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

APRIL 2010

AARHUSUNIVERSITY

Simulation of probed quantum many body systems

SØREN [email protected]

APRIL 2010DEPARTMENT 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?



Breakdown of ingredients

• Quantum many body systems• Vast Hilbert space• Strongly correlated• Just plain difficult

• Probed quantum systems• Stochastic• Non-linear



Measuring quantum systemsTextbook description

Projector Update wave function

In “practice”

More complicated update

+ normalization



Measuring quantum systems



Time evolution of probed systemMeasurement rate



The diffusion limit

Many weak interactions

Accumulated effect



ExampleSpin ½ driven by a classical field



Quantum many body systems

• One-dimensional systems

• Spin-chains, e.g.

• Bosons in an optical lattice

• Fermions in an optical lattice



Matrix product states

• Numerical method• States with limited entanglement between sites

(D dimensional)

matrices

f040805

Længde af kæde, L



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)



Calculation of operator-averages

Notation

A matrix product state

1 2 3 4 5 i L



Calculation of operator-averages(single site)

Required time:

A

f040805

Længde af kæde



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)



Time evolution for MPSTime-evolution as a variational problem:

Minimize

Quadratic form in the matrices

Minimize with respect to each matrix iteratively(alternating least squares)

Local optimization problem



Time evolution for MPSTime-evolution as a variational problem:

Minimize

We only need to calculate

efficiently

U



Stochastic evolution of MPSMeasurement as a variational problem

Minimize

Exactly the same

Provided can be calculated efficiently



Stochastic evolution of MPSFor our measurement model

is a sum of two overlaps.

If A is a sum of local operators:

Easy



Stochastic evolution of MPS



The Heisenberg Spin ½-chain

f040805

Længde af kæde




Weak measurements

L=60




Measuring the end-points

L=60




Non-local measurement

Non-local measurement long-range entanglement

L=30



Alternative MPS (tensor network) topology due to measurements



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?



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

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

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