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Dynamics of strongly driven two-level systems: analytical solutions
Superlattices and Microstructures, Vol. 23, No. 2, 1998
Dynamics of strongly driven two-level systems: analytical solutions Mathias Wagner Hitachi Cambridge Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.
P. Vasilopoulos Concordia University, Department of Physics, 1455 de Maisonneuvre Blvd West. Montr´eal, Qu´ebec, Canada, H3G 1M8 (Received 15 July 1996) The dynamics of a two-level system, strongly driven by an external sinusoidal field, is studied using a systematic iteration procedure. Analytical solutions are presented for the quasieigenenergies and Floquet states. They are valid for arbitrary strengths of the driving field and thus encompass the existing low- and high-field solutions. The higher-order corrections to the Floquet states are essential for properly describing the dynamics on the time scale of the driving field (as opposed to the time scale associated with the tunnel splitting), which is relevant for harmonic-frequency generation. The convergence of the procedure is good for driving frequencies larger than half the tunnel splitting. c 1998 Academic Press Limited
1. Introduction The high switching speed envisaged in future electronic circuits necessitates a treatment of time-dependent effects beyond the traditional adiabatic approach. As this speed is likely to be associated with high amplitudes comparable to typical energies in the devices, this renders pertinent further studies of quantum-mechanical systems driven by strong time-dependent electric fields, which are known to lead to unusual effects such as the ‘coherent destruction of tunneling’ in driven double quantum wells [1], the ‘collapse of minibands’ [2] and the absolute negative conductance in the photon-assisted tunneling current in superlattices [3–5], or a strong quenching of the transmission probability in resonant tunneling diodes [6, 7]. In the present work, the dynamics of a strongly driven two-level system, as realized, e. g. in a double quantum well with only the lowest two electronic levels occupied and driven by a strong laser field, is studied analytically and numerically. The dynamics of a driven two-level system was first considered by Shirley [8] in the framework of the Floquet-state theory. The main emphasis there was to determine the transition probabilities between two driven atomic states in weak driving fields. Later on, the field gained momentum with the discovery of ‘coherent destruction of tunneling’ in driven symmetric double-well structures by H¨anggi’s group [1]. They found that the tunneling interaction between two Floquet states in such a structure vanishes for a suitable choice of driving parameters. As a result, an electron initially prepared in one of the two wells will stay there ‘for ever’. Such a field-induced localisation, although in a different system, was actually first discussed by Dunlap and Kenkre [9]. Since then, driven two-level systems have been studied in great detail: Holthaus [10] used this approximation to explain the collapse of minibands in driven superlattices. Llorente and Plata [11] employed a two-level model to accurately describe the quasi-eigenenergies for the driven Floquet states in a quartic 0749–6036/98/020477 + 07 $25.00/0
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double well as found by Großmann et al. [1] in extensive numerical studies. The probability P(t) for an electron, initially placed in one well, to remain in that well at a later time t > 0, was investigated by H¨anggi and co-workers [12]. Dakhnovskii, Bavli, and Metiu [13–16] made extensive studies of the time-dependent dipole moment in driven two-level systems, with particular focus on harmonic-frequency generation. To the best of our knowledge, Bavli and co-workers were also the first to provide analytical solutions for the driven two-level system beyond the Holthaus solution. However, their procedure suffers from divergencies at long time scales, which necessitates a summation over the most diverging terms in the perturbation series. More recently, a formally exact, but not closed, solution for the dynamics in a driven two-level system was found [17]. It uses the zero-field solution as a starting point, and hence does not appear to be very useful for the case of strong driving. In fact, it is not clear at present whether the perturbation expansion employed in [17] actually converges in the strong-field regime. Other work on driven two-level systems includes various treatments of the effect of dissipation [18–20]. In this work, we present an analytical iterative method to solve the time-dependent Schr¨odinger equation of a driven two-level system for arbitrary strength of the driving field, provided the driving frequency ω is larger than half the tunnel splitting 10 . The procedure yields successively improved approximations not only to the quasi-eigenenergies but also to the state vectors. The limitation 10 < 2ω is the same as in most other works [10–16], but our analytical results should provide a useful complement to the existing numerical work. Moreover, they are much simpler and more compact than those of previous analytical treatments and do not suffer from any divergencies at long time scales.
2. Theory We consider a system of two levels separated by an energy 10 due to tunneling. A monochromatic external driving field of frequency ω couples these two levels via their mutual dipole moment, and is taken to be of the form 2λ cos ωt with λ as the strength of the driving force. Standard Floquet-state theory [8, 10] shows that the corresponding time-dependent Schr¨odinger equation has solutions of the form (a(t), b(t)) exp(−it), where is the quasi-eigenenergy and a(t) and b(t) are functions periodic in T = 2π/ω. The dynamics of the driven two-level system is then completely described by the equations for a(t) and b(t) [12, 17], i ∂t∂ a(t) = ( 120 − )a(t) + 2λ cos ωtb(t) i ∂t∂ b(t) = (− 120 − )b(t) + 2λ cos ωta(t).
(1)
Note that is not an independent parameter here, but rather has to be determined ‘self-consistently’ to allow for nontrivial periodic solutions to eqn (1). Our analysis starts by performing a unitary transformation to a new set of variables, √ 2λ ± (2) c (t) = exp ∓i sin ωt [a(t) ± b(t)] / 2, ω after which eqn (1) reads 10 ∂ ± (3) c (t) = exp(±iµ sin ωt)c∓ (t) − c± (t), ∂t 2 where we have defined µ ≡ 4λ/ω. A few results can already be seen simply by inspecting eqn (3): Since a(t) and b(t) are periodic in T , the same is true for c± (t); hence the time-average of its derivative over one period of the driving field must vanish, ∂ ± c (t) = 0. (4) ∂t RT (Here we have used the notation hhii ≡ (1/T ) 0 dt.) After time-averaging of (3) and normalizing such that i
Superlattices and Microstructures, Vol. 23, No. 2, 1998
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c± (t) = 1, one finds a formal solution for the quasi-eigenenergy,
10 hhexp(±iµ sin ωt)c∓ (t)ii. (5) 2 Next we set up the hierarchy of equations for generating successively improved solutions to (3) and (5). To this end we start with Holthaus’ high-field solution [10], which in our representation is given by c0± (t) ≡ 1, and use it to approximately evaluate the right-hand side of eqn (3). Integration over time then yields a correction c1± (t) and, with the help of eqn (4), an approximate value 0 for the quasi-eigenenergy. Repeating this process with c± (t) = 1 + c1± (t) yields the next higher-order corrections c2± (t) and 1 , and so on. The perturbation series thus has the form † ∞ ∞ X X ci± (t) and = i . (6) c± (t) = 1 + =
i=1
i=0
Inserting (6) into (3) results in the hierarchy of equations, i
n−2 n−1 X X 10 ∂ ± ∓ ± (t) − cn−1 (t) i − n−1 ci± (t), cn (t) = exp(±iµ sin ωt)cn−1 ∂t 2 i=0 i=0
n > 0,
(7)
with 10 n ≥ 0. hhexp(±iµ sin ωt)cn∓ (t)ii, 2 Solving (7) and (8) to lowest order, we reproduce Holthaus’ result [10] for the strong driving limit, n =
10 J0 (µ), 2 where J0 is the zeroth Bessel function of the first kind. Iterating eqn (7) once more yields 10 X exp(±inωt) , 1 = 0, Jn (µ) c1± (t) = ∓ 2ω n6=0 n c0± (t) = 1,
0 =
and in the next iteration step we get i 2 hP P exp(±imωt) exp(±inωt) 0 n6=0 Jm+n (µ)Jn (µ) + J (µ) J (µ) c2± (t) = − 1 0 n 2 n6 = 0 2ω nm m6=0 in h P P 13 1 Jn (µ)Jn+m (µ)Jm (µ) . 2 = − 8ω02 J0 (µ) n6=0 n12 |Jn (µ)|2 + n6=0 nm
(8)
(9)
(10)
(11)
m6=0
Higher orders are quite cumbersome to compute, with the number of terms in each order increasing extremely rapidly. However, we can sum up some of these higher-order terms to infinite order, yielding for the quasieigenenergy 10 2 P 1 n6=0 Jn (µ)Jn+m (µ)Jm (µ) 10 J0 (µ) − 2ω m6=0 nm . (12) ≈ 2P 2 1 2 1 + 10 2 |Jn (µ)| 2ω
n6=0 n
The parameter that determines how quickly our iteration procedure converges is 10 /2ω, which is in agreement with numerical observations [12]. Having found one solution to the time-dependent Schr¨odinger equation, one can construct a second independent solution by observing that the transformation (, a ∗ , b∗ ) → (−, b, −a) leaves eqn (1) invariant. Hence, the second solution is (−b∗ (t), a ∗ (t)) exp(it). From these two solutions one can easily compute ± † There
± is some freedom of choice in the integration constants for the corrections ci (t), which we eliminate by requiring that ci (t) ≡ 0 ∀ i > 0.
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Superlattices and Microstructures, Vol. 23, No. 2, 1998 A
0.5 εnum ε0 ε0+ε1+ε2 ε0+ε1+ε2+ε3
0.4
ε/ω
0.3
10/ω = 1.0
0.2 0.1 0.0
–0.1 –0.2
0
2
4
6
8 µ
10
12
14
16
B 0.06
10/ω = 1.0
0.04
δε/ω
0.02 0.00
–0.02
ε0–εnum ε0+ε1+ε2–εnum ε0+ε1+ε2+ε3–εnum
–0.04 –0.06 0
2
4
6
8 µ
10
12
14
16
Fig. 1. Top panel: quasi-eigenenergy as calculated numerically (full line), and to the first few orders using the analytical iteration method of eqn (7), as a function of the driving field, for 10 /ω = 1. Bottom panel: accuracy of the analytical solutions when compared with the ‘exact’ numerical solution.
the time-dependent dipole matrix elements and study the even- and odd-harmonics generation as discussed in [13–16]. According to eqn (10), this effect arises in leading order from the first-order corrections c1± (t). However, in this paper we will focus on the probability P(t) for an electron initially prepared in one of the two quantum wells at t = 0 to stay in that well, and not to tunnel into the neighbouring well. In the two-level model, a state localized in one well is realized by a linear superposition of the two Floquet states ‡ , ψ(t = 0) = α
a(0) b(0)
+β
−b∗ (0) a ∗ (0)
1 =√ 2
1 . 1
(13)
‡ Here we assume that the energy splitting between the two states is entirely due to tunneling, with the two states being completely delocalized in the two wells in the absence of driving.
Superlattices and Microstructures, Vol. 23, No. 2, 1998 This state evolves in time according to ψ(t) = α exp(−it)
a(t) b(t)
481
+ β exp(+it)
−b∗ (t) a ∗ (t)
,
(14)
and the probability of finding the electron localized in state |ψ(0)i at a later time t is then given by P(t) = | hψ(0)|ψ(t)i |2 .
3. Results Our analytical expansion of the quasi-eigenenergy and the Floquet-state vector components c± (t) is only valid for 10 < 2ω, i.e., for not too small driving frequencies compared to the tunnel splitting. Yet even though we took the high-field solution as the starting point for the iteration procedure, the result turns out to be valid for any strength λ of the driving field, as long as 10 2ω is satisfied. It is only in the range ω < 10 < 2ω that appreciable deviations appear for small driving fields. In Fig. 1 we compare our analytical solutions for the quasi-eigenenergy , as evaluated from eqns (9)–(11), with a numerical solution of eqn (3), for 10 = ω. The high-field solution 0 = (10 /2)J0 (µ) [10–12] (indicated with long dashes) turns out to be already a very good approximation to the numerical solution, except for weaker driving fields, where the higher-order corrections i presented in this paper become important† . In the right-hand panel of Fig. 1 one sees that the error of = 0 + 1 + 2 is roughly one order of magnitude smaller than that of = 0 . On the other hand, going one step further in the iteration by computing 3 does not improve the result appreciably. This is probably due to the rather large ratio chosen for 10 /ω. When evaluating the probability P(t) of finding the electron localized in one of the wells at a later time t > 0, it is crucial to include the higher-order corrections ci± for i > 0, if any fine structure on the time scale of the external driving field is to be captured in the solution: The strong-field solution {0 , c0± (t) = 1} describes only the oscillation of the (initially localized) electron back and forth between the two quantum wells on the time scale of the tunnel splitting (see the dashed curve in Fig. 2B). In Fig. 2A we plot P(t) for various values of 10 /ω and corresponding driving field strengths where suppression of tunneling can, in principle, occur. Analytical correction terms up to the second order have been included: = 0 + 1 + 2 and c± (t) = 1 + c1± (t) + c2± (t). As noted already by Großmann [12], the suppression is not complete, i.e. the minimum of P(t) is appreciably smaller than 1, when 10 /ω becomes too large. Figure 2B shows P(t) far away from the parameter region required for coherent suppression of tunneling. The slow oscillation is due to tunneling and hence given by 2π/, whilst the fast oscillation stems from the external driving field, with higher harmonics, leading to frequency generation [13–16], being clearly visible as shoulders. The phase shift seen in Fig. 2B between the high-field solution (dashed line) and our second-order solution is caused by the second-order term in the quasi-eigenenergy, 2 , which is particularly large around µ = 2 (see Fig. 1).
4. Conclusion We have presented a systematic iterative procedure for deriving analytical solutions for the quasieigenenergies and the Floquet states in a driven two-level system that go beyond the known strong-field solution, which is obtained as the lowest order result in our formalism. Comparison with exact numerical solutions shows very good agreement for arbitrary strength of the driving field, but at the same time suggests that the standard strong-field solution is already quite good, provided the tunnel splitting 10 is much smaller than the driving frequency ω. On the other hand, the corrections derived for the Floquet states are crucial when studying the time-dependent dipole moment or the probability P(t) for an electron initially localized † When evaluating eqns (10) and (11), the summation indices n and m need to run only to ±5.
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Superlattices and Microstructures, Vol. 23, No. 2, 1998 1
0.25
0.6
0.5
P(t)
0.8
0.75
0.4
∆0/ω = 1.0 0.2 A 0
0
20
40
60
80
100
120
140
ωt 1 ∆0/ω = 0.5 µ=2
0.8
P(t)
0.6 0.4 0.2 B 0
0
20
40
60
80
100
120
140
ωt Fig. 2. Probability P(t) for an electron prepared in one well at t = 0 to be still localized at t > 0, based on eqns (9)–(11). (A) On the (λ, ω) manifold where (partial) suppression of tunneling occurs, for 10 /ω = 0.25, 0.5, 0.75, and 1.0. (B) Off the manifold, with 10 /ω = 0.5 and µ = 2 (dashed line: high-field solution with = 0 and c± (t) = 1).
in one of the two quantum wells at t = 0 to be still localized at t > 0. In this case, the strong-field solution describes only the long-time behaviour due to the tunnel splitting, without any fine structure on the time scale of the external driving field. The information on the fine structure is entirely contained in the higher-order terms ci± (t), i > 0. Acknowledgements—This work was performed under the management of FED as a part of the MITI R&D program (Quantum Functional Device Project) supported by NEDO. P.V. acknowledges the support of NSERC grant OGP0121756.
References [1] [2] [3] [4]
}F. Großmann, T. Dittrich, P. Jung, and P. H¨anggi, Phys. Rev. Lett. 67, 516 (1991). }M. Holthaus, Phys. Rev. Lett. 69, 351 (1992). }S. Zeuner et al., Phys. Rev. B53, R1717 (1996). }B. J. Keay et al., Phys. Rev. Lett. 75, 4102 (1995).
Superlattices and Microstructures, Vol. 23, No. 2, 1998 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
}M. Wagner, Phys. Rev. Lett. 76, 4010 (1996). }M. Wagner, Phys. Rev. A51, 798 (1995). }M. Wagner, Phys. Rev. B49, 16544 (1994). }J. H. Shirley, Phys. Rev. B138, 979 (1965). }D. H. Dunlap and V. M. Kenkre, Phys. Rev. B34, 3625 (1986). }M. Holthaus, Z. Phys. B89, 251 (1992). }J. M. G. Llorente and J. Plata, Phys. Rev. A45, R6958 (1992). }F. Großmann and P. H¨anggi, Europhys. Lett. 18, 571 (1992). }Y. Dakhnovskii and R. Bavli, Phys. Rev. B48, 11020 (1993). }Y. Dakhnovskii and R. Bavli, Phys. Rev. A48, 2342 (1993). }R. Bavli and H. Metiu, Phys. Rev. A47, 3299 (1993). }Y. Dakhnovskii, R. Bavli, and H. Metiu, Phys. Rev. B53, 4657 (1996). }X. G. Zhao, S. J. Xu, and W. C. Cheng, Z. Phys. B98, 503 (1995). }M. Grifoni, M. Sassetti, J. Stockburger, and U. Weiss, Phys. Rev. E48, 3497 (1993). }T. Dittrich, B. Oelschl¨agel, and P. H¨anggi, Europhys. Lett. 22, 5 (1993). }Y. Dakhnovskii, Phys. Rev. B49, 4649 (1994).
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Dynamics of strongly driven two-level systems: analytical solutions Mathias Wagner Hitachi Cambridge Laboratory, Madingley Road, Cambridge CB3 0HE, U.K.
P. Vasilopoulos Concordia University, Department of Physics, 1455 de Maisonneuvre Blvd West. Montr´eal, Qu´ebec, Canada, H3G 1M8 (Received 15 July 1996) The dynamics of a two-level system, strongly driven by an external sinusoidal field, is studied using a systematic iteration procedure. Analytical solutions are presented for the quasieigenenergies and Floquet states. They are valid for arbitrary strengths of the driving field and thus encompass the existing low- and high-field solutions. The higher-order corrections to the Floquet states are essential for properly describing the dynamics on the time scale of the driving field (as opposed to the time scale associated with the tunnel splitting), which is relevant for harmonic-frequency generation. The convergence of the procedure is good for driving frequencies larger than half the tunnel splitting. c 1998 Academic Press Limited
1. Introduction The high switching speed envisaged in future electronic circuits necessitates a treatment of time-dependent effects beyond the traditional adiabatic approach. As this speed is likely to be associated with high amplitudes comparable to typical energies in the devices, this renders pertinent further studies of quantum-mechanical systems driven by strong time-dependent electric fields, which are known to lead to unusual effects such as the ‘coherent destruction of tunneling’ in driven double quantum wells [1], the ‘collapse of minibands’ [2] and the absolute negative conductance in the photon-assisted tunneling current in superlattices [3–5], or a strong quenching of the transmission probability in resonant tunneling diodes [6, 7]. In the present work, the dynamics of a strongly driven two-level system, as realized, e. g. in a double quantum well with only the lowest two electronic levels occupied and driven by a strong laser field, is studied analytically and numerically. The dynamics of a driven two-level system was first considered by Shirley [8] in the framework of the Floquet-state theory. The main emphasis there was to determine the transition probabilities between two driven atomic states in weak driving fields. Later on, the field gained momentum with the discovery of ‘coherent destruction of tunneling’ in driven symmetric double-well structures by H¨anggi’s group [1]. They found that the tunneling interaction between two Floquet states in such a structure vanishes for a suitable choice of driving parameters. As a result, an electron initially prepared in one of the two wells will stay there ‘for ever’. Such a field-induced localisation, although in a different system, was actually first discussed by Dunlap and Kenkre [9]. Since then, driven two-level systems have been studied in great detail: Holthaus [10] used this approximation to explain the collapse of minibands in driven superlattices. Llorente and Plata [11] employed a two-level model to accurately describe the quasi-eigenenergies for the driven Floquet states in a quartic 0749–6036/98/020477 + 07 $25.00/0
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double well as found by Großmann et al. [1] in extensive numerical studies. The probability P(t) for an electron, initially placed in one well, to remain in that well at a later time t > 0, was investigated by H¨anggi and co-workers [12]. Dakhnovskii, Bavli, and Metiu [13–16] made extensive studies of the time-dependent dipole moment in driven two-level systems, with particular focus on harmonic-frequency generation. To the best of our knowledge, Bavli and co-workers were also the first to provide analytical solutions for the driven two-level system beyond the Holthaus solution. However, their procedure suffers from divergencies at long time scales, which necessitates a summation over the most diverging terms in the perturbation series. More recently, a formally exact, but not closed, solution for the dynamics in a driven two-level system was found [17]. It uses the zero-field solution as a starting point, and hence does not appear to be very useful for the case of strong driving. In fact, it is not clear at present whether the perturbation expansion employed in [17] actually converges in the strong-field regime. Other work on driven two-level systems includes various treatments of the effect of dissipation [18–20]. In this work, we present an analytical iterative method to solve the time-dependent Schr¨odinger equation of a driven two-level system for arbitrary strength of the driving field, provided the driving frequency ω is larger than half the tunnel splitting 10 . The procedure yields successively improved approximations not only to the quasi-eigenenergies but also to the state vectors. The limitation 10 < 2ω is the same as in most other works [10–16], but our analytical results should provide a useful complement to the existing numerical work. Moreover, they are much simpler and more compact than those of previous analytical treatments and do not suffer from any divergencies at long time scales.
2. Theory We consider a system of two levels separated by an energy 10 due to tunneling. A monochromatic external driving field of frequency ω couples these two levels via their mutual dipole moment, and is taken to be of the form 2λ cos ωt with λ as the strength of the driving force. Standard Floquet-state theory [8, 10] shows that the corresponding time-dependent Schr¨odinger equation has solutions of the form (a(t), b(t)) exp(−it), where is the quasi-eigenenergy and a(t) and b(t) are functions periodic in T = 2π/ω. The dynamics of the driven two-level system is then completely described by the equations for a(t) and b(t) [12, 17], i ∂t∂ a(t) = ( 120 − )a(t) + 2λ cos ωtb(t) i ∂t∂ b(t) = (− 120 − )b(t) + 2λ cos ωta(t).
(1)
Note that is not an independent parameter here, but rather has to be determined ‘self-consistently’ to allow for nontrivial periodic solutions to eqn (1). Our analysis starts by performing a unitary transformation to a new set of variables, √ 2λ ± (2) c (t) = exp ∓i sin ωt [a(t) ± b(t)] / 2, ω after which eqn (1) reads 10 ∂ ± (3) c (t) = exp(±iµ sin ωt)c∓ (t) − c± (t), ∂t 2 where we have defined µ ≡ 4λ/ω. A few results can already be seen simply by inspecting eqn (3): Since a(t) and b(t) are periodic in T , the same is true for c± (t); hence the time-average of its derivative over one period of the driving field must vanish, ∂ ± c (t) = 0. (4) ∂t RT (Here we have used the notation hhii ≡ (1/T ) 0 dt.) After time-averaging of (3) and normalizing such that i
Superlattices and Microstructures, Vol. 23, No. 2, 1998
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c± (t) = 1, one finds a formal solution for the quasi-eigenenergy,
10 hhexp(±iµ sin ωt)c∓ (t)ii. (5) 2 Next we set up the hierarchy of equations for generating successively improved solutions to (3) and (5). To this end we start with Holthaus’ high-field solution [10], which in our representation is given by c0± (t) ≡ 1, and use it to approximately evaluate the right-hand side of eqn (3). Integration over time then yields a correction c1± (t) and, with the help of eqn (4), an approximate value 0 for the quasi-eigenenergy. Repeating this process with c± (t) = 1 + c1± (t) yields the next higher-order corrections c2± (t) and 1 , and so on. The perturbation series thus has the form † ∞ ∞ X X ci± (t) and = i . (6) c± (t) = 1 + =
i=1
i=0
Inserting (6) into (3) results in the hierarchy of equations, i
n−2 n−1 X X 10 ∂ ± ∓ ± (t) − cn−1 (t) i − n−1 ci± (t), cn (t) = exp(±iµ sin ωt)cn−1 ∂t 2 i=0 i=0
n > 0,
(7)
with 10 n ≥ 0. hhexp(±iµ sin ωt)cn∓ (t)ii, 2 Solving (7) and (8) to lowest order, we reproduce Holthaus’ result [10] for the strong driving limit, n =
10 J0 (µ), 2 where J0 is the zeroth Bessel function of the first kind. Iterating eqn (7) once more yields 10 X exp(±inωt) , 1 = 0, Jn (µ) c1± (t) = ∓ 2ω n6=0 n c0± (t) = 1,
0 =
and in the next iteration step we get i 2 hP P exp(±imωt) exp(±inωt) 0 n6=0 Jm+n (µ)Jn (µ) + J (µ) J (µ) c2± (t) = − 1 0 n 2 n6 = 0 2ω nm m6=0 in h P P 13 1 Jn (µ)Jn+m (µ)Jm (µ) . 2 = − 8ω02 J0 (µ) n6=0 n12 |Jn (µ)|2 + n6=0 nm
(8)
(9)
(10)
(11)
m6=0
Higher orders are quite cumbersome to compute, with the number of terms in each order increasing extremely rapidly. However, we can sum up some of these higher-order terms to infinite order, yielding for the quasieigenenergy 10 2 P 1 n6=0 Jn (µ)Jn+m (µ)Jm (µ) 10 J0 (µ) − 2ω m6=0 nm . (12) ≈ 2P 2 1 2 1 + 10 2 |Jn (µ)| 2ω
n6=0 n
The parameter that determines how quickly our iteration procedure converges is 10 /2ω, which is in agreement with numerical observations [12]. Having found one solution to the time-dependent Schr¨odinger equation, one can construct a second independent solution by observing that the transformation (, a ∗ , b∗ ) → (−, b, −a) leaves eqn (1) invariant. Hence, the second solution is (−b∗ (t), a ∗ (t)) exp(it). From these two solutions one can easily compute ± † There
± is some freedom of choice in the integration constants for the corrections ci (t), which we eliminate by requiring that ci (t) ≡ 0 ∀ i > 0.
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0.5 εnum ε0 ε0+ε1+ε2 ε0+ε1+ε2+ε3
0.4
ε/ω
0.3
10/ω = 1.0
0.2 0.1 0.0
–0.1 –0.2
0
2
4
6
8 µ
10
12
14
16
B 0.06
10/ω = 1.0
0.04
δε/ω
0.02 0.00
–0.02
ε0–εnum ε0+ε1+ε2–εnum ε0+ε1+ε2+ε3–εnum
–0.04 –0.06 0
2
4
6
8 µ
10
12
14
16
Fig. 1. Top panel: quasi-eigenenergy as calculated numerically (full line), and to the first few orders using the analytical iteration method of eqn (7), as a function of the driving field, for 10 /ω = 1. Bottom panel: accuracy of the analytical solutions when compared with the ‘exact’ numerical solution.
the time-dependent dipole matrix elements and study the even- and odd-harmonics generation as discussed in [13–16]. According to eqn (10), this effect arises in leading order from the first-order corrections c1± (t). However, in this paper we will focus on the probability P(t) for an electron initially prepared in one of the two quantum wells at t = 0 to stay in that well, and not to tunnel into the neighbouring well. In the two-level model, a state localized in one well is realized by a linear superposition of the two Floquet states ‡ , ψ(t = 0) = α
a(0) b(0)
+β
−b∗ (0) a ∗ (0)
1 =√ 2
1 . 1
(13)
‡ Here we assume that the energy splitting between the two states is entirely due to tunneling, with the two states being completely delocalized in the two wells in the absence of driving.
Superlattices and Microstructures, Vol. 23, No. 2, 1998 This state evolves in time according to ψ(t) = α exp(−it)
a(t) b(t)
481
+ β exp(+it)
−b∗ (t) a ∗ (t)
,
(14)
and the probability of finding the electron localized in state |ψ(0)i at a later time t is then given by P(t) = | hψ(0)|ψ(t)i |2 .
3. Results Our analytical expansion of the quasi-eigenenergy and the Floquet-state vector components c± (t) is only valid for 10 < 2ω, i.e., for not too small driving frequencies compared to the tunnel splitting. Yet even though we took the high-field solution as the starting point for the iteration procedure, the result turns out to be valid for any strength λ of the driving field, as long as 10 2ω is satisfied. It is only in the range ω < 10 < 2ω that appreciable deviations appear for small driving fields. In Fig. 1 we compare our analytical solutions for the quasi-eigenenergy , as evaluated from eqns (9)–(11), with a numerical solution of eqn (3), for 10 = ω. The high-field solution 0 = (10 /2)J0 (µ) [10–12] (indicated with long dashes) turns out to be already a very good approximation to the numerical solution, except for weaker driving fields, where the higher-order corrections i presented in this paper become important† . In the right-hand panel of Fig. 1 one sees that the error of = 0 + 1 + 2 is roughly one order of magnitude smaller than that of = 0 . On the other hand, going one step further in the iteration by computing 3 does not improve the result appreciably. This is probably due to the rather large ratio chosen for 10 /ω. When evaluating the probability P(t) of finding the electron localized in one of the wells at a later time t > 0, it is crucial to include the higher-order corrections ci± for i > 0, if any fine structure on the time scale of the external driving field is to be captured in the solution: The strong-field solution {0 , c0± (t) = 1} describes only the oscillation of the (initially localized) electron back and forth between the two quantum wells on the time scale of the tunnel splitting (see the dashed curve in Fig. 2B). In Fig. 2A we plot P(t) for various values of 10 /ω and corresponding driving field strengths where suppression of tunneling can, in principle, occur. Analytical correction terms up to the second order have been included: = 0 + 1 + 2 and c± (t) = 1 + c1± (t) + c2± (t). As noted already by Großmann [12], the suppression is not complete, i.e. the minimum of P(t) is appreciably smaller than 1, when 10 /ω becomes too large. Figure 2B shows P(t) far away from the parameter region required for coherent suppression of tunneling. The slow oscillation is due to tunneling and hence given by 2π/, whilst the fast oscillation stems from the external driving field, with higher harmonics, leading to frequency generation [13–16], being clearly visible as shoulders. The phase shift seen in Fig. 2B between the high-field solution (dashed line) and our second-order solution is caused by the second-order term in the quasi-eigenenergy, 2 , which is particularly large around µ = 2 (see Fig. 1).
4. Conclusion We have presented a systematic iterative procedure for deriving analytical solutions for the quasieigenenergies and the Floquet states in a driven two-level system that go beyond the known strong-field solution, which is obtained as the lowest order result in our formalism. Comparison with exact numerical solutions shows very good agreement for arbitrary strength of the driving field, but at the same time suggests that the standard strong-field solution is already quite good, provided the tunnel splitting 10 is much smaller than the driving frequency ω. On the other hand, the corrections derived for the Floquet states are crucial when studying the time-dependent dipole moment or the probability P(t) for an electron initially localized † When evaluating eqns (10) and (11), the summation indices n and m need to run only to ±5.
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0.25
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ωt Fig. 2. Probability P(t) for an electron prepared in one well at t = 0 to be still localized at t > 0, based on eqns (9)–(11). (A) On the (λ, ω) manifold where (partial) suppression of tunneling occurs, for 10 /ω = 0.25, 0.5, 0.75, and 1.0. (B) Off the manifold, with 10 /ω = 0.5 and µ = 2 (dashed line: high-field solution with = 0 and c± (t) = 1).
in one of the two quantum wells at t = 0 to be still localized at t > 0. In this case, the strong-field solution describes only the long-time behaviour due to the tunnel splitting, without any fine structure on the time scale of the external driving field. The information on the fine structure is entirely contained in the higher-order terms ci± (t), i > 0. Acknowledgements—This work was performed under the management of FED as a part of the MITI R&D program (Quantum Functional Device Project) supported by NEDO. P.V. acknowledges the support of NSERC grant OGP0121756.
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