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Simulation of probed quantum many body systems

Simulation of probed quantum many body systems. Why probe quantum many body systems?. Interactions gives rise to complex phenomena Phase-transitions Collective effects Topological states of matter Measurements can produce interesting quantum states Squeezed spins

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Simulation of probed quantum many body systems

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  1. Simulation of probed quantum many body systems

  2. Why probe quantum many body systems? • Interactions gives rise to complex phenomena • Phase-transitions • Collective effects • Topological states of matter • Measurements can produce interesting quantum states • Squeezed spins • Heralded single photon sources • Light squeezing • Measurements and feedback • High-precision measurements, atomic clocks, gravitational wave detectors • Combining measurements and interactions • Can we get the best of both worlds? • Can measurements help/stabilize complex phenomena? • Can interacting quantum systems give better/more precise measurements?

  3. Breakdown of ingredients • Quantum many body systems • Vast Hilbert space • Strongly correlated • Just plain difficult • Probed quantum systems • Stochastic • Non-linear

  4. Measuring quantum systems Textbook description Projector Update wave function In “practice” More complicated update + normalization

  5. Measuring quantum systems

  6. Measuring quantum systems

  7. Time evolution of probed system Measurement rate

  8. The diffusion limit Many weak interactions Accumulated effect

  9. Example Spin ½ driven by a classical field

  10. Quantum many body systems • One-dimensional systems • Spin-chains, e.g. • Bosons in an optical lattice • Fermions in an optical lattice

  11. Matrix product states • Numerical method • States with limited entanglement between sites (D dimensional) matrices

  12. Features of matrix product states • Efficient calculation of operator-averages • Low Schmidt-number of any bipartite cut • Ground states of nearest neighbor Hamiltonians • Low-energy excited states • Thermal states • Unitary time-evolution (Schrödinger’s equation) • Markovian evolution (master equations)

  13. Calculation of operator-averages Notation A matrix product state 1 2 3 4 5 i L

  14. Calculation of operator-averages (single site) A Required time:

  15. Features of matrix product states • Efficient calculation of operator-averages • Low Schmidt-number of any bipartite cut • Ground states of nearest neighbor Hamiltonians • Low-energy excited states • Thermal states • Unitary time-evolution (Schrödinger’s equation) • Markovian evolution (master equations)

  16. Time evolution for MPS Time-evolution as a variational problem: Minimize Quadratic form in the matrices Minimize with respect to each matrix iteratively (alternating least squares) Local optimization problem

  17. Time evolution for MPS Time-evolution as a variational problem: Minimize We only need to calculate U efficiently

  18. Stochastic evolution of MPS Measurement as a variational problem Minimize Exactly the same Provided can be calculated efficiently

  19. Stochastic evolution of MPS For our measurement model is a sum of two overlaps. If A is a sum of local operators: Easy

  20. Stochastic evolution of MPS

  21. The Heisenberg Spin ½-chain

  22. The Heisenberg Spin ½-chain

  23. The Heisenberg Spin ½-chain

  24. The Heisenberg Spin ½-chain Weak measurements L=60

  25. The Heisenberg Spin ½-chain Measuring the end-points L=60

  26. The Heisenberg Spin ½-chain Non-local measurement L=30 Non-local measurement long-range entanglement

  27. Alternative MPS (tensor network) topology due to measurements

  28. Other systems of interest • Single-site addressed optical lattice • Optical (Greiner et al. Nature462, 74) • Electron microscope (Gericke et al. Phys. Rev. Lett.103, 080404) • Interacting atoms in a cavity • Mekhov et al. Phys. Rev. Lett.102, 020403 • Karski et al. Phys. Rev. Lett.102, 053001 What is the effect of the measurement? The null-result?

  29. Summary • Measurements and stochastic evolution can be simulated using matrix product states • Local and non-local measurements on quantum many-body systems can lead to interesting dynamics • Measurements can change the topology of the matrix product state (or peps) tensor graph

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