Lecture Quantum-optical phenomena in nanophysics

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About these lectures

Lectures by Florian Marquardt (summer term 2010)

These lectures deal with the modern topic of quantum optical phenomena in nanophysics. During the past decade, a variety of solid state nanosystems have been realized which can be described using the concepts of quantum optics.

The two main topics I discuss are quantum electrodynamics in superconducting circuits of microwave resonators and qubits ("circuit cavity QED") and nanomechanical resonators interacting with light ("optomechanics"), along with smaller sections on excitons in quantum dots and nitrogen vacancy centres in diamond. In addition, there are a number of general theory chapters (marked by a T in the chapter headings). These deal with concepts of general applicability, e.g. the quantum states of a field mode (coherent states, squeezed states, Wigner density), general two-level dynamics, the Jaynes-Cummings model, noise spectra, dissipation, decoherence, and Lindblad master equations. Since most of these general concepts are naturally introduced during the first chapter on circuit QED, that chapter might seem longer than it actually is.

This course may also complement nicely a traditional quantum optics course (geared towards free optical photons and atomic physics). For example, I go through the quantization of the electromagnetic field in some detail, but for the superconducting microwave resonator on a chip, instead of the field in free space. Students might be interested to see how much overlap there is, and that the concepts learned in one domain can be immediately applied in the other domain. On the other hand, it has to be admitted that this section on field quantization is the lengthiest technical part, and it may be skipped without problems.

Who will benefit from this course? I have tried to keep the discussion self-contained, so anyone with a first course in quantum mechanics should be able to follow. Thus, the lectures are already useful to third year students, but should be especially valuable to beginning PhD students who want to work on a topic at the intersection of nanophysics and quantum optics, or even on a more "traditional" quantum optics or atomic physics project. Most of the topics are taught at a level that the tools acquired can be immediately applied in practice. Only for some of the last lectures have I chosen an "overview" style, because by that time the students have already learned how it would work in detail.

Florian Marquardt (July 25, 2010)

Brief contents

  • Lecture 1 1. Introduction: From ensembles to individual quantum systems, artificial quantum systems in nanophysics, field-matter interaction is based on oscillators (field) and two-level systems (atoms) T1. Basics of the harmonic oscillator: ladder operators, coherent states, coupling two oscillators, rotating wave approximation
  • Lecture 2 (T1, continued) many coupled oscillators and normal modes, T2. Basics of the two-level system: Pauli matrices, Bloch vector, free precession, avoided crossing in the energy spectrum, time-dependent driving (Rabi-oscillations)
  • Lecture 3 2. Quantum electrodynamics in superconducting circuits 2.1 Some basics about superconductivity: zero resistance, Meissner-Ochsenfeld effect, critical magnetic field, Cooper pairs, the BCS gap 2.2 The Cooper pair box: tunneling between two islands, the charging energy, the tunneling term, external control via a gate charge [erratum: near the end of the lecture, the arrow for the dipole moment of the Cooper pair box was drawn in the wrong direction. It should point from the negative to the positive charge, as always.]
  • Lecture 4 (2.2 continued) Energy level spectrum of the Cooper pair box, approximation as a two-level system 2.3 The microwave transmission line resonator: Cavities (optical, 3D microwave), waveguides, transmission lines, discretized circuit description of a transmission line, LC oscillator, wave equation, passage to continuum limit
  • Lecture 5 (2.3 continued) Overview: how to quantize the transmission line resonator, goal: Jaynes-Cummings model. Lagrangian and Hamiltonian description needed as steps towards quantum theory. Example: Simple LC oscillator, using flux as a variable. Lagrangian density for the transmission line. Quantizing the LC oscillator.
  • Lecture 6 (2.3 continued) Quantizing the transmission line with periodic boundary conditions. Quantizing the transmission line resonator. Coupling to a Cooper-pair box.
  • Lecture 7 2.4 The Jaynes-Cummings model: Examples: different microscopic physical models all lead to the JC model; e.g. optical cavity and atom, photonic crystal cavity and exciton, or vibrations in a lattice and a defect two-state system. T3. The JC model: Solution: Level scheme. Rotating-wave approximation. Resonant case and vacuum Rabi oscillations.
  • Lecture 8 Continued Jaynes-Cummings model. 2.5 Jaynes-Cummings model in circuit QED. Transmission spectroscopy and avoided crossing in the strong coupling regime. Dispersive regime for non-destructive qubit readout.
  • Lecture 9 T4. Dissipation in quantum systems. Relaxation. Density matrix. Lindblad Markoff master equations. Application to relaxation in a two-level system. Application to harmonic oscillator.
  • Lecture 10 (continued with Lindblad equations) Application to harmonic oscillator. Pure dephasing. Bloch equations for a dissipative two-level system. T1 and T2 times. Expression of decay rates via quantum noise spectra. Some comments on the general structure of Lindblad master equations, especially for numerical solutions.
  • Lecture 11 More detailed derivation of Fermi Golden Rule rates expressed via quantum noise spectra [note: in contrast to the announcement, there is no mistake in the previous lecture's formulas]. Numerical simulations of Bloch equations for driven two-level systems. Dissipative Rabi dynamics. Spectroscopy. Power-broadening. Multi-photon transitions. Dynamics beyond the rotating-wave approximation (Bloch-Siegert shift). 2.6 Dissipative dynamics in circuit QED.
  • Lecture 12 2.7 Quantum information processing. Quantum computation, quantum simulation, and quantum communication. Quantum bits and gates. Quantum circuits. Physical implementation in circuit QED.
  • Lecture 13 The cavity grid as a multi-qubit architecture in circuit cavity QED. Quantum error correction. Shor code for bit-flip and phase errors.
  • Lecture 14 T5. Quantum states of the field. Wigner density. Coherent states. Squeezed states. Squeezing operator. Quantum state tomography as a tool to measure the Wigner density.
  • Lecture 15 Wigner density detection via parity measurement. 2.8 Generating arbitrary field states and measuring their Wigner density. Law-Eberly protocol. Santa Barbara experiment in circuit QED. 2.9 General theory of superconducting circuits (beginning).
  • Lecture 16 Continued with general theory of superconducting circuits. Flux representation. Node variables. Lagrangian. Hamiltonian. Phase representation vs. charge representation. Phase qubit. Flux control of a Cooper-pair box via the Aharonov-Bohm effect.
  • Lecture 17 2.10 Circuit QED (recent developments). Multi-qubit architectures. Multi-qubit entanglement. Microwave photons on demand. Single Photon detection. Strongly driven artificial atoms (Autler-Townes splitting, Mollow triplet). New readout methods (Josephson bifurcation amplifier). Quantum simulation (Tavis-Cummings model, Bose-Hubbard model, Josephson arrays). Hybrid systems (Rydberg atoms, polar molecules, etc.).
  • Lecture 18 3. Optomechanics. 3.1 Introduction. 3.2 Mechanical effects of light. 3.3 Generic model of an optomechanical system. 3.4 Elementary physics (classical).
  • Lecture 19 T6. Fluctuation spectrum and fluctuation-dissipation theorem. Displacement spectrum. Wiener-Khinchin theorem. Fluctuation-dissipation theorem. Application to the mechanical harmonic oscillator in thermal equilibrium.
  • Lecture 20 3.5 Optomechanical equations of motion (classical limit). 3.6 Linearized dynamics. Effective optomechanical damping rate and effective light-induced frequency shift (optical spring effect).
  • Lecture 21 3.7 Nonlinear dynamics of optomechanical systems. Instability towards self-induced oscillations. Conditions for stable limit cycle. Power balance and force balance. Eliminating the light-field dynamics. Attractor diagram. Multistability. Possible applications.
  • Lecture 22 3.8 Quantum optomechanics. The Hamiltonian. Treating the laser drive in a rotating and displaced frame. Linearized optomechanical interaction. Quantum theory of optomechanical ground state cooling.
  • Lecture 23 Quantum-limited displacement detection. Imprecision noise and back-action noise. The Standard Quantum Limit (SQL) for displacement detection. How it works for the case of optical detection.
  • Lecture 24 3.9 Optomechanics outlook. Squeezed states (optical and mechanical). Entanglement light-mechanics. Test for new sources of decoherence, e.g. Penrose speculation about gravity-induced decoherence. Optomechanical crystals.
  • Lecture 25 4. Quantum optics with single solid-state emitters 4.1 Excitons in self-assembled quantum dots. Quantum dots and excitons. Single-photon source and photon correlations. Photonic crystal cavities, Purcell effect and strong coupling. Optical manipulation of spins in excitons and spin readout.
  • Lecture 26 4.2 Nitrogen vacancy centres in diamond. Basic structure and level scheme. Manipulation by microwaves and optical readout. Coupling to nuclear spins. Applications.
  • Lecture 27 5. Quantum hybrid systems. Nanomechanical resonator coupled to superconducting qubit. Cantilever coupled to cold atoms. Microwave resonator coupled to NV centre spins. Nanomechanical coupling of electron spins. Single atom coupled to a vibrating membrane.

Exercises (in German)

  • Blatt 1 - Gequetschter harmonischer Oszillator, Gekoppelte Oszillatoren
  • Blatt 2 - Schalten im Zweiniveausystem, Rabi-Oszillationen
  • Blatt 3 - Ladungs-Erwartungswert in der Cooper-Paar Box, Nullpunktfluktuationen im Wellenleiter
  • Blatt 4 - Eigenschaften der Dichtematrix, Schrödinger-Katze im Jaynes-Cummings Modell
  • Blatt 5 - Reine Dephasierung, Zwei-Qubit Tomographie, Wigner-Dichte
  • Blatt 6 - Statische Bistabilität in einem optomechanischen System, Zeitverzögerte Kräfte, Lichtinduzierte Kopplung mechanischer Systeme