Dynamic localization in an effective tight binding Hamiltonian model with a rapidly oscillating homogeneous electric field on a lattice
aa r X i v : . [ c ond - m a t . o t h e r] J un Dynamic localization in an effective tight bindingHamiltonian model with a rapidly oscillatinghomogeneous electric field on a lattice
L. A. Mart´ınez - Quintana
Departamento de F´ısica, Universidad Sim´on Bol´ıvar, Apartado 89000, Caracas1080A, Venezuela.
L. A. Gonz´alez - D´ıaz
Laboratorio de F´ısica Estad´ıstica de Medios Desordenados. Centro de F´ısica,Instituto Venezolano de Investigaciones Cient´ıficas, Caracas 1020 - A, Venezuela.E-mail: [email protected]
Abstract.
By the Magnus-Floquet approach we calculate the effective Hamiltonianfor a charged particle on the lattice subject to a homogeneous high frequency oscillatingelectric field. The obtained result indicate the absence of dynamic localization of theparticle for any value of the lattice constant and electric field applied, which completesthe limit results obtained by Dunlap and Kenkre.
Keywords : Dynamic localization, Effective Hamiltonian, Magnus-Floquet approach,Tight binding Hamiltonian.
1. Introduction
Dynamic localization is a versatile quantum phenomena of electrons widely studiedby its potential technological applications [1–6]. On the other hand, recent experimentsshow the analogy between photonics wave packets and the Bloch electrons on the latticesubject to electromagnetic fields [7, 8]. Based on this evidence, it has been possible toconstruct optical lattices, where the dynamic localization of photonic wave packets hasbeen observed experimentally [9]. Optical fibers arrays are the most direct applicationof these quantum systems [10, 11].It is known that in the free space, i.e., in the absence of a lattice, high frequencyelectromagnetic fields produce effects of confinements that trap particles in demarcatedregions of space [12,13]. In the literature, the theoretical approach to study the dynamicsof these systems is through the calculation of effective Hamiltonians. This technique hasbeen used with great success in areas such as high resolution NMR spectroscopy [14], the ynamic localization in an effective tight binding Hamiltonian model ... G ,of a charged particle in a one dimensional lattice subject to a homogeneous electricfield dependent on time, periodic and high frequency. We will be used the approachof Magnus-Floquet for the calculation of G and to study if the dynamic localizationeffects predicted by Dunlap and Kenkre [1] remain at high frequencies. We find, asmain result, that at high frequency the particle does not present dynamic localization.Starting from the results of Dunlap y Kenkre, we can not obtain the behavior of themean square displacement at high frequency. It is necessary to apply the formalismof Magnus-Floquet in order to obtain the behavior at high frequency. Experiments inoptical lattices would allow to verify this theoretical prediction obtained.
2. Effective Hamiltonian for periodically driven system.
Floquet’s theorem [13,14,31–37] has been applied to time-dependent quantum systems intwo ways: (a) The time-dependent propagation technique exploits that the propagator U ( t ) for a periodic Hamiltonian H ( t ) = H ( t + nT ) ( n ∈ Z ) is of the form U ( t ) = P ( t ) e − i G t , where G is a time-independent Hamiltonian and P ( t ) is cyclic: P ( t ) = P ( t + nT ). And (b) in the time-independent Floquet Hamiltonian approach [30, 38–40], thetime-dependent Hilbert-space Hamiltonian is transformed into an infinite-dimensionalHilbert space, the so-called Floquet space, where it is represented by a time-independentFloquet Hamiltonian. The time-dependent problem reduces to a time-independent one,but with infinite dimension. In this paper, we will follow the first way.We consider the coherent hopping dynamics of a quantum particle on a one-dimensional tight-binding lattice rapidly driven by an external sinusoidal field withhomogeneous hopping rates, which is described by the Hamiltonian (with ℏ = 1 and e = 1) H ( t ) = H + H ( t ) (1)with H = − A cos( ap ) (2)and H ( t ) = ǫx cos( ωt ) (3) A is the tight-binding matrix element for the hopping of the particle, a is the latticeconstant and ǫ is a perturbing force. Here, we will consider that ω ≫ ω B , with ω B = ǫa the Bloch’s frequency. ynamic localization in an effective tight binding Hamiltonian model ... G = X k =0 G ( k ) (4)with G ( k ) = 1 T Z T { H ( t ′ ) P ( k ) ( t ′ ) − k − X j =1 P ( j ) ( t ) G ( k − j ) } dt ′ , (5) G (0) = 0, where P ( t ) = X k =0 P ( k ) ( t ) (6)with P ( k ) ( t ) = − i Z t { H ( t ′ ) P ( k − ( t ′ ) − k − X j =1 P ( j ) ( t ) G ( k − j ) − G ( k ) } dt ′ . (7) P (0) ( t ) = . For the Hamiltonian (1), we have that G (1) = 1 T Z T H ( t ′ ) dt ′ (8)= H ,P (1) ( t ) = − i Z t (cid:0) H ( t ′ ) − G (1) (cid:1) dt ′ (9)= ǫ sin( ωt ) ω ∂∂p ,G (2) = 1 T Z T (cid:0)(cid:2) H , P (1) ( t ′ ) (cid:3) + H ( t ′ ) (cid:1) dt ′ (10)= 0 ,P (2) ( t ) = − i Z t (cid:0)(cid:2) H , P (1) ( t ′ ) (cid:3) + H ( t ′ ) P (1) ( t ′ ) (cid:1) dt ′ (11)= ǫω sin (cid:18) ωt (cid:19) (cid:18) i Aa sin( ap ) + 2 ǫ cos (cid:18) ωt (cid:19) ∂ ∂p (cid:19) ,G (3) = 1 T Z T (cid:0)(cid:2) H , P (2) ( t ′ ) (cid:3) + H ( t ′ ) P (2) ( t ′ ) (cid:1) dt ′ (12)= Aa ǫ ω cos( ap ) ynamic localization in an effective tight binding Hamiltonian model ... G ≈ G (1) + G (2) + G (3) = − A (cid:18) − a ǫ ω (cid:19) cos ( ap ) = J H (13)with J ≡ − a ǫ ω , which coincides with the first two terms of the Bessel function J . The result obtained in (13) is consistent up to O ( ω − ) with the exact result forthe band narrowing in the presence of a homogeneous oscillating electric field [1]. Theband narrowing effect has been observed experimentally with cold atoms in opticallattices [41]. From (6), we have that P ( t ) ≈ P (0) ( t ) + P (1) ( t ) + P (2) ( t ) = + ǫ sin( ωt ) ω ∂∂p + (14) ǫω sin (cid:18) ωt (cid:19) (cid:18) i Aa sin( ap ) + 2 ǫ cos (cid:18) ωt (cid:19) ∂ ∂p (cid:19) The wave function is given by Ψ( t, p ) = P ( t ) e − i t G ψ ( p ) (15)with ψ ( p ) a generalized eigenfunction of G .Dynamic localization can be seen through the mean square displacement, which for t ≫ πω is given by h Ψ | X | Ψ i = − Z ∞−∞ Ψ ∗ ( t, p ) ∂ ∂p Ψ( t, p ) dp ∼ − A (cid:18) a (cid:18) − (cid:16) ω B ω (cid:17) (cid:19) I + (16)+ a sin ( ωt ) (cid:16) ω B ω (cid:17) ˜ I + sin ( ωt ) (cid:16) ω B ω (cid:17) (cid:16) ˜ I + 2 a I (cid:17)(cid:19) t with I ≡ Z ∞−∞ sin ( ap ) ψ ∗ ( p ) ψ ( p ) dpI ≡ Z ∞−∞ cos(2 ap ) ψ ∗ ( p ) ψ ( p ) dp ˜ I ≡ sin ( ap ) ψ ∗ ( p ) ψ ( p ) (cid:12)(cid:12)(cid:12) ∞−∞ ˜ I ≡ sin ( ap ) ddp ( ψ ∗ ( p ) ψ ( p )) (cid:12)(cid:12)(cid:12) ∞−∞ The mean square displacement is not bounded in time, which physically means that theparticle does not exhibit dynamic localization.The dynamic localization condition of a charged particle in the presence of anoscillating and homogeneous electric field is deduced from the exact formulae of themean square displacement obtained by Dunlap and Kenkre [1]. From this formulae, itfollows that if aǫω is greater or less than a root of the Bessel function J the dynamiclocalization phenomenon disappears. In the context of high frequency, (16) shows the ynamic localization in an effective tight binding Hamiltonian model ... t ≫ πω (largestimes). The high frequency limit can not be taken from the exact result offered byDunlap and Kenkre, because the functions found in the Dunlap and Kenkre paper [1] A u and A v are not defined in that limit. Therefore, the Magnus-Floquet approach isnecessary to obtain the dynamic behavior of the particle on the lattice.
3. Conclusions
Based on the perturbative formalism introduced in [14], one obtains an averageHamiltonian equivalent to that obtained from the Magnus expansion. This formalismis very practical to calculate the effective Hamiltonian G on the lattice in the tightbinding approximation. In this sense, we obtained G up to O ( ω − ) of charged particleon a lattice subject to a homogeneous and periodically oscillating electric field at highfrequency. The effective Hamiltonian exhibits the well-known renormalization of thehopping rate by a Bessel function of the first kind, J . This renormalization is known asband narrowing effect [1]. It should be mentioned that this effect has also been obtainedby semi-classical model [28]. Following the time-independent Floquet Hamiltonianapproach, we obtained the wave-function Ψ( t, p ) associated to H ( t ) up to O ( ω − ).Finally, we calculated the mean square displacement for larges time, noting that itis not bounded in time, indicating that the particle does exhibit dynamic localization,independently of the values of a and ǫ . By virtue of this result, it could be stated thatthe effective electric field is inhomogeneous since the Bloch oscillations are not observed.The result obtained completes the limit results shown by Dunlap and Kenkre.
4. Acknowledgments
This work was supported by IVIC under Project No. 1089. [1] D.H. Dunlap and V. M. Kenkre,
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