1. January 2006Chuck DiMarzio, Northeastern University10842-1c-1 ECEG398 Quantum Optics Course Notes Part 1: Introduction Prof. Charles A. DiMarzio and Prof. Anthony J. Devaney Northeastern University Spring 2006

2. January 2006Chuck DiMarzio, Northeastern University10842-1c-2 Lecture Overview Motivation –Optical Spectrum and Sources –Coherence, Bandwidth, and Fluctuations –Motivation: Photon Counting Experiments –Classical Optical Noise –Back-Door Quantum Optics Background –Survival Quantum Mechanics

3. January 2006Chuck DiMarzio, Northeastern University10842-1c-3 Classical Maxwellian EM Waves E E E x y z H H H λ v=c λ=c/υ c=3x10 8 m/s (free space) υ = frequency (Hz) Thanks to Prof. S. W.McKnight

4. January 2006Chuck DiMarzio, Northeastern University10842-1c-4 Electromagnetic Spectrum (by λ) 1 μ10 μ100 μ = 0.1mm 0.1 μ10 nm =100Å VIS= 0.40-0.75μ 1 mm1 cm0.1 m IR= Near: 0.75-2.5μ Mid: 2.5-30μ Far: 30-1000μ UV= Near-UV: 0.3-.4 μ Vacuum-UV: 100-300 nm Extreme-UV: 1-100 nm MicrowavesX-RayMm-waves 10 Å1 Å0.1 Å Soft X-RayRFγ-Ray (300 THz) Thanks to Prof. S. W.McKnight

5. January 2006Chuck DiMarzio, Northeastern University10842-1c-5 Coherence of Light Assume I know the amplitude and phase of the wave at some time t (or position r). Can I predict the amplitude and phase of the wave at some later time t+  (or at r+  )?

6. January 2006Chuck DiMarzio, Northeastern University10842-1c-6 Coherence and Bandwidth Pure Cosine f=1 Pure Cosine f=1.05 3 Cosines Averaged f= 0.93, 1, 1.05 Same as at left, and a delayed copy. Note Loss of coherence.

7. January 2006Chuck DiMarzio, Northeastern University10842-1c-7 Realistic Example 50 Random Sine Waves with Center Frequency 1 and Bandwidth 0.8. f 012345678 -0.4 -0.2 0 0.2 0.4 012345678 -0.4 -0.2 0 0.2 0.4 Long Delay: Decorrelation Short Delay

8. January 2006Chuck DiMarzio, Northeastern University10842-1c-8 Correlation Function I 1 +I 2

9. January 2006Chuck DiMarzio, Northeastern University10842-1c-9 Controlling Coherence Making Light Coherent Making Light Incoherent Spatial Filter for Spatial Coherence Wavelength Filter for Temporal Coherence Ground Glass to Destroy Spatial Coherence Move it to Destroy Temporal Coherence

10. January 2006Chuck DiMarzio, Northeastern University10842-1c-10 A Thought Experiment Consider the most coherent source I can imagine. Suppose I believe that light comes in quanta called photons. What are the implications of that assumption for fluctuations?

11. January 2006Chuck DiMarzio, Northeastern University10842-1c-11 Photon Counting Experiment 05 Clock Gate Counter t Clock Signal t Photon Arrival t Photon Count 312 Probability Density n Experimental Setup to measure the probability distribution of photon number.

12. January 2006Chuck DiMarzio, Northeastern University10842-1c-12 The Mean Number Photon Energy is h Power on Detector is P Photon Arrival Rate is  =P/h –Photon “Headway” is 1/  Energy During Gate is PT Mean Photon Count is n=PT/h But what is the Standard Deviation?

13. January 2006Chuck DiMarzio, Northeastern University10842-1c-13 What do you expect? Photons arrive equally spaced in time. –One photon per time 1/  –Count is  T +/- 1 maybe? Photons are like the Number 39 Bus. –If the headway is 1/  5 min... –Sometimes you wait 15 minutes and get three of them.

14. January 2006Chuck DiMarzio, Northeastern University10842-1c-14 Back-Door Quantum Optics (Power) Suppose I detect some photons in time, t Consider a short time, dt, after that –The probability of a photon is P(1,dt)=  dt –dt is so small that P(2,dt) is almost zero –Assume this is independent of previous history –P(n,t+dt)=P(n,t)P(0,dt)+P(n-1,t)P(1,dt) Poisson Distribution: P(n,t)=exp(-at)(at) n /n! The proof is an exercise for the student

15. January 2006Chuck DiMarzio, Northeastern University10842-1c-15 Quantum Coherence Here are some results: Later we will prove them.

16. January 2006Chuck DiMarzio, Northeastern University10842-1c-16 Question for Later: Can We Do Better? Poisson Distribution – –Fundamental Limit on Noise Amplitude and Phase –Limit is On the Product of Uncertainties Squeezed Light –Amplitude Squeezed (Subpoisson Statistics) but larger phase noise –Phase Squeezed (Just the Opposite) Stopped here 9 Jan 06

17. January 2006Chuck DiMarzio, Northeastern University10842-1c-17 Back-Door Quantum Optics (Field) Assume a classical (constant) field, U sig Add a random noise field U noise –Complex Zero-Mean Gaussian Compute  as function of Compare to Poisson distribution Fix to Determine Noise Source Equivalent to Quantum Fluctuations

18. January 2006Chuck DiMarzio, Northeastern University10842-1c-18 Classical Noise Model Add Field Amplitudes Re U Im U UsUs UnUn 10842-1.tex:2

19. January 2006Chuck DiMarzio, Northeastern University10842-1c-19 Photon Noise 10842-1.tex:3 10842-1.tex:5 = 10842-1-5.tif

20. January 2006Chuck DiMarzio, Northeastern University10842-1c-20 Noise Power One Photon per Reciprocal Bandwidth Amplitude Fluctuation –Set by Matching Poisson Distribution Phase Fluctuation –Set by Assuming Equal Noise in Real and Imaginary Part Real and Imaginary Part Uncorrelated

21. January 2006Chuck DiMarzio, Northeastern University10842-1c-21 The Real Thing! Survival Guide The Postulates of Quantum Mechanics States and Wave Functions Probability Densities Representations Dirac Notation: Vectors, Bras, and Kets Commutators and Uncertainty Harmonic Oscillator

22. January 2006Chuck DiMarzio, Northeastern University10842-1c-22 Five Postulates 1. The physical state of a system is described by a wavefunction. 2. Every physical observable corresponds to a Hermitian operator. 3. The result of a measurement is an eigenvalue of the corresponding operator. 4. If we obtain the result a i in measuring A, then the system is in the corresponding eigenstate,  i after making the measurement. 5. The time dependence of a state is given by

23. January 2006Chuck DiMarzio, Northeastern University10842-1c-23 State of a System State Defined by a Wave Function,  –Depends on, eg. position or momentum –Equivalent information in different representations.  (x) and  (p), a Fourier Pair Interpretation of Wavefunction –Probability Density: P(x)=|  (x)| 2 –Probability: P(x)dx=|  (x)| 2 dx

24. January 2006Chuck DiMarzio, Northeastern University10842-1c-24 Wave Function as a Vector List  (x) for all x (Infinite Dimensionality) Write as superposition of vectors in a basis set.   (x)   (x) x x  (x)=  a 1  1 (x)+a 2  2 (x)+...

25. January 2006Chuck DiMarzio, Northeastern University10842-1c-25 More on Probability Where is the particle? Matrix Notation

26. January 2006Chuck DiMarzio, Northeastern University10842-1c-26 Pop Quiz! (Just kidding) Suppose that the particle is in a superposition of these two states. Suppose that the temporal behaviors of the states are exp(i  1 t) and exp(i  2 t) Describe the particle motion.   (x)   (x) x x Stopped Wed 11 Jan 06

27. January 2006Chuck DiMarzio, Northeastern University10842-1c-27 Dirac Notation Simple Way to Write Vectors –Kets –and Bras Scalar Products –Brackets Operators

28. January 2006Chuck DiMarzio, Northeastern University10842-1c-28 Commutators and Uncertainty Some operators commute and some don’t. We define the commutator as [a b] = a b - b a Examples [x p] = x p - p x = ih  x  p  h  [x H] = x H - H x = 0

29. January 2006Chuck DiMarzio, Northeastern University10842-1c-29 Recall the Five Postulates 1. The physical state of a system is described by a wavefunction. 2. Every physical observable corresponds to a Hermitian operator. 3. The result of a measurement is an eigenvalue of the corresponding operator. 4. If we obtain the result a i in measuring A, then the system is in the corresponding eigenstate,  i after making the measurement. 5. The time dependence of a state is given by

30. January 2006Chuck DiMarzio, Northeastern University10842-1c-30 Shrödinger Equation Temporal Behavior of the Wave Function –H is the Hamiltonian, or Energy Operator. The First Steps to Solve Any Problem: –Find the Hamiltonian –Solve the Schrödinger Equation –Find Eigenvalues of H *http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria* *

31. January 2006Chuck DiMarzio, Northeastern University10842-1c-31 Particle in a Box Before we begin the harmonic oscillator, let’s take a look at a simpler problem. We won’t do this rigorously, but let’s see if we can understand the results. Momentum Operator:

32. January 2006Chuck DiMarzio, Northeastern University10842-1c-32 Some Wavefunctions Eigenvalue Problem H  =E  Solution 00.20.40.60.81 -0.5 0 0.5 1 Shrödinger Equation Temporal Behavior

33. January 2006Chuck DiMarzio, Northeastern University10842-1c-33 Pop Quiz 2 (Still Kidding) What are the energies associated with different values of n and L? Think about these in terms of energies of photons. What are the corresponding frequencies? What are the frequency differences between adjacent values of n?

34. January 2006Chuck DiMarzio, Northeastern University10842-1c-34 Harmonic Oscillator Hamiltonian Frequency Potential Energy x

35. January 2006Chuck DiMarzio, Northeastern University10842-1c-35 Harmonic Oscillator Energy Solve the Shrödinger Equation Solve the Eigenvalue Problem Energy –Recall that...

36. January 2006Chuck DiMarzio, Northeastern University10842-1c-36 Louisell’s Approach Harmonic Oscillator –Unit Mass New Operators †

37. January 2006Chuck DiMarzio, Northeastern University10842-1c-37 The Hamiltonian In terms of a, a † Equations of Motion

38. January 2006Chuck DiMarzio, Northeastern University10842-1c-38 Energy Eigenvalues Number Operator Eigenvalues of the Hamiltonian

39. January 2006Chuck DiMarzio, Northeastern University10842-1c-39 Creation and Anihilation (1) Note the Following Commutators Then

40. January 2006Chuck DiMarzio, Northeastern University10842-1c-40 Creation and Anihilation (2) Eigenvalue EquationsStates Energy Eigenvalues

41. January 2006Chuck DiMarzio, Northeastern University10842-1c-41 Creation and Anihilation (3)

42. January 2006Chuck DiMarzio, Northeastern University10842-1c-42 Reminder! All Observables are Represented by Hermitian Operators. Their Eigenvalues must be Real