The Nature of Signal Velocity c Classical or Quantum Mechanical Shi-Yao ZHU and L.G. Wang

1. The Nature of Signal Velocity cClassical or Quantum MechanicalShi-Yao ZHU and L.G. Wang Department of Physics Hong Kong Baptist University Memorizing Albert Einstein at World Year of Physics Sino-German Symposium 23-27 Nov., 2005

2. The velocity of discontinuities (which carrying signals) is exact equal to c. • The nature of the signal velocity not larger than c.

3. Superluminal propagation and anomalous dispersion • Experiments on superluminal propagation • Velocity of discontinuities =c (Experiments and theory) • Classical vs Quantum Mechanical V. Numerical Examples VI. Conclusion

4. Superluminal propagation and anomalous dispersion What is the superluminal propagation? Group velocity larger than c. What is the Physical Meaning!!

5. What Causes the Superluminal Propagation of a Pulse? 1. Anomalous dispersive media (including photonic crystals, etc) 2. Maxwell EM theory.

6. Group velocity of waves Normal dispersion: Index increases with frequency Anomalous dispersion: Index decreases with frequency

7. The group velocity Effect of Dispersion Two plane waves propagate in z direction

8. ng>1 (a) n(w1)=1.4 and n(w2)=1.35; w1>w2 (a) n(w1)=1.4 and n(w2)=1.35; w1>w2

9. 1>ng>0 (b) n(w1)=1.3 and n(w2)=1.4 w1>w2

10. ng<0 (c) n(w1)=1.2 and n(w2)=1.4; w1>w2

11. Energy velocity Group velocity Various Velocities Phase velocityc/n (plane wave) n depends on the frequency Front velocity

12. n(w1)=1.4 and n(w2)=1.35; • w1>w2 (b) n(w1)=1.3 and n(w2)=1.4; w1>w2 (c) n(w1)=1.2 and n(w2)=1.4; w1>w2

13. II, Experiments on superluminal propagation Review of the Main Experimental Results

14. Schematic atomic level diagram. II. Experiment* Gain coeff. and refraction index. *L.J. Wang, et al., Nature, 406, p277(2000)

15. II. Experiment result

16. III. Velocity of discontinuities =c Can a signal be transmitted with a superluminal velocity? How to define a signal? Who carries the signal? The discontinuities ? The group velocity (Peak) ?

17. Remarkably, the signal velocity [4] of a light pulse, defined as the velocity at which the half point of the pulse front travels, also exceeds the speed of light in a vacuum, c, in the present experiment.It has also been suggested [4,16] that the true speed at which information is carried by a light pulse should be defined as the ``frontal'' velocity of a step-function-shaped signal which has been shown not to exceed c (ref. 4).The implications of the present experiment on signal propagation and its speed will be further analyzed, particularly for the case when the light pulse consists of only a few photons.

18. The velocity of a discontinuity = c It has been stated in several papers: • RY Chiao and AM Steinberg, Tunneling times and Superluminality, “progress in optics” ed. by E. wolf, 37, p345, 1997. • Y. Japha and G. Kurizki, PRA 53, 586 (1996). • JC Garrison et al., Phys. Lett. A 245, 19 (1998) However, Lack of theoretical proof.

19. Front point or a discontinuity “The front velocity,… should be identified as the true signal velocity.” “New information is communicated only when there is an unexpected change, such as a discontinuity, whose arrival time cannot be inferred from the past behavior of the wave.” “ hence the front … constitutes a genuine signal, i.e., new information” * “It is the response in the infinite-frequency limit of the system that ultimately determines the propagation speed of the points of non-analyticity, and hence of truly new information.” * “New information can only be transmitted by a nonanalyticdisturbance.”** *RY Chiao and AM Steinberg, Tunneling times and Superluminality, <<progress in optics>> ed. by E. wolf, XXXVII, p345, 2002. **Y. Japha and G. Kurizki, PRA 53, 586 (1996).

20. The propagation of a frontExperimental Evidencefrom Prof. Y.Z. WangThe front velocity is cinAnomalous and Normal medium

21. 信息传播速度vi 对超光速脉冲同样也不能超过光速 c ,非领起始点传播速度vi = c。  = -1s Start point, vi = c

22. 超光速脉冲非零起始点 vi 实验结果： 10s   -2s Start point ,vi = c

24. Recent Experimental Evidence of Discontinuity in Optical Pulses M.D. Stenner et al., Phys. Rev. Lett. 94, 053902 (2005). The velocity of a discontinuity is c, although the group velocity is0.006c. M.D. Stenner et al., Nature 425, 695 (2003) The velocity of a discontinuity is c, although the group velocity is-0.051c.

25. Anomalous medium

26. Normal medium

27. Theoretical Proof Kramers-Kronig relations Under KKRS, Causality is preserved.

28. The output-input relation is determined by the Maxwell Theory See Jackson’s book p. 336

30. is a continuous function of time, beause J(t,z) is finite. has the same discontinuity as The velocity of the discontinuity is c

31. For metal :

35. Consider the discontinuity on The n-th order derivative Similar method is used to prove discontinuity of the n-th order derivative propagates at the velocity of c

36. We prove:any discontinuity (including the front) always propagates at c. 1. The Kramers-Kronig relations. 2. Maxwell EM Theory 3. No requirement on the shape of the pulse and the medium.

37. Numerical Examples Such as the medium in the L.J. Wang’s experiments Anomalous Dispersion(1) Which satisfies the KKRS

38. Sine pulse with a Front

39. Discontiuity velocity The medium of L.J. Wang’s experiment n=1

40. First order derivative

42. Second order derivatives

44. Normal dispersion(2) usedin PRL 94, 053902 (2005) Numerical calculation confirms Discontinuity propagates at c. Two level gain

45. (2) Normal dispersion

46. (2)

48. Second Part: The nature of the signal velocity not larger than c. How to Understand The Superluminality? Someone tried to use quantum fluctuations to save the Einstein declaration (signal velocity can not be faster than c)

50. The Quantum Fluctuations limit Signal Velocity to Values Less Than c Without the Quantum Fluctuation What is the Signal Velocity ? Larger than c? Einstein needs Quantum Mechanics to made his declaration?