Dipole force and dissipation

The study of light forces can help shape intuition about atoms. Understanding them requires careful consideration of complimentarity and ???.

In this section, we'll also look at dipole forces and the dressed atom picture; these overlap with the optical Bloch equations, and provide considerable simplification for an otherwise complex and important case.

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► Category:Quantum Light

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► Category:Photon-atom interactions

  Introduction to Atomic Physics

  Quantum light: states and dynamics

  Optical Bloch equations

  Photon-atom interactions

  Frontiers in Atomic Physics

  Ion traps and quantum information

  Quantum gases

Summary of light forces

To review we have found that an atom experiences two kinds of light forces:

${\begin{array}{rcl}F_{diss}&=&-\hbar \Omega v_{st}\nabla \phi \\F_{react}&=&-\hbar u_{st}\nabla \Omega \end{array}}$

The dissipative force applies in the traveling wave configuration, and the reactive force applies in the standing wave configuration. We use this force as a friction,

$F=F_{0}-\alpha v$

where $v$ in the above is the velocity of the atom, and $\alpha$ is a fiction coefficient from the above light forces. For the reactive force, in a standing wave, there are important new physics: the friction coefficient $\alpha$ flips sign for high intensities. This happens near resonance at saturation parameter $s\geq 1$, but for $\delta \gg \Gamma$, one needs $s\sim \delta /\Gamma$.

Steady state energy exchange

Think about this: how do we get damping from the reactive force? If you are in-phase with a driving field, you just get unitary time evolution; the $u_{st}$ quadrature cannot absorb energy from the atom -- where does energy from laser cooling go? The dissipative force is a spontaneous force, and that makes sense; the reactive force is a stimulated force, which involves redistribution of photons, from one mode to another -- how can that cool? One aspect of the answer to this question is that there is a lag time on the timescale and wavelength scale of the standing wave. This is certainly part of the underlying physics. The important thing to keep in mind, in answering this question, is that there will still be spontaneous emission when an atom is in a standing wave. The force comes from $u_{st}$, which does not involve energy exchange; but the $v_{st}$ component still contributes to the process by exchanging entropy, through spontaneous emission. The $v_{st}$ component is thus essential to the energy and entropy balance of the system. Consider the following gedankenexperiment. We have a standing wave, and we pass a small beam of atoms moving with velocity $v$. Assume that $\alpha >0$, so the atoms loose energy. Where has this energy gone?

File:Dipole force and dissipation-lf-gedanken

The atom can only radiate the energ away. What is the spectrum? If the atom is driven by a strong beam, the spectrum is the Mollow triplet. The doppler shift in the intense standing wave is small, so that source of energy exchange is negligible. The centeral line stays at the same frequency, so that is not the source. The answer must be that the atoms radiate more on the blue sideband than on the red sideband; the sidebands on the Molllow triplet must therefore be asymmetric. We shall see this in detail later, in the dressed atom picture.

Transient energy exchange

The above is the CW picture; the energy is radiated away. Let us now consider a transient scenario.

Consider an optical lattice, in which atoms are captured at positions offset from the bottom of the optical potentials, as implemented in Bill Phillips' group at NIST:

File:Dipole force and dissipation-lf-lattice

Where does the force come from, and where does the energy go to? In the experiment, the two optical light beams were coupled to photodiodes, so they could be monitored. What was found was that the intensities of the two light beams oscillated, reflecting in a sense how the mechanical motion of the atoms comes from redistribution of photons from one beam to the other. This explains the force, but because the photons have the same energy, it does not explain the energy. To explain the energy transfer, we need to reconcile the above picture with the fact that the atoms' harmonic energies are very small, less than $0.1meV$? If all the atoms arrive at the bottom of their wells, and they have more energy, where does that kinetic energy come from? The fact is that sometimes it is clear where the forces come from, and sometimes it is clear where the energy transfer arises from, but this takes some thinking. Suppose the above experiment is done with far off-resonant light. Now, the explanation of asymmetric sidbands does not help us here. Consider a compass needle with an electromagnet. If you switch off the coil after driving the needle, the needle jumps. Where does the energy come from? Clearly, it comes from the electromagnetic building up the magnetic field, so the source is the power supply to the coil. Similarly, for the atom in a far-detuned lattice, the energy comes from the laser beam. The atoms are like beads -- dielectric media -- in the beam, and their energy is different whether they are in the intensity minimum or maximum. The explaination for energy transfer is really just a classical process. Obtaining a quantum description of these process, on the other hand, involves some complication, which we shall not go into here.

Applications of the optical dipole potential

Recall that we derived a potential description from which light forces can be found, known as the optical dipole potential,

$U={\frac {\hbar \delta }{2}}\log \left[{1+{\frac {I/I_{0}}{1+(2\delta /\Gamma )^{2}}}}\right]\,.$

Three applications of these are

• Localization, in optical lattices
• Atom mirrors (homework set)
• Dipole traps and optical tweezers

Localization in optical lattices

The history of this subject is interesting. In a standing wave, it was realized in the 70's and 80's that atoms experience a periodic potential. Atoms from a gas, exposed to this potential would give a temperature $k_{B}T\geq U_{0}$. In 1987, Salomon et al at the ENS, instead of doing a localization experiment, they illuminated a standing wave with a collimated atomic beam. They saw that the washboard potential gave rise to channeling -- the atoms moved preferrentially where the potential was minimum. In the late 80's and early 90's, the study of multilevel atoms lead to observation of polarization gradient cooling, such that $k_{B}T\leq U_{0}$ could be obtained. Localization of $\sim 60\%$ of the atoms in a lattice was obtained, and observed using discreteness of the Raman spectrum. Today, it has been common to use not just one, but also two and three dimensional standing waves. These are known as optical lattices. Cooling is now often done not (just) using laser forces, but also with evaporative cooling, which allows much lower temperatures to be obtained. $nK$ atoms now can be adiabatically loaded into a lattice, with very well localized distributions.

Dipole traps and optical tweezers

Two-level atoms will move to the intensity maximum of focused, red-detuned light:

File:Dipole force and dissipation-lf-dipole-trap

This process can be understood by using (1) the optical Bloch equation, and the optical dipole potential, (2) the dressed atom picture (as we shall see in the next section), or (3) classically using a picture of a dipole driven by a harmonic oscillator. In the last picture, with $W=-\mu _{u}E$. A fourth picture uses macroscopic objects for intuition (cf Art Ashkin). Consider a glass sphere in an inhomogeneous (eg Gaussian) beam, where the beam hits the glass sphere off-center. Let the index of the glass sphere $n>1$, so the sphere acts like a prism. The ray of the light is thus deflected:

File:Dipole force and dissipation-lf-glass-bead

\noindent Since, in this pictue, the intensity deflected upwards is higher than downwards, the sphere is thus sucked into the focus of the beam. If $n<1$, then in the same situtation, the strong beam is deflected in the opposite direction, so that the macroscopic force would push the bead away from the focus. This scenario captures the main physics of two-level atoms in a focused light beam, and indicates that such light forces can apply to more macroscopic objects. Experiments have been done using focused light beams to move biological molecules. These have even been used to move around objects such as bacteria inside living cells. Focused light beams can also be used to levitate objects, such as liquid fuel droplets. Such experiments have been used to study combustion. Dipole forces have also been used to levitate liquid helium droplets, to study vortices and superfluid transitions.

Dipole traps for atoms

When the detuning $\delta$ is very large, then the logarithm in the optical dipole potential can be expanded. Such a configuration is known as a FORT (far off-resonant trap), and are important today because weak dipole forces are useful when atoms are initially sufficiently cold. The potential obtained in this limit is

$U={\frac {\hbar \Omega ^{2}}{4\delta }}\,.$

When atoms are not in equilibrium it is desirable to avoid spontaneous emission. The spontaneous emission rate is

${\begin{array}{rcl}\gamma &=&\Gamma \sigma _{bb}\\&=&{\frac {\Gamma }{4}}\left({\frac {\Omega }{\delta }}\right)^{2}\,.\end{array}}$

This depends differently on $\delta$ than $U$; explicitly,

$\gamma \sim {\frac {\Gamma U}{\delta }}\,.$

It is thus advantageous to use very far off-resonant light for trapping; experiments have used one to hundred Watt lasers, with 1.06 $\mu$m YAG or 10.6 $\mu$m CO$_{2}$ lasers being common and popular for this application. Spontaneous emission lifetimes of seconds to tens of seconds have been obtained in such experiments.

Electromagnetic origin of dipole forces

What the microscopic, electriomagnetic origin of the optical dipole force we've been studying? Is it an electrical $qE$ dipole force, or a magnetic $qv\times B/c$ Lorentz force? The answer is that it depends. If you have a standing wave, and an atom inside it moves up and down on the potential, the force is a purely magnetic force. However, the force brining atoms into an intense light focus is an electrical force. Consider an atom, described as a dipole, in an inhomogenous field. Let us use DC polarizabilities, for simplicity. The atom moves inward because the force difference on the positive and negative sides moves it inward:

File:Dipole force and dissipation-lf-dipole-energy

The force is

$F=q(\nabla E)\cdot d$

which is due to $qd$, the dipole moment. The work done is $p\cdot E$. The atom is an induced moment $\alpha E$, so the electrostatic energy is $\alpha E^{2}/2$. The same thing applies when you have a standing wave laser beam in a cavity focus, and the atom moves into the focus from far away. Consideraiton of this picture gives the optical dipole potential previously derived. On the other hand, consider an atom in an infinite standing wave; let us approximate this with an extreme case, with the electric field being extremely strong:

File:Dipole force and dissipation-lf-sw-magnetic

The optical potential we should derive from this picture is $\alpha E^{2}/2$, but the origin for this is not electric. The electric force is zero in this situation, because the electric field along the direction perpendicular to the standing wave is constant. Thus, the positive and negative charges of the atomic dipole see the same field, leading to no net electric force perpendicular to the standing wave. In the same situation, the Loretnz force is

$q{\frac {v}{c}}\times B\rightarrow {\frac {p}{c}}\times B\,.$

In the standing wave,

${\begin{array}{rcl}{\vec {B}}&=&-A_{0}\sin kz\cos \omega t{\hat {e}}^{y}\\{\vec {E}}&=&A_{0}\cos kz\sin \omega t{\hat {e}}_{x}\end{array}}$

The Lorentz force is thus

${\frac {1}{c}}\alpha A_{0}\cos kz(\omega \cos \omega t){\hat {e}}_{z}(-A_{0}\sin kz\cos \omega t)\,.$

Avergaing over one period gives

${\begin{array}{rcl}{\bar {F}}&=&-{\frac {k}{2}}A_{0}^{2}\cos kz\,\sin kz\\&=&{\frac {d}{dz}}\left[{{\frac {1}{4}}\alpha A_{0}^{2}\cos ^{2}kz}\right]\end{array}}$

We can identify the term in the brackets as the average electric field $E^{2}$ over a cycle of oscillation.