Ultra-Cold Quantum Gases

Bose-Einstein Condensates

Taking a break from aerodynamics to catch up on lectures. So, I already mentioned taking some solid state physics courses this semester, against my better judgment, and one of them centers ultra-cold quantum gases. The week I'm starting this is the one that has concluded the first half of the lecture, and if I don't want to be swamped at the end of the semester, maybe I should make sure to keep up with revisions. Ultra-cold quantum gases (at around 10 - 200 nK) will discretize some of the quantum states that are usually just a goo. Ensembles will begin adhering to Fermi statistics and form a "Bose-Einstein Condensates" (BEC). It features macroscopic matter waves and the particles all occupy the same quantum state. This will give a very high control over the experimental parameters of the ensemble.

BECs are exceptional mainly because Fermions adhere to the Pauli exclusion principle. In a fermionic ensemble, this means that physically, they shouldn't be able to occupy the same quantum state. By regular solid state physics, the Pauli exclusion principle is respected. For ideal gases, for example, the Maxwell-Boltzmann distribution (the probability of finding a state at excitation i at temperature T) is

for both bosons and fermions. The implication is of course that this model breaks down at small temperatures. Such ensembles are described using the "Grand Canonical Ensemble" (GCE), in which both particles and energy can be exchanged with its environment

Most thermodynamic quantities are literal derivatives of this total energy via Legendre transformation. The probability of realizing an ensemble with exactly N' particles in a state k (with appropriate energy eigenvalue) is normalized using the partition function, since the energy of GCE carries it as a logarithmic dependency.

For other quantities:

depending on whether the ensemble is bosonic (-) or fermionic (+). In the classical limit, where the product is much larger than 1, this tends toward the Boltzman distribution. Before getting to the mathematical description of BECs, an ideal Bose gas is a bosonic ensemble. The 0-th excitation distribution should be strictly larger than 0, so the case where the chemical potential is larger than ϵ₀ is forbidden. For cases where their difference tends towards null, the distribution becomes far larger than 1.

The density of states is ultimately derived through the sum of states, pushed to a continuous integral. The energy states are still quantized depending on the potential, and the way these are quantized will provide the proportionality constant of the density of states.

With this quantity, the integral identifying the excited states can be evaluated.

The temperature at which all particles are in an excited state is the "critical temperature"

The relation between the critical temperature and the number of particles implies that BECs are high-temperature phenomena. For experimental production of BECs, laser cooling isn't quite sufficient, and particles will condense into solids before Bose-condensing. Actual BECs are always meta-stable. For a statement of atoms in the condensate, one just has to subtract the excited ones from the total number. This happens to come out to an even

As the temperature drops, the number of particles not contained within the condensate will of course tend to zero. For all particles to occupy the lowest state, the best description for the system actually begins in momentum space. The BEC should be highly localized at p = 0, which, if Fourier-transformed comes out as an even distribution in space. Each distribution comes with its own width. In real space, this is

in momentum space

As a side note, the distribution of thermal coulds in general are

Because of time-of-flight expansion, such distributions have a tendency to smear out over time, which will involve gaining momentum, which we don't want, so these particles will have to be actively trapped. Since laser-cooling runs on much higher temperatures than those at which BECs exist, scattering traps are out of the question. Instead, non-scattering traps, such as magnetic traps or optical dipole traps are chosen for this purpose. Magnetic traps work with a magnetic potential. Of course this implies the particle have a magnetic moment. some atoms are either high- or low-magnetic-seeker to be trapped at B-field maxima / minima respectively. These are determined by whether the product of mass and Zeemann factor is positive or negative. The practicle setups here are Helmholtz / anti-Helmholtz traps. To trap low-field seekers, there is a problem however, since the center of the trap will feature a spot with a net magnetic field of zero, at which point a particle might transition into other states, some might not be trapping at all, at no energy cost. These are the "Majorana losses". This point in the field must be avoided.

An alternative setup is the Ioffe-Pritchard trap, which ads four rods that runs through a Helmholtz trap setup. The rods will create a quadrupole field with tight linear trapping in the plane they run normal to, while the Helmholtz coils create an offset in the remaining coordinate with axial harmonic trapping. The radial harmonic confinement remains, and was unwanted.

Cloverleaf traps will replace the Ioffe bars with cloverleaf-shaped coils, offering better fine-tuning, optical access and increased trap stability. The Ioffe bars could also be replaced by another coil (Ioffe coil) for a Quadrupol Ioffe Configuration Trap (QUIC).

The optical traps are based in the optical dipole force, which can be adjusted using detuning.

Both options can be combined into a hybrid trap. To transfer the particles from one trap into another, the modes are matched by pre-cooling the ensemble into an almost spherical shape, and adiabatically compressing the otherwise cigar-shaped MT by increasing density (collision rate). Optical mode matching is usually just done by crossing the dipole traps. Once trapped, the ensemble must be cooled. The concept of evaporative cooling functions by selectively removing the hottest atoms. The potential is stepwise lowered, then left alone until a new thermal equilibrium is achieved and the cooling stops. The atoms don't get lost in the process, but their collisions will become more non-elastic each time. The elastic collisions don't retain total energy, so these are the ones actually evaporating away the energy. This process will come with phase space gain. The problems become creating an optical potential deep enough to hold the laser-cooled atomic sample initially, the non-linear potential for energies low enough to retain evaporation efficiency (i.e. not reheating the atoms constantly), and the trapping frequencies decreasing during evaporation, since the rate of elastic collision decreases with less available space.

Thermalization happens through a series of collisions between the particles trapped in the potential. This comes with some (relatively obvious) caveats, namely that the lifetime of the atom should be much longer than the re-thermalization rate, since the energy-mass relation will imply relatively high impulses on particle decay. The re-thermalization rate is set by atomic parameters, and the potential. For thermalization, elastic 2-body collisions are favoured, as elastic collisions feature lost energy, and the 2-body collisions might introduce a very low momentum for two scattering particles moving in one direction in exchange for relatively high momentum for the other particle, moving in the other direction, leading to a loss of particles. 2-body spin relaxation processes might relax the excitation of the particles and add momentum to a similar effect, so these collisions are also not favoured. Assume a polynomial potential in d dimensions. Then the evaporation efficiency scales

To enhance this, the elastic collision rate has to be increased. This rate can be put into ratio to the number of particles lost in that time, seeing as both good and bad collisions will always happen. They can be viewed as in competition, especially close to quantum degeneracy.

Measuring BECs will require specialized methods as well, most of which will destroy the BEC. Absorption imaging uses resonant light to excite the BEC, and the re-emitted light will be recorded. Since the optical density of BECs are very high, this method is not suited to in-situ measurements. Still, this is the basic idea behind other, better measurements. If the light is far detuned, meaning intentionally off the resonant frequency of the particle, the absorption rate is small, and while the produced signal is dispersive and prone to interference before reaching the camera, this will not completely destroy the BEC, while reducing the optical density, so this method can be used for in-situ measurements and retains at least some of the BEC. This method comes in two versions: Dark-ground imaging and phase contrast imaging. Both focus the light on the BEC. The former has the scattered light make up the image as the rest is lost, while the latter reverses the components, yielding a negative. The intensities are as follows:

For the small phase shifts, which we presuppose for a successful setup, the dark-ground imaging will have a signal quadratic in ϕ, and phase contrast imaging will have a signal linear in ϕ.

A more direct method (direct as in collecting fluorescence photons instead of imaging the shadows caused by photonic absorption) is single-photon fluorescence imaging. This is effectively non-invasive and yields the same information as measurements that would destroy the BEC. The signal is difficult to spot though, as it's a single emitted photon travelling through the detector. In addition, the photonic background radiation (whether actual radiation, or technical limitations effecting the camera) will have to be subtracted.

Up to this point, the interaction between particles in the Bose gas has been ignored. In the "weakly interacting Bose gas" this is no longer the case. Real world implications for this are that the range of interaction is much smaller than the average interatomic distance. As always in these cases, only pair-interaction is taken into consideration and the interaction distance is large enough to approximate the wave function as an asymptote. In the low temperature limit, the scattering amplitude takes the form of a negative constant, usually something in of the form of

Through the Born-approximation for low-energy scattering, an approach with an effective potential is valid

For negative scattering lengths, the interactions are attractive, for positive ones, they're repulsive. The actual wave function for the condensate is obtained through the Bogoliubov approximation, which sits on the 2nd quantization of the many-body Hamiltonian. For this, assume that below the critical temperature, many particles occupy a single wave function, then describe the BEC in the classical limit. This introduces the usual decomposition of the wave function in a Condensate wave function and the quantum fluctuations

In the ground state, the energy of the uniform weakly interacting Bose gas should have neither potential, nor kinetic energy components (those, usually noted in first and second place in the Hamiltonian), so what remains is the first order perturbation

This turns the pairwise 2-body interaction into a mean-field potential acting on the particle in question. This means the full Hamiltonian is known and can be supplemented using the Bogoliubov transformation in terms of creation/annihilation operators b, before solving

The non-diagonal terms are supposed to vanish, so that

The dynamics within a BEC are governed by the appropriate Heisenberg equation, called the "Gross-Pitaevskii equation" (GPE)

Note that GPE is non-linear and includes a the mean-field potential generated by the atoms. It only works at zero-temperature. The matrix element of the field wave function, along with the trivial time-evolution will reveal a dependency on the chemical potential which will also enter the time-independent GPE.

Note the energy components in terms of BEC size R:

for sufficient sizes, i.e. large enough interactions, the kinetic energy can be neglected. Application leads to the Thomas-Fermi approximation below critical temperature and with a chemical potential above the trapping potential.

The length at which quantum pressure (kinetic energy) and repulsive interaction are in equilibrium is called the "Healing length". The Thomas-Fermi approximation breaks down at the edge of the condensate. Above some critical particle number, the interaction will overwhelm the quantum pressure, and the condensate will collapse. It finds its way into the dimensionless form of the GPE.

Two condensates in proximity have the possibility to interfere with one another as they expand. They can be described using a wave function that is a linear combination of the two BEC wave functions, with some phase shift. Similarly, the density after time of flight will consist of the two BEC densities, and cross terms.

n(r, t) = n_a(r, t) + n_b(r, t) + 2\sqrt{n_a(r, t) n_b(r, t)} \cos(\frac{md}{\hbar t} z + \phi)

The overlap will display an interference pattern, depending on that phase shift.

Trapped BECs will similarly display interference patterns, though this is due to what's effectively reflection. The linearized solution of GPE decomposes into time independent part, and two quasi-particle amplitudes. GPE also makes visible the dispersion (from the kinetic, and standard dispersive part of the Hamiltonian) and the nonlinear interaction, which enables wave packet propagation in nonlinear media. Neither of these effects will be good for the stability of the wave packet, but the balance between focusing and defocusing forces can stabilize the wave packet propagation. Solitons are diffusionless waves. One categorizes between temporal and spatial solitons, the former being the a wave that doesn't spread over time, while the latter retains original resolution regardless of the linearity of their medium. Solitons can be effectively treated as massive particles. Dark solitons will then have highly localized effects on interaction with traditional probability distributions.

Because of the choice of model for BECs, the velocity of condensate flow is an irrotational velocity field. To model the rotation of a BEC, the adapt that of a vortex in a superfluid. In these, a rotation leads to quantized vorticity, which means that depending on the speed of the rotation, discrete holes will form in the BEC, which are arranged along axes of maximal radial symmetry.

Ultra-Cold Fermions and Ultra-Cold Collisions

As discussed, relaxing fermions to ultra-cold temperatures will run into the Pauli-exclusion principle. This will change the development of the density function, which will slowly morph into a box function as the temperature drops. In general, fermi gases house interactions only in the s-channel, since their wave-functions are antisymmetric. The contact interaction can be expressed as a potential which is proportional to the distance between the particles. In this way, spin-polarized fermions act like ideal fermi gases. The phase-space properties of BECs can be applied fermions as well, so there emerges an expression for the Fermi-Energy:

In the local density approximation, the fermi-distribution can be written as

The density distribution is barely any different to that of the BEC, and for T > 0, this difference becomes smaller. The highest degree of quantum degeneracy is thus at very small T. The basic properties for ideal fermi gases are:

where r and R are normal to the volume. The localization of the gas is only tentatively dependent on the number of particles and more dependent on trap parameters, which can be adjusted from outside the system. The momentum distribution turns out isotropic, and not dependent on the trap potential. This behavior will also be visible after a long enough time-of-flight measurement.

Introducing two different types of fermions with different spins for example, but the same mass and fermi energy are able to "double up" in the trapping potential, because the Pauli-exclusion principle doesn't hold for them. In such cases, they interact via a potential V = n·g

where the parameter k is usually meant to symbolize the strength of the interaction. Weak interaction has almost no effect on the thermodynamic properties of Fermi gases, as long as it's less than 1. In cold gases, it's usually of the significance of 0.01. For ultra-cold fermions, quantum degeneracy is required, and since laser-cooling and trap-cooling create the appropriate effects, evaporation-cooling doesn't, which is somewhat unfortunate, since that is the one that works for very low temperatures. The solution is called "sympathetic cooling" which uses bosons as thermal bath, and uses several fermionic m-states to cool the mix. The fermi-sea will gain higher density as the ensemble drops below the fermi-temperature, where Pauli-Blocking suppresses collisions. The phase space distribution also tightens, until there are no "free final states" left. The change of energy after a time of flight experiment below fermi-temperature for an ideal fermi gas is dependent on the potential. For a harmonic oscillator, this is

It's a well measureable quantity. Obviously, if the fermions are weakly interacting already, then this force has very little effect on the change of energy.

A homogeneous fermi gas can be achieved in a box trap, where the particles in the ensemble have one shared fermi energy. For the box trap, have two repellant dipole-potentials and a hollow Laguerre-Gauss beam, along with two "light sheets". The 1-dimensional momentum distribution becomes

Feshbach-Resonances are a central concept to understand resonance-scattering. Before looking at the Feshbach-resonance itself, some background on scattering theory, which I will presuppose (and loop back to on another occasion) and know that the scattering phases can be determined analytically through the radial Schroedinger Equation. For the ultra-cold scenario, the scenario l = 0 becomes very interesting. The result has the general form

where j are Bessel functions and n are Von-Neumann functions. The scattering phase is set to η = arctan B / A. The solution should be regular at r = 0, so for free particles, this phase should tend to zero. In its asymptotic form, ρ → ∞

which means that η functions as an asymptotic phase shift. For two hard spheres with diameter a, the scattering phase adheres to

For small ka, s-channel scattering dominates. For k tending toward infinity, the argument is shifted by a. For l = 0,

Using the same method, the spherical potential well can be solved to two segments. Within the pot:

and outside it:

The method of the hard sphere can be used here as well, by setting an "effective hard sphere diameter" η = -ka so that

is the scattering length. The effect of scattering is - as mentioned - a shift in energy. This can be quantified using the border condition, though in general, if the scattering length is positive, the shift is attractive, and if it's negative the shift is repulsive. Occasionally, k → 0 is not going to be a a good enough approximation. In these cases, the energy and temperature-dependency of the scattering phase will become relevant. That can be expressed by

dependent chiefly on the effective range of the potential r. The non-resonant scattering at a ≈ r₀, r = 2/3 r₀, so the k² term is only relevant for kr₀ ≥ 1. If the scattering length diverges, then r = r₀, and said term will become relevant, even for the smallest k.

While this process gives good approximations, it's not a fun method, and quite lengthy if properly done in computation. Instead, introducing a "pseudopotential" is probably more effective. The regularized delta-potential will do the trick in this application

It's the preferred method, but not as useful for keeping track of the motivations.

Feshbach Resonances

In Feshbach resonances, there are always at least two channels involved, which is somewhat contrary to the potential scattering, which only has one scattering channel. So far, the scattering considerations only included one channel, which asymptotically approaches a state for separated atoms at zero energy level. This is called the "open channel". In contrast to it, the "closed channel" is one that asymptotically approaches an energy that is forbidden. It only becomes relevant for very small atomic separation. Such channels have at least one bound state.

In the analytical model, the open channel might contain a triplet state, and the closed channel a singlet state with the corresponding wave functions, which will be different depending on the relation between r and r₀

By choosing the Hamiltonian

This should create new eigenstates with new wave orders q, The Ansatz for a solution under weak interaction introduces a mixed angle which is always normed automatically, and the general solution comes out as

with the norm-factors still unknown. While they can be fixed by choosing initial conditions, this isn't relevant for the Feshbach resonances, since they're only interested in the scattering phase. This solution can be split into singlet and triplet components to get the s-channel scattering length

where γ is the Feshbach coupling strength. The possible singularities cancel. This model is true for a fixed energy difference between open and closed channels. Using the Zeeman effect, this difference can be shifted deliberately. The resonance energy of a bound state can be written as

In actuality, the elastic scattering length is limited through the losses, which happen through 2 and 3-body collisions. For Fermions though, the Pauli-principle will suppress losses heavily. Classification of Feshbach resonances are usually centered on the l-quantum number for the closed channel. So if l = 0, it's an s-wave resonance, if l = 1, it's a p-wave resonance etc. Only channels of the same parity (of l) can couple and the strongest resonances are usually s-wave type. Whether a resonance is broad or narrow is decided by the benchmark of

This is an okay measure for classification, but does presuppose a fair bit of information. More interesting might be the checking for open-channel dominated resonances and closed channel dominated resonances. This checks whether the scattering length follows

and the bound state component is small, barely affecting the theory. This would characterize an open channel dominated resonance. The negative of these three points characterize the closed-channel dominated resonance.

Interactions are usually described using an interaction parameter, mapped against the fraction of temperature and Fermi energy. Three regimes emerge, depending on whether the inverse of the parameter is less than -1 (BCS Limit), larger than 1 (BEC Limit), or in between (unitary limit). The understanding of the latter is not very well evolved as of time of writing. Feshbach molecules obviously can't exist crossing open and closed channels. Mapping the magnetic field strength against the binding energy, will see the open and closed channels cross at some point on the B-axis. Introducing the interactions, will have two mixed states, running asymptotically along the outside of the area enclosed in both of the channels. The top of the two acts repulsively relative to the open channel, and the bottom one acts attractively relative to the open channel. The energetic minimum then exists on the lower of the two mixed states. It transforms adiabatically. Getting molecules into this state can be done through manipulation of the field into a "ramp", or binding them using 3-body collisions from the repulsive side to kick take the excess energy out of the system. There's of course always the possibility to radiating off that excess energy as well.

Feshbach-molecules are bound extremely weakly, so ideally the transfer goes directly into the ground state. Heternuclear molecules have an electric dipole moment in their ground state, introducing a more complicated system with long-range interactions. At low enough temperatures, bosonic Feshbach molecules form a BEC. It's properties are the same as those of an atomic BEC, with m → 2m, a → 0.6a. Molecular BECs experience a bimodal distribution after long enough ToF, are suprafluid, and follow the Gross-Pitaevskii equation. They are distinguished very easily from weakly interacting Fermi gases.

BCS Theory and Cold Molecules

Since the Feshbach resonances presuppose repulsive bound states, the question might arise what happens on the attractive side where no bound states exist. This is where the Bardeen-Cooper-Schrieffer (BCS) state comes in, where fermions with opposite spin and momentum create pairs under weak attractive interaction. These pairs condense, creating a supraconductive/fluid state.

Bound states in n dimensions are determined using more or less the same method each time: Solving the Schroedinger equation. There are obviously typical solutions for each of the most common dimensions (1 to 3). Generally,

For short-range potentials, some simplifications can be made, allowing for a cut-off

In 1 and 2D the integrals divert for |E| → 0, so there is always a solution for V₀, while in 3D the integral always converges. For this problem, Cooper's approach sees only fermions at the fermi-edge building pairs, so that the DOS is more or less constant. By the Pauli-principle then, the integral can be limited so that the smallest binding energy can be found through the regularized divergences. The binding energy of the Cooper pairs comes out as exponentially smaller than the Fermi energy, while the size of the Cooper pairs is exponentially larger than the mean distance between particles.

With this, two isolated fermions can be bundled up into a bound state, but suprafluidity is a many-particle state. This could be done through a macroscopic number of Cooper pairs, where the many-particle wave-function are meant to describe pairs of fermions. The BEC state of these pairs occurs a q = 0, which leads to a sharp excitation gap. The best approach to this can be found in the second quantization in momentum space.

Unfortunately, the commutators for b don't work out quite the way they should. Only at states, that are broad in momentum space is this technically correct. To fix this, consider a condensate-wave-function. Underlying this, is a coherent pair-state which features overlapping different states with different particle numbers. The BCS wave function then comes out (via forcing commutation and norming)

The transition from BCS to BEC is derived through the BCS wave-function and the density distribution. In the BCS regime, there is almost no change in the density distribution in comparison to the ideal fermi gas. Unitary fermi gases are characterized by diverging scattering length. The fermi-scale then is the only possible remaining one, and using that, and the chemical potential the density at zero temperature comes can be determined, which in turn can be compared to the BEC regime. The behavior of the pair-wave function in momentum space shows a strong localization at the fermi-edge and in the transition area, the function is smeared out. In position space, the wave function is very wide, with an oscillation with fermi-length. At high p-space localization, the oscillation is stronger, and damped. The oscillation is lost at the BEC limit. Using the Bogoliubov transformation, a quasi-particle creation operator removes a Cooper pair with momentum k from the wave function, replacing it with a single particle. From the kinetic energy and reduced pairing energy emerges the excitation energy as a sum of the two.

The excitation gap is effectively an order-parameter for the Bogoliubov transformation. The macroscopic expectation value of the pairing field can be written through the quasi-particle operators, which act fermionic. The expectation value at finite temperature can hence be expressed through the Fermi-Dirac distribution function.

Through the gap equation, the temperature can be expressed, only for vanishing gaps. The critical temperature for suprafluidity is of the same order as that for the gap. In the BEC regime, the first approximation is independent of the length scale. The cross-over regime is more or less accurate. In the BCS regime, the critical temperature is about the point at which the Cooper pairs split apart.

The condensation is characterized through the Penrose-Onsager-Criterium, where the macroscopic non-diagonal entry of the 2-particle density matrix. From this, emerges the Pairing field

The density of condensed pairs can be found through separation into center-of-mass and relative coordinates, then integrating over it. With an appropriate substitution, the molecule density becomes

For stronger interaction quantum depletion becomes relevant, reducing the condensate-component to about 91%. For a unitary regime, the BCS theory yields 70%, where Monte-Carlo simulations yield 57%. For a BCS regime, the component becomes exponentially small. In BEC, the resistance is determined through the interactions, and the thermal resistance (obviously) through the system temperature. Their ratio shows a bimodal distribution. In the crossover regime, this distribution breaks down. In the BEC regime, a strongly interacting Bose-Fermi mix can be observed. Between those atoms in the majority-density and those in the minority-density, a mexican-hat potential emerges, which acts as a signature.

In the unitary regime, the fermionic suprafluid forces all fermions to enter Cooper-pairs. At low densities, the suprafluid kernel may be surrounded of single fermions, to make sure this can be done. The density is usually already integrated into a single direction, through the detection mechanism. Usually the parts of condensate and suprafluid are not equal. There is a two-fluid description of suprafluidity by Landau, which first separates the density into two parts. The excitation energy with associated momentum in the system of suprafluidity is considered the active system. So the energy is

From this emerges a critical velocity below which excitation are impossible, as fixing the excitation energy fixes the momentum.

Excitation behaves differently in BEC and BCS regimes. In BEC, they are sound waves, analogous to atomic BEC. The splitting of molecules is impossible, and the critical velocity is

In the BCS regime, the excitation is dominated through pair-breaking. The sound waves still exist, but aren't the excitation of lowest energy. In the cross-over the critical velocity becomes maximal.

Suprafluidity in the crossover regime is characterized by quantized vortices. Wave functions of the suprafluid with velocity field shouldn't have rotational elements, but through the closed integral of the velocity field, there is a quantized rotation.

A method to measure the 1-particle excitation spectrum is the RF-spectroscopy. The idea is to incite an excitation into a third, non-interacting state.

The first bracket is said third state, the second envelops the energy of the paired state. For this method to be applicable, there needs to be a very small interaction between third state and the rest of the system (obviously), and its free path needs to be much larger than that of the atomic cloud. The part of decoupled atoms of some momentum is determined through the dispersion and the density of states. The energy required for rf-excitation is

For an ideal fermi-gas, the free dispersion shows a curve along the energy, in such a way that should be familiar. The BEC regime splits into 2 parts, depicting the free unpaired particles, and the molecules, separated by a clear gap. In the unitary regime, there is a drop relative to the ideal gas, which is the product of the pair-energy. The typical BCS bend at the Fermi-momentum is also clearly visible in the spectrum. This is an indicator for the method observing pre-formed pairs.

Optical Lattices

The number of basis states in a system with n independent spins is (trivially) the n-th exponent of 2. This makes the complexity of quantum system a problem which becomes impossible to handle relatively quickly. A method simulation is the Hubbard model. It can be translated easily into an optical lattice. The potential created by the ions in the solid state grid can be translated into the potential created by the standing light wave. In the Hubbard-model the potentials follow the Bloch-Theorem, which follows from the 1D-Schroedinger equation applied to periodic lattice structures as boundary conditions. The Brillouin zone gives the restrictions of the quasimomentum, which leads to the typical band-structure. Following the entire length of the calculation, the Fourier coefficients for optical lattice potential. After diagonalization of the energy matrix, the eigenvalues describe the band structure. In weak lattices, only one momentum exists, which is analogous to a free space solution. Deep lattice will feature many momenta and therefore be more localized in space.

Bosons tend to condense into zero-momentum Bloch-states after TOF-experiments. These states feature superpositions of plane-waves, which show up as Bragg-peaks in the data. The envelope is reflective of the Bloch-coefficients. An alternative read is charting the interference of coherent matter waves from the lattice sites. The resulting contrast reflects the coherence in the lattice, and the envelope is derived from the Wannier function.

In the band-structure, the non-interacting fermions fill up the lowest band, and excitation will move particles from that band, to higher ones. If the lowest band is (approximately) flat, the excitation energy is about equal to the band energy. The depopulation in the lowest band is measurable in the reduced zone scheme.

The alternative description has the benefit of using a basis of functions, which are maximally localized to individual lattice sites. The corresponding Wannier functions a site is given as

Which Wannier function should be chosen exactly depends on the phases used in the Bloch-functions. To get from the Bloch-Hamiltonian to the Wannier-Hamiltonian, introduce annihilation operators for the particles in both pictures. The Bloch operator a is given by the Wannier operator b

The Hamiltonian can be interpreted as an operator hopping between lattice sites with amplitudes J. J(j - j') is the matrix element of the Hamiltonian between two Wannier functions J(j). The hopping amplitude is given by the overlap of the Wannier functions, so increasing the lattice depths will drop the amplitude. If it's deep enough, only the lowest band will be active, and hopping will only be possible to directly adjacent valleys. Band Width:

In the Hubbard model then, the quantum statistics for bosons in short range can be determined by beginning with the generic pseudopotential, simplified to on-site interactions. Given the interaction energy of 2 particles on one lattice site, the Wannier function can be approximated using a Gaussian, assuming the lattice is deep enough.

The Hubbard model needs to distinguish between Bosons and Fermions (or rather their statistics)

Each has two competing energy terms: the hopping energy J and the interaction energy U. The implications (such as quantum phase transitions) of the model have been confirmed with ultracold atoms in optical lattices. In the superfluid phase, the interactions are weak and tunneling minimizes the kinetic energy. Partcles within the lattices are delocalized, and within it, BEC can be assumed. In contrast stands the Mott insulating phase, in which interactions dominate, and the interaction term of the Hamiltonian is minimized. Locally, the particle number fluctuation is reduced. The system is considered incompressible. One can get from superfluid to Mott insulator through a loss of coherence, and fixing the particle number. These things happen automatically with increasing interaction strength U/J.

At U = 0, the system enters the superfluid phase, putting the BEC into the lowest Bloch-state q=0. The coherent approximation comes from basis change, so that

For J = 0 and repulsive U, the system enters the Mott insulator state. The double and multiple occupation is energetically unfavorable, so each lattice site holds exactly n particles. The quantum-phase also exists for finite J. The mean-field solution of the Bose-Hubbard model applies mean-field theory to the Bose-Hubbard Hamiltonian. The operators are expressed through expectation values, and the local real superfluid order parameter becomes the expectation value of a bosonic operator (it doesn't matter whether it's creation or annihilation operator). The effective description of one lattice site has

couples different lattice sites for different i and j. This is not as intended, so lattice sites need to be decoupled from one another. The assumption is made, that the deviations from the order parameter are constrained to quadratic order.

Solve for the number operator and insert into the operator. The effective Hamiltonian at a site depends on the chemical potential and interaction energy, perhaps normalized with the hopping energy.

The solution for strong interactions (J → 0) has uncoupled Fock states at site i, and the Mott state as ground state.

The perturbation theory works about the same as one would expect, and delivers a quantum transition phase for the second-order energy perturbation of -1. It's solved for the chemical potential.

These boundaries are of course quantized by n, as would be expected. The coherence of the two models are taken from the contrasting behavior in ToF observation. Both are derived from the momentum density, leading to a distribution. In the superfluid, this distribution shows a clear interference pattern in the lattice peaks, while the Mott insulator only has an enveloping distribution. This coherence gets lost, when particles (in the Mott insulator) localizes. The visibility remains finite in the insulating phases.

The position of the phase transition along the frequency spectrum depends on the number of neighbors. This is a single integer depending on the lattice structure. There is usually a central peak in the normalized interaction strength. This point is used to measure the condensate fraction for long-range coherence. In the Mott insulator, the excitation spectrum is gapped by the interaction energy. Anything beyond that escapes the potential and destroys the system. The measurement is taken by applying different gradients in the insulator phase, then ramping back into the superfluid and measuring the temperature using the width of the interference peak. If excitation has taken place, then the temperature remains elevated after the peak. To avoid parity projection that can occur for atoms grouping in lattice sites, resonant light can kick them out of the minima. The atoms will disappear before contributing to the fluorescence signal, which will actually take the measurements. The detection fidelity is limited by loss- and hooping rates, determined by the vacuum lifetime and the recooling to other lattice sites. These two rates can be measured by taking two consecutive images of the same sparse clouds and taking the difference. The overall fidelity typically comes up to around 98%. Lattice sites can be resolved optically. Either by flipping the spin of a selected atom in a Mott insulator, and visualizing the spin pattern by removing one of the spin states. This can be used to initialize dynamics. Alternatively, a system can be prepared by cutting out isolated 1-dimensional systems from the initial Mott insulator, which creates barriers as required.

For inhomogeneous systems a grand-canonical description is appropriate. Locally, these systems can be approximated by a homogeneous system with local chemical potential and particle exchange. Lattice beams produce external traps which introduces an additional term into the Bose-Hubbard Hamiltonian. The local density approximation introduces the effective chemical potential. Since the different interaction regions can exchange particles (as in the grand-canonical ensemble) no exact matching of the particle numbers is necessary. The density correlation at the transition is characterized by the correlator C determined by the distributions. These change over time, so this is a statistical size. For an inverse interaction strength of 0, there is no correlation. Correlations are stronger at lower dimensions, and small inverse interaction strength, the correlation effects are dominated by coherent particle-hole pairs.

Weakly Interacting Bose Gases

In general, the interaction range of atoms is assumed to be much smaller than the average distance. This will mean that pair-interactions are the only kind that need to be taken into account and the wave function can be approximated by an asymptote. Such wave functions can then be fixed by their scattering amplitude. Usually the benefit here is that the details of the actual potential don't matter much. This is the general approach for low-energy collisions. Cold collisions can be written as a potential in the Born approximation.

A positive scattering length a describes repulsive interaction, and negative scattering lengths describe attractive interactions. The Many-body Hamiltonian in the second quantization consists of a kinetic term, an external potential and an interaction term. The wavefunction for condensates is derived from this Hamiltonian, below the Curie-temperature. This is where the Bogoliubov approximation - a type of perturbation theory - is applied. It transitions the model of mutual 2-body interaction into a mean-field theory.

In the end, the quadratic Hamiltonian is solvable using the Bogoliubov transform. The resulting reduced Hamiltonian gives rise to the Bogoliubov dispersion law for ϵ(p).

For low p the system then can't be excited by classical particles. This is the state of superfluidity. The dynamics of the condensate is determined by the Heisenberg equation of motion for the field operator. It recovers the Gross Pitaevskii equation.

Time time-dependence can be separate out of the GPE. By inspecting the matrix element, the statement for the time evolution can be fixed to be the difference of energy-eigenvalues, which, for very large N is about equal to the particle-number-derivative of the energy. This is exactly the chemical potential, so

By checking the scaling with radius R, the kinetic energy can eventually be neglected, as per the Thomas-Fermi approximation

The "Healing length" is an important length-scale in the modelling of an interacting Bose-Einstein condensate. It sits at about 0.5μm and sees the quantum pressure (effectively the kinetic energy) and the repulsive interaction balance. Because of that, the condensate wave function only varies beyond the Healing length. At this edge the Thomas-Fermi approximation is expected to break down. The competition between attractive interactions and the quantum pressure determines the stability of the gas. This is in turn determined by the particle number, which, beyond a critical value has an attractive interaction part that is large enough to collapse the condensate. This value sits at 0.0383. This implies that the ground state of weakly interacting dilute Bose gases are dominated by interactions (rather than the quantum pressure effects).

Since the BEC is considered a classical matter wave and it's equipped with a well-defined phase. This enables interference phenomena. The interference of the BEC is then simply modeled by linear combination of the wave functions. The density after TOF is given by

where ϕ is the initial phase, which fixes the pattern offset (or pattern position). λ is the pattern spacing.

From the GPE arises the description of solitons, a consequence of wave packets in which focusing and defocussing forces are balanced. In the GPE, these sources give effects in the dispersion term and the non-linear interaction term. Dark solitons can be induced to travel through an elongated BEC. It has a set theoretical (later more or less confirmed by experiments) frequency and adheres to a standard equation of motion

Artificial Gauge Potentials

For charged particles in an electric field, the quantum hall effect is a direct consequence of the minimal cyclotron orbit size and quantized energies. Because of the quantum hall effect, the Gauge transformation for the Schroedinger's equation requires a phase factor.

To justify the phase, consider the Aharonov-Bohm effect. Two path interferometer for single electrons with an infinite solenoid with an incoming B-field of B, and a vanishing outgoing B-field. Probing the magnetic field should create an interference pattern, and outside the solenoid, one would measure a zero Lorentz force. The Aharonov-Bohm phase puts into perspective the relation between local and global effects of the electromagnetic fields and gauge potentials.

with the flux quantum Φ₀. The Aharonov-Bohm phase Δφ is gauge invariant and geometric. It behaves constant under path deformation (meaning it's topologically even).

For the concept of Lorentz force for neutral particles, we require an artificial magnetic field. The resulting forces need to include an analogue for the Coriolis force and Lorentz force. This implies an external trap with a rotational element.

With the flux density n. The effective magnetic field is qB = 2MΩ. This setup introduces heating due to the rotation of circular potentials, and the trapping potential needs to balance the resulting centrifugal potential. This implies that Ω ≥ ω.

An artificial electromagnetic field can be realized using Raman coupling. The 2-photon Raman process places a 2kᵣ momentum on an atom, modifying the internal state, so internal and external d.o.f. can be coupled.

Raman-coupled BECs, feature 2-photon dressing fields to create the state, the F=1 hyperfine ground state creates a 3-Zeeman split, with a quadratic shift. The condensate frequency is detuned by some small shift from the Raman resonance. The resulting Raman dressed states are subject to spin and momentum superposition. Momentum transfer only happens along the x-axis.

The energy-momentum dispersion relation is characterized through the eigenvalues of the internal Hamiltonian. Without the coupling, each of the split Zeeman states creates a parabola.

Synthetically linking particle spin to momentum functions differently in solid-state systems and quantum gases. In the former, the relativistic effect come from the electron movement through the intrinsic material electric field. Topological insulators features strong spin-orbit coupling with results in material that is insulating and metallic at the same time. Quantum gases are engineered, so tunable as required. This includes the spin-orbit coupling. Requirements for synthetic spin-orbit coupling is opposite momentum for the up/down spin-states.

It's also realized through Raman coupling.

In a lattice structure, it's often convenient to assume the Hubbard model, which assumes a 2D square lattice with single bands and only nearest-neighbor hopping (characterized through the hopping coefficient J). The Eigenstates are written as Bloch-states, and the eigenenergies are reduced to the 1st Brillouin zone.

The Peierls substitution introduces the presence of a gauge potential via a complex tunneling matrix element. Using the resulting Peierls phase, the magnetic flux through a plaquette can be determined, which in itself will result in an analogue to the Aharonov-Bohm phase. Any particle moving through such a square lattice should experience the same flux through each plaquette. Assuming the Landau gauge and the Peierls phase will construct the Harper Hamiltonian.

The energy spectrum then is invariant under shifts α → α + 1. The magnetic field breaks the translational invariance along the x-axis, and exhibits a fractal structure. This is called the "Hofstadter butterfly". The Landau levels may be recovered for low magnetic fluxes.

Complex tunneling is a process by which a particle is excited to a state which is high enough to relax into a neighboring potential rather than its initial position. In an optical lattice, this shifts the dispersion relation. For the Floquet theorem (analogue: Bloch theorem in condensed matter physics), assume periodic linear differential equation. Floquet engineering is the method of tailoring a system through periodic driving. In optical lattices, the periodic driving of a system affects the Hamiltonian, so that H(t + T) = H(t) with the resulting eigenstates

Actual Floquet engineering may consist of lattice shaking in 1 dimension, which accelerates it in space and introduces an internal force, leading to a quasi-momentum. Another method is renomalization of the band structure, which leads to sinusoidal shaking and an effective Bans Structure with high-frequency limit.

After TOF the effective band structure sees the BEC occupying the minimal energy for the quasi-momentum k. Here, complex tunneling sees an intertial force that's asymmetric around q = 0.

Alternatively, the amplitude could be modulated, which assumes a tilted optical lattice. Let the standard lattice potential be assiciated with a tunneling constant J along the y-axis, and along the x-axis the potential be tilted via magnetic field gradient suppressing tunneling effects. This setup exhibits Raman-assisted tunneling along the x-axis. The Harper Hamiltonian is realized by tuning the alignment of the Raman beams propagating along x and y. The amplitude modulation occurs completely within Floquet theory and the Raman beams create some local optical potential, inducing a time-periodic on-site modulation of the lattice depth. The magnetic fluxes induce cyclotron motion with the orbit descriptive of mass currents around a square plaquette.