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Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector

Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector. Antony Richard Lee In collaboration with David Edward Bruschi and Ivette Fuentes University of Nottingham. University of Sydney – August 2013.

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Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector

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  1. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector Antony Richard Lee In collaboration with David Edward Bruschi and Ivette Fuentes University of Nottingham University of Sydney – August 2013

  2. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Presentation • Introduction • Relativistic quantum information, motivation, objectives • Framework • Unitary dynamics, symplectic formalism, Gaussian states • Application • Beam and down conversion Hamiltonian, quantum field theory, Unruh-DeWitt detector • Conclusions Antony Richard Lee – UoN – UoS

  3. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Relativistic quantum Information • Main questions? • How does relativity merge with quantum information? • Effects of gravity and quantum field theory on QI protocols? • New technologies? Antony Richard Lee – UoN – UoS

  4. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Relativistic quantum Information • Motivation • Implications for communication using quantum field theory • Fundamental limits on information exchange • Could help with quantum gravity experiments Antony Richard Lee – UoN – UoS

  5. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Relativistic quantum information • Objectives • Useful, tractable framework to pose and answer questions (see Nico’s talk) • Protocols that are possible only by combining relativity and QI • Use QI to provide new experiments Antony Richard Lee – UoN – UoS

  6. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Reasons for non-perturbative analysis? • How do quantum correlations evolve over time? • Interesting for QI, optics, RQI, quantum gravity, chemistry and biology • Want to examine dynamics in an exact manner in QFT • Want analytical control or few approximations • How to do this? Antony Richard Lee – UoN – UoS

  7. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Unitary Dynamics • System Hamiltonian • Schrödinger equation • Unitary equation • Formal Solution Antony Richard Lee – UoN – UoS

  8. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Explicit expression • How to compute • Use first order only? • Term by term? • Other? Antony Richard Lee – UoN – UoS

  9. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Symplectic Formulation • Quadratic Hamiltonian • Hilbert space replaced by phase space Antony Richard Lee – UoN – UoS

  10. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Symplectic Group • Structure • Can be formally extended to infinite dimensions (see Nico’s talk) Antony Richard Lee – UoN – UoS

  11. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • How to find S from U? • Symplectic dynamics • Formal solution Antony Richard Lee – UoN – UoS

  12. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Solving the time ordering • Numerically • Another way? • Decomposition Antony Richard Lee – UoN – UoS

  13. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Solving the time ordering • Evolution equation • Can be rewritten as • Notation Antony Richard Lee – UoN – UoS

  14. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Solving the time ordering • Equating coefficients • Coupled, ordinary, highly-nonlinear equations Bruschiet al 2013 J. Phys. A: Math. Theor.46 Antony Richard Lee – UoN – UoS

  15. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Framework • Different Decompositions • Polar decomposition • Bloch-Messiah/ Euler decomposition • Active • Passive Antony Richard Lee – UoN – UoS

  16. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Gaussian States • Definitions • Characteristic function • 1st and 2nd moments • Vector and matrix form Antony Richard Lee – UoN – UoS

  17. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Gaussian States • Examples • Coherent state • Two-mode squeezed state • Thermal state Antony Richard Lee – UoN – UoS

  18. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Gaussian States • Gaussian transformations • Preserved by quadratic Hamiltonians • Gaussian state evolves to Gaussian state • Can compute dynamics non-perturbativley • Models many physical situations Antony Richard Lee – UoN – UoS

  19. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Beam Splitter Hamiltonian • Lie algebra generators Antony Richard Lee – UoN – UoS

  20. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Beam Splitter Hamiltonian • Symplectic decomposition Antony Richard Lee – UoN – UoS

  21. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Beam Splitter equations • Ideal for numerical evaluation Antony Richard Lee – UoN – UoS

  22. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Parametric down conversion Hamiltonian • Lie algebra generators Antony Richard Lee – UoN – UoS

  23. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Parametric down conversion Hamiltonian • Symplectic decomposition Antony Richard Lee – UoN – UoS

  24. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Parametric down conversion equations • Simple expressions • Allows for exact solutions • Numerically stable Antony Richard Lee – UoN – UoS

  25. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application – Quantum Field Theory • Spacetime coordinates • Quantum field • Expand in basis of solutions Antony Richard Lee – UoN – UoS

  26. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Quantisation • Works for Dirac, Electromagnetic and higher spin fields Antony Richard Lee – UoN – UoS

  27. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Unruh-DeWitt detector • Comoving coordinates Antony Richard Lee – UoN – UoS

  28. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Unruh-DeWitt detector • Detector couples linearly to field operator Antony Richard Lee – UoN – UoS

  29. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector Trajectory Constant acceleration Antony Richard Lee – UoN – UoS

  30. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Unruh-DeWitt detector • Normally compute transition rate to first order in perturbation theory • Inertial trajectories cannot excite the detector from its vacuum state • Accelerated detectors predict the “Unruh effect” Antony Richard Lee – UoN – UoS

  31. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Unruh-DeWitt detector (inertial) Brown et. al. Phys. Rev. D 87, 084062 (2013) Bruschiet al 2013 J. Phys. A: Math. Theor.46 Antony Richard Lee – UoN – UoS

  32. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Application • Unruh-DeWitt detector (acc.) Brown et. al. Phys. Rev. D 87, 084062 (2013) Antony Richard Lee – UoN – UoS

  33. Non-perturbative dynamics in QFT: An application of the Unruh-DeWitt detector • Introduction • RQI • Motivation • Objectives • Framework • Unitary dynamics • Symplectic formulation • Gaussian states • Application • Beam splitter • Down conversion • UdW detector • Conclusions • Conclusions • Introduced RQI • Motivation • Objectives • Introduced a novel way to approach quantum dynamics • Analyse quadratic bosonic Hamiltonians • Can be used in quantum field theory • Powerful and efficient for Gaussian quantum information • Opens door to other questions in RQI Antony Richard Lee – UoN – UoS

  34. Thanks! (Stick around for Nico’s talk)

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