ECEG398 Quantum Optics Course Notes Part 1: Introduction

1. ECEG398 Quantum Optics Course NotesPart 1: Introduction Prof. Charles A. DiMarzio and Prof. Anthony J. Devaney Northeastern University Spring 2006 Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

3. Classical Maxwellian EM Waves v=c λ E H H x E E z H λ=c/υ y c=3x108 m/s (free space) υ = frequency (Hz) Chuck DiMarzio, Northeastern University Thanks to Prof. S. W.McKnight

4. VIS= 0.40-0.75μ γ-Ray RF Electromagnetic Spectrum (by λ) UV= Near-UV: 0.3-.4 μ Vacuum-UV: 100-300 nm Extreme-UV: 1-100 nm IR= Near: 0.75-2.5μ Mid: 2.5-30μ Far: 30-1000μ 10 nm =100Å 0.1 μ 1 μ 10 μ 100 μ = 0.1mm (300 THz) 0.1 Å 1 Å 10 Å 1 mm 1 cm 0.1 m X-Ray Soft X-Ray Mm-waves Microwaves Chuck DiMarzio, Northeastern University Thanks to Prof. S. W.McKnight

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+t (or at r+r)? Chuck DiMarzio, Northeastern University

6. 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 0 5 10 0 5 10 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 0 5 10 0 5 10 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. Chuck DiMarzio, Northeastern University

7. 0.4 0.2 0 -0.2 -0.4 0 1 2 3 4 5 6 7 8 f 0.4 0.2 0 -0.2 -0.4 0 1 2 3 4 5 6 7 8 Realistic Example Long Delay: Decorrelation 50 Random Sine Waves with Center Frequency 1 and Bandwidth 0.8. Short Delay Chuck DiMarzio, Northeastern University

8. I1+I2 Correlation Function Chuck DiMarzio, Northeastern University

9. Controlling Coherence Making Light Coherent Making Light Incoherent Ground Glass to Destroy Spatial Coherence Spatial Filter for Spatial Coherence Wavelength Filter for Temporal Coherence Move it to Destroy Temporal Coherence Chuck DiMarzio, Northeastern University

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? Chuck DiMarzio, Northeastern University

11. Clock Signal t 0 5 Photon Arrival t Photon Count 3 1 2 t Counter Gate Clock n Photon Counting Experiment Experimental Setup to measure the probability distribution of photon number. Probability Density Chuck DiMarzio, Northeastern University

12. The Mean Number • Photon Energy is hn • Power on Detector is P • Photon Arrival Rate is a=P/hn • Photon “Headway” is 1/a • Energy During Gate is PT • Mean Photon Count is n=PT/hn • But what is the Standard Deviation? Chuck DiMarzio, Northeastern University

13. What do you expect? • Photons arrive equally spaced in time. • One photon per time 1/a • Count is aT +/- 1 maybe? • Photons are like the Number 39 Bus. • If the headway is 1/a=5 min... • Sometimes you wait 15 minutes and get three of them. Chuck DiMarzio, Northeastern University

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)=adt • 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 Chuck DiMarzio, Northeastern University

15. Quantum Coherence Here are some results: Later we will prove them. Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

17. Back-Door Quantum Optics (Field) • Assume a classical (constant) field, Usig • Add a random noise field Unoise • Complex Zero-Mean Gaussian • Compute s as function of <| Unoise|2> • Compare to Poisson distribution • Fix <| Unoise|2> to Determine Noise Source Equivalent to Quantum Fluctuations Chuck DiMarzio, Northeastern University

18. Classical Noise Model Add Field Amplitudes Im U Un 10842-1.tex:2 Us Re U Chuck DiMarzio, Northeastern University

19. Photon Noise 10842-1.tex:3 10842-1.tex:5 = 10842-1-5.tif Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

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 Chuck DiMarzio, Northeastern University

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 ai in measuring A, then the system is in the corresponding eigenstate, yi after making the measurement. • 5. The time dependence of a state is given by Chuck DiMarzio, Northeastern University

23. State of a System • State Defined by a Wave Function, y • Depends on, eg. position or momentum • Equivalent information in different representations. y(x) and f(p), a Fourier Pair • Interpretation of Wavefunction • Probability Density: P(x)=|y(x)|2 • Probability: P(x)dx=|y(x)|2dx Chuck DiMarzio, Northeastern University

24. y1(x) y2(x) x x Wave Function as a Vector • List y(x) for all x (Infinite Dimensionality) • Write as superposition of vectors in a basis set. y(x)=a1y1(x)+a2y2(x)+... Chuck DiMarzio, Northeastern University

25. More on Probability • Where is the particle? • Matrix Notation Chuck DiMarzio, Northeastern University

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(iw1t) and exp(iw2t) • Describe the particle motion. y2(x) y1(x) x x Stopped Wed 11 Jan 06 Chuck DiMarzio, Northeastern University

27. Dirac Notation • Simple Way to Write Vectors • Kets • and Bras • Scalar Products • Brackets • Operators Chuck DiMarzio, Northeastern University

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 sxsp > h/2 [x H] = x H - H x = 0 Chuck DiMarzio, Northeastern University

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 ai in measuring A, then the system is in the corresponding eigenstate, yi after making the measurement. • 5. The time dependence of a state is given by Chuck DiMarzio, Northeastern University

30. Born: 12 Aug 1887 in Erdberg, Vienna, AustriaDied: 4 Jan 1961 in Vienna, Austria* 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 Chuck DiMarzio, Northeastern University

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: Chuck DiMarzio, Northeastern University

32. 1 0.5 0 -0.5 -1 0 0.2 0.4 0.6 0.8 1 Some Wavefunctions Shrödinger Equation Eigenvalue Problem Hy=Ey Solution Temporal Behavior Chuck DiMarzio, Northeastern University

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? Chuck DiMarzio, Northeastern University

34. Harmonic Oscillator • Hamiltonian • Frequency Potential Energy x Chuck DiMarzio, Northeastern University

35. Harmonic Oscillator Energy • Solve the Shrödinger Equation • Solve the Eigenvalue Problem • Energy • Recall that... Chuck DiMarzio, Northeastern University

36. † Louisell’s Approach • Harmonic Oscillator • Unit Mass • New Operators Chuck DiMarzio, Northeastern University

37. The Hamiltonian • In terms of a, a † • Equations of Motion Chuck DiMarzio, Northeastern University

38. Energy Eigenvalues • Number Operator • Eigenvalues of the Hamiltonian Chuck DiMarzio, Northeastern University

39. Creation and Anihilation (1) • Note the Following Commutators • Then Chuck DiMarzio, Northeastern University

40. Creation and Anihilation (2) Energy Eigenvalues Eigenvalue Equations States Chuck DiMarzio, Northeastern University

41. Creation and Anihilation (3) Chuck DiMarzio, Northeastern University

42. Reminder! • All Observables are Represented by Hermitian Operators. • Their Eigenvalues must be Real Chuck DiMarzio, Northeastern University