Download presentation

Presentation is loading. Please wait.

Published byAiden Billingsley Modified over 4 years ago

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

Similar presentations

OK

On the dynamics of the Fermi-Bose model Magnus Ögren Nano-Science Center, Copenhagen University. DTU-Mathematics, Technical University of Denmark. In collaboration.

On the dynamics of the Fermi-Bose model Magnus Ögren Nano-Science Center, Copenhagen University. DTU-Mathematics, Technical University of Denmark. In collaboration.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Best ppt on forest society and colonialism in india Ppt on steve jobs as entrepreneur Ppt on remote operated spy robot controller Ppt on biography of william shakespeare Ppt on conservation of land resources Ppt on zener diode voltage Ppt on 5 star chocolate chip Ppt on different types of pollution Ppt on ms excel tutorial Ppt on focus group discussion