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