APRIL 2010 AARHUS UNIVERSITY Simulation of probed quantum many body systems

SØREN GAMMELMARK 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?

SØREN GAMMELMARK 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

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems Textbook description ProjectorUpdate wave function In practice More complicated update + normalization

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Measuring quantum systems

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Time evolution of probed system Measurement rate

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The diffusion limit Many weak interactions Accumulated effect

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Example Spin ½ driven by a classical field

SØREN GAMMELMARK 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

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Matrix product states Numerical method States with limited entanglement between sites (D dimensional) matrices

SØREN GAMMELMARK 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)

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Calculation of operator-averages Notation A matrix product state iL

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Calculation of operator-averages (single site) Required time: A

SØREN GAMMELMARK 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)

SØREN GAMMELMARK 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

SØREN GAMMELMARK 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

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Stochastic evolution of MPS Measurement as a variational problem Minimize Exactly the same Providedcan be calculated efficiently

SØREN GAMMELMARK 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

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Stochastic evolution of MPS

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Weak measurements L=60

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Measuring the end-points L=60

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY The Heisenberg Spin ½-chain Non-local measurement long-range entanglement L=30

SØREN GAMMELMARK APRIL 2010 DEPARTMENT OF PHYSICS AND ASTRONOMY Alternative MPS (tensor network) topology due to measurements

SØREN GAMMELMARK 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, ) Interacting atoms in a cavity Mekhov et al. Phys. Rev. Lett. 102, Karski et al. Phys. Rev. Lett. 102, What is the effect of the measurement? The null-result?

SØREN GAMMELMARK 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