Exponential spreading and singular behavior of quantum dynamics ...

Exponential spreading and singular behavior of quantum dynamics near hyperbolic points A. Iomin Department of Physics, Technion, Haifa, 32000, Israel Quantum dynamics of a particle in the vicinity of a hyperbolic point is considered. Expectation values of dynamical variables are calculated, and the singular behavior is analyzed. Exponentially fast extension of quantum dynamics is obtained, and conditions for this realization are analyzed.

arXiv:1302.0609v1 [quant-ph] 4 Feb 2013

PACS numbers: 05.45.-a, 03.65.-w, 03.65.Nk

Hyperbolic (saddle) points are a source of instability in dynamical systems [1]. Therefore, quantum dynamics of a particle in a saddle point potential is an important problem in quantum chaos [2–4]. The hyperbolic point at the origin (x, p) = (0, 0) can be described locally by the Hamiltonian Hloc = xp. The Lyapunov exponents detecting stable Λ− and unstable Λ+ manifolds are Λ± = ±1. This leads to the exponential spreading in quantum dynamics and exponential growth of observable quantities [5, 6]. The Hamiltonian Hloc has been studied in connection with the Riemann hypothesis [7, 8], scattering of the inverted harmonic oscillator [9], and eigenstates near a hyperbolic point [10]. We consider a quantum system which is a second order polynomial of Hloc with the Hamiltonian ˆ s = 2ω x H ˆpˆ − iω~ + µ(2iˆ xpˆ + ~)2 ˆ loc − i~ ] − 4µ[H ˆ loc − i~ ]2 , ≡ 2ω[H 2 2

(1)

where ω and µ are the linearity and nonlinearity parameters correspondingly, while the coordinate and momentum operators obey the standard commutation rule [ˆ x, pˆ] = i~ with Planck’s constant ~. This system was considered in [8] in connection with the Riemann hypothesis as well. Our primary interest in this Hamiltonian is related to a problem of quantum dynamics considered in the Heisenberg picture [11] where the expectation values of the operators pˆ(t) and xˆ(t) were calculated in the coherent states [12], specially prepared at the initial moment t = 0. As shown in [11], the dynamics of these expectation values becomes singular at specific singularity times tl . For example, for the observable value of x ˆ2 these singularities occur at times π π tl = 32~µ + l 16~µ , l = 0, ±1, ±2, . . .. A remarkable property of these explosions is a pure quantum nature: in the classical counterpart this corresponds to the separatrix motion without any singularities. As follows from the analysis of Ref. [11], this explosion behavior results from the interplay between the nonlinear term and the specific choice of zero boundary conditions on infinities. When µ = 0, the singularities are shifted to infinity and expectation values of operators are finite, and this result is independent of the boundary conditions. We develop a different consideration of the problem to understand the nature of this singularities that, as will be shown, are related to the choice of the initial conditions.

First we consider the quantum dynamics of a particle of a unit mass in the saddle potential described by the Hamiltonian (1). For calculation of the expectation values, following [11], the initial wave function is chosen in the form of the coherent state Ψ0 (x) = hx|αi. In the x-representation this is the Gaussian packet [11, 12]: Ψ0 (x) = hx|αi = (~π)−1/4 e−|α|

2

√ /2~−(x2 −2 2xα+α2 )/2~

, (2) where α ∈ C. It is worth √ mentioning that in this notation the dimension of x is ~. For simplicity we calculate the expectation value of the operator x ˆ2 (t). Thus we have 2

hˆ x (t)i =

Z

∞

−∞

ˆ (t)Ψ0 (x)dx , Ψ∗0 (x)Uˆ † (t)x2 U

(3)

ˆ (t) is the evolution operator. where U The axis of integration we split into three intervals (−∞, −x0 ], [−x0 , x0 ], [x0 , ∞), and the expectation value is expressed by the following three integrals: Rx hˆ x2 (t)i = Is (t) + If− (t) + If+ (t) = −x0 0 Ψ∗s (t)x2 Ψs (t)dx R −x R∞ + −∞0 Ψ∗f (t)x2 Ψf (t)dx + x0 Ψ∗f (t)x2 Ψf (t)dx . (4) The dynamics in the finite interval [−x0 , x0 ] is considered in the framework of the truncated interaction. Near the hyperbolic point this dynamics is considered locally, such that H = Hs for x < x0 and the particle is, for example, free with H = Hf outside the interaction region |x| > x0 , where x0 > 0 determines arbitrary the interaction range. Note that we do not consider a scattering task and just ˆ s is determined by Eq. truncate the integration. Here H ˆ f = pˆ2 /2 determines free motion. Therefore, (1), while H the dynamics of an initial wave function Ψ0 is determined in these two different regions ˆf (t)Ψ0 Ψf (t) = U

ˆs (t)Ψ0 , and Ψs (t) = U h

(5)

i ˆf (t) = exp − i H ˆf t where the evolution operators U ~ h i i ˆ ˆ and Us (t) = exp − ~ Hs t describe two independent processes, and corresponding shift of the wave functions is supposed. Integrals are calculated by substituting Eqs. (5) in Eq. (4) and taking into account the explicit form of the

2 Hamiltonians. In the free motion window the evolution of the square coordinate operator is h i2 2 ˆ † (t)ˆ ˆf (t) = [ˆ xˆ2 (t) = U x U x + tˆ p] . (6) f Therefore, the free motion integrals If± (t) do not have any particular features. Their values can be expressed in the Rz 2 form of the error function Φ(z) = (2π)−1/2 −∞ eη /2 dη [13]. Then we obtain that If± (t) ∼ t2 , as expected. The saddle point integral possesses a more interesting behavior. Using calculations performed in [11][Eq. (4.16)], we arrived at the following expression: (α+α∗ )2 R x 0 2 Is (t) = (~π)−1/2 e4ωt+24i~µt e− 2~ −x0 dx x n 2 √
o
x . × exp − 2~ 1 + e32i~µt − 2 2 α∗ + αe16i~µt x (7)

First, we admit the exponential quantum growth, obtained in Ref. [11]. This quantum behavior has classical nature of the near separatrix motion, observed also for a kicked system [6]. At the singularity times π lπ tl = 32~µ + 16~µ expression (7) is simplified, and the integral is calculated exactly. It reads l

(α+α∗ )2

(−i) − 2~ Is (tl ) = (~π) eωπ/8~µ+lωπ/4~µ e3iπ/4 1/2 e   √ √ h√ 3 2ξx0 2ξx0 2~2 x0 − sinh cosh × ξ2~ 3 2 ~ ξ ~ √ i √ 2~x20 2ξx0 , − ξ sinh ~

(8)

where ξ = α∗ +i(−1)l α. This expression is finite for finite x0 . When x0 approaches infinity the integral diverges and tl are the singularity points. Note that x0 is an arbitrary defined scale [14]. To generalize the consideration of the explosion singularities, first we consider eigenvalue problem for the Hamiltonian Hs . Since the operator x ˆpˆ − i~/2 commutes ˆ with Hs , this problem is reduced to the dimensionless equation for the eigenfunctions χǫ (x):   1 d 1 x χǫ (x) = ǫχǫ (x) (9) + i dx 2 with the following solution 1 χǫ (x) = p exp[iǫ ln |x|] , Ns |x|

(10)

which satisfies the boundary conditions χǫ (x = ±∞) = 0 and Ns = 4π~1/2 [17]. For the continuous spectrum the normalization condition is Z ∞ χ∗ǫ′ (x)χǫ (x)dx = δ(ǫ − ǫ′ ) (11) −∞

(see e.g. [15]). Now, expanding Ψ0 (x) over the complete set of χǫ (x), we obtain Z Ψ0 (x) = dǫq(ǫ)χǫ (x) . (12)

Note, that the explicit form of the expansion coefficients q(ǫ) is not important, since integration over energy ǫ will be performed with exactly the same form of χǫ (x). Substituting Eq. (12) in the integral Is in Eq. (4) with x0 = ∞, we obtain Z ∞ Z ′ ∞ x2 Is (t) = 2 q ∗ (ǫ′ )q(ǫ)e−i(E − E ′ )tdǫdǫ′ 0

× χ∗ǫ′ (x)χǫ (x)dx ,

(13)

ˆ s , and we use where E = 2ωǫ − 4~µǫ2 is the energy of H that χǫ (−x) = χǫ (x). The complex Gaussian exponents are presented in the form of the Fourier integrals: Z ∞ e∓iτ ǫ dτ ±i4~µtǫ2 √ e = exp{∓iτ 2 /16~µt} . (14) ±16πi~µt −∞ Substituting these expressions in Eq. (13) and taking into account the explicit form of χǫ (x), we obtain ln |x| − 2ωt − τ = ln(|x|e−2ωt e−τ ). Then one integrates over ǫ and ǫ′ to obtain the following expression Z ∞ Z 2 −τ ′2 ) (τ +τ ′ ) dτ dτ ′ −i (τ16~µt e e−2ωt e 2 Is (t) = 2 x2 16π~µt 0  
∗ −2ωt −τ ′ × Ψ0 xe e Ψ0 xe−2ωt e−τ dx . (15)

The next step is integration over τ and τ ′ . To this end we perform the following variables change τ = u + v and τ ′ = u − v with the Jacobian of the′ transformation equaling 2. Denoting y = xe−2ωt e−(τ +τ )/2 , integration over u is exact and gives the δ function δ(v − 8i~µt). Therefore integration over v is also exact. Finally, the expectation value reads Z ∞

hˆ x2 (t)i = 2e4ωt y 2 Ψ∗0 ye8i~µt Ψ0 ye−8i~µt dy . 0

(16) Here we also use the symmetrical property of the wave function. Substituting Eq. (2) in Eq. (16), one obtains that at times t = tl this expression diverges. These are the same singularities obtained above in Eq. (7). Moreover, any “good” Gaussian and exponential functions lead to these kind of singular behavior for the expectation values of physical operators. Obviously, these singularities result from a specific preparation of the initial wave packets Ψ0 (x). Let us prepare the initial conditions “properly” to obtain the finite moments of the physical variables. Owing to Eq. (12), we present the initial condition as the spectral decompo  41 sition with the Gaussian weight q(ǫ) = 2a exp(−aǫ2 ), π where real a > 0. This yields the initial wave packet in the form of a log-normal distribution   1 1 1 Ψ0 (x) = √ exp − 2 ln2 |x| − ln |x| (17) 4a 2 N √ with N = 4a π. Using Eqs. (17) and (16), one obtains for the nth moment of x ˆ 16 2 2 2 1 hˆ xn (t)i = √ e4ωt e a2 ~ µ t 2a π

3 ∞

  1 exp − 2 ln2 x + n ln x d(ln x) 2a 0 √   16 2 exp 4ωt + a4 n2 + 2 ~2 µ2 t2 . = 2 a ×

Z

term is due to the quantum dynamics near the hyperbolic point. (18)

This behavior of the expectation values is finite and spreads exponentially for the arbitrary long time scale. This exponential increasing with time consists of two values. The first one is the above mentioned classical term e4ωt , which is due to the classical motion near the hyperbolic point. The second, pure quantum, term 16 2 2 2 a2 ~ µ t is dominant and relates to the action of the evoˆ lution operator, is a delation operator ebxp , where  which √  bxp ˆ −i ~b (see e.g., [16]). Therefore, this e f (x) = f e

[1] R.S. MacKay in Quantum Chaos, Course CXIX, edited by G. Casati, I. Guarneri, U. Smilansky (North-Holland, Amsterdam, 1993). [2] G.P. Berman, G.M. Zaslavsky, Physica A 91, 450 (1978). [3] M.V. Berry, N. Balasz, J. Phys. A: Math. Gen. 12, 625 (1979) [4] M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, Berlin, 1990). [5] G. Barton, Ann. Phys. (N.Y.) 166, 322 (1986). [6] J. Wang, I. Guarneri, G. Casati, and J. Gong, Phys. Rev. Lett. 107, 234104 (2011). [7] M.V. Berry, J.P. Keating, in Supersymmetry and Trace Formulae, eds. I.V. Lerner, J.P. Keating, D.E. Khmelnitskii, NATO ASI Series B 370 (Kluwer Academic, New York, 1999). [8] J.V. Armitage, in Number Theory and Dynamical Systems, eds. M.M. Dodson, J.A.G.Vickers (University Press Cambridge 1989). [9] R.K. Bhaduri, A. Khare, J. Law, Phys. Rev. E 52, 486 (1995). [10] S. Nonnemacher, A. Voros, J. Phys. A: Mat. Gen. 30, 295 (1997). [11] G.P. Berman, M. Vishik, Phys. Lett. A 319, 352 (2003). [12] R.J. Glauber, Phys. Rev. 131, 2766 (1963). [13] E. Janke, F. Emde, F. L¨ osh, Tables of Higher Functions (McGraw-Hill, NY, 1960). [14] The interaction range x0 is always a finite value due to the local consideration near the hyperbolic point. For example, it can be evaluated as the difference between normalization constants of the wave functions for the free motion Nf and for the saddle point motion Ns (see, Problem 133.1 in [15]). For the case of free motion the normalization constant Nf is determined from the integral [15, 18]

Both singular behavior and this, completely new, exponential quantum growrth are due to the nonlinear quantum dispersion term κ = ~µT , where T is a characteristic time scale. For example, for the explosion behavior π . For the of hˆ x2 (t)i in Eqs. (7) and (8) it is T = 32~µ quantum expansion we can define this time scale parameter as ~µ/ω to define hthe dimensionless growth of the i
κ 2 expectation values exp a ωt . This research was supported by the Israel Science Foundation.

R∞

√ ′ e−ipx/~ eip x/~ dx = 2π ~δ(p − p′ ) ≡ Nf δ(p − p′ ) . Therefore, Nf = 2π~1/2 . We also used here that x has the dimension ~1/2 . In order to find the Ns , one uses the ˆ s with an extended eigenfunction of the Hamiltonian H action on the whole x axis from minus to plus infinity. Eventually we obtain that the interaction range is x0 = Ns − Nf = 2π~1/2 . [15] L.D. Landau, E.M. Lifshitz, Quantum Mechanics (Pergamon, 1977). [16] G. Teschl, Mathematical methods in quantum mechanics: with applications to Schr¨ odinger operators (American Mathematical Society, Providence, 2009). [17] A mathematically rigorous calculation of the normalization constant for the wave function χǫ (x) can be pressented by follow the presentation in the monograph by V.A. Fock [18]. Since the operator x ˆpˆ has continuous spectrum ǫ, the eigenfunctions χ(ǫ, x) ≡ χǫ (x) are not square integrable. Therefore, the normalization condition exists not for the eigenfunction but for the “eigendifferential” [18] ∆χ(ǫ, x), which reads R ǫ+∆ǫ ∆χ(ǫ, x) = ǫ χ(ǫ′ , x)dǫ′ . Substituting here Eq. (10), one obtains |x| ∆χ(ǫ, x) = √ 2 . exp[i(ǫ + ∆ǫ) ln |x|] sin ∆ǫ ln 2 −∞

N|x| ln |x|

This solution is already square integrable and has the following normalization form √ R∞ 1 lim∆ǫ→0 ∆ǫ dx |∆χ(ǫ, x)|2 = ~ . −∞ Carrying out the following variable change z = R∞ 2 (∆ǫ/2) ln |x| and taking into account that −∞ sinz 2 z dz = π , we obtain N = 4π that coincides exactly with the dimensionless normalization constant Ns ~1/2 in Eq. (10). [18] V.A. Fock, Foundations of Quantum Mechanics, (in Russian) (Nauka, Moscow, 1976).