Resonances in quantum mechanical tunneling

Volume 163B, number 5,6

PHYSICS LETTERS

28 November 1985

R E S O N A N C E S IN Q U A N T U M M E C H A N I C A L T U N N E L I N G Michael Martin N I E T O a, Vincent P. G U T S C H I C K b, Carl M. B E N D E R c, Fred C O O P E R a and D. S T R O T T M A N a aTheorettcal Dwlswn, Los Alamos Nattonal Laboratory, Umverstty of Cahforma, Los Alamos, NM 87545,USA bHealth, Safety, and Environment Dwtswn, Los Alarnos Nanonal Laboratory, Unwerstty of Cahforma, Los Alamos, NM 87545, USA CPhystcs Department, Washmgton Unwersuy, Samt Louts, MO 63130, USA Received 19 June 1985 In asymmetric double-well potentaals, it can be tacatly assumed that a wave functmn m the hagher-energywell (false vacuum) wall always tunnel to the lower well, gwen enough Ume However,m general this as not true Whether a state can sxgmficantlytunnel to the true vacuum is a very sensatavefunctton of the shape of the potentaal We dlustrate tlus vath analytac and numerical examples Thus, ff there as not dlsslpalaon or coupling to other modes, a wave functmn may not tunnel

In discussing inflationary models of the urnverse, a quantum-mechamcal approximaUon of the complete field theory is often used to provide simple insight [1] Such models consider confining double-well potentials in which one well (say the right-hand side) is the true minimum whereas the other well is a secondary m i m m u m (false vacuum) The problem at tins stage is one of quantum mechanics, independent of the physical motwatlon b e h m d it. Historically, our intuition on tunneling comes mainly from two places. The first is from alpha decay [2]. There we know that the nucleus eventually decays; it is a resonance coupled to the continuum. The second source of our intuition is f r o m symmetric double-well potentials. There it is well known [3] that the ground and first excited states are almost degenerate. The ground state has a symmetric wave function (no nodes) and the first excited state has an anusymmetric wave function (one node). The time it takes for a wave packet originally located on one side to get to the other side is given b y ~ffi ¢ r h / ( E x - E o ) . The wave packet tunnels back and forth in multiples of this oscillation Ume. F r o m the above m t u m o n it might be assumed that g~ven any double-well potential the same

336

t h m g will be observed, the "particle" tunneling through a b a r n e r coherently (1.e., the wave pocket retains its shape). It is the purpose of this note to illustrate that such an assumption is incorrect T o do this, we consider the dimensionless Schr6dinger equataon [-d2/dx

2 + V(x)] • = l dOl/dt,

(1)

with potential (see fig. la)

V=(x+2)2+U,

= [x - ( 4 +

x<~O,

o.

(2)

(For U = 0 this is the case discussed m ref. [3].) We begin with this example since it can be treated analytically. The elgenfunctions to the SchrSdmger equation on the left and the fight are D,,_ v/2( - Vr2( x + 2)) and D,(vr2[x - (4 + U)I/2I) , where the D are parabohc cylinder functions. The elgenvalues are E , = 21, + 1, which are determined b y satisfying the boundary condition at

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Volume 163B, numbex 5,6

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U = 0 and

28 November 1985

2

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= 0 and

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X Fig 1 The three potentaals that we study m thts paper' (a) is that of eq (2), (b) is that of eq (9), and (c) is that of eq (11)

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false-vacuum state ever penetrates to the right-hand side, for any time, for arbitrary U" p~H~Xs(U) =

25

05

-0 5

i

i

i

0

I

i

2

U F~g 2 A plot of vn versus U for 0 ~
the origin. F o r U ~ 0 this condition IS

D'~_v/2(-2v/2) D~'(-v~-(4 + U) I/2) D , _ u / 2 ( - 2 V ~ ) - D~(-V~-(4 + U)X/2) "

(3)

( F o r U = 0 either D or D ' is zero, with a matching change in sign for the latter case.) The U dependence of the eIgenvalues is shown in fig. 2 The time evolution of any initial wave function q'0 is then ~o(X, t) = ~

an~nexp [ - i ( 2 v . + 1 ) / ] ,

(4)

n=0

a, = (qbnl~o),

't'o = ~ r - 1 / ' e x p [ - ½(x + 2) 2] Putting tins into eqs. (4)-(5) one can then calculate the m a x i m u m probabihty that the

m f0 dx ~/~'(x, t ) % ( x , t)

(7)

When one does tins one finds the result of fig 3a When U = 0, there is tunnehng, just as predicted for the symmetric case [3] However, going away from U = 0, there is a sharp decrease in the m a x i m u m amount of the wave packet that ever goes to the .right of x = 0 PaHs(U)maxquickly reaches a very low value, and then rises sharply again near U = 2. The process is repeated over and over. (In addition to the figure, the maxima and m i n i m a have been calculated for integers out to 12 ) T o understand what IS going on, look at the first three exgenfunctlons of eq. (2) for U = 0, 1, and 2 (figs. 4a-4c) At U = 0, the false-vacuum eIgenfunctlon of eq. (6) almost equally overlaps with $0 and $i on the left-hand side. When the tunneling time r is reached, these two eigenfunctlons are out of phase b y or, and so the overlap is on the right-hand side. The overlap of xI'0 with $2 is almost zero because of the node of ~2 near x = - 2 For U = 1, o/0 has an overlap almost exclusively with 1~1 winch now, however, is almost entirely on the left Therefore, since It'0 has no slgmficant overlap with an elgenfunctlon with large probability on the right, It can never tunnel to the nght. When U = 2, we have a situation similar to U = 0, except that the two almost degenerate elgenstates are now $i and qb2 They b o t h have nodes near x = 2.4. Tins means that when the wave packet tunnels to the right, it wtll not be a coherent tunnehng, but rather will have two humps. Tins IS shown in fig 5,1 Before going on, look agmn at fig. 2. Comparing the eigenvalues at U = 2 with those at U = 0, one

(5)

where the ~n are the normahzed D ' s , appropriately matched at the origin. N o w consider an initial wave function app r o p n a t e for the false vacuum; i.e.

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(6)

,1 The reader should note that we obtamed fig 5 both analytically and also by numerically integrating the time-dependent Schr6chnger equation Our numencal mtegrataon was a modlficatmn of that used to &scuss the time-evolution of coherent states for the hydrogen atom [4] These two methods agree to w~tlun a small fractaon of a percent (much less than the width of the hne representing the wave packet) This is a venficatlon of the txme-mtegrau o n program whach we use for our other examples

a

28 November 1985

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Volume 163B, n u m b e r 5,6

Parabolas

b

1

Polynomial I

_Y 0.

U

c

Square

d

Wells

U

Fig 3 A hne plot of the maxunum probabthty wl~ch can tunnel from the false vacuum for the potenUals (a) m eq (2), (b) m eq (9), and (c) m eq (11) The tame evoluttons of the wave funcUons were calculated for tames as large as 300 These plots have an accuracy of a few percent The tunnehng umes at the m a m m a are the followng (a) • = 38, 28, and 25 for the m a m m a at U ffi 0, 2, and 4, (b) ¢ ~ 58 and 69 for the mamma at d ffi 0 4 and I 975, (c) • ffi 44, 32, and 30 for the mamma at U ffi 0, 1 98, and 5 7 Observe that even at the resonance peaks the m a x u n u m probabthty that tunnels to the true vacuum ~s not 100% This is because there always is some fracuon of the wave packet that remains to the left of x ffi 0 339

Volume 163B, number 5,6

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U=O

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28 November 1985

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Fig 4 The first three elgenvectors ¢0, Cx, and ¢2 of eq (2) for (a) U = 0, (b) U = 1, and (c) U = 2 The ground state ¢o is represented by the sohd hne, ¢1 by the heavy dashed hne, and ¢2 by the hghfly dashed hne

Volume 163B, number 5,6

PHYSICS LETTERS

U=2 t = 0 and 0.6

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28 November 1985

both pulling the right-hand well down w~th respect to the false vacuum and also are making it wider with respect to the false vacuum for a given value of V. We now proceed to give numerical examples which do these two things separately. As another example consider the potenUal (fig. lb) V = [5/(18)2] (x + 3)2(x - 2 ) ( x - 6),

'!

X~< 9 ,

= _ (25)2/(12)(16),

9 ~ x ~
= [5/(18)2] (x - a + 3 ) 2 ( x - a - 2)

cD

x (x-d-6),

(9)

Tins potential, with d -- 0, was used in a study of the fimte element method [5]. Making d > 0 is a widening of the potenUal. Starting with the false-vacuum wavefunction 0.0

(•

~t'0(%r)-l/' exp [ - ~ ( x + 3)2],

' 2 i . . . . 22' " 6 ' " 5 ' " 4 ' " g X

Ftg 5 For U ~ 2 m the potentml of eq (2), we show the false vacuum wave packet (sohd hne), and the wave packet at t = 28 (dashed hne), when it has tunneled The tunneled packet was obtmned by two methods The small squares are from the analyUc method of eq (4) and the dashed curve xs from numencally mtegraung [4] the tune-dependent Schr&imger equauon

sees that %+I(U = 2) = % ( U = 0 ) + l .

(I0)

in fig. 3b we plot the maximum tunnehng probabxhty as a funcuon of d. We see the same phenomenon as for the parabola case. The spikes are located at d = 0.4 and at 1 975. When ~t tunnels to the right, the wave packet has three humps in the former case and four humps in the latter case. As a final example we lower the mimmum of the potential of the true vacuum. It is the potential (fig. lc)

(8)

We are clearly observing a resonance condition. It is similar to matching cavities with electromagnetic waves. When two wells are tuned correctly, the waves can pass back and forth. The tuning is caused by integrating the curvature of the wavefunction properly, tlus curvature being [E - V(x)]. For the present harmonic oseiUator-like potential, this means a jump of U = 2. Note that the spikes at U = 0, 2, 4, etc., are getting broader with increasing U. This is because, as the right-hand side of the potential becomes wider, gaps between energy levels ar6 becoming narrower and the system is behaving as if there is a "continuum" of states on the right, as m the case of alpha decay. Also, m the variable U, we are

V = 6 - x, =0, =2, -- - U , = 6 +x,

x ~ -4, -4~
(11)

Here the potential consists of square wells located inside a confining hnear potential. Startmg with the initial false-vacuum wavefunction q,0 = (~r9/2~r 2)-1/4 exp [ _ ½. ~r2( x + ~)2], (12) the maximum transition probahility ~s shown in fig. 3c. The main features of tins graph are unchanged from figs. 3a and 3b. 341

Volume 163B, number 5,6

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W e c o n c l u d e that the p r o b a b l h t y of q u a n t u m m e c h a m c a l t u n n e h n g from a relative m l m m u m to a true n u n l m u m ~s a very sensitive f u n c t m n of the shape of the p o t e n t i a l ~2 I n general the p r o b a b l h t y is very small, a n d when t u n n e h n g does occur, m general it wdl not be coherent (the wave packet wall n o t r e t a i n its shape after It tunnels) To insure decay f r o m a relative m l m m u m there must be either dlss~patlon or, in what a m o u n t s to the same t h i n g in field theory, c o u p h n g to other modes

*~- This is a completely general observation about tunnehng For example, one can consider a simple 2 × 2 matnx system, which has just two states, with a small perturbatmn and study the time dependence In general the transition probablhty to the lower state Is small Another example can be found m the work of Cnbb [6], who has used a dens~ty-mamx approach to study resonant tunnehng between asymmetric double wells m chemical systems

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28 November 1985

O n e of us ( M M N ) ".ashes to t h a n k J o h n Bell a n d Stuart R a b y for emphasizing the i m p o r t a n c e of m o d e - c o u p l i n g in the full field theory.

References [1] K Sato, m Cosmology of the early umverse, eds L Z Fang and R Ruffim (World Scientific,Singapore, 1984) p 165 [2] E Fermi, J Orear, A H Rosenfeld, and R A Sehluter, Nuclear physics, Rev Ed (Umverslty of Cbacago, Chicago, 1950) Ch III [3] E Merzbacher, Quantum mechamcs, Second Ed (Wdey, New York, 1970) p 65 [4] V P Gutschxck and M M Nleto, Phys Rev D22 (1980) 403 [Appendix B], J M Hyman, Advances m computer methods for partial d~fferenUalequatmns-III, eds R V~chevetskyand R S Stepleman (IMACS, Bethlehem, PA, 1979) p 313 [5] C M Bender, F Cooper, V P Gutsclnck, and M M Nleto, Phys Rev D, to be pubhshed [6] P H Cnbb, Chem Phys 88 (1984)47