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APRIL 2010 AARHUS UNIVERSITY Simulation of probed quantum many body systems

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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?

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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

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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

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

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

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

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

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

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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

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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

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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)

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

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

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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)

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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

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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

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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

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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

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

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

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

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

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

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

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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

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

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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?

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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

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