Laser Cooling Lab phyc30021 – Laboratory and Computational Physics 3 By Joshua P. Ellis School of Physics 21st March 2019 Contents 1 Introduction 1 1.1 Aim .................................... 1 2 Photon–Atom Interactions 1 2.1 More Complicated Atoms ......................... 2 2.2 Momentum Conservation ......................... 3 2.3 Excited Fraction .............................. 4 3 Atomic Beam 5 4 Zeeman Slower 6 5 Diffusion 7 References 8 i 1 Introduction 1 Introduction In day-to-day life, a system is cooled when it transfers thermal energy to its surrounding. The transfer in energy can be achieved in any of a number of ways, and ultimately canbe well described by Newton’s law of cooling, dQ = k»T¹tº − T ¼; dt env where Q and T are the thermal energy and temperature of the system of interest, Tenv is the surrounding temperature, and k is a proportionality constant. According to the above description, in order to achieve extremely low temperatures, an environment with an even lower temperature must be found; however, as 0 K is the lower limit1, then it appears to be very difficult to cool systems to absolute 0 as the temperature differential decreases making the heat transfer ever smaller. To make things worse, the heat gets transferred to the surrounding environment thereby raising the surrounding temperature preventing the original system from cooling. As a result, cooling systems to ever lower temperatures requires new ways of extracting energy out of the system. In 1975, Wineland and Dehmelt, and Hänsch and Schawlow simultaneously proposed the idea of using lasers in order to cool gases [1, 2]. The method relies on the high sensitivity to frequency in the atomic absorption spectra alongside with the Doppler shift introduced by the atoms’ thermal motion in order to cause a net reduction in thermal energy of the system. Three years after it was first envisaged, Wineland, Drullinger and Walls demonstrated the Doppler cooling does work [3], and ultimately work on using lasers to cool atomic gases resulted in the 1997 Nobel prize being awarded to Steven Chu, Claude Cohen-Tannoudji, and William Daniel Phillips [4]. 1.1 Aim In this lab, the physics underlying laser cooling will be investigated. 2 Photon–Atom Interactions We first begin with a simple electron around an atom: the hydrogen atom. The quantum mechanical system is well-studied and described by the Hamiltonian p2 e2 H = + : H 2m r 1Certain systems behave in un-intuitive ways which are best described as ‘negative temperature systems’, though these are rather unusual. 1 2 Photon–Atom Interactions The solutions to the hydrogen atom are well known and we will denote themas jsii with energy Ei. We will also define ~!ij := Ei − Eh. This description, however, cannot describe interactions with a photon. As a result, we must amend the Hamiltonian to describe interactions with photons: e H = H + A · p: H mc This modified Hamiltonian can be well described using perturbative techniques asthe strength of interactions with the radiation field is much smaller than interactions with the potential. One important consequence of the interactions with photons is that the original energy eigenstates jsii of the hydrogen atom are no longer exact energy eigenstates of the perturbed Hamiltonian. This should be quite obvious actually since if they were still eigenstates of the Hamiltonian, they would also be eigenstates of the time-evolution operator and thus would never transition between states. Another consequence of this perturbation is that the energy transitions !ij are no longer exact as the initial and final state energies are themselves inexact. This introduces a certain degree of ‘fuzziness’, or more formally, a width Γ. The width and lifetime of a state are related through Γ := 1/τ. Question 1: Why would you expect that Γ and τ be inversely related to each other? 2.1 More Complicated Atoms Larger atoms with more electrons have additional effects which perturb the Hamiltonian further. For example, the nucleus will often have a non-zero nuclear spin I creating a magnetic dipole moment which ultimately interacts with certain electron orbitals as it introduces a term in the Hamiltonian of the form −µI · B: There might also be some effects between the electron orbitals, though we will belookingat alkali metals (specifically sodium) for which the inner electrons can be mostly ignored. We will be interested in sodium which is essentially a hydrogenic atom as the inner electrons can be mostly ignored. The outer electron has a spin angular momentum of 1 3 jsj = 2 , and the nucleus has angular momentum jI j = 2 . In addition, the outer electron orbital has integer angular momentum, though we will be interested in the jLj = 0 and jLj = 1. All these angular momenta couple and the observable total angular momentum if F := I + L + s and we will denote the projection of the total angular momentum as mF . All of the possible ground states and first excited states are shown in fig. 1. 2 2 Photon–Atom Interactions mF −3 −2 −1 0 1 2 3 jFj = 3 jFj = 2 2p 59:6 MHz jFj = 1 jFj = 0 589 nm jFj = 2 jFj = 1 2s 1:772 GHz Figure 1: Energy levels of the sodium atom For the sodium atoms, we can assume that the ground state corresponds to the jFj = 2, mF = +2 state; and that the excited state is the jFj = 3, mF = 3 state. Note that even if the photon energy might be compatible with a transition, the photon has spin jsj = 1 and thus angular momentum conservation forbids certain transitions. For the transition above, the shift in frequency due to the Zeeman effect is ∆¹~!º = µB jBj −24 −1 where µB = 9:274 × 10 JT is the Bohr magneton. 2.2 Momentum Conservation Each photon has a tiny, but non-zero, linear momentum p = ~!/c. If an atom is initially sitting at rest and absorbs a photon with energy E = ~!, the resulting excited atom must have momentum p = ~!/c = mvr where vr is the recoil velocity and m is the atom’s mass. Similarly, an excited electron emitting a photon will also recoil slightly, though in this case the recoil is in the opposite direction to the emitted photon. Wehave no control over the direction in which the atom emits the photon, and on average the atom will emit photons isotropically. Consequently, the net velocity gained from atoms returning to the ground state is hvemissioni = 0. What we do have control over are the photons which excite the atoms. In particular, a laser will create a beam of photons which all have nearly identical momenta (and frequency) and thus hvabsorptioni , 0. The combination of absorptions from the laser and random emissions results inanet change in momentum, or in other words, resulting in a net force on the atoms. This net force is what we will ultimately use to cool the atoms in the gas. 3 2 Photon–Atom Interactions Question 2: Calculate the maximum acceleration for a sodium atom, where τ = 16:2 ns. Why is the maximum rate 1/2τ rather than 1/τ? You should find that the maximum force is ! F = ~ : max 2cτ 2.3 Excited Fraction Doppler cooling is used to cool very small systems, but still containing such large number of atoms that it is futile to look at the interaction of individual atoms. The fraction of atoms in an excited state will depend on how close the laser is to being on resonance and the rate at which atoms are being excited and also returning to the ground state. This will not be derived here, but it will suffice to know that the fraction of atoms in the excited state is: Ω2 f := 2Ω2 + 4∆2 + Γ2 p where ∆ := ! − !0 is the laser detuning from resonance, Ω := Γ I/2Is is the Rabi frequency in which I is the laser intensity and Is is the saturation intensity given by πhc !3 : ~ Is = 3 = 2 : 3λ0 12πc Note that the saturation frequency here is specific for a two-level atom. By taking into account which atoms are in the excited state, the force on the gas is F = f Fmax: 4 3 Atomic Beam Figure 2: Atomic beam source 3 Atomic Beam The experimental setup begins with hot crucible from which the sodium atoms evaporate. They stream out of the crucible and into the vacuum surrounding it as showninfig. 2, with the distribution of speeds being given by 3 m 2 mv2 f ¹vº := 4πv2 exp − 2πkBT 2kBT m is the atomic mass, kB is the Boltzmann constant, T is the temperature and v = jvj is the speed. The probability that a particular atom has speed between v and v + dv is then f ¹vº dv. Question 3: Plot the distribution of speed for sodium at 600 K. What is the most probable speed and most probable rms speed? What is the full width at half maximum (fwhm)? Question 4: Assume that the crucible has a 1 mm circular aperture and that the atomic beam is collimated with a second 1 mm aperture located 10 cm away as shown in fig. 2. What is the fwhm angular divergence of the beam and fwhm transverse velocity spread? We then place a laser in order to slow down the atoms emitted from the crucible. Question 5: Calculate the detuning required so that the maximum force is felt by those sodium atoms travelling at the most probably speed for a 600 K crucible. Question 6: What is the distribution of velocities after the atoms have gone through the laser slower? 5 4 Zeeman Slower Figure 3: Atomic beam source with a Zeeman slower which creates a magnetic field that varies along the direction of the beam of atoms.