## PHYS 570 Class Info

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

We will cover most Chapters 7 - 13 with probably some additional material at the end of the semester (EIT, Quantum Zeno, Cat States, ...?). Course description.

Syllabus

Office hours: Monday 1-2, Tuesday 4:30-5:30 (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
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/plotsChap 8; Precision papers, 2005 Nobel Prize, Hansch bio, Hall bio

Chap 9; Programs/plots optical molasses, 1997 Nobel Prize, Chu bio, Cohen-Tannoudji bio, Phillips bio

Chap 10; Programs/plots BEC, 2001 Nobel prize, Cornell bio, Ketterle bio, Wieman bio

Chap 11; Diffracting Buckeyballs, Natue diffracting molecules, Diffracting molecules, More diffracting molecules, Hanbury-Brown and Twiss, Programs/plots BEC, 2005 Nobel Prize, Glauber bio

Chap 12; 1989 Nobel Prize, Dehmelt Bio, Paul Bio,

Homework

HWK 1 (Due Wed Jan 18): Problems 1-3 are Foot Probs 7.1, 7.2, 7.3

Problem 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. 2 (for example a v

_{z}, a T, etc). Roughly determine the spread in velocities compared to the launch velocity. For the Fig. 2c, determine the time between the two Ramsey pulses. (For fun [that is, not graded], you might check the numbers in Ramsey's paper and/or look up Zacharias's paper.)

Problem 5: 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.

Do not turn in: You should play with the numerical equations for laser excitation to see how the probabilities behave with different strength interactions, different detunings, etc. If you find a cool situation, please send it to me in an email.

HWK 2 (Due Wed Jan 25): Problems 1-2 are Foot Probs 7.4 (the Omega in 7.95 should be complex conjugated), 7.6(a,b,c don't do d,e).

Problem 3: For the exponential decay model (pgs 9 and 10 of the Chap7.pdf), determine the C

_{n}(t) for t < infinity. For the case Gamma = 1E8 s

^{-1}, plot the P

_{n}(t)/Delta_omega vs. the detuning delta for t = 0.1/Gamma, 1/Gamma, 2/Gamma, 4/Gamma, and infinity. (For example, the t=infinity result would be (Gamma/2/pi)/(delta

^{2}+ (Gamma/2)

^{2}).

Problem 4: Use the density matrix differential equations on the middle of pg 11 of Chap7.pdf in terms of the tilde(rho)_x,y,z 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. 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.) For the case where the detuning is 0, get the population in each state after 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 them at 2 delta_t. At this time, a measurement of populations is performed. For this case use delta_t Omega <<< 1. 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.)

Not graded but if you work it out I will correct it. Foot problem 7.5 and 7.6(d,e).

HWK 3 (Due Wed Feb 1): Prob 1 is 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 usually done with crossed Gaussian beams. To avoid that complication, I'm using unreasonable waists in the next two problems. (For example, 2 crossed beams with 10 W with 100 micron waist and wavelength 1064 nm.)

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?

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.). (c) Estimate the temperature the atoms need to have to be in the ground state for each direction.

Prob 4: On the bottom of pg 12 of the Chap7 notes, the steady state solution for the tilde density matrix is given. The dipole operator q<i|x|j> = q X_12 if i =1 and j = 2 (and vice versa) and is zero otherwise. (a) Use the density matrix to compute the expectation value of the dipole operator in steady state. (Hint: you will need the conversions on pg 8 to get the density matrix defined from the states. Convert the density matrix with the tilde(c)_i tilde(c)_j* to the density matrix with the c_i c_j* and convert to the density matrix with the a_i a_j*.) You should find that it oscillates at the laser frequency. (b) How does the size of the dipole operator and the phase of oscillation relative to the laser change with detuning? Sketch.

HWK 4 (Due Wed Feb 8): Probs 1 & 2 are Foot 8.1 and 8.2 (use the FWHM definition (Eq. 8.6) of the Doppler width)

Problem 3: This problem is to demonstrate the effectiveness of saturation absorption spectroscopy. (a) If the strong laser is off, give the transmission vs detuning for the case that the Doppler width is much, much, much greater than the line width. (Hint: your result should be 1 minus a Gaussian.) (b) Plot the fraction of light transmitted for Na 3s to 3p transition when the Na vapor is at 1000 K as a function of the detuning. (See HWK 3 Prob 2 for relevant Na data) For clarity assume the thickness of the sample is such that 95% is transmitted when perfectly on resonance (and no strong laser) and plot over a range from about -3 to +3 of the Doppler width. (c) In a saturation absorption spectroscopy experiment, the strong laser has Omega = f_l Gamma where f_1 is sizable. The weak laser has an Omega << f_1 Gamma. Within a proportionality constant, determine the amount of weak laser transmitted as a function of detuning in the case where the Doppler width is much, much, much greater than the line-width. (Hint: only when the detuning is comparable to Gamma is there an effect and over this range the velocity distribution can be treated as a constant. You can find the convolution of a Lorentzian with a Lorentzian on line somewhere.) Do plots for a strong laser with f_1 = 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.4. Comment on the trade-off (if any) between precision and the size of the signal. (Remember, I expect informative plots! These plots should be over smaller ranges (~10 Gamma???) to show the details.)

Problem 4: See the nature21040.pdf in the Precision papers. From Fig1 caption, obtain the temperature of the antiH. The laser waist is 200 microns. If most of the linewidth is coming from the finite time for the antiH to cross the beam, what linewidth is expected? Compare to Fig. 4.

HWK 5 (Due Wed Feb 15): Probs 1-3 are Foot 9.2, 9.4, 9.8 (Note, the Omega in 9.8 is NOT the Rabi frequency. It is the angular frequency of the harmonic oscillator.)

Problem 4: This problem explores a (truly terrible) idea I had for using an optical molasses to cool the motion of a dielectric nano-sphere. For this problem, consider a 50 nm radius, silica nanosphere that is in a 6 way laser beam (+- all 3 axis) with wavelength 1064 nm. The nano-sphere Rayleigh scatters the photons with a rate that increases with a power of the frequency. This suggests the possibility for an optical molasses because the nanosphere will scatter more photons from the beam it is moving toward. (a) The photon scattering rate from one beam has the form D omega^b. Determine b. (b) Determine the light pressure force from one beam in terms of D, Omega, b and any other fundamental constants. (c) Determine the force in the x-direction from the pair of beams. (d) Redo the derivation for optical molasses using this form of scattering to obtain the steady state temperature. (I expect both a formula and a number for part (d) (Hint: The answer should not depend on D or the mass of the nanosphere. The final temperature is much higher than room temperature which is why this was a terrible idea.)

HWK 6 (Due Wed Feb 22): Probs 1-3 are 9.12 (I differ by a factor of 1/pi from the book's result, but maybe I did something wrong), .9.16, 9.17

Problem 4: You have cooled Na in an optical molasses using the 3s-3p transition. You turn off the optical molasses and in a time of ~ 1 microsec. You turn on a single laser in the vertical direction with I = I

_{sat}. (a) What detuning should you use and how long should you have the laser on so that the average velocity is 1 m/s in the vertical direction? (b) What will be the spread in velocity? (Hint: the velocity spread might be different in the x,y,z directions and don't forget about the initial spread in velocities because of the initial temperature.) (c) What would the initial temperature of the Na have to be so that the launch step contributes a substantial fraction to the final velocity spread?

HWK 7 (Due Wed Mar 1): No homework this week. I couldn't think of good problems covering the material we looked at.

HWK 8 (Due Wed Mar 8): 10.1 (just take |g_F M_F| = 1), 10.2, 10.4

Problem 4: The adiabatic approximation means the number of nodes in the wave function doesn't change as the parameter in the potential is changed. a) For a 1D harmonic oscillator show that E^a <x^2>^b is a constant as the frequency is adiabatically changed (E is the energy and <x^2> is the expectectation value [Hint: to evaluate <x^2> you might use the Schrodinger equation in scaled coordinates.] Give the value for the exponents a and b. b) Generalize your result to 3D: E^a <r^2>^b. c) Use this result to derive the T V^(2/3) from problem 10.2.

HWK 9 (Due Wed Mar 22): 10.5,

Problem 2: Experiments on BEC's quickly turn off the trapping potentials and let the BEC have a free expansion until it is much larger than the original cloud. For noninteracting atoms, this is a measurement of the velocity distribution. (a) Explain why. When T < T_c, the thermal cloud and condensate are both clearly visible. (b) For "noninteracting" atoms in a 3D, isotropic harmonic well, give the normalized velocity distribution when: T = T_c, 0.9 T_c, 0.8 T_c, and 0.7 T_c. (c) Plot these cases vs v_x. (Hint: For the ground state velocity distribution, the Fourier transform of a Gaussian is a Gaussian. For the plot, give your results in terms of scaled parameters.)

Problem 3: The alien scientest, Fluglebart, has "noninteracting" bosons in an isotropic 4D (four dimensional) harmonic trap. The density of states in 1D, 2D, and 3D is g_n = 1, (n+1)/1, (n+1)*(n+2)/(1*2). (a) Guess or derive what g_n is in 4D (if you guess the answer, you need to show your formula works for n = 0, 1, 2, 3). (b) Derive the value of k_B T_c/(hbar omega) vs. the number of bosons. (Hint: Riemann zeta function) (c) Give the ground state fraction and the chemical potential for T<T_c.

Problem 4: Derive the correction to the 3D isotropic harmonic oscillator BEC parameters (T_c and N_0 vs T) by using g_n = (n^2/2) + n in the derivation instead of only g_n = n^2/2. (Hint: x^3 + a x^2 = big can be solved iteratively x_new = (big - a x^2_old)^(1/3). One iteration is enough for big ~ 10^4). Compare to the data I plotted from the numerical calculation. Show that the corrections go to a constant as N gets large (extra points if you figure out what the constant is).

HWK 10 (Due Wed Mar 29): No homework.

HWK 11 (Due Wed Apr 5): 11.1, 11.2

In the next two problems, I left some parameters vague. Make sure you tell me what you used for them in your plots.

If you use source theory, you can use the form for psi(x,y,z) at z=0 since the distance to the detector is always >> 1/k.

Prob 3: An "atom" has 2 internal states |a> and |b> and is in a plane wave moving in the z-direction. It is originally in state |b>. It interacts with a weak potential centered at the origin that can change |a> to |b>. Just after the potential the wave has the form:

psi = C exp(i k z) [f(x,y) |a> + sqrt(1 - f(x,y)^2) |b>]

where f(x,y) = f_0 exp[-(x^2 + y^2)/D^2] with f_0 << 1. This situation only arises when k D ~ 20. Obtain the wave function in the far field. Along the line y=0, plot the intensity as a function of x when z = L >>> 1/k to be in state |a> and a separate plot for |b>. (Hint: since f_0 << 1 you might Taylor series expand the square root.)

Prob 4: A localized blob of material centered at the origin has two internal states |a> and |b> and it is 100% in |b> before a proton hits it. A proton is in a plane wave moving in the z-direction. It interacts with the blob so that just after it, the wave function of the atom + blob has the form:

psi = C exp(i k z) [f(x,y) |a> + sqrt(1 - f(x,y)^2) |b>]

where x,y,z are the proton's position, f(x,y) = f_0 exp[-(x^2 + y^2)/D^2] with f_0 << 1. This situation only arises when k D ~ 20. Obtain the wave function in the far field. Along the line y=0, plot the intensity for the blob to be in state |a> as a function of proton position x when z = L >>> 1/k and a separate plot for |b>. (Up to this point you can use many (all?) of the results from the previous problem.)

Now starts the interesting part of this problem.

Suppose there are 2 blobs. They are centered at y=0, z = 0, and x = +- S with S ~ 30 D. After the potential the wave has the form:

psi = C exp(i k z) [f(x-S,y) |ab> + f(x+S,y) |ba> + g(x,y) |bb>]

where the symbols are the same as above and

g(x,y) = sqrt(1 - f^2(x-S,y) - f^2(x+S,y)).

Why are there no terms with |aa>? Obtain the wave function in the far field. Plot the intensity for the proton to be at x (y=0, z = L>>>1/k and L>>S) and NEITHER blob to be excited (i.e. |bb>). Is there an interference pattern? Plot the intensity for the proton to be at x (y=0, z=L>>>1/k and >>S) and EITHER blob to be excited? Is there an interference pattern? Explain why one case has an interference pattern and the other does not.

HWK 12 (Due Wed Apr 12): No homework

HWK 13 (Due Wed Apr 19):

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. You should end up with something like the middle of Pg 10 notes but with some product of powers of A_1 and A_2.

Prob 2: Repeat Prob 1 but when there are 3 sources with amplitudes A_1, A_2, and A_3.

Prob 3: Repeat for N sources.

Prob 4: Use the result from Prob 3, to generalize to the case of continuous sources over a surface in x,y.

HWK 14 (Due Wed Apr 26): 12.1, .2, .3 (for a = 0 and a = -0.3), .4