* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Ultra-cold atoms - University of St Andrews
Survey
Document related concepts
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Chemical bond wikipedia , lookup
Atomic orbital wikipedia , lookup
Ferromagnetism wikipedia , lookup
Tight binding wikipedia , lookup
Rutherford backscattering spectrometry wikipedia , lookup
Electron configuration wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Ultrafast laser spectroscopy wikipedia , lookup
Hydrogen atom wikipedia , lookup
Laser pumping wikipedia , lookup
Transcript
Ultra-cold atoms •New direction in atomic physics, started about 20 years ago •A gas can be cooled from room temperature down to nK ! •Laser cooling, evaporative cooling NIST Why are cold atoms interesting? Wave-particle duality: v m λdeBroglie = h/mv Low T ( = low speed) wave-like behaviour becomes important Atom optics = optics for matter waves The motion of the atoms can be controlled using devices analogous to those in optics, e.g. mirrors, lenses, diffraction gratings, interferometers….. Applications Atom interferometry acceleration sensors Quantum computing Bose-Einstein condensation (BEC), first observed in 1995 Our plan How to achieve BEC in an atomic gas: • Laser cooling • Evaporative cooling Nobel prizes were awarded for laser cooling and BEC nλ3dB ≈ 1 N number n= V density Based on this intuitive picture we can derive an expression for the critical temperature Tc : Equipartition theorem Substituting p = h λdB 1 h2 3 = k BT 2 2m λdB 2 Substituting this in N (3mk BTC ) 3 / 2 ≈ V h3 p2 3 = k BT 2m 2 (m=atomic mass) we obtain: λdB h = 3mk BT “thermal de-Broglie wavelength” 1 N ≈ 3 we obtain: V λdB 2 N 1 h Tc ≈ 3 mk B V 23 A more rigorous calculation based on Bose-Einstein statistics leads to a result that only differs by a numerical factor: 2 h N Tc = 0.0839 mk B V 23 Below TC, we have a macroscopic occupation of the ground state given by: N BEC (T ) T N BEC (T ) = N 1 − Tc (see for instance Mandl, Statistical Physics) 3/ 2 N 0 TC T This macroscopic occupation of the ground state of a bosonic system is known as Bose-Einstein condensation. E.g.for an ultracold gas of rubidium atoms: N ≈ 10 20 m −3 V Tc = 390nK The first step towards achieving such a low temperature is laser cooling. Laser cooling (1) - Radiation pressure Transfer of momentum from a resonant laser beam to an atom A photon carries a momentum p photon = h / λ = k (k = 2π / λ ) Due to the law of conservation of momentum, when the photon is absorbed the atom receives a recoil momentum mvR = k (e.g. 23 Na : λ = 589nm vR = 3cm/s) To determine the force exerted by the laser beam, we need to know the rate at which the atom absorbs photons…. An estimate of this rate is the time the atom takes to decay back to the ground state, i.e. the lifetime τ of the excited state due to spontaneous emission: hυatom absorption of a photon atom acquires a recoil R in laser propagation direction de-excitation: spontaneous emission is isotropic and imparted momentum averages to 0 over many events result: net momentum transfer in laser propagation direction Can absorb many photons R big change in atom velocity Radiation pressure - force on atom: ∆p = mv R = h λ change of atomic momentum for an absorption/spontaneous emission event ∆p ∆p h F= = ≈ ∆t τ λτ lifetime τ of excited state due to spontaneous emission Laser cooling (2) How can radiation pressure be used for cooling? h υ atom v hυ L detuning δ = υ L − υ atom δ <0 An atom will be slowed down if it absorbs photons from a laser beam directed opposite to its velocity. For this to happen, the laser frequency υ L has to be below the atomic transition, i.e. the laser has to be “red detuned”. The condition for absorption of photons is that their Doppler shifted frequency in the atom’s frame is close to υ atom : υ observed L ≈ υ atom where υ observed L v v = υ L 1 + = (υ atom + δ ) 1 + c c Doppler effect υ atom v ≈ (υ atom + δ ) 1 + c We can solve this for δ: δ≈ υ atom v 1+ c − υ atom v − υ atom v c ≈ − υ atom = v c 1+ c can neglect if v << 1 c δ υ atom v =− c Let’s now add a second counterpropagating laser beam: atomic velocity v LASER υlaser < υ atom LASER υlaser < υ atom force F If an atom is moving to the right (as in the diagram) for the Doppler effect it will be closer to resonance with the laser on the right and further from resonance with laser on the left. If an atom is moving to the left, the opposite is true. Either way, the atom receives “kicks” opposite to its velocity, or in other words it experiences a damping force: F = − kv If v=0, F=0 because the absorption is the same from both beams. This scheme is known as Doppler cooling, as it relies on the Doppler effect. It can be extended to the 3D case simply by adding pairs of counterpropagating laser beams along orthogonal directions (“optical molasses”): The minimum temperature that can be achieved with Doppler cooling is determined by the random recoils imparted to the atoms during spontaneous emission. Although their average is 0 over many events, their standard deviation is not and this leads to diffusion in momentum space, analogous to Brownian motion. It is possible to show that: Tmin ≈ k Bτ E.g. for the sodium doublet τ=16ns, Tmin=400µK Other laser cooling mechanisms exist that can decrease the temperature even further, however they don’t achieve the critical temperature for BEC need evaporative cooling How to get BEC: • Trap a gas of atoms using laser cooling techniques • Turn off the cooling laser • Further cool the cloud with evaporative cooling until BEC is reached Magnetic trapping To implement evaporative cooling, we need to hold the cloud in a trap. Very often this is a magnetic trap. Such a trap relies on the (well known by now!) magnetic interaction in presence of a inhomogeneous magnetic field: B0 ( z ) from a set of coils z Vmag ( z ) = − µ j ⋅ B0 ( z ) = µ B m j g j B0 ( z ) dB0 ( z ) Force = − µ B m j g j dz like in the Stern-Gerlach experiment Case m j > 0 : B0 mj Vmag (z ) j µj B0 and µ j are anti - parallel z minimun stable trap atoms oscillate around the minimum of B-field Atoms in states with m j > 0 are low-field seekers. Case m j < 0 : B0 µj Vmag (z ) z j mj B0 and µ j are parallel Atoms are repelled from centre and move towards regions of high B-field Atoms in states with m j < 0 are high-field seekers. We can only trap atoms that are in the right magnetic levels! Evaporative cooling: • The velocity distribution in the trap is Maxwell-Boltzmann (as long as we are far in temperature from BEC). • Reduce trap depth faster atoms escape • If done slowly, the gas is always close to thermal equilibrium, and its temperature T is reduced trap depth T T’<T Achieving Bose-Einstein condensation Here evaporative cooling was stopped at three different temperatures. The sharp peak appearing at 230 nK is the onset of BEC: Rb-87 BEC NATOM ~ 10 5- 106 530 nK Tc = 230 nK 100 nK Oxford, 2001 These pictures represent the density profile of the atomic cloud… Detecting the atom cloud probe laser beam CCD array atom cloud shadow is imaged onto CCD The probe beam is in resonance with an atomic transition the cloud absorbs photons and produces a shadow in the probe beam. This shadow is detected by a CCD camera. We switch off the magnetic trap a short time before taking the image. The cloud has therefore expanded (hence the name “time of flight” image) due to the initial velocity distribution of the atoms. The condensed fraction is slower than the thermal atoms, hence it appears as a sharp peak in the image. Onset of BEC JILA, 1995 The atoms in a condensate share the same wavefunction, which is the ground state of the magnetic trap (essentially a simple harmonic oscillator potential). So it is a quantum object we are looking at with a simple camera! Interference of matter waves (MIT)