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Transcript
Optically polarized atoms Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley 1 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 2 Why the density matrix? No such thing as an unpolarized atom Spin ½ state: Normalized: + arbitrary phase relative phase only two free parameters relative magnitude 3 Why the density matrix? Expectation value of spin: x component: All components: 4 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 5 Definition of the density matrix Density operator: Average over all N atoms Identity operator complete set of basis states Trace of an operator Basis set can be truncated 6 Density matrix elements 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 7 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 8 Density matrix evolution Schrödinger eq.: h.c.: Time derivative of DM: “Liouville equation” 9 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 10 Example: 2-level system, subject to monochromatic light field Rabi frequency Transit rate Natural width 11 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 12 Rotations Classical rotations Commutation relations Quantum rotations Finding U (R ) D – functions Visualization Irreducible tensors Polarization moments 13 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” 14 Commutation relations Rotate green around x, blue around y From picture: For any two axes: Using Rotate blue around x, green around y Difference is a rotation around z 15 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 16 Quantum rotations Infinitesimal rotations Like classical formula, except i makes J Hermitian For small θ: minus sign is conventional gives J units of angular momentum 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: 17 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: 18 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 19 D -functions Matrix elements of the rotation operator Rotations do not change j . D-function z-rotations are simple: so we use Euler angles (zy-z): 20 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 21 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 Wigner-Eckart theorem: reduced matrix element 22 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 23 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 24