MIT OpenCourseWarehttp://ocw.mit.edu 6.453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.September 16, 2008Optical and Quantum Communications Groupwww.rle.mit.edu/qoptics6.453 Quantum Optical CommunicationLecture 4Jeffrey H. Shapiro2www.rle.mit.edu/qoptics6.453 Quantum Optical Communication — Lecture 4! Handouts! Lecture notes, slides! Quantum Harmonic Oscillator! Quantization of a classical LC circuit! Annihilation and creation operators! Energy eigenstates — number-state kets3www.rle.mit.edu/qopticsClassical LC Circuit: Undriven, Lossless Oscillation! State Variablescapacitor charge:inductor flux:! Stored Energy (Hamiltonian)! Hamilton’s Equations4www.rle.mit.edu/qopticsClassical LC Circuit: Undriven, Lossless Oscillation! Nonzero Initial Conditions:! Oscillation Frequency:! Solutions for :5www.rle.mit.edu/qopticsClassical LC Circuit: Dimensionless Reformulation! Assume! Define Complex Envelope (Phasor):! Simple Harmonic Motion at Constant Energy6www.rle.mit.edu/qopticsQuantum LC Circuit: Quantum Harmonic Oscillator! Postulate: Become Observables! Canonical Commutation Relation:! Dimensionless Reformulation:7www.rle.mit.edu/qopticsQuantum Harmonic Oscillator: Commutators! Dimensionless Reformulation of Canonical Commutators:! Hamiltonian:! Heisenberg Uncertainty Principle:8www.rle.mit.edu/qopticsClassical versus Quantum Behavior! Classical Oscillator: Noiseless ! Quantum Oscillator: Noisy9www.rle.mit.edu/qopticsEnergy Eigenvalues and Eigenkets! Notation:! Annihilation and Creation Operations! is energy eigenket with eigenvalue! is energy eigenket with eigenvalue! Minimum Energy State: such that! Energy Eigenvalues:10www.rle.mit.edu/qopticsNumber Operator and Number States! Oscillator Energy is Quantized in Increments! (Photon) Number Operator:11www.rle.mit.edu/qopticsComing Attractions: Lectures 5 and 6! Lecture 5:Quantum Harmonic Oscillator! Number measurements versus quadrature measurements! Coherent states and their measurement statistics! Lecture 6:Quantum Harmonic Oscillator! Minimum uncertainty-product states! Squeezed states and their measurement
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