- Email: [email protected]
Extended two-parameter squeezed states
2 August 1999
Physics Letters A 259 Ž1999. 7–14 www.elsevier.nlrlocaterphysleta
Extended two-parameter squeezed states Xiang-Bin Wang 1, L.C. Kwek 2 , C.H. Oh
3
Department of Physics,Faculty of Science, National UniÕersity of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore Received 3 March 1999; received in revised form 26 April 1999; accepted 24 May 1999 Communicated by P.R. Holland
Abstract In a recent paper, a new l-dependent squeezed states was proposed wJ. Beckers, N. Debergh and F.H. Szafraniec, Phys. Lett. A 243 Ž1998. 256–260x in which the usual Fock-space creation operator has been modified by introducing a simple translational term. In this paper, we generalize this approach by adding a new term proportional to the creation operator using exponential quadratic operators. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
squeezed and displacement operators, SŽ z . and DŽ a . respectively through the relations w7x
Squeezed states provide a means of challenging the theoretical and experimental limits imposed by the uncertainty principle and the zero-point fluctuations of the vacuum. Consequently, these novel nonclassical states have constantly stimulated much interest as the means for improving the signal-to-noise ratio in optical interferometry w1x, quantum teleportation w2x, Raman scattering w3,4x and a myriad of other applications such as gravitational wave detection w5x. Amplitude squeezed states have been observed in negative feedback laser with quantum nondemolition measurement of photon number w6x. They can also be generated by parametric amplifiers. Conventionally, squeezed states have been defined through the
< a , z : s D Ž a . S Ž z . <0: ,
where SŽ z . and DŽ a . are defined as the operators expŽ 12 Ž z ) a 2 y za† 2 .. and expŽ a a† y a ) a. respectively and Ž a,a† . are the usual annihilator and creator operators. In a recent paper w8x, Becker et al. proposed a new family of squeezed states by considering a modification of creation operator in Fock space using the relation a†l s a† q l ,
E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 2
Ž 2.
†
where a is the usual bosonic creation operator, a†l is called as l-parametrized bosonic creation operator. The l-parametrized Hamiltonian is given by Hl s 12 a†l ,a4 .
1
Ž 1.
Ž 3.
The eigenstates of l-parametrized Hamiltonian was investigated and it was shown that it is possible to obtain squeezed states for certain values of the parameter l w8x. Unlike usual squeezed states in which
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 4 - 0
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
8
the Hamiltonian are Hermitian, the Hamiltonian given in Eq. Ž3. is non-Hermitian and therefore these squeezed states are liable to decay processes. It has also been shown w8x that the family of squeezed states obtained through this modification constitutes new squeezed states. Moreover these l-states cannot be coherent states. Compared with the usual squeezed states w9,10x with wavefunction given by
py1 r4 Cz Ž x . s
1 2
zexp Ž x
2
.
exp
ž
z2y1 2z
2
/
x ,
py1r4
ž
(L Ž yl . 0 k
2
1 2
=exp Ž y x
2
1 y
'2
d
x q
dx
'2
k
l
/
..
s0 .
l1 2
q i l2
1
'2
1
'2
l2 x
p,
Ž 10 .
Hl1 , l 2 ,al† 1 , l 2 s al† 1 , l 2 .
Ž 7.
Ž 11 .
still hold. Indeed, non-Hermitian Hamiltonians with real positive eigenvalues have recently been studied w11x and due to their intimate relation to the transfer matrices of classical statistical mechanics and tight binding models with delocalization transition, they have also been intensively investigated in the context of random matrices w12–14x. In terms of quadratic exponential operator w15x, we can rewrite the new Hamiltonian in Eq. Ž8. as 2
2
Hl1 , l 2 s e 1r2 l1 a ql 2 a H0 e yŽ1 r2 l1 a ql 2 a. ' UH0 Uy1 .
Ž 12 . Note that the exponential operator U is not unitary hence the two Hamiltonians Hl1 , l 2 and H0 describes different physical systems, in particular unlike H0 , Hl1 , l 2 is non-Hermitian. It is not hard to show that the normalized eigenstate for the above Hamiltonian can be written as 1
(z Ž l , l . 1
s
(z
Ž 6.
The commutation relations for the new operators Ž a,a†l , l ., remain the same as the usual oscillator 1 2 algebra, namely,
w a,a x s
q i l1 xp y i
n
In this article, we generalize the transformation of the bosonic operator as
al† 1 , l 2 ,al† 1 , l 2
Ž 9.
s 12 Ž 1q l1 . x 2q 12 Ž 1 y l1 . p 2q
< n:l1 , l 2 s
2. Extended l-parameter states
s I,
s a†a q l1 a 2 q l 2 a q 12
Hl1 , l 2 ,a s ya,
Ž 5.
a†l1 , l 2 s a† q l1 a q l2 .
Ž 8.
where x and p denote the coordinate and momentum operators. This Hamiltonian is not Hermitian but the following relations
In Ref. w8x, l-dependent squeezed states are obtained through the solution of differential equations. By using exponential operator through an algebraic approach, we have generalized these l-dependent squeezed states by considering an additional term proportional to the Fock space annihilator operator. Indeed, we also show in this paper that our approach can be further generalized to include more parameters. In the next section, we introduce our modification and obtain our new squeezed states using essentially algebraic approach with the exponential operators. In Section 3, we derive some analytical and numerical results and show how this generalization adds an additional perspective to our analysis of squeezed states. Finally, in Section 4, we briefly explain how our approach can be generalized to include more parameters.
a,a†l1 , l 2
Hl1 , l 2 s 12 al† 1 , l 2 ,a4
Ž 4.
these novel squeezed states assume the form w8x
Clk Ž x . ;
Accordingly, the analogous l-parametrized Hamiltonian is
1
`
Ý n ls0
e 1r2 l1 a
1
ž
1
l! 2
Ý
n
rs0
ks0
(
n!
=
r!
Ž 13 .
l
Ý
(z
n:
l1 a2 q l2 a < n:
wŽ nyr .r2 x
s
ql2 a <
2
n
1
2
< r: .
/
1
k
ž /Ž 2
Ž 14 .
l1k l2nyry2 k n y r y 2 k . !k!
Ž 15 .
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
where znŽ l1 , l2 . s zn is the appropriate normalization factor and w m x denotes the integer part of m. It is easy to obtain the normalization factor znŽ l1 , l2 . explicitly as < zn < Ž w nyr . r2 x
n
s
Ý rs0
ž
1
k
ž /
Ý
2
ks0
l1k l2nyry2 k
n!
k! Ž n y r y 2 k . !
( /
n
'
Ý
n!
2 Tny r
r!
rs0
Using Ž21., we can easily calculate the expectation values and the variances. In particular, we see that for the position operator, x s 1r '2 Ž a q a† ., the expectation values of x and x 2 for the eigenstate < n:l1 , l 2 are
2
² x :l1 , l 2 s
'2
r!
Ž 16 . ,
n
zn
rs1
1
1
² x 2 :l1 , l 2 s
Ž 17 .
n
Ts s
Ý ks0
ž / 2
l1k l2sy2 k k! Ž s y 2 k . !
.
Ž 18 .
a < n:l1 , l 2 s
zny 1
al† 1 , l 2 < n:l1 , l 2 s
'n < n y 1:l , l 1
zn znq 1
,
2
'n q 1 < n q 1:l , l 1
. 2
Ž 20 .
min Ž m , n .
s
Ý
m zn
=
ž
/
,
Ž 23 .
² p 2 :l1 , l 2 s
Ž 24 .
1
n
1 q
zn
2 n
y
Ý rs2
ž
2 Tny r n!
Ý rs1 Ž r y 1 . !
Tny rq2 Tnyr n!
Ž r y 2. !
/
.
Ž 25 .
Using the results above, we can easily compute the variances 2
2
2
2
Ž D x . l1 , l 2 s ² x 2 :l1 , l 2 y Ž ² x :l1 , l 2 .
Ž 26 .
and
Ž D p . l1 , l 2 s ² p 2 :l1 , l 2 y Ž ² p :l1 , l 2 .
Ž 27 .
and show that these quantities satisfy the inequality 2
2
Ž D x . l1 , l 2 Ž D p . l1 , l 2 ) 14 .
Ž 28 .
lŽ2mqny2 r . r!
r
3. Results and numerical simulation
w myrr2 x w nyr x r2
=
Ž r y 2. !
Indeed, these states are not minimum uncertainty states Žexcept for n s 0..
² m < n:l1 , l 2
l1 , l 2
(z
Tny rq2 Tnyr n!
Ý
For the momentum operator, p s 1ri'2 Ž a y a† ., the expectation values of p and p 2 are
Ž 19 .
It is easy to see that, in the limit l1 ™ 0, we recover the features of the l-parametrized states in Ref. w8x and that in the double limit in which both l1 , l2 ™ 0, we get the usual features of the Fock states under the action of the operators a and a†. As for the l-parameter states in Ref. w8x, our two-parameter l states are normalized but not orthogonal
'm!n!
Ž r y 1. !
² p :l1 , l 2 s 0 ,
In the limit l1 ™ 0, the normalization factor reduces to the generalized Laguerre polynomial, L0nŽyl22 .. The Fock space representation of the operators Ž a,a†l , l . can be expressed as 1 2
zn
rs1
rs2
s k
žÝ
zn
Ž 22 .
2 Tny r n!
n
q 2
where the partial sum, Ts is defined as 1
n!
Ý Ž Tnyrq1Tnyr . Ž r y 1. ! ,
q
2
9
Ý
Ý
ks0
js0
1
kq j
ž / 2
Ž kqj . l1kq jly2 2
Ž m y r y 2 k . ! Ž n y r y 2 j . !k! j!
/
.
Ž 21 .
In this Section, we discuss our results and numerical simulation of the l-dependent squeezed states. Indeed, it is instructive to consider our results for different values of n. For n s 1, it is obvious that one of the parameters, l1 , does not contribute to the
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
10
Fig. 1. Three-dimensional plot of Ž D x .l21, l 2 as a function of l1 and l2 for ns 2. Note that for squeezed states, we need to demand that Ž D x .l21, l 2 - 0.5.
expectation values and our results are the same as those obatined in Ref. w8x, namely ² x :l1 , l 2 s
'2 l2 l2 q 1
² x 2 :l1 , l 2 s
,
Ž 29 .
l22 q 3 2 Ž l22 q 1 .
l42 q 3
2
Ž D x . l1 , l 2 s
2 Ž l22 q 1 .
,
Ž 30 . ,
Our two-parameter formulation inevitably provides a fuller picture for the phase space of the l i Ž i s 1,2. parameters. It is possible to plot the variance Ž D x .l21 , l 2 with different values of l1 and l2 . This plot is shown in Fig. 1. To locate the existence of l-parametrized squeezed states, we seek for values of the parameters l1 and l2 satisfying Ž D x .l21 , l 2 - 12 . We note that in the one-parameter limit in which l1 ™ 0, the constraint that Ž D x .l21 , l 2 - 21 yields < l 2 < ) 0.938744 in this case of n s 2, which is consistent with the results in Ref. w8x. To provide more insight on the three dimensional plot, it is often useful to investigate the density plot. In Fig. 2, we have plotted the density plot for the l1 y l2 parameter space. The dark area, R, in the middle of the density plot in Fig. 2 shows the region in which the variance Ž D x .l21 , l 2 exceeds 1r2 while the lighter regions correspond to regions in which we can get the l-parametrized squeezed states. Note that in the limit l 2 ™ 0, squeezed states are only possible for l1 - y2 in this case Ž n s 2.. We have also numerically simulated the three dimensional plots of Ž D x .l21 , l 2 as a function of l1
Ž 31 .
² p :l1 , l 2 s 0 ,
Ž 32 .
² p 2 :l1 , l 2 s Ž D p . 2l1 , l 2 s
l22 q 3 l22 q 1
2Ž
.
.
Ž 33 .
However, for n s 2, our results generalize those given in Ref. w8x so that in the limit l1 ™ 0, the results reduce to those given in Ref. w8x. The expectation values in general depend on both parameters, l1 and l2 . ² x :l1 , l 2 s
2'2 l2 Ž 2 q l1 q l22 . 2 q 4l22 q Ž l1 q l22 .
² x 2 :l1 , l 2 s
,
Ž 34 .
2 Ž 2 q l1 q 3 l22
1 q 2
2
2 q 4l22 q Ž l1 q l22 .
2
,
² p :l1 , l 2 s 0 ,
Ž 36 .
² p 2 :l1 , l 2 s Ž D p . 1 s
q 2
Ž 35 .
2 l1 l 2
4 y 2 l1 q 2 l22 2 q 4l22 q Ž
2 l1 q l22
.
.
Ž 37 .
Fig. 2. Density plot of Ž D x .l21, l 2 for different values of l1 and l 2 for ns 2. The dark region in the center corresponds to the regions in which the variance Ž D x .l21, l 2 exceeds 0.5.
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
11
Fig. 3. Three dimensional plots of Ž D x .l21, l 2 as a function of l1 and l2 for values of n corresponding to ŽA. n s 3, ŽB. n s 4, ŽC. n s 6 and ŽD. n s 10. Note that the peak value of the plot increases with increasing values of n and the plots have been rescaled accordingly.
Fig. 4. Graphs of l1 against l2 for different values of n showing the shaded region Žwith dots. in which Ž D x .l21, l 2 ) 0.5.
12
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
Fig. 5. Graph of Ž D x .l21, l 2 as a function of l2 for the section of three-dimensional plot in which l1 s y7. The ripples are clearly shown here.
and l2 for other values of n. It is interesting to note that as n increases, the peak at the center becomes narrower and sharper. Moreover, we note that as n increases, the more ripples appear and these ripples fanned out ‘spokes’ in the region l1 ,0. The three-di-
mensional plots for n s 3,4,6 and 10 are illustrated in Fig. 3. We have also plotted the regions in the l1 y l2 plane in which the three dimensional plot corresponds Ž D x .l21 , l 2 ) 0.5. These regions are plotted in Fig. 4 and shaded with little dots. These plots
Fig. 6. Plots of Ž D x .l21, l 2 across the section l 2 s 0 for even values of n, namely n s 2,4 and 6. These plots cut the dotted line for Ž D x .l2 , l s 0.5 at l1 s y2,y 2.95066 and y3.83009 respectively. 1 2
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
13
Fig. 7. Plots of Ž D x .l21, l 2 across the section l2 s 0 for odd values of n, namely n s 3,5 and 7. Note that these graphs do not cross the dotted line at Ž D x .l21, l 2 s 0.5 and we therefore cannot obtain squeezed states conditions.
further confirm the existence of n ‘spokes’ for eigenstate < n:l1 , l 2 with the line l2 s 0 as a line of symmetry. To illustrate the ripples more clearly, we have also considered the variation of Ž D x .l21 , l 2 across the section of the three-dimensional plot in which l1 is fixed at y7. The graph in Fig. 5 clearly shows that for large < l 2 <, the curve asymptotically approaches 0.5 from below and that the peaks in the ripples are generally much less than the peak at l1 s l2 s 0. Our extension naturally allows us to investigate Ž D x .l2 , l as a function of l1 by setting l2 s 0. We 1 2 have already seen that in this limit, l-dependent squeezed states are only possible for l1 - y2 for n s 2. We observe that for odd n, the line segment corresponding to l2 s 0 passes right through one of the spokes. This observation means that there are no roots to the equation Ž D x .l21 ,0 s 0.5, which is consistently with the plots in Fig. 7. Indeed, it is interesting to note that these novel squeezed states can only be achieved for even n in this limit. A plot of Ž D x .l21 , l 2 for even n and odd n has been plotted in Figs. 6 and 7 respectively. The graphs clearly show that squeezed states are not possible for odd values of n, whereas, for even values of n, we can achieved squeezed state conditions for l1 - N Ž n. where N Ž n. depends in general on n. The values of N Ž n. for n s 2,4,6,8
and 10 are y2,y 2.95066,y 3.83009,y 4.663 and y5.46237 respectively.
4. Discussion In this paper, we have shown that by using exponential operator w15x, we can easily generate squeezed states for non-hermitian Hamiltonian. Indeed, our approach can often yield results more concisely and it can also be generalized to analyze more complicating and more general nonlinear, non-Hermitian Hamiltonian through a general modification of the form s
a† ™ a† q
Ý l sy iq1 a i ,
Ž 38 .
is0
where the set l1 , l2 , . . . , l sq1 4 are a finite set of parameters which can be solved for squeezed states conditions. It is not hard to see that the appropriate exponential operator, U , should be l1 sq1 l s s U s exp a q a q PPP sq1 s ls 2 q a q l sq 1 a Ž 39 . 2
½
5
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
14
and the unnormalized eigenstate of the modified Hamiltonian is U < n: s exp q
½
ls 2
l1 sq1
a sq1q
5
ls s
n. Therefore, it would be interesting to apply our method to the situation in Ref. w16x.
a sq PPP
a 2 q l sq 1 a < n: ,
References
Ž 40 .
which, apart from a normalization constant, is < n:Ž l1 , l 2 , . . . , l sq 1 . Recently, Beckers et al. w16x considered a modification of the form a† ™ a† q c1 a q c 2 a† q c 3 ,
Ž 41 .
a ™ a q c 4 a q c5 a† q c6 ,
Ž 42 .
in which one could restore the hermiticity of the Hamiltonian by a specific choice of the parameters c1 ,c 2 , . . . ,c6 4 . They obtained squeezed states by explicitly solving Schrodinger equation. Moreover, they ¨ have computed the exact expressions of squeezing properties for the case of n s 0. However, they have not provided analytical expressions for the case of non-zero n. In this letter, using exponential quadratic operator, we have seen that it is often computationally easy and straightforward to compute explicit expressions of squeezing properties given arbitrary
w1x R.E. Slusher et al., Phys. Rev. Lett. 55 Ž1985. 2409; L. Wu et al., Phys. Rev. Lett. 57 Ž1986. 2520. w2x A. Furusawa, Science 282 Ž1998. 706. w3x X. Hu, F. Nori, Phys. Rev. Lett. 76 Ž1996. 2294. w4x G.A. Garrett, Science 275 Ž1997. 1638. w5x E. Giacobino, C. Fabre ŽEds.., Appl. Phys. B 56 Ž1992.. w6x Y. Yamamoto, S. Michida, N. Imoto, M. Kitagawa, G. Bjork, J. Opt. Soc. Am. B 4 Ž1987. 1645. w7x M.S. Kim, F.A.M. de Oliveiro, P.L. Knight, Phys. Rev. A 40 Ž1989. 2494. w8x J. Beckers, N. Debergh, F.H. Szafraniec, Phys. Lett. A 243 Ž1998. 256, and reference therein. w9x V.A. Kostelecky, M.H. Nieto, R. Traux, Phys. Rev. A 48 Ž1993. 1045. w10x Y.S. Kim, M.E. Noz, Phase space picture of quantum mechanics: a group theoretical approach, World Scientific, Singapore, 1991. w11x C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 Ž1998. 5243. w12x N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77 Ž1996. 570. w13x J. Feinberg, A. Zee, Nucl. Phys. B 504 Ž1997. 579. w14x N.M. Shnerb, D.R. Nelson, Phys. Rev. Lett. 80 Ž1998. 5172. w15x X. Wang, C.H. Oh, L.C. Kwek, J. Phys. A: Math. Gen. 31 Ž1998. 4329; quant-phr9802048. w16x J. Beckers, N. Debergh, F.H. Szafraniec, quant-phr9903078.
Physics Letters A 259 Ž1999. 7–14 www.elsevier.nlrlocaterphysleta
Extended two-parameter squeezed states Xiang-Bin Wang 1, L.C. Kwek 2 , C.H. Oh
3
Department of Physics,Faculty of Science, National UniÕersity of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore Received 3 March 1999; received in revised form 26 April 1999; accepted 24 May 1999 Communicated by P.R. Holland
Abstract In a recent paper, a new l-dependent squeezed states was proposed wJ. Beckers, N. Debergh and F.H. Szafraniec, Phys. Lett. A 243 Ž1998. 256–260x in which the usual Fock-space creation operator has been modified by introducing a simple translational term. In this paper, we generalize this approach by adding a new term proportional to the creation operator using exponential quadratic operators. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
squeezed and displacement operators, SŽ z . and DŽ a . respectively through the relations w7x
Squeezed states provide a means of challenging the theoretical and experimental limits imposed by the uncertainty principle and the zero-point fluctuations of the vacuum. Consequently, these novel nonclassical states have constantly stimulated much interest as the means for improving the signal-to-noise ratio in optical interferometry w1x, quantum teleportation w2x, Raman scattering w3,4x and a myriad of other applications such as gravitational wave detection w5x. Amplitude squeezed states have been observed in negative feedback laser with quantum nondemolition measurement of photon number w6x. They can also be generated by parametric amplifiers. Conventionally, squeezed states have been defined through the
< a , z : s D Ž a . S Ž z . <0: ,
where SŽ z . and DŽ a . are defined as the operators expŽ 12 Ž z ) a 2 y za† 2 .. and expŽ a a† y a ) a. respectively and Ž a,a† . are the usual annihilator and creator operators. In a recent paper w8x, Becker et al. proposed a new family of squeezed states by considering a modification of creation operator in Fock space using the relation a†l s a† q l ,
E-mail: [email protected] E-mail: [email protected] 3 E-mail: [email protected] 2
Ž 2.
†
where a is the usual bosonic creation operator, a†l is called as l-parametrized bosonic creation operator. The l-parametrized Hamiltonian is given by Hl s 12 a†l ,a4 .
1
Ž 1.
Ž 3.
The eigenstates of l-parametrized Hamiltonian was investigated and it was shown that it is possible to obtain squeezed states for certain values of the parameter l w8x. Unlike usual squeezed states in which
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 3 5 4 - 0
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
8
the Hamiltonian are Hermitian, the Hamiltonian given in Eq. Ž3. is non-Hermitian and therefore these squeezed states are liable to decay processes. It has also been shown w8x that the family of squeezed states obtained through this modification constitutes new squeezed states. Moreover these l-states cannot be coherent states. Compared with the usual squeezed states w9,10x with wavefunction given by
py1 r4 Cz Ž x . s
1 2
zexp Ž x
2
.
exp
ž
z2y1 2z
2
/
x ,
py1r4
ž
(L Ž yl . 0 k
2
1 2
=exp Ž y x
2
1 y
'2
d
x q
dx
'2
k
l
/
..
s0 .
l1 2
q i l2
1
'2
1
'2
l2 x
p,
Ž 10 .
Hl1 , l 2 ,al† 1 , l 2 s al† 1 , l 2 .
Ž 7.
Ž 11 .
still hold. Indeed, non-Hermitian Hamiltonians with real positive eigenvalues have recently been studied w11x and due to their intimate relation to the transfer matrices of classical statistical mechanics and tight binding models with delocalization transition, they have also been intensively investigated in the context of random matrices w12–14x. In terms of quadratic exponential operator w15x, we can rewrite the new Hamiltonian in Eq. Ž8. as 2
2
Hl1 , l 2 s e 1r2 l1 a ql 2 a H0 e yŽ1 r2 l1 a ql 2 a. ' UH0 Uy1 .
Ž 12 . Note that the exponential operator U is not unitary hence the two Hamiltonians Hl1 , l 2 and H0 describes different physical systems, in particular unlike H0 , Hl1 , l 2 is non-Hermitian. It is not hard to show that the normalized eigenstate for the above Hamiltonian can be written as 1
(z Ž l , l . 1
s
(z
Ž 6.
The commutation relations for the new operators Ž a,a†l , l ., remain the same as the usual oscillator 1 2 algebra, namely,
w a,a x s
q i l1 xp y i
n
In this article, we generalize the transformation of the bosonic operator as
al† 1 , l 2 ,al† 1 , l 2
Ž 9.
s 12 Ž 1q l1 . x 2q 12 Ž 1 y l1 . p 2q
< n:l1 , l 2 s
2. Extended l-parameter states
s I,
s a†a q l1 a 2 q l 2 a q 12
Hl1 , l 2 ,a s ya,
Ž 5.
a†l1 , l 2 s a† q l1 a q l2 .
Ž 8.
where x and p denote the coordinate and momentum operators. This Hamiltonian is not Hermitian but the following relations
In Ref. w8x, l-dependent squeezed states are obtained through the solution of differential equations. By using exponential operator through an algebraic approach, we have generalized these l-dependent squeezed states by considering an additional term proportional to the Fock space annihilator operator. Indeed, we also show in this paper that our approach can be further generalized to include more parameters. In the next section, we introduce our modification and obtain our new squeezed states using essentially algebraic approach with the exponential operators. In Section 3, we derive some analytical and numerical results and show how this generalization adds an additional perspective to our analysis of squeezed states. Finally, in Section 4, we briefly explain how our approach can be generalized to include more parameters.
a,a†l1 , l 2
Hl1 , l 2 s 12 al† 1 , l 2 ,a4
Ž 4.
these novel squeezed states assume the form w8x
Clk Ž x . ;
Accordingly, the analogous l-parametrized Hamiltonian is
1
`
Ý n ls0
e 1r2 l1 a
1
ž
1
l! 2
Ý
n
rs0
ks0
(
n!
=
r!
Ž 13 .
l
Ý
(z
n:
l1 a2 q l2 a < n:
wŽ nyr .r2 x
s
ql2 a <
2
n
1
2
< r: .
/
1
k
ž /Ž 2
Ž 14 .
l1k l2nyry2 k n y r y 2 k . !k!
Ž 15 .
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
where znŽ l1 , l2 . s zn is the appropriate normalization factor and w m x denotes the integer part of m. It is easy to obtain the normalization factor znŽ l1 , l2 . explicitly as < zn < Ž w nyr . r2 x
n
s
Ý rs0
ž
1
k
ž /
Ý
2
ks0
l1k l2nyry2 k
n!
k! Ž n y r y 2 k . !
( /
n
'
Ý
n!
2 Tny r
r!
rs0
Using Ž21., we can easily calculate the expectation values and the variances. In particular, we see that for the position operator, x s 1r '2 Ž a q a† ., the expectation values of x and x 2 for the eigenstate < n:l1 , l 2 are
2
² x :l1 , l 2 s
'2
r!
Ž 16 . ,
n
zn
rs1
1
1
² x 2 :l1 , l 2 s
Ž 17 .
n
Ts s
Ý ks0
ž / 2
l1k l2sy2 k k! Ž s y 2 k . !
.
Ž 18 .
a < n:l1 , l 2 s
zny 1
al† 1 , l 2 < n:l1 , l 2 s
'n < n y 1:l , l 1
zn znq 1
,
2
'n q 1 < n q 1:l , l 1
. 2
Ž 20 .
min Ž m , n .
s
Ý
m zn
=
ž
/
,
Ž 23 .
² p 2 :l1 , l 2 s
Ž 24 .
1
n
1 q
zn
2 n
y
Ý rs2
ž
2 Tny r n!
Ý rs1 Ž r y 1 . !
Tny rq2 Tnyr n!
Ž r y 2. !
/
.
Ž 25 .
Using the results above, we can easily compute the variances 2
2
2
2
Ž D x . l1 , l 2 s ² x 2 :l1 , l 2 y Ž ² x :l1 , l 2 .
Ž 26 .
and
Ž D p . l1 , l 2 s ² p 2 :l1 , l 2 y Ž ² p :l1 , l 2 .
Ž 27 .
and show that these quantities satisfy the inequality 2
2
Ž D x . l1 , l 2 Ž D p . l1 , l 2 ) 14 .
Ž 28 .
lŽ2mqny2 r . r!
r
3. Results and numerical simulation
w myrr2 x w nyr x r2
=
Ž r y 2. !
Indeed, these states are not minimum uncertainty states Žexcept for n s 0..
² m < n:l1 , l 2
l1 , l 2
(z
Tny rq2 Tnyr n!
Ý
For the momentum operator, p s 1ri'2 Ž a y a† ., the expectation values of p and p 2 are
Ž 19 .
It is easy to see that, in the limit l1 ™ 0, we recover the features of the l-parametrized states in Ref. w8x and that in the double limit in which both l1 , l2 ™ 0, we get the usual features of the Fock states under the action of the operators a and a†. As for the l-parameter states in Ref. w8x, our two-parameter l states are normalized but not orthogonal
'm!n!
Ž r y 1. !
² p :l1 , l 2 s 0 ,
In the limit l1 ™ 0, the normalization factor reduces to the generalized Laguerre polynomial, L0nŽyl22 .. The Fock space representation of the operators Ž a,a†l , l . can be expressed as 1 2
zn
rs1
rs2
s k
žÝ
zn
Ž 22 .
2 Tny r n!
n
q 2
where the partial sum, Ts is defined as 1
n!
Ý Ž Tnyrq1Tnyr . Ž r y 1. ! ,
q
2
9
Ý
Ý
ks0
js0
1
kq j
ž / 2
Ž kqj . l1kq jly2 2
Ž m y r y 2 k . ! Ž n y r y 2 j . !k! j!
/
.
Ž 21 .
In this Section, we discuss our results and numerical simulation of the l-dependent squeezed states. Indeed, it is instructive to consider our results for different values of n. For n s 1, it is obvious that one of the parameters, l1 , does not contribute to the
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
10
Fig. 1. Three-dimensional plot of Ž D x .l21, l 2 as a function of l1 and l2 for ns 2. Note that for squeezed states, we need to demand that Ž D x .l21, l 2 - 0.5.
expectation values and our results are the same as those obatined in Ref. w8x, namely ² x :l1 , l 2 s
'2 l2 l2 q 1
² x 2 :l1 , l 2 s
,
Ž 29 .
l22 q 3 2 Ž l22 q 1 .
l42 q 3
2
Ž D x . l1 , l 2 s
2 Ž l22 q 1 .
,
Ž 30 . ,
Our two-parameter formulation inevitably provides a fuller picture for the phase space of the l i Ž i s 1,2. parameters. It is possible to plot the variance Ž D x .l21 , l 2 with different values of l1 and l2 . This plot is shown in Fig. 1. To locate the existence of l-parametrized squeezed states, we seek for values of the parameters l1 and l2 satisfying Ž D x .l21 , l 2 - 12 . We note that in the one-parameter limit in which l1 ™ 0, the constraint that Ž D x .l21 , l 2 - 21 yields < l 2 < ) 0.938744 in this case of n s 2, which is consistent with the results in Ref. w8x. To provide more insight on the three dimensional plot, it is often useful to investigate the density plot. In Fig. 2, we have plotted the density plot for the l1 y l2 parameter space. The dark area, R, in the middle of the density plot in Fig. 2 shows the region in which the variance Ž D x .l21 , l 2 exceeds 1r2 while the lighter regions correspond to regions in which we can get the l-parametrized squeezed states. Note that in the limit l 2 ™ 0, squeezed states are only possible for l1 - y2 in this case Ž n s 2.. We have also numerically simulated the three dimensional plots of Ž D x .l21 , l 2 as a function of l1
Ž 31 .
² p :l1 , l 2 s 0 ,
Ž 32 .
² p 2 :l1 , l 2 s Ž D p . 2l1 , l 2 s
l22 q 3 l22 q 1
2Ž
.
.
Ž 33 .
However, for n s 2, our results generalize those given in Ref. w8x so that in the limit l1 ™ 0, the results reduce to those given in Ref. w8x. The expectation values in general depend on both parameters, l1 and l2 . ² x :l1 , l 2 s
2'2 l2 Ž 2 q l1 q l22 . 2 q 4l22 q Ž l1 q l22 .
² x 2 :l1 , l 2 s
,
Ž 34 .
2 Ž 2 q l1 q 3 l22
1 q 2
2
2 q 4l22 q Ž l1 q l22 .
2
,
² p :l1 , l 2 s 0 ,
Ž 36 .
² p 2 :l1 , l 2 s Ž D p . 1 s
q 2
Ž 35 .
2 l1 l 2
4 y 2 l1 q 2 l22 2 q 4l22 q Ž
2 l1 q l22
.
.
Ž 37 .
Fig. 2. Density plot of Ž D x .l21, l 2 for different values of l1 and l 2 for ns 2. The dark region in the center corresponds to the regions in which the variance Ž D x .l21, l 2 exceeds 0.5.
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
11
Fig. 3. Three dimensional plots of Ž D x .l21, l 2 as a function of l1 and l2 for values of n corresponding to ŽA. n s 3, ŽB. n s 4, ŽC. n s 6 and ŽD. n s 10. Note that the peak value of the plot increases with increasing values of n and the plots have been rescaled accordingly.
Fig. 4. Graphs of l1 against l2 for different values of n showing the shaded region Žwith dots. in which Ž D x .l21, l 2 ) 0.5.
12
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
Fig. 5. Graph of Ž D x .l21, l 2 as a function of l2 for the section of three-dimensional plot in which l1 s y7. The ripples are clearly shown here.
and l2 for other values of n. It is interesting to note that as n increases, the peak at the center becomes narrower and sharper. Moreover, we note that as n increases, the more ripples appear and these ripples fanned out ‘spokes’ in the region l1 ,0. The three-di-
mensional plots for n s 3,4,6 and 10 are illustrated in Fig. 3. We have also plotted the regions in the l1 y l2 plane in which the three dimensional plot corresponds Ž D x .l21 , l 2 ) 0.5. These regions are plotted in Fig. 4 and shaded with little dots. These plots
Fig. 6. Plots of Ž D x .l21, l 2 across the section l 2 s 0 for even values of n, namely n s 2,4 and 6. These plots cut the dotted line for Ž D x .l2 , l s 0.5 at l1 s y2,y 2.95066 and y3.83009 respectively. 1 2
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
13
Fig. 7. Plots of Ž D x .l21, l 2 across the section l2 s 0 for odd values of n, namely n s 3,5 and 7. Note that these graphs do not cross the dotted line at Ž D x .l21, l 2 s 0.5 and we therefore cannot obtain squeezed states conditions.
further confirm the existence of n ‘spokes’ for eigenstate < n:l1 , l 2 with the line l2 s 0 as a line of symmetry. To illustrate the ripples more clearly, we have also considered the variation of Ž D x .l21 , l 2 across the section of the three-dimensional plot in which l1 is fixed at y7. The graph in Fig. 5 clearly shows that for large < l 2 <, the curve asymptotically approaches 0.5 from below and that the peaks in the ripples are generally much less than the peak at l1 s l2 s 0. Our extension naturally allows us to investigate Ž D x .l2 , l as a function of l1 by setting l2 s 0. We 1 2 have already seen that in this limit, l-dependent squeezed states are only possible for l1 - y2 for n s 2. We observe that for odd n, the line segment corresponding to l2 s 0 passes right through one of the spokes. This observation means that there are no roots to the equation Ž D x .l21 ,0 s 0.5, which is consistently with the plots in Fig. 7. Indeed, it is interesting to note that these novel squeezed states can only be achieved for even n in this limit. A plot of Ž D x .l21 , l 2 for even n and odd n has been plotted in Figs. 6 and 7 respectively. The graphs clearly show that squeezed states are not possible for odd values of n, whereas, for even values of n, we can achieved squeezed state conditions for l1 - N Ž n. where N Ž n. depends in general on n. The values of N Ž n. for n s 2,4,6,8
and 10 are y2,y 2.95066,y 3.83009,y 4.663 and y5.46237 respectively.
4. Discussion In this paper, we have shown that by using exponential operator w15x, we can easily generate squeezed states for non-hermitian Hamiltonian. Indeed, our approach can often yield results more concisely and it can also be generalized to analyze more complicating and more general nonlinear, non-Hermitian Hamiltonian through a general modification of the form s
a† ™ a† q
Ý l sy iq1 a i ,
Ž 38 .
is0
where the set l1 , l2 , . . . , l sq1 4 are a finite set of parameters which can be solved for squeezed states conditions. It is not hard to see that the appropriate exponential operator, U , should be l1 sq1 l s s U s exp a q a q PPP sq1 s ls 2 q a q l sq 1 a Ž 39 . 2
½
5
X.-B. Wang et al.r Physics Letters A 259 (1999) 7–14
14
and the unnormalized eigenstate of the modified Hamiltonian is U < n: s exp q
½
ls 2
l1 sq1
a sq1q
5
ls s
n. Therefore, it would be interesting to apply our method to the situation in Ref. w16x.
a sq PPP
a 2 q l sq 1 a < n: ,
References
Ž 40 .
which, apart from a normalization constant, is < n:Ž l1 , l 2 , . . . , l sq 1 . Recently, Beckers et al. w16x considered a modification of the form a† ™ a† q c1 a q c 2 a† q c 3 ,
Ž 41 .
a ™ a q c 4 a q c5 a† q c6 ,
Ž 42 .
in which one could restore the hermiticity of the Hamiltonian by a specific choice of the parameters c1 ,c 2 , . . . ,c6 4 . They obtained squeezed states by explicitly solving Schrodinger equation. Moreover, they ¨ have computed the exact expressions of squeezing properties for the case of n s 0. However, they have not provided analytical expressions for the case of non-zero n. In this letter, using exponential quadratic operator, we have seen that it is often computationally easy and straightforward to compute explicit expressions of squeezing properties given arbitrary
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