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Exercises Light-Matter Interaction 2012

Repetition Quantum Mechanics / Atomic Physics 1. We consider a particle in an infinitely deep box, which extends from 0 to L.

a) Write down the wavefunctions and energies We assume that the particle is in the lowest level of the box. At time t=0, the box is suddenly expanded to twice its original width, and now extends from 0 to 2L. (At t=0, the system can be described by the wavefunction from 1).) b) Write the wavefunction as a superposition of the eigenstates of the new box (Hint: use the following formula):

a

0 sin(bx)sin(cx)dx = -1/2[sin{(b+c)a}/(b+c) - sin{(b-c)a}/(b-c)] c) Describe what will happen to this system as a function of time – will the particle start moving? (Will the wavefunction be time-dependent?). This is a good example of a wavepacket. Calculate at what times the wavepacket will come back to its original state. This is called revival of the wavepacket.

2. Consider a hydrogen atom. At time t=0, the wavefunction is the following superposition

of energy eigenfunctions nlm(r): (r,t=0) = N[2100(r) - 210(r) + 311(r)] a) Normalize the wavefunction b) Is the wavefunction of even or odd parity? c) What are the probabilities of finding the system in the ground state? The state (200)? Or (311)? Another eigenstate? d) What is the expectation value of the total energy? Of the operators L2, and Lz?

3. Classically-forbidden region in hydrogen

Classically, any region of space where the kinetic energy of a particle is negative is forbidden. a) Show that the classically forbidden region for the ground state of hydrogen is r > 2a0, where a0 is the Bohr radius. b) Calculate the probability of finding the electron in this region.

hint:

b

abar eabaabdrer /322/2 )22(

4. Fine structure in Hydrogen We want to calculate the fine structure for the n=2 level in H. We consider the energies and wavefunctions corresponding to the unperturbed Hamilton operator H0, which is the sum of the kinetic energy operator and the Coulomb potential. The electron has a ½.spin. a) What is the degeneracy of the n=2 level? b) Which orthonormal basis can be used to describe level 2? Which quantum numbers are

necessary to characterize the wavefunctions?

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c) The fine structure is described with the Hamilton operator Wf=Wmv+Wso+WD, where Wmv corresponds to the relativistic mass increase, WD is the Darwin term, and Wso describes the spin-orbit interaction. Wmv is proportional to 4p

, WD depends only on r,

and Wso= VLS(r) SL

. The hyperfine interaction is neglected. Show that Wso does not commute with Lz. What does it imply for the matrix that describes Wso in the basis

smmn for n=2?

d) Show that Wso commutes with 2L

and that the matrix consists of a 2x2 matrix for 2s and a 6x6 matrix for 2p.

e) To simplify the problem, consider the total angular momentum operator SLJ

. Use the triangle rule to determine which j, and s are possible for n=2. These states are labelled j

s 12 .

f) Show that SL

can be written as )( 22221 SLJ

.

g) Show that 2J

and Jz commute with H0 and Wso.

h) In which basis are 2J

, Jz , H0, Wso , 2L

, 2S

diagonal? i) Write down Wso in the above matrix using the integrals 222 )( rVLS where

2 is the radial wavefunction for 2. How many energy levels are there?

5. The anomalous Zeeman effect in alkalis (a) Give the value of gJ for the one-electron levels 2S1/2,

2P1/2 and 2P3/2. (b) Show that the Zeeman pattern for the 3s 2S1/2- 3p 2P3/2 transitions in sodium have six equally-spaced lines when viewed perpendicular to a weak magnetic field. Find the spacing (in GHz) for a magnetic flux density of 1 T. Sketch the Zeeman pattern observed along the magnetic field. (c) Sketch the Zeeman pattern observed perpendicular to a weak magnetic field for the 3s 2S1/2- 3p 2P1/2 transition in sodium. (d) The two fine-structure components of the 3s-3p transition in sodium in parts (b) and (c) have wavelengths of 589.6 nm and 589.0 nm, respectively. What magnetic flux density produces a Zeeman splitting comparable with the fine structure?

Chapter 7 6. Ramsey fringes.

A beam of atoms with velocity v is sent through a chamber of length L, in which they interact with a radio-frequency field Ecos(t) of frequency . The atoms are initially in state i, and the frequency is close to resonant with an atomic transition i->f (frequency 0). The perturbation from the rf-field is described by H’ = -E*Dcos(t), where D is the dipole operator.

v

L

3

a) Express the flux F1 of atoms in the excited state at the exit of the chamber (the incoming flux is F0) – making the rotating wave approximation and using time-dependent perturbation theory. b) A second chamber of length L is added to the path of the atoms. The two chambers are separated by L1 > L. Find the flux F2 of atoms in the excited state after the second chamber. Express F2 as a function of F1.

c) Plot (approx.) the variation of F1 and F2 as functions of (0-). 7. Hydrogen atoms in a time-dependent electric field An ensemble of hydrogen atoms in the ground state is placed in a time-dependent electric field E=0, t<0, E(t)=E0 exp(-t/) t>0. a) Use time-dependent perturbation theory to calculate what is the fraction of atoms in

the 2p state after a long time? Express this fraction as a function of a0 (Bohr radius), , (1s-2p transition frequency), , e (electron charge) and E0.

Hint: consider only the 1s, 2p states and express the variation of the coefficients of the wave function in time using the Schrödinger equation. Solve the equation for c2p assuming that the probability to go from 1s to 2p is very small.

b) Assume that = 1 ps. Which strength should the electric field have to move 0.1% of the atom population in the 2p state? (Such fields can be achieved in laser-plasma interactions or in atom-ion collisions).

c) What approach should be used to calculate the electric field strength required to move 0.1% of the atom population in the 2s state?

8. Coherent excitation of Mg atoms – steady state A Mg26 atom in a trap is excited by a laser tuned close to the 3s2 1S0 - 3s3p 1P1 transition ((laser) = (3s3p 1P1) - (3s2 1S0) + . It is assumed that the system can be treated as a two-level system and that it can be fully described by the Bloch equations for a two-level system. The 3s3p 1P1 state in Mg has the lifetime =2 ns. a) Calculate the laser power required for obtaining a Rabi frequency, , equal to =1/(3) if the laser beam has a top hat intensity profile and a diameter of 1 mm. b) Calculate the upper state probability for =0 och =1/ at steady state. (The energy of the magnesium 3s3p 1P1 state is 35051 cm-1)

Hint: As shown in Section 7.2 there is a relation between the Einstein coefficient (A) for a transition and the transition dipole moment. Write this relation, and consequently the relation between A and the Rabi frequency. In the particular case above one may (correctly) assume that the Mg 3s3p 1P1 state predominantly decays to the 3s2 1S0 state.

v L1

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Chapter 8 9. Laser spectroscopy of Cs The naturally-occurring isotope of caesium (133Cs) has a nuclear spin of I = 7/2. Draw a diagram showing the hyperfine sub-levels, labelled by the appropriate quantum numbers, that arise from the 6 2S1/2 and 6 2P3/2 levels in caesium, and the allowed electric dipole transitions between them. The figure below shows the saturated absorption spectrum obtained from the 6 2S1/2-6 2P3/2 transition in a vapour of atomic caesium, including the cross-over resonances which occur midway between all pairs of transitions whose frequency separation is less than the Doppler width. The relative positions of the saturated absorption peaks within each group are given below in MHz. A B C D E F 0 100.7 201.5 226.5 327.2 452.9 a b c d e f 0 75.8 151.5 176.5 252.2 353.0 Using these data and the information in the diagram: (a) determine the extent to which the interval rule in the hyperfine levels is obeyed in this case and deduce the hyperfine parameter Anlj for the 6 2S1/2 and 6 2P3/2 levels (b) estimate the temperature of the caesium vapour. (The wavelength of the transition is 852 nm.)

10. Two-photon experiment The experimental scan besides comes from a two-photon experiment like that shown in Fig. 8.8. The transition from the 5p6 1S0 ground level of xenon to a J = 0 level of the 5p56p configuration was excited by ultraviolet radiation with a wavelength of 249 nm and the scale gives the (relative) frequency of this radiation. This J = 0 to J' = 0 transition has no hyperfine structure and the peak for each isotope is labelled with its relative atomic mass. The xenon gas was at room temperature and a pressure of 0.3 mbar. Light from a blue laser with a frequency jitter of 1 MHz was frequency doubled to generate the ultraviolet radiation and the counter propagating beams of this radiation has a radius of 0.1 mm in the interaction region. Estimate the contribution to the linewidth from the (a) transit time (b) pressure broadening, (c) the instrumental width, and (d) the Doppler effect.

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Chapter 9 11. Laser cooling of atoms with hyperfine structure a) The treatment of Doppler cooling in the course assumes a two-level atom. In real experiments with a MOT, the hyperfine structure of the ground configuration causes complications. We consider Sodium atoms with I=3/2. Draw approximately an energy level diagram of the hyperfine structure of the 3s 2S1/2 and 3p 2P3/2 levels and indicate the allowed electric dipole transitions. b) In a laser cooling experiment, the transition 3s 2S1/2, F=2, to 3p 2P3/2, F'=3 is excited by light that has a frequency detuning of = -/2= -5 MHz (to the red of the transition). Selection rules dictate that the excited state decays back to the initial state so that there is a nearly closed cycle of absorption and spontaneous emission. However, there is some off-resonant excitation to the F'=2 state which can decay to the F=1 state and be "lost" from the cycle. The F'=2 level lies 60 MHz below the F'=3 level. Estimate the average number of photons scattered by an atom before it falls into the lower hyperfine level of the ground configuration. (Assume that the transitions have similar strengths). To counteract the leakage out of the laser cooling cycle, experiments use an additional laser beam that excite atoms out of the F=1 level (so that they come back to F=2). Suggest a suitable transition for this repumping process. 12. Zeeman slower Radioactive 21Na atoms (I=3/2) are produced by bombarding Mg atoms in an oven at 500 oC with a beam of protons. A fraction of the Na atoms are subsequently cooled and trapped in a MOT. First the atoms are cooled transversally with laser beams. Then they travel a region where they are slowed down to zero longitudinal velocity by interaction with a counter propagating laser beam. The slowing laser beam is tuned near the F=2 to F'=3 hyperfine component of the 3 2S1/2 to 3 2P3/2 transition (D2 line, =589 nm, lifetime 16 ns, nuclear spin 3/2) and has + polarization. The resonance frequency of the atoms in the beam is Doppler shifted. As the atoms slow down, the Doppler shift decreases. In order to keep the atoms in resonance with the laser beam used for slowing, a spatially varying magnetic field is produced by a solenoid with non-uniform winding. a) Calculate the most probable initial velocity of the Na atoms by assuming that the velocity distribution in the oven is given by the Maxwell-Boltzmann distribution:

222

ve)v(n Tk/mv B and that the flux of

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atoms leaving the oven is Av)v(n)v( where A is the dimension of the hole. We now assume that the atoms have the same initial velocity equal to the most probable one calculated above. We assume furthermore that I=Isat and =0.

b) Calculate the deceleration due to the scattering force. Estimate the time it takes for the atoms to stop.

c) Calculate the distance of the slow-down region so that the final velocity of the atoms is equal to zero. Express the velocity in the slow-down region as a function of distance from the trap.

d) In which state will the atoms be optically pumped when excited with + polarized field? Express the Zeeman energies for the initial and final states of the cooling transition as a function of the magnetic field.

e) The laser is tuned near the zero-field resonance frequency. Calculate the size and longitudinal coordinate dependence of the magnetic field that needs to be produced by the slow-down solenoid.

Chapter 10 13. Magnetic trapping of sodium atoms a) Sketch the energy of the hyperfine levels of the 3s 2S1/2 ground state of sodium as a function of the applied magnetic field strength (assumed to be much weaker than the hyperfine structure). Give the energy difference between the hyperfine levels at zero magnetic field. (The hyperfine structure constant is A= 886 MHz; sodium has a nuclear spin equal to 3/2). Show that for both hyperfine levels the magnetic levels have a splitting of 7 GHz/T. b) The sodium atoms are placed in a magnetic trap trapping the atoms radially, (see Fig. 10.2) with a radius 10 mm, and a field gradient b'=1.5 Tm-1. Which magnetic sub-levels will be trapped? (These should be "low-field-seeking" states, i.e. such that the magnetic (Zeeman) energy decreases as the atom moves into a lower magnetic field). Estimate the maximum temperature of the atoms that can be trapped for the relevant magnetic sub-levels. c) A radio frequency radiation field with frequency 70 MHz is applied. At which distance from the center of the trap will atoms move to other magnetic states and cease to be trapped? What is the new maximum temperature of the trap?

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Answers

1. 2)

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124,

2

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1

2

nnaa

n

n (n odd)

2. 63

2899

6

1

6

10

3

2

6

1 2 ,,eV.,,,,,No,

3. 0.238 4. e) =0,1, j=1/2, 3/2 i) 3 energy levels: 0, - p2 , p2 /2

5. a) 2, 2/3, 4/3 b) 9.3 GHz c) 4 lines separated by 2/3, 4/3, 2/3 BB d) 50 T

6. 2

0

02

2

2

01 )(

2)(sin.

v

LDE

FFfi

;

v

LLFF

2)(cos4 1

02

12

7. a)

222

20

20

2

1

55.0

aEe b) 8.5 GV/m

8. a) 0.78 mW b) 9% ,2% 9. a) 50 MHz, 2300 MHz b) 400 K 10. a) 1 MHz b) 20 MHz c) 4 MHz d) 500 Hz 11. 122, F=1 to F’=1

12. a) 957 m/s b) 5 105 m/s2, 1.9 ms, c) 0.91 m, ax2 e) 0.12 x T 13. b) 5 and 10 mK c) 6.7 mm, 0.667x Tb