Gaussian Klauder coherent states of general time-dependent harmonic oscillator

Physics Letters A 325 (2004) 1–8 www.elsevier.com/locate/pla

Gaussian Klauder coherent states of general time-dependent harmonic oscillator Jeong-Ryeol Choi Department of New Material Science, Division of Natural Sciences, Sunmoon University, Asan 336-708, South Korea Received 10 October 2003; received in revised form 18 December 2003; accepted 11 March 2004 Communicated by C.R. Doering

Abstract We investigated the Gaussian Klauder coherent states for general time-dependent harmonic oscillator. We confirmed that the uncertainty product of general time-dependent harmonic oscillator in Gaussian Klauder coherent state is same as the minimum uncertainty product in number state. This is exactly agree with the theory for the standard harmonic oscillator developed by Fox et al. We applied our study to a damped harmonic oscillator and a harmonic oscillator with time-variable frequency that can be applied to a motion of an ion in a Paul trap. The uncertainty product in Gaussian Klauder coherent state fluctuated more or less as time goes by.  2004 Elsevier B.V. All rights reserved. PACS: 03.65.-w; 42.25.Kb Keywords: Gaussian Klauder coherent states; Time-dependent harmonic oscillator; Uncertainty product; Paul trap

1. Introduction For several decades, the study of exact quantum mechanical solutions for general time-dependent harmonic oscillator has been attracted outstanding interest in the literature. The standard coherent states for harmonic oscillator, that is the prototype for most kind of coherent states are proposed by Glauber [1,2]. The coherent states of the simple harmonic oscillator that considered by Schrödinger [3] are nearly same as the classical wave packet. Coherent states of a charged particle in a time-dependent electromagnetic fields are investigated by Malkin et al. [4,5]. The coherent states of the damped harmonic oscillator and harmonic oscillator with time-dependent frequency are also studied in the literatures [6–8]. The coherent states for general potentials are extensively investigated by Nieto et al. [9] and can be extended to nonstationary systems [10,11]. Several years ago, significant progress has been made in obtaining generalized coherent states for the Coulomb potential problem by Klauder [12] and by Majumdar et al. [13]. The genuine Gaussian generalized coherent states, so-called Gaussian Klauder coherent states, were established and analyzed for hydrogen atom by Fox [14]. E-mail address: [email protected] (J.-R. Choi). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.03.025

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J.-R. Choi / Physics Letters A 325 (2004) 1–8

The existence of Gaussian Klauder coherent states for the Coulomb problem suggests that they can be applied to many other systems as well such as standard harmonic oscillator, planar rotor and the particle in a box [15]. Thus, we expect that the Gaussian Klauder coherent states to be extendable for time-dependent systems such as time-dependent harmonic oscillator. The main aim of this Letter is to investigate Gaussian Klauder coherent states in general time-dependent harmonic oscillator. The quantum and classical correspondence can be shown using Gaussian Klauder coherent states which provide a general means of construction for Husimi–Wigner distributions [15]. In Section 2, we review the quantum-mechanical solution of the general time-dependent harmonic oscillator. We investigate in Section 3 for Gaussian Klauder coherent states in general time-dependent harmonic oscillator. We applied Gaussian Klauder coherent states to a damped harmonic oscillator and a harmonic oscillator with timevariable frequency in Section 4. Finally, in Section 5, we summarized our development in the previous sections.

2. General time-dependent harmonic oscillator We start from the review [16] of the quantum mechanical solution for general time-dependent harmonic oscillator that the Hamiltonian can be written as ˆ + C(t)pˆ + D2 (t)xˆ 2 + D1 (t)xˆ + D0 (t), Hˆ (x, ˆ p, ˆ t) = A(t)pˆ 2 + B(t)(xˆ pˆ + pˆ x)

(1)

where A(t)–C(t) and Di (t) (i = 0, 1, 2) are time-dependent coefficients. These coefficients are real and differentiable with respect to time. Note that A(t)
= 0. The introduction of invariant operator which is a powerful tool to derive exact solutions of time-dependent quantum mechanical Hamiltonian systems may save the labor that finding quantum mechanical solution of the system. The invariant operator corresponds to Eq. (1) is given by [16]
2 



 Iˆ(t) = α1 (t) pˆ − pp (t) + α2 (t) xˆ − xp (t) pˆ − pp (t) + pˆ − pp (t) xˆ − xp (t)
2 + α3 (t) xˆ − xp (t) , (2) where xp (t) and pp (t) are particular solutions of the classical equation of motion of the system in x and p space, respectively, and α1 (t) = c1 ρ12 (t) + c2 ρ1 (t)ρ2 (t) + c3 ρ22 (t),  1 
2 α2 (t) = 4 c1 ρ1 (t) + c2 ρ1 (t)ρ2 (t) + c3 ρ22 (t) B 4A
 − 2c1 ρ1 (t)ρ˙1 (t) + c2 ρ˙1 (t)ρ2 (t) + c2 ρ˙2 (t)ρ1 (t) + 2c3 ρ2 (t)ρ˙2 (t) ,   1 1
2 α3 (t) = c1 ρ˙1 (t) + c2 ρ˙1 (t)ρ˙2 (t) + c3 ρ˙22 (t) 2A2 2 
− B 2c1 ρ1 (t)ρ˙1 (t) + c2 ρ˙1 (t)ρ2 (t) + c2 ρ˙2 (t)ρ1 (t) + 2c3 ρ2 (t)ρ˙2 (t) 
 + 2B 2 c1 ρ12 (t) + c2 ρ1 (t)ρ2 (t) + c3 ρ22 (t) .

(3)

(4)

(5)

In Eqs. (3)–(5), c1 –c3 are arbitrary constants and ρ1,2 are two independent classical solutions of the following auxiliary differential equation [17]  ˙  A˙ 2AB − 2B˙ − 4B 2 + 4AD2 ρ1,2 (t) = 0. ρ¨1,2 (t) − ρ˙1,2 (t) + (6) A A

J.-R. Choi / Physics Letters A 325 (2004) 1–8

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We can write the eigenvalue equation for Iˆ(t) as Iˆ(t)|n(t) = λn |n(t).

(7)

This can be easily solved by introducing appropriate lowering and raising operators so that we can express the eigenstate and eigenvalue in x-space ˆ as [16,17] 

   1/4  

Ω i 1 Ω 1 Ω 2 x|n(t) = √ (xˆ − xp ) exp pp xˆ exp − Hn + iα2 (xˆ − xp ) , h¯ 2α1 h¯ π 2α1 h¯ 2α1 h¯ 2 2n n! (8)   1 , λn = h¯ Ω n + (9) 2 where Ω2 =

  1 (ρ1 ρ˙2 − ρ˙1 ρ2 )2 4c1 c3 − c22 = const. 2 4A

(10)

The wave function in x-space, ˆ x|ψn (t), satisfying Schrödinger equation related to the Hamiltonian, Eq. (1), is same as the eigenstate of Iˆ, except for some time-dependent phase factor, exp[in (t)] [18]:
 x|ψn (t) = exp in (t) x|n(t). (11) By substitution of Eq. (11) into Schrödinger equation, we can derive the phase as  t 

t   C 2 (t  ) A(t  )  1 1     + D0 (t ) dt  , n (t) = −Ω n + Lp xp (t ), x˙ p (t ), t − dt − 2 α1 4A(t  ) h¯ 0

(12)

0

where Lp (xp (t), x˙p (t), t) is defined as   Lp xp (t), x˙p (t), t =

  1 B(t) B 2 (t) 2 2 xp (t). x˙ (t) − xp (t)x˙p (t) − D2 (t) − 4A(t) p A(t) A(t)

(13)

We can use Eq. (11) to evaluate various expectation values in number state. For example, the uncertainty product in number state can be calculated as
 2 1/2
 2 1/2 ˆ n = ψn (t)|xˆ 2 |ψn (t) − ψn (t)|x|ψ ˆ n (t) ˆ n (t) (x) ˆ n (p) ψn (t)|pˆ 2 |ψn (t) − ψn (t)|p|ψ

 2 h¯ α2 = 1+4 (2n + 1). (14) 2 Ω Although this is deviated from the standard uncertainty product, it is always larger than h¯ /2 as expected.

3. Gaussian Klauder coherent states The coherent state of the time-dependent harmonic oscillator can be described in terms of the eigenstate of the invariant operator [19]. Gaussian Klauder coherent states for harmonic oscillator are given by [15]

∞  1 (n − n0 )2 inφ0 exp − |n(t), |n0 , φ0  = √ (15) e 4σ 2 N(n0 ) n=0

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J.-R. Choi / Physics Letters A 325 (2004) 1–8

where n0 , φ0 and σ are c-number parameters and N(n0 ) is determined from normalization condition, n0 , φ0 |n0 , φ0  = 1,

∞  (n − n0 )2 N(n0 ) = (16) exp − . 2σ 2 n=0

We abbreviate the Gaussian factor in Eq. (15) as [15]

(n − n0 )2 . G(n) ≡ exp − 4σ 2

(17)

By using the lowering and raising operators, the following identities can be established  ∞  √ h¯ α1 2 cos φ0 n0 , φ0 |x|n (18) ˆ 0 , φ0  = n + 1 G(n)G(n + 1) + xp , Ω N(n0 ) n=0

∞ 1 √ h¯ (Ω sin φ0 − 2α2 cos φ0 ) n0 , φ0 |p|n (19) ˆ 0 , φ0  = n + 1 G(n)G(n + 1) + pp , α1 Ω N(n0 ) n=0   ∞  ∞   h¯ α1 1 2 2 n0 , φ0 |xˆ |n0 , φ0  = 2 cos(2φ0 ) (n + 1)(n + 2) G(n)G(n + 2) + (2n + 1)G (n) Ω N(n0 ) n=0 n=0  ∞  √ 4xp h¯ α1 cos φ0 n + 1 G(n)G(n + 1) + xp2 , + (20) N(n0 ) Ω n=0  

 1 h¯ Ω2 − α22 cos(2φ0 ) + 2α2 Ω sin(2φ0 ) n0 , φ0 |pˆ 2 |n0 , φ0  = − 2 α1 Ω N(n0 ) 4    2 ∞  ∞  Ω 2 2 + α2 × (n + 1)(n + 2) G(n)G(n + 2) − (2n + 1)G (n) 4 n=0 n=0

∞  √ 2pp h¯ (Ω sin φ0 − 2α2 cos φ0 ) n + 1 G(n)G(n + 1) + pp2 . + (21) N(n0 ) α1 Ω n=0

Since G(n) is largest at n = n0 , we can and will expand all factors around n = n0 so that the sums can be replaced by Gaussian integrals. And this leads to the approximations of Eqs. (21)–(23) in Ref. [15]. Then, by performing integration after replacing sums with Gaussian integrals, the Eqs. (18)–(21) can be written as  hα ¯ 1 cos φ0 f1 + xp , n0 , φ0 |x|n (22) ˆ 0 , φ0  = 2 Ω

h¯ (Ω sin φ0 − 2α2 cos φ0 )f1 + pp , ˆ 0 , φ0  = n0 , φ0 |p|n (23) α1 Ω   h¯ α1
hα ¯ 1 2 cos φ0 f1 + 2 cos(2φ0 )f2 + f3 + xp2 , n0 , φ0 |xˆ |n0 , φ0  = 4xp (24) Ω Ω   2   

 2 Ω h¯ Ω 2 − α22 cos(2φ0 ) + 2α2 Ω sin(2φ0 ) f2 − + α22 f3 n0 , φ0 |pˆ 2 |n0 , φ0  = − α1 Ω 4 4

h¯ (Ω sin φ0 − 2α2 cos φ0 )f1 + pp2 , + 2pp (25) α1 Ω

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where

  2  

σ2 1 1 1 1 σ 1 1 σ2 n0 + 1 1 − 2 exp + − − + − , 1− 2 2 4σ 2 4n0 2n0 4n20 4σ 4 2n0 σ 2 2n30 2n20 σ 2   2

 σ2 σ 3 σ2 1 1 1 σ4 3σ 4 3σ 2 1 − 2 exp , − − − + − + − f2 = n0 + 2 2n0 2n20 2n30 2σ 2 n0 n20 2n40 2n40 2n50  2   σ σ2 σ2 1 1 − . f3 = (2n0 + 1) 1 − 2 exp − n0 2n0 2n20 n20

f1 =



(26) (27) (28)

Now, using Eqs. (22)–(25), it is possible to derive the variances for canonical variables as  h¯ α1
−4 cos2 φ0 f12 + 2 cos(2φ0 )f2 + f3 , Ω  h¯ 2 (p) ˆ c= −(Ω sin φ0 − 2α2 cos φ0 )2 f12 α1 Ω     2

 2 Ω Ω 2 2 − α2 cos(2φ0 ) + 2α2 Ω sin(2φ0 ) f2 + + α2 f3 . − 2 4 4 (x) ˆ 2c =

(29)

(30)

The uncertainty products in standard coherent states are minimum value of the uncertainty products in number state. Let us investigate whether this relation also exist or not between the Gaussian Klauder coherent state and the number state for the general time-dependent harmonic oscillator under some restriction or approximation. To do this, we set σ 2 = n0 . Then, the leading order of Eqs. (26)–(28) becomes √ f2  n0 , f3  2n0 + 1. f1  n0 , (31) By substitution of the above equations into Eqs. (29) and (30), we obtain that h¯ α1 , Ω  2  Ω h¯ 2 2 + α2 . (p) ˆ c= α1 Ω 4 (x) ˆ 2c =

And the uncertainty product in Gaussian Klauder coherent states is easily obtain that

 2 h¯ α2 (x) ˆ c (p) ˆ c= 1+4 . 2 Ω

(32) (33)

(34)

This is exactly same as the minimum value of uncertainty product in number state, which is given by Eq. (14).

4. Applications We apply our theory to the damped harmonic oscillator. For this system, the Hamiltonian is given by pˆ 2 1 Hˆ = e−βt + eβt mω02 xˆ 2 , 2m 2 where m is a mass and β is a damping constant. The corresponding equations in Eq. (6) are ρ¨1,2 + β ρ˙1,2 + ω02 ρ1,2 = 0.

(35)

(36)

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J.-R. Choi / Physics Letters A 325 (2004) 1–8

Fig. 1. Uncertainty product in Gaussian Klauder coherent states for a damped harmonic oscillator as a function of time t. Thick solid line is our result Eq. (40) while dotted line is Um and Yeon’s result Eq. (41), with h¯ = 1, ω0 = 1, and β = 1.

The two independent c-number solutions of the above equations are ρ1 (t) = ρ1 (0)e−βt /2eiωt , ρ2 (t) = ρ2 (0)e where ω=

−βt /2 −iωt

e

(37) ,

(38)

ω02 −

β2 . 4

(39)

Since the classical equation of motion of the system has no driving term, the two particular solutions are zero, i.e., xp = 0 and pp = 0. Therefore, the system can be completely described in terms of Eqs. (37) and (38). If we choose ρ1 (0) = ρ2 (0) = 1, c1 = c3 = 1 and c2 = 3 as an example, Eq. (34) becomes 

 h¯ 1 3β 2 1/2 1 + 2 β cos(2ωt) + 2ω sin(2ωt) + (x) ˆ c (p) (40) ˆ c= . 2 2 5ω This is somewhat different from Um and Yeon’s result [6] which is given by   3 

2 1/2 β h¯ β β2 2 1+ sin ˆ c= + (ωt) + sin(2ωt) . (x) ˆ c (p) 2 8ω3 ω 8ω2

(41)

From Fig. 1, we can see that the uncertainty product in Gaussian Klauder coherent state varies periodically with time. We now apply the theory of the Gaussian Klauder coherent states to a harmonic oscillator with time-variable frequency. In this case, the Hamiltonian is given by 1 pˆ 2 + mω2 (t)xˆ 2 . Hˆ = 2m 2 We choose the time-variable frequency of the system as

h1 (1 − h22 ) ω2 (t) = 1 + ω2 (0 < h2 < 1), (1 + h2 cos(2ω0 t))2 0

(42)

(43)

where h1 , h2 and ω0 are real constants. Then, the frequency varies periodically: ω2 (t + T ) = ω2 (t),

(44)

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Fig. 2. Uncertainty product in Gaussian Klauder coherent states for a time-dependent harmonic oscillator whose frequency is given by Eq. (43) as a function of time t. We used Eq. (34) with m = 1, h¯ = 1, n0 = 50, ω0 = 1, h1 = h2 = 0.5, ρ0 = 1, c1 = c3 = 0.1, and c2 = 1.

where period T is π/ω0 . If h2 1, Eq. (43) can be approximated to ω2 (t)  a + 2b cos(2ω0 t),

(45)

where a = [1 − h1 (h22 − 1)]ω02 and b = h1 h2 (h22 − 1)ω02 . This is important to the quantum motion of ions in a Paul trap [20–22]. One can trap charged particles, ions or electrons, without material walls by injecting them into the plane electric and magnetic multipole fields. The Paul trap supplies an oscillating quadrupole potential so that each equation of motion of the three rectangular coordinate for a charged particle represented by that of the harmonic oscillator with time-dependent frequency in Eq. (45). The trap enables us the investigation of isolated particles with extremely high accuracy according to the Heisenberg’s uncertainty principle [20]. In this case, Eq. (6) becomes ρ¨1,2 (t) + ω2 (t)ρ1,2 (t) = 0. The exact c-number classical solutions of the above equation are given by [23]

1 + h2 cos(2ω0 t) iΘ(t ) e , ρ1 (t) = ρ0 1 + h2

1 + h2 cos(2ω0 t) −iΘ(t ) e , ρ2 (t) = ρ0 1 + h2 where ρ0 is integral constant and  √ 1 − h2 sin(2ω t) 0 1 + h1 −1 2 sin Θ(t) = . 2 1 + h2 cos(2ω0 t)

(46)

(47)

(48)

(49)

Two particular solutions xp and pp are zero because of the same reason as the previous case. We depicted the uncertainty product in Gaussian Klauder coherent state for this system in Fig. 2. This is also fluctuated periodically with time.

5. Summary The coherent states that introduced by Schrödinger in 1926 [3] are distinguished set of states that can be used to show the quantum and classical correspondence in the harmonic oscillator. We used the invariant operator method

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J.-R. Choi / Physics Letters A 325 (2004) 1–8

in order to derive the Gaussian Klauder coherent states of the general time-dependent harmonic oscillator. This state is based on a procedure of Gaussian generalization of the coherent state by Fox [14], that was further applied to the particle in a box [15]. The Gaussian Klauder coherent states of the system described in terms of the four classical solutions, xp (t), pp (t), ρ1 (t), and ρ2 (t). This coherent states are wave packets that can be made so that they retain sharp Gaussian distributions for a long time, are overcomplete like standard kind of coherent states and allow a resolution of the identity operator [15]. We investigated the uncertainty products in Gaussian Klauder coherent state by setting σ 2 = n0 . We showed that the uncertainty product in Gaussian Klauder coherent state is same as the minimum uncertainty product in number state by considering the leading order of Eqs. (26)–(28). This is exactly agree with the theory developed for the standard harmonic oscillator [15]. We applied the time-dependent theory of the Gaussian Klauder coherent state to a damped harmonic oscillator and a time-variable harmonic oscillator. If we choose the time-variable frequency as Eq. (43), our development can be applied to the Paul trap for an ion, which is important for the observation of isolated ions. The uncertainty product in Gaussian Klauder coherent state fluctuated periodically as time goes by.

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