Ultracold Bosons

Superfluid to Mott Insulator Transition

Sound propagation in Bose-Einstein condensates

We've seen two general cooling methods so far: Doppler cooling and, on trapped ions, sideband cooling. Last time: Bogolubov transform to diagonalize interacting Bose Einstein condensate.

$E_{k}={\sqrt {\left({\frac {\hbar ^{2}k^{2}}{2m}}\right)^{2}+(\hbar ck)^{2}}}\,.$

This dispersion relation shows us that the low lying excitaitons are phonons. At $k\rightarrow 0$, $E_{k}=\hbar ck$ that of sound, while at $k\rightarrow \infty$, $E_{k}=\hbar ^{2}k^{2}/2m$, a free particle. Free particles start with a quadratic dispersion relation, while phonons and other Bose systems start with a linear dispersion relation.

File:Superfluid to Mott insulator transition-bec3-sound1

File:Superfluid to Mott insulator transition-bec3-sound2

The Bogolubov solution has a great deal of physics in it. It gives the elementary excitation, and the ground state energy. In the simple model that we have a mean field, the ground state energy is

$E_{0}={\frac {U_{0}h}{2}}\left(1+{\frac {128}{15}}{\sqrt {na^{3}/\pi }}\right)\,.$

The extra correction term on the right is a small term, recently observed by the Innsbruck group, due to collective effects. The Bogolubov solution also gives the ground state wavefunction,

$|\psi _{0}\rangle =|0{\rangle }^{\otimes N}+\epsilon \,,$

where $\epsilon$ is the quantum depletion term, which makes the wavefunction satisfy

$\langle n_{k}\rangle ={\frac {v_{k}^{2}}{1-v_{k}^{2}}}$

and

$\langle n_{0}\rangle =N-\sum \langle n_{k}\rangle =N\left[{1-{\frac {8}{3}}{\sqrt {na^{3}/\pi }}}\right]\,.$

The quantum depletion term, which arises from the fact that the gas is weakly interacting, has now been experimentally observed. Recall that in the Bogolubov approximation, the original interaction

$H'=U_{0}\sum a_{p}^{\dagger }a_{q}^{\dagger }a_{r}a_{s}$

is approximated by

$H'=U_{0}a_{0}a_{0}\sum a_{p}^{\dagger }a_{-p}^{\dagger }$

The quantum depletion this leads to is very small. The effect can be more readily experimentally observed by increasing the mass of the particle, and this can be done by placing the particles in a lattice. Plotting the quantum depletion which can be obtained as a function of lattice depth, in such an experiment, one gets:

File:Superfluid to Mott insulator transition-bec3-qdepletion

Beyond a quantum depletion fraction of $0.6$, the Bogolubov approximation breaks down, as the condensate goes through a superfluid to Mott-insulator transition.

The inhomogeneous Bose gas

File:Superfluid to Mott insulator transition-bec3-first-expt

The physics of a BEC happens not just in momentum space, but also in position space, and it is useful to analyze it accordingly. With a trapping potential applied, the Hamiltonian is

$H=\int d^{3}r\psi ^{\dagger }(r)[{\frac {-\hbar ^{2}}{2m}}+V_{trap}]\psi (r)+{\frac {1}{2}}\int d^{3}r\int d^{3}r'+....$

This must be approximated, in the spirit of Bogolubov's momentum space approximation, to obtain a useful solution. We thus replace

${\hat {\psi }}(r,t)=\psi (r,t)+{\tilde {\psi }}(r,t)\,,$

where $\psi (r,t)$ is an expectation, and ${\tilde {\psi }}(r,t)$ captures the quantum (+ thermal) fluctuations. The resulting equation is a nonlinear Schr\"odinger equation (also known as a Gross-Pitaevskii equation):

$i\hbar {\frac {\partial \psi }{\partial t}}=\left[{-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{trap}+U_{0}N|\psi (r,t)|^{2}}\right]\psi (r,t)$

The $|\psi (r,t)|^{2}$ term captures a potential proportional to the density. In the mean field approximation, it is determined by the trapping potential. This equation can now be solved. In the Thomas-Fermi approximation, with positive (repulsive) interactions, there is a characteristic length $\xi$ which arises, known as the healing length,

$\xi =(8\pi an)^{-1/2}\,,$

arising from

${\frac {\hbar ^{2}}{2m\xi ^{2}}}=U_{0}h\,.$

File:Superfluid to Mott insulator transition-bec3-healing

If the interactions are really strong, the kinetic energy term can be neglected, because the interactions will keep the density constant in its spatial distribution. Such an approximation is the Thomas-Fermi approximation, giving an equation for the wavefunction,

$[V_{ext}+U_{0}|\psi (r,t)|^{2}-\mu ]\psi (r)=0\,,$

giving the solution

$|\psi (r,t)|^{2}={\frac {1}{U_{0}}}(\mu -V(r)\,.$

The wavefunction is essentially just the potential filled up to the chemical potential level, inverted. For a quadaratic potential, $V(r)=m\omega ^{2}r^{2}/2$, the chemical potential is

$\mu ={\frac {\hbar \omega }{2}}(15N_{a}/a_{osc})^{2/5}\,,$

where $x=15N_{a}/a_{osc}$ is a common term worth identifying, and $a_{osc}={\sqrt {\hbar ^{2}/m\omega }}$ is a characteristic length scale of the oscillator, its zero point motion. Defining $R=a_{osc}x^{1/5}$, we may find $\mu =m\omega ^{2}R^{2}/2$. This explains the profile of the condensate data obtained in experiments:

File:Superfluid to Mott insulator transition-bec3-bec-peak

\noindent Note that the size of the ground stat BEC is much larger than the zero-point motion of the harmonic oscillator. This is due to the pressure of the repulsive interactions. The Gross-Pitaevskii interaction gives not only the ground state wavefunction, but also the dynamics of the system. For example, it predicts soliton formation: stable wavefunctions with a size scale determined by a balance of the kinetic energy and the internal interactions. This requires, however, an attractive potential. Such soliton formation can nevertheless be seen in BEC's, with tight traps (see recent Paris experiments).

Length and energy scales in BEC

\begin{itemize}

• Size of atom: $a=3$ nm
• Separation between atoms $n^{1/3}=200$ nm
• Matter wavelength $\lambda _{dB}=1$ $\mu$m
• Size of confinement $a_{osc}=30$ $\mu$m

Note that

$a\ll n^{1/3}\leq \lambda _{dB}<2\pi \xi <a_{osc}\,.$

For a gas, $a\ll n^{1/3}$. For a BEC, $\lambda _{dB}\geq n^{1/3}$ in addition. The corresponding energy scales are also useful to identify. Let $\hbar \omega =a_{osc}$. Then:

$k_{B}T_{s-wave}\gg k_{B}T_{c}\geq k_{B}T>U_{int}>\hbar \omega$

The interaction energy scale $U_{int}\sim (h^{2}/m)na$, corresponding to the healing length.

Vortices in a BEC

The Gross-Pitaevskii equations also predict the formation of topological defects in the condensate, such as vortices. These form with a chacteristic vortex diameter of $\xi$, the healing length.

File:Superfluid to Mott insulator transition-bec3-vortex-creation

File:Superfluid to Mott insulator transition-bec3-vortex-movie

Question: do you get the best votrices by stirring gently, or vigorously?

Hydrodynamics

We may transform the GPE into a hydronamic equation for a superfluid,

${\frac {\partial |\psi |^{2}}{\partial t}}+\nabla {\frac {\hbar }{2mi}}\left({\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}}\right)\,,$

by introducing flow, from current $j$,

$v={\frac {j}{n}}={\frac {\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}}{2mi|\psi |^{2}}}\,.$

This gives the continuity equation

${\frac {\partial n}{\partial t}}+\nabla (nv)=0\,.$

Writing $\psi =fe^{i\phi }$, and noting that the gradient of the phase gives us the velocity field, we get equations of motion for $f$ and $\phi$,

$-\hbar {\frac {\partial \phi }{\partial t}}=-frac{\hbar ^{2}}{2mf}\nabla ^{2}f+{\frac {1}{2}}mv^{2}+V(r)+U_{0}f^{2}\,.$

This reduces to

${\begin{array}{rcl}m{\frac {\partial v}{\partial t}}&=&-\nabla (\delta \mu +{\frac {1}{2}}mv^{2}\\\delta \mu &=&v+U_{0}n{\frac {\hbar ^{2}}{2m{\sqrt {n}}}}\nabla ^{2}{\sqrt {n}}-\mu _{0}\,.\end{array}}$

The Thomas-Fermi approximation is now applied, neglecting $\nabla f$, but keeping $\nabla \phi$, giving

$m{\frac {\partial ^{2}\delta n}{\partial t^{2}}}=U_{0}\nabla (n_{0}\nabla (\delta n))\,,$

a wave equation for the density. For $n_{0}$ constant, $U_{0}$ is the speed of sound squared, $c={\sqrt {U_{0}/m}}$. The Thomas-Fermi solution for $n_{0}$ gives collective modes of the condensate. A droplet of condensate can have shape resonances, waves, and many other physical behaviors, captured by these solutions.

Superfluid to Mott-Insulator transition

A condenstate in a shallow standing wave potential is a BEC, well dessribed by a Bogolubov approximate solution. As the potential gets deeper, though, eventually the system transitions into a state of localized atoms, with no long-range coherence, known as a Mott insulator. These physics are important in a wide range of condensed matter systems, and can be explored deeply with BECs. The ultracold atoms are trapped in a periodic potential,

$V(x,y,z)=V_{0}\left({\sin ^{2}kx+\sin ^{2}ky+\sin ^{2}kz}\right[\,,$

and we would like to know what happens when neutral atoms are in this potential. Traditionally, this is studied in condensed matter physics in the context of atoms in a periodic potential, but many anologies exist for the neutral atom system. The Hamiltonian is

$H={\frac {\hbar ^{2}}{2m}}\nabla ^{2}-V_{0}\sin ^{2}kz\,,$

in one dimension, giving the Bloch ansatz wavefunction solution

$\psi _{q,n}(x)=e^{iqx/\hbar }U_{q}(x)\,,$

where $q$ is a quasi-momentum labeling the eigenstate, and $U_{q}(x)$ is a periodic function. For a given $q$ there are several solutions, labeled by $n$, the band index. To solve this equaiton, a Fourier expansion of $U_{q}(x)$ is used, with elements $c_{l}^{a,n}e^{i2lkx}$. The potential is also expanded as $V_{r}e^{i2lkx}$. The solutions to the Schrodinger equation,

$H\psi _{q,n}=E_{a,n}\psi _{a,n}\,,$

give

$\sum _{l'}e^{i(q+2l'kx)}\left[{\left({{\frac {(q+2l'\hbar k)^{2}}{2m}}-E_{q,n}}\right)c_{l}^{q,n}+\sum _{v}c_{l',r}^{q,n}}\right]=0\,.$

Graphically, these solutions are as follows. If energies are plotted as a function of $\hbar k$, we get:

File:Superfluid to Mott insulator transition-bec3-smott-dispersion

With no interations, we get a free particle dispersion diagram. But with the periodic potential, a Brilloiuin zone appears, forbidding momentum beyond $\pm \hbar k$, giving bands. The potential couples the bands; for example, at $V\sim 3E_{r}$ (where $E_{r}=k^{2}/2m$), the band degnericies are lifted to become:

File:Superfluid to Mott insulator transition-bec3-3er

\noindent At higher potentials, the bands become even flatter. We will be interestd in the limit in which $V_{r}/E_{r}$ is large, and we shall focus on the physics of the lowest, $n=1$ band, in which in the limit of tight binding, the energy is

$E_{1}(q)={\frac {3}{2}}\hbar \omega _{0}\,,$

becoming constant with respect to quasi-momentum. The derivation of the energy as a function of quasi-momentum gives

$-2J(\cos q_{x}a+\cos q_{y}a+\cos q_{z}a)\,,$

where $a={\frac {\lambda }{2}}=\pi /k$, and the paramter $J$ tells us how wide the band is and how large the dispersion region is.

BEC to BCS crossover transition

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-

File:Superfluid to Mott insulator transition-bec3-