When the defor- mation parameter q being a root of unity, the beam quality factor M2 ... theoretical value of M2 can be calculated by second moment me...

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The deformed uncertainty relation and the

arXiv:quant-ph/9603029v1 27 Mar 1996

corresponding beam quality factor∗ Kang Li, Dao-Mu Zhao, and Shao-Min Wang Department of Physics, Hangzhou University Hangzhou,310028, P.R. China (Received 23 March, 1996)

Abstract By using the theory of deformed quantum mechanics, we study the deformed light beam theoretically. The deformed beam quality factor Mq2 is given explicitly under the case of deformed light in coherent state. When the deformation parameter q being a root of unity, the beam quality factor Mq2 ≤ 1. PACS number(s): 42.60Jf, 42.65Jx, 03.65.-w

Typeset using REVTEX

∗ The project

supported by the National Natural Science Foundation of China and Zhejiang Provin-

cial Natural Science Foundation of China.

1

I. INTRODUCTION

The beam quality factor M 2 is a very important concept in laser physics, which expresses directly the goodness of a laser beam. So the theoretical analysis and experimental measurement of the beam quality factor is a very interesting and hot subject recently. It is known that the beam quality factor can be defined as[1] : M2 =

Real beam space − beam width product . Ideal Gaussian space − beam width product

Its mathematical expression is M2 =

π θ.ω0 , λ

(1)

where θ is the far-field angular spread of the beam, ω0 is the beam waist radius. The theoretical value of M 2 can be calculated by second moment method as: θ = 2π

RR

ω0 = 2

RR

ˆ x , sy )dsx dsy (sx − s¯x )I(s RR ˆ x , sy )dsx dsy I(s

(2)

(x − x¯)I(x, y, z)dxdy RR |z=z0 I(x, y, z)dxdy

ˆ x , sy ) is a spatial frequency diswhere I(x, y, z) is a time-average intensity profile and I(s tribution, z0 is the beam waist location. In the far-field condition, the spatial frequency and the divergence θ have the relation s = λθ . Using the uncertainty relation of quantum mechanics, the authors in reference [2] proved that for any beam its beam quality factor defined by the second moment method can not be less than 1. On the other hand, if a beam propagates through a self-focusing nonlinear medium, its effective divergence angle is[3][4] 2 2 θef f = θ − βJ,

(3)

where θ is the divergence angle for same beam propagating through a linear medium, β is a constant depending on the medium, J is a quantity related to the critical power of the beam. From equation (1) we have: 2

2 4 Mef f = (M −

1 π2 2 ω0 βJ) 2 . 2 λ

(4)

2 2 Obviously when βJ positive, Mef f < M , namely when a beam propagates through a self-

focusing nonlinear medium, the beam quality factor will reduce. If the beam is the ideal Gaussian beam, then in the nonlinear medium, its quality factor will less than 1. Recently, a new CO2 laser with equivalent beam quality factor Me2 = 0.3 is realized experimentally by the Hangzhou University and the 12th Institute of National Electronic and Industry Department of China[5] . According to the reference [2], these results of M 2 ≤ 1 can not be analyzed by the theory of quantum mechanics. We call the beam being the deformed beam if the beam quality factor is less than 1. In this paper we will use the theory of q-deformed quantum mechanics[7,8] to disciple the deformed beam, and give out the generalized beam quality factor Mq2 explicitly , where the deformation parameter q plays a row of a bridge connecting the usual beam and deformed beam. The paper is organized as following: In second part, we give out the deformed uncertainty relation in the q-deformed quantum mechanics. In third part, we discuss the uncertainty and the deformed beam quality factor in the deformed coherent state. The deformed wave functions of deformed coherent states are given in the part four. There are some results and discussions given in the last part.

II. THE DEFORMED UNCERTAINTY RELATION IN Q-DEFORMED QUANTUM MECHANICS

In recent years, on the base of quantum group there developed a new quantum mechanics theory– q-deformed quantum mechanics[8] . In the deformed quantum mechanics, the derivative is replaced by a parametric derivative, and the Hamiltonian has quantum group symmetry. In the one dimensional case, the coordinate and momentum of quantum mechanics (QM) and that of deformed quantum mechanics (DQM) have the following relation ship 3

QM Coordinate x

DQM Relation xq = x

xq

d Momentum −i¯h dx −i¯h ddqqx

dq dq x

d = x1 [x dx ]q

From the table above, we can see that in the q-deformed quantum mechanics the coordinate does not deformed, but the momentum is deformed into pq = −i¯h

dq 1 d = −i¯h [x ]q dq x x dx

(5)

where the q-bracket is defined by [A]q =

q A − q −A . q − q −1

(6)

It can be proven easily that the [A]q=1 = A, thus in this case (q = 1) the deformed quantum mechanics comes back to quantum mechanics. Introduce operators: dq 1 ) ¯ aq = √ (xq + h dq x 2¯h

(7)

1 dq a+ ), (xq − h ¯ q = √ dq x 2¯h then xq =

s

h ¯ (aq + a+ q ), 2

s

pq = −i

h ¯ (aq − a+ q ). 2

(8)

The operator aq , a+ q as well as another operator Nq form a deformed harmonic oscillator algebra: + + −Nq + [aq , a+ , [Nq , a+ q ]q = aq aq − qaq aq = q q ] = aq , [Nq , aq ] = −aq .

(9)

Nq and aq , a+ q have the following relations: + aq · a+ q = [Nq + 1]q , aq · aq = [Nq ].

From the equations (8),(9),(10), we obtain the commutator between xq and pq as 4

(10)

[xq , pq ] = i¯h{[Nq + 1]q − [Nq ]q }.

(11)

(∆xq )2 = (xq − x¯q )2 = x2q − xq 2 , (∆pq )2 = (pq − p¯q )2 = p2q − pq 2 ,

(12)

If we define

where the bars over the operators represent the quantum mechanics average values. Then from the general principle of uncertainty relation, we have (∆xq )2 · (∆pq )2 ≥

h ¯2 2 ([Nq + 1]q − [Nq ]q ) 4

(13)

Reference [8] pointed out that, for the consistent q-deformed quantum mechanics, the deformation parameter can only take the value in the range: (1) q being positive real number, (2) q being a root of unity. When q being positive, the deformed uncertainty is greater 2

than that of undeformed case; When q being a root of unity, ([Nq + 1]q − [Nq ]q ) ≤ 1, the corresponding uncertainty is weaker than that of the quantum mechanics in general. In order to describe the deformed beam, we are interested very much in the case of q being a root of unity.

III. THE UNCERTAINTY AND THE BEAM QUALITY FACTOR IN THE DEFORMED COHERENT STATE iπ ), then the Hilbert In the sequels, we choose q being a root of unity, i.e., q = exp( p+1

space corresponding to the deformed harmonic oscillator algebra has finite dimensions–(p+1) dimensions, its basis is n (a+ q ) |n >= q |0 >, [n]q

n = 0, 1, 2, · · · , p

(14)

We can obtain easily that:

a+ q |n >= aq |n >=

q

[n + 1]q |n + 1 >

q

[n]q |n − 1 >

aq |0 >= 0, a+ q |p >= 0, Nq |n >= n|n > . 5

(15)

Or aq , a+ q , Nq can be represented as aq = a+ q = Nq =

Pp

n=1

Pp

q

n=1

Pp

[n]q |n − 1 >< n|

q

n=1

[n]q |n >< n − 1|

(16)

n|n >< n|.

The coherent state |α > is defined as the eigenstate of aq , i.e: aq |α >= α|α >

(17)

which can be expanded by the basis of the Hilbert space: |α >=

p X

n=0

cn |n > .

(18)

Applying aq to both sides of (18), we can get the recurrence relation of cn , and then obtain cn as αn c0 , cn = q [n]q !

(19)

where c0 is the normalization factor which is given by c0 = (

p X

|α|2n − 1 ) 2. [n] ! q n=0

(20)

Set |αm >= (

p X

p |αm |2n − 1 X αn q m |n >, ) 2 [n]q ! n=0 [n]q ! n=0

then from the knowledge of linear algebra, we know that there are (p+1) independent |αm > which satisfy p X

m=0

|αm >< αm | = 1.

The relation between space frequency s and momentum p is s= So ∆s =

∆px 2π¯ h

p . 2π¯h

(21)

. Then from equations (1) and (2), we get that (∆x)2 · (∆px )2 = 6

h ¯2 2 M . 4

(22)

In the following, we calculate the (∆x)2 and (∆px )2 for the deformed coherent state. From equations (8) and (17), we have: s

xq =< α|x|α >=

h ¯ ∗ (α + α) 2

1 h ¯ x2q = α∗2 + α2 + α∗ α + (q + q −1 )|α|2 + 2 2 s

pq =< α|px |α >= −i

s

1 1 + (q − q −1 )2 |α|4 4

h ¯ − ∗ (α α ), 2

(∆xq ) = x2q − xq

(∆pq )2 = p2q − pq 2 So:

(26)

s

(27)

s

(28)

2

(29)

h ¯ 1 = (q + q −1 − 2)|α|2 + 2 2 h ¯ 1 = (q + q −1 − 2)|α|2 + 2 2

h ¯2 1 (∆xq ) · (∆pq ) = (q + q −1 − 2)|α|2 + 4 2 2

2

(25)

1 1 + (q − q −1 )2 |α|4 . 4

Using the equation (12), we get 2

(24)

s

h ¯ 1 p2q = − (α − α∗ )2 + α∗ α − (q + q −1 )|α|2 − 2 2

2

(23)

1 1 + (q − q −1 )2 |α|4 . 4

1 1 + (q − q −1 )2 |α|4 . 4

s

1 1 + (q − q −1 )2 |α|4 . 4

From the equations (22) and (29), we find that the beam quality factor M 2 of deformed coherent state reads: Mq2

1 =| (q + q −1 − 2)|α|2 + 2

s

1 1 + (q − q −1 )2 |α|4 | . 4

(30)

Obviously, when q = 1, Mq2 = 1. Namely, the coherent in quantum mechanics is the ideal π π Gaussian beam. When q being a root of unity, q − q −1 = 2i sin p+1 , q + q −1 = 2 cos p+1 , then

Mq2

s

= | 1 + |α|4 sin2

π π − (1 − cos )|α|2|. p+1 p+1

Now we discuss two cases as following: 7

(31)

1), for a given coherent state, i.e. the value of α is given, in order to keep the Mq2 being real, the integer p must be constrained as: π

p≥

(32)

arcsin |α|1 2 − 1

When |α| = 1, p can be any positive integers, under this case we have Mq2 = |2 cos

1 π − 1| = p+1 2 cos

p=1 π p+1

(33)

−1 p≥2

The table below shows that Mq2 ≤ 1. p

123

4

5

6

7

··· ∞

M 2 1 0 0.414 0.618 0.732 0.802 0.848 · · · 1 2), for a given integer p (i.e. the deformation parameter q is given). For example, for iπ

p = 3, namely q = e 4 , equation (31) becomes s √ 1 2 − 2 |α|2 − 1 − |α|4 |, Mq2 = | 2 2

(34)

and the constraint for |α| is |α|2 ≤

√

2.

(35)

Under the constraint of (35), the Mq2 given by equation (35) takes values in the region from 0 to 0.414 which is less than 1.

IV. THE EXPRESSION OF DEFORMED WAVE FUNCTION IN COHERENT STATE

From equations (18),(19) and (20), we know that, in the coordinate representation, the wave function of deformed coherent state is Ψ(x) =< x|α >= (

p αn |α|2n − 1 X q ) 2( ) < x|n > . [n]q ! n=0 n=0 [n]q ! p X

8

(36)

Define ψn =< x|n >,

(37)

dq ψ0 + β 2 xψ0 = 0, dq x

(38)

then from aq |0 >= 0, we have

where β 2 = ¯h1 . The solution to equation (38) reads: ψ0 = C0

∞ X

(−1)n

n=0

∞ β 2n β 2n 2n X (−1)n x C1 x2n+1 [2n]q !! [2n + 1] !! q n=0

(39)

where [2n]q !! = [2n]q [2n − 2]q · · · [4]q [2]q , [2n + 1]q !! = [2n + 1]q [2n − 1]q · · · [3]q [1]q , C0 and C1 are constants to be chosen by physics or mathematical conditions. When q = 1

1(p → ∞), ψ0 ∼ e− 2 β

2 x2

, which is just the result in quantum mechanics. When p finite,

[p + 1]q = 0, [n(p + 1)]q = 0. So, when p being an even number, the second term of ψ0 divergences, we can choose C1 = 0 to overcome the divergence. when p being an odd number, the first term of ψ0 divergences, so we can choose C0 = 0 in this case. Now we have: ψ0 (x) =

C0 C1

n β 2n 2n n=0 (−1) [2n]q !! x

P∞

P∞

n=0 (−1)

n

p being an even number

β 2n x2n+1 [2n+1]q !!

(40)

p being an odd number

From |n >= √1 a+ q |n − 1 > we have [n]q

1 a+ ψn = q q ψn−1 [n]q

(41)

From this recurrence relation, we can obtain ψn = (

1 1 1 dq n 2 (βx − ) ) ψ0 (x), n = 0, 1, · · · , p. 2n [n]q ! β dq x

9

(42)

V. CONCLUSION REMARKS AND DISCUSSIONS

In this paper, we give out the beam quality factor in deformed coherent state from the q-deformed uncertainty relation. When q = 1, Mq2 = 1, this is the result of quantum iπ mechanics. When q = exp( p+1 ), (p is finite integer), Mq2 ≤ 1. So the deformed coherent

state corresponding to a deformed beam. The ideal Gaussian beam propagating in the self-focusing nonlinear medium is just a deformed beam. If the deformation part of Mq2 of deformed Gaussian beam comes only from the divergence angle θq , namely: Mq2 =

π ω0 θq , λ

(43)

then from the equations (31) and (43), we have s

λ π π θq = | 1 − |α|4 sin2 − 2|α|2 sin2 |. πω0 p+1 2(p + 1)

(44)

From the equations (3) and (44), the βJ coefficient of the medium reads 4 π π π π λ2 + sin2 − 2 sin2 ) 1 − |α|4 sin2 βJ = 2 2 |α|4 (sin2 2 π ω0 p + 1 |α| p+1 p+1 p+1 s

(45)

Different medium corresponds to different deformed divergence angle θq , that is to say, different medium corresponds to different deformation parameter. For the self-focusing medium, 0 ≤ βJ ≤ 1, so when a beam propagates in the medium, the beam quality factor will decrease. Using the q-deformed quantum mechanics to study the deformed beam which provides an effective method to study the diffraction free beam in general, and make us know better about the inherent characteristics of the beam propagating theory. Acknowledgement The authors would like to thank Q. Lin and J.M. Xu for useful discussions.

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