Abstract
The first stage of producing an ultracold atomic gas is to vaporize a solid (with an “oven” or other source) to create a hot gas, whose center of mass is then slowed sufficiently to be trapped. The slowing is frequently accomplished by a “Zeeman slower”: a series of current carrying coils that generate magnetic fields, and which decrease in intensity over the length of the slower, to create a magnetic field gradient. Atoms used in this technique possess an electron spin degree of freedom so that the internal energy of the atom depends on the applied magnetic field and thus the magnetic field gradient introduces a potential energy gradient that slows the atoms.
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Notes
- 1.
Stark decelerators—very roughly speaking replacing magnetic fields with electric fields—are another option when the particles being decelerated possess an electric dipole moment, often used with molecules
- 2.
I will go between setting \(\hbar\) and \(k_B\) to one and showing them explicitly throughout this thesis.
- 3.
Note that to get the new Hamiltonian governing the system in the transformed basis, we need to ensure that its solution give the same equations of motion (EOM) in the original basis as the original Hamiltonian (this is inequivalent to simply substituting the definition of the new basis states in the old Hamiltonian). The EOM of \({\vert{g}\rangle}\) is unchanged. The EOM for \({\vert{e}\rangle}\) is, from the original Hamiltonian,
$$ i\partial_t {\vert{e}\rangle} = H{\vert{e}\rangle} = E_e^{(0)} {\vert{e}\rangle} + E_0 \cos(\omega t){\vert{g}\rangle}. $$(2.9)Substituting the definition of \({\vert{\tilde e}\rangle}\), we have
$$ \begin{aligned} i\partial_t e^{-i\omega t}{\vert{\tilde e}\rangle} = H{\vert{\tilde e}\rangle} = E_e^{(0)} e^{-i\omega t}{\vert{\tilde e}\rangle}+E_0 \cos(\omega t){\vert{g}\rangle}\\ \Rightarrow i\left [e^{-i\omega t}\partial_t {\vert{\tilde e}\rangle} -i\omega e^{-i\omega t}{\vert{\tilde e}\rangle} \right ] +{\frac{{E_0}}{{2}}} \left ( e^{i\omega t} + e^{-i\omega t}\right ) {\vert{g}\rangle} = E_e^{(0)} e^{-i\omega t}{\vert{\tilde e}\rangle}. \end{aligned} $$(2.10)Thus
$$ i \partial {\vert{\tilde e}\rangle}+{\frac{{E_0}}{{2}}}\left ( e^{2i\omega t} + 1\right ) {\vert{g}\rangle} = (E_e^{(0)}-\omega){\vert{\tilde e}\rangle}, $$(2.11)exactly the EOM obtained from the new Hamiltonian given in Eq. 2.12.
- 4.
Had we directly done an appropriate time-dependent perturbation theory, we could have done this directly and bypassed the rotating wave approximation—it would have emerged automatically.
- 5.
This is another reason for the prominence of time-of-flight imaging, since it expands the imaged cloud’s size.
- 6.
The pixel size is slightly smaller. At the moment, other imaging elements limit the resolution, but may be removed if one can properly account for the point-spread function of the optics.
- 7.
This criterion follows from a simple argument: basically by dimensional analysis (assuming that the interaction energy is proportional to \(a\), motivated later), the kinetic energy per particle is \(E_k \sim n^{2/3} \) and the interaction energy per particle is \(E_i \sim a n\). Requiring \(E_i \ll E_k\) gives \(na^3 \ll 1\).
- 8.
A warning about notation is in order. Ostensibly, this argument would preclude experiments from studying many-body physics in the Bose–Hubbard model, since these experiments are in a regime \(na^3 \ll 1\)—in fact, this is a necessary criterion for the applicability of the Bose–Hubbard model to describe optical lattice experiments (see Sect. 2.2.3). However, the “\(a\)” appearing in the text’s criterion is the scattering length for two particle problem in whatever external potential is applied, including any optical lattice. The effective scattering length in a lattice—call it \(a_L\)—is much larger than the free space \(a\). Indeed, for strong lattices is on the order of the lattice spacing \(d\). Thus, the scattering length \(a_L\) describing two low energy particles in a lattice is comparable to the density in the Bose–Hubbard regime as one approaches the Mott insulator, and one is indeed in a strongly correlated regime.
- 9.
This depends somewhat on particle statistics and conventions; the result is given for bosons. For single particle scattering from a potential or distinguishable particles, one obtains \(\sigma=4\pi a^2\).
- 10.
To see this, rewrite the Schrödinger equation as \((E-H_0 ){\vert{\psi}\rangle}=V{\vert{\psi}\rangle}\). Then an equation of the form \(\hat L {\vert{\psi}\rangle}= {\vert{\phi}\rangle}\) with \(\hat L\) a linear operator has the general solution \({\vert{\psi}\rangle}={\vert{\psi_0}\rangle}+\hat G_0 {\vert{\phi}\rangle} \) where \(\hat G_0\equiv {\hat L}^{-1}\) is the Green’s function associated with \(\hat L\) and \({\vert{\psi_0}\rangle}\) is the general solution of \(\hat L {\vert{\psi_0}\rangle}=0\). [[This is a generalization of the usual statement for differential equations, that (1) the general solution is the general solution of the homogeneous equation plus any particular solution of the inhomogeneous equation and (2) an inhomogeneous solution is given by summing over the Green’s functions times the source function.]]
- 11.
There are other, equivalent, definitions of the \(T\)-matrix, but this has relevant physical content: it allows one to obtain exact quantities by using the \(T\)-matrix acting on the unperturbed wavefunction.
- 12.
The definition of “long ranged" is dependent on context: whether a given potential counts as long ranged depends on the physics one is interested in. For example, specifying to three dimensions, there are at least two senses in which a \(1/r^3\) potential is long ranged: (1) in the thermodynamic limit, the interaction energy grows faster than extensively for a uniform system and (2) it is indescribable by a contact interaction at low energies (note it is the \(fastest\) decaying interaction indescribable by a contact interaction at low energies, since all \(1/r^\alpha\) with \(\alpha>3\) may be described as contact interactions at low energies). However, it is \(not\) long ranged in the sense that it decays faster than the \(1/(\hbox{length})^2\) characteristic of kinetic energy. Consequently, the high (low) density limit is still strongly (weakly) interacting, in contrast to say the Coulombic interaction, where this is switched (e.g., one forms a Wigner crystal at low density and a Fermi liquid at high density). A final subtlety is that although some other interactions decay slower, for example the Coulomb interaction, they can still have extensive interaction energy in the thermodynamic limit because they are compensated systems: positive and negative charges screen each other at long distance.
- 13.
If we tried the same trick expanding perturbatively in \(g(\Uplambda=\infty)\), the second order correction would diverge.
- 14.
Beyond the fairly trivial transitions from finite density to the vacuum that occur even in non-interacting continuum systems.
- 15.
Just called “The Hubbard model” in virtually all other areas of physics.
- 16.
Bloch functions are just the eigenstates of a non-interacting periodic system. Symmetry dictates that they are periodic under translations by lattice vectors \({\mathbf{R}}\) up to a phase, from which the rest of the properties follow.
- 17.
An exciting alternative perspective to understand both regimes is to do perturbation theory around the phase transition point, and this (much more difficult) perturbation theory is essentially the basis of describing quantum critical systems.
- 18.
Although this holds generally in classical mechanics, the issue in quantum mechanics is somewhat trickier. Thinking of a path integral description, one needs to worry about the time dimension. The proper description of the infinite spatial dimension (but still one real or imaginary time dimension) requires dynamical mean field theory, which for bosonic systems is presented in Ref. [24].
- 19.
Even symmetry breaking causes some difficulties, since the vacuum is altered, but these can generally be alleviated by perturbing around the quadratic theory applicable near the new symmetry broken solution.
- 20.
Not to be confused with the much more sophisticated local density approximation used in the electronic structure community. Only the \(corrections\) to the non-interacting quantum mechanics problem are treated with a local approximation there.
- 21.
For many cases of interest, the value of the chemical potential is also on the order of all the interesting scales in the problem, e.g. the tunneling rate \(t\) and interaction energy \(U\). Other circumstances may be treated analogously.
- 22.
Once one understands the physics of this regime, one can extrapolate to infinitely large \(\xi\) to obtain a complete picture of behavior throughout a system’s phase diagram.
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Hazzard, K.R.A. (2011). Theoretical and Experimental Techniques Used to Explore Many-Body Physics in Cold Atoms, Especially Optical Lattices. In: Quantum Phase Transitions in Cold Atoms and Low Temperature Solids. Springer Theses. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8179-0_2
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