Tomographic approach to quantum states of electromagnetic radiation and spin states
description
Transcript of Tomographic approach to quantum states of electromagnetic radiation and spin states
Tomographic approach to
quantum states of electromagnetic
radiation
and spin states
Sergey FilippovMoscow Institute of
Physics and Technology
Outline
• Accuracy and operational use of optical homodyne tomograms
• Towards microwaves• Evolution and – product• Spin tomography and
MuSR
Outline
• Accuracy and operational use of optical homodyne tomograms
• Towards microwaves• Evolution and – product• Spin tomography and
MuSR
Homodyne tomography
Homodyne tomography
†ˆ ˆ ˆ ˆ2 2
i i
L
N ae a eX
Homodyne tomography
†ˆ ˆ ˆ ˆ2 2
i i
L
N ae a eX
X
Homodyne tomography
†ˆ ˆ ˆ ˆ2 2
i i
L
N ae a eX
X
( , )h X ( , )h X
Homodyne tomography
X
Homodyne tomography
X
0
Tomography in phase spaceWigner function
Experimental data: how to get the probability density correctly?
Experimental data: example of a coherent state
Experimental data: example of a SPACS
Detector efficiency
• Coherent:• SPACS:
Purity: how to calculate?
• Tomographic approach:
Accuracy
Experimental data: mismatch• Coherent
• SPACS
Reasons and Consequences
Further frontiers
• Checking uncertainty relations with definite precision
• Purity-dependent URs• State-extended URs• Entropic enequalities
Towards microwaves
“Heterodyne” detection
Moments’ calculation
Linear amplifier
Calculation of moments: noise influence
Revealing true moments
Relations with the Wigner function
Relation between the tomogram and the ordered moments
• State purity
Uncertainty relations
Two phase spaces: the relation
[Phys. Rev. A, 2011]
State evolution: an example
“Lattice” phase space
Star product on the “lattice” phase space
Star product kernel
Evolution in the “lattice” phase space
[J. Phys. A, 2012]
Spin systems
Muon
• Charge • Mass • Spin• Magnetic moment• Mean decay time• Decay channels
Directional diagram of decay positrons
Spin tomogram
• Stern-Gerlach (1922)
• Probability
43
Muon spin tomography
• Spin• Spin projection• Angular moment operators
, • Tomogram
• “Dequantizer”
Decay diagram and tomogram
Experimental setups
Muons in matter
Two-spin tomography
• Unitary spin tomogram
• Two-spin tomogram
• Reconstruction procedure
Reduced tomogram
Hyperfine interaction
• Initial state• Initial tomogram
• Tomogram evolution
• Evolution of the reduced tomogram
Muonium-like system 2х3
Muonium in quartz, magnetic field is perpendicular to z
Anomalous muonium in silicon
Summary
• Tomograms provide the primary information about quantum systems
• Tomographic analysis of the data allows operational extraction of desired quantities and determines their accuracy
• Tomography opens new vistas toward high-precision experiments and checking the fundamental laws of quantum physics