R. Y. Chiao; J. M. Hickmann; C. Ropers; D. Solli Faster-than-light propagations, and their applications. In: N. Bigelow (Org.). Coherence and Quantum Optics VIII. 2002.
Faster-than-light propagations and their applications Raymond Y. Chiao, Claus Ropers, Daniel Solli, and Jandir M. Hickmann Department of Physics, University of California Berkeley, California 94720-7300 phone: (510)642-7166, fax: (510)643-8497 e-mail:
[email protected] Abstract: Recent experiments have conÞrmed our predictions that the group velocity of light pulses propagating in transparent optical media can exceed c. In electronics, the causality principle does not forbid negative group delays of analytic signals in electronic circuits, in which the peak of an output pulse leaves the exit port of a circuit before the peak of the input pulse enters the input port. In condensed matter physics, negative transmission times of atoms are possible through superßuid helium slabs and through atomic BECs. Relativity is not violated by these phenomena. 1
INTRODUCTION
In this paper, we discuss some recent experimental and theoretical developments concerning faster-than-light propagation phenomena in optics, electronics, and condensed matter physics. For an earlier critical review of this subject, see [1], in which the phenomena of superluminal (i.e., faster-than-c) tunneling times of photons, and the theoretical predictions of superluminal light pulse propagation in transparent media, were reviewed. We hope here to be able to help and correct the commonly held, but mistaken, belief that only the phase velocity in a medium can exceed c, but not the group velocity. We discuss here several situations in which the group velocity in fact exceeds c, and point out some applications of these phenomena, such as to speed-up of computer circuits by using negative group delays to cancel out deleterious positive group delays. We also apply these ideas to novel situations in condensed matter physics, such as to the transmission of atoms through a slab of superßuid helium, or through an atomic Bose-Einstein condensate. The transmission times of such atoms, under the appropriate experimental conditions, can be superluminal, or even negative. 2
PHASOR DESCRIPTION OF SUPERLUMINAL PROPAGATION
A simple phasor picture (see Figure 1) helps explain how such superluminal group velocities can occur for all kinds of waves (e.g., for light and for matter waves), in complete generality. The peak of a Gaussian wave packet, indicated by point B in Figure 1, is the moment when all the phasors which represent the various Fourier components of the wave packet, line up in a straight line, so that the resultant phasor has a maximum amplitude. Therefore, the peak of the wave packet represents a point of maximum constructive interference. By the same token, a point in the early part of the wave packet where the amplitude is small, represented by point A in Figure 1, is a moment of almost total destructive interference, in which the same phasors curl up to form a polygon which almost closes in on itself. Hence, the resultant amplitude at point A in the early tail of the Gaussian wave packet, is small. After propagation through a dispersive medium, there is no physical reason why the phasors at the early point A cannot precess due to dispersion, in such a way that the curled-up polygon becomes uncurled, forming a large resultant phasor at point A0 . The phasors at this point are now aligned in constructive interference, and the peak occurs earlier in time in the wave packet compared with what would have happened in the vacuum. By the same reasoning, the aligned phasors at point B now curl up to form a polygon which almost closes in on itself, leading to almost total destructive interference at point B 0 in the trailing tail of the wave packet. The result is an advancement of the entire Gaussian wave packet relative to vacuum propagation; this occurs whenever the spectrum of this wave packet lies in a region of negative wave-number dispersion relative to that of the vacuum. In the case of optics, this can occur in a spectral region of anomalous dispersion.
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Fig. 1. Phasor description of superluminal propagation.
Furthermore, if the dispersion in the group velocity vanishes at the carrier frequency, the entire wave packet can propagate superluminally with negligible change in shape. Moreover, by the superposition principle, one can add up many such wave packets to form an arbitrary, analytic waveform, which can propagate faster than c with negligible distortion. Much more commonly, the medium possesses a positive dispersion, and the phasors precess in the opposite sense from the above case of negative dispersion. This results in a retardation of the Gaussian wave packet relative to vacuum propagation, rather than an advancement. For example, in optical media such as ordinary glass, one normally observes subluminal pulse propagation in spectral regions of normal dispersion. Nevertheless, the underlying physical mechanisms for both the retardation and the advancement of the wave packet are identical, apart from the sign of the precession of the phasors, and both are possible in principle. However, it should be emphasized that there are conditions on the above phasor-precession processes, which must be fulÞlled before they can be physically signiÞcant. In general, the group advancements or retardations of an incident wave packet will be destroyed by the decoherence of the system. For example, in the optical case it is necessary to maintain the phase coherence of a light pulse during the phasor-precession process which produces the advanced peak; otherwise, the constructive interference will be destroyed. In particular, for an optical medium with gain, stimulated emission must always be accompanied by spontaneous emission. Hence, spontaneous emission will lead to phasor decoherence, and this will limit the maximum possible amount of superluminal advancement [2]. One possible deÞnition of a signal velocity in terms of a signalto-noise ratio which is limited by ampliÞed spontaneous emission, leads to a velocity which is less than c [3]. Thus, signals so deÞned cannot propagate faster than c. Why do such superluminal pulse propagation phenomena not violate relativity? The answer can be seen in Figure 2, where the Gaussian wave packet is multiplied by a step function, such that there results a f ront in its early tail, ahead of which the electromagnetic Þeld is strictly zero. Here, all phasors representing the Þeld ahead of the front have strictly zero length. Since the discontinuity at the front contains Fourier components of inÞnite frequency, where the index of refraction of any dispersive medium approaches unity, it follows that the front velocity is exactly c, as was pointed out by Sommerfeld and Brillouin [4]. Although the peak of the Gaussian wave packet moves at the group velocity, which may exceed c, this peak can never overtake the front. The reason is that there is no way that phasors of zero length ahead of the front can ever produce a resultant phasor of Þnite length. Thus, no signal can overtake the front, and this sets a fundamental limit
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Fig. 2. Front deÞned by discontinuity.
on superluminal pulse propagation, in agreement with the principle of causality in special relativity. It is important to distinguish between two kinds of signal waveforms in physics: analytic waveforms, such that of a Gaussian wave packet, and nonanalytic wave forms, such as that of a step function. Any small but Þnite piece of the the early part of an analytic wave form can be extrapolated into the future by means of a Taylor series; therefore, there is no surprise associated with the arrival of any of the features of the subsequent waveform, including its peak. Conversely, the same extrapolation of the signal ahead of the Sommerfeld front would predict that the waveform would remain identically zero for all time. When the front arrives, there is a genuine surprise associated with its arrival [5]. This suggests another deÞnition of signal velocity in terms of discontinuities [1]. Again, signals so deÞned cannot travel faster than c. An important special case of superluminal propagation occurs when the group velocity becomes negative. Recent optical experiments at Princeton NEC [6] have veriÞed the prediction by one of the authors and his co-workers that superluminal pulse propagation, in particular, propagation with a negative group velocity, can occur in transparent media with optical gain [7]. These experiments have shown that a laser pulse can indeed propagate with little distortion in an optically pumped cesium vapor cell with a negative group velocity: The peak of the output laser pulse left the output face of the cell bef ore the peak of the input laser pulse entered the input face of the cell. This implies a negative time of ßight of the pulse through the cell. A negative-time-of-ßight process is represented in Figure 3 by a zigzag world-line for the motion of the peak of the wave packets. Note that a time slice through the middle of the zigzag world-line shows that at that moment, three wave packets are simultaneously present in the system: the input wave packet propagating to the right, whose peak is about to enter the input face of the cell, a wave packet propagating to the left, which is in the middle of the cell, and an output wave packet propagating to the right, whose peak has already left the output face of the cell. The question immediately arises as to how energy can be conserved in such a process. One answer is that there can be optical gain, for example, through the population inversion of an atomic system: the atoms can then possess enough extra energy which can then be “loaned” to the light in order to produce the two extra pulses at point β at the output face of the medium, in a process reminiscent of “pair creation” (in the sense of the creation of a pair of wave packets). The borrowed energy is later restored to the inverted atomic system at point α at the input face of the medium, in the process reminiscent of “pair annihilation” (in the sense of the annihilation of a pair of wave packets). Note that for this zigzag process, if one were to double the cell length, the output laser pulse would have to come out twice as early. This process is reminiscent of the zigzag world line introduced by Feynman in quantum electrodynamics shown in Figure 4(b). In this diagram, Feynman interpreted the backwards-in-time propagation of a particle (an electron) as the forwards-in-time propagation of an antiparticle (a positron). Point b, which corresponds to electron-positron pair creation in the vicinity of a nucleus of charge +Ze, occurs earlier in time than point
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Fig. 3. zigzag world-line diagram.
a, which corresponds to pair annihilation in the vicinity of the nucleus. The net result of this Feynman process is that the peak of the scattered electron wave packet comes out earlier from the scattering region than when it entered [8]. This is in contrast to the more intuitive process shown in Figure 4(a), where the electron wave packet comes out later from the scattering region than when it entered. Both processes are possible, and must be added coherently when the Þnal states are indistinguishable. However, there is a fundamental limit on the magnitude of the wave packet advancement associated with the counter-intuitive zigzag diagram. Note that the intermediate state consisting of the positron portion of the zigzag is a virtual state. The energy-time uncertainty principle ∆E∆t ∼ ~, (1) allows the temporary borrowing of the energy ∆E from the nuclear Coulomb Þeld at point b, as long as the borrowed energy is paid back later at point a after a short time ∆t, where ∆t is the duration of the intermediate state during which the positron exists. The resulting negative group delay for the scattered electron wave packet is plotted schematically in Figure 5. Note that there results a “negative Hartman effect,” in which a negative group delay saturates at a value given by the above uncertainty principle. This behavior is analogous to the positive Hartman effect seen for wave packets which tunnel through a barrier. In this effect, the tunneling particle experiences a positive group delay, which saturates for thick barriers at a positive value given by the uncertainty principle. The Hartman effect [9] represents the thick-barrier limit of the Wigner tunneling time [10], which is also plotted in Figure 5, for the sake of comparison. We performed some early experiments on the speed of the quantum tunneling process [11]. We found that a photon tunneled through a barrier at an effective group velocity which was faster than c. In these experiments, spontaneous parametric down-conversion was used as a light source which emitted randomly, but simultaneously, two photons at a time (i.e., photon “twins”). These photons were detected by means of two equidistant Geiger counters (silicon avalanche photodiodes), so that the time at which a “click” was registered was interpreted as the time of arrival of the photon. Coincidence detection was used to detect these photon twins. One photon twin traverses a tunnel barrier, whilst the other traverses an equal distance
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Fig. 4. Feynman processes in QED for multiple scattering of an electron from a nuclear (+Ze) Coulomb Þeld. In the zigzag process (b), pair creation occurs at point b before pair annihilation occurs at point a. Lorentz transformations leave the interval between a and b space-like, and hence “superluminal.”
in the vacuum. The idea of the experiment was to measure the time of arrival of the tunneling photon with respect to its twin, by measuring the time difference between the two “clicks” of their respective Geiger counters. (We employed a two-photon interference effect in the Hong-Ou-Mandel interferometer [12] in order to achieve sufficient time resolution.) The net result was surprising: On the average, the Geiger counter registering the arrival of the photon which tunneled through the barrier clicked earlier than the Geiger counter registering the arrival of the photon which traversed the vacuum. This indicates that the process of tunneling in quantum physics is superluminal. The earliest experiment to demonstrate the existence of faster-than-c group velocities was performed by Chu and Wong at Bell Labs. They showed that picosecond laser pulses propagated superluminally through an absorbing medium in the region of anomalous dispersion inside the strong optical absorption line [13]. This experiment was reproduced in the millimeter range of the electromagnetic spectrum by Segard and Macke [14]. These experiments veriÞed the prediction of Garrett and McCumber [15] that Gaussian-shaped pulses of electromagnetic radiation could propagate with faster-than-c group velocities in regions of anomalous dispersion associated with an absorption line. Negative group velocities were also observed to occur in these early experiments. However, these kinds of superluminal pulse propagation phenomena were not known to occur in transparent optical media at the time. Subsequently, we observed these counter-intuitive pulse sequences in experiments on electronic circuits [16]. In the Þrst of these experiments, we used an electronic circuit which consisted of an operational ampliÞer with a negative feedback circuit containing a passive RLC network. This circuit produced a negative group delay similar to that observed in the recent optical pumping experiment [6]: The peak of the output voltage pulse left the output port of the circuit bef ore the peak of the input voltage pulse entered the input port of the circuit. Such a seemingly anti-causal phenomenon does not in fact violate the principle of causality, since there is sufficient information in the early portion of any analytic voltage waveform to reproduce the entire waveform earlier in time. We showed that causality is solely connected with the occurrence of discontinuities,
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Fig. 5. Positive group delays in Wigner tunneling time lead to a positive Hartman effect; negative group delays (see Figure 4(b)) lead to a negative Hartman effect.
such as “fronts” and “backs” of signals, and not with the peaks of voltage waveforms, and, therefore, that causal loop paradoxes could never arise [17]. Since there was gain in these electronic circuits, the output signal was not strongly attenuated, in contrast to the earlier optical experiments. We believe that these counter-intuitive ideas can be applied to the design of microelectronic devices, in particular, computer chips [18]. This is timely, since it is widely believed that Moore’s law for microprocessor performance will fail to hold in the next decade due to a “brick wall” arising from fundamental physical limitations [19]. Therefore, there have been many proposals for new transistor technologies to try to solve this problem [20][21]. At the present time, the “transistor latency” problem is one of the main factors limiting computer performance, although the “propagation delays” due to the RC time constants in the interconnects between individual transistors on a computer chip are beginning to be another serious limiting factor. As the scale of microprocessor circuits fabricated on a silicon wafer is reduced to become ever smaller in size, the transistor switching time becomes increasingly faster, but the propagation delay from transistor to neighboring transistor becomes increasingly longer [22]. This will still be true even after new technologies to replace MOSFETS with faster devices are implemented. 3
GENERAL PRINCIPLES FOR GENERATING NEGATIVE GROUP DELAYS IN ELECTRONICS
We now begin our discussion of superluminal effects in electronic circuits, using the concept of negative group delays as the starting point. Electronic circuits are usually very small in size compared with the wavelengths corresponding to the typical frequencies of operation of these circuits; thus, the retardation due to the speed of light across these circuits is usually negligible. Nevertheless, a concatenation of such negative-delay circuits interspersed periodically along a transmission line, could lead to superluminal propagation of pulses with a negative group velocity. Hence, we focus here only on how a negative group delay can be generated in general. 3.1
Negative group delays necessitated by the “Golden Rule” for operational ampliÞer circuits with negative feedback
In Figure 6, we show an operational ampliÞer with a signal entering the noninverting (+) port of the ampliÞer. The output port of the ampliÞer is fed back to the inverting (−) port of the ampliÞer by means of a black box, which represents a passive linear circuit with an arbitrary complex transfer function Fe (ω) for a signal at frequency ω. We thus have a linear ampliÞer circuit with a negative feedback loop containing a passive Þlter. In general, the transfer function of any passive linear circuit, such as a RC low-pass Þlter, will always lead to a positive propagation delay through the circuit.
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Fig. 6. Operational ampliÞer circuit with negative feedback.
However, for operational ampliÞers with a sufficiently high gain-feedback product, the voltage difference between the two input signals arriving at the inverting and noninverting inputs of the ampliÞer must remain small at all times. The operational ampliÞer must therefore supply a signal with a negative group delay at its output, such that the positive delay from the passive Þlter is exactly canceled out by this negative delay at the inverting (−) input port. The signal at the inverting (−) input port will then be nearly identical to that at the noninverting (+) port, thus satisfying the “Golden Rule” which demands small voltage differences at all times. The net result is that this negative feedback circuit will produce an output pulse whose peak leaves the output port of the circuit bef ore the peak of the input pulse arrives at the input port of this circuit. In Figure 7, we show experimental evidence for this counter-intuitive behavior for the special case of an RLC tuned bandpass circuit in the negative feedback loop [16]. The peak of an output pulse is advanced approximately by 12 milliseconds relative to the input pulse. The output pulse has obviously not been signiÞcantly distorted with respect to the input pulse by this linear circuit, apart from a slight ampliÞcation factor. Also, note that the size of the advance of the output pulse is comparable in magnitude to the width of the input pulse. At these very low frequencies, the role of spontaneous emission is entirely negligible, and the pulse advance can obviously satisfy Rayleigh’s criterion for pulse resolution, as can be seen by inspection of the data shown in Figure 7. That causality is not violated is demonstrated in a second experiment, in which the input signal voltage is very suddenly shorted to zero the moment it reaches its maximum. The result is shown in Figure 8. By inspection, we see that the output signal is also very suddenly reduced to zero voltage at essentially the same instant in time that the input signal has been shorted to zero. This demonstrates that the circuit cannot advance in time truly discontinuous changes in voltages: These are the only points on the signal waveform which are directly connected by the principle of causality [17]. However, for the analytic changes of the input signal waveform, such as those in the early part of the Gaussian input pulse which we used, the circuit evidently has the ability to extrapolate the input waveform into the future, in such a way as to reproduce the output Gaussian pulse peak bef ore the input pulse peak has arrived. In this sense, the circuit anticipates the arrival of the Gaussian pulse. 3.2
The “Golden Rule” and the inversion of the transfer function of any passive linear circuit
Now we shall analyze under what conditions the “Golden Rule” holds and negative group delays are produced. e (ω) denotes the complex amplitude of an input signal of frequency ω into the noninverting In Figure 6, A e (ω) refers to that of the feedback signal into the inverting (−) port of the ampliÞer. The (+) port and B e (ω) is then related to the feedback signal B e (ω) by means of the complex linear feedback output signal C
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Fig. 7. Experimental results showing the pulse advancement.
Fig. 8. Experimental results showing that discontinuities cannot be advanced.
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Fig. 9. Circuit with RC Þlter placed before the negative feedback circuit.
transfer function Fe (ω) (the black box) as follows:
e (ω) = Fe (ω) C e (ω) . B
(2)
The voltage gain of the operational ampliÞer is characterized by the active complex linear transfer function e (ω), which ampliÞes the difference of the voltage signals at the (+) and (−) inputs to produce an output G signal as follows: ³ ´ e (ω) = G e (ω) A e (ω) − B e (ω) . C (3)
e (ω) /A e (ω) as the ratio of the output signal C e (ω) to DeÞning the total complex transfer function Te (ω) ≡ C e input signal A (ω), we obtain for the total transfer function, Te (ω) =
e (ω) G . e (ω) 1 + Fe (ω) G
If the gain-feedback product is very large compared to unity, i.e., ¯ ¯ ¯e e (ω)¯¯ >> 1, ¯F (ω) G
(4)
(5)
we see that to a good approximation this leads to the inversion of the transfer function of any passive linear circuit by the negative feedback circuit, i.e., ³ ´−1 . (6) Te (ω) ≈ 1/Fe (ω) = Fe (ω) This also implies through Eq. (2), that the “Golden Rule,”
e (ω) ≈ B e (ω) , A
(7)
holds under these same conditions. Equation (6) also implies that the negative feedback circuit shown in Figure 6 can completely undo any deleterious effects, such as propagation delays, produced by a linear passive circuit (whose transfer function is identical to Fe (ω)) when it is placed before this active device.
In Figure 9, we show one example, where an RC low-pass Þlter is placed before the negative feedback circuit. The positive propagation delay τFe (ω) due to this RC low-pass circuit, can in principle be completely canceled out by the negative group delay produced by the active circuit with the same RC circuit in its feedback loop. This will be true in general for any linear passive circuit, if an identical copy of the circuit is placed inside the negative feedback loop of the active device. The group delay of the negative feedback circuit in the high gain-feedback limit is then be given by ³ ´ e (ω) d arg 1/ F e d arg T (ω) d arg Fe (ω) τTe (ω) = (8) ≈ =− = −τFe (ω) . dω dω dω
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Fig. 10. V out with (upper) and without (lower) the negative feedback.
This shows that the positive group delay from any linear passive circuit can in principle be completely canceled out by the negative group delay from a negative feedback circuit. It is important to note that this negative feedback scheme places a requirement on the gain-bandwidth product of the ampliÞer. For this active circuit to advance the waveform, it must have a large gain at all of the frequency components present in the signal. In particular, if we want to counteract a particular RC time delay, the ampliÞer must have a large gain at frequencies greater than 1/RC. 3.3
Data demonstrating the possibility of the elimination of propagation delays from RC time constants by means of negative group delays
In a recent, simple experiment with the circuit shown in Figure 9, we obtained the data shown in Figure 10, of the outputs from a square wave input into an RC low-pass circuit, with (in the upper trace), and without (in the lower trace) negative feedback. It is clear by inspection of the data in Figure 10, that the propagation delays due to the RC time constant on both the rising and falling edges of the square wave have been almost completely eliminated by means of the negative feedback circuit. However, there is a ringing or overshoot phenomenon accompanying the restoration of the rising and falling edges. Since the CMOS switching levels between logic states occur within 10% of zero volts for LO signals, and within 90% of volt-level HI signals [18], the observed ringing or overshoot phenomenon is not deleterious for the purposes of computer speedup. It is clear from these data that not only the RC time constants associated with transistor gates (the “latency” problem), but also the RC “propagation delays” from the wire interconnects between transistors on a computer chip, can in principle be eliminated by means of the insertion of negative feedback elements. In particular, the Þnite rise time of a MOSFET arising from its intrinsic gate capacitance can be eliminated.
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3.4
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Energy transport by pulses in the optical and electronic domains
In the optical domain, there has been a debate concerning whether or not the velocity of energy transport by the wave packet can exceed c when the group velocity of a wave packet exceeds c. In the case of anomalous dispersion inside an absorption line, Sommerfeld and Brillouin showed that the energy velocity deÞned as follows:
venergy ≡
hSi , hui
(9)
where hSi is the time-averaged Poynting vector and hui is the time-averaged energy density of the electromagnetic wave, is dif f erent from the group velocity [4][26]. Whereas the group velocity in the region of absorptive anomalous dispersion exceeds c, they found that their energy velocity is less than c. Experiments on picosecond laser pulse propagation in absorptive anomalous dispersive media, however, show that these laser pulses travel with a superluminal group velocity, and not with the subluminal energy velocity of Sommerfeld and Brillouin [13]. Hence, the physical meaning of this energy velocity is unclear. When the optical medium possesses gain, as in the case of laser-like media with inverted atomic populations, there arises ambiguities as to whether or not to include the energy stored in the inverted atoms in the deÞnition of hui or not [27][28]. However, in regions of anomalous dispersion well outside of the gain line, and, in particular, in a spectral region where the group-velocity dispersion vanishes, a straightforward application of Sommerfeld and Brillouin’s deÞnition of the energy velocity would imply that the group and energy velocities both exceed c. The equality of these two kinds of wave velocities arises because the pulses of light are propagating inside a transparent medium with little dispersion. In particular, in the case when the energy velocity is negative, the maximum in the pulse of energy leaves the exit face of the optical sample bef ore the maximum in the pulse of energy enters the entrance face. A recent paper deÞned the energy velocity in terms of a time expectation integral over the Poynting vector without any use of the concept of “energy density,” and therefore avoids the above ambiguities associated with the deÞnition of the energy density of the optical medium [29]. The result is that the energy velocity so deÞned can be superluminal. In the case of the electronic circuit with negative feedback which produces negative group delays, the question of when the peak of the energy arrives, can be answered by terminating the output port of Figure 6 by a load resistor, which connects the output to ground. The load resistor (not shown) will be heated up by the energy in the output pulse. It is obvious that the load resistor will then experience the maximum amount of heating when the peak of the Gaussian output pulse arrives at this resistor, and that this happens when the peak of the output voltage waveform arrives. For negative group delays, the load resistor will then heat up earlier than expected. However, there is no mystery here: The operational ampliÞer can supply the necessary energy to heat up the load resistor ahead of time. The negative group delay and the negative energy delay are identical to each other in this case, and likewise the negative group and energy velocities are equal. 3.5
Kramers-Kronig relations necessitate superluminal group velocities, and the Bode relations necessitate negative group delays
These counter-intuitive results also follow quite generally from the Kramers-Kronig relations for optical media [23], and the Bode relations for electronic circuits [24]. For optical media, we have proved two theorems starting from the principle of causality, along with the additional assumption of linearity of the media, that superluminal group velocities in any optical medium must generally exist in some spectral region, and that for an amplifying medium, this spectral region must exist outside of the regions with gain, i.e., in the transparent regions outside of the gain lines [25]. Negative group delays for electronic circuits similarly follow quite generally from the Bode relations. These dispersion relations can also be generalized to apply to the transmission of atoms through the quantum many-body systems which we shall discuss below. Negative group delays in this case means negative transmission times of atomic wave packets through the many-body system. Thus, causality itself necessitates the existence of these counter-intuitive, superluminal phenomena.
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Fig. 11. Condensate-mediated processes associated with the coherent transmission of an incoming atom.
4
ANOMALOUS TRANSMISSION TIMES IN CONDENSED MATTER SYSTEMS
So far, we have considered examples from optics and electronics. The quantum-mechanical tunneling process is another example of a superluminal phenomenon; however, this example involved only single-particle propagation. Here we consider other quantum mechanical examples taken from condensed matter physics. Unusual many-body systems, such as superßuid helium and atomic Bose-Einstein condensates (BECs), might also exhibit anomalous transmission times. These effects arise from Bose exchange symmetry and macroscopic quantum coherence. In such systems, one cannot know, even in principle, which identical particle of the many-body system was involved in a collision process. The Bose symmetrization of the total wavefunction leads to a long-range, macroscopic entangled state of the entire many-body system, and thus to off diagonal long range order. 4.1
Condensate-mediated transmission of helium atoms through superßuid helium slabs
We Þrst focus on superßuid helium. Solving the scattering problem of 4 He atoms from a superßuid helium surface is not an easy task. Several approaches have been taken to reproduce the experimental data on quantum evaporation and condensation obtained by Wyatt and Tucker [30], and others. These approaches include semiclassical treatments (Mulheran and Inkson [31]), time-dependent density functional theory using a phenomenological density functional (Dalfovo et al. [32]) and the correlated basis function method used by Campbell et al. [33]. The difficulty of the problem arises from the necessity of a self-consistent many-body solution for a system of strongly interacting bosons. Circumventing all these difficulties, we give here some general arguments for why there can in principle be anomalous transmission times in the transmission of a helium atom through a superßuid helium slab, if measurements with sufficiently high time resolution are performed. Halley et al. [34] proposed the possibility of the transmission of helium atoms through a superßuid helium slab due to a condensate-mediated process with transmission time delays independent of the slab thickness. Consider a superßuid helium slab in an N particle ground state. In the Halley et al. process, the transmission of an incoming helium atom through the slab occurs via a virtual transition of the N particle ground state plus the incident particle, to the N + 1 particle ground state, followed by the coherent reemission of one particle on the opposite side of the superßuid (see Figure 11(a)). Following the steps of the paper Þrst suggesting this process [34], we consider the transfer Hamiltonian
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HT =
X (TkL0 bk0 ,L a†0 + TkR0 b†k0 ,R a0 + h.c.)
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(10)
k0
The b† and b operators are creation and annihilation operators of particles with momentum k0 at the left (L) and right (R) surfaces; the a†0 and a0 operators create and annihilate the particles in the condensate part of the ßuid. The coefficients T depend on the microscopic structure of the ßuid surface and must be determined by experiment. We assume that |lef ti = |N, N0 i|kiL |vaciR is the initial state of the system with a total number of N atoms in the ßuid, out of which N0 particles are in the condensate; one helium atom with momentum k is incident from the left. A transmission of an atom from the left to the right side of the slab can occur in the second-order process shown in Figure 11(a), for which the transition matrix element is the following: hright|HT |iihi|HT |lef ti . E − Ei
(11)
Here |ii = |N +1, N0 +1i|vaciL |vaciR is the intermediate state, Ei its energy, and |righti = |N, N0 i|vaciL |kiR the Þnal state. Using the standard deÞnitions for the creation and annihilation operators, it is easy to show that the numerator of Eq. (11) is (N0 + 1)TkL TkR .
(12)
In this scattering process, the energy in the initial and the Þnal state is given by the sum of the kinetic energy of the incoming particle plus the energy of the N particle ground state E = ²k + EN
(13)
Ei = EN+1 .
(14)
The energy in the intermediate state is
This leads to an energy denominator in second-order perturbation theory of ∆E = ²k + EN − EN +1 = ²k + |µ|.
(15)
Here, µ is the chemical potential (−7.16 K in the case of superßuid helium). Thus, the transition matrix element for this process is (N0 + 1)TkL TkR . ²k + |µ|
(16)
If the intermediate state used above were the only one coupling the initial and Þnal states, a transition rate in the form of Fermi’s golden rule could be given by squaring the transition amplitude [34]. However, a coherent sum of the transition amplitudes over all possible intermediate states has to be performed, and there is another process which was not taken into account in Ref. [34]. This process is equally important and leads to negative time delays in transmission. It is pictured in Figure 11(b), and we will refer to it as the primed process. A helium atom approaches the slab, and, before the atom reaches the superßuid, the condensate coherently emits another atom on the other side of the slab. In this intermediate state there are two atoms outside of an N − 1 particle ground state. Finally, the atom on the left side of the superßuid gets absorbed into the condensate, and the ßuid is brought back into the N particle ground state. The energy for the intermediate state in the primed process is
Chiao et.al., Faster-than-light propagations and their applications
Ei0 = 2²k + EN−1 .
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(17)
The energy difference between the initial and the intermediate state is ∆E 0 = ²k + EN − (2²k + EN−1 ) = −²k − |µ| = −∆E.
(18)
The numerator of Eq. (11) is now N0 TkR TkL ,
(19)
so that the transition matrix element for the primed process is
−
N0 TkR TkL . ²k + |µ|
(20)
For plane waves, the Þnal states of these two condensate-mediated processes are indistinguishable. Therefore the two transition matrix elements must be added up coherently. Comparison of Eq. (16) with Eq. (20) shows that the two processes destructively interfere with each other, and that they almost cancel out in the large N0 limit. This implies that the transmission probability for very long, single-particle wave packets via a condensate-mediated process becomes vanishingly small. The energy-time uncertainty relation restricts the durations of the intermediate states, so that the transmission times will be of the order of ±~/|∆E| for the unprimed and primed processes, respectively. For low energy incident atoms the transmission times are approximately given by ±~/|µ|, which is of the order of a picosecond in the system considered. In order to detect either one of these processes experimentally, the Þnal states have to be made distinguishable. This could be achieved by temporal resolution, i.e., the formation of atomic wave packets with a duration comparable to the lifetimes of the respective intermediate states, so that the transmitted wave packets would be Rayleigh resolved, and an actual measurement of the time sequence of events could be performed. A wave packet this short in time will have an energy uncertainty comparable to the size of the chemical potential, so that Rayleigh resolution will only barely be possible. It should be noted that both of the processes considered here depend in the same way on the macroscopic coherence and on the off diagonal long range order of this quantum system. Both processes will lead to faster-than-light effects. In the primed process this is due to the intrinsic negativity of the transmission time. In the unprimed process faster-than-light effects can be achieved by the choice of a sufficiently thick slab, since the transmission time, although positive, is independent of the slab thickness, similar to the case of the Hartman effect in tunneling discussed earlier. 4.2
Positive and negative transmission times of atoms through an atomic BEC
In connection with condensate-mediated transmission of atoms, it is simpler to consider the recently observed atomic BECs [35], which are weakly-interacting Bose gases, than superßuid helium, which is a stronglyinteracting Bose liquid. The microscopic theory of a dilute, weakly-interacting Bose gas is well characterized in terms the Bogoliubov theory [36], which has been veriÞed by experiment. In the case of the atomic BECs, the chemical potential sets the energy scale for the uncertainty-principle lifetime or duration of the intermediate state, and hence for the transmission times (either positive or negative) through the condensate of low-energy incident atoms identical to the atoms of the BEC. The chemical potential µ in the condensate can be calculated in the Bogoliubov approximation, and is given by µ ≈ N0
2π~2 a , mV
(21)
where N0 is the number of atoms in the condensate, a is the S-wave scattering length, m is the mass of the atom, and V is the volume of the condensate. The typical experimental parameters for a BEC consisting
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of sodium atoms are a = 2.75 nm, N0 /V = 1.5 x 1014 cm−3 , and D = 10 µm for the typical size of the condensate [37]. The speed of sound for such a condensate has been observed to be vs ≈ 1.0 mm/s, which is close to that predicted by the Bogoliubov theory. From these numbers, we infer that the chemical potential for an atom in the typical sodium BEC is µ ≈ 7.4 x 10−24 erg ≈ 54 nK. The typical time scale associated with condensate-mediated processes is therefore ~/µ, which is two orders of magnitude shorter than the transmission time due to sound wave propagation across the condensate. Hence it should be easy to distinguish the condensate-mediated processes from sound-wave mediated processes. Note that the time scale ~/µ is independent of the size D of the condensate, and leads to a Hartman-like effect, which resembles the Hartman effect in tunneling. The positive and negative condensate-mediated transmission times, or group delays, which correspond to the two processes depicted in Figure 11(a) and (b), respectively, are τ± ≈ ±
~ ~ . ± ≈ ±0.14ms. εk + µ µ
(22)
These times are much larger than those observed in the photon tunneling experiments. Again, as in the case of superßuid helium, it should barely be possible to distinguish, by Rayleigh’s resolution criterion, a negative group delay from a positive group delay, for cold atoms whose incident kinetic energy is comparable to the chemical potential. Nevertheless, this effect should be experimentally observable with a sufficiently high signal-to-noise ratio, as has already be done in the case in the measurement of the tunneling times for photons. Furthermore, each atomic transmission event detected in an experiment would result in a deÞnite sign for the transmission time for that event. The post-selection of rare, condensate-mediated transmission events leads to a “weak measurement” of this post-selected subensemble in the sense of Aharonov [38], and results, as in the case of tunneling [39], in surprising “weak values,” such as negative transmission times. Thus, not only can light pulses be slowed to a stop in atomic BECs [40], but atoms can also be transmitted superluminally through such BECs. 5
CONCLUSIONS
There is a widespread view among electrical engineers and physicists that although the phase velocity can exceed the vacuum speed of light, the group velocity can never do so. Otherwise, signals would be able to propagate faster than light, since conventional wisdom equates the group velocity with the signal velocity. From this conventional point of view, the group velocity is essentially the same as the energy velocity in transparent media, and the latter could never exceed c. Several generations of students have been taught this. Many of the standard textbooks and handbooks state this conventional viewpoint, some, however, with qualiÞcations which unfortunately are not strong enough, so that the net result is still misleading. For example, The Electrical Engineering Handbook in its discussion concerning the group velocity states the following [41]: “When traveling in a medium, the velocity of energy transmission (e.g. a light pulse) is less than c, and is given by [the group velocity].” This statement, and other similar statements in many of the early standard textbooks in optics and classical electrodynamics, are misleading. As a result, we have been blinded by our misconceptions, and thereby have been prevented from exploring and discovering many new, interesting, and possibly important, phenomena, which could have been discovered long ago. Some of these are only now being uncovered, and some of these phenomena may in fact lead to important applications, such as the speed-up of computers. The effects reported in this paper do not violate relativity. The front velocity of Sommerfeld and Brillouin, which is strictly c, is the only velocity which is relevant to relativity. However, the group, the energy, and Brillouin’s “signal,” velocity can all exceed c, without violating the principle of relativistic causality [1] 6
ACKNOWLEDGMENTS
This work was supported by the ONR, NSF, and NASA. We thank S. Coen, M. Haakestad, C. McCormick and M. Mojahedi for helpful discussions. J. M. H. is on a sabbatical leave from the Department of Physics,
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