Coherent states and squeezed states of realq-deformed quantum oscillators

PRAMANA journal of physics

© Printed in India Vol. 42, No. 4, April 1994 pp. 299-309

Coherent states and squeezed states of real q-deformed quantum oscillators

G VINOD, K BABU JOSEPH and V C K U R I A K O S E

Department of Physics, Cochin University of Science and Technology, Kochi 682 022, India MS received 24 August 1993; revised 20 November 1993

Abstract. A detailed physical characterisation of the coherent states and squeezed states of a real q-deformed oscillator is attempted. The squeezing and q-squeezing behaviours are illustrated by three different model Hamiltonians, namely i) Batemann Hamiltonian ii) harmonic oscillator with time dependent mass and frequency and iii) a system with constant mass and time-dependent frequency.

Keywords. Quantum groups; quantum oscillators; q-coherent states; q-squeezed states.

PACS Nos 02-20; 03-65; 42-50 1. Introduction

Quantum groups and quantum oscillators have been widely studied recently.

Biedenharn [1] and Macfarlane [2] discussed q-oscillators and realized the quantum algebra su~(2) in terms of these oscillators. Many other versions of q-oscillator have appeared [3, 4]. Quantum oscillators have already found applications in diverse fields such as molecular spectroscopy [5], condensed matter physics [6], quantum optics [4, 7] and many-body theory [8]. An anharmonic version of q-oscillator with quartic interaction has been discussed recently [9].

Coherent states and squeezed states of the electromagnetic field have been the subject of several investigations. Sehrodinger [10] introduced a system of wavefunctions to describe the nonspreading wave packets for the quantum harmonic oscillator. Later Glauber [11] called these states coherent states (CS) and applied them to the radiation field. Squeezed states were first introduced by Yuen 112] and are of great importance in quantum optics. Coherent states and squeezed states can be defined for the q-oscillators also and are of current interest [4, 7]. Chaturvedi ^{et al} [4] have constructed the coherent states and squeezed states corresponding to the commutation relation aa + - q a + a = 1. In this paper we compute the variances associated with the coherent and squeezed states and attempt to distinguish between squeezing and q-squeezing for real q-deformed oscillators.

Section 2 is a brief review of the spectrum of a real q-deformed oscillator. In § 3 a general treatment of the coherent states and squeezed states is given, emphasising the physical aspects. Section 4 deals with these model Hamiltonians that illustrate the ideas and formalism of squeezing and q-squeezing under the SU(1, 1) pattern.

The results of numerical computations of the probability for observing a definite number of quanta in q-squeezed states are presented and some concluding remarks are offered in § 5.

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G Vinod et al

2. Spectrum of a

q-deformed

Here we recapitulate the basic facts about real q-deformed oscillators [4]. Let us consider the q-commutation relation (q-CR)

aa + - q a + a = 1 (1)

where q is real. We demand that [ N , a + ] = a +,

IN, a] = - a. (2)

If we take

aa + = I N + 1]q

a + a = [N]q (3)

with [X]q = (qX _ l)/(q - 1), (1) is readily satisfied. With the help of (3), (1) can be rewritten as

[a, a + ] = qS. (4)

The operators a +, a and N can be thought of as creation, annihilation and number operators respectively for a q-deformed H.O. with the Hamiltonian

Hq p2/2m ..1.- 1 - - 2 V 2

where

Xq = , f h / 2 m w ( a + a + ), Pq = - i /m~-~ -~ (a - a + ),

(5)

(6)

Xq and P~ are hermitian operators satisfying the CR

IX., pq] = ihq N. (7)

The uncertainty relation corresponding to this is AXqAPq >I ~{qN}.

AS q ~ t, one recovers the corresponding operators of ordinary quantum mechanics namely X and P.

The q-oscillator Hamiltonian can be written as hoJ +

H~ = --~ (aa + a + a),

= ~ - ( [ N ] ~ + IN + 1].). (8)

Z

300 Pmnmn. - J. Phys., VoL 42, No. 4, April 1994

Real q-deformed quantum oscillators The energy eigenvalues are

E. = ½hm([n]q + In + 114) (9)

where n is an eigenvalue of the operator N:

Nln)~ = nln)q. (10)

In general, any state fn)q can be built up from the q-vacuum (a+)"

In)q = ~/[n][ 10)q (11)

where [hi! = [n] [ n - 1] ... [2][1] and the q-vacuum [0)q is identified with ordinary vacuum 10).

3. Coherent states and squeezed states

A normalized q-coherent state (q-CS) is defined as

I~)q = [expq(l~12)] -1/2 ~ cx"

exp,ixffi ~ X"

where

(12) (13)

The q-CS can be obtained from the q-vacuum by the action of a q-displacement operator Dq(~):

I~)q - D~(u)10)q (14)

where

D~(0t) ffi [expq]~,[ 2 ] - 1/2 expq(~a + ).

Note that the displacement operator that generates a q-CS from the vacuum is not equal to expq(~a + - 0c'a), which is the q analogue ofexp(ua + - oc*a), the displacement operator in the q-- 1 theory.

The probability of measuring n quanta in the q-CS is p = [exp,(10cl2)]- ' (l~12"'~

\ [hi! ]

= [exp,(< [n] ) ) ] - ' \ [--~].~--.w / (15) where we have made use of the fact that

I~l 2 = <[n] >. (16)

Equation (i5) represents a q-Poisson distribution. Variances of Xq and Pq defined

Pramana - J. Phys., Vol. 42, No. 4, April 1994 301

G V i n o d e t a l

by (7) are calculated:

VarXq = ~ h (l~tl2(q - 1) + 1),

VarPq = ~ (10tl2(q - 1) + I). (17)

Equation (17) fixes an upper limit for ~t i.e. I~1 ~< 1/(1 - q).

The uncertainty product for a q-CSIn>q is expressed in the form

(AXq)tAPq) = h/2(Ictl2(q - 1) + 1) (18)

while the corresponding relation for the state In>~ is

(AX,).tAP,). = ( h / 2 ) ( [ n ] + [ n + 1]). (19) The q-vacuum state satisfies the relation

(AX,)o(AP,)o = h/2 (20)

which coincides with the position-momentum uncertainty relation for the ordinary vacuum state J0>. Thus the uncertainty product for q-CS is different from that of the q-vacuum state, which is h/2. Also the uncertainty product depends on the parameter ct.

Let us introduce two self-adjoint operators A~ and At, called the quadrature components, such that

a = A1 + i A 2

a + = A 1 - i A 2 (21)

then it can be seen that

V a r A l = (¼)((q - 1)l=l 2 + 1),

V a r A 2 - - ( ¼ ) ( ( q - 1)1~12 + 1),

( A A I ) ( A A 2) = l ( ( q _ 1)latl2 + 1). (22)

The quantum noise energy [12] is zero for q-CS also:

< = l ( A a ) + (Aa)l~t>, = 0. (23)

Let us now define two new operators b and b+:

With

I~l ~ - I v l ~ = 1

it follows that

b b ÷ - b ÷ b = q ~.

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(24) (25)

(26)

Real q-deformed quantum oscillators

The SU(I, 1) squeezed states of the q-oscillator are defined as follows:

bLt)q

W e can express It)q as a linear combination of the number states [n)q [131:

It),l-- ~ C,,In>q,

n = O

b l t ) q = ~ a + va +) ~ C,,In)q,

n = O

= t ~ C,[n>~.

n = O

It follows that

Cl = t C o / #

C2 = (tiC, - vCo)/x/([2])#

or in general,

C N =

(27)

(2s)

(29)

It can easily be shown that

( A t ># = 1/2(fl(#* - v*) + fl*(# - v)),

<A2 )# = 1/2(fl*(# + v) + t(#* + v*)), Var a i = ¼1# - vl 2 ~(t),

VarA2 = ¼1# + vl2a(t).

Also (AAx) (AA2) --- ¼l# 2 - v2lo'(t) where

~(t) = ~ q'lC.[ 2.

n = O

Pramana - J. Phys., VoL 42, No. 4, April 1994

(31)

(32)

(33) (34)

(35)

303 H = 7 ( 2 # * v ' b 2 - 2#vb +2 + bb + + b + b).

so that the Hamiltonian (8) becomes

tc,_,

vx/t[n-

#~/[nl

Although, in general # and v are arbitrary, here we consider the evolution of a state which is initially a q-coherent state (i.e. #(0) = 1, v(0) = 0) into a q-squeezed state It)q at t. This procedure is motivated by similar considerations made in references [141 and [15] in the context of ordinary coherent states and squeezed states.

Inverting (24), we have a = #*b - vb + a + = v*b + #b +

G Vinod et al In the present work we normalization

restrict q to the domain 0~< q ~< 1, and assume the

[C,[ 2--- 1. (36)

n = O

This yields the inequality o(t) ~< 1.

At this point we wish to draw a distinction between squeezing and q-squeezing.

By squeezing we mean that the variance goes below the value of the uncertainty product in the q-vacuum state, which is same as that for the ordinary state. On the other hand, q-squeezing implies that the variance is less than the uncertainty product for the q-CS state. For an undeformed oscillator the uncertainty product is the same for vacuum as well as for CS. But when q # 1, the uncertainty product has different values corresponding to q-vacuum and q-CS.

From (35),

o'(0) = expq(q 1ilia) (37)

exp¢([~[2) "

We assume that [ff)¢ is prepared initially as a q-CS. Hence it follows from (22), (34) and (37) that

(q - l)la~l 2 + 1 = expq(ql/~[2) (38)

exp,([/~l z)

Thus in the q ~ 1 theory, u and ~ are interrelated. Squeezing means (AA 1) or (AA2) ~< ¼ and q-squeezing means (AAI) or (AA 2) ~< 0"(0)/4. We may define the degree of squeezing and the degree of q-squeezing respectively as

S(A) = 2((AA) 2 ) 4 1

~ , (39)

4

5~) = 2 ( ( A A ) 2 ) - (¼) e(0) (40)

¼o(0)

Squeezing corresponds to S A' < 0, or SA2"< O, while q-squeezing implies Sff' < 0 or S~" < O.

4. SU(1, 1) squeezed states of real q-deformed Hamiltoaiaas 4.1 Batemann Hamiltonian

Baseia et al [14] studied the appearance of squeezed states for the Batemann Hamiltonian where the mass of the oscillator changes suddenly. In this section we discuss the squeezing and q-squeezing properties associated with SU(1, 1) squeezed states of a real q-deformed Batemann Hamiltonian, defined by the relation

where

e2 ~

H(t) = 2M(t---)) + M(t)°~2X2 (41)

M(t) = Mo eat.

304 Pramana - J. Phys., VoL 42, No. 4, April 1994

Real q-deformed quantum oscillators

W e take the corresponding q-deformed Hamiltonian in the form:

Setting

p2 ₁

Hq(t) = + ¼M(t)f~2Xa z (42)

2M(t) ^{L -}

#(t) = cosh(gt/2) = ~ \ ~ - ~ o +

v(t)=sinh(gt/2)ffil( M~(t)- ^{M~(t) )}

2 \ ~ / M o

(43) the condition Igl 2 - [ v l 2 = 1 is easily satisfied. Now consider the case where mass changes suddenly as

M o - - * M ~ at t m t l :

M(t) = MoO(t -

where 0 is the Heaviside step-function.

The quadrature components A 1 and A2 have equal variances:

(AA 1 )~ = (AA2) ~ =

¼o(t)

Thus IP)q behaves as a q-CS.

For t > tt,

(45)

where

Then

M ( t ) = M o + A M -- Mo(l + 6) 6 = - - , AM A M = M I - M o

Mo (AA~)2 = ~(t)

4(1

a)'

(AA 2)2 -- (~-~--~) or(t),

(AA1) = (AA2) = ~°'t---2". (46)

4

This implies that the uncertainty product can go below ¼. For ( ¢ ( t ) - 1 ) < 6 <

(1/{'¢(t)'1- 1), both (AA~) 2 and (AA2) z fall below this value. In this region one can in principle measure both A~ and A2 with uncertainties less than that predicted by Heisenberg's uncertainty principle. Also in the region ([¢(t)]/['¢(0)] - 1) < 6 < ([~r(0)l/

[a(t)] - 1), both A1 and A2 have variances below that in q-CS. The squeezing pattern is as follows:

AI is squeezed if 6 > ( ¢ ( t ) - 1),

A 1 is q-squeezed if 6 > (¢r(t)/~r(0) - 1). (47) Similar remarks apply to squeezing and q-squeezing of the A2 component.

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G Vinod et al

4.2

Harmonic oscillator with time dependent mass and frequency

Consider the Hamiltonian

H = e2/2M(t) + (1/2)M(t)to2(t)X 2

where

M(t) = Mo e;~, to(t) = COo e-°'.

Squeezing properties of this Hamiltonian have been studied earlier [15]. We can q-deform it as

Hq-

+ (1/2)M(t)to2(t)X 2.

- 2M(t)

The coefficients/z(t) and

v(t)

#(t) = cosh (-A-2-P-) t,

v(t) f sinh ( ~ - ~ )t.

The variances of the quadrature components are calculated as:

(AAI)2 = ¼e-<a-m,o(t), (AA 2)2 = ¼e<~- p), o(t),

(AA,)(AA2)

(51)

Here also the uncertainty product may decrease below ¼.

1 l n ( ° ( 0 ) ' ~ . - 1 l n ( o ( t ) ) a n d q squeezed if (3. - p) < t \ o ( 1 ) ] AI is squeezed if (3. - p) < t

Similar conditions can be obtained for A 2 also.

4.3

System with constant mass and time dependent frequency

Consider the q-deformed Hamiltonian

D 2

n = ~ + ½Mto2[1 + Zcos(2to o +

e)t]X~.

The squeezing properties of the corresponding q = 1 Hamiltonians have been studied by Geethakumari

et al

We put

#(t)=~[( l+c°s(2to°+8)t'l'/21+-~ ]

X+e)t)l/2], v(t)=½[(l+cos(2too+e)t)'/2 (

I + Z - l+cos(2to o + e ) t }

_]"

306 Pramana ^- J. Phys., VoL 42, No. 4, April 1994

Then

Real q-deformed quantum oscillators

I 1 + Z ] a(t), (AAI)2=¼ 1 + c o s ( o ~ o + O t

[ l + x ]a(t), (AA2)2 = ¼ 1 + cos(2oJ 0 + e)t (AA1)(AA2) = ¼a(t).

1 + X 1

A~ is squeezed if <

1 + cos(2a) o + ~)t a(t)

1 + X ~r(0)

q-squeezed if 1 + cos(2oJ o + e)t < e(t---)"

and

(54)

5. Results and discussion

We have calculated the probability distribution P , for various values of q keeping fixed. In figure 1, logioP,(q) is plotted against n for four different values of q, namely 1, 0-9, 0"8 and 0 for ~ = 0.5. Figure 2 gives the corresponding graph for ~ = 0.9. It is clear from these that the graphs for a given q value for different ~, values are similar.

In figure 3, logtoPo(q)/Po(1) is plotted against q where Po(q) is the probability of the ground state corresponding to q # 1 and P0 that corresponding to q = 1. These graphs plotted for the values, namely, ~ = 0.5 and 0"9, show marked variation in their behaviour except near the standard value, namely, q = 1, where they converge.

Starting from a commutation relation for the q oscillator, coherent states of the oscillator have been constructed and the variances of the quadrature components calculated. The q analogues of bosonic squeezed states have been defined in two ways and are illustrated using three model Hamiltonians. It has been found that if we could achieve a condition which pushes q below unity, we could measure Xq or Pa (or both) with uncertainties smaller than h/2. But ultimately, it is experiment that ought to determine the physical realizability of such states.

0~

- 5

I -10

~ --15

- -20

-250

Figure 1.

D) q = 0 .

¢o ;s /o 2s

n

Variation of logloP.(q) with n, for • = (~5: A) q - 1; B) q = 0-9; C) q = 0"8;

Pramana - J . Phys., Vol. 42, No. 4, April 1994 307

O Vinod et al

0 ~

-- 5

- - 1 0

- - 1 5

- 2 0

- - - 2 5 - - 3 0

- 3 5 0

Figure 2, D) q = 0 .

D

A

I I I /

5 10 15 20 25

Variation oflogtoP.(q) with n, for u -- 0-9: A) q = 1; B) q ffi 0-9; C) q ffi 0-8;

0.~

- 0 - I

° ~ - 0 , 3

- 0 . 4 B

I , ! |

o,2 o'•, de o-e

q

Figure 3.

1.2 Variation of loglo[Po(q)/Po(l)] with q: A) a ffi 0-5; B) a = 0-9.

A c k n o w l e d g e m e n t s

The authors are grateful to Prof. S Chaturvedi, School of Physics, Central University of H y d e r a b a d for a careful reading of the manuscript and useful suggestions. One of us (GV) thanks U G C for the award of a junior research fellowship.

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