Explanation of superluminal phenomena based on wave-particle duality and proposed optical experiments

Transcription

1 Exlanation of suerluminal henomena based on wave-article duality and roosed otical exeriments Hai-Long Zhao * Jiuquan satellite launch center, Jiuquan, 73750, China Abstract: We suggest an exlanation for suerluminal henomena based on wave-article duality of hotons. A single hoton may be regarded as a wave acket, whose satial extension is its coherence volume. As hoton roagates as a wave train, its velocity is just the seed of light in vacuum. When it tunnels through a barrier as a article, its wave function collases and it travels faster than light. But suerluminal roagation can only occur within the coherence length, and the duration is constrained by uncertainty rincile. On the other hand, a article with non-vanishing mass cannot travel faster than light. So suerluminal henomena do not violate causality. We exlain the rinciles of existing suerluminal exeriments and roose three tyes of exeriments to further verify suerluminal henomena. The first is to show that a single hoton is equivalent to a wave acket, which occuies certain satial volume. The second demonstrates that suerluminal henomena can only occur within the coherence length. The third indicates that negative and suerluminal grou velocity in anomalous disersion medium is merely a reshaing henomenon of the ulse, and it will become subluminal at large distances. Keywords: suerluminal; negative grou velocity; barrier tunneling; uncertainty rincile 1 Introduction Various exeriments have been erformed on suerluminal henomena in recent years, which include barrier tunneling of hotons [1,], microwave tunneling through undersized waveguide [3,4], suerluminal exeriments of microwave in oen sace [5-8], microwave traversing double-rism [9-11], light or electromagnetic ulses traveling through anomalous disersion region [1-16]. For a review of suerluminal issue, see [17]. The suerluminal henomena in barrier tunneling are usually exlained with evanescent waves, which are accelerated through otential barriers and therefore travel faster than light. The negative grou velocity in anomalous disersion region is believed to be related to reshaing of the ulse. Both of the suerluminal henomena are further analyzed in the aer. The exlanation of barrier tunneling is based on the wave-article duality of hotons. We think that a hoton can tunnel through a barrier as a article, and the tunneling is actually a collasing rocess of the wave function. The traversal velocity during the tunneling rocess may be suerluminal. However, the advancement shift of the eak of a ulse in anomalous disersion region is not the true suerluminal roagation, it is only the ulse shae deformation arising from the different frequency (or olarization) comonents traveling with different velocities in medium. The roagation seed of each hoton of the ulse cannot exceed the seed of light in vacuum. The negative grou velocity can be obtained only when the satial length of the ulse is much larger than the medium thickness. A coule of exeriments are roosed to further verify suerluminal henomena. Exlanation for suerluminal henomena.1 The suerluminal tunneling of hotons through a barrier Let s start with the satial extension of a single hoton. We know that the energy of a hoton is E = hω and the wavelength is λ = πc / ω with c the seed of light in vacuum. In fact, a monochromatic hoton with an exact angular frequency ω does not exist according to uncertainty rincile. For a hoton with energy E = hω, we should regard that its frequency fluctuates around ω. Accordingly, a hoton should be viewed as a wave acket instead of a oint-like article and its satial * address: zhlzyj@16.com 1

2 extension is within a limited region. Then how large is its satial extension? It cannot be the scale of one wavelength, otherwise we can hardly understand the single-hoton interference exeriment [18]. As interference can only occur when the otical ath length difference is smaller than the coherence length, we may think that the extension of a single hoton in roagation direction is its coherence length. For a ulse, its coherence length is l = λ / Δλ with Δ λ the sectral width. For a single hoton, Δ λ should be understood as the uncertainty of the wavelength λ. The coherence length of a hoton is usually larger than its wavelength, it may comrise several to thousands of wavelengths. Similarly, the whole satial extension of a hoton is its coherence volume. The wave acket of a single hoton is in essence equivalent to that of a ulse excet that there are large numbers of hotons within the coherence volume due to their bosonic roerty. In order to have an intuitive understanding of suerluminal roagation in tunneling rocess, we give an analogous examle in our life. Suose a walker is walking with the ste of s and the frequency of f, and there is a road-block with the height of H and width of a over the way. The height that the walker each time raises his feet is H 0. If H 0 > H, the walker can waltz through the road-block in most of the cases. But in a secial case, for examle, the walker s foot just touches the edge of the road-block, he would stumble or even fall down. So the walker may ass through or fall down based on the different start oints. This examle corresonds to the situation where a beam of light is incident on a medium. Some of the hotons will ass through and the other will be reflected based on their different initial hases or quantum states. Now let H 0 < H. If the walker still walks in his initial manner, he certainly cannot ass through. But he can lea over the road-block given his leaing distance is larger than a. This situation corresonds to the barrier tunneling of hotons. Suose the energy of the hoton is E and the height of the barrier is V0. By analogy with the above examle, the hoton must borrow at least the energy ΔE = V0 E to traverse the barrier. Then another question arises: how far can the hoton tunnel through the barrier each time? The tunneling distance cannot be arbitrary long. As interference can only occur within the coherence length, we assume it to be the coherence length. In fact, when we use Schröedinger equation to solve the roblem of barrier tunneling, we have imlied that the tunneling should occur within the coherence length, which can be seen from the wave function in the barrier region ix / h ix / h ψ = Ae + Be, (1) which means that the incoming and reflected waves are coherent, otherwise the subsequent calculations of coherent suerosition are invalid. So tunneling as well as resonance transmission must occur within the coherence length. To ensure that the forward and reflected waves in the barrier region are coherent, the coherence length of the waveacket must be at least twice the barrier thickness. We may continue to imagine: what haens during the tunneling rocess? We see that tunneling is unlike wave roagation. For the instance of the walker leaing over the road-block, both of the initial ste and velocity do not have any sense. Similarly, we think that when a hoton traverses a barrier, it is as a article instead of as a wave train. In other words, the wave function of the hoton collases. c is only the seed limit of light roagating in vacuum, but for the instance of hoton tunneling through a barrier, its velocity can exceed c, just like the instance that the walker s leaing seed is greater than his walking seed. Besides, we see that when a hoton tunnels through the barrier as a article, its hase remains unchanged, which just exlains the rincile that the hase of evanescent wave remains constant during the tunneling rocess. A hoton needs to borrow energy Δ E to tunnel through a barrier, which we think can be borrowed from vacuum given that it can be returned within the time interval Δt. In this case, the

3 uncertainty relation can be written The borrowed energy should at least be h ΔE Δt. () ΔE = V0 E. According to above relation, we have h Δt. (3) ( V 0 E) It can be seen that when the height of the otential barrier is fixed, the tunneling time of the hoton Δ t is no longer than h / ( V0 E0 ). This is the uer limit for a article to traverse a barrier, i.e., for a fixed article energy and barrier height, the tunneling time of the hoton cannot exceed the value given in Eq. (3), otherwise the article cannot tunnel through the barrier. If we only increase the thickness of the barrier and do not change the height of the barrier, Δ t will tend to the saturation value h / ( V0 E0 ). This is the saturation effect for a article to tunnel through a barrier. It is obtained directly from the uncertainty rincile and similar to Hartman effect. But the exression is different from the transmission time of Hartman effect, which is [19] h τ =. (4) E( V 0 E) Eq. (4) is obtained from Schröedinger equation, so it holds only for non-relativistic articles. While Eq. (3) holds for all the articles, including relativistic and non-relativistic articles. We summarize as follows. When the barrier height V0 is less than the hoton energy E, most of the hotons can ass through the barrier in wave state with a seed less than c. If V0 > E, only a small fraction of the hotons can tunnel through the barrier in article state by borrowing energy from vacuum with a seed greater than c. In this case, the coherence length of the hotons must be two times larger than the barrier thickness and the tunneling time must be within the time interval allowed by uncertainty rincile.. The suerluminal roagation in anomalous disersion media The roagation rocess of light in medium is much more comlicated than that of in vacuum. In addition to the wave roagation of hotons, there exist interactions between hotons and electrons. When the hotons ass through the medium, they may interact with electrons and be absorbed and then be reemitted. It should be noted that the absortion and emission rocesses are also the collasing rocess of the wave functions. Both of the rocesses may roceed at a suerluminal velocity. But there will be a break between the two rocesses, so the whole roagation rocess is subluminal. Only the fraction of the hotons that do not interact with the electrons can travel with the seed of c, and these hotons form the wavefront. As there are large numbers of electrons in medium, the robability that the hotons do not interact with any electron is the smallest and so is the amlitude of the wavefront. The second smallest amlitude is the hotons that interact with only one electron and the next is the hotons that interact with two electrons. Certainly, the robability that the hotons interact with all the electrons is also the smallest. For a light or electromagnetic ulse which comrises large numbers of hotons, we should also take into account the shae deformation arising from the different roagation velocities of the hotons with different frequencies, i.e. reshaing henomenon. This can be exlained with the suerosition of waves or interference effect. We have suosed that a hoton occuies certain satial volume and its extension in roagation direction is its coherence length. When a large number of hotons with different roagation velocities coexist within the coherence length in disersion medium, the interference effect of these hotons will result in the shae deformation comared to its 3 V 0

4 initial shae. For normal disersion medium, the interference effect will result in the delay of the eak of the ulse. While for anomalous disersion medium, the interference effect will result in the advancement shift of the eak. We may understand intuitively the reshaing henomenon as follows. The hotons with higher frequency have faster roagation velocity in anomalous disersion medium, while those with lower frequency travel slowly. As the hotons with higher frequency concentrate mostly in the rising and falling edges of the ulse, the rising and falling edges of the ulse will shift forward comared to the initial ulse. In this case, only the shae of the ulse changes. The wavefront does not advance, i.e. its velocity is still c. In the exeriment of Wang and co-workers [1,13], the time required for hotons to traverse the 6cm length of the cesium vaor cell is 0. ns, while the advancement shift of the rising and falling edges of the ulse is 6 ns. Even if the ulse roagation in the cell did not require time, the advancement shift of the eak would be 0. ns. This is the uzzle brought out by the exeriment. In order to solve this aradox, let s start with the exression of grou velocity dv vg = v λ, (5) d λ where v is hase velocity, which is the average velocity of the whole ulse after taking into account the interactions between hotons and electrons and is always smaller than c. The second term in the above equation is the deformation velocity of the ulse arising from the interference effect during the roagation rocess. For anomalous disersion medium, we have dv / dλ < 0 and the eak of the ulse will shift forward. The reshaing rocess is also a rocess of wave acket collase and may roceed at suerluminal seed. For anomalous disersion medium, it can be seen from Eq. (5) that v g is always ositive. Then how is negative grou velocity obtained? We start with the definition of grou velocity v g = dω / dk, where ω is angular frequency and k = π / λ is wave number. With ω = v k, we have dω d( v k) dv dv v g = = = v + k = v λ, (6) dk dk dk dλ which is just the formula derived by Rayleign. On the other hand, we can also get v g dω dω dω = = = dk d( ω / v d ) ω ω dv v v v c / n c = = = ω dv f d( c / n) 1 1 n + f v dω c / n df dn df. (7) For anomalous disersion medium, we have fdn / df < 0. If n + fdn / df < 0, negative grou velocity will aear, and Eq. (7) is the theoretical foundation of the existence of negative grou velocity. For examle, the discussions of negative grou velocity in [13] and [15] are based on Eq. (7). We see that both Eqs. (6) and (7) are derived from the definition of grou velocity. They are correct from the oint of view of mathematical exression. For anomalous disersion medium, Eq. (6) is always ositive while Eq. (7) can be both ositive and negative. Then which exression is more reasonable? We know that a ulse has a certain satial extension. For examle, in the exeriment of Wang and co-workers, the ulse width is 3.7 us, which corresonds to the satial length of about 1 km. As the motion of the eak involves large numbers of hotons at different locations of the wave acket, i.e., the motion of the eak is the collective motion of lots of hotons, we must measure the roagation velocity of the eak in the medium in a long eriod of time, e.g. longer than the ulse duration, to get the grou velocity. As both Eqs. (6) and (7) are instantaneous velocities measured in an infinitesimal time interval, they cannot describe correctly the roagation velocity of the eak of the ulse. Relatively seaking, Eq. (6) is a little more reasonable. 4

5 We further analyze the grou velocity from the actual measuring rocess. Suose two identical ulses start out simultaneously, one traverses an anomalous disersion medium, and the other roagates in the air. If we record the instants when the eak of the ulse enters and exits the medium to obtain the time needed for the ulse to traverse the medium, and then we divide the length of the medium by the time, we get the traversal velocity of the ulse, which is always ositive and may be suerluminal. But we cannot record the instants when the eak of the ulse enters and exits the medium, so we have to make use of another measuring method, just as Wang and co-workers did in the exeriment of [1]. After both of the ulses have crossed a same distance, we comare their waveforms on the oscilloscoe to determine the time difference between the arrival instants of the eaks. This measuring method will bring confusion when the length of the ulse is much longer than that of the medium. In order to exlain this henomenon, we illustrate with an aroriate examle. Suose a erson cometes with a giant of 500 m stature. The seed of the erson is 10 m/s, and that of the giant is 9 m/s. During the racing rocess, the giant can tilt 30 degrees forward, and the tilting seed is much larger than the racing seed. Let the racing distance be 100 m. If we record the arrival instants of the racers, we find that the giant s seed is much larger than that of the erson due to the tilting of his body, but it is always ositive. Now we switch to another recording manner. We comare the distance difference between them after they have both comleted the race. We find it to be 50-11=39 m, that is to say, the giant arrives about 4 s earlier than the erson. From the viewoint of the erson, this situation is inconceivable. Even if the seed of the giant were infinity and the time required to traverse the course were zero, the giant would arrive 10 s earlier than the erson. So the erson can only image that the giant has a negative grou velocity. But this is only an illusion. Now we extend the racing distance to m, the time needed for the giant to comlete the race is ( )/ s, so the erson will arrive at the end oint 83 s earlier than the giant. We see from this examle that suerluminal roagation in anomalous disersion medium arises from the fact that the satial length of the ulse is much longer than the medium length, and the negative grou velocity is due to inaroriate measuring method. In the exeriment of Wang and co-workers, the ulse width is 3.7 us, which corresonds to a giant of over 1 km stature traversing a distance of 6 cm. In the exeriment, the temoral shift of the eak of the ulse is determined by comaring the waveforms of the two ulses on the oscilloscoe. Under these conditions, it s not difficult to understand negative grou velocity observed in the exeriment. It should be noted that in the resence of negative grou velocity, each hoton of the ulse cannot travel faster than c. In order to rove that negative grou velocity can only occur at a short distance, we may increase the length of the cell. During the roagation rocess, there exists a limit for the deformation of the ulse (the advancement shift of the eak), which cannot exceed an half of the whole ulse duration, as shown in Fig. 1. In other words, the eak of the ulse cannot arrive earlier than the wavefront. Then when increasing the length of the cell, we will observe that the roagation velocity of the eak of the ulse varies from negative grou velocity to suerluminal velocity, and finally becomes subluminal. When the medium is adequately long, the grou velocity will aroach hase velocity, i.e., the increased velocity arising from the reshaing effect of the ulse can be ignored. As it s difficult to build a large scale of atomic cell, we will suggest a relatively simler exeriment to test this hyothesis in the next section. 5

6 Fig. 1 The shae deformation of the ulse roagating in anomalous disersion medium varies with the increasing length of the medium..3 Exlanation of existing suerluminal exeriments There are mainly three tyes of barriers at resent. One is eriodical dielectric hetero structure (hotonic crystal). When a ulse exeriences eriodical otential interaction in medium, the frequency of the hotons that can ass through the medium must lie within the allowed energy bands. The hotons of frequency within the forbidden energy bands cannot traverse the medium as wave roagation. But a small fraction of the hotons can tunnel through the medium as articles. For examle, the suerluminal tunneling has been realized in [1,] in this way. The roagation velocity observed in [1] is 1.7 c, and the exerimental result in [] has verified the Hartman effect. This tye of exeriment also includes the barrier tunneling in fiber using Brag reflection [0,1]. The second tye of barrier is undersized waveguide, which corresonds to a rectangular barrier. The height of the barrier is the energy of the hoton with cutoff frequency. Microwave with a frequency below the cutoff frequency cannot ass through the waveguide in wave state. Only a small fraction of the hotons can tunnel through the waveguide in article state. The tyical one is Nimtz s exeriment [4], where a roagation velocity of 4.7 c was obtained for microwave. We have assumed that tunneling of a article must occur within its coherence length, which should be twice larger than the barrier thickness. Now we test this hyothesis. In the exeriment of [1], the coherence time of the hotons is about 0 fs, so the coherence length is 6 um, while the barrier thickness is 1.1 um. In the exeriment of [0,1], the central wavelength of the laser ulse is 1.5 um, and the sectral ulse bandwidth is ~ GHz, so the coherence length is l = λ / Δλ =150 mm, while the maximum barrier thicknesses are 0 mm and 64 mm for single barrier and double-barrier, resectively. In the exeriments of [3,4], the central frequency of the microwave is 8.7 GHz, and the frequency range of the microwave ulse is 8.~9. GHz, so the coherence length of the microwave ulse is l = λ / Δλ =300 mm, while the maximum length of the undersized waveguide is 114. mm. In other exeriments, the coherence lengths are unavailable from the original aers. Our another assumtion is that the tunneling of a article must satisfy the time limit allowed by the uncertainty rincile. In the general case, the height of the barrier cannot be determined. For the undersized waveguide whose barrier height is known, we can make a calculation. In the exeriments of Nimtz [3,4], the cutoff frequency is 9.49 GHz, and the frequency range of the microwave ulse is 8.~9. GHz. According to Eq. (3), we get Δ t =6~74 s. If the central frequency 8.7 GHz of the ulse is adoted, we have Δt =100 s. The time interval observed in exeriment for the microwave ulse to traverse the 100 mm length of the undersized waveguide is τ =130 s, and for the 114. mm length of the undersized waveguide the time interval is τ =81 s. These results are in agreement with Eq. (3). If Hartman s transmission time is used, we obtain Δ t =49~97 s for the whole sectral width of the ulse, and Δt =61 s for the central frequency of the ulse. It can be seen that we cannot decide whether Eq. (3) or (4) is correct by Nimtz s exerimental results, so more exerimental data are needed in order to judge which exression is more reasonable. It should be noted that besides Eqs. (3) and (4), there are also other formulas for tunneling time, such as semiclassical time, comlex time, et al. Among them the most aealing ones are the universal tunneling times roosed by Haibel and Nimtz in [,3], and Esosito in [4]. Haibel and Nimtz roosed a simle tunneling time formula which equals the recirocal of the hoton frequency. Esosito added a correction factor to the formula. Their formulas agree with the exerimental results well in most cases. Those who are interested in this toic may see [-4]. 6

7 The third tye of barrier is just contrary to the first tye. In this case the roagation of a beam of light or electromagnetic ulse in medium is searated by an air ga, which forms a barrier. This tye of exeriment may date back to Bose s microwave tunneling through double-rism exeriment [5]. The recent frustrated total internal reflection exeriments (see [9-11]) are basically similar to that of Bose, so we discuss Bose s exeriment in detail. The schematic diagram is shown in Fig.. A l θ d O B D Fig. Illustration of microwave tunneling through double-rism. We may understand the tunneling rocess as follows. When the angle of incidence θ is greater than the critical angle, the total reflection will take lace. But a Goos-Hänchen transversal shift aears for the reflected ray, i.e. the incident osition is at A while the reflected osition is at B. In order to exlain this henomenon, we suose that the turning oint of the incident ray is at oint O. Because the hotons have satial extension, the wave ackets of the hotons actually enter the air ga and their distributions are within the coherence length of l. If the entrance thickness d is smaller than the thickness of the air ga D, the hotons will be totally reflected at oint O. If d D, the wave ackets of the hotons reach the second rism, then a small fraction of the hotons will be induced out and roagate in the second rism. From Fig., we have d l =. (8) cosθ Now we analyze the exerimental data of Bose. Table 1 is the relation between the angle of incidence and the minimum thickness of the air ga required to generate total reflection. Table is the coherence length calculated according to the data in Table 1. The critical angle in the exeriment is 9. Table 1 Relation between angle of incidence Table The coherence length calculated and minimum thickness of the air ga according to the data in Table 1 θ ( ) d (mm) 30 13~ ~ ~7.6 θ ( ) l (mm) ~ ~ ~15.0 It can be seen from Table that the calculated l is 14~16 mm. Taking into account the sensitivity of the measuring device and the exerimental uncertainties, we may think that the data in Table agree with Eq. (8). Bose s exeriment also indicated that when the air ga is thick sufficiently for total reflection, a ortion of the microwave will still be induced out if a thin iece of cardboard or any other refracting substance is inserted into the air ga, which shows that there exist two tunneling rocesses for the incident microwave, one from the first rism to the inserted substance, the other from the inserted substance to the second rism, and the induced substance is not necessarily the same as the substance uon which the microwave is incident. Similar exeriment is the one carried out by Mugnai [6], where the microwave is incident uon a diffraction grating made of metal stris while it is induced out by a araffin rism. Besides the above exeriments, the suerluminal roagation of microwave near the transmitting 7

8 antennae also belongs to the third tye of barrier tunneling, where the air between the transmitting and receiving antennae forms a barrier. When the distance between the two antennae is smaller than half of the coherence length of the microwave, there exists microwave tunneling from the transmitting antenna to the receiving antenna. If the distance is larger than half of the coherence length, suerluminal roagation disaears. It should be noted that there exist synchronously evanescent wave and radiant comonent near the transmitting antenna. The radiant comonent must be suressed in order to make suerluminal henomenon obvious. In the exerimental setu of [5], mis-alignment of the receiving antenna makes the evanescent wave comonent dominated, as shown in Fig. 3. In the case of Fig. 3(b), the receiving antenna can receive more tunneled microwave signal, so the suerluminal henomenon is more obvious. On the other hand, the distance between the two antenna walls is smaller comared to the case of Fig. 3(a), so suerluminal henomenon can be observed at a larger distance of d, just as indicated by the exerimental results in [5]. In fact, when the two horn antennae are laced face to face, there will also exist tunneling of evanescent wave. But in this case the radiant comonent dominates, so suerluminal henomenon is too weak to be observed. A similar exeriment was carried out by Mugnai [8], where the evanescent wave is first induced out by a reflecting mirror and then received by a receiving antenna, as shown in Fig. 4. Due to the time delay caused by the reflection of the mirror, the suerluminal henomenon is not obvious with only a result of c obtained. In fact, this exeriment can be divided into two stages of barrier tunneling, i.e. one from the slit in front of the transmitting antenna to the mirror and the other from the mirror to the receiving antenna. If the receiving antenna is laced at a larger distance, there is only one tunneling rocess from the slit to the mirror. The rincile of this exeriment is the same as that of in [6] in essence. In both cases the microwaves are induced out through a narrow slit. The suerluminal henomenon was exlained in [8] with X wave, but we think it s more reasonable to exlain with evanescent wave. d Fig. 3(a) Giakos-Ishii s exerimental setu (1). Fig. 3(b) Giakos-Ishii s exerimental setu (). Fig. 4 The exerimental setu of Mugnai et al. If we try to understand the above exerimental henomena according to classical electromagnetic theory, it can be regarded that there exist oscillating standing waves near the transmitting antenna, or we start with the electromagnetic fields of diole antenna [7] 8

9 ck e H = ( n ) 4π r ikr 1 e 1 ik E = k ( n ) n + [3n( n ) ]( ) e 3 4πε 0 r r r In the near zone, the electromagnetic fields aroach ikr 1 (1 ), (9) ikr ikr. (10) iω 1 H = ( n ), (11) 4 π r 1 1 E = [3n( n ) ], (1) 3 4 πε r 0 which indicate that the there is energy of electromagnetic fields stored near the antenna besides the energy radiated. In the general case, the stored energy does not radiate outwards and there exists only electromagnetic energy current loo near the antenna. If there is a conductor or other substances within half of the coherence length of the electromagnetic ulse, the stored energy will be induced out. The rincile is the same as that of the energy exchange in fiber couler. The only difference is that the coherence length of the microwave is larger than that of the light, so energy exchange effect can be observed within a larger distance. Because the energy exchange is realized by evanescent wave, suerluminal henomenon will only aear near the transmitting antenna. We have suosed above that even when two horn antennae are laced face to face, there will also exist tunneling of evanescent wave. But the suerluminal henomenon is not obvious. In order to suress the radiant comonent, we may relace the horn antennae with waveguides, and then lace them within a short distance in order that the tunneling ower is larger than the radiant ower. The exerimental setu is shown in Fig. 5, where a, b and c are waveguides and the length of the waveguide c is equal to the sum of the length of a and b. The two microwave signals are sent to dual channel oscilloscoe to comare the arrival instants of the eaks of the two ulses. In this exeriment, we exect that suerluminal tunneling can be observed. a b Microwave ulse c Oscilloscoe Fig. 5 Suerluminal tunneling between two waveguides. During the tunneling rocess, if there exist several times of interactions between a hoton and a coule of electrons, the whole traversal velocity of the hoton through the barrier may be slower than c. For examle, in the exeriment of [8], an undersized waveguide was filled with carbon loaded urethane foam. During the tunneling rocess, a hoton may interact with a coule of electrons of the foam. Due to the time delay caused by the absortion-emission rocess, the tunneling velocity of the hoton will slow down. In Nimtz s exeriment, a traversal velocity of 0.7 c was obtained. As the tunneling rocess requires time, negative grou velocity cannot be resent in tunneling exeriments. It was redicted in [9,30] that negative grou velocity could be obtained in barrier tunneling using X wave, but the authors only gave out the theoretical results and there are no exerimental verifications thus far. It was reorted in [31] that negative grou velocity was observed exerimentally in the near zone of the antenna. The exeriment setu is shown in Fig. 6, where two 9

10 antennae are laced very close (less than 10mm). The exerimental result can be exlained as follows. As the two antennae are laced very close, they form a couling caacitor. In this situation, there exists couling wave comonent besides the radiant and evanescent mode comonents near the antenna, and the couling wave comonent dominates. The equivalent electric circuit on the right-hand side can be regarded as a RC circuit, as shown in Fig. 7. Oscilloscoe U i C R U o Fig. 6 The exerimental setu of Budko. Fig. 7 The equivalent electric circuit of Fig. 6. The voltage U o of the load R is R ω R C ωrc U o = U i = ( + j ) U i, (13) ω R C 1 + ω R C + R jωc The grou delay is 1 φ = arctan. (14) ωrc dφ RC τ = =. (15) dω 1 + ω R C We see from Eq. (15) that as the frequency of the signal increases, the transmission time through the load R decreases, which agrees with the feature of anomalous disersion. Thus the exerimental setu of Fig. 6 can indeed obtain negative grou velocity. In fact, a simle RLC bandass amlifier has been used to demonstrate negative grou delay in [3]. From above calculation we see that a simler RC electric circuit can also obtain negative grou velocity. A detailed analysis and exerimental demonstration of negative grou velocity and suerluminal velocity with RC circuit may see [33]. The suerluminal velocity and negative grou velocity in anomalous disersion media include many exeriments, most of which use the roerty of different roagation velocities of hotons for different frequencies, i.e., the higher the frequency of the hotons, the larger the roagation velocity. Thus the eak of the ulse will shift forward. The exeriments using transmitted ulses to demonstrate negative grou velocity may see [1,13,15,34-37], and those using reflected ulses may refer to [1,38,39]. In the case of reflected ulses, the higher frequency of the hotons, the less time needed for the reflecting rocess. This results in the reshaing of the ulses. There are also other exeriments using the roerty of different roagation velocities for different olarization comonents of the ulse to obtain reshaing effect (see [40,41]). The rincile of these exeriments has been exlained in the receding section. They are not the true suerluminal roagations. We have discussed two tyes of suerluminal effects, i.e. barrier tunneling and ulse reshaing. In fact, the two effects may coexist in a same rocess. For examle, there exists barrier tunneling as well as reshaing for a ulse to ass through an undersized waveguide. A microwave ulse comrises large numbers of hotons with different frequencies. The higher frequency of the hoton, the less energy it needs to borrow from the vacuum, and the longer duration it may ossess to traverse the waveguide. 10

11 On the contrary, a hoton with a lower frequency must traverse the waveguide more quickly. Such interactions lead to the normal disersion effect, which results in the backward shift of the eak of the ulse. So for an undersized waveguide with small length, the traversal velocity of the microwave ulse may be less than c. Only when the waveguide is long enough for the tunneling effect to dominate can suerluminal roagation be obvious. In fact, Hartman has redicted theoretically that the traversal velocity of a ulse would be less than c for thin barrier [19], and the exerimental verification may see [4]. Our above exlanation is easier to understand comared to Hartman s calculation..4 The relationshi between suerluminality and causality In the receding discussions we suose that suerluminal henomena can exist under certain conditions. Does this violate causality and secial relativity? Here it is necessary to distinguish between the seed of signal (information) and that of the individual hotons. There are lots of hotons with various frequencies in a ulse. The suerluminal motion of a ortion of the hotons will not lead to the suerluminal roagation of the whole ulse. In the case of barrier tunneling, only a small fraction of the hotons can traverse the barrier. Thus as a ulse asses through a barrier, attenuation of the signal is inevitable. On the other hand, the robabilities of the hotons with different frequencies tunneling through the barrier are different. So the signal will be distorted. The more obvious the suerluminal motion, the more attenuated and distorted the signal. This is the rice we must ay for suerluminal roagation. The seed of an attenuated and distorted signal cannot reresent the true seed of the signal. In addition, suerluminal roagation can only occur within a short range and a short term. So the suerluminal motion of a small fraction of the hotons does not violate causality. We now turn to the roblem of the tunneling of electron. According to our assumtion, once the coherence length of the electron is smaller than twice the barrier thickness, the electron cannot tunnel through the barrier, so it s imossible to use Hartman effect by increasing the barrier thickness to obtain suerluminal velocity of electron. On the other hand, suose the initial velocity of the electron is v 0. If its traversal velocity v through a barrier could reach c, the borrowed energy Δ E = m0 c / 1 ( v / c) m0c / 1 (v0 / c) would be infinite, which is imossible. So the traversal velocity cannot reach c, and the exression m = m0 / 1 ( v / c) will not be imaginary in any case. In fact, the tunneling of electron can also be regard as the rocess of waveacket collase, and the electron traverses the barrier as a article. In [3], the tunneling articles are described by virtual articles. For an electron, if its tunneling velocity is larger than c, then we may regard it as a virtual article. Now we see the exerimental results. In the exeriment of [43], the ionization and tunneling delay time of electrons in helium were measured. The exerimental results ut an uer limit for the tunneling time of 34 as. If a weighted intensity-averaged method is adoted, the tunneling time is 6.0 ± 5.6 as. Let s first estimate the tunneling time according to Eq. (3). The ionization energy of helium is 4.59 ev, which can be regarded as the borrowed energy Δ E of electron to escae from helium atom, so we get Δt 85 as. The next ste is to calculate the tunneling velocity. There is one roblem remains unsolved in the exeriment: the tunneling distance is unknown. Let s make a rough estimate. Let the tunneling distance be the distance between the orbit of energy level n = and that of n =1, for the tunneling time we take the minimum of the exerimental value, that is 0.4 as. The orbital radius of the electron in helium atom is r = n r0 /, where r0 is Bohr radius. Then the tunneling velocity is 10 8 m/s. In the exeriment of [44], the semiclassical traversal time in Josehson junction was measured to be the order of 100 s. Suose the thickness of the Josehson junction to be the order of 10 nm. Then the tunneling velocity of electron is the order of 100 m/s. In the exeriment of [45], the traversal time for electron tunneling in water was measured. For distances of the order of 1 11

12 nm the tunneling times comuted with Büttiker-Landauer aroach are in the range of 0.1~1 fs. Then the tunneling velocity of the electron is the order of 10 6 ~10 7 m/s. These exerimental results all demonstrate a subluminal velocity of electron tunneling. We think that more recise exerimental dada are needed in order to judge whether electron can travel faster than light during the tunneling rocess. 3 Proosed otical exeriments 3.1 Exerimental verification of the satial extension of a single hoton wave acket If we suose that a single hoton is a wave acket occuying certain satial volume, it s easy to understand the single hoton double-slit interference exeriment, i.e. one half of the wave acket asses through one slit and the other half through the other slit. Now we test this hyothesis. We make the exeriment based on the exerimental setu of Asect and co-workers [18], as shown in Fig. 8, where the single hoton was obtained from the cascade radiation in calcium. We first verify the fact that the interference oututs of D 1 and D will disaear when the otical ath length difference is larger than the coherence length of the hoton. We then increase the distance L between the beam slitter (BS) and the reflecting mirror M. We exect that when L is larger than the transverse extension of the hoton wave acket (this is equivalent to increase the width of double-slit in interference exeriment), the interference henomenon will also disaear. In fact, such results might be conceived even if we don t make the exeriment. What is imortant is that it gives us the clear evidences that a single hoton has certain satial extension. Similarly, we can imagine that a single electron also has satial extension, and then it s easy to understand the double-slit interference exeriment of single electron in [46]. D Single hoton L M BS BS M 1 D 1 Fig.8 Exerimental test of satial extension of a single hoton. We can also erform other exeriment to test the satial extension of single hoton. Let a highly attenuated laser source emit one hoton each time. The single hoton is then incident uon a thin glass late or other medium. When the thickness of the glass late is less than half of the coherence length of the hoton, interference will be observed. If the thickness of the glass late is roerly adjusted, all the hotons will be transmitted. We then increase the thickness of the glass late to let it be larger than half of the coherence length of the hotons, interference will disaear, and the behavior of the hotons obeys Fresnel s law. This exeriment tests the assumtion that the extension of the hoton in roagation direction is its coherence length. 3. Exerimental test of suerluminal roagation within coherence length According to our assumtion, no hotons can tunnel through a barrier if the coherence length is smaller than twice the barrier thickness. While in terms of evanescent wave theory, the tunneling robability of hotons decreases exonentially with tunneling distance but will not be zero at large distances. In order to test which theory is correct, we make the following exeriment. Let a beam of microwave or light ulse be incident uon a hotonic crystal or a double-rism, as shown in Fig. 9. We then gradually increase the thickness of the barrier and measure the ower of the tunneled microwave 1

13 to see whether it decreases exonentially with tunneling distance or the signal is undetectable beyond a certain distance no matter how we increase the ower radiated. If it turns out that the latter case is correct, then our assumtion holds. In Fig. 9(a), the hotonic crystal may also be relaced with an undersized waveguide. Fig.9(a) Microwave tunnels through hotonic crystal. Fig. 9(b) Microwave tunnels through double-rism. However, the above exeriments are difficult to realize, for the measuring devices must be very sensitive to low ower signal. Here we roose an otical exeriment which is relatively easier to realize. We use the exerimental setu in [0], as shown in Fig. 10, where fiber Bragg gratings are emloyed to form a hotonic barrier and a laser ulse is used to traverse the barrier. Fiber Bragg gratings are otical fiber devices in which the refractive index of the core is modulated along the longitudinal axis with an almost sinusoidal rofile of submicrometric eriod. Comared with Bragg mirrors, the weak Bragg scattering rovided by the fiber Bragg gratings enables the use of long barriers. A continuous-wave laser is externally modulated at a reetition frequency of 1 GHz to obtain reeated ulses, and the central wavelength of the laser ulse is 1.5 um. We have calculated in the revious section that the coherence length of the laser ulse is about 150 mm. We may increase the thickness of the barrier to let it be larger than half of the coherence length of the ulse. In this case, the ulses cannot tunnel through the barrier according to our assumtion, or even if a small fraction of the hotons can traverse the barrier by multi-tunneling rocess, their traversal velocities will be subluminal. So the suerluminal tunneling will disaear according to our exectation. While in terms of the resent theory, a larger suerluminal velocity should be observed. Thus from the exerimental result we can determine which assumtion is correct. Laser ulses Fig. 10 Laser ulses tunnel through fiber Bragg gratings. It should be noted that ractical signals are never genuinely bandwidth limited, there always exist non-evanescent waves traveling through the barrier. In fact, it was ointed out earlier by Hartman that when the barrier is very thick, the sectral comonents just above the cutoff frequency will dominate and the transit time is aroximately equal to time required for the incident wave acket to traverse a distance of the barrier thickness [19]. So we must ensure that the tunneling comonents dominate in order to verify the above assumtion, and a low-ass filter may be used to suress the sectral comonents above the cutoff frequency of the barrier. In order to eliminate the influence of the hotons with energy higher than the barrier height, another method is to use barrier tunneling of single hoton, as shown in Fig. 11. A air of linearly olarized hotons with the same energy are generated by tye-i non-collinear down-conversion. One hoton travels in the air and the other tunnels through a barrier. Two interference filters (IF) are used to ick out the hoton airs with aroriate frequency. The hotons are detected by single hoton detectors D1 and D, resectively. The oututs of the detectors are then sent to coincidence detection 13

14 circuit. We exect that when the thickness of the barrier is larger than half of the coherence length of the hoton, the coincidence counting rate is zero. While in terms of the resent theory, the coincidence rate decreases exonentially with the barrier thickness but will not be zero. D1 Pum laser NC IF Coinc counter Tye-I IF Barrier D Fig. 11 Coincidence detection of single hoton tunneling through a barrier. 3.3 Exerimental test of subluminal roagation in anomalous disersion medium We have suosed that the suerluminal henomena in anomalous disersion media are the consequence of reshaing of the ulse, and suerluminal roagation will disaear at large distances. As the exeriment of Wang and co-workers is difficult to realize on a large scale, while it s easy to realize by using tunneling of electric ulse through a coaxial hotonic crystal, we adot the methods in [35] and [36], where the exerimental setus are similar. The main difference is the use of a coaxial hotonic crystal in [36] with a higher imedance mismatch that ermits access to a negative grou velocity of -1. c, while the exeriment in [35] only obtained the results of ~3.5 c. In terms of our theory, when the length of the coaxial cable increases, the roagation velocity of the eak of the ulse will vary from negative grou velocity to suerluminal velocity, and finally becomes subluminal. Suose we adot the exerimental setu of [35], whose simlified sketch is shown in Fig. 1. One outut of the signal generator is connected directly to the oscilloscoe as the reference signal, and the other outut asses through a hotonic crystal made of alternating quarter-wavelength segments of two different imedance coaxial cables. Now we estimate the length L required for the hotonic crystal to obtain subluminal velocity of the electric ulse. The hase velocity of the electric ulse in both segments is 0.66 c in [35]. Suose the eak of the ulse can shift at most earlier to the wavefront. For a ulse with duration of us adoted in the exeriment of [35], we have L L c c 6. (16) Signal Generator Oscilloscoe Fig. 1 Exerimental test of subluminal roagation of electric ulse at large distances. It follows L 600 m. By contrast, the length of the coaxial cable used in [35] is 10 m. In fact, the eak of the ulse cannot shift to the wavefront, so it s likely that we can observe subluminal roagation at a length less than 600 m. As the signal amlitude attenuates exonentially with the 14

15 length of the cable, the ulse signal must be amlified in order to be observed on the oscilloscoe. It should be noted that above rediction only holds for the situation where the length of the hotonic crystal is larger than the coherence length of the ulse, otherwise the grou velocity will fluctuate with the length of the hotonic crystal. This can be seen from the exression of index of refraction [35] cφ n r =, (17) ω D where ω is the angular frequency of the ulse, and φ is the overall hase shift accumulated through the crystal of length D, and we have [35] Im( t) φ = arctan + mπ, m = 0,1,L L, (18) Re( t) where t is the comlex coefficient of electric field transmission through the whole crystal. As t is also deendent on the length of the crystal, the grou velocity of the ulse will fluctuate with D, just as indicated in the exeriment of [47], where the ulse duration is 10 us, and the coherence length is much larger than the crystal length. Another imortant oint for the above verification exeriment is that we must rule out the influence of the sectral comonents beyond the anomalous disersion region. As ointed out in [35], a very short ulse or a ste function is not exected to roagate at suerluminal seeds because of its excessive sectral width. Furthermore, it was ointed out in [48] that even when the sectral width of the initial ulse lies well within the anomalous disersion region, the strong attenuation of these sectral comonents will result in the dominance of normally disersive sectral comonents that suffer from lower dissiation. So even though subluminal roagations of ultra-short ulses through anomalous disersion media have been observed in [49,50], they cannot be used as the convincing evidences for our assumtion. In [49], an electric ulse with the duration of.4 ns was used to ass through a coaxial hotonic crystal, the grou velocity slowed down to 0.67 c, only a little larger than the normal roagation velocity of 0.66 c. While in [50], a suerluminal to subluminal transition has been observed with femtosecond laser ulses in an absorbing dye solution through a short to a long range of roagation distance. In order to unambiguously verify our assumtion, a band-ass filter must be used to suress the sectral comonents beyond the anomalous disersion region, which also alies to our above roosed exeriment. In the exeriment of [40], reshaing effect is obtained by the different roagation velocities of different olarization comonents of linearly olarized light in birefringent otical fiber. If we increase the length of the fiber, similar exerimental result will be obtained. But in that exeriment, the length of the fiber must be chosen so that the roagation time difference between the two olarization comonents in the fiber is less than the coherence time of the ulse, thus its length is limited. 4 Conclusion When light roagates as a wave train in vacuum or medium, its velocity cannot be suerluminal. In the resence of a barrier, a small fraction of the hotons can tunnel through the barrier as articles (evanescent wave) and their velocities can exceed c. But this suerluminal state can only last within the coherence length and a short term. For a article with non-vanishing mass, its velocity cannot be suerluminal whether in wave or article state due to the uncertainty rincile. Suerluminal roagation is always accomanied with attenuation and distortion of the signal. As for the suerluminal roagation in anomalous disersion medium, we may regard it as the consequence of 15