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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

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

January 2006Chuck DiMarzio, Northeastern University c-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+  )?

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

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

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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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?

January 2006Chuck DiMarzio, Northeastern University c-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.

January 2006Chuck DiMarzio, Northeastern University c-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?

January 2006Chuck DiMarzio, Northeastern University c-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.

January 2006Chuck DiMarzio, Northeastern University c-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

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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

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

January 2006Chuck DiMarzio, Northeastern University c-19 Photon Noise tex: tex:5 = tif

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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)+...

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

January 2006Chuck DiMarzio, Northeastern University c-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

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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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

January 2006Chuck DiMarzio, Northeastern University c-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 * Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria* *

January 2006Chuck DiMarzio, Northeastern University c-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:

January 2006Chuck DiMarzio, Northeastern University c-32 Some Wavefunctions Eigenvalue Problem H  =E  Solution Shrödinger Equation Temporal Behavior

January 2006Chuck DiMarzio, Northeastern University c-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?

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

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

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

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

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

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

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

January 2006Chuck DiMarzio, Northeastern University c-41 Creation and Anihilation (3)

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