PHYS 570 Class Info

This page contains links to information for the PHYS 570 course (AMO Physics I).

The text book for the class is Atomic Physics by Christopher J. Foot.

We will cover Chapters 1-7 with additional material at the end of the semester on scattering. The class notes below was covered in 2022 & 24. Course description.

Syllabus

Office hours: Wednesday 2:00-3:00, Thursday 2:00-3:00 (Contact me by email if you want to arrange a special time.)

Homework will be due by midnight on Thursdays. You.You must show enough work on the homework so the grader can understand what you're doing. Ideally you should work the problems yourself without help. Any collaborative effort in doing the homework should be explicitly acknowledged, include the names of people collaborating with on each problem set. Acknowledge all sources from whom material is taken in homework solutions.

So that no one has a weird advantage, the solutions are at
https://users.physics.ox.ac.uk/~Foot/Book/index.shtml

Chap 1;  Dipole Blockade info, Laser wakefield info, High Harmonic Generation info, Above Threshold Ionization info, 2018 Nobel Prize, 2023 Nobel Prize
Chap 2;  Legendre Polynomials, Associated Legendre Polynomials, Spherical Harmonics, Radial energies/wave functions, Radail wavefunction image, Radial wavefunction, Clebsch-Gordon pdf, Clebsch-Gordon calculator
Chap 7;  Ramsey papers, 1989 Nobel Prize, Ramsey bio, Programs 2 state, plots of 2 state calculations (pdf, pptx), Programs Exponential Decay, Programs Density Matrix, Quantum Zeno Effect programs/plots
Chap 3; Papers and plots
Chap 4; Papers and plots
Chap 5; Papers and plots
Chap 6; Papers and plots
Chap 10Griffiths; Papers and plots
Chap 11; Papers and plots
Chap 12; Papers and plots


HWK 1 (Due Thu Jan 15; 3 problems):
1) Foot (1.1)
2) Foot (1.3)
3) Foot (1.8)

HWK 2 (Due Thu Jan 22; 4 problems):
Prob 1) Foot (1.10) AND For the fields in this problem, estimate the size of the diamagnetic energy shift  (Chap 1 notes bottom pg 3) for n = 1, 2, 4, 8, 16, 32. Use MHz for the energy units. Compare the size of the maximum Zeeman shift to the diamagnetic shift estimate for these values of n and B.
Prob 2) For HHG, using the 3 step model (Chap 1 notes pg 6 Example 2) find the maximum KE of the returning electron if the Coulomb potential is neglected. Also, give the value of v_o/v_Q which gives this max KE. (Hint: you can write your own code or use the spread sheet or Mathematica notebook in High Harmonic Generation info).
Prob 3) Foot Prob 2.4 (this value goes into calculating the energy shift due to the finite size of the proton.).
Prob 4) For all of the following, only consider the real part of the hydrogen eigenstate wave function. Find the state using knowledge of how many zeros there are in each coordinate. For r, ignore the zero at r = 0 (if it exists) and infinity. For theta, ignore the zero at theta = 0 or pi. For phi, the nodes are between phi = 0 and pi. Determine, n, L (ell), m when: a) the function has 2 zeros in r, 0 zeros in theta, and 0 zeros in phi; b) the function has 0 zeros in r, 2 zeros in theta, and 0 zeros in phi. c) the function has 0 zeros in r, 0 zeros in theta, and 2 zeros in phi. d) there are 2 zeros in r, 1 zero in theta, and 1 zero in phi. e) there are 1 zeros in r, 2 zeros in theta, and 3 zeros in phi.

HWK 3 (Due Thu Jan 29; 4 problems):
Prob 1) On the bottom of page 4 of Chap 2, I gave the angular and spin part of the wave function for L=1, s=1/2. Repeat for L=2,s=1/2. You only need to write out the states with positive total M.
Prob 2) Use the expression on page 3 of Chap 2 for the cos(theta) matrix element and the states from Prob 1) to get the cos(theta) matrix element with the L=1,j=3/2,M=3/2 state in the ket. That is find
<1,1/2,3/2,3/2|cos(theta)|2,1/2,j,M>
for the states of Prob 1). (Hint: show only one value of M gives non-zero results.)
Prob 3) Foot 7.1 ((a) means integrate |X₁₂|² over all solid angles and divide by 4 pi)
Prob 4) Foot 7.2 (Hint: Try c₂(t) = A exp(i B t) with A and B constants,  find the two allowed values of B, remember you can add or subtract the two solutions)

HWK 4 (Due Thu Feb 5; 4 problems):
Prob 1) Foot 7.3 (a,b,c,d)
Prob 2) Foot 7.3 (e,f) (Hint for e: it helps to figure out what a pi/2 pulse does to |2>), (Hint for f: the phase shift is applied to state 2)
Prob 3) The Ramsey separated oscillation technique has been used in atomic fountains. In the A. Clairon et al paper (in the Ramsey papers set above), they have data in Fig. 2b (for example, v_z, and T, etc). Roughly determine the spread in velocities compared to the launch velocity. For Fig. 2c, determine the time between the two Ramsey pulses and is that value consistent with the reported v_z? Be careful to distinguish between frequency and angular frequency! (For fun [that is, not graded], you might check the numbers in Ramsey's paper and/or look up Zacharias's paper.)
Prob 4) Foot 7.6 (a,b,c) [don't do (d,e)]

HWK 5 (Due Thu Feb 12; 4 problems):
Prob 1) Foot 7.4 (Eq. (7.95) assumes the Rabi frequency is real)
Prob 2) For a 2X2 case, which of the following density matrices are allowed and which are not allowed. For the ones that are not allowed, give the reason they are not allowed. For the ones that are allowed, what physical case do they correspond to?
The numbers correspond to (rho11, rho12, rho21, rho22)
a) (1/2, 0   , 0   , 1/2)
b) (1/2, 1/2, 1/2, 1/2)
c) (1/2, 1/2, 0   , 1/2)
d) (   1, 1/2, 1/2, 0   )
e) (  -1,  0  , 0  , 2   )
f)  (1/2, i/2,  i/2, 1/2)
g) (3/4, 1/4, 1/4, 1/4)
Prob 3) Go to Gaussian beam to get the intensity of a beam as a function of r, z, power, ... A laser with 1064 nm wavelength has a total power of 100 mW and a waist of 10 micrometers. Give numbers for your answers. (a) Calculate the peak intensity. (b) The sodium 3p state is 2.1 eV higher energy than the 3s state. Use a reasonable estimate for the 3s-3p dipole matrix element to compute Omega for the intensity in (a). (c) The sodium 3p lifetime is approximately 16 ns. What is the rate that photons are scattered by 1 sodium atom at the peak intensity? (Remember the light in this problem is not close to resonant. Far off resonant light is often used for trapping atoms in a region of space.) (The results in this problem will be in a problem next week so save these results.)
Prob 4) Foot 7.12 (Hint: Write H as H_atom + H_I where H_atom is diagonal and H_I only is off diagonal.)

HWK 6 (Due Thu Feb 19; 4 problems):
Prob 1) (a) For the parameters of the Gaussian beam problem of HWK 5 (Prob 3), compute the energy depth of the dipole trap. Atoms bound to the trap execute harmonic oscillations in space if they are very cold.  (b) Using the expression for the intensity as a function of r,z, find the trap frequencies. (Hint: to get the x-frequency, take the derivative of the potential energy with respect to x and show Force = - k x for small x. Do the same for y and z.). (c) Estimate the temperature the atoms need to have to be mostly in the ground state for each direction.
Prob 2)  Use the Table in Foot Prob (7.6). A hydrogen atom is in a laser field that is resonant (or nearly resonant) with the 1s-3p transition and is linearly polarized in the z-direction. Have Omega be the Rabi frequency for the 1s-3p transition; take Omega to be real. a) Why are the only relevant states the 1s, 2s, and 3p(m=0) states? b) Why can you drop the Rabi frequency for the 2s-3p transition? c) For this 3X3 case, determine the number of independent density matrix elements? (Hint: don't forget condition on trace.) d) In terms of the 1s, 2s, 3p(m=0) states, give the differential equation for the 6 independent density matrix elements. (note to self: 6 is typo that needs to be fixed since the answer to  part c is 5!) For example, I want
d rho_11(t)/dt = ...
(Don't do the tilde density matrix like pg 11 of Chap 7 but the plain density matrix like pg 8) Hint: Ordering the states 1s, 3p, 2s give the least change from the notes. Write the rho_ij = a_i a_j^* and use the equations for the a_i and a_j^* to get the Hamiltonian part, then use logic like pg 12 to get the other terms.
Prob 3) Use the density matrix differential equations on the middle of pg 11 of Chap7.pdf in terms of the tilde(rho)_x,y,z (or pg 12 with Gamma=0). A model of the Quantum Zeno effect is to set the off-diagonal elements of the density matrix to 0 at a short time delta_t, and again at 2 delta_t, and again at 3 delta_t, etc. Start 100% in state 1. (a) What happens to the tilde(rho)_x,y,z when the off diagonal elements of the density matrix are set to 0? (Hint: determine which are set to 0 and which are unchanged.) (b) For the case where the detuning is 0, get the population in each state at the time delta_t. At this time, a measurement of populations is performed (equivalent to setting the offdiagonal elements of the density matrix to 0). Use the  tilde(rho)_x,y,z at this time to find the density matrix at 2 delta_t. At this time, a measurement of populations is performed.  For this case use delta_t Omega <<< 1. (c) Obtain the density matrix when this is repeated N times. Suppose the final time is pi/Omega, what are the populations in each state as the number of measurements, N, gets large (delta_t = (pi/Omega)/N)? (Hint: first show that the solution for tilde(rho)_y and tilde(rho)_z is a sin() and a cos() respectively when the y-component is 0 at the initial time.)
Prob 4) Although it isn't a great approximation, treat the alpha particle mass to be infinite and use a non-relativistic Hamiltonian for this problem. Muonic helium is the exotic atom made with one muon and one electron bound to an alpha particle. a) Write down an approximate wave function for the ground state and for the lowest two excited states. Make sure to clearly distinguish the muon and electron coordinates and masses. b) Why are the singlet and triplet energies the same at this level of approximation while normal He has substantial energy difference at this level of approximation? c) Is there a direct and an exchange integral for the energy? Explain your answer. d) For the second excited state, write down the direct and/or the exchange integral in terms of constants (which you should give) and an integral over r_mu and r_e.

HWK 7 (Due Thu Feb 26; 4 problems):
Use the NIST web page to get energies. Putting He I in the "Spectrum" box gives neutral He energies. Putting He II in the "Spectrum" box gives He+ energies.
Prob 1) You can use level units of cm-1 or eV. IMPORTANT: make sure to use enough significant digits that the energy differences are meaningful. a) What energy is required to remove an electron from a He atom in its ground state? b) If the direct and exchange integrals were 0, what would be the energy of 1s3p where the energy of 1s^2 is 0. c) What is the difference of this energy from the actual singlet and triplet P energies? d) Repeat for 1s3d. (Hint: The answers to c) and d) should be much less than 1 eV.)
Prob 2)  (a) Use the PhysRevLett.66.1306.pdf (in the Chap3Papers and plots) Fig 1a, what is the 2s2p 1P energy relative to the 1s^2 state? (b) The autoionization rate hbar Gamma = 0.042 eV. What is Gamma in units of s^-1? (c) The 2s2p 1P state can also emit a photon. When it emits a photon, why does it mostly go to the final state 1s2s 1S instead of 1s^2 1S? (d) What energy photon is emitted when the final state is 1s2s 1S? (e) For a reasonable estimate of the dipole matrix element, get the photon emission rate of the 2s2p 1P state. (f) Compare the photon emission rate and the autoionization rate.
Prob 3) Helium is in its ground state. (a) What is the least photon energy needed to ionize He? (b) What is the least photon energy needed to ionize He and leave the He+ ion in the 2s or 2p excited state?
Prob 4) The lifetime of the Rb 5p state is 26 ns. In class, I said something like the dipole matrix element is roughly 1 angstrom. Using 1 angstrom for <1|z|2>, what is the photon emission rate of the 5p state? How close is it to the physical answer?

HWK 8 (Due Thu Mar 5; 4 problems):
Prob 1)  Use PhysRevA.67.052502.pdf (in the Chap4 Papers and plots) Table VI and Eqs. (5,6). To get accurate answers, you need to have the R_Rb and c correct to enough significant digits. (a) Calculate the np_1/2 and np_3/2 quantum defects for n=28 and 33. (b) Calculate the n=28 and n=33 Fine-structure intervals. (c) Compare your result to that in Table III. Make sure you keep enough significant digits. Your error should be less than 1 MHz.
Prob 2)  This question is about Rb. Use PhysRevA.67.052502.pdf (in the Chap4 Papers and plots) Table VI and Eq. (6). To get accurate answers, you need to have the R_Rb correct to enough significant digits (below Eq. (5)). For the energy unit, use cm^-1.  (a) Use the data from the NIST web page to get energy of the 20p J=1/2. (b) Use the quantum defects from PhysRevA.67.052502.pdf to get the energy of the 20p J=1/2 state. (c) How big is the error? (d) Repeat for the 30p J=1/2 state. (Hint: your errors should be much less than 1 cm-1.)
Prob 3) This question is about Rb. a) Use the data from the NIST web page to get the frequency you need to drive transitions between the 40s and 40p state. (You should end up with an answer in the microwaves between 1 and 100 GHz.) b) Using a reasonable estimate for the dipole matrix element, what electric field do you need for these microwaves to get a Rabi frequency of 2 pi 1 MHz. c) What intensity does this correspond to? (Hint: your answer should be in W/m^2.)
Prob 4) Use the data from the NIST web page for the Phosphorus ion (one electron removed); type P II in the "spectrum" slot. For all of the states with energy less than 100,000 cm^-1, determine whether the spin orbit splitting is roughly what is expected from the "interval rule".

HWK 9 (Due Thu Mar 12; 4 problems):
Prob 1) Use the data from the NIST web page for the Phosphorus ion (one electron removed). a) For all of the states with energy less than 100,000 cm^-1 that have the Lande-g factor (that is g_J) listed, determine whether the spin-orbit estimate of g_j matches the experimental values. b) Which states have the largest error?

Questions below may use PhysRevA.67.052502.pdf (in the Chap4 Papers and plots) Table VI and PhysRevA.87.042522.pdf (in the Chap6 Papers and plots) Fig 3 (inset).
Prob 2)  This question is about 87Rb. a) What are the hyperfine states for an ns state? b) On pg 1 of the Chap 6 notes, I give the 5s splitting as 6.834... GHz h. Give a reference where this value is measured. c) Using the data from the two PhysRevA calculate the hyperfine splitting of 5s. (You won't get a very accurate value because the large extrapolation.)
Prob 3)  This question is about 87Rb. In a recent discussion, an experimentalist said they measured a hyperfine splitting of 24 MHz for the 70s state. a) Can this value be correct? b) Using the data from the two PhysRevA calculate the hyperfine splitting of 70s.
Prob 4) For this problem, you will need to decide whether to use Eq. (6.31) or (6.33). The magnetic field is 8.5 Gauss. Ignore the magnetic moment of the nucleus. a) For the 5s state, what are the energies of all of the states? I expect the answer in Hz, but it is OK to give it in a formula (for example 34.51 kHz times ?? with ?? equal to ???).  b) Repeat for the 70s state. c) In the experiment mentioned in problem 2, the B-field was 8.5 G. Was there confusion between hyperfine and Zeeman splitting?