Quantization of intrinsic localized modes: Effective linear lattice method

Physics Letters A 359 (2006) 438–444 www.elsevier.com/locate/pla

Quantization of intrinsic localized modes: Effective linear lattice method Shozo Takeno  Institute for Innovative Science and Technology, Nagasaki Institute of Applied Sciences, Nagasaki 851-0193, Japan Received 2 March 2006; accepted 3 March 2006 Available online 5 July 2006 Communicated by A.R. Bishop

Abstract A working method is proposed to quantize intrinsic localized modes (ILMs) in lattices with hard quartic anharmonicity in the framework of the rotating-wave approximation. This is done by reducing a nonlinear eigenvalue problem to a linear one by averaging slowly varying reduced effective force constants over frequencies, enabling us to quantize the ILM in a straightforward manner. Such an effective linear lattice (ELL) method is first applied to an analytically tractable d-dimensional cubic lattice to show that the concept of the ELL holds exactly in the strong localization limit. Next, general lattice models are investigated to achieve quantization of the ILM in an approximate manner. The obtained analytical results are tested by solving numerically a model one-dimensional lattice to show that phase-space trajectory of an ILM-bearing atom is of elliptic type with finite but small width. The numerical result confirms the validity of the ELL leading to its semi-classical quantization. On the other hand, orbits of all the remaining atoms exhibits complex non-periodic trajectory to which a direct application of the semi-classical quantization rule appears impossible. © 2006 Published by Elsevier B.V.

1. Introduction In his classic paper published in 1917, Einstein examined the condition by which mechanical systems are subject to the Bohr– Sommerfeld–Epstein rule of quantization [1], pointing out that the absence of invariant tori in the phase-space prevents any use of the quantization rule. Finding effective methods for quantizing classical dynamical systems has since remained one of the central problems in modern physics. The underlying fact behind this is the disparity between the nonlinearity of classical dynamics and the linearity of quantum mechanics. Meantime, intrinsic localized modes (ILM) in classical nonlinear lattices [2], which are strongly localized in space and periodic in time nonlinear modes, have received particular attention in recent years. Being fairly well-defined modes in non-integrable lattices, the ILMs occupy a unique position among various nonlinear modes in the ubiquity of their existence and persistence against disturbances [3,4]. A natural question is whether or not is it possible 

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to quantize the ILMs. This can be considered as an issue to be addressed in the following two respects: One is its conceptual interest and importance in theoretical physics and the other the existence of abundant experimental data for the existence of the ILMs. Examples of the latter are magnons in antiferromagnets [5], Raman spectra in charge-transferred solids [6, 7], coupled Josephson elements [8,9], micro-mechanical oscillators [10], optical wave guides [11] and photonic crystals [12, 13], optical lattices [14,15] and so on. Two kinds of approaches have so far been used for the quantization of the ILMs. The first is the case of weak nonlinearity in which excitations can be treated as envelope solitons [16]. The second is the use of the so-called Einstein–Keller–Brillouin quantization on the KAM torus [17]. Quantum aspects of the ILMs have also been studied by several researchers [18–20]. The lastest result in this problem showed that for specific onedimensional (1D) lattice models quantum ILMs are stable in spite of the existence of small tunneling effects [21]. It is the purpose of this Letter to present a method of effective linear lattices (ELL) for quantization of the ILMs. The underlying physics behind this is the observation that a strongly

S. Takeno / Physics Letters A 359 (2006) 438–444

localized ILM looks something like a defect (ILM defect) in the corresponding linear lattices. The difference of the ILMdefect from conventional defects in harmonic lattices is that it can spontaneously appear at any site in pure crystal lattices where an atom makes large excursion in comparison with the other atoms. Our basic theoretical procedure in the one-ILM problem is to introduce a concept of the ELL associated with a given ILM. A key point here is to show that a fluctuating force constant is a slowly varying quantity, being divided into two parts, a mean value and small deviations around it. This naturally leads to the result that a phase-space orbit of the ILM is elliptic with small fluctuations. In comparison with the previous methods, this is a working method for quantization of the ILMs in the sense that, once established, it provides us with a straightforward quantization procedure in line with the conventional quantum theory. This Letter is organized as follows. In Section 2, a theory of the ILM using the lattice Green’s function method is formulated. An analytically solvable model is then adopted to get a hint on achieving quantization of the ILM. In Section 3, the concept of the ELL is introduced. Quantization of the ILM can be done exactly and approximately for the exactly solvable model in the strong localization limit and general cases, respectively. In Section 4 the validity of the ELL method is tested by solving numerically equations of motion for a model one-dimensional nonlinear lattice equation. The last section is devoted to concluding remarks.

We consider a pure mon-atomic lattice with the hard quarn) ≡ u(x) be the αth (α = 1, 2, 3) tic anharmonicity. Let uα ( component of the displacement vector u( n) of an atom with atomic mass m, where x ≡ ( n, α) stands for a pair of symbols , n2 , n3 ) is a vector, and the position of ( n, α). Here n = (n1 the atom is given by 3j =1 nj aj , nj and aj being integers and the lattice constants, respectively. In terms of K2 (x1 , x2 ) and K4 (x1 , x2 , x3 , x4 ), harmonic and hard quartic force constants, respectively (these quantities are invariant under permutation of the indices), the Hamiltonian H describing the atomic displacement field of the lattice can be written in the form H=

 p(x)2 x

2m

+

2

1  + K4 (x1 , x2 , x3 , x4 )u(x1 )u(x2 )u(x3 )u(x4 ), 4 x ,x ,x ,x 1

2

3

4

(1) with  K2 (x1 , x2 ) = 0, x2



K4 (x1 , x2 , x3 , x4 ) = 0.

K2 (x1 , x2 ) , m K4 (x1 , x2 , x3 , x4 ) J4 (x1 , x2 , x3 , x4 ) = , m equations of motion are written as  J2 (x, x1 )u(x1 ) u(x) ¨ +

J2 (x1 , x2 ) =

(2)

x2 ,x3 ,x4

Here p(x) = mu(x) ˙ is the momentum associated with u(x), and the periodic boundary condition is assumed for the lattice. In

(3)

x1

+



J4 (x, x1 , x2 , x3 )u(x1 )u(x2 )u(x3 ) = 0.

(4)

x1,x2,x3

The coefficient Ji (·) satisfy the same symmetry relations as those satisfied by the Ki (·). We are concerned with stationary intrinsic localized modes (ILM) associated with Eq. (4). This amounts to seeking solutions to Eq. (4) in the form u(x, t) = φ(x) cos(ωt),

φ(x): time-independent,

(5)

where φ(x) is an envelope function and ω is the frequency. Inserting Eq. (5) into Eq. (4) and using the rotating-wave approximation (RWA) lead to  J2 (x, x1 )φ(x1 ) x1

+



3 J4 (x, x1 , x2 , x3 )φ(x1 )φ(x2 )φ(x3 ) = ω2 φ(x). 4

(6) Eq. (6) poses a nonlinear eigenvalue problem. We are con2 of the cerned with those ω2 ’s which exists above the top ωM 2 frequency band ω(λ) of the harmonic lattice due to the intrinsic nonlinearity, which is the eigenvalue of the dynamical matrix J2 (x, x1 ), i.e.  (7) J2 (x, x  )ϕλ (x  ) = ω(λ)2 ϕλ (x), x

where ϕλ (x) is its eigenfunction. Then, we introduce the lattice Green’s functions (LGF) g(x, x  ; ω) =

 ϕ  (x)ϕλ (x  ) λ

λ

1  K2 (x1 , x2 )u(x1 )u(x2 ) 2 x ,x 1

terms of mass-reduced force constants

x1,x2,x3

2. Analytical solutions for intrinsic localized modes in strong localization limits

439

ω2 − ω(λ)2

,

(8)

to rewrite Eq. (6) as  φ(x) = g(x, x1 ; ω)J4 (x1 , x2 , x3 , x4 )φ(x1 )φ(x2 )φ(x3 ). x1 ,x2 ,x3 ,x4

(9) 2 , the g(x, x  )s are rapidly decreasWe note that for ω2 > ωM ing functions of |x − x  |. Let us consider physical situations in which several atoms in the system, which are well separated from one another, spontaneously make large excursion compared with their surrounding atoms. Let  be the number of such atoms undergoing large-amplitude motion. Then, the nonlinear lattice under consideration can be divided into  independent ILM regions and the remaining, and in each of the ILM region we need only consider one-ILM problem.

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The eigenfrequencies and eigenfunctions of the ILMs are in principle obtainable from Eq. (9). Without introducing a concrete model, however, we cannot proceed any further. To obtain concrete results for the ILMs and their quantization, we introduce an analytically, rather than numerically, solvable model. For this purpose, we consider n-dimensional pure cubic lattices (n = 1, 2, 3) with nearest neighbor harmonic force constant K2 and nearest-neighbor quartic force constant K4 . The atomic mass is taken to be m. In the 2- and 3-dimensional cubic lattices, each component of the displacement vector of an atom is independent of one another, and φ(x) is rewritten as φ( n), where n − m;  ω) n is the position vector. The LGF g(x, x  ; ω) → g( takes the form n − m)]  1  exp[i q · ( , g( n − m;  ω) = N ω2 − ω0 ( q )2 q   d  2 q ) = 2J2 d − cos(qj ) , ω0 ( j =1

K2 K4 J2 = (10) , J4 = , m m where q is the wave vector playing now the role of the parameter λ in Eqs. (7) and (8) and N is the total number of atoms. We assume that an ILM is located at n = 0. For simplicity, we limit our discussion to the strong localization limit where only the site n = 0 and its nearest neighbors are directly affected by the presence of the ILM. Then, Eq. (9) takes the form d   3   − φ(ej ) 3 g( n; ω) φ(0) φ( n) = J 4 4 j =1    − φ(ej ) 3 + φ(0) d  3 3  + J4 g( n − ej ) φ(ej ) − φ(0) 4 j =1  3 + g( n + ej ) φ( n − ej ) − φ(0) ,

(11)

d    ω2 − 2dJ2 g( n ) + J2 g( n + ej ; ω) + g( n − ej ; ω) j =1

(12)

where ( n) is the Kronecker delta, we can write equations corresponding to Eq. (11) in the form φ( n) =

3J4 (φ0 − φ1 4J2

ω2 g( n, ω) − ( n) (φ0 − φ1 ).

2  ωM 1 + z(ω) + · · · , 2 2ω

2 ω 1 2 ω2 g1 = − M2 , − z(ω) . . . , for ω2  ωM 2ω 2d

ω 2 g0 = 1 +

where z(ω) =

2 ωM 1 . 1 + 2d 2ω2

1 3 ] 3J4 [1 + 2d ω2 = φ02 2 ωM 8J2 [1 + z(ω)]2



)2 

Combining these two equations and treating φ02 , the square of the amplitude of the ILM, as a parameter lead to an equation which determines its eigenfrequency. We note in passing that here g1 can be written entirely in terms of g0 by using Eq. (12). The above procedure completely solves the one-ILM problem. To obtain a concrete result for this, we are concerned with analytical expressions, rather than numerical results, for the eigenfrequency and the eigenfunctions of the ILM. This can be achieved by using the following asymptotic expressions for 2 ≡ 4dJ : ω2 g0 and ω2 g1 for ω2  ωM 2

(17)

(18)

Thus, inserting Eqs. (17) into Eqs. (15) and (16) yields

where ej (j = 1, 2, 3) is the unit vector in the j th axis. We pay particular attention to an s-like mode having the symmetry relation φ(ej ) = φ(−ej ) ≡ φ1 for all j = 1, 2, 3 and we set  ≡ φ0 . Using the identity relation satisfied by the LGF φ(0)

= ( n),

Corresponding to φ0 and φ1 , let us introduce a compact notation  ≡ g0 and g(ej ) = g(−ej ) ≡ g1 for all j . Then, a pair of g(0) equations useful for studying the properties of the ILMs are written as 3J4 (φ0 − φ1 )2  2 ω (g0 − g1 ) − 1 = 1, (15) 4J2 φ1 ω 2 g1 . (16) = 2 φ0 ω g0 − 1

(13)

2 for ω2  ωM ,

(19)

1 − z(ω) 1 φ1 2 = − 2d . →− for ω2  ωM (20) φ0 1 + z(ω) 2d We note that Eq. (20) is a d-dimensional version of the wellknown relation φ1 /φ0 = −1/2 for the 1D case.

3. Effective linear lattice and quantization The result obtained in the preceding section provides us with a hint on how to achieve quantizing of the ILM. To do this, we go back to Eq. (6) and introduce the concept of an effective linear lattice (ELL). This amounts to modifying Eq. (6) as  J2 (x, x1 )φ(x1 ) x1

+

3  J4 (x, x1 , x2 , x3 )C(x2 , x3 )φ(x1 ) 4 x1 x2,x3 2

= ω φ(x),

(21)

where

From the above equations, the localization properties of the ILM are given in terms of g( n, ω) as follows

  C(x2 , x3 ) = φ(x2 )φ(x3 ) ,

n, ω) ω2 g( φ( n) = , 2   φ(0) ω g(0, ω) − 1

where the symbol φ(x2 )φ(x3 ) ω is an average which will be specified later. It is a straightforward matter to show that applying this procedure to Eq. (11) and using the same procedure as

 n = 0.

(14)

(22)

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of f (y) , the eigenfrequency ω0 of the ILM is given by the equation

 3J4 1 3 2 2 f (y) φ02 ωM ≡ ω02 , 1+ ω0 = (30) 8J2 2d

Fig. 1. Plot of f (y) as a function of y. Top, middle and bottom curves correspond to 3d, 2d and 1d cases, respectively.

before gives a modified version of Eq. (13) φ( n) =

3J4 C(φ0 , φ1 )  2 ω g( n, ω) − ( n) (φ0 − φ1 ), 4J2

(23)

where   C(φ0 , φ1 ) = (φ0 − φ1 )2 .

(24)

Then, Eq. (16) holds as it stands, while Eq. (15) takes the form 3J4 C(φ0 , φ1 )  2 ω (g0 − g1 ) − 1 = 1. 4J2

(25)

Using Eq. (20) gives

   1 2 2 2 φ0 f (y) , (φ0 − φ1 ) ω = 1 + 2d where



f (y) =

2

1 1 + (1 +

1 1 2d ) 2y

,

with y =

(26)

ω2  1. 2 ωM

(27)

The quantity f (y) defined by Eq. (27) is plotted in Fig. 1. Several interesting features are observed: (a) It is a slowing varying and monotonically increasing function of y( 1); For example, it increases monotonically from 0.7 to 0.85 for d = 1 and from 0.75 to 0.88 for d = 2, 3 in the region 3 < y < 10. (b) Increasing is smaller for higher dimensionality of the lattice. (c) The quantity f (y) is itself an effective force constant, ensuring our reasoning of the effective linear lattice by taking its average, i.e. f (y) . Here, we take the average in the form, 

 f (y) =

1 y 2 − y1

=1−

y2 f (y) dy y1

c y 2 − y1



    1 1 y2 + c +c − . 2 ln y1 + c y2 + c y1 + c (28)

with 1 1 (29) y1  1, y2  1. + , y1 < y2 , 2 4d Setting, for example, y1 = 3, y2 = 10, we obtain f (y) = 0.812, 0.840, 0.850 for d = 1, 2, 3, respectively. Thus, in terms c=

where the square φ02 of the amplitude of the localized mode is taken as a parameter. Thus, the ELL method developed so far in this section can  in contrast to be summarized as follows. Eq. (23) for n = 0, Eq. (13), in conjunction with Eq. (16), shows that an atom located at the site 0 can be regarded as a single harmonic oscillator with its eigenfrequency ω0 determined by Eq. (30). Then, quantization of the ILM located at the site 0 proceeds as follows. Let Hˆ 0 be the quantum-mechanical Hamiltonian for this single ILM. Then, it is a straightforward matter to show that

1 † ˆ H0 = h¯ ω0 aˆ 0 aˆ 0 + , (31) 2   h¯

aˆ 0 + aˆ 0† , φ0 → qˆ0 = 2m0 ω0   hm ¯ 0 ω0 1

pˆ 0 = (32) aˆ 0 − aˆ 0† . 2 i Here qˆ0 and pˆ 0 are a quantum-mechanical counterpart of φ0 and its conjugate momentum, respectively, in which aˆ 0 , and aˆ 0† are an annihilation operator and a creation operator associated with them. Here, the eigenfrequency ω0 of the ILM depends on the square φ02 of its amplitude φ0 . So, in the quantum regime, re-consideration is required for its meaning. A straightforward procedure here is to consider replacing φ02 by its average with √ respect to the eigenstates |n = (1/ n!)(aˆ 0† )n |0 (a|0 = 0) of Hˆ 0 , i.e. h¯  2n|aˆ † aˆ 0 |n + 1 2m0 ω0 h¯ (2n + 1). = 2m0 ω0

n|φ02 |n ⇒ n|qˆ02 |n =

(33)

Gerally speaking, the ILMs results from large-amplitude atomic motion in nonlinear lattices where large number of quanta are involved. It would be a rare event in which an ILM with a definite quanta is generated. Thus, much more adequate here is to consider number-indefinite multi-quanta states, replacing φ02 by its expectation value with respect to coherent states |α = exp[−(1/2)|α|2 ] exp(α aˆ † )|0 , i.e. α|φ02 |α ⇒ α|qˆ02 |α ≡ q02 . q02

(34)

can be considered as a classical Physically, the quantity counterpart of qˆ02 . Keeping the above concrete result obtained for the specific lattice model in mind, we go back to the general lattice models described by Eq. (21) and consider a physical situation in which there exist several ILMs, well separated from each another. Then, the system under consideration can be mainly divided into two parts: a set of ILM region Rj enumerated by the index j , and the other the remaining. In Rj , the ILM is assumed

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S. Takeno / Physics Letters A 359 (2006) 438–444

to exist at the site x0j for which we have     φ(x0j )2  φ(x)2 for x = x0j .

(35)

In quantizing the ILMs associated with Rj , we define an effective dynamical matrix as Dj (x, x1 ) = J2 (x, x1 ) +



J4 (x, x1 , x2 , x3 )C(x2 , x3 ),

x1 x2,x3

x, x1 , x2 , x3 ∈ Rj .

(36)

It is assumed that the quantity C(x2 , x3 ) defined by Eq. (22) has a well-defined mean value having the form    Cj φj20 fj (y) , C(x2 , x3 ) = (37) j

where Cj , φj 0 , and fj (y) are a constant, the amplitude of a j th ILM, and a slowly varying function of y, respectively, all associated with Rj . Let us consider a linear eigenvalue problem associated with D(x, x1 ). Because the concept of the effective linear lattice is assumed to hold for each of Rj , the eigenvalue equation is actually of the form  2 (38) Dj (x, x1 )ψμj (x1 ) = ω0j ψμj (x) for x, x1 ∈ Rj , x1

where the symbol μj stands for the identification of eigen2 and the eigenfunction states associated with the eigenvalue ωoj ψμj (x) belonging to Rj . The eigenfunctions ψn (μj ) make up a complete set and are chosen to be orthonormalized. With  ∗ ψμj (x)ψμj (x  ) = (x, x  ), μj



∗ ψμj (x)ψμ j (x) = (μj, μ j  ).

(39)

x

Here, the asterisk is the complex conjugate. The aim of our calculations now is to quantize the field φ(x). To this end we observe that the eigenvalue problem (36) can be associated with a lattice governed by the Hamiltonian

 1  2 1  Dj (x1 , x2 )q(x ˜ 1j )q(x ˜ 2j ) , p˜ j (x) + Heff = 2 x 2 x ,x j

1

2

(40) with p˜ j (x) = q˜˙j (x). Such a lattice will be referred to as the effective lattice. Next, quantization of effective linear lattice is achieved in the conventional way. Namely, we introduce a pair of canonical coordinates (qμ j, pμj ) by the equations   qj ψμj , p˜ j (x) = pj ψμj , q˜j (x) = (41) μj

μj

thereby reducing the effective Hamiltonian Heff to the form Heff =

1  2 pj pj∗ + ω0j qj qj∗ . 2 j

(42)

As is well known, quantization of the system described by Eq. (21) can now be achieved by introducing annihilation operators aμ and creation operators aμ† in such a way that 

 h¯ qˆj = aˆ j + aˆ j† , 2mj ω0j   hm ¯ j ω0j 1

aˆ j − aˆ j† , pˆ j = (43) 2 i thus reducing the (qj ) and (pj ) from c-number quantities to q-number ones (qˆj ) and (pˆ j )’ which satisfy the canonical commutation relation [pˆ j , pˆ j  ] = [qˆj , qˆj  ] = 0, [pˆ j , qˆj  ] = −i hδ ¯ j,j  ,

3  3J4 1  2 2 f (y) qj2 ωM = , with 1+ ω0j 8J2 2d     qj2 = αj qˆj2 αj ,

(44)

(45)

where αj is a coherent state associated with the j th ILM using the ELL method. Finally, the Hamiltonian of the quantum effective lattice takes the form

 1 † Hˆ eff = (46) a ˆ + . a ˆ hω ¯ 0j j j 2 j

Thus, within the framework of the ELL method, the original nonlinear lattice can be regarded as a set of quanta with energy h¯ ω0j [aˆ j† aˆ j + 12 ]. 4. Numerical test of the ELL: phase-space orbits To examine the validity of the concept of the ELL, we performed numerical experiments of the time-evolution of the phase-space orbits of an ILM and those of its surrounding atoms for a 1D lattice with hard quartic nonlinearity governed by the equation u¨ n = J2 [un+1 + un−1 − 2un ]  + J4 (un+1 − un )3 − (un − un−1 )3 ,

J4 = 0.2, J2

(47)

where J2 > 0 and J4 > 0 are mass-reduced harmonic and quartic anharmonic force constants, respectively. The lattice constant of the 1D lattice is taken to be unity. The numerical calculations were performed for a model 1D lattice composed of N = 100 atoms with n ranging from n = 0 to n = 99 to which the periodic boundary condition was applied, N being the total number of atoms. The initial condition was chosen as un (0) = 0 for all n,  v0 for n = 50, −v0 u˙ n (0) = 0 for n = 50, 51.

(48) for n = 51,

(49)

As an example, the initial velocity v0 was taken to be v0 = 6.0 to get the well-defined ILM. Firstly, the space–time evolution of the initial condition for the dynamics of all the atoms in the system is shown in Fig. 2. It is seen that the ILMs are located at the sites n = 50 and n = 51 while all the remaining atoms

S. Takeno / Physics Letters A 359 (2006) 438–444

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undergo small oscillatory motion. Keeping this in mind, we performed numerical calculations of phase-space orbits ((v ≡ u˙ vs u curves) of atoms in the system, and the result is shown for atoms located at the sites n = 50, 49, 20 in Fig. 3(a)–(c). Case (a) and case (b) correspond to the orbit of the ILM-bearing atom and that of its nearest neighbor, respectively, while case (c) depicts orbits of an atom away from the ILM site. Similar results were obtained for atoms far separated from the ILM site. It is seen that the phase-space orbit of the ILM is nearly elliptic. On the other hand, those of all the other atoms are fairly complicated to which application of the semi-classical quantization rule due to Einstein and Epstein does not appear possible. The elliptic orbit of the ILM as shown in Fig. 3(a) is quite consistent with the concept of the ELL, according to which the effective Hamiltonian Hce of a single ILM with eigenfrequency ω0 , coordinate q0 and its conjugate commentum p0 is written as Fig. 2. Space–time evolution of the initial condition, Eqs. (48) and (49) with v0 = 6, for the 1D nonlinear lattice equations governed by Eq. (47).

1 2 mω02 2 (50) p + q , 2m 0 2 0 where m is the atomic mass. The phase-space orbit is elliptic, and an application of the classical quantization rule yields  2πH p0 dq0 = = nh, or H = nh¯ ω0 , ω0 n = 1, 2, 3, . . . . (51)

Hce =

Acknowledgements The author would like to express his sincere thanks to Professor V.V. Konotop for his continued interest in this work. References (a)

(b)

(c) Fig. 3. Phase-space orbits of atoms at: (a) the ILM site (n = 50), (b) its nearest neighbor n = 49 and (c) n = 20.

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