Optically polarized atoms Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley

surface for J=2, octupole, Atomic density matrix Why the density matrix? Definition of the density operator Density matrix elements Density matrix evolution Angular-momentum probability surface for J=2, octupole, in z-directed E-field

Why the density matrix? No such thing as an unpolarized atom Spin ½ state: only two free parameters Normalized: + arbitrary phase relative phase relative magnitude

Why the density matrix? Expectation value of spin: x component: All components:

Why the density matrix? Spin “points” in the (θ,Ф) direction An unpolarized sample has no preferred direction state of atom i We can't use a wave function to describe the average state of an unpolarized sample

Definition of the density matrix operator: Average over all N atoms Identity operator complete set of basis states Trace of an operator Basis set can be truncated

Density matrix elements Expansion coefficients Diagonal matrix elements are real: “population” of state n Off-diagonal matrix elements average to zero if atoms are uncorrelated

Density matrix elements Unpolarized sample in state with angular momentum J: Equal probability to be in any sublevel No correlation between the atoms Trace is 1 For J=1: Total number of states

Density matrix evolution Schrödinger eq.: h.c.: Time derivative of DM: “Liouville equation”

Density matrix evolution In practice, there are other terms not described by the (semiclassical) Hamiltonian Repopulation matrix Relaxation matrix These terms describe, e.g., spontaneous decay and atom transit

Example: 2-level system, subject to monochromatic light field Rabi frequency Transit rate Natural width

Rotating wave approximation We would like to get rid of the time-dependence at the optical frequency Use With unitary transformation conserves total probability drop off-resonant terms

Rotations Classical rotations Quantum rotations Visualization Commutation relations Quantum rotations Finding U (R )‏ D – functions‏ Visualization Irreducible tensors Polarization moments

Classical rotations Rotations use a 3x3 matrix R: position or other vector Rotation by angle θ about z axis: For θ=π/2: For small angles: For arbitrary axis: Ji are “generators of infinitesimal rotations”

Commutation relations Rotate green around x, blue around y From picture: Rotate blue around x, green around y For any two axes: Using Difference is a rotation around z

Quantum rotations Want to find U (R) that corresponds to R U(R) should be unitary, and should rotate various objects as we expect E.g., expectation value of vector operator: Remember, for spin ½, U is a 2x2 matrix A is a 3-vector of 2x2 matrices R is a 3x3 matrix

Quantum rotations Infinitesimal rotations Like classical formula, except i makes J Hermitian For small θ:  gives J units of angular momentum minus sign is conventional The Ji are the generators of infinitesimal rotations They are the QM angular momentum operators. This is the most general definition for J We can recover arbitrary rotation:

Quantum rotations Determining U (R)‏ Start by demanding that U(R)‏ satisfies same commutation relations as R The commutation relations specify J, and thus U(R)‏ That's it! E.g., for spin ½:

Quantum rotations Is it right? We've specified U(R), but does it do what we want? Want to check J is an observable, so check Do easy case: infinitesimal rotation around z Neglect δ2 term Same Rz matrix as before

D -functions Matrix elements of the rotation operator Rotations do not change j . D-function z-rotations are simple: so we use Euler angles (z-y-z):

Visualization Angular momentum probability surfaces “probability to measure m=j along quantization axis” rotate basis set to measure along arbitrary axis: ρjj(θ,Φ) contains all the information of the DM Can be expanded in spherical harmonics

Irreducible tensors rotation of basis kets: rotation of spherical harmonics: these are irreducible tensors: rank κ, components -κ<q<κ for irreducible tensor operators: generalizes reduced matrix element Wigner-Eckart theorem:

Polarization operators define irreducible tensor operators with reduced matrix element W-E theorem: # of operators: complete basis expand DM: The PM's are physically significant and have useful symmetries

Visualizing polarization operators Calculate ρjj(θ,Φ) for polarization operator: (rotation of irr. tensor)‏ (matrix elem. of pol. op.)‏ Each polarization moment corresponds to a spherical harmonic