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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?
SØREN [email protected]
APRIL 2010DEPARTMENT 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 [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Measuring quantum systemsTextbook description
Projector Update wave function
In “practice”
More complicated update
+ normalization
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Time evolution of probed systemMeasurement rate
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The diffusion limit
Many weak interactions
Accumulated effect
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
ExampleSpin ½ driven by a classical field
SØREN [email protected]
APRIL 2010DEPARTMENT 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 [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Matrix product states
• Numerical method• States with limited entanglement between sites
(D dimensional)
matrices
SØREN [email protected]
APRIL 2010DEPARTMENT 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 (Schrödinger’s equation)
• Markovian evolution (master equations)
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Calculation of operator-averages
Notation
A matrix product state
1 2 3 4 5 i L
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Calculation of operator-averages(single site)
Required time:
A
SØREN [email protected]
APRIL 2010DEPARTMENT 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 (Schrödinger’s equation)
• Markovian evolution (master equations)
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
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
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Time evolution for MPSTime-evolution as a variational problem:
Minimize
We only need to calculate
efficiently
U
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Stochastic evolution of MPSMeasurement as a variational problem
Minimize
Exactly the same
Provided can be calculated efficiently
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Stochastic evolution of MPSFor our measurement model
is a sum of two overlaps.
If A is a sum of local operators:
Easy
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The Heisenberg Spin ½-chain
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The Heisenberg Spin ½-chain
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The Heisenberg Spin ½-chain
Weak measurements
L=60
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APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The Heisenberg Spin ½-chain
Measuring the end-points
L=60
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
The Heisenberg Spin ½-chain
Non-local measurement
Non-local measurement long-range entanglement
L=30
SØREN [email protected]
APRIL 2010DEPARTMENT OF PHYSICS AND ASTRONOMY
Alternative MPS (tensor network) topology due to measurements
SØREN [email protected]
APRIL 2010DEPARTMENT 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?
SØREN [email protected]
APRIL 2010DEPARTMENT 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
