqSo) is the coherent state [-- 2/x//2). F r o m (18) it follows that

(llq~m)= . exp L~-" , l ~ m , (19)

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where L~(x) are the Laguerre polynomials. The states (19) cannot be generated by a hamiltonian linear in the harmonic oscillator operators.

3.2 M i n i m u m uncertainty states o f spins

Consider a system of spins described by the spin operators S - (S,, Sy, Sz) whose components obey the angular m o m e n t u m commutation relation

[a.S, b . S ] = i(a x b).S, (20)

where a and b are arbitrary three dimensional vectors. The state space of the operators is spanned by the eigenstates Ira, S ) - tin) of the Casimir operator 2 S x + Sr + S, and S~ 2 2 with eigenvalues S ( S + 1) (S = i/2, 1, 3/2 . . . . ) and m (m = - S, - S + 1 .. . . . S) respectively. The operators S+ = S x +_ iSy are the raising and lowering operators on those eigenstates such that S + iS) = S_ [ - S ) = 0. There is clearly only one pair-class in this case which is generated by any two, say (S,,, Sy), of the three generators. The minimum uncertainty states of a hamiltonian linear in spin operators is given by (8) where kb) is the solution of

[ sx + i,~s,] l ~ ) = z lck), (21)

which for 2 = + 1 is solved only by z = 0 with [~b) = I - S). Since [ + S) is an eigenstate of Sz, the transformation Ug in (8) is generated by S x, Sy leading to the following well-known expression for the spin coherent states [15, 16]

[0, 4~) = exp [o {exp(-- ic~)S+ - exp(i~b)S_ }][ __ S). (22) Thus the spin coherent states are minimum uncertainty states with equal variance in two orthogonal components U S x U t and USrU* of spin. The variance in each of them is given by [ ( U S z U*)[/2 i.e. by half the absolute value of the average of the component orthogonal to both of them. Note that

US~ U t -- sin(0) cos(40S~ + sin(0) sin(~b)Sy + cos(0)S~ -- e(0, 4~).S, (23) where e(0, ~b)-(sin(0cos(~b), sin(0)sin(~b),cos(0)) is the unit vector in the direction (0, 4~)- That is also the direction of the average spin vector ( S ) as can be seen by evaluating

(S~) = S sin(0) cos(~b), ( S t ) = S sin(0) sin(~b), (Sz) = S cos(0). (24) Hence U S , U t is in the average direction of the spin. A spin coherent state can therefore be defined as the state in which the variance in any spin c o m p o n e n t orthogonal to the average direction of spin is S/2.

The solution of (21) for ). ~ _+ 1 m a y be obtained by rewriting it as (exp(/3) = , / ( 1 + ,~)/(1 - ,~)).

[exp(fl)S+ + exp(-- fl)S_]I~b) -- exp(flSz)(S+ + S_) exp(-/~Sz)[~b ) --2 exp(/~S~, e x p ( - i 2 S y ) S z e x p ( i 2 S t ) e x p ( - f l X z , l~b)

=zJ~), (25)

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Coherent and squeezed states on physical basis

which is evidently solved by z = 2m with the solution given by [17, 18]

[d?,.) = A . exp(flSz) exp(-- i~ ~ )lm) - A , exp(flS,_)lm),

(26) where [m) is the eigenstate of S x. By c o m p a r i n g (26) and (12) the similarity between the m i n i m u m uncertainty states of the harmonic oscillator and those of spins is clearly b r o u g h t out. Recall that in the case of a harmonic oscillator there exists a non-linear unitary t r a n s f o r m a t i o n (16) equivalent to the linear n o n - u n i t a r y t r a n s f o r m a t i o n (12).

There is, however, no k n o w n analogous non-linear transformation equivalent to the linear n o n - u n i t a r y one given by (26) except for S = 1. The equivalent non-linear t r a n s f o r m a t i o n for S = 1 is e x p [ ~ S 2 - ~*$2_] with ~ appropriately related with the parameters in (26). F o r higher spins, the equivalent non-linear unitary t r a n s f o r m a t i o n seems to involve higher powers of spin operators.

We will see that the states (26) are the squeezed spin states.

4. Spin squeezing

The n o t i o n of squeezing was introduced in the context of the study of precision of m e a s u r e m e n t of a force carried by determining the change that it induces in the state of a h a r m o n i c oscillator. T h e precision of measurement depends, amongst o t h e r things, on the dynamical variable of the system to which the force is coupled, on the variable on which the m e a s u r e m e n t is carried and on the initial state of the system. T h o s e states of the oscillator for which the precision of measurement carried by determining the change in the position of the oscillator coupled linearly to the force is m o r e c o m p a r e d with that o b t a i n e d by preparing it in a coherent state are called the squeezed states.

Those states turn out to be the ones for which the variance in a generator; q or p; of the canonical algebra is less than its value 1/2 in a coherent state.

T h a t n o t i o n of squeezing has since been extended to non-canonical systems by generalizing the a p p r o a c h based on the uncertainty relation. The s t a n d a r d a p p r o a c h [15] for identifying the squeezed states is to proceed from the definition of c o h e r e n t state for two observables A and B. In that state the uncertainty relation (1) is satisfied with equality with ( F ) = 0; AA 2 = AB 2 = I(C)1/2. A squeezed state is then defined as the one in which AA 2 or AB 2 is less than

I(C)1/2.

T h a t definition leads to an u n a m b i g u o u s identification of squeezed states in a canonical system because in that case [ ( C ) I is the same constant for all the states in the set of the canonical c o h e r e n t states. In the case of non-canonical operators,

I(C)l

depends on the p a r a m e t e r s of the c o h e r e n t state. T h e r e may, therefore, be states in the set of coherent states which satisfy the a b o v e m e n t i o n e d condition of squeezing for some pair of observables. T h e class of squeezed and the coherent states so constructed are then no longer mutually exclusive.

In the case of a system of spins, for example, it can be seen that the c o h e r e n t states 10, nrc), where n is an integer, satisfy the condition of squeezing with A, B taken as

S x, S r

[15].

T h e u n c e r t a i n t y relation based a p p r o a c h to squeezing in the case of spins can be made u n a m b i g u o u s by noting, following Kitagawa and U e d a [ 19], that the variance in any c o m p o n e n t in a direction o r t h o g o n a l to the average direction of the spin is

S/2.

O n e can, therefore, examine the noise in the c o m p o n e n t s o r t h o g o n a l to the average Pramana - J. Phys., Vol. 48, No. 3, March 1997 793

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direction of the spin in the state under investigation. If a and b are two directions orthogonal to each other such that a x b=S/ISI then in a coherent state (Aa.S) 2 = (Ab.S) 2 ~- (ASu~) 2 = ](S)1/2 i.e.

S 1

(AS ) = = ~ I ( S ) I- (27)

A squeezed state is then defined [19] as the state for which

(AS~) 2 < I(S)I/2, SQ(I) (28)

It is straightforward to see that the minimum uncertainty states (21) satisfy the condition (28) of spin squeezing.

The criterion SQ(I), based on the uncertainty relation, makes the sets of coherent and squeezed states mutually exclusive. A criterion based on the requirement of improvement of sensitivity over what is obtained by using the coherent states in the measurements with spins is derived in the next section. It is shown that the parameter

¢ = 2 S ( A S ~ ) 2 / ( S u ) z, (29)

where S~ is a spin component perpendicular to S u, is a measure of the sensitivity involving spins. By using (24) and

AS~ = S ( 1 - sin2(0)cosZ(¢)), ASr 2 = S ( 1 - sin2(0)sin2(~)),

AS~2 = 5 sin2(0)' S (30)

it can be shown that the minimum value of ~ in a spin coherent state is 1 i.e.

~mi . . . . h = 1. (31)

A spin squeezed state is defined as a state for which ~ for some spin component is less than unity i.e. a state for which

=- 2 S ( A S ~ ) 2 / ( S u ) 2 < 1 SQ(II). (32)

It is easy to see that SQ(II) holds only if SQ(I) does. The criterion SQ(II) reduces to the one derived in [20] in the context of improvement of sensitivity in Ramsey spectroscopy. The condition SQ(II) in that context has been referred to as spectroscopic squeezing [ 18].

As shown in [18], the minimum uncertainty states (26) for 2 v ~ + 1 satisfy the condition SQ(II).

A distinguishing feature of the states satisfying (28) and (32) is that the spins in those states are correlated [18, 19]. That follows from the fact that, by way of their construction, the squeezed spin states and the spin coherent states constitute mutually exclusive classes and it can be shown that an uncorrelated state of spins is a spin coherent state. A detailed discussion of the relationship between the degree of correlation and the squeezing parameter (32) for the states (26) is given in [18]. Those correlations can be shown to be quantum mechanical in nature [21]. Note that quantum correlations is a distinguishing characteristic of the canonical squeezed states as well.

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In the next section we show that the squeezing p a r a m e t e r (32) is a measure o f the sensitivity in the process of measurement involving spins.

5. Sensitivity o f m e a s u r e m e n t

A force is measured by determining the changes it induces in an observable of a dynamical system. T h e observables of a dynamical system described q u a n t u m mechanically are, however, inherently statistical in nature thereby limiting the accuracy of measurement. G i v e n the coupling and the observable to be measured, the accuracy depends u p o n the state in which the measuring system is prepared. In this section we derive the p a r a m e t e r which determines the precision of m e a s u r e m e n t performed by coupling the force linearly to a system.

Let the hamiltonian of a system coupled to an external force be given by (5) in which one or m o r e a i are p r o p o r t i o n a l to the external forces to be measured. Let us suppose that ak is p r o p o r t i o n a l to the strength of the force and that it is to be determined by relating it with the m e a n value ( X j ) of the observable X j after the unitary e v o l u t i o n generated by the hamiltonian. The mean is determined with an accuracy which is p r o p o r t i o n a l to the variance AX 2. The inaccuracy in the d e t e r m i n a t i o n of the m e a n leads to an e r r o r A ~ k in the inference o f c t k. Since ( X j ) is a function of {ai} it follows that

A~t k = A X j / 1 ~ [ " (33)

E q u a t i o n (33) determines the error Aa k in the estimation of ~k from the measurements carried on Xj. Smaller the value of Aa k better is the precision of measurement. If A~mi ... h. is the m i n i m u m value of Aa k in a coherent state of a pair class containing X~, then the states for which Aa k < A~mi ... h. may be called squeezed states.

It is straightforward to see that the parameter ~ in the case of a m e a s u r e m e n t of a force by means of a H O leads to the same expression for the m i n i m u m measurable force and hence to the same criteria of squeezing as the ones arrived at by o t h e r m e t h o d s [ 10, 11]. Here we consider a process of m e a s u r e m e n t by means of a spin. T h e hamiltonian in that case is given by H = a.S. Suppose that one of the c o m p o n e n t s ak of a is required to be determined by measuring the spin c o m p o n e n t St(a) o r t h o g o n a l to a k after the unitary transformation generated by H. Hence, invoking (32), the e r r o r in the estimate of a k is

Aa k = ASl(a) / 8 ( S t ) (34)

Since St(a ) is obtained from S z by a unitary transformation generated by the hamiltonian, it follows that t~St(a)/aa k is expressible as a c o m m u t a t o r of St(a ) with a spin c o m p o n e n t , say, e.S. T h e c o m m u t a t o r , by virtue of (20), is a spin c o m p o n e n t o r t h o g o n a l to S z. Hence, the measure of error in the estimate of a force acting on a spin c o m p o n e n t is

Aak ~: ASt(a)/l(S'(a))l, (35)

where S±(a) is a c o m p o n e n t orthogonal to S(a). Hence the squeezing p a r a m e t e r defined by (32) is a measure of the sensitivity of m e a s u r e m e n t by means of a system of Pramana - J. Phys., Voi. 48, No. 3, March 1997 795

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spins. In terms of ¢, (35) reads

AS k ~ x / ~/2S. (36)

The expression (36) reduces to the ones derived in [-20] and [22] in the context of Ramsey spectroscopy and S U (2) interferometry respectively. Since the minimum value of ~ in a spin coherent state is 1, it follows that the maximum sensitivity in that case is of the order x//T/2s. O n the other hand, there are squeezed spin states for which ~ goes as 1/S [18, 20, 22]. In those states the error in the measurement, therefore, is of the order of 1/S which is evidently a marked improvement over the performance of a spin coherent states.

An experimental verification of theoretically predicted q u a n t u m noise has been carried in the experiments on trapped ions [23].

6. C o n c l u s i o n

A definition of the coherent states is proposed based on dividing the observables belonging to the Lie algebra of the hamiltonian into classes of pairs of observables such that the pairs in a class can be transformed into each other by a unitary transformation generated by the given Lie algebra. A state generated by a unitary transformation of the state which is a m i n i m u m uncertainty state with equal variance in a pair of observables in a pair class is a coherent state of that class. The set of coherent states of each such class completely determines the coherent states of the given hamiltonian. T h a t ap- p r o a c h applied to the hamiltonian of a h a r m o n i c oscillator gives, besides the well- k n o w n canonical coherent states, a new set of c o h e r e n t states. In the case of spins there is only one pair-class leading to the k n o w n spin c o h e r e n t state. T h e n o t i o n of squeezing is similarly sought to be based on a physical requirement by defining a squeezed state as the one p r e p a r e d in which the given system gives better precision than obtained by preparing it in a c o h e r e n t state in the measurement of a force coupled linearly to it. T h e condition of squeezing based on that criterion is derived for a system of spins.

R e f e r e n c e s

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[2] J R Klauder and Bo-Sture Skagerstam (eds), Coherent states (World Scientific, Singapore, 1985)

[3] Wei-Min Zhang, D H Feng and R Gilmore, Rev. Mod. Phys. 62, 867 (1990) [4] A O Barut and C Girardello, Comrnun. Math. Phys. 21, 41 (1971)

[5] G S Agarwal, J. Opt. Soc. Am. B5, 1940 (1988)

[6] A M Perelomov, Commun. Math. Phys. 26, 222 (1975); Generalized coherent states and their applications (Springer, Berlin, 1986)

[7] M M Nieto and D R Truax, Phys. Rev. Lett. 71, 2843 (1993) [8] R R Puri, Phys. Rev. AS0, 5309 (1994)

[9] R R Purl, Phys. Rev. A49, 2178 (1994) [10] H N Hollenhorst, Phys. Rev. D19, 1669 (1979)

[11] R S Bondurant and J H Shapiro, Phys. Rev. D30, 2548 (1984) [12] R R Puri and G S Agarwal, Phys. Rev. A53, 1786 (1996) [13] R R Puri and G S Agarwal, Int. J. Mod. Phys. BI0, 1563 (1996)

[14] P Shanta, S Chaturvedi, V Srinivasan, G S Agarwal and C L Mehta, Phys. Rev. Lett. 72, 1447 (1994)

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[15] K Wodkiewicz and J H Eberly, J. Opt. Soc. Am. B2, 458 (1985) [16] J M Radcliffe, J. Phys. A4, 313 (1971)

F T Arecchi, E Courtens, R Gilmore and H Thomas, Phys. Rev. A6, 2211 (1972) [17] M A Rashid, J. Math. Phys. 19, 139t (1978)

[18] G S Agarwal and R R Puri, Phys. Rev. A49, 4968 (1994)

R R Purl and G S Agarwal, in Coherent states: Past, present and future edited by J R Klauder and D H Feng (World Scientific, 1994) p. 447

[19] M Kitagawa and M Ueda, Phys. Rev. A47, 5138 (1993)

[20] D J Wineland, J J Bollinger, W M Itano and F L Moore, Phys. Rev. A46, 6797 (1992) [21] R R Puri, J. Phys. A29, 5719 (1996)

[22] B Yurke, S L McCall and J R Klauder, Phys. Rev. A33, 4033 (1986) B Yurke, Phys. Rev. Lett. 56, 1515 (1986); Physica BI51, 286 (1988)

[23] W M Itano, J C Bergquist, J J Bollinger, J M Gilligan, D J Heinzen, F L Moore, M G Raizen and D J Wineland, Phys. Rev. A47, 3554 (1993)

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Coherent and squeezed states on physical basis

Coherent and squeezed states on physical basis

Coherent and squeezed states on physical basis

2 =i2(F),

1(C)1/2.

H = ~ ~iXi,

U=exp[i ~

UP1 U*, Q2

P2)

Q z)

SU(1,

U+l ~),

[ ~ ]1/2

(14) L2",,/'~ m~ J

m=O

I ~)

=zJ~), (25)

Coherent and squeezed states on physical basis

[d?,.) = A . exp(flSz) exp(-- i~ ~ )lm) - A , exp(flS,_)lm),

I(C)1/2.

I(C)l

S x, S r

S/2.

UP1 U, Q2*