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Dynamics of Bose-Einstein Condensates in
Trapped Atomic Gases at Finite Temperature
Lecture 4: Applications of the ZNG Equations
Eugene Zaremba
Queen’s University, Kingston, Ontario, Canada
Financial support from NSERC
Outline, cont’d
Lecture 3: Solution of the ZNG Equations
• numerical solution of the GGP, split-operator method
• numerical solution of the kinetic equation, test-particle method,
collision algorithm
• model of condensate growth
• the moment method, scissors mode and ZNG simulations
• the static thermal cloud approximation, dissipative GP equations
Lecture 4: Application of the ZNG Equations
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quadrupole modes: JILA and ENS experiments
Landau damping: example of monopole (breathing) mode
introduction to vortices
stirring, rotating frames of reference
vortex energetics
phenomenological nucleation and equilibration of vortices
ZNG simulations of vortex precession and relaxation
Quadrupole Modes in Axisymmetric Traps
• as our first example we will consider the experimental results obtained
by the JILA group for the quadrupole collective modes
• a trap anisotropy couples the lowest monopole (l = 0, m = 0) and
quadrupole (l = 2, m = 0) modes of an isotropic trap, giving rise to two
m = 0 modes with frequencies (Stringari, 1996)
E: Derive this result from the quantum hydrodynamic equations in the
TF limit using the moment method. Hint: Consider the moment
<x2 + y2>.
The lower frequency ω− mode corresponds to the m = 0 mode of
interest in the experiments. For the experimental value of λ = √8, this
mode has the frequency
. The other mode of interest is the
m = 2 quadrupole mode with frequency
.
• above TBEC the thermal cloud undergoes a breathing-like oscillation at
the frequency
.
Visualizing the Modes
m=0
m=2
Exciting the Modes in the ZNG Simulations
• as a model, we start with the equilibrium configuration and impose
velocities on the two components:
m=0
m=2
• for the condensate, this is done by multiplying the equilibrium
wavefunction by an appropriate phase factor; for the thermal cloud we
simply add a small position-dependent velocity to each test particle
Model Results
• the points with error bars are
experimental results
• the blue squares are the
results obtained when only
the condensate is excited
T/TBEC
• the triangles are the results
for excitation of both the
condensate and thermal
cloud. The blue triangles are
for the condensate and the
red triangles are for the
thermal cloud.
• the results for m = 0 are poor
at the higher temperatures
Experimental Excitation Scheme
• we now show what happens when the actual experimental excitation
scheme is used for the m = 0 mode: the transverse frequency is
modulated harmonically for a short interval
• this panel shows the condensate
quadrupole moment; the vertical lines
indicate the experimental observation
window
• this panel shows the thermal cloud
quadrupole moment
 = 1.95 
T/TBEC = 0.8
Frequencies and Damping Rates
Drive Frequencies

T/TBEC
• when the drive frequency
approaches the thermal cloud
resonance at 2, the thermal
cloud mean field drives the
condensate at its frequency and
leads to the upward shift of the
condensate frequency at the higher
temperatures
The ENS Experiment
• “top hat” excitation scheme of the
m = 0 transverse breathing mode
Transverse Breathing Mode
• for the pure condensate at T = 0, this mode approaches 2 as
• for T > TBEC, the thermal cloud in the near-collisionless regime has a
transverse-breathing-mode also at 2. This near degeneracy of the
condensate and thermal cloud modes has an important consequence.
T = 125 nK
• shown are the results of a
ZNG simulation with the tophat excitation; the red curve is
for the condensate and the
blue curve is for the thermal
cloud
Comparison with Experiment
• there is overall good
agreement between
experiment and theory for
both the frequency and
damping of the mode
• note in particular that the
damping rate is an order of
magnitude smaller than found
in the JILA experiments; the
obvious question is: Why?
Further Simulation Results
• in this simulation, all collisions
are switched off; evidently,
damping is a collisional effect
• this is a simulation of the m = 2
mode; the damping is much
stronger than for the transverse
breathing mode. In addition, the
damping persists even when
collisions are switched off. This
damping mechanism – Landau
damping – is usually the
dominant damping mechanism
Landau Damping
• we have seen earlier that the source of damping in the static thermal
cloud approximation is the collisional exchange of atoms between the
condensate and thermal cloud
• in the case of the transverse breathing mode in elongated traps,
collisions within the thermal cloud are responsible for the damping
• however, in many cases, condensate modes decay predominantly as a
result of Landau damping; in the context of the GGP equation, it arises
from the mean-field interaction with the thermal cloud
• this mean-field interaction effectively exerts a force on the condensate;
conversely, the condensate exerts a force on the thermal cloud. This
force can do work on the thermal cloud and in this way transfer energy
to it. Interestingly, in order for the effect of the thermal cloud on the
condensate to be accounted for properly, the detailed dynamical
evolution of the thermal cloud density is required.
Comparison of Damping Mechanisms: The Monopole Mode
• in these simulations we consider 5 × 104 87Rb atoms in an isotropic
trap; the monopole, or radial breathing mode is excited by a radial
scaling of the equilibrium condensate wavefunction:
T /TBEC = 0.65
• the solid curve is the full simulation; the
dashed curve neglects collisions
• Landau damping dominates up to
T ≅ 0.6 TBEC
•
C12 = C22 = 0 (solid circles); C22 only
(open circles); all collisions (triangles)
Vortices
• one of the most exciting developments in the study of cold atomic
gases was the observation of vortices. The reason for this was that the
appearance of a vortex was a direct manifestation of the superfluid
nature of a Bose-condensed gas. Of course, superfluidity was evident
in many other situations not involving vortices (for example,
condensate collective modes) but somehow the observation of vortices
just had a lot more impact on the imagination of physicists!
• one of the early challenges in the cold atom community was creating
vortices in the first place – it was a kind of holy grail! But ironically,
once the tricks had been learned, vortices could be created at will and
in abundance.
• although much has been learned, there are still unanswered questions
regarding the creation of vortices, how they equilibrate and how they
behave. I will be addressing some of these questions in the following.
GP Description of Quantized Vortices
• to begin, let’s consider a line vortex in a uniform condensate at zero
temperature. What we are looking for is a specific solution of the timeindependent GP equation having the form
• as opposed to the earlier examples we have considered, this stationary
state is complex and thus has a velocity field associated with it:
• this velocity field is irrotational except along the line of the vortex
where
• as a result there is a quantized circulation given by
• the integer l is sometimes referred to as the ‘charge’ of the vortex; the
most stable vortex is singly-quantized
GP Description of Quantized Vortices, cont’d
• because the velocity field diverges at ρ = 0, the radial part of the
wavefunction must vanish – the vortex has a core
• the size of the core is determined by
the healing length
• the vortex state is an excited state solution of the GP equation and this
state has a higher energy. One would therefore not expect to see
vortices in an equilibrium situation. This is true unless the condensate
is stirred.
Stirring
• in order to create vortices, one must impart some angular momentum
to the system; this can be done by imposing a rotating anisotropic
external potential
• in the rotating frame
• in the lab frame
E: Show that this potential is time independent if ε = 0.
• the potential is static in the rotating frame, but this frame is a
noninertial frame of reference – both classical and quantum equations
of motion are different in the two frames of reference
Rotating Frames of Reference: Classical Treatment
• the classical equation of motion for a particle in the rotating frame is
• this equation corresponds to the Hamiltonian
• the distribution function in this frame is then found to satisfy
• the steady-state equilibrium solution is
with
Rotating Frames of Reference: Classical Treatment, cont’d
• since p’ = p and v = v’ + Ω x r’, we see that p’ – mvrb = mv’, that is
the velocity distribution is isotropic in the rotating frame
• this same distribution in the lab frame is
• this shows that the velocity field in the lab frame is v(r) = vrb = Ω x r
which implies that the density distribution rotates rigidly with angular
velocity Ω. The centrifugal potential Vcent causes the density to bulge
out in the radial direction
Rotating Frames of Reference: Quantum Treatment
• similar results are found in the quantum treatment. Without going into
details (see Ch. 9 of GNZ), one finds that the state vector in the lab
frame is given by
where
• this state vector is defined by a time-independent trapping potential but
now the angular velocity appears together with the angular momentum
operator. The physical state in the lab frame is obtained by performing
an active rotation of the state through an angle Ωt.
• if we look for stationary states in the rotating frame, these states are
affected by the angular momentum term in the Hamiltonian. The
ground state will depend on the magnitude of Ω and if Ω is sufficiently
large, the ground state will contain vortices.
Vortex Energetics at the GP Level
• we now consider the relative stability of a single axial vortex in a
trapped gas at T = 0 relative to the vortex-free state. In the rotating
frame the GP energy functional is
• the energy difference between the vortex-free state and the singlevortex configuration is
where one finds in the TF approximation that
• this defines a critical angular velocity above which the vortex is
thermodynamically stable
Vortex Energetics at the GP Level, cont’d
• similar calculations can be carried out for an off-centre vortex (Fetter
and Svidzinsky, 2001)
• for Ω = 0 (a), the energy of the vortex decreases as it moves to the
edge of the cloud; if there were some way to dissipate the energy, the
vortex would move to the edge and be expelled from the condensate.
The curve (c) corresponds to the situation where the vortex at the
centre of the trap is thermodynamically stable.
GP Solutions for a Rotating Anisotropic Trap
• at low rotation rates, the flow is
irrotational: (a) flow in lab
frame, (b) flow in rotating
frame (Feder et al., 1999)
• the simulations use imaginary time
propagation to find the ground state
in the rotating frame; the number of
vortices is seen to increase with Ω
• in the lab frame these images would
rotate with an angular velocity Ω
Vortex Nucleation: 2D Simulations
• the stirring of a nearly pure condensate leads to the formation of
vortices. The nucleation process is of considerable interest and can be
simulated using a phenomenological dissipative GP equation of the
kind developed earlier:
• these 2D simulations were
performed by Tsubota et
al., 2002
• starting with an isotropic
state, a rotating anisotropic
potential is switched on;
one observes a quadrupole
distortion (b), excitation of
surface modes (c),
penetration of vortices (d-e)
and equilibration (f)
Vortex Nucleation: 3D Simulations
• the following animation was provided by Naramisa Sasa
• the same dissipative GP
equation was used for these
simulations with a constant
dissipation constant ; as we
have seen, this parameter can
be related to the collisional
exchange of atoms between
the condensate and thermal
cloud
• all these results are at the
level of the static thermal
cloud approximation which
ignores the dynamics of the
thermal cloud
Vortex Precession
• these experimental images (and fits) show the precession of an offcenter vortex, Anderson et al., PRL 85, 2857 (2000)
• the sense of precession can be understood by appealing to image
theory of superfluid flow
• the image vortex has the
opposite sense of
circulation to ensure that
the normal current
vanishes at the boundary
• the flow field of the image
sweeps the vortex along
the surface in the direction
shown
Vortex Precession, cont’d
• at T =0, the dynamics is conservative and the vortex precesses at a
constant radius
• a Lagrangian analysis (Lundh and Ao, Fetter and Svidzinsky) gives a
precession frequency of
• Bogoliubov theory for a condensate with a central vortex gives
condensate modes with (Dodd et al., PRA 56, 587 (1997))
• the so-called anomalous m = −1 mode has a negative frequency and
• this mode corresponds to the precession of an off-centre vortex (Fetter
and Svidzinsky) and has the same sense of precession as that of the
vortex
Vortex Precession at Finite T in the HFBP Theory
• shown below is the condensate and thermal cloud densities as
calculated in the HFBP approximation
• the m = −1 mode has a frequency −1
> 0: thus the sense of precession is
opposite to the T = 0 case
• dynamics is equivalent to the
condensate moving in the static
external potential
• the thermal cloud density acts as a pinning centre and causes the
opposite sense of precession: this is analogous the the violation of the
Kohn theorem in the HFBP for the c.m. dipole mode
• to restore the proper behaviour one must treat the dynamics of the
thermal cloud in a consistent fashion; this is what the ZNG theory
provides
ZNG Simulation of Vortex Precession
• starting with a vortex-free equilibrium state at a given temperature T,
the vortex state is obtained by multiplying the condensate
wavefunction by the phase factor exp[iS(r)] where
• this imprints the velocity field of a straight vortex at (x0, y0)
• the GP equation is then evolved in imaginary time for a short period
until a fully-developed vortex state in the condensate is formed
• this initial state is a quasi-equilibrium state containing one off-centre
vortex
• the dynamics of the vortex is then obtained by evolving the full set of
ZNG equations in real time
Vortex Decay*
• at finite temperatures, an off-centre
vortex spirals outwards as its energy
is dissipated due to friction with the
thermal cloud
• the decay rate is found by fitting the
data to an exponential function
*Jackson, Proukakis, Barenghi and
Zaremba, Phys. Rev. A 79, 053615 (2009)
Condensate and Thermal Cloud Densities
• the top panel shows the condensate density; the vortex core with its
depleted density appears as a blue dot
• the lower panel shows the thermal cloud density; the density has a
peak (red) in the vortex core due to the reduced mean-field repulsion
from the condensate
• the sense of precession is the same as at T = 0
Rotating Thermal Clouds
• we now consider a situation in which the thermal cloud undergoes
solid-body rotation with angular velocity th. This is achieved by
adding vn = thrv to the velocity of each thermal atom in the
equilibrium state. There is now a motion of the vortex relative to the
thermal cloud.
• radial position vs. time: th/v = 0.2 (black), 0 (red), 0.2 (green)
0.37 (blue); T/Tc = 0.7
• decay rate vs. th
Vortex Lattice Formation and Decay*
• following an initial stirring phase, the vortices are seen to equilibrate
into a triangular lattice; the number of vortices then decays as a
function of time
*Abo-Shaeer et al., Science 292 (2001)
Decay of Vortex Lattices*
• rotating (a) condensate and (b)
thermal cloud
• aspect ratio for a rotating cloud is
• (a) aspect ratio vs. time and (b)
rotation rate vs. time
*Abo-Shaeer et al., Science 292 (2001)
Decay of Vortex Lattices: Simulations
• the figure below shows the evolution for two different initial vortex
structures containing 7 vortices (T/Tc = 0.7)
• the top panel shows that the outer vortices are shed while the central
vortex persists for much longer times; this is in accord with
observations
• in the lower panel, all vortices are shed; however, for this initial
configuration a central, long-lived vortex may still appear
Overview
• the ZNG equations provide a reasonably complete description of the
dynamics of a Bose-condensed gas at finite temperature
• the theory is based on the idea of Bose broken symmetry – it assumes
that the condensate exists as a well-defined physical entity even in
highly nonequilibrium situations
• the thermal cloud can be viewed in (semi-)classical terms as a
collection of particle-like excitations
• this picture conforms to what we SEE in experiments!
• the theory of course is approximate and has its limitations. It is
‘correct’ to the extent that it explains experimental observations.
• it is not a theory one would want to apply to highly correlated systems
such as atoms in an optical lattice with a few particles per site
Quo vadis ZNG?
• perhaps Bose mixtures and spinor condensates, dipolar systems,
strongly perturbed Bose condensates, …?