1. APRIL 2010 AARHUS UNIVERSITY Simulation of probed quantum many body systems

2. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY 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. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Breakdown of ingredients Quantum many body systems Vast Hilbert space Strongly correlated Just plain difficult Probed quantum systems Stochastic Non-linear

4. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems Textbook description ProjectorUpdate wave function In practice More complicated update + normalization

5. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems

6. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems

7. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Time evolution of probed system Measurement rate

8. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The diffusion limit Many weak interactions Accumulated effect

9. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Example Spin ½ driven by a classical field

10. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Quantum many body systems One-dimensional systems Spin-chains, e.g. Bosons in an optical lattice Fermions in an optical lattice

11. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Matrix product states Numerical method States with limited entanglement between sites (D dimensional) matrices

12. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY 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 (Schrödingers equation) Markovian evolution (master equations)

13. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Calculation of operator-averages Notation A matrix product state 12345 iL

14. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Calculation of operator-averages (single site) Required time: A

15. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY 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 (Schrödingers equation) Markovian evolution (master equations)

16. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY 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. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Time evolution for MPS Time-evolution as a variational problem: Minimize We only need to calculate efficiently U

18. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Stochastic evolution of MPS Measurement as a variational problem Minimize Exactly the same Providedcan be calculated efficiently

19. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Stochastic evolution of MPS For our measurement model is a sum of two overlaps. If A is a sum of local operators: Easy

20. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Stochastic evolution of MPS

21. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

22. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

23. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

24. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Weak measurements L=60

25. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Measuring the end-points L=60

26. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Non-local measurement long-range entanglement L=30

27. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Alternative MPS (tensor network) topology due to measurements

28. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Other systems of interest Single-site addressed optical lattice Optical (Greiner et al. Nature 462, 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. SØREN GAMMELMARK gammelmark@phys.au.dk APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY 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