Ultracold Bosons

Ultracold Bosons

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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = m\omega^2 r^2/2} , the chemical potential is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu = \frac{\hbar\omega}{2} (15 N_a/a_{osc})^{2/5} \,, }

where $x=15N_{a}/a_{osc}$ is a common term worth identifying, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{osc} = \sqrt{\hbar^2/m\omega}} is a characteristic length scale of the oscillator, its zero point motion. Defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 3} nm
• Separation between atoms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{1/3} = 200} nm
• Matter wavelength $\lambda _{dB}=1$ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} m
• Size of confinement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{osc} = 30} $\mu$m

Note that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\ll n^{1/3} \leq \lambda_{dB} < 2\pi\xi < a_{osc} \,. }

For a gas, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\omega = a_{osc}} . Then:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_BT_{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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial |\psi|^2}{\partial t} + \nabla \frac{\hbar}{2mi} \left( { \psi^*\nabla \psi - \psi \nabla\psi^* } \right) \,, }

by introducing flow, from current $j$,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{j}{n} = \frac{\psi^*\nabla \psi - \psi \nabla\psi^*}{2m i |\psi|^2} \,. }

This gives the continuity equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial n}{\partial t} + \nabla(n v) = 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_0} constant, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_0} is the speed of sound squared, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = \sqrt{U_0/m}} . The Thomas-Fermi solution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{\hbar^2}{2m} \nabla ^2 - V_0 \sin^2 kz \,, }

in one dimension, giving the Bloch ansatz wavefunction solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{q,n}(x) = e^{iq x/\hbar} U_q(x) \,, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} is a quasi-momentum labeling the eigenstate, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_q(x)} is a periodic function. For a given $q$ there are several solutions, labeled by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , the band index. To solve this equaiton, a Fourier expansion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_q(x)} is used, with elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_l^{a,n} e^{i2l k x}} . The potential is also expanded as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r e^{i2lkx}} . The solutions to the Schrodinger equation,

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

give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm \hbar k} , giving bands. The potential couples the bands; for example, at $V\sim 3E_{r}$ (where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_r/E_r} is large, and we shall focus on the physics of the lowest, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=1} band, in which in the limit of tight binding, the energy is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = \frac{\lambda}{2} = \pi/k} , and the paramter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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-