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Tunneling with dissipation
G. Nimtz, H. Spieker, H. Brodowsky
To cite this version:
G. Nimtz, H. Spieker, H. Brodowsky. Tunneling with dissipation. Journal de Physique I, EDP Sciences, 1994, 4 (10), pp.1379-1382. �10.1051/jp1:1994194�. �jpa-00246999�
Classification Physics Abstracts
03.40K 42.50 73.40G
Tunnefing with dissipation
G. Nimtz, H. Spieker and H-M- Brodowsky
II.Physikalisches Institut, Universitàt zu KôIn, D-50937 KôIn, Germany
(Received 30 June 1994, received in final form 2 July1994, accepted 8 July1994)
Abstract. Dissipative tunneling was studied with electromagnetic waves. Experiments were carried eut with undersized waveguides loaded with a lossy dielectric medium which corresponds
ta a complex potential barrier. With increasing dissipation the Hartman elfect disappears, 1-e-, the traversal time becomes dependent on barrier length. The results are in agreement with a
recent phase time approach on complex tunneling.
1. Introduction.
Experimental data on tunnehng times of charged particles such as electrons are rather difficult to interprete. Coulomb interactions with the environment are unavoidable and may domi- nate experimental tunneling data. However, it has been shown recently that one-dimensional partiale tunneling is formally identical to the propagation of evanescent modes in waveguides (il-
The analogy has been evidenced in several microwave experiments [2-4] and in an optical experiment [Si. The microwave data are in agreement with Hartman's calculations for partiale tunneling [6]. In a phase time approach Hartman has shown that the tunneling time of wave packets is independent of the barrier length for opaque barriers (where kL < 1 holds, with k the central wave number of the tunneling wave packet and L the barrier length) [6, ii. Such
a behaviour eventually results in superluminal group velocities traversing opaque barriers iii.
This 'Hartman eifect' was confirmed by Enders and Nimtz recently in an analogous tunneling experiment with electromagnetic wave packets [2].
In this paper we report on traversai times of a complex barrier, i-e-, the barrier potential has
an imaginary component, so within trie barrier region dissipation takes place.
In the following section we present the dispersion of a waveguide loaded with a dielec- tric medium, which causes both dispersion and dissipation of the evanescent (tunneling) fre-
quency regime of the wavegmde. This complex evanescent frequency regime corresponds to
one-dimensional quantum mechanical tunneling through a complex potential barrier.
Experimental data on the dissipative tunneling are presented in section 3. They show that the Hartman elfect disappears with increasmg dissipation. Obviously the dissipative interaction
1380 JOURNAL DE PHYSIQUE I N°10
à
(a)
(b) i~'
w
PotenÙal 12 (C) ~l
Fig. 1. (a) Rectangular waveguide with a part of reduced cross section in the center. (b) Rect-
angular waveguide filled with dielectric media with refractive indexes ni and n2. (c) Sketch of the
corresponding photonic potential.
is time consuming and lengthens the barrier traversai time (of the signal maximum).
In the final section the experimental data are compared with two quite recently performed
calculations and they agree with the extended Hartman phase time approach of Raciti and Salesi [8].
2. Dissipative photonic tunneling.
The analogy to one-dimensional quantum mechanical tunneling cari be established with an
electromagnetic waveguide and frequencies such that the propagation has an imaginary wave
number. For instance, using a rectangular waveguide at microwave frequencies, the tunneling
or evanescent region is realized either by reducing the size of a part of the waveguide as shown
in figure 1a or by changing the refractive index inside a waveguide of uniform cross section as illustrated in figure 1b.
In the first case the dispersion relation is given by
k~
= (2irU/C)~(i (Uc/U)~)
where u is the frequency, c is the vacuum velocity of light, and uc
= c/2b
= c/Àc is the
cut-off frequency with b the waveguide width and Àc the cut-off wavelength. For u < uc the wavenumber becomes imaginary and we are dealing with the so-called evanescent or tunnel
mode, characterized by an exponential decay of the fields along the tunnel axis.
The evanescent regime of undersized waveguides has been successfully used to study the tun-
neling process [2, 3, ii. However, instead of changing the waveguide cross section, an evanescent
~2 ~2
200 o
_
0 , ~
ÎÎ c
-
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, , ~
j ~~°
~ Ù~~,,"
~
~' -200 '".,_ ',
, 1,
j ,"Î:"
,"
/
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,~_:" f -400 c/._
',',,~
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' [,
~
"§~ 600 ".._ '
-60 ~ ".
b ""
-800
6 8 10 12 6 8 0 2
Frequency (GHz) Frequency (GHz)
(a) (b)
Fig. 2. Transmission amplitude vs frequency (a) and phase vs frequency (b) for two barrier lengths
a, c = 49.0
mm and b,d = 64.7 mm. c and d are filled with a complex dielectric medium characterized by n2.
region and a corresponding potential barrier may also be constructed by loacling hollow rect-
angular waveguide sections homogeneously with dielectrics of diiferent refractive indices ni, n2
as displayed in figure 1b. For purely real (non-dissipative) dielectrics the evanescent frequency
regime is given by uci < u < uc2 with the cut-off frequencies uci := c/(2bnz) for1
= 1,2. The
corresponding photonic tunneling barrier is sketched in figure 1c. The dispersion relations are
k)
=
(2xunz/c)~(1- (ucz/u)~) for 1= 1,2
With the experimental arrangement of figure 1b one can also study the influence of dissipation
on barrier traversai times. The above dispersion relations still hold for the dissipative case, but
as n2 gets complex (non-zero imaginary part) the same is true for uc2 and so the evanescent
frequency regime is no longer well defined.
3. Experimental.
Experiments were carried ont with the set-up of figure 1b. The complex transmission function
was measured with a network analyzer as described in [2]. The connecting waveguides were
filled with a dielectric material (paraflin) with ni
" 1.50. Their cut-off frequency constitutes a
lower bound of uci " 6.33 GHz for the operating frequency. The center part of figure 1b, which acts as the tunnel region, was filled either with air (n2 " 1) or with a dielectric characterized
by the complex refractive index n2
" 1.16 +10.07 (carbon loaded urethane foam). The cut~olf
frequency of the empty waveguide was uc2
= 9.49 GHz. Measurements were made with two
dilferent barrier lengths of 49.0 mm and 64.7 mm.
The amplitude and the phase shift of the transmission as a function of frequency are shown
in figure 2. The experimental data show m consequence of the complex refractive index an increased transmission and phase shift in the tunneling regime of the empty waveguide below
1382 JOURNAL DE PHYSIQUE I N°10
9.49 GHz. For instance, at a frequency of1 GHz, the phase time (dq7/dw) yields group
velocities of 1.1,1.5, o-1, and o-1 times the vacuum velocity of light for the configurations a to
d, respectively. For the non~dissipative barrier the traversai times were again independent of the length, so that superluminal traversai velocities were observed. In the dissipative case the velocities stayed well below the vacuum velocity of light, so that the Hartman elfect disappears.
4. Discussion.
The experimental investigation has shown that with increasing dissipation the traversal time of the wave packets is increased, too. The phase shift becomes dependent on bottier length,
the Hartman elfect disappears.
Raciti and Salesi [8] have calculated the phase time for an electron traversing a complex
square barrier. Their approach is based on that of Hartman [6] but taking into consideration the imaginary potential comportent. They found that the Hartman elfect, 1-e-, a traversai time
independent of barrier length, is suppressed in the presence of an imaginary comportent in the potential. Recently the influence of a complex tunnel barrier was investigated by a path- integral solution of the telegrapher's equation by Mugnai et a1. [9]. This approach yielded
a decreasmg traversal time with increasing dissipation, m contrast to both to the phase time calculation by Raciti and Salesi [8] and to the experimental data presented here.
References
[1] Martin Th. and Landauer R., Phys. Rev. A 45 (1992) 2611; see aise Rev. Med. Phys. 66 (1994)
217.
[2] Enders A, and Nimtz G., J. Phys. I France 2 (1992) 1693; Phys. Rev. E 48 (1993) 632.
[3] Enders A. and Nimtz G., J. Phys. I France 3 (1993) 1089.
[4] Nimtz G., Enders A. and Spieker H., J. Phys. I France 4 (1994) 565.
[5] Steinberg A.M., Kwiat P-G- and Chiao R-Y-, Phys. Rev. Lent. 71 (1993) 708.
[6] Hartman T., J. Appt. Phys. 33 (1962) 3427.
[7] Olkhovsky V.S- and Recami E., Phys. Rep. 214 (1992) 339.
[8] Raciti F. and Salesi G., preprint INFN/BE-94/01 "Complex Barrier Tunnelling Times".
[9] Mugnai D., Ranfagm A., Ruggeri R. and Agresti A., Phys. Rev. E 49 (1994) 1771.
