Tunneling dynamics in a perturbed triple-well potential: Enhancement versus quenching of tunneling

Chemical Physics Letters 424 (2006) 218–224 www.elsevier.com/locate/cplett

Tunneling dynamics in a perturbed triple-well potential: Enhancement versus quenching of tunneling Kaushik Maji, S.P. Bhattacharyya

*

Department of Physical Chemistry, Indian Association for the Cultivation of Science, 2A&2B Raja. S. C. Mallick Road, Jadavpur, Calcutta 700 032, West Bengal, India Received 20 March 2006; in final form 8 April 2006 Available online 22 April 2006

Abstract We show that perturbation produced by coupling a triple well tunneling system to a harmonic mode can generally reduce tunneling duration as the coupling strength increases, although in the weak coupling regime, it may also cause tunneling-delay. A periodically varying well-depth is predicted to enhance tunneling duration in a triple well and create a ‘resonance-like’ suppression of tunneling at a critical frequency (xc) of oscillation.  2006 Elsevier B.V. All rights reserved.

1. Introduction Tunneling is a purely quantum mechanical phenomenon involving under-barrier transmission or over-barrier reflection. It was first invoked to explain the alpha-decay of a radioactive nucleus (1930). The discovery of Esaki (tunnel) diode and Josephson effect (1960) brought into focus the importance of tunneling phenomenon in the electronics industry. The importance of tunneling in chemistry has long been identified, explored and debated [1–3]. The atom tunneling reactions and tunneling diffusion of atoms were experimentally confirmed in the 1990s [2–6]. A vast majority of examples of tunnel effects in chemistry involve tunneling in a symmetric or asymmetric double-well potential [5,6]. Relatively, little seems to be known, however about tunneling dynamics in symmetric or asymmetric triple-well potential [7] although the triple-well potential has been practically realized in semiconductor super lattices with different (compound) compositions and has found applications in electronic devices relating to lasers, specially the intersubband semiconductor lasers [8,9]. In fact, the generic structure that is assumed to form the *

Corresponding author. Fax: +91 33 473 2805. E-mail address: [email protected] (S.P. Bhattacharyya).

0009-2614/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.04.036

building blocks of the active layers of the electronically pumped intersubband laser is a coupled triple quantum well element with an injector well, a central laser well and an extractor well. The carrier transport between the wells is characterized by a tunneling time s12 from the injector well to the lasing well and a tunneling time s23 from lasing well to the extractor well. The laser dynamics which is critically determined by the carrier transit time (st) through the structure has been modulated to achieve terahertz modulation bandwidth [8]. Since st depends upon other things (like ss, the intersubband radiative relaxation time, ssc, the time delay caused by intersubband scattering), it is important to understand typical features of tunneling dynamics – specially the characteristic tunneling times in a triple-well potential. The triple-well potential has also been invoked in the study of reaction dynamics of isomerization amongst three stable species A, B, and A 0 , where A and A 0 denote equally stable species represented by two outer wells while B represents a slightly less stable species represented by the central well. The intermediate isomeric state has been shown to modify the flow of phase points from A ! A 0 thereby also modifying the rate of A ! A 0 isomerization [10]. It would be interesting to investigate if there could be any role of the intermediate B state if the reaction A ! A 0 takes place

K. Maji, S.P. Bhattacharyya / Chemical Physics Letters 424 (2006) 218–224

by tunneling. Indeed, one of the new directions in the exploration of reactions by tunneling has been towards study of tunneling time (duration) that characterizes such processes. There has been claims both on theoretical and experimental grounds, that faster-than light signalling is possible in tunneling raising the specter of an apparent breakdown of causality [11–14]. Recently however, it has been stressed that propagation of information does not exceed the speed of light in the relevant experiments and there is no violation of relativistic causality [15,16]. The objective of the present communication has been to investigate the model dynamics of an isomerization process that occurs purely through the under-barrier process in a triple well in which all the well-depths are equal. Tunneling rate constants and the tunneling time (duration) are both computed. We also analyze the effects of perturbations on the computed tunneling time and their possible implications in atom or molecule transfer reactions occurring by tunneling in a triple-well potential. 2. Methodology and model Let us consider a triple-well potential of the form [17] 1 2 V ðx1 Þ ¼ m1 x21 x21 ðx21  a2 Þ 2

ð1Þ

which has three equally deep wells L, C, R (left, central and right) located at x1 = a, 0, +a, respectively. The harmonic frequency for the central well is x1 while it is 2x1 for both the left and the right wells. We propose to use the timedependent Fourier grid Hamiltonian method [18,19]] for following the dynamics of the system. The Hamiltonian of the system is given in the form H ðx1 Þ ¼ 

2 o2 h þ V ðx1 Þ 2m ox21

ð2Þ

‘m’ being the mass of a proton in atomic units. The coordinate space is uniformly discretized with a spacing Dx1 so that x1i = iDx1. The state vector jw(x1)æ in this discretized coordinate space is written as n X wp ðtÞjx1p iDx1p ð3Þ jwðx1 Þi ¼

219

The evolution equations can be easily and efficiently integrated numerically (by the 4th or 6th order Runge–Kutta method) once wq(t = 0), q = 1, 2, . . ., n are specified. If the wq(0)s are so chosen that w(x1, t = 0) is initially entirely localized in the well L and its energy is much below the barrier separating L from C or C from R, the initial state can delocalize into the well R only by tunneling . We may follow the process by computing hÆx1(t)æi from t = 0 to t = T and estimate the average slope

dhx1 ðtÞi dt

av

¼ hviav . If the sep-

aration between the minima at L and R is l0, the tunneling duration or tunneling time may be defined to be [21] sLR ¼

l0 ¼ sLC þ sRC ¼ 2sLC hviav

ðby symmetryÞ

ð7Þ

The reciprocal of sLR may be taken as the rate constant for isomerization by tunneling from the well L to R in the triple-well potential V(x1). Let us now modify our system by coupling it to a harmonic oscillator described by a potential V ðx2 Þ ¼ 12 m2 x22 x22 , where x1 and x2 are mutually orthogonal coordinates. The specific form of coupling chosen is V(x1,x2) = kx1x2, k being the coupling strength. The total Hamiltonian now reads p21 p2 1 1 þ 2 þ m1 x21 x21 ðx21  1Þ2 þ m2 x22 x22 þ kx1 x2 2m1 2m2 2 2 ¼ Hðx1 Þ þ H ðx2 Þ þ kx1 x2

Hðx1 ; x2 Þ ¼

ð8Þ The time-dependent states of the system are now described by superposing the products of the eigenfunctions of H(x1) and H(x2): XX wðtÞ ¼ cij ðtÞ/i ðx1 Þvj ðx2 Þ ð9Þ i

j

The time evolution of w(t) is then described by evolution equations for the amplitudes cij(t) dcij X X ¼ ih H kl;ij ckl ðtÞ; k ¼ 1; 2; . . . ; n1 ; dt ij kl l ¼ 1; 2; . . . ; n2

for each i; j

ð10Þ

where

These equations are integrated numerically to get w(t) and Æx1(t)æ. The tunneling times required are then determined as already described. The computed values of sLR for different values of k and x2 are expected to yield information on how the perturbation provided by an external agent could affect the rate of isomerization by tunneling. A third model to be investigated concerns the hamiltonian H(x1) when the well-depth or barrier heights are made to oscillate periodically, without causing any change in the location of the minima/maxima of the triple-well potential. An appropriate hamiltonian to describe such a system may be constructed as follows:

p2 1 2 H ðx1 Þ ¼ 1 þ m1 x21 x21 ðx21  a2 Þ 2m1 2

  p21 1 2 H ðx1 ; tÞ ¼ m1 x1 þ Dc sin ðxtÞ x21 ðx21  a2 Þ2 þ 2 2m1

p¼1

The orthonormality condition on the grid is [20] hx1p jx1q iDx1 ¼ dpq

ð4Þ

The use of Dirac–Frenkel variational principle leads then to the evolution equations for the grid amplitudes [18,19] i h

n dwp X ¼ hxq1 jH jxp1 iwq ðtÞ; dt q¼1

for p ¼ 1; 2; . . . ; n

ð5Þ

ð6Þ

ð11Þ

220

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The frequency(x) and amplitude Dc of the fluctuations are so chosen that no over-barrier excitation takes place. The transfer from the well L to R can therefore only take place by tunneling. We may adopt the method outlined in Section 3.1 to follow the dynamics and compute the tunneling time or rate as function of x or Dc.

Table 1 Eigenstate

Energy (a.u.)

Ground 1st(+) 1st() 2nd

0.009389061 0.0184633413 0.0184653835 0.0268931952

3. Results and discussion We begin by considering an initial state w(0) that is practically localized entirely in the left well (Fig. 1(a)). The state has an energy E < V0, where V0 is the barrier height. If w(0) is allowed to evolve, it may get delocalized into the central as well as the right well. The snapshots of the evolving jwj2 at later times are displayed in Fig. 1(b,c). There is hardly any probability of finding the proton in the central well. The initially localized state (the wave packet) appears to move back and forth from the left to the right well, avoiding, so to say, the central well, altogether. Since the energy throughout remains conserved and below the barrier top, this passage from the left to the right well can take place only by tunnel-

3.1. Tunneling in an isolated triple-well potential We have taken the distance between the twoqouter well ffiffiffiffi minima (RL) to be equal to 1.9 a.u. and x ¼ 0:5 a:u:, m m being the proton mass. A simple variational argument would suggest that the lowest energy eigenstate would be symmetric under inversion and non-degenerate while the first and second excited states should have opposite parities and be close in energy (not degenerate). Table 1 reports the first three energy levels computed by the FGH method(grid length = 6.0 a.u. and spacing = 0.025 a.u.) which reflect the pattern expected.

0.15 (a)

0.20

(b)

0.15 Probability

0.10

0.10

0.05

0.05

0.00 -3

-2

-1

0 1 Distance (a.u.)

2

0.00

3

-3

-2

-1

0

1

2

3

Distance (a.u.) 1.0

(c)

0.20

(d)

0.5

(a.u.)

0.15

0.10

0.0

-0.5

0.05 -1.0

0.00 -3

-2

-1

0 Distance (a.u.)

1

2

3

0

6

1x10

6

6

2x10 3x10 Time (a.u.)

6

4x10

6

5x10

Fig. 1. (a–c) The snapshots of evolving probability distribution at different instants of time in a triple well system. (d) Average position (Æxæ a.u.) versus time (a.u.) plot during tunneling in a triple well.

K. Maji, S.P. Bhattacharyya / Chemical Physics Letters 424 (2006) 218–224

221

L to the R well. The average speed Ævæ has been computed from the average slope of Æx(t)æ versus t plots. The distance separating the minima of the L and R well is 1.9 a.u. which yields an estimated tunneling duration (time) s = 1.5 · 106 a.u. and an average tunneling rate constant ks = 2.8 · 1010 s1. We may note here that the WL and WR (states localized in the left and the right wells, respectively) can be viewed as ‘in’h and ‘out’ of phase combinationsi of /þ and 1 þ  þ  1ffiffi 1ffiffi p p W ¼ ð/ þ / Þ; W ¼ ð/  / Þ . One may try / L R 1 1 1 1 1 2 2

ing. Since L and R wells are equivalent, we may interpret the passage of the packet from the L to the R state as an example of ‘tunneling isomerization’. We note here that the term ‘tunneling isomerization’ was specifically used for the first time to describe the spontaneous conversion of cis-formic acid to trans-formic acid by phonon-assisted tunneling in solid rare gases at low temperature [22]. Tunneling isomerization has also been observed between cis- and trans- hydroquinone at 16–75 K [23]. The ‘trajectory’ of tunneling isomerization process in the present example can be computed by plotting Æx(t)æ as a function of time elapsed (Fig. 1(d)). The ‘quantum-phase space’ accessed by the system during the tunneling isomerization is depicted in Fig. 2(a). The ‘quantum-phase space trajectory’ is closed and extends from the

to compute the transition rate from WL ! WR by invoking Fermi Golden rule which predicts that it would be proportional to (DEi/2)2, DEi being the magnitude of tunnel splitting of the states concerned.

0.08 (a)

0.06 0.04

(a.u.)

0.02 0.00 -0.02 -0.04 -0.06 -1.0

-0.5

0.0 (a.u.)

0.5

1.0

Probability 0.05 0.04 0.03 0.02 0.01 0

(b)

-2

-1

0

1

2 -2

-1

0

1

2

Y (a.u.)

X (a.u.) Probability 0.04 0.03 0.02 0.01 0 -2

Probability 0.04 0.03 0.02 0.01 0

(c)

-1

0 X (a.u.)

1

2 -2

-1

0

1

(d)

2 -2

-1

0

Y (a.u.)

1

2 -2

-1

0

1

2

Y (a.u.)

X (a.u.)

Fig. 2. (a) Quantum-phase space of the tunneling particle in triple well. (b–d) Snapshots of the evolving probability distribution of the perturbed system at different instants of time.

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K. Maji, S.P. Bhattacharyya / Chemical Physics Letters 424 (2006) 218–224

3.2. Tunneling isomerization in the presence of external perturbation Let us assume that our system is described by the hamiltonian of Eq. (8). Physically, it means that our tunneling system or the tunneling mode (characterized by a triple well) is coupled to another mode which is modeled by a harmonic oscillator, k being the strength of the coupling. The coupling with a harmonic oscillator may provide either a ‘sink’ or a ‘source of energy’ to the tunneling process. For the specific form of coupling chosen, the tunneling time for the transition from the L to the R well may therefore be expected to depend not only upon the strength of coupling, but also on the harmonic frequency of the oscillator, for the oscillator may either deliver energy to the tunneling system or take energy away from it depending upon the energy available to it. Fig. 2(b)–(d) display snapshots of the evolving probability distribution of the composite system at three instants of time. It is clear that the system has passed from an initially localized state in the left well to one on the right well. Fig. 3(a,b) display the dependence of the computed tunneling time s on the coupling strength k for different harmonic frequencies (force constants) of the perturbing oscillator. Generally, s is seen to decrease as k increases from 0 to 0.075 a.u. The coupling with the harmonic mode is therefore expected to cause a faster tunneling isomerization in a triple well. Since total energy is conserved, energy must have been transferred from the harmonic perturbing mode to the tunneling mode. It appears that higher the k, the more efficient is the transfer of energy to the tunneling mode. For k = 0.25 a.u., there is an initial increase in s in the low k region(0.0–0.01 a.u.) followed by a sharp decrease as k increases further. For a low frequency perturbing oscillator, therefore weak coupling may cause a

slowing down of the tunneling isomerization. From Fig. 3(a,b), it is clear that for 0.01 < k < 0.075, s(k = 0.25 a.u.) > s(k = 0.5 a.u.), which means coupling with a harmonic mode enhances the rate of tunneling isomerization more efficiently for an oscillator of higher harmonic frequency (force constant). However, for k > 0.075 a.u., the situation is reversed. The harmonic perturbing mode may therefore either quench or promote tunneling isomerization depending upon its frequency and the strength of coupling. Fig. 4(a,b) show how the computed ‘tunneling time’ varies as the harmonic frequency (force constant) of the oscillator increases for different strengths of coupling. For k = 0.05 a.u., there is a sharp decrease in s as the force constant (k) increases (Fig. 4(b)). Beyond k = 0.5 a.u. there is no further decrease in s. For k = 0.01 a.u., virtually no effect is detected on s as k increase (Fig. 4(a)). A high frequency mode coupled to a triple well tunneling mode could therefore enhance the rate of isomerization by tunneling. Proton-tunneling in symmetric or asymmetric double-minimum potential wells have been experimentally investigated for a long-time. Two of the most thoroughly investigated systems are malonaldehyde (MA) and tropolone (TRN). The central quantity of interest has been the magnitude of tunneling splitting (Dm). It has been experimentally observed that the magnitude of the tunneling splitting depends remarkably upon the excited vibronic state [24]. In the specific case of TRN, it has been experimentally observed that there are: (i) tunneling suppressing modes (m11, m25 and m26); (ii) tunneling promoting modes (m13 and m14) and (iii) modes having little or no effect on tunneling [24]. Our theoretical investigation with a triple well system also predict the possible existence of similar modes. It would be interesting to have experimental confirmation of our findings in a real life triple well tunneling system.

Force Constant=0.25 a.u. Force Constant=0.50 a.u.

1.6

Coupling 0.01 a.u. Coupling 0.05 a.u.

1.7

1.5

(b)

1.2 1.1 1.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Coupling (a.u.)

Fig. 3. (a–b) Tunneling time (s a.u.) versus coupling strength (k a.u.) for coupling with different harmonic modes corresponding to force constants 0.25 a.u. and 0.50 a.u., respectively.

1.5

6

1.3

(a)

Time (X10 )(a.u.)

6

Time (X10 )(a.u.)

1.6 (a)

1.4

1.4 (b)

1.3 1.2 1.1 0.1

0.2

0.3 0.4 Force Constant (a.u.)

0.5

Fig. 4. (a–b) Tunneling time (s a.u.) versus force constant plots shown for different strengths (k) (0.01 a.u. and 0.05 a.u.) of coupling, respectively.

K. Maji, S.P. Bhattacharyya / Chemical Physics Letters 424 (2006) 218–224

3.3. Tunneling in an oscillating triple well When the tunneling mode is coupled to an external timevarying field, the system may be described by the hamiltonian H(x,t) of Eq. 11 in which the triple-well potential is assumed to have been dressed by the external field such that the well-depths oscillate periodically without any change in the locations of minima/maxima. The dressed potential is shown in Fig. 5(a), where the unperturbed potential is also displayed (circular dots). The fluctuations 0.05

(a)

Potential (a.u.)

0.04

0.03

0.02

0.01

0.00 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

223

may now set up Rabi like oscillations among the low-lying states of the unperturbed system, thereby increasing the duration of tunneling. Since , however, there could be a resonance-like situation for a specific frequency of the welldepth oscillation, the tunneling time may reveal a resonant-behavior as a function of the parameter of the perturbing field. Fig. 5(b) shows the behavior of s(computed) as function of log x. There is a sharp (narrow) maximum at x = xc(say). To understand the nature of the resonance-like behavior of s, we have plotted in Fig. 5(c) hx/jH22  H33j against log x. H22, H33 being the energies of the first and second excited states of the unperturbed triple-well oscillator. It is clearly seen that the resonance-like exaltation in s versus log x plot takes place for x = xc for which hxc = jH22  H33j. It appears that oscillations in the well-depth can cause tunneling-delay by inducing transitions among the eigenstates of the unperturbed triple well, at a matching frequencies of oscillation. An appropriately designed external time-varying field may therefore be used to modulate the potential seen by the tunneling system and control the rate of isomerization by tunneling. It would be interesting to have experiments designed to test the prediction in a triple-well proton-tunneling system.

Distance (a.u.)

4. Conclusions 1.0

(b)

Time (a.u.)X10

5

0.9 0.8 0.7 0.6 0.5 0.4 -5.5

-5.0

-4.5

-4.0

-3.5

-3.0

log (ω)

(c)

4

A simple systematism for computing tunneling time (duration) proposed earlier by us [21] is shown to provide meaningful interpretation of the dynamics of tunneling in a triple-well potential. It is predicted that perturbation produced by coupling the system to a harmonic mode may either promote or quench tunneling isomerization rate in a triple well depending upon the frequency of the perturbing mode and the strength of coupling. Barrier oscillation can cause enhancement of tunneling duration revealing a resonance-like suppression of tunneling rate. It is curious that the particle practically does not appear to spend any time in the central well during tunneling from the left to the right well. It would be interesting to investigate the dynamics of tunneling in a triple-well potential in which well-depth fluctuates randomly due to thermal perturbations. We hope to return to the problem in the near future.

h ω/|H22–H11|

3

Acknowledgement 2

One of us (K.M.) thank the CSIR, Government of India, New Delhi, for the award of a Senior Research Fellowship.

1

0 -5.6

-5.4

-5.2

-5.0

log (ω)

-4.8

-4.6

Fig. 5. (a) Fluctuating triple-well potential with fixed minima/maxima. The unperturbed potential is displayed by circular dots. (b) Tunneling time (s a.u.) versus log x plot. x is the frequency of the external field. (c) Plot of hx/jH22  H33j versus log x. H22, H33 being the energies of the first and  second excited states of the unperturbed triple-well oscillator. x is the external field frequency.

References [1] R.P. Bell, Tunnel Effect in Chemistry, Chapman and Hall, London, 1980. [2] T. Miyazaki (Ed.), Atom Tunneling Phenomena in Physics Chemistry and Biology, Springer, Berlin, 2003, and references cited therein. [3] V.I. Goldanski, L.I. Trakhtenberg, V.N Fleurov, Tunneling Phenomena in Chemical Physics, Gordon and Breach Science Publishers, New York, 1989.

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[4] H. Yamamoto, K. Miyamoto, T. Hayashi, Phys. Stat. Sol 209 (1998) 305. [5] D. Muganai, A. Ranfagni, L. Ronchi, Phys. Lett. A 247 (1998) 281. [6] G.K. Ivanov, M.A. Kozhushner, L.I. Trakhtenberg, J. Chem. Phys. 113 (2000) 1992. [7] S. Bu, B.-H. Wang, P.-Q. Jiang, Int. J. Mod. Phys. B 18 (2004) 2669. [8] W.M. Lee, K.A. Shore, E. Scho¨ll, Appl. Phys. Lett. 63 (1993) 1089. [9] N. Mustafa, L. Pesquera, C.Y.L. Cheung, K.A. Shore, IEEE Photonic. Technol. Lett. 11 (1999) 527. [10] S. Jang, M. Zhao, S.A. Rice, J. Chem. Phys. 97 (1992) 527. [11] E.H. Hauge, J.A. Støvneng, Rev. Mod. Phys. 61 (1989) 917. [12] R. Landauer, Th. Martin, Rev. Mod. Phys. 66 (1994) 217. [13] A.M. Steinberg, Phys. Rev. Lett. 74 (1995) 2405. [14] A.M. Steinberg, P.G. Kwait, R.Y. Chiao, Phys. Rev. Lett. 71 (1993) 708.

[15] A. Ranfugni, P. Fabeni, G.P. Pazzi, D. Mugnai, Phys. Rev. E 48 (1993) 1453. [16] P.C. Davies, Am. J. Phys. 73 (2005) 23. [17] S.-Y. Lee, J.-R. Khang, S.-K. Yoo, D.K. Park, C.H. Lee, C.-S. Park, E.-S. Yim, Mod. Phys. Lett. A 12 (1997) 1803. [18] S. Adhikari, P. Dutta, S.P. Bhattacharyya, Chem. Phys. Lett. 199 (1992) 574. [19] S. Adhikari, S.P. Bhattacharyya, Phys. Lett. A 172 (1992) 155. [20] C.C. Marston, G.G. Balint Kurti, J. Chem. Phys. 91 (1989) 3571. [21] C.K. Mondal, K. Maji, S.P. Bhattacharyya, Int. J. Quantum Chem. 94 (2003) 94. [22] M. Petterson, E.M.S. Macoas, L. Khriachtchev, M. Ra¨sa¨nen, J. Chem. Phys. 117 (2002) 9095. [23] N. Akai, S. Kudoh, M. Takayanagim, M. Nakata, Chem. Phys. Lett. 356 (2002) 133. [24] H. Sekiya, Atom tunneling and Molecular Structure, pp. 201–231 in Ref. 2.