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Rabi oscillations between Bloch bands beyond weak interband coupling regime
Physica B 291 (2000) 299 }306
Rabi oscillations between Bloch bands beyond weak interband coupling regime Wei Zhang*, X.-G. Zhao LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009 (26), Beijing 100088, People's Republic of China Received 3 January 2000
Abstract Within a two-band tight-binding model, we investigate Rabi oscillations of electrons in superlattices driven by electric "elds beyond the weak interband coupling regime. By the analytical and numerical studies, conditions for Rabi oscillations and oscillation period are obtained. A picture showing the in#uence of electric "elds on the dynamics of Bloch electrons is given. The possibility of experimental observations is discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 71.70.Ej; 73.40.Gk; 73.20.Dx Keywords: Rabi oscillation; Interband coupling
1. Introduction Recently, there are a lot of theoretical and experimental researches on semiconductor superlattices [1], which provide us the possibility of testing quantum mechanical properties of Bloch electrons [2]. The main features of superlattices, large lattice constant and narrow `minibanda, make it easier to observe many quantum transport phenomena, for example, Bloch oscillations. One of the important aspects of the dynamics of Bloch electrons is their behavior in the superlattices driven by electric "elds. In these cases, people "nd some new phe-
* Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (X.-G. Zhao).
nomena, such as negative di!erential conductivity [3], fractional Wannier}Stark ladders [4], dynamical localization [5}9], band suppression [10,11], band collapse [12], etc. Most of the researches on the e!ects of the electric "elds are based on the single-band approximation. In some situation, for instance, when the frequency of electric "elds is high enough (photo energy is much larger than energy gap), the interband transition becomes important. Many interesting e!ects will be induced by the interband tunneling [13}16]. Zhao et al. studied the coherent interband transitions or Rabi oscillations in the weak interband coupling regime [17]. They found that Rabi oscillations can occur under some speci"c conditions. Their results were experimentally veri"ed by the use of ultra-cold atoms in an accelerating optical lattice, which can mimic the status of Bloch electrons in external electric "elds [18].
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 2 8 2 - 9
300
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
In the present paper, we will consider Rabi oscillations beyond the weak interband coupling regime. Our analytical and numerical results will show that when the interband coupling is strong enough, Rabi oscillations always happen; when the interband coupling decreases to the intermediate regime, Rabi oscillations can be quenched under some conditions. Some issues on the experimental observation of the above e!ects are addressed.
2. Theoretical treatment We consider the two-band tight-binding model with nearest-neighbor interaction in an external "eld E(t)"E #E cos(ut). The Hamiltonian can be written as H(t)"D a>a #D b>b ? K K @ K K K K ! R (a>a #h.c.) ? K K> K #R (b>b #h.c.) @ K K> K # edE(t) m(a>a #b>b ) K K K K K #kE(t) (a>b #h.c.). K K K
* a(k, t)"+D !2R cos [kd#A(t)],a(k, t) ? ? *t #kE(t)b(k, t),
* b(k, t)"+D #2R cos [kd#A(t)],b(k, t) @ @ *t #kE(t)a(k, t),
(1)
(2)
where A(t)"u t#(edE /u) sin(ut) and u " edE is the Bloch frequency. We assume that the initial condition is a(k, 0)"1, b(k, 0)"0, and rewrite the above equations in the dimensionless form i
* k edE(q) a(k, q)" (k, q)a(k, q)# b(k, q), ? *q ed u
i
k edE(q) * b(k, q)" (k, q)b(k, q)# a(k, q), @ ed u *q
(3)
where the dimensionless time q is de"ned by q" ut, and (k, q)"D /u!2(R /u) cos [kd#A(q)], ? ? ?
(k, q)"D /u#2(R /u) cos [kd#A(q)]. For the @ @ @ physical situation under consideration, we have " ";1 and " ";1, which are the consequences ? @ of narrow minibands and high frequency of the applied electric "eld. When the e!ective interband coupling kE /u or kE /u is large enough, the terms containing
a(k, q) and b(k, q) in Eq. (3) can be neglected. ? @ Thus, we obtain the "rst-order approximate solution of Eq. (3) as follows:
a(k, q)"cos
The integer m labels the lattice sites, and a (b ) is K K the destruction operator for band a (b); D and ? D are the site energies belonging to the two bands; @ e and d are, respectively, the electronic charge and the lattice period; R and R are the nearest-neigh? @ bor hopping matrix elements and k describes the interband coupling. The probability amplitudes a(k, t) for band a and b(k, t) for band b are determined by the following SchroK dinger equations in the momentum k space i
i
k A(q) , ed
b(k, q)"!i sin
k A(q) . ed
(4)
The probability P for the Bloch electron staying in ? band a is
k u edE sin(q) P (k, q)"cos q# ? ed u u
.
(5)
For the matching ratio ku /edu"p/q, P (k, q# ? q ) 2p)"P (k, q). When u "0, P (k, q#p)" ? ? P (k, q). That is, the period of Rabi oscillations ? ¹ is p. (When it is written in the dimensional form, 0 ¹ "¹, where ¹"2p/u is the period of the ap0 plied electric "eld.) When the interband coupling kE /u (or kE /u) decreases, the higher order correction from terms
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
a(k, q) and b(k, q) is necessary. By some kind of ? @ gauge transformation, we can show that
a(k, q)
p "exp+ig(k, q), exp !ip W 4 b(k, q)
;exp i
"
a(k, 0)
b(k, 0)
,
O
(6)
(7)
* ;(k, q)"H (k, q);(k, q) ' *q
(8)
with the initial condition ;(k, 0)"I. H (k, q) is ' de"ned by H (k, q)"X(k, q)p #>(k, q)p , ' V W where
>(k, q)" (k, q) sin 2
k A(q) , ed
and
(9)
k A(q) ed
(10)
1 (D !D ) (R #R ) @ ? # ? @ cos[kd#A(q)].
(k, q)" 2 u u (11) Under the conditions D,"D !D ";u and @ ? R,"R #R ";u, the solution of Eq. (8) can be ? @ given by ;(k, q)"exp+!i[XK (k, q)p #>K (k, q)p ], V W with XK and >K being de"ned by
XK (k, q)"
"
O O
dq X(k, q )
dq (k, q ) cos 2
(13)
>K (k, q)"r(k, q) sin[B(k, q)].
(14)
After some algebraic calculation, one can arrive at the following result:
dq cos(kd#A(q ))
X(k, q)" (k, q) cos 2
k dq (k, q ) sin 2 A(q ) . ed
k P (k, q)"cos[r(k, q)] cos A(q) ? ed
and the matrix ;(k, q) satis"es the SchroK dinger equation in the operator form i
O
dq >(k, q )
XK (k, q)"r(k, q) cos[B(k, q)],
1 (D #D ) (R !R ) ? @ q# ? @ g(k, q)" 2 u u ;
O
We rewrite XK (k, q) and >K (k, q) as
k A(q)p ;(k, q) X ed
p ;exp ip W4 where
>K (k, q)"
301
k A(q ) , ed
(12)
k #sin[r(k, q)] cos B(k, q)# A(q) , ed
k P (k, q)"cos[r(k, q)] sin A(q) @ ed
k #sin[r(k, q)] sin B(k, q)# A(q) . ed (15) When the interband coupling is strong enough, one has XK ,>K &0. This leads to strong coupling result (5). When the interband coupling is in the intermediate regime, we expect that the Rabi oscillations will be the dominant behavior in contrast to the situation with the small interband coupling. Under some speci"c conditions, Rabi oscillations may be quenched, i.e., the electron stays in one band during the whole driving process. From the fact that in general r(k, q)"cq#2 and A(q)"(u /u)q#(edE /u) sin(q), a necessary quenching condition (P ;1) is u "0. When @ u "0, sin[(k/ed)A(q)] is small in the intermediate regime, we have k P &sin[r(k, q)] cos A(q) sin[B(k, q)]. (16) @ ed
So the quenching conditions are u "0, and B(k, q)"0. We now come to calculate B(k, q). When u "0, X(k, q) and >(k, q) are periodic, and therefore have the Fourier expansions XK (k, q)"C q# (C cos(nq)#CI sin(nq)), V LV LV L >K (k, q)"C q# (C cos(nq)#CI sin(nq)). (17) W LW LW L
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W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
The long time behavior of XK and >K is dominated by the linear term in q. In this case we obtain
where J is the 0th order Bessel function. Thus, the quenching conditions are
XK "C q, V
u "0,
>K "C q W
(18)
and
B(k, q)"arctan(C /C ). W V
(19)
C and C can be calculated as V W p C " dqX(k, q) V
1 k edE 1 # R cos(kd) " *J 2 2 ed u 2
C " W
p
2k edE #J #1 ed u
2k edE !1 ed u
,
dq>(k, q)
1 " R sin(kd) J 2 !J
2k edE !1 ed u
2k edE #1 ed u
2k edE !J !1 ed u
2k edE "0. #1 ed u (21)
When B(k,q)O0, cos[(k/ed)A(q)]&1 (in the intermediate regime)
r(k, q)"(C #C q"r q, V W
; J
J
,
(20)
P &sin(r q) sin(B) @ 1 & sin(B)(1!cos(2r q)). 2 The period of Rabi oscillation ¹ is 0 ¹ /¹"u/2r . 0
(22)
(23)
3. Discussion Before going to the discussions of some physical issues, let us give some numerical results to check our analytical treatment. We "rst compare our analytical results (15) with the exact results, which are obtained by numerical method, in the strong coupling regime (kE /u or kE /u'1). Figs. 1 and (corresponding to the exact numerical solution) 2 (corresponding to the analytical results from
Fig. 1. The probability for the electron staying in band a in strong coupling regime kE /u"1.5 (exact solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"0, edE /u"6.0, k/ed"0.25, k"p/2. ? @ ? @
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
303
Fig. 2. The probability for the electron staying in band a in strong coupling regime kE /u"1.5 (analytical solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"0, edE /u"6.0, k/ed"0.25, k"p/2. ? @ ? @
Eq. (15)) show the probability for the electron staying in band a. One can see that the analytical results "t quite well with the exact results in the strong coupling regime (kE /u"1.5). From Figs. 1 and 2, one can also see that the period for Rabi oscillations is ¹/2 (for u "0), which coincides with the analytical result. Next, we check our analytical treatment in the intermediate coupling regime (kE /u or kE /u&
0.5) by Figs. 3 and 5 (exact numerical results) and Figs. 4 and 6 (analytical results). In Figs. 3 and 4, the dashed lines correspond to quenched state. We choose the parameters so that interband coupling is in the intermediate regime (kE /u"0.5) and the quenching conditions (21) are satis"ed. While the parameters of solid lines are chosen so that the interband coupling is in the intermediate regime (kE /u"0.77) and the quenching conditions
Fig. 3. The probability for the electron staying in band a in intermediate coupling regime (exact solution). The parameters for the superlattice are D /u"0.1, D /u"0.4, R "R "0.05, k/ed"0.129. The parameters for the dashed line (corresponding to quenched ? @ ? @ state) are chosen as edE /u"0, edE /u"3.88, k"p/2 such that quenching conditions are satis"ed. The parameters for the solid line (corresponding to oscillated state) are chosen as edE /u"0, edE /u"6.0, k"p/2 such that quenching conditions are not satis"ed.
304
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
Fig. 4. The probability for the electron staying in band a in intermediate coupling regime (analytical solution). The parameters for the superlattice are D /u"0.1, D /u"0.4, R "R "0.05, k/ed"0.129. The parematers for the dashed line (corresponding to quenched ? @ ? @ state) are chosen as edE /u"0, edE /u"3.88, k"p/2 such that quenching conditions are satis"ed. The parameters for the solid line (corresponding to oscillated state) are chosen as edE /u"0, edE /u"6.0, k"p/2 such that quenching conditions are not satis"ed.
Fig. 5. The probability for the electron staying in band a in the presence of AC}DC "elds (in intermediate coupling regime) (exact solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"2.5, ? @ ? @ edE /u"3.88, k/ed"0.129, k"p/2.
are not satis"ed. Figs. 3 and 4 show that Rabi oscillations do happen in this case and the analytical results "t well with the exact results, except that there are small shifts in the positions of some valleys and peaks. It is due to the perturbative nature of our criteria (21) that the probabilities P s (for ? solid lines) do not reach zero. From the parameters of Figs. 3 and 4, we can obtain the period of Rabi
oscillations ¹ /¹"6.0 from Eqs. (19)}(23). This is 0 almost the same as that shown in Figs. 3 and 4. Figs. 5 and 6 show the behavior of the electrons in the presence of AC}DC "elds. Rabi oscillations are realized just as our theoretical prediction. Combining our results with those obtained by Zhao et al. [17] leads to a nice picture of the dynamics of electrons in superlattices driven by
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
305
Fig. 6. The probability for the electron staying in band a in the presence of AC}DC "elds (in intermediate coupling regime) (analytical solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"2.5, ? @ ? @ edE /u"3.88, k/ed"0.129, k"p/2.
electric "elds, which is consistent with our physical intuition. We can see that when the e!ective interband coupling kE/u&0, there is little chance to realize Rabi oscillations. With the increase of kE/u, Rabi oscillations become possible. From the results of Ref. [17], when kE/u is small, the dominant behavior of the electron is quenching in one band, and Rabi oscillations will happen only under some speci"c conditions. In other words, the probability to realize quenched states is much bigger than that of oscillated states. When the interband coupling increases further to the intermediate regime kE/u&0.5, Rabi oscillations become dominant. Quenched state can be obtained under some speci"c conditions (21). When interband coupling is strong enough, say kE/u'1.0, Rabi oscillations always happen. Finally, we brie#y discuss the possibility of experimental observation of our theoretical predictions. As discussed in Ref. [17], we may choose a system of superlattices whose unit cell consists of two quantum wells. By taking l/d"0.72, we can get k/ed"0.129, where l is the total width of a double well and the coupling dipole moment is given by k"16el/9p. To obtain a system of narrow bands and small band gap, we take the superlattice parameters as D"D !D "6 meV, R "R " @ ? ? @ 1 meV (i.e., the band widths are 4R "4R " ? @
4 meV). We apply on the superlattice a high-power terahertz AC "eld with frequency u/2p" 4.6 THz (20 meV), which can be generated in a free-electron laser, with tunable frequency from 120 GHz to 4.8 THz [19]. To probe the results in Figs. 3 and 4, we can take the "eld strength as E "7.7;10 V/m (for dashed lines) or E "1.2;10 V/m (for solid lines). Of course, in a real system, there are also incoherent e!ects due to lattice imperfection, which prevent the observation of Rabi oscillations. However, for a highquality superlattice, such e!ects should be small. Therefore one may still observe Rabi #ops (not full Rabi oscillations) in those systems [20]. In summary, we have considered Rabi oscillations of the electrons in superlattices driven by electric "elds beyond weak interband coupling regime. The analytic and numerical results supply us a clear picture of the behavior of Bloch electrons in the presence of strong interband tunneling.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant No. 19725417.
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W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
L. Esaki, R. Tsu, IBM J. Res. Dev. 14 (1970) 61. E.E. Mendez, G. Bastard, Phys. Today 46 (1993) 34. R. Tsu, G.H. Dohler, Phys. Rev. B 12 (1975) 680. X.-G. Zhao, R. Jahnke, Q. Niu, Phys. Lett. A 202 (1995) 297. D.H. Dunlap, V.M. Kenkre, Phys. Rev. B 34 (1986) 3625. D.H. Dunlap, V.M. Kenkre, Phys. Lett. A 127 (1988) 438. X.-G. Zhao, Phys. Lett. A 155 (1991) 299. X.-G. Zhao, Phys. Lett. A 167 (1992) 291. B.J. Keay, S. Zeuner, S.J. Allen Jr., K.D. Maranowski, A.C. Gossard, U. Bhattacharya, M.J.W. Rodwell, Phys. Rev. Lett. 75 (1995) 4102. [10] X.-G. Zhao, J. Phys.: Condens. Matter 6 (1994) 2751. [11] K.W. Madison, M.C. Fischer, P.B. Diener, Q. Niu, M.G. Raizen, Phys. Rev. Lett. 81 (1998) 5093.
[12] M. Holthaus, Phys. Rev. Lett. 69 (1992) 351. [13] X.-G. Zhao, Q. Niu, Phys. Lett. A 222 (1996) 435. [14] J. Rotvig, A.-P. Jauho, H. Smith, Phys. Rev. Lett. 74 (1995) 1831. [15] J. Rotvig, A.-P. Jauho, H. Smith, Phys. Rev. B 54 (1996) 17 691. [16] X.-G. Zhao, W.-X. Yan, D.W. Hone, Phys. Rev. B 57 (1998) 9849. [17] X.-G. Zhao, G.A. Georgakis, Q. Niu, Phys. Rev. B 54 (1996) R5235. [18] M.C. Fischer, K.W. Madison, Q. Niu, M.G. Raizen, Phys. Rev. A 58 (1998) R2648. [19] H. Drexler, J.S. Scott, S.J. Allen, K.L. Campman, A.C. Gossard, Appl. Phys. Lett. 67 (1995) 2816. [20] S.-Q. Duan, W. Zhang, X.-G. Zhao, Rabi oscillations between dissipative Bloch bands, unpublished.
Rabi oscillations between Bloch bands beyond weak interband coupling regime Wei Zhang*, X.-G. Zhao LCP, Institute of Applied Physics and Computational Mathematics, P.O. Box 8009 (26), Beijing 100088, People's Republic of China Received 3 January 2000
Abstract Within a two-band tight-binding model, we investigate Rabi oscillations of electrons in superlattices driven by electric "elds beyond the weak interband coupling regime. By the analytical and numerical studies, conditions for Rabi oscillations and oscillation period are obtained. A picture showing the in#uence of electric "elds on the dynamics of Bloch electrons is given. The possibility of experimental observations is discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 71.70.Ej; 73.40.Gk; 73.20.Dx Keywords: Rabi oscillation; Interband coupling
1. Introduction Recently, there are a lot of theoretical and experimental researches on semiconductor superlattices [1], which provide us the possibility of testing quantum mechanical properties of Bloch electrons [2]. The main features of superlattices, large lattice constant and narrow `minibanda, make it easier to observe many quantum transport phenomena, for example, Bloch oscillations. One of the important aspects of the dynamics of Bloch electrons is their behavior in the superlattices driven by electric "elds. In these cases, people "nd some new phe-
* Corresponding author. E-mail addresses: [email protected] (W. Zhang), [email protected] (X.-G. Zhao).
nomena, such as negative di!erential conductivity [3], fractional Wannier}Stark ladders [4], dynamical localization [5}9], band suppression [10,11], band collapse [12], etc. Most of the researches on the e!ects of the electric "elds are based on the single-band approximation. In some situation, for instance, when the frequency of electric "elds is high enough (photo energy is much larger than energy gap), the interband transition becomes important. Many interesting e!ects will be induced by the interband tunneling [13}16]. Zhao et al. studied the coherent interband transitions or Rabi oscillations in the weak interband coupling regime [17]. They found that Rabi oscillations can occur under some speci"c conditions. Their results were experimentally veri"ed by the use of ultra-cold atoms in an accelerating optical lattice, which can mimic the status of Bloch electrons in external electric "elds [18].
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 2 8 2 - 9
300
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
In the present paper, we will consider Rabi oscillations beyond the weak interband coupling regime. Our analytical and numerical results will show that when the interband coupling is strong enough, Rabi oscillations always happen; when the interband coupling decreases to the intermediate regime, Rabi oscillations can be quenched under some conditions. Some issues on the experimental observation of the above e!ects are addressed.
2. Theoretical treatment We consider the two-band tight-binding model with nearest-neighbor interaction in an external "eld E(t)"E #E cos(ut). The Hamiltonian can be written as H(t)"D a>a #D b>b ? K K @ K K K K ! R (a>a #h.c.) ? K K> K #R (b>b #h.c.) @ K K> K # edE(t) m(a>a #b>b ) K K K K K #kE(t) (a>b #h.c.). K K K
* a(k, t)"+D !2R cos [kd#A(t)],a(k, t) ? ? *t #kE(t)b(k, t),
* b(k, t)"+D #2R cos [kd#A(t)],b(k, t) @ @ *t #kE(t)a(k, t),
(1)
(2)
where A(t)"u t#(edE /u) sin(ut) and u " edE is the Bloch frequency. We assume that the initial condition is a(k, 0)"1, b(k, 0)"0, and rewrite the above equations in the dimensionless form i
* k edE(q) a(k, q)" (k, q)a(k, q)# b(k, q), ? *q ed u
i
k edE(q) * b(k, q)" (k, q)b(k, q)# a(k, q), @ ed u *q
(3)
where the dimensionless time q is de"ned by q" ut, and (k, q)"D /u!2(R /u) cos [kd#A(q)], ? ? ?
(k, q)"D /u#2(R /u) cos [kd#A(q)]. For the @ @ @ physical situation under consideration, we have " ";1 and " ";1, which are the consequences ? @ of narrow minibands and high frequency of the applied electric "eld. When the e!ective interband coupling kE /u or kE /u is large enough, the terms containing
a(k, q) and b(k, q) in Eq. (3) can be neglected. ? @ Thus, we obtain the "rst-order approximate solution of Eq. (3) as follows:
a(k, q)"cos
The integer m labels the lattice sites, and a (b ) is K K the destruction operator for band a (b); D and ? D are the site energies belonging to the two bands; @ e and d are, respectively, the electronic charge and the lattice period; R and R are the nearest-neigh? @ bor hopping matrix elements and k describes the interband coupling. The probability amplitudes a(k, t) for band a and b(k, t) for band b are determined by the following SchroK dinger equations in the momentum k space i
i
k A(q) , ed
b(k, q)"!i sin
k A(q) . ed
(4)
The probability P for the Bloch electron staying in ? band a is
k u edE sin(q) P (k, q)"cos q# ? ed u u
.
(5)
For the matching ratio ku /edu"p/q, P (k, q# ? q ) 2p)"P (k, q). When u "0, P (k, q#p)" ? ? P (k, q). That is, the period of Rabi oscillations ? ¹ is p. (When it is written in the dimensional form, 0 ¹ "¹, where ¹"2p/u is the period of the ap0 plied electric "eld.) When the interband coupling kE /u (or kE /u) decreases, the higher order correction from terms
W. Zhang, X.-G. Zhao / Physica B 291 (2000) 299 }306
a(k, q) and b(k, q) is necessary. By some kind of ? @ gauge transformation, we can show that
a(k, q)
p "exp+ig(k, q), exp !ip W 4 b(k, q)
;exp i
"
a(k, 0)
b(k, 0)
,
O
(6)
(7)
* ;(k, q)"H (k, q);(k, q) ' *q
(8)
with the initial condition ;(k, 0)"I. H (k, q) is ' de"ned by H (k, q)"X(k, q)p #>(k, q)p , ' V W where
>(k, q)" (k, q) sin 2
k A(q) , ed
and
(9)
k A(q) ed
(10)
1 (D !D ) (R #R ) @ ? # ? @ cos[kd#A(q)].
(k, q)" 2 u u (11) Under the conditions D,"D !D ";u and @ ? R,"R #R ";u, the solution of Eq. (8) can be ? @ given by ;(k, q)"exp+!i[XK (k, q)p #>K (k, q)p ], V W with XK and >K being de"ned by
XK (k, q)"
"
O O
dq X(k, q )
dq (k, q ) cos 2
(13)
>K (k, q)"r(k, q) sin[B(k, q)].
(14)
After some algebraic calculation, one can arrive at the following result:
dq cos(kd#A(q ))
X(k, q)" (k, q) cos 2
k dq (k, q ) sin 2 A(q ) . ed
k P (k, q)"cos[r(k, q)] cos A(q) ? ed
and the matrix ;(k, q) satis"es the SchroK dinger equation in the operator form i
O
dq >(k, q )
XK (k, q)"r(k, q) cos[B(k, q)],
1 (D #D ) (R !R ) ? @ q# ? @ g(k, q)" 2 u u ;
O
We rewrite XK (k, q) and >K (k, q) as
k A(q)p ;(k, q) X ed
p ;exp ip W4 where
>K (k, q)"
301
k A(q ) , ed
(12)
k #sin[r(k, q)] cos B(k, q)# A(q) , ed
k P (k, q)"cos[r(k, q)] sin A(q) @ ed
k #sin[r(k, q)] sin B(k, q)# A(q) . ed (15) When the interband coupling is strong enough, one has XK ,>K &0. This leads to strong coupling result (5). When the interband coupling is in the intermediate regime, we expect that the Rabi oscillations will be the dominant behavior in contrast to the situation with the small interband coupling. Under some speci"c conditions, Rabi oscillations may be quenched, i.e., the electron stays in one band during the whole driving process. From the fact that in general r(k, q)"cq#2 and A(q)"(u /u)q#(edE /u) sin(q), a necessary quenching condition (P ;1) is u "0. When @ u "0, sin[(k/ed)A(q)] is small in the intermediate regime, we have k P &sin[r(k, q)] cos A(q) sin[B(k, q)]. (16) @ ed
So the quenching conditions are u "0, and B(k, q)"0. We now come to calculate B(k, q). When u "0, X(k, q) and >(k, q) are periodic, and therefore have the Fourier expansions XK (k, q)"C q# (C cos(nq)#CI sin(nq)), V LV LV L >K (k, q)"C q# (C cos(nq)#CI sin(nq)). (17) W LW LW L
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The long time behavior of XK and >K is dominated by the linear term in q. In this case we obtain
where J is the 0th order Bessel function. Thus, the quenching conditions are
XK "C q, V
u "0,
>K "C q W
(18)
and
B(k, q)"arctan(C /C ). W V
(19)
C and C can be calculated as V W p C " dqX(k, q) V
1 k edE 1 # R cos(kd) " *J 2 2 ed u 2
C " W
p
2k edE #J #1 ed u
2k edE !1 ed u
,
dq>(k, q)
1 " R sin(kd) J 2 !J
2k edE !1 ed u
2k edE #1 ed u
2k edE !J !1 ed u
2k edE "0. #1 ed u (21)
When B(k,q)O0, cos[(k/ed)A(q)]&1 (in the intermediate regime)
r(k, q)"(C #C q"r q, V W
; J
J
,
(20)
P &sin(r q) sin(B) @ 1 & sin(B)(1!cos(2r q)). 2 The period of Rabi oscillation ¹ is 0 ¹ /¹"u/2r . 0
(22)
(23)
3. Discussion Before going to the discussions of some physical issues, let us give some numerical results to check our analytical treatment. We "rst compare our analytical results (15) with the exact results, which are obtained by numerical method, in the strong coupling regime (kE /u or kE /u'1). Figs. 1 and (corresponding to the exact numerical solution) 2 (corresponding to the analytical results from
Fig. 1. The probability for the electron staying in band a in strong coupling regime kE /u"1.5 (exact solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"0, edE /u"6.0, k/ed"0.25, k"p/2. ? @ ? @
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Fig. 2. The probability for the electron staying in band a in strong coupling regime kE /u"1.5 (analytical solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"0, edE /u"6.0, k/ed"0.25, k"p/2. ? @ ? @
Eq. (15)) show the probability for the electron staying in band a. One can see that the analytical results "t quite well with the exact results in the strong coupling regime (kE /u"1.5). From Figs. 1 and 2, one can also see that the period for Rabi oscillations is ¹/2 (for u "0), which coincides with the analytical result. Next, we check our analytical treatment in the intermediate coupling regime (kE /u or kE /u&
0.5) by Figs. 3 and 5 (exact numerical results) and Figs. 4 and 6 (analytical results). In Figs. 3 and 4, the dashed lines correspond to quenched state. We choose the parameters so that interband coupling is in the intermediate regime (kE /u"0.5) and the quenching conditions (21) are satis"ed. While the parameters of solid lines are chosen so that the interband coupling is in the intermediate regime (kE /u"0.77) and the quenching conditions
Fig. 3. The probability for the electron staying in band a in intermediate coupling regime (exact solution). The parameters for the superlattice are D /u"0.1, D /u"0.4, R "R "0.05, k/ed"0.129. The parameters for the dashed line (corresponding to quenched ? @ ? @ state) are chosen as edE /u"0, edE /u"3.88, k"p/2 such that quenching conditions are satis"ed. The parameters for the solid line (corresponding to oscillated state) are chosen as edE /u"0, edE /u"6.0, k"p/2 such that quenching conditions are not satis"ed.
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Fig. 4. The probability for the electron staying in band a in intermediate coupling regime (analytical solution). The parameters for the superlattice are D /u"0.1, D /u"0.4, R "R "0.05, k/ed"0.129. The parematers for the dashed line (corresponding to quenched ? @ ? @ state) are chosen as edE /u"0, edE /u"3.88, k"p/2 such that quenching conditions are satis"ed. The parameters for the solid line (corresponding to oscillated state) are chosen as edE /u"0, edE /u"6.0, k"p/2 such that quenching conditions are not satis"ed.
Fig. 5. The probability for the electron staying in band a in the presence of AC}DC "elds (in intermediate coupling regime) (exact solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"2.5, ? @ ? @ edE /u"3.88, k/ed"0.129, k"p/2.
are not satis"ed. Figs. 3 and 4 show that Rabi oscillations do happen in this case and the analytical results "t well with the exact results, except that there are small shifts in the positions of some valleys and peaks. It is due to the perturbative nature of our criteria (21) that the probabilities P s (for ? solid lines) do not reach zero. From the parameters of Figs. 3 and 4, we can obtain the period of Rabi
oscillations ¹ /¹"6.0 from Eqs. (19)}(23). This is 0 almost the same as that shown in Figs. 3 and 4. Figs. 5 and 6 show the behavior of the electrons in the presence of AC}DC "elds. Rabi oscillations are realized just as our theoretical prediction. Combining our results with those obtained by Zhao et al. [17] leads to a nice picture of the dynamics of electrons in superlattices driven by
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Fig. 6. The probability for the electron staying in band a in the presence of AC}DC "elds (in intermediate coupling regime) (analytical solution). The parameters for the superlattice and applied "elds are D /u"0.1, D /u"0.4, R "R "0.05, edE /u"2.5, ? @ ? @ edE /u"3.88, k/ed"0.129, k"p/2.
electric "elds, which is consistent with our physical intuition. We can see that when the e!ective interband coupling kE/u&0, there is little chance to realize Rabi oscillations. With the increase of kE/u, Rabi oscillations become possible. From the results of Ref. [17], when kE/u is small, the dominant behavior of the electron is quenching in one band, and Rabi oscillations will happen only under some speci"c conditions. In other words, the probability to realize quenched states is much bigger than that of oscillated states. When the interband coupling increases further to the intermediate regime kE/u&0.5, Rabi oscillations become dominant. Quenched state can be obtained under some speci"c conditions (21). When interband coupling is strong enough, say kE/u'1.0, Rabi oscillations always happen. Finally, we brie#y discuss the possibility of experimental observation of our theoretical predictions. As discussed in Ref. [17], we may choose a system of superlattices whose unit cell consists of two quantum wells. By taking l/d"0.72, we can get k/ed"0.129, where l is the total width of a double well and the coupling dipole moment is given by k"16el/9p. To obtain a system of narrow bands and small band gap, we take the superlattice parameters as D"D !D "6 meV, R "R " @ ? ? @ 1 meV (i.e., the band widths are 4R "4R " ? @
4 meV). We apply on the superlattice a high-power terahertz AC "eld with frequency u/2p" 4.6 THz (20 meV), which can be generated in a free-electron laser, with tunable frequency from 120 GHz to 4.8 THz [19]. To probe the results in Figs. 3 and 4, we can take the "eld strength as E "7.7;10 V/m (for dashed lines) or E "1.2;10 V/m (for solid lines). Of course, in a real system, there are also incoherent e!ects due to lattice imperfection, which prevent the observation of Rabi oscillations. However, for a highquality superlattice, such e!ects should be small. Therefore one may still observe Rabi #ops (not full Rabi oscillations) in those systems [20]. In summary, we have considered Rabi oscillations of the electrons in superlattices driven by electric "elds beyond weak interband coupling regime. The analytic and numerical results supply us a clear picture of the behavior of Bloch electrons in the presence of strong interband tunneling.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant No. 19725417.
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