imported>Hutzler |
imported>Ichuang |
| Line 4: |
Line 4: |
| | material serves as an "appetizer" for the rigorous treatment which | | material serves as an "appetizer" for the rigorous treatment which |
| | comes later. | | comes later. |
| − | == The spontaneous light force ==
| |
| − | Consider a two-level atom with energy spacing <math>\omega_0</math>,
| |
| − | interacting with a single mode laser beam:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-twolevel.png|thumb|408px|none|]]
| |
| − | Let the laser intensity be <math>I</math>, and the interaction matrix element
| |
| − | between atom and light be <math>\hbar\omega_R = \vec{E}\cdot
| |
| − | | \langle 1|e\vec{r}|2{\rangle}|</math>, where <math>\omega_R</math> is known as the Rabi frequency,
| |
| − | <math>\vec{E}</math> is the electric field strength, and <math>e\vec{r}</math> is the dipole
| |
| − | moment of the atom.
| |
| − | It is useful to define a quantity known as the saturation intensity
| |
| − | <math>I_0</math> as the intensity of light at which the rabi frequency becomes
| |
| − | <math>\omega_R = \Gamma/\sqrt{2}</math>, where <math>\Gamma</math> is the spontaneous
| |
| − | emission rate (the natural decay rate of the atom from <math>|2{\rangle}</math> to
| |
| − | <math>|1{\rangle}</math>. This gives
| |
| − | :<math>
| |
| − | \frac{I}{I_0} = \frac{2\omega_R^2}{\Gamma^2}
| |
| − | \,.
| |
| − | </math>
| |
| − | The rate at which photons are scattered from the atom is known to be
| |
| − | :<math>
| |
| − | \gamma_s = \frac{\Gamma}{2} \frac{I/I_0}{1+\frac{I}{I_0} +
| |
| − | (2\delta/\Gamma)^2}
| |
| − | \,,
| |
| − |
| |
| − | </math>
| |
| − | where <math>\delta</math> is the frequency detuning of the laser from the center
| |
| − | of resonance of the atom. Two useful limits of this scattering rate
| |
| − | are
| |
| − | :<math>\begin{array}{rcl}
| |
| − | \lim_{I\rightarrow\infty} \gamma_s &=& \frac{\Gamma}{2}
| |
| − | = \frac{1}{2\tau}
| |
| − | \\
| |
| − | \lim_{I\rightarrow I_0} \gamma_s &=& \frac{\Gamma}{4}
| |
| − | \,,
| |
| − | \end{array}</math>
| |
| − | where we have assumed <math>\delta=0</math> (resonant light). These expressions
| |
| − | have a natural physical interpretation: in the limit of infinite
| |
| − | intensity, the atomic levels become equally populated between the
| |
| − | excited and ground state, and thus only half the atoms (the excited
| |
| − | ones) can scatter light. Thus, the scattering rate is <math>\Gamma/2</math> in
| |
| − | that limit.
| |
| − | Suppose the force imparted by light on the atom is given by the recoil
| |
| − | of photons spontaneously emitted from the atom. This force would then
| |
| − | be
| |
| − | :<math>
| |
| − | F = \hbar k \gamma_s
| |
| − | \,,
| |
| − | </math>
| |
| − | where <math>\hbar k</math> is the momentum of each photon. This expression makes
| |
| − | several assumptions: that is is the net momentum transfer in absorption,
| |
| − | that there is no "stimulated" force, and that <math>\hbar k/m k \ll
| |
| − | \Gamma</math>, meaning that the jump in the Doppler shift is less than the
| |
| − | natural linewidth.
| |
| − | Typically, for alkali atoms, this force is <math>F\leq 10^5 g</math> times the
| |
| − | mass of an atom (<math>g</math> being the acceleration due to gravity). This
| |
| − | means that light can stop a sodium atom going at <math>1000</math> m/s in one
| |
| − | millisecond, or about half a meter. In comparison to electrostatic
| |
| − | forces on ions, this is very small, however: it is comparable to the
| |
| − | force exerted by an electric field of 1 millivolt/cm on an ionized
| |
| − | sodium atom.
| |
| − | Moving atoms experience a Doppler shift, which we can model as a
| |
| − | frequency dependent force, based on Eq.(\ref{eq:ci:lorentzian}), as
| |
| − | :<math>
| |
| − | F = \hbar k \frac{\Gamma}{2} \frac{I/I_0}{1+\frac{I}{I_0} +
| |
| − | \left[\frac{2(\delta+kv)}{\Gamma}\right]^2}
| |
| − | \,,
| |
| − |
| |
| − | </math>
| |
| − | where <math>v</math> is the velocity of the atom. The effect of a fixed laser
| |
| − | frequency on an ensemble of atoms is to modify their Maxwell-Boltzmann
| |
| − | thermal velocity distribution:
| |
| − | ::[[Image:chapter1-intro-to-cooling-l1fig2.png|thumb|408px|none|]]
| |
| − | Note how the initial distribution changes to one with atoms piling up
| |
| − | below the velocity group resonant with the laser. The atoms bunch.
| |
| − | Historically, this is the first method that was done to cool atoms to
| |
| − | Kelvin temperatures.
| |
| | | | |
| − | == One-dimensional optical molasses ==
| + | {{Applications of the spontaneous light force}} |
| − | Let us now turn to a method which allows cooling of atoms to zero
| |
| − | velocity. Consider two laser beams incident on an atom from opposite
| |
| − | directions. We assume that the total force is the sum of the two
| |
| − | forces, ignore standing wave effects, and take the laser intensity to
| |
| − | be low compared with the saturation intensity, <math>I\ll I_0</math>. Taking the
| |
| − | force to be the sum of two forces described by Eq.(\ref{eq:ci:vdf}),
| |
| − | we find that the two lorentzians sum to give the following force as a
| |
| − | function of velocity:
| |
| − | ::[[Image:chapter1-intro-to-cooling-l1fig3.png|thumb|200px|none|]]
| |
| − | The velocity dependent force is positive from one light beam, and
| |
| − | negative from the other. With a detuning chosen such that force is
| |
| − | zero at zero velocity, the force around <math>v=0</math> can be expanded
| |
| − | linearly, giving
| |
| − | :<math>
| |
| − | F(v) = -\alpha v
| |
| − | \,,
| |
| − | </math>
| |
| − | where <math>\alpha</math> describes the viscosity imparted by the light
| |
| − | force to the atom, reflecting the restoring force applied when the
| |
| − | atom is not at zero velocity. This configuration is known as an {\em
| |
| − | optical molasses}, because of this restoring force, which makes the
| |
| − | light behave like a thick, viscous medium for the atoms in it.
| |
| − | The damping coefficient <math>\alpha</math> can be calculated to be
| |
| − | :<math>
| |
| − | \alpha = 2\hbar k^2
| |
| − | \frac{(2I/I_0)(2\delta/\Gamma)}{\left(1+\left(\frac{2\delta}{\Gamma}\right)^2\right)^2}
| |
| − | \,.
| |
| − | </math>
| |
| − | == The Doppler cooling limit ==
| |
| − | We have seen that the spontaneous light force, characterized by the
| |
| − | Lorentzian response of an atom to light, together with the Doppler
| |
| − | shift due to movement of the atom, gives a velocity dependent force,
| |
| − | which can be zero at zero velocity. Does this mean that the atoms can
| |
| − | be cooled to zero temperature?
| |
| − | If the rate of energy loss due to cooling is
| |
| − | :<math>
| |
| − | \dot{E}_{\rm cool} = F v = -\alpha v^2 = -\frac{2\alpha}{M} E
| |
| − | \,,
| |
| − | </math>
| |
| − | then we should reach zero velocity, and zero temperature. Indeed, the
| |
| − | kinetic energy decays exponentially. However, the spontaneous force
| |
| − | has a random character, and thus has fluctuations which limit the
| |
| − | minimum temperature achievable.
| |
| − | This limit is determined by momentum diffusion. The force imparted
| |
| − | can be described by a random walk. The final momentum is
| |
| − | :<math>
| |
| − | p_{\rm final}^{RMS} = \hbar k \sqrt{N}
| |
| − | \,,
| |
| − | </math>
| |
| − | on average, due to the random walk. Note that the momentum spread is
| |
| − | :<math>\begin{array}{rcl}
| |
| − | \langle p^2 \rangle &=& (\hbar k)^2 v
| |
| − | \\
| |
| − | \frac{d \langle p^2{\rangle}}{dt} &=& (\hbar k)^2 \gamma_s
| |
| − | \,.
| |
| − | \end{array}</math>
| |
| − | This describes heating which arises due to photons randomly scattering
| |
| − | in all directions, such that the net momentum almost adds up to zero,
| |
| − | but not quite. There is also a similar term due to absorption: some
| |
| − | atoms will absorb more or less photons, due to the Poissonian
| |
| − | statistics of absorption.
| |
| − | Thus, the time variation of the kinetic energy due to the fluctuating
| |
| − | forces is
| |
| − | :<math>
| |
| − | \dot{E}_{\rm heat} = \frac{2\hbar^2 k^2\gamma_s}{2M} = \frac{D}{M}
| |
| − | \,,
| |
| − | </math>
| |
| − | where <math>D</math> is the momentum diffusion coefficient
| |
| − | :<math>
| |
| − | D = \frac{ \langle p^2{\rangle}}{2}
| |
| − | \,,
| |
| − | </math>
| |
| − | which we'll later see is a correlation function of the fluctuation
| |
| − | forces.
| |
| − | Let us now derive the Doppler limit for cooling. In equilibrium,
| |
| − | <math>\dot{E}_{heat} = \dot{E}_{cool}</math>. This means
| |
| − | :<math>
| |
| − | \frac{D}{M} = \frac{2\alpha}{M}E
| |
| − | \,.
| |
| − | </math>
| |
| − | The heating rate is independent of kinetic energy, whereas the cooling
| |
| − | rate is a function of kinetic energy. So as the atoms cool down, the
| |
| − | cooling rate slows down, resulting in a final temperature equilibrium
| |
| − | being reached:
| |
| − | :<math>
| |
| − | 2E_{final} = kT_{\rm doppler} = \frac{D}{\alpha}
| |
| − | </math>
| |
| − | <math>\alpha</math> is a viscosity parameter: it reflects transport. <math>D</math>
| |
| − | reflects mobility. Thus, this is an Einstein relation, a universal
| |
| − | expression in statistical mechanics resulting from the fundamental
| |
| − | theorem which relates dissipation to fluctuations.
| |
| − | We've now obtained an expression for the Doppler limit temperature, a
| |
| − | limit on the temperature an ideal two-level atom can be cooled to by
| |
| − | laser beams,
| |
| − | :<math>
| |
| − | k T_{\rm doppler} = \frac{\hbar\Gamma}{2}
| |
| − | \,.
| |
| − | </math>
| |
| − | This optimal temperature is achieved for <math>I\ll I_0</math>, and detuning of
| |
| − | <math>\delta = -\Gamma/2</math> (half a linewidth). Physically, at low
| |
| − | temperatures, the atom cannot determine whether the photon comes from
| |
| − | left or right; at higher temperatures, the atom can discriminate
| |
| − | whether photons come from left or right, thus cooling. For sodium,
| |
| − | this temperature is <math>240</math> <math>\mu</math>K, corresponding to a velocity of <math>30</math>
| |
| − | cm/s.
| |
| − | == Beam slowing ==
| |
| − | Let us now go back to slowing and atomic beams, in which we have many
| |
| − | atoms and only one laser beam. If we have a Maxwell-Boltzmann
| |
| − | distribution with atoms moving counterpropagating to the laser, the
| |
| − | atoms' velocity distribution is pushed to lower velocities:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-beam-slowing.png|thumb|612px|none|]]
| |
| − | Note that in the Doppler cooling picture, we can only talk to atoms
| |
| − | which are already pretty slow. From an atomic beam, however, there
| |
| − | are hardly any atoms within this capture velocity range. It would be
| |
| − | much nicer to collect atoms from the whole MB distribution; this is
| |
| − | called beam slowing. Two stages are typically used: beam slowing,
| |
| − | then Doppler cooling.
| |
| − | We studied Doppler cooling first, because it was useful to get the
| |
| − | concepts of the friction force and momentum diffusion.
| |
| − | Beam slowing uses only one laser beam, but involves an extra twist in
| |
| − | that we must first transform into a decelerating frame. Conceptually,
| |
| − | the idea is to change the frequency of light applied as the atoms slow
| |
| − | down, so that atoms continually experience a negative force. One way
| |
| − | to accomplish this is to use multiple laser frequencies (white light
| |
| − | slowing). Another is "diffuse light slowing" which uses light
| |
| − | covering a large spectrum. Most efficient is to use a single
| |
| − | frequency, and change the detuning of atoms with a magnetic field (re
| |
| − | homework). We'll discuss "chirped slowing," because of its
| |
| − | conceptual simplicity. It has little experimental use today, but was
| |
| − | important in the mid 90's.
| |
| − | === Chirped slowing ===
| |
| − | The force on an atom in the beam due to the light is
| |
| − | :<math>
| |
| − | F = -\hbar k \frac{\Gamma}{2} \frac{I/I_0}{1+I/I_0 + \left[
| |
| − | \frac{2(\delta+k v)}{\Gamma} \right]^2}
| |
| − | </math>
| |
| − | Let us assume a frame of reference and experimental setup such that
| |
| − | <math>a<0</math>, <math>v>0</math>, <math>k>0</math>, <math>a_{max}>0</math>. We can call <math>\hbar k \Gamma/2 = M
| |
| − | a_{max}</math>, where <math>M</math> is the atom's mass.
| |
| − | The scheme begins by selecting the deceleration desired, some <math>a<0</math>.
| |
| − | Then set <math>F = -Ma</math>, and look for a <math>\delta' = \delta + kv</math> to obtain
| |
| − | this desired force. This will exist if
| |
| − | :<math>
| |
| − | |a| < \frac{I/I_0}{1+I/I_0} a_max
| |
| − | \,.
| |
| − | </math>
| |
| − | Next, select an initial velocity <math>v_0</math> such that <math>v(t) = v_0 + a t</math>.
| |
| − | <math>\delta'</math> is the detuning for this "targeted" velocity group, so we
| |
| − | must provide a laser with frequency in the lab frame of <math>\delta(t) =
| |
| − | \delta' - k v(t)</math>. The atom's velocity will differ from the desired
| |
| − | target group by <math>v' = v-v(t)</math>. With these definitions, we now have
| |
| − | :<math>
| |
| − | F = -\hbar k \frac{\Gamma}{2} \frac{I/I_0}{1+I/I_0 + \left[
| |
| − | \frac{2(\delta'+k v')}{\Gamma} \right]^2}
| |
| − | \,,
| |
| − | </math>
| |
| − | in the frame of reference of the atoms in the target velocity group.
| |
| − | Transforming into this decelerating frame, we get a fictitious force
| |
| − | with is <math>F_{fict} = -Ma</math>, and
| |
| − | :<math>
| |
| − | F'(v') = M a_{max} \left[
| |
| − | \frac{I/I_0}{1+I/I_0 + \left[
| |
| − | \frac{2(\delta'+k v')}{\Gamma} \right]^2 }
| |
| − | + \frac{I/I_0}{1+I/I_0 + \left[
| |
| − | \frac{2(\delta')}{\Gamma} \right]^2 }
| |
| − | \right]
| |
| − | </math>
| |
| − | This second term has the same structure as the first, but it is
| |
| − | velocity independent. All we've done is to substitute definitions, so
| |
| − | far, but they provide useful intuition.
| |
| − | In the lab frame, we have a force which is a positive Lorentzian. In
| |
| − | the decelerating frame, we had to add <math>-Ma</math>, so this Lorentzian shifts down:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-beam-slowing-lock.png|thumb|510px|none|]]
| |
| − | Therefore, there is now a stable "lock" point, where <math>F=0</math> as a
| |
| − | function of <math>v'</math>. Thus, we may write, as we did with the molasses, an
| |
| − | expression for the linearized force around this point, <math>F(v') =
| |
| − | -\alpha v'</math>, in which <math>\alpha_{beam} =
| |
| − | \alpha_{molasses}/2</math>. We may also calculate a momentum diffusion
| |
| − | coefficient, and we'd find that <math>D_{beam} = D_{molasses}/2</math>, so that
| |
| − | the final temperature limit of the beam is actually the same as that
| |
| − | achievable with a molasses: <math>kT_{beam} = kT_{molasses}</math>.
| |
| − | We've seen that one laser can bunch up atoms from a beam at a single
| |
| − | velocity. Physically, what happens is that if the atoms fall behind,
| |
| − | the light does not interact with them, but if the atoms are too fast,
| |
| − | the laser cools them, much like in the molasses case.
| |
| − | Here is a graphical summary of what we've learned about beam slowing.
| |
| − | In the decelerating frame, this is the situation. Change sign, so
| |
| − | that in the frame the decelerating force is positive, for this graph:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-bs-velocity.png|thumb|408px|none|]]
| |
| − | Initially, our zero force point is at the targeted velocity <math>v_0</math>.
| |
| − | All atoms at larger velocity experience a constant positive force,
| |
| − | accelerating them. After a certain time <math>t</math>, the tail of the
| |
| − | maxwell-Boltzmann distribution is pushed to higher velocities:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-bs-lab-frame.png|thumb|408px|none|]]
| |
| − | similarly, all the atoms at lower velocities are pushed up in velocity
| |
| − | until they stack up at <math>v_0</math>, producing a narrow distribution around
| |
| − | <math>v_0</math>. The width of this narrow velocity distribution is proportional to
| |
| − | <math>\sqrt{kT_{beam}}/M</math>.
| |
| − | Beam cooling is actually the simplest and cleanest example of laser
| |
| − | cooling, because in the two-beam molasses case, one should really
| |
| − | consider interference effects.
| |
| − | == Energy versus momentum picture ==
| |
| − | === Energy ===
| |
| − | Where did the energy go, in cooling the atoms? The energy was
| |
| − | radiated away by spontaneous emission, as we shall now see.
| |
| − | Light emitted by the atom is at the resonant energy <math>\omega_0</math>, but
| |
| − | can be absorbed when the photon is just slightly less than <math>\omega_0</math>.
| |
| − | The emission is isotropic, whereas the incident light is directed and
| |
| − | Doppler shift dependent.
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-energy.png|thumb|204px|none|]]
| |
| − | Doppler cooling can be explained in this picture. Laser light is
| |
| − | detuned below <math>\omega_0</math> ("red detuned").
| |
| − | The same intuition can be applied to solids and liquids. Phonon
| |
| − | assisted absorption is balanced against emission, resulting in cooling:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-cooling-solid.png|thumb|408px|none|]]
| |
| − | How hard is it to cool liquids and solids? Consider a system at
| |
| − | <math>T=1</math>K; that gives the phonon energy. Then <math>\hbar\omega = 25,000</math>
| |
| − | Kelvin. In practice, there is a lower than unity fluorescence quantum
| |
| − | yield, because there are non-radiative ways to exit the excited
| |
| − | state. The cooling will be efficient, however, only when the quantum
| |
| − | yield is higher than <math>1-1/25000</math>, which is typically unrealistic.
| |
| − | Cooling with laser light is therefore not typically practical, for
| |
| − | systems other than atoms, which have a unity fluorescence quantum
| |
| − | yield. Molecules are hard, because they have non-radiative
| |
| − | de-excitation pathways.
| |
| − | == 3D molasses, high intensities ==
| |
| − | To cool atoms along not just one axis, but along three axes, use six
| |
| − | counter-propagating laser beams. This configuration is called a 3D
| |
| − | molasses. Everything we've discussed in one dimension can be applied;
| |
| − | just sum up the forces. Some care must be taken, however, if
| |
| − | interference patterns are created between the beams. As long as the
| |
| − | atoms move a distance greater than the wavelength, interference may be
| |
| − | neglected. But large field gradients can add extra forces and
| |
| − | heating.
| |
| − | One can also alternate between the six beams, but having simultaneous
| |
| − | beams actually turns out to be good; it gives polarization gradients
| |
| − | and other subtle effects which provide extra cooling.
| |
| − | Landmark: in 1985, Steve Chu used chirped slowing and a 3D molasses
| |
| − | configuration to obtain atoms colder than <math>1</math> mK, for the first time.
| |
| − | === Cooling at high intensities ===
| |
| − | Keep in mind that laser cooling works because <math>F=-\alpha v</math> and
| |
| − | <math>\alpha>0</math>. Assume we have a detuning of about one linewith,
| |
| − | <math>\delta=-\Gamma</math>. Now plot <math>\alpha</math> as a function of intensity:
| |
| − | ::[[Image:chapter1-intro-to-cooling-Lec1-high-inten.png|thumb|408px|none|]]
| |
| − | First, <math>\alpha</math> increases as a function of intensity. Don't be
| |
| − | confused by the fact that the Doppler limit is achieved at low
| |
| − | intensities. The diffusion coefficient is also linear in intensity at
| |
| − | low intensity. <math>\alpha</math> increases with <math>I/I_0</math> at first, and peaks
| |
| − | around <math>0.5</math>, but above the saturation limit <math>\alpha</math> actually changes
| |
| − | sign and starts heating. When <math>\alpha<0</math> then, counter-intuitively,
| |
| − | blue detuned light can be used to cool atoms. This is a
| |
| − | non-trivial result, which we'll return to later, and understand in the
| |
| − | context of the optical Bloch equation and the dressed atom model.
| |
| − | == Momentum and spatial diffusion ==
| |
| − | === Momentum diffusion ===
| |
| − | First, consider diffusion of the momentum of an atom being cooled.
| |
| − | The momentum diffusion coefficient is defined as
| |
| − | :<math>\begin{array}{rcl}
| |
| − | 2D^p = \frac{d}{dt} \langle (p- \langle p{\rangle}^2)^2{\rangle}
| |
| − | \,.
| |
| − | \end{array}</math>
| |
| − | This can be directly calculated if we have a fluctuating force, using
| |
| − | the fact that <math>d\vec{p}/dt = \vec{f}</math> is a force:
| |
| − | :<math>\begin{array}{rcl}
| |
| − | 2D^p &=& \frac{d}{dt} \left[
| |
| − | {\langle}\vec{p}\cdot\vec{p}{\rangle}-{\langle}\vec{p}{\rangle}{\langle}\vec{p} \rangle \right]
| |
| − | \\
| |
| − | &=& 2 \left[
| |
| − | {\langle}\vec{p}\cdot\vec{f}{\rangle}-{\langle}\vec{p}{\rangle}{\langle}\vec{f} \rangle \right]
| |
| − | \\ &=& 2 \int^0_{-\infty}
| |
| − | {\langle}\vec{f}(0) \cdot \vec{f}(t) \rangle - \langle \vec{f}(0) {\rangle}{\langle}
| |
| − | \vec{f}(t) \rangle \, dt
| |
| − | \,,
| |
| − | \end{array}</math>
| |
| − | showing that the diffusion is given by the integral of the force-force
| |
| − | correlation function. Essentially:
| |
| − | :<math>\begin{array}{rcl}
| |
| − | 2D^p
| |
| − | &=& 2 \int^0_{-\infty} \langle \vec{f}(t) \cdot \vec{f}(0) \rangle \, dt
| |
| − | \,.
| |
| − | \end{array}</math>
| |
| − | This results due to the fluctuation-dissipation theorem.
| |
| − | === Spatial diffusion ===
| |
| − | This is less frequently discussed in the literature compared with
| |
| − | momentum diffusion, but it is of practical importance in
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| − | experiments. Suppose the atoms start in a single point, embedded in a
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| − | 3D optical molasses. How does the point distribution expand? On the
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| − | time scale determined by <math>\alpha</math>, the atoms loose their memory of
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| − | their original velocities. The molasses has a nearly perfect thermal
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| − | distribution, despite atoms in the cloud never interacting with each
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| − | other, because they thermalize to the laser beam.
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| − | The damping time is
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| − | :<math>
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| − | \frac{1}{\gamma} = \frac{M}{\alpha}
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| − | \,.
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| − | </math>
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| − | Spatial diffusion can be described by a random walk (in space), with a
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| − | step size <math>L</math> given by the RMS velocity of the atoms and the damping
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| − | time,
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| − | :<math>
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| − | \ell = 2 \frac{v_{rms}}{\gamma}
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| − | \,,
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| − | </math>
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| − | where the extra factor of <math>2</math> comes from a more rigorous treatment.
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| − | Thus, starting from a point distribution, by the standard random walk
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| − | result, after time <math>t_d</math>, we obtain
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| − | :<math>
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| − | \langle r^2 \rangle = 2 \ell^2 t_d \gamma
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| − | \,,
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| − | </math>
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| − | where the number of steps is <math>t_d\gamma</math>. This is
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| − | :<math>\begin{array}{rcl}
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| − | \langle r^2 \rangle &=& 2 \frac{v_{rms}^2}{\gamma} t_d
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| − | \\
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| − | \frac{D^p t_d}{\alpha^2}
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| − | \,.
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| − | \end{array}</math>
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| − | Now recall the definition
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| − | :<math>
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| − | \langle r^2 \rangle = 2 D^x t_d
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| − | \,,
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| − | </math>
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| − | where <math>D^x</math> is the spatial diffusion coefficient.
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| − | This gives a relation between the spatial and momentum diffusion
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| − | coefficients,
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| − | :<math>
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| − | D^x = \frac{D^p}{\alpha^2} = \frac{KT}{\alpha}
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| − | \,.
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| − | </math>
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| − | Note the similarity of this expression with the Einstein relation for
| |
| − | carriers in semiconductors, <math>D/\mu = kT/q</math>.
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| − | These expressions are useful in the laboratory context, as an example
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| − | illustrates. How long does a typical trapped alkali atom (eg cesium
| |
| − | or sodium) take to diffuse out by <math>0.5</math> cm at the Doppler temperature?
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| − | Using the formulas above, we get <math>t_d = 1</math> second. This is very
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| − | accessible in the laboratory, and is one of the reasons why optical
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| − | molasses are so useful in practice.
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