9/7/16 September 20, 2016 6.453 Quantum Optical Communication - PDF document

9/7/16 September 20, 2016 6.453 Quantum Optical Communication Lecture 4 Jeffrey H. Shapiro Optical and Quantum Communications Group www.rle.mit.edu/qoptics 6.453 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 kets 2 www.rle.mit.edu/qoptics 1

9/7/16 Classical LC Circuit: Undriven, Lossless Oscillation § Stored Energy (Hamiltonian) § State Variables § Hamilton ’ s Equations capacitor charge: inductor flux: 3 www.rle.mit.edu/qoptics Classical LC Circuit: Undriven, Lossless Oscillation § Nonzero Initial Conditions: § Oscillation Frequency: § Solutions for : 4 www.rle.mit.edu/qoptics 2

9/7/16 Classical LC Circuit: Dimensionless Reformulation § Assume § Define Complex Envelope (Phasor): § Simple Harmonic Motion at Constant Energy 5 www.rle.mit.edu/qoptics Quantum LC Circuit: Quantum Harmonic Oscillator § Postulate: Become Observables § Canonical Commutation Relation: § Dimensionless Reformulation: 6 www.rle.mit.edu/qoptics 3

9/7/16 Quantum Harmonic Oscillator: Commutators § Dimensionless Reformulation of Canonical Commutators: § Hamiltonian: § Heisenberg Uncertainty Principle: 7 www.rle.mit.edu/qoptics Classical versus Quantum Behavior § Classical Oscillator: Noiseless § Quantum Oscillator: Noisy 8 www.rle.mit.edu/qoptics 4

9/7/16 Energy 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: 9 www.rle.mit.edu/qoptics Number Operator and Number States § Oscillator Energy is Quantized in Increments § (Photon) Number Operator: 10 www.rle.mit.edu/qoptics 5

9/7/16 Coming 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 statistics 11 www.rle.mit.edu/qoptics 6

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