## PHYS 570 Class Info

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. Course description.

Syllabus

Office hours: Monday 1:30-2:30, Tuesday 1:30-2:30 (Contact me by email if you want to arrange a special time.)

Homework will be due by midnight on Wednesdays.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

Class
Notes

Chap 1; Dipole Blockade info,
Laser
wakefield info, High
Harmonic Generation info, Above Threshold Ionization info, 2018 Nobel Prize, 2023 Nobel PrizeChap 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

Homework

HWK 1 (Due Wed Jan 17):

1) Foot (1.1)

2) Foot (1.3)

3) Foot (1.8)

4) Foot (1.10)

5) For the fields in Prob 4), 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.

6) 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).

IMPORTANT: Starting HWK 2 you will submit your homework as a single pdf file in Brightspace. To make sure there isn't a problem, you should log in to brightspace and make sure you see how to upload the homework well before the due time because it's unlikely I'll be checking my email late Wed.

HWK 2 (Due Wed Jan 24):

1) Foot (2.4)

2) Foot (2.9)

3) 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.

4) From the pattern in Table 2.2 for L=n-1 (look at R

_{1,0}(r), R

_{2,1}(r), R

_{3,2}(r)), determine R

_{n,n-1}(r). (Don't worry about the normalization constant, but beware rho depends on n.) At what value of r, is |psi |^2 maximum when L=n-1? How does this correspond with the expectation from Eq. (1.8)?

5) 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.

6) Use the expression on page 3 of Chap 2 for the cos(theta) matrix element and the states from Prob 5) 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 5). (Hint: show only one value of M gives non-zero results.)

HWK 3 (Due Wed Jan 31):

1) Foot 7.1 ((a) means integrate |X

_{12}|

^{2}over all solid angles and divide by 4 pi)

2) Foot 7.2 (Hint: Try c

_{2}(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)

3) Foot 7.3 (a,b)

4) Foot 7.3 (c,d)

5) Foot 7.3 (e) (Hint: it helps to figure out what a pi/2 pulse does to |2>)

6) Foot 7.3 (f) (the phase shift is applied to state 2)

HWK 4 (Due Wed Feb 14):

Prob 1) Foot 7.4 (Eq. (7.95) assumes the Rabi frequency is real)

Prob 2) Foot 7.6 (a,b,c) [don't do (d,e)]

Prob 3) 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 4) 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.)

HWK 5 (Due Wed Feb 21):

Prob 1) Foot 7.12 (Hint: Write H as H_atom + H_I where H_atom is diagonal and H_I only is off diagonal.)

Dipole traps are often done with crossed Gaussian beams to give tight traps in all directions. To avoid complicating the math, I'm using a single beam in the next two problems.

Prob 2) 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.)

Prob 3) (a) For the parameters of the previous problem, 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.). (c) Estimate the temperature the atoms need to have to be in the ground state for each direction.

Prob 4) 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, why are there only 5 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. For example, I want

d rho_11(t) = ...

(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.

HWK 6 (Due Wed Feb 28):

Prob 1) This problem does a special case where the laser intensity is not constant. (a) For 0 detuning, obtain the equations like Foot 7.25 but for a laser with a time dependent amplitude F(t) (to see the definition of F(t) look at the top of page 4 of the notes). (b) Solve this equation (Hint: define a C

_{+}= C

_{2}+ C

_{1}and a similar "minus" equation.) (c) Find the population in the excited state (generalizes Eq. 7.27). (Hint: you should find that Omega X t gets replaced by an Omega X integral F(t) dt.) (d) This trick does not work for nonzero detuning. Explain why.

Prob 2) 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 3) 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.

Prob 4) Use the NIST web page to get the energies. In "Spectrum" type: He i 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.)

HWK 7 (Due Wed Mar 6):

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) (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 2) 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 3) 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?

Prob 4) 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.

HWK 8 (Due Wed Mar 20):

Prob 1) 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 2) 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 3) 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".

Prob 4) 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?

HWK 9 (Due Wed Mar 27):

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 1) 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 2) 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 3) 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?

Prob 4) Light with an intensity of 2.0 microW/cm^2 is resonant with the 5s F=2 to 5p F=3 transition in 87Rb. The lifetime of Rb 5p states is 26 ns. a) What is the J of the initial and final state? b) What is the flux of photons? Flux is photons per area per time. c) If the cross section were lambda^2/(2 pi), what would be the rate of photons scattered by the atom? d) From Chap 7 notes, what is the maximum rate of scattered photons? e) Comment on the comparison of c) and d)?

HWK 10 (Due Wed Apr 3):

Prob 1) For classical scattering, cos(theta) = -1 + 2 b/b

_{0}for b < b

_{0}and 1 otherwise; remember b is only defined for b>0. a) Compute the differential cross section d sigma/d Omega vs cos(theta) for this example. b) Sketch the cross section vs. theta. c) Compute the total cross section.

Prob 2) The phase shift for a spherical potential as a function of L is delta_L = 5 exp(-L). Give the value for the total cross section in units of 1/k

^{2}to (at least) 3 significant digits plot the differential cross section vs. cos(theta) in units of 1/k

^{2}. (Hint: If you don't have access to a Legendre polynomial function use P

_{0}(x) = 1 and P

_{1}(x) = x and P

_{n+1}(x) = [(2 n +1) x P

_{n}(x) - n P

_{n-1}(x)]/(n+1).) The plot below is for delta_L = 7 exp(-L) which has a total cross section of 70.17/k

^{2}.

Prob 3) a) Repeat Prob 2) but for delta_L = exp(- (L/5.5)

^{2}). Do two plots of the differential cross section, one with a linear scale for the y-axis and one with a log scale for the y-axis.

Prob 4) For 3D scattering where the particle has initial velocity in the +z direction, the wave function for large r = sqrt(x

^{2}+ y

^{2}+ z

^{2}) has the form

psi(x,y,z) = e

^{i k z}+ f(theta,phi) (1/r) e

^{i k r}

at large r.

a) For 2D scattering where the particle has initial velocity in the +x direction, give the form of the wave function at large rho = sqrt(x

^{2}+ y

^{2}). b) Explain why it has to have that form.

HWK 11 (Due Wed Apr 10):

Prob 1) P_L(x) is the Legendre polynomial. a) On separate graphs, plot P_L(cos (theta)) vs. L from L = 0 to 20 for cos(theta) = 1, 1/2, 0, -1/2, -1. b) For cos(theta) = 1 - epsilon, with epsilon a positive number much less than 1, what is the smallest value of L where P_L(cos(theta)) is negative? (Hint: you might do b) experimentally by doing calculations for different values of epsilon (say 0.01, 0.01/4, and 0.01/9).)

Prob 2) For a spherical potential, relate the imaginary part of the scattering amplitude, Im[f(theta)], for theta=0 to the total cross section. Use the partial wave expansion. (This is a special case of the optical theorem which relates the total scattering to the amount scattered out of the forward direction.)

Prob 3) The phase shift for a spherical potential as a function of L has the form

delta_L = A/L

^{B}for large L. a) For what values of B is the total cross section finite? b) Determine the values of B that give a finite total cross section but give an infinite differential cross section at a particular value of theta. (Hint: Examine the forward scattering direction.)

Prob 4) Suppose the potential is the sum of two potentials: V(vec{r}) + V(vec{r}-vec{a}). At the 1st Born level, show that the resulting differential cross section is the product of the differential cross section for one potential and an interference term. Generalize to the case where there are N identical potentials at N different places.

HWK 12 (Due Wed Apr 17):

Use source theory and |vec{r}-vec{r}'| = r - hat{r} dot vec{r}' for r>>r' when necessary.

Prob 1) A plane wave traveling in the z-direction hits a screen at z=0 that has a circular hole of radius R. a) Give the integral that gives the probability for finding the particle at (x,y,z) with z>0. You don't need to evaluate the integral but it needs to be clear, including limits of integration. b) For R and |x| and |y| much smaller than |z|, evaluate the integral to obtain the probability vs. x,y,z. (Not for points: If you are interested, you could show that the result is a proportional to a cylindrical Bessel function if you don't do the limit in part (b)).

Prob 2) A particle with wave number of magnitude k has positive velocity in the z-direction and amplitude(x,y) = C exp(- [x

^{2}+ y

^{2}]/D

^{2}) at z=0. Do not worry about the overall magnitude in parts a) & b). a) What is the probability to be at x,y and z=L with L >> D and L >>> 1/k? Take |x|,|y| to be much smaller than L. b) For D much larger than 1/k but much smaller than L, how does the radius of the probability depend on L? (Tell me how you chose the radius of the probability.)

Prob 3) Use Fig. 11.1: the figure on pg 2 of Chap 11. Your answer can be in terms of l_1 and l_2. Get the probability for the electron to be at the point x,z if the electron emerges from the upper hole with spin wave function (|up> + |down>)/sqrt(2) and from the lower hole with spin wave function (|up> + i |down>)/sqrt(2).

Prob 4) Electrons with energy 1 eV traveling in the z-direction go through narrow slits (widths much smaller than the wavelength) in a foil at z=0. The slits are very long in the y-direction. a) For one slit at x=0 and the screen at z = 0.01 m, plot the probability to hit the screen as a function of x between x=-2 mm and 2 mm. b) Repeat but for three slits at x=0, and 20 nm, and -20 nm. c) Repeat for 7 slits with 20 nm spacing (-60 nm, -40 nm, ... 40 nm, 60 nm). d) Explain why the pattern changes the way it does for the three plots.

HWK 13 (Due Wed Apr 24):

Prob 1) For the Hanbury-Brown and Twiss with 2 detectors, derive the

<(I_1 - <I_1>)(I_2 - <I_2>)>

when there are 2 sources AND source 1 has amplitude A_1 and source 2 has amplitude A_2 with A_1 not equal to A_2. You should end up with something like the middle of Pg 12 notes but with a product of powers of A_1 and A_2. (Hint: Start by calculating <I_1> and <I_2>.)

Prob 2) To simplify this problem take all sources to be in a plane perpendicular to the direction to the detectors. a) Repeat Prob 1 but when there are 3 sources with amplitudes A_1, A_2, and A_3. b) Repeat for N sources. c) Use this form to generalize to the case of continuous sources over a surface in x,y. (Hint: treat each source has having its own phase phi_i. Then average over every one of these phases.)

Prob 3) A

^{40}Ca

^{+}ion is in a Paul trap with K = 0.22, V_end = 75 V, V_RF = 150 V, z_0 = 2 mm, r_0 = 1.25 mm, and RF frequency of 4 MHz (from units you should be able to figure out whether this is Omega_RF [Hint: it's proportional to Omega_RF]). a) Compute the 3 frequencies associated with this trap. b) Approximately, what range of V_RF will give a stable trap?

Prob 4) An 85 nm radius sphere of amorphous silica has a positive charge of 200e equally distributed over the surface of the sphere. Your Paul trap has K = 0.22, z_0 = 2 mm, r_0 = 1.25 mm, and an RF frequency of 4 kHz. a) What V_end and V_RF are needed so that the z frequency is 500 Hz and the x,y frequencies are 800 Hz? b) If the center of mass motion of the sphere has a temperature of 300 K, what is <x^2>, <y^2>, <z^2>? c) Roughly what temperature for the center of mass motion needs to be reached, so that the sphere is mostly in the motional ground state?