Preface
I would like to briefly discuss the motivation of the thesis in order to set the scene for the reader.
This thesis explores Feynman’s idea of quantum simulations by using ultracold quantum gases.
The concept of quantum simulations was introduced by Feynman as a way to avoid the difficulty
of simulating quantum phenomena with classical computers. To appreciate the limiting power of
classical computers, we refer to page 1 of the book ‘‘Quantum Field Theory of Many-Body
Systems’’ (2004) by Xiaogang Wen:
‘‘In the 1980s, a working station with 32Mbyte RAM could solve a system of eleven interacting
electrons. After twenty years the computing power has increased by 100-fold, which allows us to
solve a system with merely two more electrons. The computing power required to solve a typical
system of 10 23 interacting electrons is beyond the imagination of the human brain. A classical
computer made by all the atoms in our universe would not be powerful enough to handle the
problem. Such an impossible computer could only solve the Schrödinger equation for merely
about 100 particles…’’
So if we can design a quantum system where the Hamiltonian of the designed quantum system is
the same as a target quantum system which is difficult to handle, then by observing the designed
quantum system we can get useful information about the target quantum system. A quantum
simulator is such a quantum device which mimics the dynamics of another quantum system. The
rapid development of experimental techniques of ultracold quantum gases in recent years makes
these systems attractive as quantum simulators.
The thesis explores this idea of quantum
simulations by using ultracold quantum gases. In the first part of this thesis, we develop a general
1
method which is applicable to a wide class of species, ranging from atoms to molecules and even
to nanoparticles, and show how to decelerate a hot fast gas beam to zero velocity. From this part
of the thesis, we give the reader a flavour of how to obtain these cold gases. In the second part of
the thesis, we will illustrate what one can do with these cold quantum gases. In particular, we
design special Hamiltonians that can be realised with multi-component ultracold fermionic atoms
in optical lattices by controlling the spin-dependent hopping and on-site spin flipping with
Raman lasers. We demonstrate the power of quantum simulations which are relevant to both
condensed matter physics and high energy physics. For the quantum simulations of condensed
matter physics, we show how to design a Hamiltonian which will give Dirac-Weyl fermions with
any arbitrary spin by properly tailoring the spin-dependent hopping. This generalization to
arbitrary spin goes beyond the spin ½ Dirac fermion scenario found in graphene and topological
insulators which are currently two actively researched topics in condensed matter physics. We
also show what new physics we can learn from these high spin Dirac-Weyl fermions. For the
quantum simulations of high energy physics effects, by turning on the on-site spin flipping, we
further show how to simulate topics such as modified dispersion relations and neutrino
oscillations, which are concepts beyond the Standard Model of particle physics. This thesis
demonstrates the important role ultracold quantum gases play in terms of quantum simulations in
order to address some of the most challenging topics in modern physics.
2
PART I:
Decelerations due to Cavity-Induced Phase Stability
3
Chapter One: Introduction
1.1 Background
Since the first generation of atomic Bose Einstein Condensate (BEC) [1] in 1995, the field of
cold and ultracold quantum gases has achieved remarkable progress, such as the observation of
Degenerate Fermi Gas (DFG) [2], the realization of superfluid to Mott-insulator transition with
ultracold atoms in optical lattice [3], and most recently, the demonstration of Dicke quantum
phase transition with a superfluid gas in an optical cavity [4]. (For a comprehensive discussion
regarding ultracold atomic Bosonic/Fermionic gases, see the review articles [5, 6]). Ultracold
atomic gas is not the only species scientists are interested in, ultracold molecules, especially
polar ultracold molecules with long-range and anisotropic electric dipole-dipole interaction, have
also stimulated great interests in the community [7, 8]. Ultracold molecules can be used for a
variety of purposes, like, as qubits in quantum computation [9], to constrain the time variation in
fine structure constant [10], search for parity violation [11] and test physics beyond the standard
model by measuring the electric dipole moment (EDM) of electron with great precision[12,13].
Moreover, ultracold molecules are also anticipated to be important in chemistry, as resonance
and tunnelling phenomena could be dominating effects at ultracold temperatures, with reaction
rates predicted to be many orders of magnitude larger than at room temperature for some species
[14, 15].
In view of the importance of cold or ultracold quantum atomic and molecular gases in modern
science, one may wonder how to really get these cold species. While the traditional method to
cool atoms uses many consecutive absorption-emission cycles in a closed multilevel system to
4
extract kinetic energy from the atoms [16], the method is not universal in that it cannot generally
be applied to molecular species, except for a few special cases [17], due to their complex energy
structures that preclude the closed-cycling transitions. Nonetheless, ultracold molecules can be
created from association of laser cooled atomic species by photoassociations or on magnetic
Feshbach resonances at microkelvin temperatures [18, 19]. Progress has been made recently to
transfer these molecules in high vibrational levels to absolute ground states [20]. Buffer-gas
cooling is a more general method, which can dissipatively cool atomic or complex molecular
species by use of thermalizing collisions with buffer gas in a cryogenic cell [21]. Buffer-gas
cooled BEC has been reported recently [22]. Optical cavity cooling is another general scheme
independent of the specific internal energy structure of the species. It cools particles by a
dissipative optical dipole force arising from the nonadiabatic dynamics between the optical field
and particles in the cavity [23-28]. An advantage of this method is the low temperatures it can
achieve, which are limited by the cavity linewidth and can be much lower than the Doppler limit
set by the atomic linewidth. Optical cavity cooling of atoms has been demonstrated
experimentally [29-31]. In general, cavity cooling can be taken as a secondary cooling scheme
which is employed to reduce further the temperature of a primary cold sample, typically in the
order of tens to hundreds of millikelvin, to the ultracold regime at submillikelvin temperatures.
The first part of this thesis explores the feasibility to create such a primary cold sample from a
hot fast gas beam within an optical cavity. Traditional methods to get such a primary cold
sample relying largely on phase space filtering technique, where conservative electrostatic [32],
magnetic [33, 34] or optical potentials [35] are used to filter out a narrow velocity distribution of
a hotter gas and then transfer them from the moving frame in the gas beam to zero velocity in the
laboratory frame (see figure1.1 for the setup of the Stark, Zeeman and optical Stark decelerators).
5
Figure 1.1: Schematic of the setup for the Stark decelerator, Zeeman
decelerator and optical Stark decelerator (taken from Nat. Phys. 4, 595(2008)).
6
For the electrostatic Stark deceleration, the dipole of a polar molecule is acted on by the electric
field gradients. For molecules in low-field-seeking states and with even-numbered electrodes
switched to high voltage and odd-numbered electrodes grounded initially, the molecules will
experience the increasing electric field as a potential hill when approaching the plane of the first
electrodes, and thus lose kinetic energy on the upward slope of the potential hill. If we switch off
the electric field when the molecules have reached the top of the potential hill, the acceleration
on the downward slope of the hill can be avoided. At the same time, if we switch the electrodes
that were grounded to high voltage, the molecules will find itself again in front of a potential hill
and will again lose kinetic energy when climbing this hill. By repeating this process many times,
the velocity of the molecules can be reduced to desired value. This process can be described as
the trapping of a packet of molecules in a travelling potential well and thus the molecules can be
slowed down when the velocity of the traveling potential well is decreased by computer
controlled sequences of switching [32]. Also since the electric field is always lower on the axis
than on the electrodes, the transverse confinement keeping the molecules together in the
transverse direction can be achieved for molecules in low-field-seeking states. Gas of polar
molecules in a single quantum state at 10 mK were reported [32]. Recently, the magnetic
analogue of the Stark decelerator, the so called Zeeman decelerator that relies on the interaction
between magnetic dipole and the magnetic field, has also been developed [33, 34]. Furthermore,
optical fields can also provide a general method to manipulate the motion of neutral molecules
since an intense optical field will polarize and align molecules, thus the polarized molecules will
experience a force that is proportional to the gradient of the laser intensity. By carefully
controlling the frequency difference between the two lasers that create the optical lattice, the
velocity of the lattice can be lowered and thus the molecules that are trapped by the lattice can be
7
decelerated to any given velocity. P. Barker et al [35] have demonstrated experimentally that
with a suitable choice of parameters, the molecules can make exactly a half oscillation within the
optical potential where NO molecules were decelerated from 400 to 270 m/ s . Following the
success of these decelerators, other deceleration schemes have also been studied theoretically. A
microwave Stark decelerator was proposed to slow a hot polar molecular beam by using a timevarying standing-wave in a cavity that is created by timing the external pump source in a similar
way as in electrostatic Stark decelerator [36]. Deceleration of a particle in a bistable optical
cavity is another scheme in which the deceleration force is induced by feedback-controlled
switching of the optical pumps between a high and a low state [37]. A setback for this scheme is
that the cooling effect seems to be washed out quickly with the increase of the particle number
due to lack of collective motion of particles in the cavity.
In the first part of this thesis, we will demonstrate that new deceleration schemes beneficial from
the strong-correlation dynamics of the molecules induced by optical cavity are feasible.
Specifically, we identify a novel phase stability mechanism from the intracavity field induced
self-organization of a fast-moving gas beam into travelling packets in the bad cavity regime,
which is then used to decelerate the beam by properly introducing the decelerating force. Since
this new phase stability mechanism stems from the collective particle-field dynamics of all the
particles in the beam, this mechanism ensures the phase stability of the majority of the particles
in the cavity rather than a small fraction determined by the acceptance volume as in phase space
filtering techniques. It should be pointed out that the deceleration methods studied in this thesis
are in principle applicable to a wide class of species, ranging from atoms to molecules or even to
nanoparticles, though we use molecules as examples in the following.
8
1.2 Model
Figure 1.2: Schematic of the optical cavity based decelerator where the
feedback for controlling the pump may or may not be used for different
deceleration schemes.
We consider a fast molecular beam travelling along the axis of an optical cavity that supports a
standing-wave mode of the form cos( kx) exp( ict ) , where k is the wavenumber and c the bare
cavity resonance frequency (see figure 1.2). The cavity is pumped by laser beams transverse to
the axis of the cavity with pump frequency  p such that the photons in the cavity are created
from scattering of the pump photons by the molecules in the cavity and we consider a scheme
where the pumps are far-off-resonance from any electronic transitions of the molecules in the
cavity. At low saturation where the spontaneous emissions of the molecules are negligible, we
can adiabatically eliminate the internal dynamics of the molecules and treat them as classical
polarizable point objects. Therefore, the deceleration methods studied in this thesis are in
principle applicable to a wide class of species, ranging from atoms to molecules or even to
nanoparticles. A moving molecule inside the cavity serves as effective refractive index which
9
shifts the cavity resonance frequency in a position dependent way U ( x)  U 0 cos 2 (kx) , where
U 0   p Re[ ( p )] /( 0V ) [37, 38],  ( p ) is the polarizability of the molecule, V the mode
volume of the cavity and  0 the permittivity of free space (The scattering loss resulting from the
imaginary part of  ( p ) has been neglected due to the far-off-resonance scheme). In the
semiclassical limit, the combined system dynamics of the intracavity field amplitude and the
centre-of-mass motions of the molecules can be described by the following coupled differential
equations in one dimension [25, 28, 38] (also see Appendix A for details):
  (i c   )  iU 0  cos 2 (kx j )  i  cos( kx j )
j
xj 
j
k
[U 0 |  |2 sin( 2kx j )  2 Re{ }sin( kx j )]
m
,
(1.1)
where  is the amplitude of the photon number in the cavity,  c   p  c the detuning of the
pump lasers with respect to the cavity resonance,  the cavity decay rate, x j and m the position
and mass of the jth molecule, j  1, 2, ..., N , where N is the total number of the molecules and 
the effective pump amplitude by a molecule. In equation (1.1) noise terms are neglected,
because when the molecular beam is fast-moving, the noise terms have a negligible effect on the
system dynamics, which is supported by work of H. Ritsch et al as in [38].
Equation (1.1) is a discrete description of the system which is suitable for numerical simulation
but not suitable for theoretical analysis. In order to establish a model which is suitable for
theoretical analysis, we reformulate equation (1.1) in the following way: when the molecular
number in the beam is sufficiently large, the molecules can be described by the position and
10
velocity distribution function f ( x, v, t ) [28], then the position-related summations reduce to
 cos(kx )  N  f ( x, v, t ) cos(kx)dxdv and  cos (kx )  N  f ( x, v, t ) cos (kx)dxdv .
2
j
2
j
j
j
Consequently, the first equation in equation (1.1) can be rewritten as
  (i c   )  iNU 0  f ( x, v, t ) cos 2 (kx)dxdv  iN  f ( x, v, t ) cos( kx)dxdv ,
(1.2)
while the distribution function f ( x, v, t ) obeys the collisionless Boltzmann equation
f ( x, v, t )
f ( x, v, t ) F ( x, t ) f ( x, v, t )
v

 0,
t
x
m
v
(1.3)
F(x, t) is the force exerted on the molecules with
F ( x, t )  kU0  sin( 2kx)  2k Re( ) sin( kx) .
2
(1.4)
Equations (1.2)-(1.4) form the statistical description of the system. Thus the above two
descriptions serve for different purposes in this thesis, while the statistical description is mainly
used for theoretical analysis, the discrete description is used for numerical simulations directly.
The remaining chapters of the first part of the thesis are a demonstration of finding a new work
window of the semiclassical equation (1.1) which is suitable for molecular deceleration based on
optical cavity and the chapters are organized as following. In chapter two, we will explain how
new phase stability mechanism can emerge from the bad cavity limit. In chapter three, three
different deceleration schemes based on this new phase stability mechanism will be
demonstrated. In chapter four, practical issues of the deceleration schemes and outlook will be
discussed.
11
Chapter Two: Cavity-induced Phase Stability
An effective deceleration needs two ingredients: one is the phase stability and the other is the
deceleration force. While the requirement of deceleration force is obvious, phase stability means
the particles should be bunched together stably during each stage of the deceleration process, so
the deceleration process can be repeated stage by stage till a desired final velocity is achieved.
While the traditional phase stability mechanism, like the one used in Stark deceleration [32], is
imposed by external source, the phase stability mechanism used in this thesis is self-emerging
spontaneously from the collective particle-field dynamics. This new phase stability is motivated
by the self-organization-like phase transition predicted in [25] and realized experimentally
recently [4]. It is predicted [25] that when a stationary cold atomic cloud is placed in a standingwave cavity pumped by a laser in a direction perpendicular to the cavity axis, the initial
homogeneous atomic distribution evolves to a regular patterned state which maximally scatters
the pump photons into the cavity by atomic crystallization at either the even or the odd antinodes
of the cavity mode. This self-organization-like phase transition occurs above certain pump
intensity (threshold) and spontaneously breaks a discrete translational symmetry of the system.
When a fast-moving molecular beam as in the deceleration studies is considered instead of the
stationary cold atomic cloud, a new parameter, the central velocity of the beam v0 , is introduced
into the system, which brings in new physics that has no counterpart as in the case of stationary
cold atomic cloud. Phase transition of a fast-moving gas beam in a ring cavity pumped by two
counter-propagating laser fields through the cavity mirrors has been studied in [39], and it shows
the phase transition in a ring cavity occurs only when the frequency shift induced by the particles
12
is larger than the cavity linewidth, which implies a large ensemble of particles or a high Q cavity.
In the ring cavity, which is the simplest multimode cavity supporting two counter-propagating
modes, the locations of the antinodes are collectively determined by the particles moving in the
cavity, instead of self-emergent as in the standing-wave cavity, so the translational symmetry
breaking of the system is continuous rather than discrete as in the standing-wave cavity. This
collective determination of the antinodes in a ring cavity results in a shift of peak density
position of the particles from the optical field minima in the cavity and thus the system cannot
reach a time independent self-consistent particle-field steady state [39]. In our setup which uses a
standing-wave cavity, we expect a self-consistent particle-field steady state because the
antinodes of the standing-wave cavity are fixed by the cavity geometry thus when the peak
density position of the particles moves in the cavity, it will function as a dynamic Bragg grating
that modulates the intracavity field periodically by superradiant scattering of the pump photons
into the cavity. Meanwhile, such steady state is also expected to achieve in the bad cavity regime
since in this regime there is no dissipative factor to destroy its stability. This steady state will be
the base for the new phase stability mechanism to be implemented for different deceleration
schemes investigated in chapter three.
In this chapter, we will demonstrate this new phase stability mechanism both analytically and
numerically. First we will derive the threshold for this self-organization-like phase transition of a
fast-moving molecular beam within a standing-wave cavity in the adiabatic limit analytically.
Then numerical investigations will be carried out to study the dynamical interplay between the
moving molecules and the intracavity field, particularly focusing on the self-consistent moleculefield steady state.
13
2.1 Linear stability analysis and phase transitions
We consider the phase transition as a linear stability problem of the solution of the coupled
intracavity field and Boltzmann equations (1.2-1.4). To obtain the threshold pump for the onset
of the phase transition in the adiabatic limit, we linearize the coupled equations around the trivial
solution, i.e., initial conditions, and then solve the linearized equations as an eigenvalue problem.
The parameter dependence on the threshold gives the scaling laws of the system. To do so, we
expand the variables
 (t )  0  (t ) , f ( x, v, t )  f0  f ( x, v, t ) ,
(2.1)
where  0  0 is the initial photon number amplitude in the cavity, and f 0 is the initial positionvelocity distribution function of the molecular beam, assumed to be uniform in space and
Gaussian in velocity, f 0  f x  f v  1 / L  exp[ (v  v0 ) 2 / 2 2 ] / 2 2 , where v0 is the central
velocity,  the velocity spread and L the length of the beam, respectively. Substituting (2.1) into
equations (1.3), (1.4) and keeping only linear terms, we obtain the linearized Boltzmann equation
f
f 2k
f
v

Re( ) sin( kx) 0  0 .
t
x
m
v
(2.2)
We then express the trial solution of equation (2.2) in the form of a travelling density wave with
velocity v0 , i.e.
f  et f v [ A sin( kx  kv0t )  B cos( kx  kv0t )] ,
(2.3)
where A , B are constants and  is to be determined by the system parameters. This trial solution
of travelling first harmonic wave with velocity v0 is based on two facts: the beam is travelling
14
with central velocity v0 and the source term in its parent equation (2.2) is a first harmonic wave.
For convenience, we recast the trial solution as
f  e t f v [ A(t ) sin kx  B(t ) cos kx] ,
(2.4)
where A(t )  A cos kv0t  B sin kv0t and B(t )  B cos kv0t  A sin kv0t are two orthonormal bases.
In the adiabatic limit, the intracavity field follows the change of the molecular distribution
instantaneously, so from equation (1.2) we get
Re( )  et N c B(t ) /[ 2( c2   2 )] ,
(2.5)
where  c  (c  U 0 N / 2) is the modified cavity detuning. Substituting the trial solution (2.4)
together with the expression (2.5) into equation (2.2), we obtain
et [ A(t ) sin kx  B(t ) cos kx]  et kv0[ B(t ) sin kx  A(t ) cos kx]
 et kv[ A(t ) cos kx  B(t ) sin kx]  et k (v  v0 ) [ B(t )] sin kx  0
,
(2.6)
where   N c 2 /[ m 2 ( c2   2 )] . The two Fourier components in equation (2.6), sin kx and
cos kx , must equal to zero separately, which leads to the eigenvalue equation
 
kv  kv
0

kv0  kv  k (v  v0 )  A(t )
  B(t )  0 ,



(2.7)
with solutions 2  [k (v  v0 )]2 (   1) . When   0 , i.e.,
   thr 
m 
 N
( c2   2 )
,
( c )
(2.8)
the trivial solution becomes unstable (phase transition), which leads to the exponential growth of
a travelling density wave in the form of equation (2.3) until saturation occurs from the nonlinear
effects. The expression (2.8) defines the threshold pump for the phase transition and also gives
15
the scaling laws with respect to the parameters of the system. We note that in the adiabatic limit,
where the intracavity field always follows the molecular motion instantaneously, the threshold is
independent on the central velocity of the beam but proportional to the velocity spread. Also as
2
 thr
 1 / N , the phase transition is more likely to be observed for a large ensemble of molecules.
Equation (2.8) is consistent with the mean-field approximation [40] and our previous work [28]
under the relation m 2  k BT / 2 .
The physical mechanism underlying the self-organization of the fast molecular beam in the
adiabatic limit is similar to that for a stationary cold atomic cloud in standing-wave cavity [25].
The travelling molecules being transversally pumped by the lasers scatter photons into the cavity
according to the source term i  j cos( kx j ) in the first equation of (1.1). Molecules in the nodes
of the standing-wave cavity mode do not make a contribution, whereas those in the antinodes
scatter maximally. The photons scattered by molecules separated by half a wavelength have
opposite phase and interfere destructively, so preventing the buildup of the intracavity field for
the uniform spatial distribution of the beam. However, due to density fluctuations of the
molecules, small intracavity field can emerge momentarily which, for red-detuned laser pumps,
creates an attractive optical potential to pull molecules to every other antinodes of the cavity
mode. When the pump intensity exceeds a certain level (threshold), this induced molecular
redistribution within wavelength-spaced wells at every other antinodes can strongly enhance the
Bragg-type scattering of the pump photons into the cavity, which in turn further deepens the
optical potential and traps more molecules in a runaway process. In the initial stage, the
modulation on the molecular distribution function grows exponentially in the form as given in
equation (2.3), evidenced by direct simulations of equation (1.1) as shown in figure 2.1(a), where
16
the position-velocity snapshots in the initial stage are given at moments (1)-(4). The intracavity
intensity also grows exponentially in this period as shown in figure 2.1(b) where the
corresponding moments (1)-(4) are also marked.
Figure 2.1. Caption overleaf.
17
Figure 2.1: (a) Phase-space snapshots of the molecular beam at different
moments (1)-(6) from the simulation of equation (1.1), where the
corresponding intracavity intensities at moments (1)-(6) are marked in (b). The
molecules marked with red are those outside the separatrix (green) at moment
(5), which is determined by the intracavity intensity at this moment. These
molecules are tracked to help understand the self-organization and evolution of
the beam and the slow oscillation of the intensity profile. (b) The evolution of
the intracavity intensity with time. Parameters used in the simulation,
k 2 / m  106 , U 0  10 7  ,  c  10 , N  10 4 ,   2.5th and the initial
distribution of the beam is Gaussian in velocity, with kv0 /   0.1 ,
k /   0.01 and homogeneous in space within the length of five wavelengths
(only three are shown in (a)). Periodic boundary conditions are used in the
simulation.
2.2 Formation of travelling molecular packets
The runaway process is eventually saturated by the nonlinear effects of the system when the
amplitudes of the intracavity field and the travelling molecular wave have grown sufficiently
strong (moment (4) in figure 2.1). Figure 2.1 further shows the long-term time evolution of the
molecular distribution and intracavity intensity, where the intracavity intensity exhibits two
characteristic oscillations after the initial exponential growth (see figure 2.1(b)). As we will see
below, whereas the period of the fast oscillation corresponds to the time for the trapped majority
of the molecules by a moving optical lattice to travel through a cycle of the standing-wave cavity
18
mode, the slow oscillation is transient and related to the motions of the minority molecules that
are untrapped by the moving lattice.
After the saturation, the majority of the molecules are found to be bunched into packets and
move synchronically with central velocity v0 in the cavity, thus in order to illustrate the dynamics
of the system, the distribution function of the molecules can be approximately written as
f ( x  v0t ) . Under this approximation, the last term on the right hand of equation (1.2) can be
rewritten as
N  cos( kx) f ( x  v0t )dx  N cos( kv0t )  cos( kx) f ( x)dx  N sin( kv0t )  sin( kx) f ( x)dx   N eff cos( kv0t ) , (2.9)
where we have set moment (4) as t=0 where the maximum values of molecular position
distribution f (x) are positioned at x  ...  3 ,   ,  ,3 ... , so the integral  sin( kx) f ( x)dx  0 ,
and N eff   N  cos( kx) f ( x)dx is the effective number of the molecules. Since NU0   the
second term in right hand side of equation (1.2) and first term in right hand side of equation (1.4)
can be neglected. Therefore, the important dynamics of the system in adiabatic limit can be
approximately expressed from equation (1.2) and (1.4) as
|  |2  I 0 cos 2 (kv0t )
,
mx  F0 [sin( kx  kv0t )  sin( kx  kv0t )]
where
2
I 0   2 Neff
/( 2c   2 )
is
the
(2.10)
amplitude
of
the
intracavity
intensity
and
F0  kc Neff 2 /( 2c   2 ) is the amplitude of the dipole force acting on the bunched molecular
packets from the standing-wave potential, which consists of two counter-propagating optical
19
lattices with the same velocity v0 as the bunched molecular packets. As discussed in electrostatic
Stark deceleration [41], the lattice whose velocity comes close to the bunched molecular packets
interacts more significantly with it, so the lattice that propagates oppositely to the bunched
molecular packets can be neglected. As such, the important system dynamics of the bunched
molecular packets moving in the standing-wave cavity mode is reduced to bunched molecular
packets travelling within an optical lattice of the same velocity. So the bunched dynamics of the
molecular packets comes essentially from the trapped dynamics of the packets by the potentials
of the moving lattice, equivalent to the transportation scheme as in Stark deceleration. The phase
stability in our scheme thus results from the cavity-induced collective behavior of all the
molecules in the beam. This mechanism ensures the phase stability of majority of the molecules
in the cavity rather than a small fraction determined by the acceptance volume as in phase space
filtering techniques.
After illustrating the trapped dynamics of the molecular packets by the moving lattice, we now
turn to the intracavity intensity. The first equation of (2.10) captures well the fast oscillatory
behavior of the intensity profile, which stems from the fact that bunched molecular packets travel
along each cycle of the standing-wave cavity mode with period of  / kv0 and thus switch the
intracavity intensity on and off dynamically with the same period  / kv0 . The slow oscillatory
behavior of the intensity profile is related to the motions of the minority molecules that are
untrapped by the moving lattice. These untrapped molecules (marked with red in figure 2.1(a))
are best identified at the first dip of the intensity profile (moment (4) in figure 2.1(a)) when they
move to the space between the bunched molecular packets. Molecules within the separatrix
(green curve, which is determined from the height of the potential with the moving lattice), are
trapped while these outside the separatrix are untrapped. These untrapped molecules are then
20
tracked at different moments (1)-(6) as shown in figure 2.1(a) to help understand the selforganization and evolution of the beam and the slow oscillatory behavior of the intensity profile.
Since the untrapped molecules are travelling within the moving lattice from one potential well to
another, when they travel to the crests (troughs) of the potential, which correspond to the
minimums (maximums) of the spatial molecular distribution, they mainly serve as ‘defects’
(‘gains’) which scatter photons with opposite (same) phase to the trapped majority of the
molecules and thus undermine (enhance) the intracavity intensity slightly. Since the dynamics of
the intracavity intensity is determined mainly by the trapped molecular packets, the trajectories
of the untrapped molecules are constantly modified by the intracavity field in an uncorrelated
manner. And also the trapped and untrapped molecules near the interface of the separatrix can
switch their roles as the intensity varies. As a result, the correlation between the untrapped
molecules is eventually washed out, leading to the disappearance of the slow oscillatory behavior
in figure 2.1(b).
In the moving frame with the lattice, the trapped molecules are circulating approximately along
closed-orbits in phase-space within the lattice potential, so the velocity distribution of the trapped
molecules is determined by the amplitude of the lattice potential. An increase of the pump
intensity will lead to the increase of the amplitude of the lattice potential, which in turn will
widen the velocity distribution of the trapped molecules. In the same moving frame, the
untrapped molecules are travelling along the lattice potential, so the travelling period, i.e., the
time required by the untrapped molecules to travel through a cycle of cavity mode, is also
determined by the amplitude of the lattice potential. In a similar way, an increase of the pump
intensity will lead to the increase of the amplitude of the lattice potential, which in turn will
21
shorten the travelling period of the untrapped molecules along the lattice potential, thus
accelerating the disappearance of the slow oscillatory behavior of the intensity profile.
Figure 2.2. Caption overleaf.
22
Figure 2.2: Self-consistent molecule-field steady state of the system. (a) The
phase-space snapshots of the molecular distribution at three moments A, B and
C; also shown is the standing-wave potential proportional to cos 2 (kx) . At
moments A and C, the centers of the molecular packets are located at the
troughs of the standing-wave potential, while at moment B, the centers are
located at the crests. (b) The dynamical interplay between the intracavity
intensity and the central velocity of the molecular packets.
After the disappearance of the slow oscillatory behavior of intensity profile, the system
approaches its self-consistent molecule-field steady state, the main characteristic of which is the
fast oscillation of the intracavity intensity correlated with the travelling of the bunched molecular
packets along the standing-wave potentials (stage (6) in figures 2.1). At this stage, the system
dynamics is more accurately described by equation (2.10). In the above analysis to illustrate the
phase stability mechanism, we have neglected the relative minor effects of the optical lattice in
second equation of (2.10) that has the opposite velocity to the bunched molecular packets. This
lattice has however a perburbative effect to the motion of the bunched molecular packets,
inducing a weak periodic oscillation to the central velocity of the travelling molecular packets
(see figure 2.2(b)). This weak oscillation can be understood from the ascending and descending
processes of the bunched molecular packets in the standing-wave potential, as evidenced from
figure 2.2, where from time A to B (or from time B to C), the molecular packets climb up (or
down) the standing-wave potential thus its central velocity decrease (or increase). These main
characteristics of the self-consistent molecule-field steady state form the foundation of the
multistage decelerations to be discussed in the next chapter.
23
Chapter Three: Deceleration schemes
The self-consistent molecule-field steady state as discussed in the previous chapter implies a new
phase stability mechanism. Since the motions of the stably bunched molecular packets in the
standing-wave cavity are the climbing-up and down behaviors within the potential wells of the
standing-wave cavity mode, a proper modification of the cavity setup can conveniently introduce
the deceleration force required to slow down the molecules. In this chapter we will demonstrate
the feasibility of this new phase stability mechanism for effective multistage decelerations.
Specifically, we will show three schemes for the effective decelerations. In the first scheme, the
cavity is the same as in chapter two, i.e., the cavity is still working in the adiabatic limit. The
deceleration force in this case comes from the switched pumps between a high pump intensity
and a low pump intensity as in electrostatic Stark deceleration, but in our scheme, the switching
dynamics is controlled by feedback automatically rather than the timing sequences externally as
in electrostatic Stark deceleration [32]. In the second scheme, the deceleration force comes from
the nonadiabatic setup of the cavity, i.e., the cavity is not working in the adiabatic limit. In a
nonadiabatic cavity, there is a delay response of the cavity field to the motion of the particles in
the cavity, and this delay dynamics will give the deceleration force needed to slow down the
molecules. As we will see later on, in this scheme, the nonadiabaticity will undermine the phase
stability, so we need to seek a balance between phase stability and deceleration force. Finally,
we study a composite scheme where the above two schemes can be combined together to
perform more efficiently.
24
3.1 Bad cavity regime
At the adiabatic limit, since the position information of the bunched molecular packets is
encoded in the output of the intracavity intensity instantaneously, we can modulate automatically
the pump intensity of the lasers by using the output of the intracavity intensity via feedback
mechanism, which will create a deceleration force to slow down the bunched molecular packets
in a similar way as in electrostatic Stark decelerator. In the following, we will first introduce the
principle of this deceleration scheme in section 3.1 then simulation results and analysis will be
presented in section 3.2.
3.1.1 The deceleration principle
Since the bunched molecular packets behave like a single molecule modulated by the effective
number N eff , we use the single molecule model to illustrate the idea for deceleration. In the
adiabatic limit, the intracavity field follows the motion of the molecule instantaneously, as given
by equation (1.1)
|  |2 
cos 2 (kx) 2
2

 2  [ c  U 0 cos 2 (kx)]2 K ( x)
k c 2 sin( 2kx)
d
mx 
  [V ( x)]
2
2
2
[ c  U 0 cos ( x)]  
dx
,
(3.1)
where K ( x)  [ 2  ( c  U 0 cos 2 (kx)) 2 ] / cos 2 (kx) is position-dependent intensity modulation
parameter and V ( x)   c 2 [arctan(  c /   U 0 cos 2 (kx) /  )] / U 0 is the potential. As  c  U 0 ,
we expand cos 2 (kx) to its leading order and neglect the constant potential term,
V ( x)   c 2 cos 2 (kx) /( 2  2c ) . When the pump is constant (no feedback), the molecule
25
travels along the cosine-squared conservative potential and there will be no net force to
accelerate or decelerate the molecule.
Figure 3.1: The working principle of the optical cavity based molecular
decelerator with feedback-controlled time-varying optical pumps. (a) The
operation of the feedback loop. (b) The evolution of system quantities with the
time-varying optical pumps.
Now we introduce the time-varying optical pumps in the following way. When the molecule is
about to move down the potential hill as shown by point 1 in figure 3.1, the pump is switched
from the high intensity level H to the low intensity level  L (jump from point 1 to 2 in figure 3.1).
The molecule gains kinetic energy during the moving-down process, which corresponds to the
potential difference between points 2 and 3 in V (x ) . The pump is then switched back to the high
level H , when the molecule has arrived point 3 and starts to climb up the potential hill. It will
26
lose kinetic energy during the climbing-up process, which equals the potential difference
between points 4 and 1 in V (x ) . After the completion of a full cycle in the cavity mode, the
molecule will lose energy that equals the amount between points 1 and 2 in V (x ) . This
deceleration scheme presents an optical version of the Sisyphus cooling, where the conservative
motion of the molecule is interrupted by sudden transitions between high and low pump
intensities. In this way, the molecule is slowed down as it travels along the standing-wave
potential.
The switching of the pump in the above process can be controlled automatically by the output of
the intracavity intensity via a feedback loop. The two jumps in each cycle occur at the time when
the intracavity intensities are at I1 and I 2 , as shown in figure 3.1. For the case of a single
molecule I1  0 and I 2   L2 [ 2  ( c  U 0 ) 2 ] . Relevant issues of setting the two values for the
deceleration of the travelling molecular packets will be discussed below.
3.1.2 Numerical simulations
In this scheme, the phase stability is maintaining well at adiabatic limit, while the deceleration
force is deriving from the time-varying pumps mechanism as described above. We find that in
order to ensure the phase stability of the bunched molecular packets, at least one of the pump
intensities should be kept above the threshold pump for phase transition as described by
expression (2.8). Figure 3.2 and figure 3.3 show the simulation results.
As seen from figure 3.2, the average velocity of the travelling molecular packets decreases
linearly (constant deceleration) and the switching intervals of the pumps increase because the
molecular packets spend more time in one cycle of the standing-wave cavity mode due to the
27
reduced average velocity. The deceleration process stops at the time when the average velocity
of the molecular packets is reduced to the point that significant amount of molecules no longer
travel synchronously with the rest. The reduction of molecular number of the bunched molecular
packets leads to the decrease of the intracavity intensity which will eventually be lower than the
threshold I 2 for switching. As a result, the pump no longer switches and stays in the low intensity
level as figure 3.2 shows.
Figure 3.2: Deceleration of the travelling molecular packets by time-varying
optical
pumps.
The
following
parameters
are
used,
 L  0.8th ,  H  4th , I min  100, I max  10000 , while other parameters are the
same as in figure 2.1.
28
Figure 3.3: (a) The evolution of the phase-space plots of the molecular packets
at different times. (b) The velocity distributions corresponding to the phasespace plots in (a). Parameters used are the same as in figure 3.2.
29
Figure 3.3 (a) plots the position-velocity distributions of the travelling molecular packets at
different times, which show the stability of the bunched molecular packets during the
deceleration process. Figure 3.3 (b) plots the velocity distributions corresponding to figure 3.3(a).
The initial half-width of the velocity distribution of the molecular packets is v  0.039( / k ) as
marked in figure 3.3 (b). The calculation based on the trapped molecules by the optical potential
of the moving lattice is v  2U / m  2 c Neff /[ m( 2  2c )]  0.034( / k ) , with
N eff  4560 estimated from the phase-space plot ( t  0(1 /  ) in figure 3.3 (a)). The half-width of
the velocity distribution in the deceleration process is v  0.055( / k ) , which is widened
slightly (see figure 3.3(b)) compared with the initial distribution. The slight widening of the
velocity distribution is accompanying with the narrowing of the position distribution due to the
conservation of the phase space distribution, which is evidenced by the increased effective
number of the molecules N eff  6700 in the deceleration process (estimated from the simulation
results). Compared to the deceleration scheme will be shown in section 3.2, there is no extra
widening factor to the velocity distribution at the end of the deceleration process in the present
scheme due to the fact that when the pump intensity ceases to jump between the two states,
molecular velocity distribution does not spread under the adiabatic condition.
Using the single molecule approximation for the bunched molecular packets with the effective
molecular number of N eff , the energy extracted from the molecular packets each cycle, as
discussed in figure 3.1(b) is expressed as
W  V ( x1 )  V ( x2 )  c Neff (H  L )[cos 2 (kx2 )  cos 2 (kx1 )] /( 2  c ) .
2
2
2
Since
30
(3.2)
I   2 N eff cos 2 (kx) /( 2c   2 )
2
2
cos 2 (kx2 )  cos 2 (kx1 )  [( κ 2  2c ) / N eff
]( I max /  L  I min /  H ) ,
2
(3.3)
2
the expression (3.2) can be simplified to
 (  L ) I max I min
W  c H
( 2  2).
Neff
L H
2
2
(3.4)
Due to I min  0 , I max  L the energy extracted in each cycle is proportional to ( H   L ) . This
2
2
2
implies that the deceleration force is constant, which in turn explains a constant deceleration of
the molecular packets as in figure 3.2. The deceleration is 0.85 104 ( 2 / k ) from numerical
simulation (figure 3.2), while the theoretical value from equation (3.4) is 1.1 104 ( 2 / k ) which
shows a good qualitative agreement considering the simple single-molecule treatment.
Normally, at the end stage of the deceleration process, some molecules stop moving collectively
with the molecular packets as they are decelerated to near zero velocities, so the effective
number of the molecules N eff will decrease. Then the intracavity intensity will drop. In order to
keep the jumps work at the end stage of the deceleration process, we can set the jump threshold
I 2 lower than the maximum that can be achieved. This setup does not change the picture of the
deceleration scheme but only makes the deceleration process not at its maximal efficiency
because the energy extracted from the molecular packets each cycle is not at its maximum.
Another benefit of this setup is that it will make the bunched molecular packets stay most of their
time at the high pump state in each deceleration cycle, which will further guarantee the stability
of the packets, because there will be not enough time for the packets collapse during its stay at
the low pump state.
31
3.2 Intermediate cavity regime
In section 3.1, we discuss a deceleration scheme based on modifying the pump intensity which
will give the deceleration force. However, one can ask is there any ‘‘simpler’’ scheme where a
constant pump will do the job in case there is no feedback apparatus available. In this section we
will introduce such a scheme but with a different cavity setup.
3.2.1 The deceleration principle
First, we will explain the principle of the deceleration scheme based on the nonadiabatic
dynamics
Figure 3.4: Illustration of the deceleration force from the nonadiabatic
dynamics. Due to the nonadiabatic dynamics, the average intracavity intensity
during the climbing-up process of the particle is stronger than the climbingdown process, thus resulting in deceleration.
32
From chapter two we know when the system approaches its steady state, the intensity of the
cavity field follows adiabatically the motion of the particles in the potential wells, see figure
2.2(b). We represent this fact in figure 3.4 to illustrate how the deceleration force will emerge
when the setup of the system is nonadiabatic. As shown in figure 3.4, when the system is
adiabatic, the field I a during the climbing-down (from time 1 to 2) and climbing-up (from time 2
to 3) is the same, so the particles will regain exactly the same amount of energy during the
climbing-down process as they lose during the climbing-up process, resulting in vanishing
deceleration force. However, when the system is not adiabatic, i.e., there is a delay response
between the cavity field and the motion of the particles, there will emerge a deceleration force
for the following reasons.
When the particles are at the crests of the potential wells, see for
example moment ‘1’ in figure 3.4, for adiabatic system, the intensity of the cavity field is at its
minimum, but for nonadiabatic system, this intensity minimum is achieved at some time later
due to the delay dynamics, for example at moment ‘a’ when the particles have passed the crests.
Now, due to this delay dynamics, an interesting thing happens, i.e., the cavity field during time 1
to 2 (the particles are climbing down the potential wells during this period) is lower than that
from time 2 to 3 (the particles are climbing up the potential wells during this period), see figure
3.4. The implication of this is that, the particles lose more energy during the climbing-up process
than they regain during the climbing-down process, precisely because they experience a stronger
cavity field during the climbing-up process than the climbing-down process.
Due to this
deceleration force, the particles will gradually lose their energy and slow down after many cycles
of the cavity mode. Since the delay and thus the deceleration force depends on the velocity of the
particles, as the particles are slowed down, the deceleration force will gradually decrease.
33
In the following, we will demonstrate the validity of this deceleration scheme by directly
numerical simulations where the decay rate of the cavity is chosen comparable to the central
velocity of the beam.
3.2.2 Numerical simulations
Since the deceleration force comes from the nonadiabatic effects of the cavity dynamics while
the phase stability requires the adiabatic response of the cavity field to the motion of the particles,
a compromise is needed in the choice of the cavity parameters if we require both phase stability
and molecular deceleration. We have extensively studied the operation conditions of the system
by analyzing the numerical results of Eq. (1.1) and found that the requirement can be met by
appropriately setting the ratio r  kv0 / ( 2  2c ) , where 1 / kv0 is the time for molecules to
travel one wavelength and 1 /  2  2c the detuning-enhanced cavity life time. A smaller (larger)
r means faster (slower) cavity response to the dynamics of molecules, which leads to better
(poorer) molecular spatial organization but weaker (stronger) deceleration.
We find that
deceleration works within the window 0.1  r  0.6 . When r  0.1 the deceleration force is too
weak for the deceleration and when 0.6  r , the phase stability is gradually lost. We note that the
value of  c we have chosen is very different from that for observing cavity cooling of atoms [25]
and no effective cooling occurs for the parameters we set here.
34
Figure 3.5: Position-velocity distributions of the molecules at the initial stage
(a) and after spatial self organization (b). (c) Rapidly switching optical field
intensity in the cavity as the molecular beam travels along the cavity axis, the
insert shows the evolution of the intensity for the initial period. The parameters
used are k 2 / m  1.16  10 4 , U 0  2.88  10 5  ,  c  10 , N  10 4 ,
  2.4 , and the initial distribution of the beam is Gaussian in velocity, with
kv0 /   3 , k /   0.3 and homogeneous in space within the length of five
wavelengths.
35
Figure 3.6: (a) Position-velocity snapshots of a travelling molecular beam at
different times (traces (a)-(e)), the intracavity field intensities of which are
marked in Figure 3.5(c). (b) Velocity distributions of the molecules
corresponding to trace (a)-(e) in (a). The parameters used are the same as in
Figure 3.5.
36
Fig. 3.5 and 3.6 are the numerical simulations of deceleration of the molecular beam for a pump
level of   2.4 , which is some 30% above the threshold. Spontaneous emission from the
excited states is weak (the saturation parameter as defined in [40] ~ 1%) at this pump level and
can be neglected. The traces (a)-(e) in figure 3.6 show the evolution of the molecules in the
space-velocity space. As observed, the phase stability is well maintained when the molecules are
decelerated until they touch zero velocity (traces (a-c)), evident by the vertical shape of the
bunched molecules in the beam and the nearly constant amplitude of the intracavity field during
the period as shown in Fig. 3.5(c). We note that to avoid spatial overlapping in the display of the
molecular beam, Figure 3.5 and 3.6 is the simulation for a short molecular beam of only 5
wavelengths. The results for a long beam of hundreds of wavelengths remain essentially the
same, as the boundary effects at the two ends of the beam play little role. Further slowing-down
of the molecules from Figure 3.6 trace (c) leads to the reduction of molecular number in the
beam (traces (d-e)), which in turn decreases the intracavity field intensity, as shown in Figure
3.5(c). The molecules gradually lose phase stability during this period. This process continues
until the intensity drops to zero, when the external optical pump field can be switched off. The
velocity distributions at different stages are given in Figure 3.6 (a)-(e). The final distribution has
three peaks. The main peak consists of the molecules that have been synchronously slowed to a
zero central velocity and the velocity half width is approximately twice of the initial value  0 .
We note that the half width of the slowed molecules depends on the pump intensity and increases
(decreases) with the increase (decrease) of the pump. The two smaller peaks are for nonsynchronous molecules close to velocity v0 . Our simulation shows the decelerated molecules
travel a distance of around 100 and for duration of 1000  1 before the central velocity is
37
reduced to zero. In general, this decelerator requires much shorter deceleration time and
travelling distance compared to the electrostatic Stark decelerator.
3.3 Composite scheme
In this section, we will show that the above two schemes as discussed in section 3.1 and section
3.2 can be combined together, where the deceleration scheme based on time-varying laser pumps
as discussed in 3.1 can be used to compensate the reduced deceleration force at the end stage of
the deceleration scheme discussed in section 3.2.
We note that since the level of nonadiabaticity depends not only on the cavity lifetime but also
on the average velocity of the molecular packets, as the molecular packets are slowed down
along the cavity axis as discussed in section 3.2 where the system approaches the adiabatic limit,
the intracavity field tends to follow the changes of the molecular motion better. Consequently,
the deceleration force from nonadiabatic dynamics will drop. However, as discussed in section
3.1, the deceleration force from time-varying optical pumps works well in the adiabatic limit, so
the deceleration scheme by the time-varying optical pumps would be complementary to the
deceleration scheme via nonadiabaticity of the cavity as discussed in section 3.2 in the sense that
the latter works better than the former when the velocity of the molecular packets is higher and
vice versa when the molecular packets have been slowed down. In the following we will show
that the decelerator in section 3.2 would be more efficient when combining with the deceleration
mechanism from time-varying optical pumps as discussed in section 3.1.
38
Figure 3.7: (a) The evolution of average molecular velocity with time by
constant pump (black line, which is the same as in section 3.2) and timevarying pumps (red). Parameters used in the simulation, k 2 / m  1.16 104 ,
U 0  2.88  10 5  ,  c  10 , N  10 4 , and initial condition of the beam:
Gaussian velocity distribution with kv0 /   3 , k /   0.3 and homogeneous
39
spatial distribution in five wavelengths with periodic boundary conditions.
  2.4
for
the
constant
pump,
L  2.2 ,H  2.6
,
I min  60000, I max  120000 for the time-varying pumps. The insert shows the
pump strength with time. Panel (b) shows the velocity distributions of the
molecules for the two deceleration schemes at different times in the process.
Figure 3.7 shows the simulation results in the intermediate cavity regime with a constant pump
(   2.4 ) and time-varying pumps controlled automatically by the feedback mechanism
between two intensity levels ( L  2.2 ,H  2.6 ). As shown in the figure, in the initial stage
of the deceleration, the molecular packets move fast, so the intracavity field cannot follow the
motion of the packets instantaneously. The deceleration force from time-varying laser pumps
plays no role and it mainly comes from the nonadiabaticity, as evidenced by the constant pump
in this period shown in the insert of figure 3.7 (a) and the overlapping of the two velocity curves
in the initial stage. When the travelling molecular packets are slowed, the response of the
intracavity field becomes better to the motion of the packets, thus the deceleration force from
nonadiabaticity deceases. Meanwhile, as the system approaches the adiabatic limit, the
deceleration force from time-varying pumps steps in (around t  100(1 /  ) as shown in figure 3.7
(a)), which results in a constant deceleration compared with the constant pump case. Figure 3.7(b)
shows the evolution of the velocity distributions of the molecules in the deceleration process.
Because of constant deceleration from time-varying pumps in the adiabatic limit, the decelerator
in section 3.2 when combining with time-varying pumps in section 3.1 would be more efficient
than the scheme without the time-varying pumps.
40
Chapter Four: Practical issues and outlook
In the previous chapters, the dynamical behaviours of the system are described by normalized
parameters so the model can describe all polarisable particles. The cavity-induced phase
transition and deceleration can therefore be observed for these particles as long as they fall
within the parameter regions. In this chapter, we will discuss in detail the practical issues of our
proposed deceleration scheme. To do so, we will convert the normalised parameters to those of a
well-defined physical system, which is expressed in optical wavelength, intensity, atomic density
and velocity, etc., for light-atom interactions and in cavity length and finesse in terms of cavity
configuration. This allows us to discuss in detail how a specific molecular species can be
decelerated in realistic laboratory conditions. Finally we present our conclusions and outlook for
the first part of this thesis.
4.1 Practical issues
The recent experimental efforts of self-organization of BEC in optical cavity [4], cavityenhanced Rayleigh scatting [42] and the real-time feedback control of a single atom trajectory in
cavity [43] put our proposal within the reach of current technology.
As discussed in previous chapters, a bad cavity is needed for the realization of the cavity-induced
phase stability, which is different from the cavity cooling studies where a high-finesse cavity is
preferred. This is because cavity cooling needs friction-like force from the nonadiabatic response
of the cavity to condense the phase space of the cold molecular sample, whereas cavity
deceleration requires an immediate response of the cavity to achieve a self-consistent molecule41
field steady state. Therefore, for deceleration the cavity response time must be significantly
shorter than the characteristic time of the system dynamics induced by the travelling molecular
beam. For a fast molecular beam with central velocity v0 ~ 100m / s and pump lasers with
wavelength  ~ 1m , the time for the molecular beam to travel one deceleration stage is around
t c   /( 2v0 )  5 10 9 s , which corresponds to the rate  c  0.2GHz , thus a cavity with decay
rate of  2GHz would meet the bad cavity criterion (   1 / tc ) in this case. The decay rate for a
Fabry-Perot cavity with length L  1cm and reflectance of cavity mirrors R  90% is
  c ln( 1 / R) / L  3.2GHz which readily meets the bad cavity criterion. For the deceleration
scheme working in the nonadiabatic regime, the time for the molecular beam to travel one
deceleration stage is tc ~  c , in this case the cavity length can be much longer ( L  10cm for the
same cavity discussed above). Self-organization of the fast molecular beam takes place around
tens of nanoseconds with a cavity decay rate of 10GHz as can be inferred from figure 2.1 in this
case (note the time needed for the self-organization also depends on the pump intensity, with the
intensity at the level of the threshold pump, critical slowdown dominates). As shown in the
numerical simulations earlier, the deceleration process typically requires tens to hundreds of
stages, so the molecular packets travel typically tens to hundreds of micrometers for duration
about one microsecond before it stops. Therefore, a cavity with length of one centimeter would
be sufficient theoretically for the deceleration process in our proposal. We note in our theoretical
description, transverse confinement perpendicular to the cavity axis is not discussed explicitly,
but since the stopping distance (tens of micrometers) is much shorter than the transversal
dimension of the mirrors (several millimeters) and meanwhile, we use standing-wave transverse
pump scheme, where in the third remaining direction molecules are confined by the transverse
42
envelope of the cavity and pump field as discussed in [25], we think our description is valid in
this aspect.
We now discuss the pump threshold power required to trigger the phase transition. For a
Gaussian laser beam with waist wL , the pump strength  can be expressed in terms of the laser
power P as |  | [ ( p ) /  0 ]  p P /( cVwL2 ) [38]. Substituting this expression to the pump
threshold (2.7), we have,
P  m 2
 c2   2  0 2 N 1 wL2c
[
]( )
.
( c )  ( p ) V
p
(4.1)
For convenience, we introduce two frequency shift parameters rc and ra to describe the pumpcavity detuning c  rc and the maximum shift of the empty cavity resonance frequency
induced by the molecules NU0  ra . While rc  ra  1 is used in cavity cooling scheme [25],
where the cavity is in resonance and thus the nonadiabatic effect is dominant, the conditions of
rc  1 and ra  1 ,which essentially avoids this resonance region, are required for the effective
operation of the cavity in the deceleration regime as the reasons discussed above. Normally,
rc  5 and ra  0.5 work for the deceleration setup as we found in the numerical simulations.
Using
the
relation
U 0   ( p ) p /( 0V )
,
equation
(4.1)
is
simplified
to
P  m 2 (rc / ra )[ 0 /  ( p )]wL2c under the conditions of rc  1 and ra  1 .The potential energy
of a molecule in a far-off-resonant optical field in free-space is U  2I /  0 c [35]. By using the
relation I  P / wL2 and m 2  kBT / 2 , the simplified pump threshold condition can be rewritten
as 2I /  0c  kBT (rc / ra ) , the meaning of which is clear compared with the case in free space: in
43
order to trigger the cavity-induced phase transition, the potential depth generated by the intensity
of the pump lasers should be larger than the transverse beam temperature modified by the cavityrelated parameter (rc / ra ) . We note from equation (4.1) that the number of molecules enters only
in the form of atomic density N / V , which shows the scaling invariance of the system as long as
N / V is constant. Such invariance can indeed be obtained from equation (1.1) under the
condition of

j
cos( kxj )  N , which is only valid when the molecules are spatially organized.
Since the coupling constant g   c /( 2 0V ) , where  is the electric dipole transition moment,
thus N / V  Ng 2 , which implies that the smaller coupling constant with a bigger cavity can be
compensated by increasing the molecular number.
The validity of the pump threshold (4.1) can be tested by the experimental data from [4] where
the self-organization-like phase transition has been demonstrated with
87
Rb BEC. The
parameters used in the experiment are: ( g ,  ,  )  2  (10.6, 1.3, 3.0) MHz , cavity length of
178m ,waist radius of 25m , pump-atom detuning  a of 4.3nm from the atomic D2
line( a  2100GHz) and NU0  6.5 , where U 0  g 2 /  a [25] in this case. If we choose one set
of parameters ( c , P)  (2  20MHz , 400W ) from their phase diagram, and using the BEC
temperature of ~ 100nK , pump area of 70m  70m ,the calculated pump threshold power
according to expression (4.1) is ~ 250w ,which is close to the real experimental value of
400 W . A further simple extrapolation of the above results to a fast
87
Rb gas beam with
transverse temperature of ~ 100mK in a bad cavity (rc / ra  10) would require a pump threshold
power of 2kW . Currently, single mode Ytterbium doped fiber lasers with an output power of
44
2kW are commercially available and some 5kW has been demonstrated in laboratory
environment [44].
We then consider an example of benzene molecules, which have been used in the experiment of
single-stage optical Stark deceleration [35]. The molecule has an average polarizability of
 ( p )  11.6 1040 Cm2 / V at the pump laser wavelength of   1064nm . For a pulsed fast beam
of molecular benzene with velocity spread of   10m/ s , pump laser waist (also the molecular
beam length) of 1mm , and the cavity-related parameter rc / ra  10 , the required pump threshold
intensity would be I  9.3  1010W / cm2 , which can be readily achieved by the pulsed laser used
in [35] if stretching it to a microsecond-duration one. Since the pump detuning is in the order of
1014 Hz , the saturation parameter (to be discussed in the following) is calculated to be 0.01%
which is far below the acceptable level of 1% where the population excitation and thus
spontaneous emission is negligible. The using of pulsed laser is justified by the short time scale,
typically within one microsecond, of the deceleration process. To produce a time-varying optical
pump field in the deceleration scheme discussed in section 3.1, the pulsed laser can be modulated
by electro-optic modulators [45] or fiber modulators [27] with bandwidths at tens of GHz which
are much larger than the required cavity decay rate of several GHz. For the deceleration scheme
working in the nonadiabatic regime as discussed in section 3.2, where a constant pump is
sufficient, no further modulation by electro-optic or fiber modulators is needed. The molecular
density in this case is about N / V  1015 cm 3 , and such intense low-energy molecular beam can
be readily obtained by the method of ‘‘pressure shock’’ as described in [46].
45
Since the pump lasers are far-off-resonance from all electronic transitions in the above analysis,
we can safely neglect spontaneous emissions in our analysis [4, 42]. However, the operation
conditions can be significantly relaxed if one makes use of the effects of resonantly enhanced
dipole moment. The parameter setting in this case depends on the chosen molecule and the
number of open transitions, however, if the pump frequency detuning from the transitions is
much larger than the energy splitting of the allowed transitions, then an approximated two level
model is valid [28].
In order to suppress spontaneous emission, the saturation parameter
s |  |2 g 2 / 2a [40] should be negligible in this case, where |  |2 is the intracavity photon
number and  a the pump-atom detuning (see Appendix A). By inserting the expression (2.10) for
|  |2 and after some algebra, the saturation parameter is simplified to s  m 2 (ra / rc ) /  a , which
means that in order to avoid significant population excitation, the energy associated with the
detuning should be much larger than the transverse temperature of the beam modified by the
cavity-related parameter (rc / ra ) . Such operation has been discussed in detail in our previous
work [28], which shows to be feasible by using a pump source far-detuned from the allowed
optical transitions (hundreds to thousands of GHz) to suppress spontaneous emission. Recently, a
‘‘supersonic electric conveyor belt’’ experiment, where metastable CO molecules are trapped
and transported in travelling potential wells at constant velocities on a chip, has been
demonstrated [47]. Our self-consistent molecule-field steady state described in chapter two can
be taken as a cavity-based version of this transportation experiment. We consider the pump
threshold power to realize this ‘‘conveyor belt’’ experiment with a bad cavity of rc / ra  10 . For
pulsed beam ( 1mm long) of CO molecules at transverse temperature of 20mK with Q2 (1)
transition of a 3  X 1  , which has a transition dipole moment of 1.37 Debye, as used in [47],
46
to avoid significant population excitation( s ~ 0.01) , the detuning is then  a ~ 1.3  1010 Hz . The
pump threshold power at this case is around 1kW , which is within touch for experiments [44].
Other methods to reduce the pump threshold power, such as seeding the cavity and pump power
recycling with a second cavity are also available [38].
4.2 Conclusions and outlook
In the first part of this thesis, we have explored deceleration schemes based on a new phase
stability mechanism. In chapter two we explored in detail the dynamical interplay between a fast
molecular beam moving along the axis of a standing-wave cavity and the intracavity field formed
from the scattering of the transversal pump photons by the molecules in the beam. We found that
in the adiabatic limit, a phase transition, from which travelling molecular packets are formed
from the initial spatially homogeneous fast molecular beam above some threshold pump, results
in a well-defined self-consistent molecule-field steady state that can be used for multistage
decelerations. This phase stability mechanism from the cavity-induced collective behaviour of
molecules ensures the phase stability of majority of the molecules in the cavity rather than a
fraction (small acceptance volume) as in phase space filtering techniques. In chapter three, we
discussed three schemes which can decelerate the molecules to zero velocities efficiently when
proper deceleration forces are introduced. The deceleration force can be introduced by using the
time-varying pumps in a similar way to electrostatic Stark deceleration by introducing sudden
switching between two levels of the pump intensities, which are in synchronous to the climbing
up and down processes of the molecular packets in the standing-wave potential. However, in our
scheme the switching sequence of the pumps is achieved automatically by feedback, rather than
timed externally as in electrostatic Stark deceleration. The deceleration force can also be
47
introduced by simply using a nonadiabatic cavity. For the deceleration scheme based on timevarying optical pumps, due to no extra widening factor to the velocity distribution at the end of
the deceleration process compared with the deceleration scheme based on nonadiabaticity, the
scheme based on time-varying pumps can maintain the low transverse temperature and high
density of the beam while it requires only tens of deceleration stages. The two deceleration
schemes based on time-varying pumps and nonadiabaticity can also be combined together to
form a composite deceleration scheme where the deceleration process is more efficient than the
two schemes alone. We also discussed the practical issues of our deceleration proposals and
demonstrate they are promising under the current experimental technologies.
An important difference between our method and electrostatic Stark deceleration is that while the
phase stability and deceleration force in the latter are interweaved, i.e., at higher deceleration
rates, only smaller amount of stably bunched molecules can be handled, vice verse [41], our
method allows the engineering of phase stability and deceleration force separately. Our method
is also different from both the single and multistage optical Stark decelerators in free space based
on optical lattices as demonstrated or proposed previous [35, 45]. For the experimentally
demonstrated single-stage optical Stark decelerator for molecules with pulsed optical lattice [35],
since there is no bunching effect due to the single-time interaction of the molecules with the laser,
the width of the slowed molecules in the velocity space is quite broad. For the multistage optical
Stark decelerator in free space as proposed in [45], where bunching effect is present in obtaining
a narrowed velocity distribution, the energy extracted from the molecular packet of each period
is small due to the lack of cavity-induced collective effect, and the stages required for the
deceleration process is tens of thousands of stages [45]. By using the collective enhancement
effect with a cavity, the deceleration process needs only tens of stages while keeping the initial
48
low transverse temperature and high density of the beam. We note that the work in [27] was
based on the assumption that a molecular sample below 1K has been prepared by using a
decelerator technique, before both external and internal degrees of freedom of these molecules
can be further cooled by a cavity. Our proposed method can serve for this purpose by providing a
high density molecular beam at the required temperature. Since deceleration or cooling of
external motion is a relatively fast stage, in which the internal motion is not affected by the
scattering process, the present work together with previous cavity cooling studies [25] show the
feasibility to bring a hot molecular beam into the ultracold regime with only cavity setup where
high density and low temperature can be achieved at the same time in principle. Thus this thesis
contributes to the present methods of getting cold or ultracold quantum gaseous sample and will
stimulate experimental efforts toward this exciting new possibility. Finally we would like to
point out that while this thesis has focused only on deceleration schemes based on the selfconsistent molecule-field steady state discussed in chapter two; this steady state may be used for
other applications as well, such as supersonic conveyor belt [47].
49
PART II:
Quantum Simulations of Multiple-layered Dirac
Cones in Optical Lattices
50
Chapter Five: Dirac cones and Dirac fermions
5.1 Background
Graphene and topological insulators are two hot topics in condensed matter physics which have
stimulated an enormous interest at both theoretical and experimental levels (see [48, 49] and
references therein). These two systems share an attractive and unique property, namely that the
electronic transport at low energies is not governed by the usual Schrödinger equation, but rather
by its relativistic counterpart-the Dirac equation. In graphene, which is a single layer of carbon
atoms densely packed in a honeycomb lattice [48], the band structure corresponds to a semimetallic phase whose Fermi surface consists of an even number of isolated points. The lowenergy excitations around these points display a relativistic dispersion relation, and thus can be
described by the two-dimensional Dirac Hamiltonian for massless fermions. Moreover, in threedimensional topological insulators, which are semiconducting alloys with a strong spin-orbit
coupling [49], the bulk band structure corresponds to a gapped insulating phase. Nonetheless,
these materials also support a robust surface conductivity which can be described by an odd
number of two-dimensional massless Dirac fermions. The recent interest in both graphene and
topological insulators is two-fold. On the one hand, they provide concrete platforms where to
realize interesting phenomena originally predicted in a high-energy context, e.g., the Klein
paradox [50], or axion electrodynamics [51]. On the other hand, they also foresee novel and
useful device applications (see for example Ref. [52]). Needless to say, the emergence of
massless Dirac fermions in solid-state materials is developing into an exciting area of condensed
matter.
51
In solid-state materials, the pseudospin of an emergent relativistic fermion depends on the
geometry of the underlying lattice. For example in graphene, where the unit cell of the
honeycomb lattice has two sites, the emergent massless Dirac fermions present a pseudospin-1/2
structure. It has been thought that the usual electron spin and the pseudospin indexing the
graphene sublattice state are merely analogues. Recently, researchers have shown that the
pseudospin is also a real angular momentum [53], the implication of which is that half-integer
spin like that carried by the quarks and leptons can derive from hidden substructure, not of the
particles themselves, but rather of the space in which these particles live. Also recently, some
researchers have studied alternative lattices, e.g., the T3 lattice [54], the line-centered-square
(Lieb) lattice [55-57], and the Kagome lattice [58], where emergent massless relativistic
fermions present a pseudospin-1 structure due to the three-site nature of the unit cell of these
lattices. These higher-spin relativistic fermions show some distinctive features compared to
graphene's spin ½ Dirac fermions, such as all-angle perfect tunnelling [57], particle localization
[56], and the absence of the anomalous quantum Hall Effect [55]. In view of these results, a
natural question arises: could we engineer an experimental setup where massless relativistic
particles with any arbitrary spin emerge? Moreover, would these high spin fermions host novel
effects that have no counterpart in the low-spin cases? While the approach based on complicated
lattice geometries to create these high spin fermions is difficult to implement [59, 60], here we
resort to the idea of a quantum simulator.
Quantum simulators have recently attracted a great interest [61]. The concept of a quantum
emulator was first introduced by Feynman as a way to avoid the difficulty of simulating quantum
phenomena with classical computers [62]. The idea was to use one controllable system to
simulate another, possibly computationally intractable system. Feynman’s intuition is today
52
being implemented in various setups and among them, cold gases of neutral atoms play a central
role [63]. In 2002 [3], I. Bloch’s group realized the superfluid to a Mott insulator phase transition
by loading a Bose Einstein Condensate (BEC) into a three-dimensional optical lattice when
increasing the depth at the individual lattice sites. They found that in the superfluid phase, each
atom is spread out over the entire lattice with long-range phase coherence while in the insulating
phase, exact numbers of atoms are localized at individual lattice sites. Recently, the classic Klein
paradox, in which a relativistic particle seems to be transmitted unhindered into a potential
barrier was reproduced with trapped ions [64]. Systems of trapped ultracold atoms in optical
lattices have proven to be a remarkable tool for simulating a vast range of physical phenomena
known from condensed matter and lately also high energy physics. In chapter six, we address the
task of achieving arbitrary large spin Dirac-Weyl fermions by implementing the idea of quantum
simulation, i.e., we engineer the Hamiltonian of an optical lattice populated by multicomponent
ultracold fermionic atoms with specific state-dependent hopping. This implementation is based
on the fact that these pseudospin structures, which arise from complex lattice geometries, can be
reproduced by a standard square lattice with a matrix hopping [65]. We find that such a setup
presents a rich playground where the low-energy excitations can be described by massless
relativistic particles with arbitrary spin. Interestingly, these fermions are governed by the Weyllike Hamiltonian H   F S  p , which is the massless version of the Dirac Hamiltonian but can
have any spin, and whose eigenstates we refer to as Dirac-Weyl fermions with arbitrary spin. In
chapter seven, we use these ultracold systems to further simulate some advanced topics from
Standard Model Extensions (SMEs), such as Modified Dispersion Relations (MDRs), and
neutrino oscillations (NOs), thus demonstrating the important role these ultracold systems play in
addressing some challenging topics in modern physics. In the remaining part of this chapter, we
53
will introduce the Hamiltonian considered in this part of the thesis, and then briefly discuss the
experimental implementation of the Hamiltonian in an optical superlattice populated with multicomponent ultracold fermionic atoms.
5.2 The Hamiltonian
In order to realize the spin-s Dirac-Weyl fermions, we consider a system of N  2s 1 component
ultracold fermionic atoms trapped in an optical square lattice with lattice spacing of a in the
noninteracting limit, which can be realized by means of Feshbach resonance [65]. We assume the
depth of the lattice in both the x and y directions is so large that hopping in these directions due
to kinetic energy is prohibited. The dynamics of the atoms in the lattice is controlled by Raman
transitions, which allows us to consider the following model Hamiltonian [66] (see figure 5.1),
H  H t  H o   t v [v ] ` cr , `cr  H .c.  [O] ` cr `cr ,
r ,  `
r
(5.1)
 `
where H t denotes the hopping dynamics and H o the on-site dynamics. For the hopping term H t ,
the hopping amplitude along the   x, y direction is modified by a spin-dependent operator v
which is associated with the link connecting the lattice points Rr  Rr  , and cr (cr ) is the
fermionic creation (annihilation) operators, see figure 5.1. For the on-site term H o , the spins can
flip by the on-site microwave Raman transitions. This optical lattice simulator relies on the
ability to control the form of v and O by Raman lasers as discussed in the following section.
54
Figure 5.1: Schematic description of the laser-assisted hopping of
multicomponent ultracold fermionic atoms in an optical square lattice.
5.3 Experimental setup
In this section we outline the main ingredients for implementing the Hamiltonian (5.1) in an
optical lattice. For a detailed discussion we refer the reader to the paper [66]. The key idea is
easily captured by the cartoon as in figure 5.2 where the hopping of atoms in the ground state
L  0, F  9 / 2 manifold is assisted by the auxiliary states in a different hyperfine manifold
F  M  7 / 2 . Consider for instance a cloud of ultracold 40K atoms described by the fermionic
field operators cx (cx ) , where   {1...N } is the internal index that labels a particular subset of
Zeeman sublevels in the ground state L  0, F  9 / 2 , and x labels for the sites of a twodimensional square superlattice. This particular superlattice follows from the optical potential
created by two pairs of counter-propagating lasers along each axis  ,
V ( x )  V1  cos 2 (k L x )  V2  cos 2 (2k L x ) ,
j
(5.2)
j
55
where V1  V2 represent the lattice depth, and k L the optical wavevector. As shown in figure 5.2,
the atoms are trapped in a periodic structure of primary and secondary minima, which shall be
referred to as sites r , and links l henceforth.
Figure 5.2: (a) Superlattice scheme for a laser-assisted hopping. Atoms in the
ground state L  0, F  9 / 2 manifold are coupled via a two-photon Raman
transition
to
neighboring
atoms
in
a different
hyperfine manifold
L  0, F  7 / 2 . This Raman transition takes place through an intermediate
excited state L  1 , such that for large enough detuning  , with | 1 | , | 2 |  ,
where 1 and  2 are Rabi frequencies for each laser, one readily arrives at the
Hamiltonian in (5.3). (b) In the cartoon analogue, atoms in the ground state
L  0, F  9 / 2 manifold are hopping along the primary sites of a two-
dimensional lattice assisted by the ‘‘steppingstone’’ states in the hyperfine
manifold L  0, F  7 / 2 .
56
For a deep optical superlattice, the atomic tunneling between neighboring sites due to kinetic
energy will be completely suppressed. The fundamental idea to engineer a spin-dependent
hopping is to assist these tunneling using additional lasers that drive a Raman transition to an
excited state in the hyperfine manifold F  M  7 / 2 residing on the links, hence represented by
the fermionic operators d x (d x ) . The use of a pair of lasers in a Raman configuration is two-fold.
On the one hand, the effective frequency can be tuned to the microwave transitions
F  9 / 2  F  7 / 2 . On the other hand, the effective wavevector can be large so that the lasers
impart enough momentum to the atoms to tunnel between neighboring lattice sites. As customary,
such a two-photon transition is obtained after the adiabatic elimination of a higher excited state
in the fine structure L  1 . The Raman lasers aligned along a particular hopping direction not
only drive the transition between the hyperfine levels, but also transfer a finite momentum to the
atoms which allows them to tunnel to a neighboring site. This assisted-hopping [67], i.e.,
hopping from one side of the river to the steppingstone as in the cartoon analogue, is described
by the following Hamiltonian
H R   eff S xx `d x cx ' ei R t  H .c. ,
(5.3)
x `, x
where eff is
the two-photon
Rabi
frequency driving
the
transition | F  9 / 2, 
ik  r
ik  r
3
*
 | F  7 / 2,7 / 2  , and S xx `  x | e R | x'   d rwx (r )e R wx ` (r ) determines the momentum
transfer that the Raman lasers impart on the atom, thus assisting the transition between
neighboring superlattice sites. The parameters of this two-photon Raman transition R  1  2
and kR  k1  k2 , follow from each laser frequency and wavevector. Here we have introduced the
corresponding Wannier function wx (r ) . In the expression of S xx ` , one sees the importance of
57
using a two-photon Raman scheme rather than a simple microwave, since the integral between
neighboring Wannier functions will only be finite when the imparted momentum is large, i.e., the
effective wavelength is on the order of the lattice spacing R  a , typically a few hundred
nanometers. As shown in [66], by selecting an appropriate detuning, Zeeman splitting, and lattice
staggering, it is possible to perform an adiabatic elimination of the auxiliary states d x (d x ) that
reside on the links, and thus obtain an effective Hamiltonian that describes the hopping of
F  9 / 2 atoms along the primary sites of a two-dimensional lattice. Therefore, the auxiliary link
serves as a bus that allows us to assist the tunneling, and one obtains the effective Hamiltonian of
H t   tv [v ] ` cr , `cr  H .c. where the tunneling strengths tv now depend on the fourr ,  `
photon Rabi frequencies (hopping twice as in the cartoon analogue, thus the two-photon Raman
scheme becomes an effective four-photon one) . We note that the particular matrices v can in
principle be designed at will, although the experimental requirements are certainly challenging.
The on-site spin flipping O can be performed with standard microwave transitions, or Raman
transitions carrying negligible momentum.
For the simplest situation with s  1 / 2 , one selects a pair of Zeeman sub-levels M  9 / 2,7 / 2
1
1
where the hopping operators would correspond to Tx   x , Ty   y . Each of these can be
2
2
engineered with a single pair of Raman beams, once their frequencies are tuned to the resonance
given by the Zeeman splitting and the lattice staggering. To account for the real and complex
matrix elements, one should control the laser phases appropriately, which can be accomplished
by means of acusto-optical modulators. The scheme gets more complicated for s  1, where the
hopping operators have four non-vanishing elements, and thus double the number of beams
58
required. Let us note, however, that since the difference in resonance frequencies rely on the
Zeeman splitting and staggering, which is a small fraction of the laser frequency, the desired
frequencies can be obtained from the same source once the beams are split and their frequencies
tuned by an acusto-optical modulator. For arbitrary s , the scheme is more involved, yet benefits
from the fact that the hopping matrices are sparse. It should be noted that there might be more
clever schemes that take advantage of the light polarization to select the different hopping
elements.
59
Chapter Six: Hopping dynamics and multiple-layered Dirac cones
In this chapter, we will focus on the hopping dynamics governed by H t and consider a regime
where the hopping matrix is tuned according to the representations of the su(2) Lie algebra. More
explicitly, we demonstrate that a spin-dependent hopping, tuned according to the (2s+1)dimensional representations of the su(2) Lie algebra, directly leads to a regime where the lowenergy excitations around the band-touching points are Dirac-Weyl fermions with pseudospin s.
We study the properties of this exotic Dirac-Weyl fermion in the presence of a synthetic
magnetic field (see [69] and references therein). We find that the Weyl-Landau problem can be
mapped onto a Dicke Hamiltonian [70], a well-known model in quantum optics which describes
the interaction of an ensemble of two-level atoms with a quantized mode of the electromagnetic
radiation. From this insightful mapping, we derive the exact solution of the Hamiltonian, and
predict the interesting consequences of an anomalous half-integer quantum Hall effect [71].
These predictions are confirmed by the numerical evaluation of the topological Chern numbers
[72] and edge-states [73], which directly give the quantum Hall sequence. We obtain a general
rule describing the quantum Hall effect for Dirac-Weyl fermions with arbitrary spin structures,
thus generalizing the anomalous quantum Hall effect for pseudospin 1/2 Dirac fermions in
graphene [48, 71]. Additionally, we show that this platform hosts two different phases at halffilling: a semi-metallic phase that occurs for half-integer s, and a metallic phase that contains a
flat zero-energy band at integer s. In the semi-metallic phase, we show that a Dirac-Weyl
fermion with high spin can be described as a collection of spin 1/2 counterparts, where each
species has a different effective speed of light. Accordingly, the low-energy transport is
characterized by spin 1/2 Dirac-Weyl fermions moving at different velocities, an effect known as
60
multi-refringence in optics [68]. As a consequence, we find exotic tunnelling properties across a
potential barrier, which shows a remarkable multi-refringent Klein paradox. Finally, we show
how to get rid of the structure of su(2) algebra, and thus allowing us to consider any arbitrary
effective speeds of light.
6.1 Hopping matrices according to the representations of the su(2) Lie algebra
We find that when the hopping operators are described by an N-dimensional representation of
the su(2) algebra, namely Tx  S x and Ty  S y , which fulfill the corresponding algebra
[Sz , S ]  S and [S , S ]  2Sz , where S  S x  iS y , then the low-energy excitations around the
band-touching points are Dirac-Weyl fermions with pseudospin s . In the following, we only
consider a two-dimensional lattice, but we would like to point out that the setup can also be
implemented in a three-dimensional lattice by setting Tz  S z . Transforming the hopping
Hamiltonian to momentum space by applying the Fourier transformation cr 
1
L
e
 ikr
ck ,
k
where L is the number of lattice sites and k  [ ,  )  [ ,  ) lies within the first Brillouin zone,
one obtains the following N-band model,
H     (k ) H (k ) (k ) H (k )   2tv Sv cos( kv ) ,
k
(6.1)
v
where the spinor (k )  (ck1,..., ckN ) contains the fermionic operators and k is in units of the
lattice spacing a . Using the properties of the su(2) algebra, the Hamiltonian can be readily
diagonalized, giving the following spectrum,
Em (k )  m (k )  m (2t x cos k x )2  (2t y cos k y )2 ,
61
(6.2)
where m   s,..., s  1, s . We show in figure 6.1 the resulting band structures with N  2,3,4,5
internal components. Further band structures, corresponding to configurations using more
internal atomic states ( N  5) , are easily derived from this figure. As can be observed, for N  2 ,
one recovers the familiar Dirac cones that arise in graphene. Conversely, for N  3 the Dirac
cones are accompanied by a zero-energy flat band. When N  4 and N  5 , we get interesting
double-layered cones that host excitations with pseudospin 3/2 and 2 respectively. These doublelayered cones correspond to different effective speeds of light, therefore leading to the
birefringence phenomenon.
Figure 6.1: Energy bands and Dirac points. Energy spectrum E  E (k x , k y ) for
(a) N  2 , (b) N  3 , (c) N  4 , (d) N  5 . The wavevector k  (k x , k y ) belongs
to the first Brillouin zone.
62
In general, we may conclude that N energy bands touch at four highly-symmetric momenta


K Dd  ( d x , d y ) , where dv  {1,  1} , which we shall refer to as Dirac points. At half-filling,
2
2
the low-energy excitations can be described by the following Hamiltonian,
H eff     ( p) H Dd ( p ) ( p ) , H Dd ( p)   cv dv Sv pv ,
d, p
(6.3)

where p  (k  K Dd ) is the momentum around each Dirac point, and we define cv  2t v /  . We
stress that in the isotropic regime cx  c y , the effective Hamiltonian for the excitations around
 
K D  ( , ) corresponds to that of the usual Weyl Hamiltonian for arbitrary spin. One finds
2 2

p
H D  cp s , with the helicity operator  s  S .  , i.e., the projection of the spin along the
| p|
direction of the linear momentum. The excitations described by equation (6.3) can therefore be
interpreted as ND  4 relativistic Dirac-Weyl fermions in an underlying anisotropic spacetime
with an effective speed of light cx  c y .
6.2 Anomalous Hall effects
In the following we will investigate the anomalous quantum Hall effect when the Dirac-Weyl
fermions are subject to an external synthetic magnetic field. Since the ultracold atoms are neutral,
one needs to mimic the effects of an external magnetic field in order to perform a quantum
simulation of the quantum Hall effect. A simple way to introduce a magnetic field is to rotate the
system where the Coriolis force in the rotating frame plays the same role as the Lorentz force on
a charged particle in a uniform magnetic field [69]. Alternatively, optically induced gauge
63
potentials can be used which rely on laser assisted tunnelling [69]. When the system is subject to
a magnetic field, the hopping is modified according to the so-called Peierls substitution
T  T e
i
e
dr  A
 

, where A is a synthetic gauge potential giving rise to an effective magnetic field
B    A , and ‘ e ’ is an effective charge defined by the laser parameters. In the following, we
study the interplay between the Hall plateaus and the underlying spin structure, which offers a
generalization of the anomalous quantum Hall effect observed in graphene where zero energy
modes contribute in a fundamental manner. We demonstrate how this peculiar Hall sequence is
indeed related to the number of Dirac points and zero energy modes for the general case of
arbitrary spin structure. The study is performed both for the lattice and for the continuum limit of
our model.
6.2.1 Continuum description: Weyl-Landau levels
For a small enough magnetic field, one can show that the Peierls substitution leads to the usual
minimal coupling performed in the effective Hamiltonian (6.3). The standard procedure is to
replace the canonical momentum by a gauge-invariant quantity, p    p  eA , whose
components no longer commute [ x ,  y ]  i2 / lB2 , where lB   / eB is the magnetic length.
By introducing the bosonic creation and annihilation operators, a  
a
lB
( x  i y ) and
2
lB
( x  i y ) , the effective Weyl-like Hamiltonian in equation (6.3) is recast into
2
HWd L  HWd ( p  eA)  g d  aS  g d  aS  H .c. ,
(6.4)
64
where we have introduced g d   (cx d x  cy d y ) /( 2 2lB ) . Interestingly, the problem of DiracWeyl fermions subject to a magnetic field, hereafter referred to as the Weyl-Landau levels
(WLLs), can be exactly mapped onto the so-called Dicke model which is well known from
quantum optics [70]. This model, which describes the interaction between a collection of twolevel atoms and a single mode of the quantized electromagnetic field, displays a wide range of
interesting phenomena.
In the following, we will derive the exact solution of the Hamiltonian (6.4) for the Dirac point
d  (1,1) in the isotropic regime cx  c y . To do this, we write out the matrix elements
explicitly as
H Dd  g [ Enm n,m1 n a   Enm n,m1 n a] ,
(6.5)
nm
where n  n(2s  1  n) and Enm is the (2s  1)  (2 s  1) matrix with matrix elements equal to
one in row n and column m and zero otherwise. The eigenvector can be expressed as
  (1 , 2 ,..., 2 s 1 )T . The eigen-equation then reduces to the following operator equation:
i 1a i 1  i  i ai 1  0 ,
(6.6)
where i  1, 2,..., 2 s  1. We solve the Weyl Landau levels by successive substitutions, i.e., we
substitute the second equation into the first one, and then substitute the third one to the result
obtained by the previous substitution and repeat the process. After r substitutions, we get
Ar ,i (aa )i 1r  r Br , j (aa ) j 1 ar 1 ,
(6.7)
65
where Ar ,i and Br ,i are energy-dependent constants and the Einstein summation convention is
used for repeated indexes i and j . Using r a r  r 1  r 1ar  2 , we get
Ar ,i (a a)i 1 (r 1  r 1ar  2 )  Br , j r (a a) j r 1 ,
(6.8)
and
 k  1
 k 
(1)k i  r2
(1)k i 1
Ar , k 
Br , k 

[( r  2 ) / 2 ] k  i
[( r 1) / 2 ] k  i 1
 i 1 
 i  1
,
 k  1
k i

 Ar ,k  i  1 (1)
[( r  2 ) / 2 ] k  i


Ar 1,i  
Br 1,i

(6.9)
n
where   is the binominal coefficient and [r ] is the integer part of r . By using the cutoff
m
condition  2 s 1  0 , the Weyl-Landau levels are obtained compactly as
[ s  3 / 2]
A
i 1
2 s 1, i
(n  1)i 1  0 ,
(6.10)
where A2 s 1,i are determined by the recursion relation in (6.9) with initial values A1,1   and
B1,1  1 . Here n is the eigenvalue of the number operator a  a for the Fock state | n  . The
eigenvector is then expressed as
ns  { f 2 s (n,  ) | n  2s , f 2 s 1 (n,  ) | n  2s  1 ,..., f 0 (n,  ) | n }T ,
(6.11)
where fi (n,  ) are determined by the algebraic equation,
i 1 f2s  2i n  2s  i  1  f2s 1i  i f2s i n  2s  i  0 ,
(6.12)
with i  1,2,...,2s  1 . For s  1 / 2 and s  1 , the results confirm the ones in [48, 57]. However, the
general expression here is applicable to any arbitrary spin. The Weyl-Landau states of these high
spin particles are a mixture of successive non-relativistic Landau levels, i.e., the spin and orbital
66
degrees of freedom of ns are highly entangled. We next explicitly derive the zero-energy modes
of the Weyl-Landau Hamiltonian in equation (6.4) for d  (1,1) in the isotropic regime cx  c y .
The complete Hilbert space is H  F  C N , where F  span{| n , n  0,1...} is the Fock space of
the cyclotron mode, and C N  span{| s, m , m  s,..., s} is the angular momentum space.
Nonetheless, the action of the Hamiltonian decomposes into a set of invariant subspace
H  n  0 H n , where each H n is spanned by a different combination of the Fock and spin states.
In particular, we find that for n  2s , there are certain special subspaces with odd dimensionality:
H 0  span{| 0 | s, s }
H 2  span{| 2 | s, s },..., | 0 | s, s  2 }
.
...
(6.13)
H 2 s 1  span{| 2s  1 | s, s },..., | 0 | s, s  1 }
Each of these subspaces hosts a zero-energy mode of the Weyl-Landau Hamiltonian. By
introducing the following quantities
f n, s , m  g n( s( s  1)  m(m  1)) ,
(6.14)
where g is the coupling constant introduced in the Weyl-Landau Hamiltonian of equation (6.4),
one finds the following zero-energy modes for which H | E0  0
| E0(1) | 0 | s, s 
| E0( 2)  f1, s ,  s 1 | 2 | s, s   f 2, s ,  s | 0 |  s  2 
...
, (6.15)
2s2
2s4
| E0( N z )   f n `, s , s 1 n` | 2s  1 | s, s   f 2 s 1, s ,  s  f n `, s , s 1 n ` | 2s  3 | s, s  2   ... 
n `odd
n `odd
2 s 3
2 s 1
n `even
n `even
(1) s 3 / 2 f1, s , s  2  f 2 s 1 n `, s ,  s  n ` | 2 | s, s  3   (1) s 1 / 2 f1, s , s  2  f 2 s 1 n`, s ,  s  n ` | 0 | s, s  1 
67
where we have omitted an irrelevant normalization factor. By simple counting, we find that the
Weyl-Landau Hamiltonian hosts N z  s  1 / 2 zero-energy modes for half-integer spin.
Interestingly, the number of zero-modes coincides with the number of the spin ½ Dirac-Weyl
fermions. Therefore, one may argue that each underlying spin ½ Dirac-Weyl fermion hosts a
single zero-energy mode, which shall be responsible for a half-integer anomaly in the quantum
Hall response of the system. Conversely, the number of zero modes for integer spin is not
bounded.
This solution (6.9) is iterative in nature and allows us to take the analytical expressions of the
WLLs from pseudospin s to s  1/ 2 . From this method, one can derive the exact energy
spectrum for different pseudospin, such as s  {1 / 2, 1, 3 / 2, 2} presented in table 6.1. In this table,
we find also that the energy spectrum of the s  1 / 2 WLL is analogous to the relativistic Landau
levels in graphene [48].
We also observe the characteristic dependence of the energies,
E  g  B which is a hallmark that guarantees the relativistic nature of the particles. As can
be seen in the table 6.1, this peculiar dependence, E  B , is also fulfilled in the higher
pseudospin cases. In the s  1 case, we observe a couple of particle-hole symmetric levels with
analogous properties, but also a novel zero-energy Landau level which is completely absent in
the half spin case. The presence of this particular zero-energy WLL will have important
consequences in the quantum Hall response of the system. Finally, for s  3 / 2 , and s  2 , we
observe two pairs of particle-hole symmetric levels, which is a consequence of the two
underlying species of spin ½ Dirac-Weyl fermions assigned to each Dirac point. Note the
dependence on the number of index n gets more involved as the pseudospin is increased.
68
Table 6.1: Analytical expressions for the energies of the Weyl-Landau levels.
The energies corresponding to the Landau levels of the Dirac-Weyl fermions
with different pseudospin s  {1 / 2, 1, 3 / 2, 2} are expressed as a function of the
coupling strength g  c / 2l B and the number of motional bosonic quanta n .
In addition to the WLLs presented in table 6.1, we also find certain special topological solutions
that occur at zero energy. These solutions, the so-called zero-energy modes, play a key role in
the quantum Hall response of the sample and give rise to the half-integer anomaly [71]. The
underlying topological modes contribute with a fractional transverse conductivity (in units of
e 2 / h) , even in the absence of interactions. In table 6.2, we show that Dirac-Weyl fermions with
69
a half-integer spin s support N z  s  1 / 2 topological zero-modes which are protected by a
topological Atiyah-Singer index theorem [74-75] (see Appendix B).
Table 6.2: Analytical expressions of the zero-energy modes. The Weyl-Landau
Hamiltonian in equation (6.4) also yields certain zero-energy modes whenever
the constraints over the number of motional quanta presented in table 6.1 are
not fulfilled. In this table, we list these topological zero-energy modes for
pseudospin s  {1 / 2, 1, 3 / 2, 2} , where | s, m  refers to the spin state and | n 
to the motional Fock state.
Besides, they present half the degeneracy of higher Landau levels, and thus lead to a half-integer
anomaly in the quantum Hall response. On the other hand, for integer spin s , there are also nontopological zero-energy Landau levels that arise from the highly-degenerate flat band. As argued
in Appendix B, these zero modes are not related to an index theorem and thus, are not protected.
This highly-degenerate zero-energy band characterizing the integer-spin case is responsible for
70
the vanishing of the half-integer anomaly. As confirmed numerically in the following, the Hall
conductivity of the system fulfills
e2
N
NW (v  z ), s  half integer
h
2
,
2
e
 xy  NW v, s  integer
h
 xy 
(6.16)
where v  0,1,... determines the different plateaus, and NW  4 is the number of Dirac points.
Therefore, as stated above, the half-integer anomaly is only valid for the half-integer spin DiracWeyl fermions. It is also important to note that due to the fermion doubling [76], where NW  4
in our case, the fractional character of the Hall sequence is lost.
6.2.2 Lattice description: computing the Chern number.
The Hall conductivity can be evaluated numerically by diagonalizing the full tight-binding
Hamiltonian after having made the Peierls substitution T  T e
i
e
dr  A
 

. Considering the standard
Kubo formula, the Hall conductivity is given by the TKNN expression [77] as
H 
e2
h

E  EF
i
2
 dk 
T2
u u
u u
e2
|
    |  
 Nch (band ) ,
k x k y
k y k x
h occ.bands
(6.17)
where | u  are single-particle eigenstates, the integration is taken over the 2-torus T 2 and where
the Chern number associated to each occupied band, N ch (band), can be efficiently computed
using the method of [72]. Here EF denotes the Fermi energy, which can be tuned in our setup by
varying the atomic filling.
71
Figure 6.2: Hofstadter-like fractal butterflies in the Dirac-Weyl fermions
system. The energy spectrum E  E ( ) as a function of the magnetic flux
shows a fractal butterfly structure: (a) N  2( s  1 / 2) , (b) N  3( s  1) , (c)
N  4( s  3 / 2) , and (d) N  5( s  2) . The parabolic dependence at low
magnetic fluxes can be related to the underlying relativistic fermions. Note also
that for integer spin, one gets a flat band exactly located at zero energy.
Before analyzing the specific Hall plateaus of the systems associated to different spin structures,
let us draw their energy spectrum E  E ( ) as a function of the dimensionless magnetic flux
  2Ba 2 . These computed spectra generalize the famous Hofstadter butterfly [78]. The fractal
butterfly spectra corresponding to the cases N  2,3,4,5 are illustrated in figure 6.2. For N  2 ,
one recovers the spectrum of the  -flux model [73]. For N  3 one observes a spectrum similar
72
to the Lieb lattice [55], which highlights the similarity between these two models which both
share a spin-1 configuration. We note the existence of a highly-degenerate flat band lying exactly
at zero energy. For N  3 , the spectra become more complex and show complicated overlaps
between butterfly-like substructures. In particular, one identifies two overlapping butterflies in
figure 6.2 (c)-(d), each of which belongs to one of the two species of spin ½ Dirac-Weyl
fermions for s  3 / 2 and s  2 . We note that for integer spin (N odd), the central flat band at
E  0 remains robust for all flux  .
The Hall plateaus corresponding to N  2,3,4,5 are illustrated in figure 6.3. We expect the Hall
sequences to be compatible with the continuum analysis in the low flux regime, and we therefore
set   1 / 51 in this analysis. We also focus on the low-energy regime, where the description in
terms of the Weyl-Landau levels is valid. First we note that for all cases in figure 6.3 (a)-(d)
steps of NW  4 are observed in the ranges EF  0 (hole) and EF  0 (particle). The difference
between these Hall sequences occurs at half-filling ( EF  0 ), where the flat band and the number
of zero modes play a fundamental role. For N  2( s  1 / 2) , one observes a central step of
NW N Z  4 1  4 , while for N  4( s  3 / 2) , one gets a central step of NW N Z  4  2  8 . These
numerical results confirm the prediction of equation (6.16) based on the analytical expressions
for the Dirac points, and the zero-energy modes derived earlier.
73
Figure 6.3: The quantum Hall effects in the Dirac-Weyl fermions system. The
transverse Hall conductivity as a function of the Fermi energy  H ( EF ) displays
a sequence of plateaus that are associated to an underlying topological order. (a)
N  2( s  1 / 2) , (b) N  3( s  1) , (c) N  4( s  3 / 2) , and (d) N  5( s  2) .
Here we set the magnetic flux to   1 / 51 .
For general even N (half-integer spin), one indeed obtains that the two central plateaus around
half-filling are given by  xy   NW N Z / 2 (in units of e 2 / h) , therefore giving a fundamental
signature of the number of Dirac points and zero modes. For odd N (integer spin), one finds a
vanishing contribution of the zero modes to the Hall conductivity. A Hall plateau corresponding
to  xy  0 is clearly observed in the vicinity of EF  0 . One can understand the vanishing of the
74
half-integer anomaly as a consequence of the lack of an index theorem of the zero modes. Note
that this general result is in perfect agreement with the Hall sequences computed for the T3 [54]
and Lieb lattice [55], i.e., a lattice with pseudospin 1.
In order to deepen our understanding of the different Hall conductivity plateaus for odd and even
N, we further investigate the associated edge-states, which can be obtained by diagonalizing the
system in a cylindrical geometry. In other words, the topological properties hidden in the bulk
(i.e., the Chern number) may be visible around the boundaries by the holographic bulk-to-edge
correspondence [73]. This correspondence is based on the fact that the edge-states carry the Hall
current along the boundaries of the system. In figure 6.4 we show the corresponding energy
spectra E  E (k ) , where k is a Bloch parameter, for N  2,3,4,5 . Note that figure 6.4 (a) is
similar to the spectrum of graphene, and highlights the anomalous quantum Hall effect that
occurs for half-integer spins where the contribution of the zero modes confirms the
aforementioned results for which  xy ( EF  0 )  NW NZ / 2 in units of e 2 / h . In figure 6.4 (a) and
(c), we observe that each boundary is populated by two edge-states. This is due to the presence
of four Dirac points in the first Brillourin zone (in contrast with the two Dirac points of graphene
that leads to a single edge-state). Figure 6.4 (b) and (d) show the absence of gapless edge-states
in the first gap above E  0 , as observed for all odd N. This analysis confirms the general result
presented by equation (6.16).
The absence of an anomalous quantum Hall effect for integer s is certainly interesting. As
discussed above, this effect is also manifested by the zero Hall plateaus around EF  0 (cf. figure
6.3 (b) and (d)) or by the absence of visible edge-states stemming from the zero-modes. It is
therefore reasonable to argue that they cannot contribute to the Hall conductivity since this
75
physical observable is topologically protected. Furthermore, the localized properties of the states
rooted in the flat band could also explain why their associated zero-modes potentially contribute
to edge-states with zero velocity.
Figure 6.4: Energy spectrum and current-carrying edge states. Energy spectrum
E  E (k ) for (a) N  2( s  1 / 2) , (b) N  3( s  1) , (c) N  4( s  3 / 2) , and (d)
N  5( s  2) . Here we set the magnetic flux to   1 / 51 and the length of the
cylinder equal to 100 lattice sites.
76
6.3. Multi-refringent Klein tunnelling
It has been shown [50] that due to the coupling of positive and negative chirality channels
outside and inside a potential barrier, quantum tunnelling of Dirac particles in graphene becomes
highly anisotropic, where the barrier remains perfectly transparent for normal incidence.
Evidence for Klein tunnelling has been obtained from graphene p-n junctions [79-80] and most
recently been simulated using trapped ions [62]. The different helicities carried by the DiracWeyl fermions can naturally couple inside the barrier, but there is also a possibility to transform
one helicity to another outside the barrier, resulting in a remarkable multi-refringence
phenomenon familiar from optics [68]. In the following, we first present a detailed treatment of
the Klein tunnelling of Dirac-Weyl fermions with arbitrary spin, and then a numerical
investigation of the double-layered cone structure of spin 3/2 particles which shows Kleinbirefringence is carried out.
We shall first give a general description of the Klein multi-refringence of the Dirac-Weyl
fermions with arbitrary spin. We consider the isotropic Weyl-like Hamiltonian with cx  c y in
equation (6.3), which is valid for low-momentum excitations around the Dirac point d  (1,1) ,
i.e., HWd (k )   ckSv kv . For simplicity, we consider a spin-s Dirac-Weyl particle tunnelling
v
through a rectangular potential barrier with potential V0 in the interval 0  x  D and zero
elsewhere. The particle is incident on the interface at x  0 at an angle  from the interface
i(k x xk y y)
normal (see figure 6.5). The plane-wave part of the solution is e
, where k x  k cos  and
k y  k sin  . Due to the particle-hole symmetry, the helicities form [s+1/2] pairs, where [s]
stands for the integer part of s.
77
Figure 6.5: Klein birefringent tunnelling. Schematic of a spin 3/2 Dirac-Weyl
fermion incident on a potential barrier (a) while following the outer cone (b).
For incident particles with energy V0  E  0 , and helicity h0 , i.e., the helicity determines the
corresponding energy E  ckh0 , all negative-helicity channels are coupled inside the potential
barrier with the relation ( E  V )  ckinh h , where kinh , x  kinh cos inh and kinh , y  kinh sin inh . Outside the
potential barrier, the helicity h0 is allowed to convert into other positive values under energy and
h
h
h
h
h
h
h
h  kh0 , kout
momentum conservation, i.e., kout
, x  kout cos out and kout, y  kout sin out . Due to the
conservation
of
parallel
wavevectors
in
the
tunnelling
process
one
has
h
h
kinh sin inh  k sin   kout
sin out
. Note that a nonzero helicity is not allowed to convert into a zero
helicity due to the violation of energy conservation. The wave function in the three regions is:
78
I  Rh0 eik x x 
II 
III 
h  1 / 2 or 1

hS
hs
r e
h
h  ikout, x x
h L
h 1 / 2 or 1
AhRhe
hs
t  e
ikinh , x x
h
h ikout, x x
h R
h 1 / 2 or 1
, x0
 BhLhe
 ikinh , x x
, 0 x D
,
(6.18)
,D  x
where the spinor  Rh ( Lh ) is the eigenvector of H Wd with helicity h for the right (left) moving
h
wave. Note that  inh and out
for left moving waves are obtained by using the (  inh ) and
h
(  out
) angles in the solution for the right moving wave. Here, rh , Ah , Bh and t h are 4[ s  1 / 2]
unknown parameters to be determined. By integrating the equation HWd   E over an interval
in the vicinity of the interface, the boundary conditions are obtained. For half-integer spin, the
boundary conditions require the continuity of each component of the (2s+1)-component spinor at
the two boundaries of x  0 and x  D . This gives 4s  2 equations which are equal to the
number of unknowns, 4[ s  1 / 2]  4s  2 for half-integer spin. However, for integer spin where
even and odd components of the spinor are decoupled, the boundary conditions require the
continuity of each even spinor component, together with the continuity of a sum of two
neighboring odd components at x  0 and x  D . These conditions give 4s equations which are
also equal to the number of unknowns, 4[ s  1 / 2]  4s for integer spin. Thus, in principle, the
tunneling properties can be completely determined by the wavefunction and boundary conditions.
Since different helicity spinors carry different currents, the transmission and reflection
coefficients
have
to
be
renormalized
with
respect
to
the
incident
current,
i.e.,
Th  th2[Rh  S xRh ] /[Rh0  S xRh0 ] and Rh  th2 [Lh  S xLh ] /[Lh0  S xLh0 ] according to the current density
j    S x  of the Weyl-like Hamiltonian. Consequently, conservation of the current requires
79
s
 (R
h 1 / 2 or 1
h
 Th )  1 . In general, for incident particles of spin-s Dirac-Weyl fermions and fixed
helicity, the transmission shows [s+1/2]-fringence.
We next discuss the role evanescent waves play in the multirefringent tunneling. In graphene,
evanescent waves don’t play a role in the Klein tunneling due to the single-layered cone structure.
For multiple-layered cones, however, evanescent waves will play an important role in the
coupling and transformation of one helicity to another both inside and outside the barrier, by
exerting cutoff conditions for each helicity component. We first consider the coupling of the
incident helicity to the ones inside the barrier. For a low enough barrier, there are negative
helicities inside the barrier which are not coupled. They are evanescent. Specifically, for an
incident wave beyond the critical angle c  arsin ((V0  E ) / ckm) , where m  1 / 2 or 1 , all
helicities are uncoupled, thus there is no transmission. For a higher barrier, beyond the critical
angle c  arsin ((V0  E ) / ckhc ) , only helicities with h  hc are coupled in the form of
propagating waves. However, the transmission properties would be modified dramatically in this
regime. Besides the cutoff conditions inside the barrier, the evanescent waves also exert cutoff
conditions for different helicities outside the barrier when a small helicity is transformed into a
h
 sin h / h0 , for incident waves beyond the critical angle
larger one. Since sin out
c  arcsin( h0 / h) , the transmitted wave with helicity h becomes evanescent. This gives a cutoff
condition for each helicity component which is larger than the helicity of the incident particle. It
is clear that if the incident particle follows the outermost cone of the multicone structure, there
will be no cutoff for any helicity outside the barrier.
80
We present next a numerical calculation confirming the above theoretical analysis. The incident
energy of the particle is E , the barrier width is D and height V0 , respectively. The incident
particle is chosen to follow the outer layer cone. As such, the evanescent wave does not cause a
cutoff for any helicity outside the barrier, see figure 6.5 (b). Figure 6.6 shows the transmission
of spin 3/2 Dirac-Weyl fermions when the width and the height of the barrier changes. For
normal incidence, the barrier is perfectly transparent as in grapheme [50], see figure 6.6 (a). The
component following the inner cone shows periodic peaks in the transmission, which is the
hallmark of birefringence. There are no resonant conditions where the barrier would become
perfectly transparent at certain incident angles, apart from normal incidence. For lower barriers,
evanescent waves play an important role when coupling helicities outside and inside the barrier,
as shown in the two lowest panels in figure 6.6(a) and figure 6.6(b). For a low enough barrier,
and for an incident wave beyond the critical angle c1  arsin ((V0  E ) / ckh1 , with h1  1 / 2 , all
the helicities coupled inside the barrier are evanescent waves, hence there is no transmission for
any component as shown in figure 6.6 (b). For a combination whose critical angle is
c 2  arsin ((V0  E ) / ckh2 with h2  3 / 2 there is only one negative helicity h1  1 / 2 , inside the
barrier which is coupled in the form of a propagating wave. The transmission properties are
consequently dramatically modified in this regime, see figure 6.6 (b). These two boundaries
h1  1 / 2 and h2  3 / 2 are clearly visible in figure 6.6(b). Herein lays the paradox: for low
barriers, there is no transmission, while for high barriers, the transmission is high. For the case of
an incident particle following the inner cone, there is an additional cutoff condition exerted by
evanescent waves for helicity h2  3 / 2 outside the barrier, i.e., beyond the critical angle
c  arcsin( h1 / h2 )  arcsin( 1 / 3)  0.34 . A particle with helicity h2  3 / 2 can therefore not be
transmitted.
81
Figure 6.6: Klein birefringent tunnelling. Double transmissions (Birefringence)
of spin 3/2 Dirac-Weyl particles from the double-layered cone structures as the
width and height of the barrier change. In (a), the transmission diagram as a
function of the width D (top two panels) and height (middle two panels) of the
potential barrier is shown. Parameters used in the simulation: top two panels,
2(V0  E ) / ck  5 ; middle two panels kD  50 . (b) The initial increase in
height of the barrier (see also the middle two panels in (a)) illustrates the role
of the evanescent waves, where the cutoff conditions for helicity h1  1 / 2 and
h2  3 / 2 are clear visible.
82
For n-layered cones we obtain similar results, where inside the barrier the presence of evanescent
waves will exert a cutoff for each helicity, where with increasing barrier height transmission is
gradually allowed for the different helicities outside the barrier. Outside the barrier, the
evanescent waves only exert a cutoff when a helicity is transformed into a larger one. The
transmission spectrum typically shows n-refringence, which we refer to as Klein multirefringent
tunneling. This multi-refringence is a result of the rather unique and non-trivial helicity of these
Dirac-Weyl fermions.
6.4. Generalizations to tunable effective speeds of light
In order to go beyond the structure of the su(2) algebra, we first write out explicitly the
representations of the su(2) algebra. A simple representation of S x and S y for spin s with
n  2s  1 components which generalizes the Pauli representation for spin ½, can be expressed in
terms
of
a
(n-1)-vector
n  ( 1, 2 ,..., n 1 )
S x  {Superdiag { j }, Subdiag{ j }}
and
where
j 
j (n  j )
S y  {Superdiag {i j }, Subdiag{i j }}
such
,
that
where
j  1,..., n  1 . However, it is worth noticing that for any real vector  , S x and S y , the location of
the Dirac points of H (k ) does not change because the spectrum is E (k )    | g k | , where   are
the eigenvalues of the Tv matrix with v  x, y and g k  {2t x cos k x ,2t y cos k y ,0} . This observation
is crucial as the goal is to preserve the stability of the Dirac points while relaxing the integer or
half-integer spin structure such that we can construct arbitrary effective speeds of light in the
emergent quasi-relativistic scenario. To this aim it is essential to keep the off-diagonal shape of
the hopping matrixes while allowing  j to take any real value.
83
Proof: for S x  {Superdiag { j }, Subdiag{ j }} with any real vector  , its spectrum is
determined by the zeros of det( S x  I ( n ) )  Pn ( ) where  is the eigenvalue, i.e., the
eigenvalues are determined by the characteristic polynomial Pn ( ) . By using the
n
Laplace’s formula det( A)   (1)i  j aij M ij for the determinant of a n  n matrix in
i 1
terms of its minors M ij which is defined to be the determinant of the (n  1)  (n  1)
matrix that results from A by removing the i -th row and the j -th column, we can
get the following recurrence relation:
Pn1 ( )  Pn ( )   n2 Pn1 ( ), (n  1) .
(6.19)
For example,

P2 ( )  det(
 1


1 
2
2
)    1 , P3 ( )  det( 1
 

 0
1

2
0 

 2 )   ( 12   22   2 )
  
with the initial conditions P0  1 and P1   (determined from the expression of
P2 ( ) ). By induction, in general, Pn ( ) can be expressed as Pn ( )  f n ( 2 ) for even
integer n and Pn ( )  g n ( 2 ) for odd integer n . Thus this demonstrates the particlehole symmetry of S x , i.e., when  is the eigenvalue of S x , then   is also an
eigenvalue of it. And for odd integer n , there is an additional zero-energy flat band. It
is straightforward to show that S y has the same eigenvalues as S x as they share the
same characteristic polynomial.
84
With this generalization, properties that depend on the topological aspects of the system such as
the Hall plateaus will consequently not change, while properties depending on local aspects of
the system such as the butterfly nature of the spectrum and the Klein multirefringent tunnelling
will change. The resulting spectrum at each Dirac point will be a collection of Dirac fermions
with tunable effective speeds of light and a topological charge N equal to the number of the
layers.
As an example, we consider a double layered Dirac cone structure with a tunable Fermi velocity
for each component. We parametrize 4 as  (sin  cos  , sin  sin  , cos  ) .The corresponding
spectrum is of the form    1   / 2 / 2 , where   3  cos 2  cos 4  cos 2 cos 4 .
Interestingly, a birefringent breakup of the doubly degenerate Dirac cones into cones with
different speeds of light has also been discussed in quite a different setup with cold atoms in a
square optical lattice [81]. The authors show that when there is an average flux of half a flux
quantum per plaquette, the spectrum of low-energy excitations can be described by a four-band
model which gives the birefringent breakup of Dirac fermions. It is worth stressing that our setup
allows a breakup, or indeed a coalition, of any number of Dirac fermions. For example, when
  0 the two layered cones collapse into a single degenerate one with topological charge N=2.
This is also the mechanism for mixing the different Dirac species when on-site dynamics is
introduced. The N=3 situation, on the other hand, can be used to mimic the three families of
fermions in particle physics. These hopping matrices considered above can also be seen as
generalized spin-orbit coupled systems (see [82] for a recent realization of spin-orbit coupling in
Bose-Einstein condensate).
85
Chapter Seven: On-site dynamics and mixing of Dirac species
In the previous chapter we considered only the hopping dynamics. It is also possible to
manipulate
the
on-site
spin
flipping
dynamics
described
by
the
Hamiltonian
H o  [O] ` cr `cr . Since no momentum transfer is required, this term can be engineered by
r
 `
the standard technology based on microwave transitions [66]. It turns out the Hamiltonian
H  H t  H o allows us to simulate high energy phenomena, such as modified dispersion relations,
and neutrino oscillations [83] in Lorentz and CPT breaking no breaking scenarios [84-86].
7.1. Exotic particle dispersions
Modifications of dispersion relations have attracted a great interest in both condensed matter and
high energy physics. The Dirac fermions in graphene and semi-Dirac fermions with linear
dispersion in one direction and quadratic dispersion [87] in the other are examples of engineered
dispersion relations in a condensed matter setting. On the other hand, in high energy physics,
modifications of the energy momentum dispersion relations at the Planck scale are suitable, for
instance, to reconcile Lorentz symmetry and finite resolution of the spacetime points in quantum
gravity models [88]. As an alternative to the splitting of the multiple degenerate Dirac point, it is
possible to preserve the Dirac point and to engineer exotic particle dispersions which can have
any power of momentum. This Dirac-point preserving mixing is described by the Hamiltonian
H ( p)  {Superdiag{g  }, diag{ p   }, Subdiag{g }} , which mixes the N Dirac species and
gives the spectrum E  p N [89]. Our setup also allows us to investigate the mixing of DiracWeyl fermions with high spin beyond the above spin 1/2 case, for instance the fate of the flat
86
band with integer spin-s after mixing. Furthermore, disorder can also be introduced to the optical
lattice to simulate spacetime fluctuations at Planck scale, thus giving an opportunity to resolve
possible conundrums in quantum gravity models by a combination of concepts, such as
modifications of dispersion relations, spacetime fluctuations, and Lorentz violations discussed in
the following section.
7.2 Neutrino oscillations
The hopping Hamiltonian H t in (5.1) with the generalization discussed in section 6.4 allows us
to create a collection of spin ½ Dirac fermions at the same location in momentum space with
tunable Fermi velocities. A constant on-site term H o on the other hand gives an effective mixing
of the different Dirac species. This mixing mechanism is fundamentally different compared to
the mixing mechanism in graphene. In graphene the Dirac fermions are located at different sites
K  in the momentum space. A commensurate perturbation G  K  K introduced by lattice
distortions is needed to mix the Dirac fermions (see [90-91] for a discussion of the chiral mixing
in graphene by a Kekule texture when constructing the chiral gauge theory of graphene).
Furthermore, opening a gap in monolayer graphene is difficult too. Our setup allows mixing of
any number of Dirac fermions of the same chirality, and to open up gaps by the on-site
microwave Raman transitions.
We now consider the mixing of triple degenerate Dirac cones with a topological charge N  3 at
a single Dirac point K D 

2
(1, 1) , i.e., the Dirac cones are at the same location in momentum
space. We write the constant mixing term O in (5.1) in the form of hi   where hi  (hxi , hyi , hzi ) ,
87
Figure 7.1: (a) Triple-layered Dirac cones with three different effective speeds
of light. (b) The splitting of triple-degenerate Dirac cones. (c) Gap opening of
triple degenerate Dirac cones. The band structures in (b) and (c) are used to
simulate the exotic and traditional neutrino oscillations as discussed in the text.
i  1, 2, 3 are constant vectors, and   ( 1 ,  2 ,  3 ) is the vector of Pauli matrices. The
unperturbed case hi  0 corresponds to 6  (1, 0, 1, 0, 1) . The result is a combination of mixing
and splitting of the Dirac points, which are the paradigm of topological quantum phase
transitions (TPT). The corresponding Hamiltonian near the Dirac point is
( g k  h1 )  

H m (k )  
0

0

0
( gk  h2 )  
0

0

0
.
2
( g k  h )   
(7.1)
The spectrum is given by Ei   | g k  hi | . This concise expression provides a convenient way to
control the positions of the split Dirac points in the Brillouin zone. While hxi and hyi control the
positions of the split Dirac points, hz3 control the gap the energy spectrum. By requiring E  0
and setting hzi  0 for the time being, the split Dirac points are determined by the conditions
88
2t x cos k x  hxi  0 and 2t y cos k y  hyi  0 . In principle, h i may be tuned to appropriately large
values to create marginal Dirac points with topological charge N  0 by considering the Dirac
point of opposite chirality. The applications of such a scenario go beyond the realm of condensed
matter. However, in what follows we will focus on perturbative splitting.
Neutrino oscillations (NO) are considered by many as a possible window to physics beyond the
Standard Model. The most accepted mechanism to explain such oscillations is in terms of the
nontrivial mass term matrix. The Lagrangian density describing the flavor states vTf  (ve , v , v )
with a mass term M is L  v f ( x)(i     M )v f ( x) . The flavor of the neutrino oscillates as it
propagates since the flavor eigenstates, which are the eigenstates of the weak interaction in
conjunction with a particular flavor of a charged lepton, do not coincide with their mass
eigenstates with a definite mass and energy. In general, the weak interaction flavor eigenstates
can be represented as a coherent linear superposition of the mass eigenstates, v f  Uvm , where
vmT  (v1 , v2 , v3 ) describes the state with definite masses and U is the unitary transform action that
diagonalizes the mass matrix M, known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
lepton mixing matrix [83]. For all currently observed NO, the corresponding masses are less than
1eV and the energies are at least E  1MeV , with a Lorentz factor greater than 106 in all cases. In
this ultrarelativistic limit the energy is given by
Ei  ( p 2c 2  mi2c 4 )1 / 2  E  mi2c 4 / 2 E .
(7.2)
The time-dependent mass states are | vi (t )  exp[ iEit / ] | vi  , thus
v f (t )  UTU v f (0) ,
(7.3)
89
where T  diag{eiE1t /  , eiE2t /  , eiE3t /  } is the time evolution matrix. This leads to the transition
probability
P(v  v ) | U ajU e
*
j

imi2 c 4
t
2
2 E
| ,
(7.4)
j
which shows that the neutrino flavour changes with time. The NO are attributed to the mass
difference between the mass states. The mechanism to generate such mass - or any other mixing
term - implies an extension of the Standard Model. The question of which mechanism is the best
is still not completely settled.
An analogue of neutrino oscillations can be engineered in optical lattices. The simple model of
Eq. (7.1) is the starting point to capture the essence of 3 species' mixing. Interestingly, our setup
allows us to simulate both traditional NO and exotic anisotropic NO. In the following, we will
consider the ultra-relativistic limit for coupled pseudo-particles for which | hi || g k | cl p
where p  ( HW  k ) . For the traditional NO, the analogue of a mass term can be reproduced by
setting hi  (0,0, hzi ) for which the spectrum Ei   | g k  hi | reduces to
Ei  cl | p | (hzi ) 2 /( 2cl | p |) ,
(7.5)
thus hzi represents the mass of the neutrinos. Moreover, our setup also allows us to simulate
exotic anisotropic NO where a direction dependent mass can be achieved in the limit of small
hi  (hxi , hyi ,0) which we shall focus on in this study. In this case, the spectrum of Ei   | g k  hi |
reduces to
Ei  cl (| p |  pˆ  hi ) ,
(7.6)
90
 
where pˆ  p / | p | . For sake of simplicity, we consider h1  0 and h 2  h3 . The H m (k ) of Eq.
(7.1) therefore plays the role of the block diagonal Hamiltonian in the mass basis. In fact, to
observe NO in the lab we have to implement the Hamiltonian H f (k ) governing the flavor
pseudo-particles. In terms of the PMNS mixing matrix U
H f (k )  (U   I 2 ) H m (k )(U  I 2 )  I3  gk    M  h   ,
(7.7)
where I n is the n n identity matrix, and M  U  diag{0,1,1}U , i.e., M ij  U3 jU3*i  U 2 jU 2*i . In
Lagrangian terms, our 2+1 model is
L  f ( x)(i     M  h )f ( x) ,
(7.8)
near the Dirac point. Thus the flavour Hamiltonian fits in the family of H f (k )  H t (k )  H o (h) ,
with H t (k )  I 3  g k   and H o (h)  M  h   . Remarkably, while H t (k ) can be implemented
by the triple-layered degenerate Dirac cones as discussed above, the PMNS mixing matrix is
completely encoded by the on-site Hamiltonian H o (h) , which allows us in principle to simulate
any mixing angles and CP-violating phase. The standard parametrization of the PMNS can be
found for instance in [83] and can be created by the on-site microwave Raman transitions [66].
As an explicit example, we discuss the simplified case when all the mixing angles are equal to
 / 4 with
H exf


( g k  cos 2 h)  


  i sin
h 
2


1
h 

2

 i sin

( g k  cos
2
h 

2
1
 h 
2
h)  
1

h  
2

1
 h   ,
2


gk   

91
(7.9)
Figure 7.2: Anisotropic quasi-neutrino oscillations and T-violations in an
optical lattice with mixing angles (12 ,13 , 23 )  (1 / 4, 1 / 4, 1 / 4) and splitting
h  0.01(2 )(1,0,0) . (a) The oscillations of the probability P ( e    ) with the
directional vector p̂ of the momentum where the CP-violation phase is   0 .
(b) The T-violation behavior with    / 2 (red), 3 / 2 (red), 0,  (green),
where T 
P( e    )  P(    e )
P( e    )  P(    e )
.
92
where  is the CP-violating phase. The quasi-neutrino oscillations are shown in figure 7.2. In
figure 7.2 (a), the transition probabilities for   0 are plotted against p̂ , showing anisotropic
and energy-independent behaviors. The Hamiltonian of Eq. (7.9) is real for   0 , hence invariant
under time-reversal symmetry T. Since the oscillation period is inversely proportional to the
splitting, one obtains for a small splitting parameter h  0.01(2 )(1,0) , a period which is of the
size of typical lattices. In high energy physics, evidence for NO is only presented over a distance
L  100 km and no evidence for oscillations for L  1 km [83]. In figure7.2 (b), the effect of
  0 is considered. The Hamiltonian becomes complex and T is violated. Due to CPT
invariance, T-violation is equivalent to CP-violation. The Time-violation behaviour of the
transition probability
T  [ P( e    )  P(    e )] /[ P( e    )  P(    e )] ,
(7.10)
is shown for   {0,1 / 2,1,3 / 2} . The signature of CP-violation is rather spectacular. In an optical
lattice experiment, direct evidence of the above phenomena can be obtained by measuring the
different populations in different points of the lattice by the colour resolutions or individual atom
detection techniques [92, 93].
Our model is Lorentz and CPT conserving if h is a Lorentz covariant vector. The role of
symmetries in the analogue simulation is indeed subtle as they are artificial. The Hamiltonian
engineered in the lattice is specific for a certain reference frame (or gauge). To consider
transformed Hamiltonians in other frames implies physical modifications of the experimental
apparatus. For similar discussions of gauge symmetries see [94]. In practice, we can choose how
h transforms. For instance, we can treat it as a constant vector and reproduce one of the Lorentz
and CPT violating NO terms, as discussed in extensions of the Standard Model [84]. The
93
simulation of NO in optical lattices is not only a relevant exercise in order to show the vast
possibilities of ultracold atom technology, but it also opens the way to study the effect of novel
physics. In particular, optical lattices can be ideal for simulating new flavour couplings
originated from strong coupling in a controllable way.
94
Chapter Eight: Experimental detections and outlook
In chapter six and seven we have discussed several phenomena, such as anomalous quantum Hall
effects, Klein multi-refringent tunnelling and neutrino oscillations. In this chapter we will briefly
discuss how to detect these phenomena by virtue of the recent experimental techniques for
ultracold quantum gases.
8.1 Experimental detections
The experimental realisations of the proposed scenarios and the detections of the resulting effects
are certainly challenging but should still be within experimental reach with state of the art
trapping and manipulation of atomic ensembles. We point out that any detection scheme capable
of resolving the effects discussed in this thesis will typically have to be able to distinguish
between spin states, particle density and momentum distribution. We briefly outline some of the
possible techniques one can use.
Regarding the number of Dirac points we note that this can be addressed by measuring the atom
density close to a zero chemical potential [95]. To obtain the number of zero modes and the
number of Dirac points one can evaluate, or indeed measure, the Hall conductivity around halffilling. This could be achieved using the Streda formula based on the atomic density which can
be measured precisely by the in situ individual atom detection as in [92, 93]. In addition, a
measurement of the atom density as a function of the chemical potential allows us to map the
density of states (DOSs), where the Van Hove singularities in the DOSs are directly related to
the number of layers of the cone structure, see [65]. By mapping out the momentum distribution
using atomic angle-resolved photoemission spectroscopy (ARPES) [96], momentum-resolved
95
Bragg spectroscopy [97], or adiabatic release [98], allows us to map the Fermi surface and thus
the location of each Dirac point.
The existence of a flat band with integer spin can be detected by its energy dispersion and its
related wave function. The flat band gives a peak in the DOSs [54], which can be detected by
measuring the atomic density. More importantly, the localization properties resulting from the
flat band could also be detected [56]. This localization can be observed after the weak harmonic
confinement of the atoms is removed but with the optical lattice kept in place: the atoms
occupying the Dirac cones will fly away fast while the atoms occupying the flat band will remain
stuck in the immobile flat band states as shown in [56]. In our system, this localization property
should be observed for N odd and can be understood by observing the vanishing of the DiracWeyl spinor components corresponding to odd cyclotron modes (see equation (6.15))-in direct
analogy with the flat band wave function of the Lieb lattice. The vanishing of the components
corresponding to the odd cyclotron modes of the Dirac Weyl spinor can also be confirmed by the
colour resolution strategies summarized in Ref. [99].
The presence of edge states in the bulk gap above half-filling [100-102] is an intriguing concept.
For this undoubtedly challenging endeavour, one would need to engineer a sharp boundary, with
a characteristic length of the order of the lattice constant, in order to stabilize the presence of
topological edge-states within the center of the trap [102]. Loading the atoms into the edge
states can be achieved with external light pulses [101,102]. In addition, the dynamical structure
factor S ( q,  ) from light Bragg scattering can also provide a direct way to observe the edge
states and bulk states as demonstrated in [100]. The lack of edge-states in the bulk gaps around
half-filling could be an important indicator of the existence of a flat band.
96
To detect the Klein tunnelling, the most natural approach seems to be designing a potential
barrier for the atoms by optical or magnetic means, and preparing the atoms with a well-defined
momentum, then see if particles have tunnelled through the barrier to the other region. However,
a direct confirmation of the Klein multirefringent tunnelling would require the launch of a multicomponent mass current. To launch such a mass current, several schemes can be used. For
example, one can connect the optical lattice to two reservoirs with different chemical potentials
as in [103,104], or exert a static force from a tilted optical lattice [105], or by an effective electric
field on the atoms from the optical dipole force [106]. Measuring the mass current across the
barrier will consequently reveal the intricate dependence on helicity for the tunnelling dynamics.
8.2 Conclusions and Outlook
In the second part of this thesis, we studied the quantum simulations of the multiple-layered
Dirac cones and related phenomena by using multi-component (colour) ultracold fermionic
atoms in an optical square lattice. This implementation relies on the ability to control the spindependent hopping and the on-site spin flipping by Raman lasers. In chapter six, we have
proposed the quantum simulations of Dirac-Weyl fermions with any arbitrary spin by tuning the
spin-dependent hopping according to the representations of the su(2) algebra. In this way we can
assign any arbitrary spin s to these fermions, and thus go beyond the standard spin1/2 regime of
both high energy physics and condensed matter physics. We have also presented a detailed study
both analytically and numerically, of several striking aspects of the Dirac-Weyl fermions. In
particular, our system hosts two different phases: a semi-metallic phase for half-integer s and a
metallic phase that contains a flat zero-energy band for integer spin s . In the presence of a
synthetic magnetic field, we have connected the Weyl-Landau problem to the Dicke model
known from quantum optics. The corresponding Hamiltonian presents a rich structure of Weyl97
Landau levels and zero-energy modes whose robustness can be related to an index theorem for
half-integer s, which also induces an anomalous half-integer quantum Hall effect. We have also
shown that the low-energy transport of a high spin Dirac-Weyl fermion is characterized by
multiple spin ½ Dirac-Weyl fermions moving at different speeds. As a consequence, we also
found an exotic Klein multi-refringent tunnelling across a potential barrier. Finally, we showed
how to get rid of the limitations of su(2) algebra and thus obtaining multiple-layered Dirac cones
with arbitrary tunable different speeds of light. In chapter seven, we focused on an analogue
between the three-layered cones scheme and the three-family of fermions in particle physics.
Specifically, we showed how to simulate some topics from Standard Model Extensions, such as
modified dispersion relations which are common approaches to reconcile the paradox between
Lorentz symmetry and finite resolution of the spacetime points in quantum gravity models, and
neutrino oscillations where the mass of neutrino is not included in the Standard Model of particle
physics. We also discussed how to experimentally detect the various phenomena emerging from
the optical lattice. The second part of this thesis contributes to the design of a fully fledged
quantum simulator capable of addressing challenging new topics in modern physics.
98
Conclusions
In this thesis, we have explored the idea of quantum simulations by using ultracold quantum
gases. In the first part of the thesis, we have developed a general method, applicable to a wide
class of species, ranging from atoms to molecules and even to nanoparticles (though we used
molecules as an example in the study), to decelerate a hot fast gas beam to zero velocity by using
optical cavity. The novelty of the deceleration method is the new phase stability mechanism
found in the bad cavity regime which is very different from the traditional cavity cooling studies
where a good cavity is needed, thus allowing us to uncover the new physics hidden in the bad
cavity regime. This new phase stability is derived from the intracavity field induced selforganization of a fast-moving molecular beam into travelling molecular packets where all the
molecules in the beam are interacting collectively. This mechanism ensures the phase stability of
the majority of the molecules in the cavity rather than a small fraction determined by the
acceptance volume as in phase space filtering techniques (e.g. electrostatic Stark decelerations).
Based on this new phase stability mechanism, we proposed several schemes to decelerate the
beam to zero velocity when properly introducing the deceleration force. In the first scheme we
used a bad cavity and since at steady state the position information of the bunched molecular
packets is encoded in the output of the intracavity intensity instantaneously, then by using the
output of the intracavity intensity via a feedback loop we modulated automatically the pump
intensity of the lasers between a high pump state and a low pump state which are correlated with
the climbing-up and climbing-down processes of the molecular packets in the standing wave
potential of the cavity. In this way, when the molecular packets climb up the potential they
experience a strong slope and when the molecular packets climb down the potential they
99
experience a weak slope, such that they lose energy after a cycle of the cavity mode and slow
down after many cycles. In the second scheme, we tuned the cavity away from the bad cavity
regime to an intermediate cavity regime in order to gain a deceleration force from the
nonadiabatic response of the intracavity field to the molecular motion. However, since the phase
stability mechanism works well in the bad cavity regime and the stability of the molecular
packets decreases when the nonadiabatic effect is introduced, we found that a balance between
the phase stability and deceleration force has to be achieved. By properly balancing the two
effects, we also demonstrated an effective deceleration. We further showed the above two
schemes can be combined to give a more effective deceleration though this technique is more
complicated. We also discussed in detail the practical issues to realize the proposed deceleration
schemes which show they are feasible under the present experimental techniques. Thus the first
part of this thesis contributes to the methodological developments of getting cold and ultracold
gaseous samples by using cavity setup and will stimulate the experimental efforts toward this
new possibility.
In the second part of the thesis, we have showed what can be done with these cold quantum
gases in terms of quantum simulations. We considered a setup with multi-component ultracold
fermionic atoms in optical square lattices with both spin-dependent hopping and on-site spin
flipping controlled by Raman lasers and designed Hamiltonians that are both relevant to
condensed matter physics and high energy physics. For the quantum simulations of condensed
matter physics we considered a topic related to Dirac fermions from graphene and topological
insulators which are currently two actively researched topics in condensed matter physics. Our
goal was to generalize the spin ½ Dirac fermions in graphene to any arbitrary spin. We achieved
this by tuning the spin-dependent hopping according to the representations of su(2) algebra. In
100
this scheme, we found there are four band-touching points, around which the low energy
excitations are Dirac-Weyl fermions with any pseudospin defined by the number of internal
levels of the atom being addressed. We further investigated what we can learn from this
generalization and found very rich anomalous quantum Hall effects and remarkable Klein multirefringent tunnelling. For the quantum simulations of high energy physics where we focused on
topics beyond the Standard Model, we first showed how to get rid of the su(2) algebra restriction
and then by implementation of an analogue between three-family fermions of particle physics
and a three-layered cones scheme, we showed how to design Hamiltonians that can simulate
neutrino oscillations both in Lorentz and CPT breaking, or no breaking scenarios, when turning
on the on-site spin flipping. We also showed how to simulate exotic dispersion relations, so
called modified dispersion relations, which are useful ways in order to reconcile the paradox
between Lorentz symmetry and finite resolution of the spacetime points in quantum gravity
models at the Planck scale. We believe these ultracold systems are attractive for building a
quantum theatre where we may create not only existing physics that is difficult to reproduce but
also physics that may not exist in Nature. For example, we may build a simulator of lattice QCD
by using fermionic atoms as quarks and bosonic atoms as gluons within specially designed
optical lattices and fine-tune interactions between different species. We may also create a whole
new “universe” where extra spatial dimensions [110] from string theory landscapes or even extra
temporal dimensions from 2-T physics [111] may be studied by utilizing the rich energy
structures of ultracold atomic gases. This thesis demonstrates the important role ultracold
quantum gases play in terms of quantum simulations in order to address some challenging topics
in modern physics at the frontiers of both condensed matter physics and high energy physics.
101
Appendix A
In this Appendix, we will give a derivation of the semiclassical equation (1.1) used in the first
part of the thesis in detail. For simplicity, we start from a two-level atom, where the extension to
many particles and beyond the two-level approximation are straightforward. The atom-field
dynamics is described by the following Hamiltonian in a frame rotating with pump frequency  p ,
H
p2
 C a  a   A    ig ( x)(  a  a   )  i (     ) .
2m
(A.1)
The first term is the kinetic energy of the atom, the second term is the free energy of the photon
in the cavity, the third term is the internal energy of the two-level atom, the fourth term is the
Jaynes-Cummings (J-C) coupling between the atom and photon, and the last term is the pump
term of the cavity by the atom, where  C   p  c and  A   p   A . The operators p and x are
associated with the atomic momentum and position, respectively and g ( x)  g0 cos( kx) is the
position-dependent atom-field coupling, while  the pump strength. The field is described by
the annihilation and creation operators a and a  , while the two-level atom is described by the
lowering and raising operators   and   . From the Heisenberg equation
dA i
 [ H , A] , we can
dt 
get:
a  i C a  g ( x)   a
   i A   g ( x)a z   z   
x  p / m
,
(A.2)
p  i(  a  a   )g / x
102
where we have also included the damping dynamics of the photon and the atom from the decay
rate of the cavity  and the decay rate of the atom  . When the internal atomic dynamics  
evolve on a much more rapid time scale than the external dynamics p , either due to the large
detuning  A   p   A or due to the large damping rate  , the population in the excited atomic
state is negligible, i.e., we consider the low saturation regime, which allow us to adiabatically
eliminate the internal atomic dynamics, i.e., setting    0 in the second equation of (A.2) and
noticing  z  1in this case, we can get   
g ( x)a  
. Substituting this expression to the
i A  
other equations in (A.2) and assuming that the coherent field amplitude   a  is much bigger
than one, then we can treat the quantum operators as their classical counterparts, a   ,
a    * , then we get,
  (iC   )  (iU 0  0 ) cos 2 (kx)  eff cos( kx)
x  p / m
,
(A.3)
*
p  2k |  |2 sin( kx) cos( kx)U 0  ik(eff
   *eff ) sin kx
where U 0 
g 0
g02 A
g 02


,
and eff 
.
0
2
2
2
2
 i A  
A  
A  
Equation (A.3) is easily to be generalized to many-particle case, which gives,
  (iC   )  (iU 0  0 )  cos 2 (kxj )  eff  cos( kx j )
j
j
x j  p j / m
.
*
p j  k |  |2 sin( 2kx j )U 0  ik(eff
   *eff ) sin kx j
103
(A.4)
When the detuning is large, we can neglect  , rewriting 
g 0
A
  , then (A.4) reduces to
  (iC   )  iU 0  cos 2 (kx j )  i  cos( kxj )
j
j
x j  p j / m
.
p j  k |  |2 sin( 2kx j )U 0  k 2 Re( ) sin kx j
Equation (A.5) is just equation (1.1) used in the thesis.
104
(A.5)
Appendix B
In this appendix, we describe the Atiyah-Singer theorem which governs the robustness of the
zero energy modes discussed in chapter six. The Chern number which determines the quantum
Hall response of the system as in chapter six is a manifestation of topology. In this Appendix, we
describe yet another manifestation of topology: the relation of the zero modes to the AtiyahSinger theorem [74]. This famous theorem, which relates the analytical and topological features
of differential operator, has important consequences for the properties of Dirac fermions subject
to external gauge fields [107]. Graphene therefore has turned out to be an excellent platform to
understand this relationship both from a theoretical [108] and experimental viewpoint [107]. We
describe how these concepts can be generalized to Dirac-Weyl fermions of arbitrary spin s. We
find that only the zero modes of half-integer spin Dirac-Weyl fermions are protected by the
topological features of the system. This justifies the absence of the half-integer anomaly for
integer-spin Dirac-Weyl fermions.
The Weyl-Landau Hamiltonian in equation (6.4) for d  (1,1) in the isotropic regime cx  c y
presents the following particle-hole symmetry
{HWd L, s }  0 ,
s  ei ( S z  s ) ,
(B.1)
which fulfills s2  I , s  s . This operator, known as an involution [109], allows us to
decompose the Hilbert space as H  H  H , where H  follow from the orthogonal projections
105
P 
1
( I  s ) associated to the  eigenvalues of the involution. With this formulation, the
2
Weyl-Landau Hamiltonian can be rewritten as a supercharge C
d
WL
H
 0 D 
,
 C  

D 0 
(B.2)
where the differential operators D  P HWd LP : H   H  , and D   P HWd LP : H   H  , join the
orthogonal subspace. In this language, the analytical index of the supercharge can be expressed
as:
indC  v  v  dim(ker D)  dim(ker D ) .
(B.3)
For elliptic operators [109], this index can be related to the topological features of the system via
the famous Atiyah-Singer theorem. This relationship not only gives insight into the number of
zero modes in the system, but also pinpoints their robustness with respect to local perturbations
of the Hamiltonian. From the results in table 6.1 and 6.2, we observe that v  s  1 / 2 and v  0
when s is half integer. The total number of zero modes determines the index ind C  s  1 / 2 ,
which is related to the total magnetic flux that pierces the system. Therefore, these zero modes
are extremely robust with respect to local perturbations of the Hamiltonian. Conversely, the
number of zero modes for integer spin is unbounded. In this case, the differential operator D is
not an elliptical operator, and thus the Index theorem does not apply. Accordingly, the zero
modes for an integer-spin Dirac-Weyl fermions are not topologically protected.
106
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