“Quantum optics” - course 2800 rub. from MSU, training 15 weeks. (4 months), Date: December 5, 2023.

1. Introduction to statistical optics.
Analytical signal, complex amplitudes, coherent and thermal states of light. Moments of the field. Correlation functions. Properties of Gaussian fields. Wiener-Khinchin theorem. Van Zittert-Zernike theorem. Mach-Zehnder interferometer.
Young's interferometer.

2. The concept of optical mode.
Michelson stellar interferometer. Brown-Twiss stellar interferometer.
Spectral brightness. Energy in one mode. Primary quantization. Volume of fashion. The energy of fashion. Definition of fashion. Detection volume. Number of registered modes. Multimode coherent and thermal state of light.

3. Quantization of the electromagnetic field.
The connection between Hamiltonian formalism and the formalism of quantum mechanics.
Quantization of a mechanical harmonic oscillator. Transition from the Hamiltonian function to the Hamiltonian. Dimensionless variables and their commutator. Properties of a quantum harmonic oscillator, uncertainty relation, minimum energy, discrete spectrum. Primary and secondary quantization. Field quadratures and their physical meaning for traveling and standing waves. Operators of photon creation and annihilation. Transition to continuous variables: single-photon wave packet. Uncertainty relations for a single-photon wave packet. Vacuum fluctuations.

4. Bases of the Hilbert space of quantum states of light.

Description of an arbitrary state of light in the basis of Fock states. Dynamics of Fock states. Period of oscillation. Quadrature states. Representations of Q- and P-, quadrature wave functions of Fock states. Dynamics of creation and annihilation operators. Dynamics of quadrature operators and quadrature distributions.

5. Phase space of quadratures P-Q.
Joint distribution over quadratures P and Q. Wigner function. Its definition and key properties. Wigner functions of quadrature and Fock states. Minimum volume of phase space. Coherent states. Their representation in the Fock and quadrature basis. Dynamics of coherent states. Dynamics of Wigner functions.

6. Tomograms and Wigner functions.
Description of the beam splitter, Hong-Ou-Mandel interference. Homodyne detection. Tomogram. Wigner function. Examples of tomograms and Wigner functions of superpositions of Fock states. Schrödinger's cats and kittens. Their quadrature distributions, Wigner functions and tomograms.

7. Representations of coherent states and their transformations.
Representations of coherent states. Their characteristic functions, convolution properties. Transformations of quasi-probability functions on a beam splitter, joint measurement of P and Q, description of losses, shift of the Wigner function. Shift operator. Shifted states. Examples of tomograms and Wigner functions.

8. Quadrature compression.
Odomode quadrature compression in a nonlinear medium. Hamiltonian, Bogolyubov transformation, quadrature transformation. Tomograms of compressed states. Nonclassicality of compressed states. Compressed vacuum. Its expansion into Fock states. Compressed states and Schrödinger's kittens

9. Nonclassical states of light.
Thermal states, Lee's measure of non-classicality, Factorial moments, signs of non-classicality, measurement of factorial moments. Grouping and antibunching of photons. Semiclassical theory of photodetection.

10. Changing photon statistics at the beam splitter.
Hamiltonian of the beam splitter, implementation of the annihilation and creation operators. How can the detachment of a photon lead to an increase in the average number? Conversion of photon statistics at the beam splitter. Example for Fock, coherent and thermal states. Entanglement of modes by the number of photons. Distinguishing entanglement from correlation.

11. Polarization qubit.
Sources of single photons. Polarization. Basis of polarization states. Bloch sphere and Poincaré sphere. Polarizers, phase plates, polarization beam splitters. Stokes parameters and their measurement. Tomography of quantum states. Tomography of quantum processes.

12. Measurements on a polarization qubit. POVM decomposition. Weak measurements. Detector tomography.

13. Different types of qubit encoding and their application in quantum cryptography.
Spatial, phase-temporal, frequency coding. Quantum cryptography. BB84 protocol, its various implementations. Using coherent states instead of Fock states.

14. Quantum computing. Lots of mixed up qubits.
Conditional preparation of entangled states. Measurement in Bell basis. Quantum teleportation and entanglement exchange. Nonlinear and conditional two-qubit gates. Cluster computing concept. Boson-sampling.

15. Dual-mode quadrature compression in nonlinear media.
Confusion by quadratures and number of photons. Schmidt decomposition. Polarization compression. Converting dual-mode compression to single-mode compression on a beam splitter.

16. Spontaneous parametric scattering (SPR).
History of discovery. Phase synchronism. Perestroika curves. Width of frequency and angular spectra. Confusion in frequencies and wave vectors. Isolation of Schmidt modes. Conditional preparation of a pure one-photon state. Relationship between correlation and spectral properties. Dispersion compensation.

17. Application of SPR and compressed states in metrology.
Standard-free calibration of detectors. Hidden (ghost) images. Two-photon interference, edge optical coherence tomography, remote synchronization
hours. Breaking the standard quantum limit using squeezed states of light.

18. Violation of Bell's inequality.
The principle of determinism and its role in the history of science. Proof of Bell's inequality based on the classical description. Proof of violation of Bell's inequality based on quantum description. Experimental tests of violation of Bell's inequality.

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