- Email: [email protected]
Dynamics of a superlattice with an impurity in a dc–ac electric field
7 January 2002
Physics Letters A 292 (2002) 275–280 www.elsevier.com/locate/pla
Dynamics of a superlattice with an impurity in a dc–ac electric field Ai-Zhen Zhang a,b,∗ , Ping Zhang c , Duan Suqing c , Xian-Geng Zhao c , Jiu-Qing Liang a,b a Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China b Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China c Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
Received 22 February 2001; received in revised form 26 September 2001; accepted 21 November 2001 Communicated by A. Lagendijk
Abstract Within a single-band tight-binding model, we investigate the dynamics of a one-dimensional semiconductor superlattice with an impurity driven by a dc–ac electric field. By numerically calculating the quasienergy spectrum and the probability propagators, we find that the dynamic localization condition is not destroyed by the impurity, but the local dynamics of the system is sensitively dependent on the relation of the impurity potential, the Bloch oscillation frequency and the ac field frequency. In appropriate circumstances, the dynamics of the system is dominated by resonant oscillations between the impurity and it nearest-neighbor sites. Such an effect might find important application for generating electromagnetic radiation and systematic study of localization phenomena. 2002 Published by Elsevier Science B.V. PACS: 73.22.-b; 72.15.Rn; 63.20.Kr Keywords: Fractional Wannier–Stark ladders; Dynamic localization; Impurity
1. Introduction Since the development of semiconductor superlattices, the study of dynamic effects of charged carriers in such novel quantum structures subject to timedependent electric fields has attracted increasing attention. In the investigation of electron state evolution in a pure ac field E1 cos(ωt) within the tightbinding approximation, it is found [1] that an ini-
* Corresponding author.
E-mail address: [email protected] (A.-Z. Zhang).
tially localized particle will remain localized when J0 (eaE1/ω) = 0, where e, a and J0 are the electronic charge, lattice constant and zeroth-order Bessel function, respectively. This phenomenon has been called dynamic localization and was later found to be related to the effect of band collapse of a quasienergy miniband [2], which has been verified experimentally in an optical lattice [3]. The effects produced by a dc– ac field E0 + E1 cos(ωt) show more fascinating aspects [4]. It is found that the spectral and dynamical properties depend very sensitively on the matching ratio ωB /ω, where ωB = eaE0 is the frequency of Bloch oscillations under the dc electric field E0 . For an integer matching ratio ωB /ω = n, an ini-
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 7 8 9 - 7
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A.-Z. Zhang et al. / Physics Letters A 292 (2002) 275–280
tially localized particle will remain localized only if Jn (eaE1/ω) = 0 (the pure ac situation is a special case for n = 0). If ωB /ω = p/q with p and q being relatively prime integers, the dynamic localization can occur whatever the value of eaE1 /ω is. The corresponding quasienergies become fractional Wannier–Stark ladders, i.e., a parent band will split into q subbands if the quasienergy is restricted to a Brillouin zone of length ω, which have been theoretically demonstrated to arise in a simulation by using an ultra-cold atom ensemble driven by an accelerated optical potentials [5]. However, since fabrication is always somewhat imperfect, the successive wells are not identical to each other in size, shape and separation, various defects are inevitably introduced into the actual superlattice systems. It is, therefore, essential to investigate the effects of disorder on the dynamics of these systems. This fact has stimulated a great interest in the studies of superlattices which contain different kinds of disorders [6–8]. Recently, a very interesting phenomenon has been discussed by Nazareno et al. [9]. With a dc electric field E0 imposed on a superlattice with an impurity, described within a single-band tight-binding model, they found that if the impurity potential ε0 coincides with multiples of the Bloch frequency, i.e., ε0 = neaE0 , the dynamics of the system is dominated by resonant oscillations between the impurity and the particular site n. In particular when n = 1, resonant oscillations between the impurity and its nearestneighbor site occur. In the present work, we investigate the dynamics of this system under the action of a dc–ac electric field. Exactly as the periodicity in space gives rise to quasimomenta, the periodicity in time leads to quasienergies which play a role similar to that of energies in the time-independent problems. By means of numerically calculating the quasienergy spectrum with the help of the Floquet formalism, we find that the dynamic localization condition is not destroyed by the impurity, but the local dynamics of the system is tuned dramatically by the relation of the impurity potential, the frequency of the Bloch oscillations and the ac field frequency. Especially if appropriate conditions are satisfied, resonant oscillations can take place between the impurity and its two nearest-neighbor sites which is manifested as an avoided crossing in the quasienergy spectrum.
2. Model and method For a single-band tight-binding model of a onedimensional system with an impurity at site 0 in a dc– ac electric field E(t) = E0 + E1 cos(ωt), the Hamiltonian in the Wannier representation can be written as V |nn + 1| + |n + 1n| H= n
+ ε0 |00| − eaE(t)
n|nn|,
(1)
n
where V is the nearest-neighbor intersite hopping matrix element, ε0 is the impurity potential, and |n represents a Wannier state localized at the lattice site n. In our calculations, we choose n = 0, ±1, . . . , ±25, i.e., the size of the lattice is 51. The size is one readily achieved in actual superlattices. The time periodicity of Hamiltonian (1) enables us to describe the system in terms of a quasienergy spectrum [10]. For a numerical calculation of quasienergies it is convenient to introduce the time-evolution operator U (t, 0) which satisfies the equation (we put h¯ = 1 throughout this Letter) i
∂ U (t, 0) = H (t)U (t, 0), ∂t
(2)
with the initial condition U (0, 0) = 1. In combination with the Floquet theorem, the quasienergies can be obtained by numerically integrating Eq. (2) and diagonalizing U (T , 0) with T = 2π/ω. Given that the electron is initially localized at the impurity site |0, we can therefore investigate its subsequent dynamic evolution. By expressing the particle state |φ(t) as a linear combination of Wannier states |n, φ(t) = cn (t)|n, (3) n
where cn (t) are the time-dependent amplitudes n|φ(t), one obtains from Eq. (1) the following evolution equation for the amplitudes cn (t): i
d cn (t) = V cn+1 (t) + cn−1 (t) + ε0 c0 (t)δn,0 dt − neaE(t)cn (t),
(4)
with the initial condition cn (0) = δn,0 .
(5)
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277
By numerically solving Eqs. (4) and (5) with a fourthorder Runge–Kutta integration, we can obtain the probability propagator ψn (t) (= |cn (t)|2 ) for any site n at time t.
3. Results and discussion In the case of a perfect lattice, the interplay of the Bloch frequency ωB and the ac frequency ω enlarges greatly the physics of the field-driven superlattice. When the ac field is tuned to an “n-photon resonance” with the Wannier–Stark ladder, i.e., ωB = nω, the quasimomentum k remains a good quantum number and the quasienergies of the Floquet states are [11] eaE1 ε(k) = (−1)n Jn (6) E(k) mod ω, ω with unperturbed energy-momentum dispersion E(k) = 2V cos(ka). We can see that if eaE1/ω equals a zero of Jn , the quasienergy-quasimomentum dispersion disappears and the initially localized wave packet will be dynamically localized. In the situation of the fractional ratio ωB /ω = p/q, with p and q being relatively prime integers, k is no longer a good quantum number, there appears a dynamic fractional ladder and correspondingly, the initially localized particle will be localized no matter what the value eaE1/ω is [4]. In the following, we will discuss the effects of the impurity on the dynamics of the system in accordance with the two cases. Integer matching ratio. Consider first the particular case of integer matching ratio ωB = ω. Fig. 1 shows the numerical result of the quasienergy band ε/ω as a function of eaE1 /ω, where other parameters are chosen to be ε0 /ω = 0.2 and V /ω = 0.05. We can see because of the influence of the impurity, there is one quasienergy that behaves differently from the rest of the miniband which, with increasing value of eaE1/ω, oscillates in width and collapses at values of eaE1/ω equal to a zero of the Bessel function J1 just as the case of perfect lattice. This one special quasienergy physically is a localized impurity state. So, despite the existence of the impurity Fig. 1 suggests the same dynamic localization condition of a superlattice with an impurity as that of perfect lattice. In order to see clearly the effect of the impurity on the quasienergy spectrum, we plot in Fig. 2 one Brillouin
Fig. 1. Quasienergy band ε/ω plotted versus eaE1 /ω with ε0 /ω = 0.2, ωB /ω = 1 and V /ω = 0.05.
zone −1/2 ε/ω 1/2 of quasienergies versus ε0 /ω with V /ω = 0.05 and dynamic localization condition satisfied eaE1/ω = 3.8317 (the first root of J1 ). It shows in Fig. 2 that avoided crossings between two quasienergies occur at integer values of ε0 /ω. As we shall show below these avoided crossings lead to profound effects on the dynamics of the system. To illustrate the essential dynamic changes at these avoided crossings indicated in Fig. 2, we show in Figs. 3(a)–(c) the time evolution of the probability amplitudes for sites 0, −1 and 1 (respectively represented by ψ0 (t), ψ−1 (t) and ψ1 (t)) for different values of ε0 /ω, where other system parameters are the same as that used in Fig. 2. Fig. 3(a) shows the case for ε0 /ω = 1, corresponding to the first avoided crossing of quasienergies. We can see that the system is dominated by resonant oscillations between the impurity and its two nearest-neighbor sites, which bears analogy to the case under a pure ac electric field [12]. However, with increasing value of ε0 /ω, the tunneling to site 1 is suppressed and the dynamics of the system is dominated by the resonant oscillations between the impurity and its left nearest-neighbor site, as shown in Fig. 3(b) (ε0 /ω = 3) and Fig. 3(c) (ε0/ω = 6), which is different from that of the pure ac case.
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Fig. 2. One Brillouin zone −1/2 ε/ω 1/2 of quasienergies as functions of ε0 /ω with eaE1 /ω = 3.8317, ωB /ω = 1 and V /ω = 0.05.
Fractional matching ratio. Without loss of generality, we choose the value of the ratio ωB /ω = 1/3. The quasienergies as functions of ε0 /ω is shown in Fig. 4 with eaE1 /ω = 3 and V /ω = 0.05. We notice that in addition to the fractional Wannier–Stark ladder, avoided crossings occur at the values of ε0 /ω = 1/3 and 2/3, but without occurring at the value of ε0 /ω = 1. The corresponding time evolution of the probability propagators for sites 0, 1 and −1 for the three values of ε0 /ω are shown in Figs. 5(a)–(c). Fig. 5(a) shows the case for ε0 /ω = 1/3, the particle oscillates between site 0 and its nearest-neighbor site −1, and the tunneling to the site 1 is prohibited entirely. Whereas in Fig. 5(b) for ε0 /ω = 2/3, we can see the resonant oscillations take place between the impurity and its nearest neighbor site 1 and the tunneling to the site −1 is fully prohibited. This oscillation selection between the impurity site and its different nearest neighbors can be understood in terms of one-photon resonance. When ε0 /ω = 1/3, the on-site energy mismatch between the impurity and its nearest neighbor in the left is ω, thus this one-photon resonance induces the strong participation of the site state |−1 in the time evolution of the initially localized state |0,
Fig. 3. ψ0 (t), ψ−1 (t) and ψ1 (t) are plotted as functions of dimensionless time ωt for different values of ε0 /ω with eaE1 /ω = 3.8317, V /ω = 0.05 and ωB /ω = 1, (a) ε0 /ω = 1, (b) ε0 /ω = 3 and (c) ε0 /ω = 6. The solid line corresponds to site 0, the dotted line corresponds to site −1, and the dashed line corresponds to site 1.
leading to the complete resonant oscillations between these two sites. On the other hand, when increasing the value of the impurity potential to satisfy ε0 /ω = 2/3, the on-site energy mismatch between the states |0 and |1 is ω. In this case, we have the full resonance be-
A.-Z. Zhang et al. / Physics Letters A 292 (2002) 275–280
279
Fig. 4. Quasienergies ε/ω as functions of ε0 /ω, where eaE1 /ω = 3, V /ω = 0.05 and ωB /ω = 1/3.
tween the impurity site and its nearest neighbor on the right as shown in Fig. 5(b). The disappearance of the avoided crossing at the value of ε0 /ω = 1 is reflected by the fact that no photon resonance occurs in this situation and, consequently, the initially localized particle will remain localized at its initially occupied site during the time evolution, as shown in Fig. 5(c).
4. Conclusions In summary, within a single-band tight-binding model, by means of the analysis of quasienergies we have studied the effects of an impurity on the dynamics of a superlattice driven by a dc–ac electric field. We find that the dynamic localization condition is not destroyed by the impurity, but the quasienergy spectrum and local dynamics of the system are tuned dramatically by the relation of the impurity potential, the Bloch oscillations frequency, and the ac field frequency. In the case of the integer matching ratio ωB /ω = n, we have the avoided crossings at the values of impurity potential ε0 = nω, which is responsible for resonant oscillations between the impurity and its
Fig. 5. ψ0 (t), ψ−1 (t) and ψ1 (t) are plotted as functions of dimensionless time ωt for different values of ε0 /ω with eaE1 /ω = 3, V /ω = 0.05 and ωB /ω = 1/3, (a) ε0 /ω = 1/3, (b) ε0 /ω = 2/3 and (c) ε0 /ω = 1. The solid line corresponds to site 0, the dotted line corresponds to site −1, and the dashed line corresponds to site 1.
neighboring sites. In the case of a fractional matching ratio ωB /ω = p/q, where p, q are relatively prime numbers, the avoided crossings occur at values of the impurity potential ε0 /ω = np/q when np/q is not an integer and for different values of n, the resonance
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alternatively happens between the impurity and one of its two nearest neighbors. Otherwise, no resonance occurs and the particle is almost localized at its initial site. Briefly, we find that if the dynamic localization condition is satisfied and the system parameters are in certain ratios to each other, the Floquet states of the system undergo a series of avoided crossing, at which remarkable changes in the dynamic behavior of the system occur, as reflected by the resonant oscillations between the impurity and its nearestneighbor sites. We expect the present results are useful in illustrating the dynamics of the impurity-doped superlattices driven by the external electric fields.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grant No. 19725417, the National PAN-DENG Project under grant No. 95-YU-41, and a grant of the China Academy of Engineering and Physics.
References [1] D.H. Dunlap, V.M. Kenkre, Phys. Rev. B 34 (1986) 3625. [2] M. Holthaus, Phys. Rev. Lett. 69 (1992) 351; M. Holthaus, Z. Phys. B 89 (1992) 251. [3] K.W. Madison, M.C. Fischer, R.B. Diener, Q. Niu, M.G. Raizen, Phys. Rev. Lett. 81 (1998) 5093. [4] X.-G. Zhao, Q. Niu, Phys. Lett. A 191 (1994) 181; X.-G. Zhao, R. Jahnke, Q. Niu, Phys. Lett. A 202 (1995) 297. [5] Q. Niu, X.-G. Zhao, G.A. Georgakis, M.G. Raizen, Phys. Rev. Lett. 76 (1996) 4504. [6] D.W. Hone, M. Holthaus, Phys. Rev. B 48 (1993) 15123. [7] Z.-Q. Zhang, X.-D. Jing, Phys. Rev. B 39 (1989) 11428. [8] P.E. de Brito, C.A.A. da Silva, H.N. Nazareno, Phys. Rev. B 51 (1995) 6096. [9] H.N. Nazareno, C.A.A. da Silva, P.E. de Brito, Phys. Rev. B 50 (1994) 4503. [10] J.H. Shirley, Phys. Rev. 138 (1965) B979; V.I. Ritus, Zh. Eksp. Teor. Fiz. 51 (1966) 1544, Sov. Phys. JETP 24 (1967) 1041. [11] M. Holthaus, D.W. Hone, Phys. Rev. B 47 (1993) 6499; J. Zak, Phys. Rev. Lett. 71 (1993) 2623. [12] A.-Z. Zhang, P. Zhang, D. Suqing, X.-G. Zhao, J.-Q. Liang, Phys. Rev. B 63 (2001) 045319.
Physics Letters A 292 (2002) 275–280 www.elsevier.com/locate/pla
Dynamics of a superlattice with an impurity in a dc–ac electric field Ai-Zhen Zhang a,b,∗ , Ping Zhang c , Duan Suqing c , Xian-Geng Zhao c , Jiu-Qing Liang a,b a Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China b Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China c Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China
Received 22 February 2001; received in revised form 26 September 2001; accepted 21 November 2001 Communicated by A. Lagendijk
Abstract Within a single-band tight-binding model, we investigate the dynamics of a one-dimensional semiconductor superlattice with an impurity driven by a dc–ac electric field. By numerically calculating the quasienergy spectrum and the probability propagators, we find that the dynamic localization condition is not destroyed by the impurity, but the local dynamics of the system is sensitively dependent on the relation of the impurity potential, the Bloch oscillation frequency and the ac field frequency. In appropriate circumstances, the dynamics of the system is dominated by resonant oscillations between the impurity and it nearest-neighbor sites. Such an effect might find important application for generating electromagnetic radiation and systematic study of localization phenomena. 2002 Published by Elsevier Science B.V. PACS: 73.22.-b; 72.15.Rn; 63.20.Kr Keywords: Fractional Wannier–Stark ladders; Dynamic localization; Impurity
1. Introduction Since the development of semiconductor superlattices, the study of dynamic effects of charged carriers in such novel quantum structures subject to timedependent electric fields has attracted increasing attention. In the investigation of electron state evolution in a pure ac field E1 cos(ωt) within the tightbinding approximation, it is found [1] that an ini-
* Corresponding author.
E-mail address: [email protected] (A.-Z. Zhang).
tially localized particle will remain localized when J0 (eaE1/ω) = 0, where e, a and J0 are the electronic charge, lattice constant and zeroth-order Bessel function, respectively. This phenomenon has been called dynamic localization and was later found to be related to the effect of band collapse of a quasienergy miniband [2], which has been verified experimentally in an optical lattice [3]. The effects produced by a dc– ac field E0 + E1 cos(ωt) show more fascinating aspects [4]. It is found that the spectral and dynamical properties depend very sensitively on the matching ratio ωB /ω, where ωB = eaE0 is the frequency of Bloch oscillations under the dc electric field E0 . For an integer matching ratio ωB /ω = n, an ini-
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 7 8 9 - 7
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A.-Z. Zhang et al. / Physics Letters A 292 (2002) 275–280
tially localized particle will remain localized only if Jn (eaE1/ω) = 0 (the pure ac situation is a special case for n = 0). If ωB /ω = p/q with p and q being relatively prime integers, the dynamic localization can occur whatever the value of eaE1 /ω is. The corresponding quasienergies become fractional Wannier–Stark ladders, i.e., a parent band will split into q subbands if the quasienergy is restricted to a Brillouin zone of length ω, which have been theoretically demonstrated to arise in a simulation by using an ultra-cold atom ensemble driven by an accelerated optical potentials [5]. However, since fabrication is always somewhat imperfect, the successive wells are not identical to each other in size, shape and separation, various defects are inevitably introduced into the actual superlattice systems. It is, therefore, essential to investigate the effects of disorder on the dynamics of these systems. This fact has stimulated a great interest in the studies of superlattices which contain different kinds of disorders [6–8]. Recently, a very interesting phenomenon has been discussed by Nazareno et al. [9]. With a dc electric field E0 imposed on a superlattice with an impurity, described within a single-band tight-binding model, they found that if the impurity potential ε0 coincides with multiples of the Bloch frequency, i.e., ε0 = neaE0 , the dynamics of the system is dominated by resonant oscillations between the impurity and the particular site n. In particular when n = 1, resonant oscillations between the impurity and its nearestneighbor site occur. In the present work, we investigate the dynamics of this system under the action of a dc–ac electric field. Exactly as the periodicity in space gives rise to quasimomenta, the periodicity in time leads to quasienergies which play a role similar to that of energies in the time-independent problems. By means of numerically calculating the quasienergy spectrum with the help of the Floquet formalism, we find that the dynamic localization condition is not destroyed by the impurity, but the local dynamics of the system is tuned dramatically by the relation of the impurity potential, the frequency of the Bloch oscillations and the ac field frequency. Especially if appropriate conditions are satisfied, resonant oscillations can take place between the impurity and its two nearest-neighbor sites which is manifested as an avoided crossing in the quasienergy spectrum.
2. Model and method For a single-band tight-binding model of a onedimensional system with an impurity at site 0 in a dc– ac electric field E(t) = E0 + E1 cos(ωt), the Hamiltonian in the Wannier representation can be written as V |nn + 1| + |n + 1n| H= n
+ ε0 |00| − eaE(t)
n|nn|,
(1)
n
where V is the nearest-neighbor intersite hopping matrix element, ε0 is the impurity potential, and |n represents a Wannier state localized at the lattice site n. In our calculations, we choose n = 0, ±1, . . . , ±25, i.e., the size of the lattice is 51. The size is one readily achieved in actual superlattices. The time periodicity of Hamiltonian (1) enables us to describe the system in terms of a quasienergy spectrum [10]. For a numerical calculation of quasienergies it is convenient to introduce the time-evolution operator U (t, 0) which satisfies the equation (we put h¯ = 1 throughout this Letter) i
∂ U (t, 0) = H (t)U (t, 0), ∂t
(2)
with the initial condition U (0, 0) = 1. In combination with the Floquet theorem, the quasienergies can be obtained by numerically integrating Eq. (2) and diagonalizing U (T , 0) with T = 2π/ω. Given that the electron is initially localized at the impurity site |0, we can therefore investigate its subsequent dynamic evolution. By expressing the particle state |φ(t) as a linear combination of Wannier states |n, φ(t) = cn (t)|n, (3) n
where cn (t) are the time-dependent amplitudes n|φ(t), one obtains from Eq. (1) the following evolution equation for the amplitudes cn (t): i
d cn (t) = V cn+1 (t) + cn−1 (t) + ε0 c0 (t)δn,0 dt − neaE(t)cn (t),
(4)
with the initial condition cn (0) = δn,0 .
(5)
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By numerically solving Eqs. (4) and (5) with a fourthorder Runge–Kutta integration, we can obtain the probability propagator ψn (t) (= |cn (t)|2 ) for any site n at time t.
3. Results and discussion In the case of a perfect lattice, the interplay of the Bloch frequency ωB and the ac frequency ω enlarges greatly the physics of the field-driven superlattice. When the ac field is tuned to an “n-photon resonance” with the Wannier–Stark ladder, i.e., ωB = nω, the quasimomentum k remains a good quantum number and the quasienergies of the Floquet states are [11] eaE1 ε(k) = (−1)n Jn (6) E(k) mod ω, ω with unperturbed energy-momentum dispersion E(k) = 2V cos(ka). We can see that if eaE1/ω equals a zero of Jn , the quasienergy-quasimomentum dispersion disappears and the initially localized wave packet will be dynamically localized. In the situation of the fractional ratio ωB /ω = p/q, with p and q being relatively prime integers, k is no longer a good quantum number, there appears a dynamic fractional ladder and correspondingly, the initially localized particle will be localized no matter what the value eaE1/ω is [4]. In the following, we will discuss the effects of the impurity on the dynamics of the system in accordance with the two cases. Integer matching ratio. Consider first the particular case of integer matching ratio ωB = ω. Fig. 1 shows the numerical result of the quasienergy band ε/ω as a function of eaE1 /ω, where other parameters are chosen to be ε0 /ω = 0.2 and V /ω = 0.05. We can see because of the influence of the impurity, there is one quasienergy that behaves differently from the rest of the miniband which, with increasing value of eaE1/ω, oscillates in width and collapses at values of eaE1/ω equal to a zero of the Bessel function J1 just as the case of perfect lattice. This one special quasienergy physically is a localized impurity state. So, despite the existence of the impurity Fig. 1 suggests the same dynamic localization condition of a superlattice with an impurity as that of perfect lattice. In order to see clearly the effect of the impurity on the quasienergy spectrum, we plot in Fig. 2 one Brillouin
Fig. 1. Quasienergy band ε/ω plotted versus eaE1 /ω with ε0 /ω = 0.2, ωB /ω = 1 and V /ω = 0.05.
zone −1/2 ε/ω 1/2 of quasienergies versus ε0 /ω with V /ω = 0.05 and dynamic localization condition satisfied eaE1/ω = 3.8317 (the first root of J1 ). It shows in Fig. 2 that avoided crossings between two quasienergies occur at integer values of ε0 /ω. As we shall show below these avoided crossings lead to profound effects on the dynamics of the system. To illustrate the essential dynamic changes at these avoided crossings indicated in Fig. 2, we show in Figs. 3(a)–(c) the time evolution of the probability amplitudes for sites 0, −1 and 1 (respectively represented by ψ0 (t), ψ−1 (t) and ψ1 (t)) for different values of ε0 /ω, where other system parameters are the same as that used in Fig. 2. Fig. 3(a) shows the case for ε0 /ω = 1, corresponding to the first avoided crossing of quasienergies. We can see that the system is dominated by resonant oscillations between the impurity and its two nearest-neighbor sites, which bears analogy to the case under a pure ac electric field [12]. However, with increasing value of ε0 /ω, the tunneling to site 1 is suppressed and the dynamics of the system is dominated by the resonant oscillations between the impurity and its left nearest-neighbor site, as shown in Fig. 3(b) (ε0 /ω = 3) and Fig. 3(c) (ε0/ω = 6), which is different from that of the pure ac case.
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Fig. 2. One Brillouin zone −1/2 ε/ω 1/2 of quasienergies as functions of ε0 /ω with eaE1 /ω = 3.8317, ωB /ω = 1 and V /ω = 0.05.
Fractional matching ratio. Without loss of generality, we choose the value of the ratio ωB /ω = 1/3. The quasienergies as functions of ε0 /ω is shown in Fig. 4 with eaE1 /ω = 3 and V /ω = 0.05. We notice that in addition to the fractional Wannier–Stark ladder, avoided crossings occur at the values of ε0 /ω = 1/3 and 2/3, but without occurring at the value of ε0 /ω = 1. The corresponding time evolution of the probability propagators for sites 0, 1 and −1 for the three values of ε0 /ω are shown in Figs. 5(a)–(c). Fig. 5(a) shows the case for ε0 /ω = 1/3, the particle oscillates between site 0 and its nearest-neighbor site −1, and the tunneling to the site 1 is prohibited entirely. Whereas in Fig. 5(b) for ε0 /ω = 2/3, we can see the resonant oscillations take place between the impurity and its nearest neighbor site 1 and the tunneling to the site −1 is fully prohibited. This oscillation selection between the impurity site and its different nearest neighbors can be understood in terms of one-photon resonance. When ε0 /ω = 1/3, the on-site energy mismatch between the impurity and its nearest neighbor in the left is ω, thus this one-photon resonance induces the strong participation of the site state |−1 in the time evolution of the initially localized state |0,
Fig. 3. ψ0 (t), ψ−1 (t) and ψ1 (t) are plotted as functions of dimensionless time ωt for different values of ε0 /ω with eaE1 /ω = 3.8317, V /ω = 0.05 and ωB /ω = 1, (a) ε0 /ω = 1, (b) ε0 /ω = 3 and (c) ε0 /ω = 6. The solid line corresponds to site 0, the dotted line corresponds to site −1, and the dashed line corresponds to site 1.
leading to the complete resonant oscillations between these two sites. On the other hand, when increasing the value of the impurity potential to satisfy ε0 /ω = 2/3, the on-site energy mismatch between the states |0 and |1 is ω. In this case, we have the full resonance be-
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279
Fig. 4. Quasienergies ε/ω as functions of ε0 /ω, where eaE1 /ω = 3, V /ω = 0.05 and ωB /ω = 1/3.
tween the impurity site and its nearest neighbor on the right as shown in Fig. 5(b). The disappearance of the avoided crossing at the value of ε0 /ω = 1 is reflected by the fact that no photon resonance occurs in this situation and, consequently, the initially localized particle will remain localized at its initially occupied site during the time evolution, as shown in Fig. 5(c).
4. Conclusions In summary, within a single-band tight-binding model, by means of the analysis of quasienergies we have studied the effects of an impurity on the dynamics of a superlattice driven by a dc–ac electric field. We find that the dynamic localization condition is not destroyed by the impurity, but the quasienergy spectrum and local dynamics of the system are tuned dramatically by the relation of the impurity potential, the Bloch oscillations frequency, and the ac field frequency. In the case of the integer matching ratio ωB /ω = n, we have the avoided crossings at the values of impurity potential ε0 = nω, which is responsible for resonant oscillations between the impurity and its
Fig. 5. ψ0 (t), ψ−1 (t) and ψ1 (t) are plotted as functions of dimensionless time ωt for different values of ε0 /ω with eaE1 /ω = 3, V /ω = 0.05 and ωB /ω = 1/3, (a) ε0 /ω = 1/3, (b) ε0 /ω = 2/3 and (c) ε0 /ω = 1. The solid line corresponds to site 0, the dotted line corresponds to site −1, and the dashed line corresponds to site 1.
neighboring sites. In the case of a fractional matching ratio ωB /ω = p/q, where p, q are relatively prime numbers, the avoided crossings occur at values of the impurity potential ε0 /ω = np/q when np/q is not an integer and for different values of n, the resonance
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alternatively happens between the impurity and one of its two nearest neighbors. Otherwise, no resonance occurs and the particle is almost localized at its initial site. Briefly, we find that if the dynamic localization condition is satisfied and the system parameters are in certain ratios to each other, the Floquet states of the system undergo a series of avoided crossing, at which remarkable changes in the dynamic behavior of the system occur, as reflected by the resonant oscillations between the impurity and its nearestneighbor sites. We expect the present results are useful in illustrating the dynamics of the impurity-doped superlattices driven by the external electric fields.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China under grant No. 19725417, the National PAN-DENG Project under grant No. 95-YU-41, and a grant of the China Academy of Engineering and Physics.
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