(a) (b) (c) ω= ω= T t=0 t= T t=0 t= V(x), au x, au x, au x, au

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1 EUROPHYSICS LETTERS 15 January 1998 Europhys. Lett., 41 (), pp (1998) Field-induced barrier transparency I. Vorobeichik 1, R. Lefebvre and N. Moiseyev 1 1 Department of Chemistry and Minerva Center for Non Linear Physics Technion Israel Institute of Technology - Haifa 3, Israel Laboratoire de Photophysique Moleculaire - Campus d'orsay, 9145 Orsay and UFR de Physique Fondamentale et Appliquee, Universite Pierre et Marie Curie 753, Paris, France. (received 6 August 1997; accepted in nal form November 1997) PACS. 3.65?w { Quantum mechanics. PACS. 73.4Gk { Tunneling. Abstract. { We show that the transition of free particles (regardless of their mass) through a periodically driven potential barrier can be almost 1%, although the probability of tunneling through the static potential barrier is almost zero. The conditions for this to happen are discussed and physical candidates for eld-induced transparency are examined. There is an increasing interest in photo-induced dynamics in strong laser elds which has stimulated numerous experimental and theoretical investigations. One example is the suppression of tunneling through a potential barrier in a driven double-well potential [1]. Another example which is relevant to the subject discussed in this letter is the chaos-assisted tunneling phenomenon []. In our work we address ourselves to a scattering of particles through a driven potential barrier. This case is very dierent from the previous studied problems since the energy spectrum is continuous (rather than discrete as in the driven double-well potential problem). For the sake of clarity and simplicity and without loss of generality let us consider a case where a ux of particles with mass m approaches a rectangular potential barrier where V and a are, respectively, the height and the width of the potential barrier. The Hamiltonian is given by where bh = ^p x + V (x); (1) m 8 >< V ; if jxj < a= ; V (x) = >: ; elsewhere : () c EDP Sciences

2 11 EUROPHYSICS LETTERS V(x), au (a) (b) (c). ω=.3 ω=.3.15 T t= t= T t= t= x, au 4 4 x, au 4 4 x, au 1..8 T(E) E, au..1. E, au E, au Fig. 1. { (a) Upper part: the static eld-free rectangular potential barrier, V (x) (see eq. ()); lower part: the corresponding transition probability, T (E). The width of the barrier a = 8 au (4:38 A), the height V = :147 au (.4 ev), the mass m = :1 au (m = :1m e) and h = 1. (b) Upper part: driven potential barrier V (x; t) = V (x + cos(!t)) (see eq. (3)) at time t = and t = T = = =!, where = au and! = :3 au ( 1 1 Hz); lower part: the corresponding transition probability T (E) obtained using the full Hamiltonian (eq. (3)). Here, T (E) stands for the case where the energy of the transferred particle (E) is exactly the same as the energy of the incoming particle (E ). The cases in which E = E + h!m; m = 1; ; : : : were found to be negligible for the parameters in this gure. (c) The same as (b) for! = :3 au ( 1 14 Hz). The transition probability given in g. 1 (a) was calculated using the analytical expression given in Landau and Lifshitz textbook [3]. As one can see for the chosen potential parameters (which correspond to transition through a potential barrier in semiconductors), the transition probability for a ux of particles with energy E = :1 au < V is almost equal to zero. (Atomic units are dened as h, electron charge, electron mass and Bohr radius equal to 1.) Let us now assume that the potential barrier periodically moves forward and backward. At t = the potential barrier is centered at x =? and at t = T= its center is at x = +. The frequency of the oscillations is! = =T. The time-dependent Hamiltonian is given by ^H = ^p x m + V (x + cos(!t)): (3) In the case that the Hamiltonian is driven by a laser eld the value of the parameter is determined by the maximum eld amplitude and the laser frequency, i.e. = E =m!, and ^p the Hamiltonian in eq. (3) is obtained from ^H = x m + V (x) + E x cos(!t) by an unitary transformation (known as the Kramers-Hennerberger transformation [4] or the acceleration frame representation). In this representation pure time-dependent terms (which physically are not important since they only yield an overall phase factor) are dropped. (For dierent gaugeto-gauge transformations and the transformation to the Kramers-Hennerberger representation see, for example, ref. [5].) The probability that a ux of free incoming particles with energy E will get through the driven potential barrier, V (x; t) = V (x+ cos(!t)), and will be found on the other side of the oscillating barrier with the same initial energy E has been calculated by

3 i. vorobeichik et al.: field-induced barrier transparency 113 the use of the time-independent scattering theory for time-dependent Hamiltonian developed by Peskin and Moiseyev [6]. The results presented in g. 1 (c) clearly show that when the period of the oscillations of the driven rectangular potential barrier is! = :3 au there are four peaks in the spectra which correspond to almost 1% transition probability, although the energy E is much less than the height of the potential barrier. When the potential barrier oscillates we may consider the eective height of the barrier as V a=( + a), where a is the width of the static rectangular barrier and ( + a) is about equal to the width of the time-averaged potential, V e (x) = 1 T Z T V (x; t)dt: (4) Therefore, the periodic motion of the barrier reduces the minimal energy of the free particles which is required for a classical transition. However, the almost 1% transmission associated with the four peaks in g. 1 (c) occurs at energies in which the classical transition is forbidden. When the frequency of the driven force is reduced by two orders of magnitude, i.e. to! = :3 au, the transition probability (calculated by the use of the time-independent scattering theory for time-dependent Hamiltonians [6]) is very similar to the analytical results obtained for the static potential barrier (compare g. 1 (a) and (b) when E < : au ' 3 V ). As we will prove below, the almost 1% transition which is due to the periodically motion of the potential barrier is obtained only when the time period, T, is suciently small (i.e.! is suciently large). We will show that the reference frequency,, is the vibrational frequency of the particles which are temporarily trapped by the oscillating potential barrier. If! >, then the particles (regardless of their mass) can get through the oscillating barrier although this transition is classically forbidden. The explanation of the enhancement of the transition when the potential barrier oscillates in time is based on Gavrila's proof [7] that shows that, in the limit of high-eld frequency, V e gives a good insight into the properties of the time-dependent system. In our case of a driven rectangular potential barrier we proved that 8 x? a= x + a= arccos arccos ; if?(? a=) x? a=; x? a= >< arccos ; if? a= x + a=; V e (x)= V arccos? x + a= ; if?( + a=)x?(? a=); >: ; otherwise: As one can see from the plot of V e (x) given in g. (a), the averaging of the driven barrier over one optical cycle results in a double-barrier potential. We marked the position of the rst four lowest resonances (we calculated them by the use of the complex coordinate method [8]). The vibrational frequency of these lowest resonances,, can be estimated from the harmonic expansion of the eective potential, V e (x) m x. For a driven rectangular barrier the harmonic approximation for is given by HO = s V a m(? a =4) 3= ' The lowest resonant energies can be approximated as (5) s V a m 3 : (6) E n = V e (x = ) + h HO (n + 1=); n = ; 1; : : : ; (7)

4 114 EUROPHYSICS LETTERS Fig.. { (a) The averaged potential, V e (x), given by eq. (4) and eq. (5). (b) The solid line stands for the transition probability through the averaged potential V e (x); the dashed line stands for the transition probability through the driven potential with! = :3 au (the same as in g. 1 (c)). where V e (x = ) = V (arccos(?a= )? arccos(a= )). In our studied case ' HO ' 1?3 au. The position of the rst two resonances, E n= = :383 au and E n=1 = :338 au is in a good agreement with the results presented in g. (a). Our numerical results presented in g. 1 (c) show that indeed for! = :3 au the transition probability for the static V e (x) presented in g. (b) (calculated by using Seideman-Miller expressions [9]) is in a remarkable agreement with the results obtained for the oscillating potential barrier. We should stress that the new physical phenomenon we present here is a drastic enhancement of tunneling due to the periodic motion of the potential barrier and its association to the enhancement of the tunneling through a static symmetrical double-barrier potential. The phenomenon of almost total transition through two identical potential barriers at specic energies (the resonance energies of the well as shown in g. (a)) is known. This phenomenon stands behind the Fabry-Perot interferometer [1], the Ramsauer- Townsend eect [11] and resonant tunneling in semiconductors double barriers [1]. This is essentially an interference eect: the reected wave, which is formed when the incoming wave (electro-magnetic or quantum) rst meets the potential barrier, interferes destructively with the spray of the reected wavelets which are created by the part of the wave which is trapped inside the potential well and oscillates periodically between the two potential barriers. A simple way to see it is from the simplication of the Lippman-Schwingwer expression for the transition probability when the scattered particles have energy which is equal to one of the resonance energies. The Lippman-Schwinger expression for the transition probability is T = jhjv + V G(E)V jij, where is a plane wave with positive momentum [11]. When E is equal to the real part of the energy-pole of the scattering matrix (i.e. E? i? is the resonance pole of the S-matrix, where? =h is the rate of decay of the metastable resonance state), then the rst term in the Lippman-Schwinger expression can be neglected and T ' jhjv G(E)V jij ' jhjv j res ih res jv ji(=? )j. res is the resonance wave function and therefore jhjv j res ij is the probability that the trapped particle inside the potential well shown in g. (a) will tunnel through the right potential barrier [13]. jhjv j res ij =h is the decay rate through the right potential barrier. Since?=h is the total decay rate and the quantum particle can tunnel out also through the left potential barrier, then jhjv j res ij =?= and therefore T ' 1. Using the (t; t ) formalism [14] the extension of the proof to the time-dependent Hamiltonian is very similar. In the time-dependent case: the Green operator should be taken as the inverse of

5 i. vorobeichik et al.: field-induced barrier transparency 115 E? H(x; t ) rather than of E? H(x), where H(x; t ) is the Floquet operator, H(x; t + V (x; t ); (8) and t serves as an additional coordinate [14]; is an eigenfunction of H with V = ; and res is the resonant eigenfunction of H (since! these resonances are very similar to those obtained as eigenfunctions of H when the time-averaged potential V e (x) is used). For more detailed derivation see ref. [15]. We can conclude that the eect where the tunneling through potential barriers is enhanced to almost 1% transition probability when the energy of the incoming particles is equal to the resonance energy is rst shown here to be valid for driven systems. It occurs when the frequency of the eld acting on a particle is larger than the vibrational frequency corresponding to temporary trapping between the two static potential barriers obtained upon cycle averaging. The non-trivial result is that even for relatively large values of the strong coupling between the dierent Floquet channels does not destroy the eect. Our numerical studies indicate that even when!, an enhancement of the tunneling through the driven potential barrier is obtained as well. The almost 1% transition probability through the driven potential barrier requires that the standard deviation of the energy of the incoming particles should be smaller than? which is the width of the corresponding resonance energy. Since? gets smaller as the particles mass is increased, it is clear that the 1% transition probability of heavy particles requires preparation of a very broad initial wavepacket. If the Hamiltonian is driven by the electromagnetic eld, the large nuclear mass implies that the eld intensity should be extremely high and we can exclude application of the present scheme to chemical reactions. As an example, for the barrier of height.45 ev opposing the transfer of hydrogen in a collinear model of the H+H! H +H reaction [9], with a mass m = 16 au and a frequency of the order of the barrier height, the intensity is I 1 17 W=cm when is 1 au. At such an intensity photoionization and photodissociation will \win" over a soft rearrangement such as atom exchange. On the other hand, it may be relevant to nuclear motions in very high eld intensities in the study of the possibilities to enhance nuclear fusion. Moreover, as was discussed recently by Coalson and co-workers [16] for a long-range molecular energy transfer (ET) and similarly for quantum wells in semiconductors, the giant-dipole character of these systems can provide strong-eld eects with weaker electric elds. Probably one could design a semiconductor system to exploit the same principle. Using short pulses (duration of a laser pulse which supports 15-5 oscillations of the eld is sucient to make our arguments which are based on time periodic eld to be valid [17]) denitely will help to prevent destruction of the ET material, whatever its detailed composition. Another factor that enables to get high values of for not too high intensity elds is the fact that in semiconductors the eective mass of the electron is about.1 times smaller than its mass in vacuum, m = :1m e. It is possible by nowadays technique of molecular-beam epitaxy to produce layers of materials with dierent conductivities so that the potential in a direction perpendicular to the layers is stepwise, with a series of rectangular barriers and wells [18]. Many electronic devices have been proposed and realized to take advantage of this situation [19]. An example [] is the use of layers of GaAs to produce wells and layers of AlGaAs or more generally Al x Ga 1?x As to build barriers. It is even possible to grade x to produce a potential of the desired form [1]. Our rectangular barrier is built with parameters which are all compatible with those met in this area. This work was supported in part by the Israel-US Binational Science Foundation. ***

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