Atomic physics (2005)

•Atoms emerge from an oven to form an atomic beam in an evacuated chamber. The atoms have a mean free path much greater than the length of the apparatus (i.e. there are no collisions).

•The deflection of atoms in the A and B regions depends on MJ . If these two regions have magnetic field gradients in the same direction,

as indicated in Fig. 6.15, then atoms only reach the detector if their

MJ quantum number changes in the C region, i.e MJ = + 21 ↔ MJ = −21 . This is known as the flop-in arrangement.37

•As the atoms travel from A into the C region their state changes adiabatically to that in a low magnetic field as shown in Fig. 6.16 (and similarly as the field changes between the C and B regions). The transitions in the low-field region that can be observed are those that connect states with di erent values of MJ in the high-field regions, e.g. the transitions:

low frequency (∆F = 0):

F = 1, MF = 0 ↔ F = 1, MF = −1 with ∆E = gF µBB ; higher frequency (∆F = ±1):

In this example, the MF = 1 ↔ MF = 0 change between the Zeeman sub-levels of the upper F = 1 level cannot be detected, but this

37The flop-out arrangement is shown in Fig. 6.14.

38Such quantum metrology has been used to define other fundamental constants, with the exception of the kilogram which is still defined in terms of a lump of platinum kept in a vault in Paris.

39The next chapter gives a complete treatment of the interaction of atoms with radiation.

6.4.2Atomic clocks

An important application of the atomic-beam technique is atomic clocks, that are the primary standards of time. By international agreement the second is defined to be 9 192 631 770 oscillation periods of the hyperfine frequency in the ground state of 133Cs (the only stable isotope of this element). Since all stable caesium atoms are identical, precise measurements of this atomic frequency in the National Standards Laboratories throughout the world should agree with each other, within experimental uncertainties.38

The definition of the second is realised using the hyperfine frequency of the F = 3, MF = 0 ↔ F = 4, MF = 0 transition in caesium. This transition between two MF = 0 states has no first-order Zeeman shift, but even the second-order shift has a significant e ect at the level of precision required for a clock. The apparatus can be designed so that the dominant contribution to the line width comes from the finite interaction time τ as atoms pass through the apparatus (transit time). Fourier transform theory gives the line width as39

where vbeam is the typical velocity in the beam and l is the length of the interaction region. Therefore the atomic beams used as primary standards are made as long as possible. An interaction over a region 2 m long gives a line width of ∆f = 100 Hz.40 Thus the quality factor is f /∆f 108; however, the centre frequency of the line can be determined to a small fraction of the line width.41 As a result of many years of careful work at standards laboratories, atomic clocks have uncertainties of less than 1 part in 1014. This illustrates the incredible precision of radio-frequency and microwave techniques, but the use of slow atoms gives even higher precision, as we shall see in Chapter 10.42 There is a great need for accurate timekeeping for the synchronisation of global telecommunications networks, and for navigation both on Earth through the global positioning system (GPS) and for satellites in deep space.

The atomic-beam technique has been described here because it furnishes a good example of the Zeeman e ect on hyperfine structure, and it was also historically important in the development of atomic physics. The first atomic-beam experiments were carried out by Isador Rabi and he made numerous important discoveries. Using atomic hydrogen he showed that the proton has a magnetic moment of 2.8 µN, which was about three times greater than expected for a point-like particle (cf. the electron with µB, i.e. one unit of the relevant magnetic moment). This was the first evidence that the proton has internal structure. Other important techniques of radio-frequency spectroscopy such as optical pumping are described elsewhere, see Thorne et al. (1999) and Corney (2000).

40Caesium atoms have velocities of vCs = (3kBT /MCs)1/2 = 210 m s−1 at T = 360 K.

41In a normal experiment it is hard to measure the centre of a line with an uncertainty of better than one-hundredth

of the line width, i.e. a precision of only 1 in 1010 in this case.

42The hydrogen maser achieves a long interaction time by confining the atoms

in a glass bulb for τ 0.1 s to give a line width of order 10 Hz—the atoms bounce o the walls of the bulb without losing coherence. Thus masers can be more precise than atomic clocks but this does not mean that they are more accurate, i.e. the frequency of a given maser can be measured to more decimal places than that of an atomicbeam clock, but the maser frequency is shifted slightly from the hyperfine frequency of hydrogen by the e ect of collisions with the walls. This shift leads to a frequency di erence between masers that depends on how they were made. In contrast, caesium atomic clocks measure the unperturbed hyperfine frequency of the atoms. (The use of cold atoms improves the performance of both masers and atomic clocks—the above remarks apply to uncooled systems.)

Exercises

(6.1) The magnetic field in fine and hyperfine structure

Calculate the magnetic flux density B at the centre of a hydrogen atom for the 1s 2S1/2 level and also for 2s 2S1/2.

Calculate the magnitude of the orbital magnetic field experienced by a 2p-electron in hydrogen (eqn 2.47).

(6.2) Hyperfine structure of lithium

The figure shows the energy levels of lithium involved in the 2s–2p transitions for the two isotopes 6Li and 7Li. (The figure is not to scale.)

Explain in simple terms why the hyperfine splitting is of order me/Mp smaller than the finestructure splitting in the 2p configuration of lithium.

Explain, using the vector model or otherwise, why the hyperfine interaction splits a given J level into 2J + 1 hyperfine levels if J I and 2I + 1 levels if I J. Hence, deduce the nuclear spin of 6Li and give the values of the quantum numbers L, J and F for all its hyperfine levels. Verify that the

6Li

interval rule is obeyed in this case.

Determine from the data given the nuclear spin of 7Li and give the values of L, J and F for each of the hyperfine levels on the figure. Calculate the hyperfine splitting of the interval marked X.

(For the hyperfine levels a to d the parameter Anlj is positive for both isotopes.)

(6.3) Hyperfine structure of light elements

Use the approximate formula in eqn 6.17 to estimate the hyperfine structure in the ground states of atomic hydrogen and lithium. Comment on the di erence between your estimates and the actual values given for hydrogen in Section 6.1.1 and for Li in Exercise 6.2.

(6.4) Ratio of hyperfine splittings

The spin and magnetic moment of the proton are (1/2, 2.79µN ), of the deuteron (1, 0.857µN ) and of 3He (1/2, −2.13µN). Calculate the ratio of the ground-state hyperfine splittings of (a) atomic hydrogen and deuterium and (b) atomic hydrogen and the hydrogen-like ion 3He+.

7Li

122Hyperfine structure and isotope shift

(e)Define what is meant by a ‘strong field’ when considering hyperfine structure. Give an approximate numerical value for the magnetic field at which the cross-over from the weakfield to the strong-field regime occurs in this example.

(6.9) Isotope shift

Estimate the contributions to the isotope shift between 8537Rb and 8737Rb that arise from the mass and volume e ects for the following transitions:

trons has a kinetic energy T given by

where pN is the momentum of the nucleus and pi is the momentum of the ith electron. The total of these momenta is zero in the centre-of-mass frame of the atom:

N

pN + pi = 0 .

To describe the interaction of a two-level atom with radiation we shall use a semiclassical treatment, i.e. the radiation is treated as a classical electric field but we use quantum mechanics to treat the atom. We shall calculate the e ect of an oscillating electric field on the atom from first principles and show that this is equivalent to the usual time-dependent perturbation theory (TDPT) summarised by the golden rule (as mentioned in Section 2.2). The golden rule only gives the steady-state transition rate and therefore does not describe adequately spectroscopy experiments with highly monochromatic radiation, e.g. radio-frequency radiation, microwaves or laser light, in which the amplitudes of the quantum states evolve coherently in time. In such experiments the damping time may be less than the total measurement time so that the atoms never reach the steady state.

From the theory of the interaction with radiation, we will be able to find the conditions for which the equations reduce to a set of rate equations that describe the populations of the atomic energy levels (with a steady-state solution). In particular, for an atom illuminated by broadband radiation, this approach allows us to make a connection with Einstein’s treatment of radiation that was presented in Chapter 1; we shall find the Einstein B coe cient in terms of the matrix element for the transition. Then we can use the relation between A21 and B21 to calculate the spontaneous decay rate of the upper level. Finally, we shall study the roles of natural broadening and Doppler broadening in the absorption of radiation by atoms, and derive some results needed in later chapters such as the a.c. Stark shift.

7.1Setting up the equations 123

7.2The Einstein B

7.3Interaction with monochromatic radiation 127

2Note that E0 cos (ωt) is not replaced by a complex quantity E0e−iωt, because a complex convention is built into quantum mechanics and we must not confuse one thing with another.

3This dipole approximation holds when the radiation has a wavelength greater than the size of the atom, i.e. λ a0, as discussed in Section 2.2.

4Section 2.2 on selection rules shows how to treat other polarizations.

functions of H0; the unperturbed eigenvalues and eigenfunctions of H0 are just the atomic energy levels and wavefunctions that we found in previous chapters. We write the wavefunction for the level with energy En as

For a system with only two levels, the spatial wavefunctions satisfy

H0ψ1 (r) = E1ψ1 (r) ,

(7.4)

H0ψ2 (r) = E2ψ2 (r) .

These atomic wavefunctions are not stationary states of the full Hamiltonian, H0 + HI (t) , but the wavefunction at any instant of time can be expressed in terms of them as follows:

or, in concise Dirac ket notation (shortening c1 (t) to c1, ω1 = E1/ , etc.),

Normalisation requires that the two time-dependent coe cients satisfy

7.1.1Perturbation by an oscillating electric field

The oscillating electric field E = E0 cos (ωt) of electromagnetic radiation produces a perturbation described by the Hamiltonian

This corresponds to the energy of an electric dipole −er in the electric field, where r is the position of the electron with respect to the atom’s centre of mass.2 Note that we have assumed that the electric dipole moment arises from a single electron but the treatment can easily be generalised by summing over all of the atom’s electrons. The interaction mixes the two states with energies E1 and E2. Substitution of eqn 7.6 into the time-dependent Schr¨odinger eqn 7.1 leads to

The electric field has almost uniform amplitude over the atomic wavefunction so we take the amplitude |E0| outside the integral.3 Thus, for radiation linearly polarized along the x-axis, E = |E0|ex cos (ωt), we obtain4

To solve the coupled di erential equations for c1 (t) and c2 (t) we need to make further approximations.

7.1.2The rotating-wave approximation

When all the population starts in the lower level, c1 (0) = 1 and c2 (0) = 0, integration of eqns 7.9 and 7.10 leads to

This gives a reasonable first-order approximation while c2 (t) remains small. For most cases of interest, the radiation has a frequency close to the atomic resonance at ω0 so the magnitude of the detuning is small, |ω0 − ω| ω0, and hence ω0 + ω 2ω0. Therefore we can neglect the term with denominator ω0 + ω inside the curly brackets. This is the rotating-wave approximation.5 The modulus-squared of the co-rotating term gives the probability of finding the atom in the upper state at time t as

(7.15)

or, in terms of the variable x = (ω − ω0)t/2,

The sinc function (sin x) /x has a maximum at x = 0, and the first minimum occurs at x = π or ω0 − ω = ±2π/t, as illustrated in Fig. 7.1; the frequency spread decreases as the interaction time t increases.

7.1 Setting up the equations 125

5This is not true for the interaction of atoms with radiation at 10.6 µm from a CO2 laser. This laser radiation has a frequency closer to d.c. than to the resonance frequency of the atoms, e.g. for rubidium with a resonance transition in the near infra-red (780 nm) we find that ω0 15ω, hence ω0 +ω ω0 −ω. Thus the counter-rotating term must be kept. The quasi-electrostatic traps (QUEST) formed by such long wavelength laser beams are a form of the dipole-force traps described in Chapter 10.

Fig. 7.1 The excitation probability function of the radiation frequency has a maximum at the atomic resonance. The line width is inversely proportional to the interaction time. The function sinc2 also describes the Fraunhofer diffraction of light passing through a single slit—the di raction angle decreases as the width of the aperture increases. The mathematical correspondence between these two situations has a natural explanation in terms of Fourier transforms.