DARK SOLITONS

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DARK SOLITONS IN ATOMIC BOSE-EINSTEIN CONDENSATES
Nick Proukakis
U.K. EPSRC
Nick Parker - Charles Adams
PHYSICS DEPT.
http//massey.dur.ac.uk
Collaboration
Carlo Barenghi (Newcastle)
Dimitri Frantzeskakis (Athens)
Panos Kevrekidis (Amherst)
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OUTLINE
1 Intro to Dark Solitons In Bose-Einstein
Condensates
2. Quasi-1D Systems Soliton
Decays via Sound Emission
KEY POINT Soliton Dynamics Are Sensitive
to Soliton Sound Interactions
3. How to Control Soliton Dynamics ?
Modify Soliton-Sound Interactions
By Parametric Driving

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THE SYSTEM
Gas of Dilute, Weakly-Interacting Bosonic Atoms
Confined in (Anisotropic) 3D Harmonic Traps
GROSS-PITAEVSKII EQUATION
Atom-Atom Interactions
Longitudinal
Transverse
CONFINEMENT
g lt 0
g gt 0
BRIGHT SOLITONS
DARK SOLITONS
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DARK SOLITONS
? Density Dips (Kinks) Propagating without
Dispersion
1D Solution (NO TRAP)
Dimensionless
Healing Length
density
Density Dip
v0 v0.5 v0.9
?m
Speed of Sound
Phase Slip
phase
mm s-1
z
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DARK SOLITON CREATION BY PHASE IMPRINTING
NIST Science 287, 97 (1999)
Aspect Ratio 2
LASER
Hannover PRL, 83, 5198 (1999)
Aspect Ratio 30
NIST EXPERIMENT
THEORY
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DARK SOLITON DECAY MECHANISMS
Negligible for
THERMAL INSTABILITIES
TRANSVERSE INSTABILITIES
Feder et al. PRA 62, 053606 (2000)
Brand Reinhardt PRA 65, 043612 (2002)
Komineas Papanikolaou, PRA 68, 043617 (2003)
2D DENSITY CROSS SECTION
SOLITON CLOSE-UP
Vortex Rings (Spherical Traps)
Snake Instability
3D
Dynamically Unstable ? Decay Into
Experiment PRL, 86, 2926 (2001)
Vortex Lines (Elongated Traps)
Quasi-1D
(1D) gtgt (3D)
Enhanced Stability
Muryshev PRL 89, 110401 (2002)
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EFFECT OF LONGITUDINAL CONFINEMENT
(I) HOMOGENEOUS 1D SYSTEM
RENORMALIZED DENSITY PLOTS
Sound Emission
t
Decreasing V0
z
cf. Interface Between Optical Media of Different
Refractive Indices (Kosevich-Kivshar-Aceves et
al.)
zSOL
SOLITON ENERGY
Emitted Sound Energy Up to 50 of Initial
Soliton Energy
N.G. Parker, N.P. Proukakis, M. Leadbeater, and
C.S. Adams, J. Phys. B 36, 2891 (2003).
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EFFECT OF LONGITUDINAL CONFINEMENT (ctd.)
SINGLE HARMONIC TRAP
(T. Busch G. Huang Frantzeskakis Konotop)
Soliton Oscillates in Trap ?sol ?trap /
v2
z
t
Emitted Sound Fully Confined Soliton Reabsorbs
Sound Energy Periodically (No Net Decay)
ESOL
Beating Effect
Time
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How Can We Control Emission / ReAbsorption of
Sound By Dark Soliton?
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CONTROLLING SOLITON SOUND INTERACTIONS
DAMP OUT EMITTED SOUND FROM TRAP REGION
Parker, Proukakis, Leadbeater Adams, PRL 90,
220401 (2003)
SOUND
INFINITE TRAP
-Emitted Sound Fully Confined -Continuous
Soliton-Sound Interactions
SOUND
SOUND
DOUBLE TRAP WITH LOW CUT-OFF
- Emitted Sound Escapes Inner Trap - Soliton
Decays
ON-AXIS SOLITON ENERGY
SOUND FULLY TRAPPED
SOUND ESCAPES
BEATING
DECAY
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
SOUND ALLOWED TO ESCAPE TO OUTER TRAP

? sound ( ve amplitude )
z
? background
? sound ( -ve amplitude )

? soliton

non-dissipative trajectory

t
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
z


Time
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
Determining Rate of Sound Emission
z
From Energy Functional


Soliton Acceleration
For Homogeneous Optical Fibers
Ns Soliton Atom Number
Ss Phase Across Soliton
Pelinovsky et al., PRE 54, 2015 (1996)
Time
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
Determining Rate of Sound Emission
z
From Energy Functional


Soliton Acceleration
Soliton Deformation
For Homogeneous Optical Fibers
Ns Soliton Atom Number
Ss Phase Across Soliton
Pelinovsky et al., PRE 54, 2015 (1996)
Time
(? constant )
Parker, Proukakis, Leadbeater, Adams PRL 90,
220401 (2003)
Parker et al., J Phys B 36, 2891 (2003)
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
z


Soliton Acceleration
Soliton Deformation
SOLITON PROFILES
Background Density
Deformed Soliton Profile
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QUANTIFYING SOUND EMISSION IN DOUBLE TRAP
z


Soliton Acceleration
Soliton Deformation
SOLITON PROFILES
Background Density
Deformed Soliton Profile
RENORMALIZED PROFILES
Deformed Soliton Profile
Undeformed Soliton Profile
Soliton Locally Confined Sound (within zSOL)
OUR SOLITON DEFINITION (Perturbed Soliton)

• NOT a soliton in strict mathematical sense
• BUT This IS what is experimentally relevant !

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EXPERIMENTAL SIGNATURE OF SOUND EMISSION
Observe Change in Soliton Oscillations for
Different Cut-offs
Experimental Handle Dimple Depth V0
V0
DECAY
ANTI-DAMPING
Busch Anglin PRL 84, 2298 (2000)
SHALLOW DIMPLE
DEEP DIMPLE
V0 m
V0 5m
Sound Escapes
Sound Reabsorbed
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CONTROLLING DARK SOLITON DYNAMICS
(A) Manipulation of Interactions Between
Soliton Background Sound Field
Longitudinal Density Inhomogeneity Provides ONLY
Source of Instability
Soliton Dynamics Controlled By
Confining Geometry
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CONTROLLING DARK SOLITON DYNAMICS
(A) Manipulation of Interactions Between
Soliton Background Sound Field
(B) Engineering / Enhancing Background
Sound Field (Parametric Driving)
Longitudinal Density Inhomogeneity Provides ONLY
Source of Instability
Can Completely Stabilize Soliton Against Decay In
Dissipative Systems
Soliton Dynamics Controlled By
Confining Geometry
External Drive
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PARAMETRICALLY DRIVEN DARK SOLITON
Dark Solitons ? Generally Prone to Dissipative
Losses
COMPENSATE AGAINST LOSSES BY
Periodically Pumping Energy Into System
(Parametric Driving)
e.g. Barashenkov et al., PRL 90, 054103 (2003)
ATOMIC CONDENSATES ?
? Pump Energy Into Soliton By Manipulating Sound
Excitations
Artificially Create Sound Field which Enhances
Soliton Energy
Proukakis et al. PRL (In Press)
cond-mat/0403566 (2004)
CREATE DENSITY PERTURBATION AT SOLITON FREQUENCY
HARMONICTRAP
DENSITY
DRIVE POTENTIAL
CHOOSE
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(I) EFFECT OF DRIVING NO DISSIPATION
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DISSIPATIONLESS REGIME (OPTIMIZED CASE)
DRIVING ON UNTIL
DRIVING ON (all times)
NO DRIVING
Soliton Becomes Stationary (Black Soliton)
Time
(Homogeneous NLSE)
Energy Flow Sensitive to Phase Difference Between
Drive Soliton Oscillations
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(II) EFFECT OF DRIVING WITH DISSIPATION
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DISSIPATIVE REGIME
? Introduce Phenomenological Damping
Choi, Morgan Burnett PRA 57, 4057 (1998)
Tsubota, Kasamutsu Ueda PRA 65, 023603 (2003)
DAMPED DRIVEN CASE
UNDRIVEN CASE
NO DAMPING
DAMPED
Time
(Quasi-1D BEC )
Decay Constant Defines Soliton Lifetime
Muryshev-Fedichev-Shlyapnikov et al. PRA 60,
3220 (1999) PRL 89, 110401 (2002)
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(1) SOLITON STABILIZATION AGAINST DECAY
For optimum stabilization, consider slow soliton
Speed of Soliton at Trap Centre (Locally
Homogeneous Region)

NO DRIVE
CONTINUOUS DRIVING
NO DRIVE
Time
? Soliton STABILIZES Against Decay at Reduced
Energy
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(2) SOLITON STABILIZATION AT CONSTANT ENERGY
Potential Applications? Stabilize Solitons at
Constant Energy
? Achieved by Rephasing Drive Relative to Soliton
Oscillations
REPHASING OPERATIONS
DRIVING (rephasing)
Proukakis et al. PRL (In Press)
cond-mat/0403566 (2004)
NO DRIVE
Time
? Rephasing Operations Can Be Repeated
Indefinetely (In Principle)
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VORTEX SOUND INTERACTIONS
VORTEX DECAY BY SOUND EMISSION
VORTEX RING COLLISIONS
PRL 92, 160403 (2004)
PRL 86, 1410 (2001)
Vortex follows line of constant potential
i.e. precess around trap
INFINITE TRAP
DOUBLE TRAP
SOUND PULSE (x-z plane)
VORTEX TRAJECTORY
Dipolar Radiation Pattern
( Vortex Motion)
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CONCLUSIONS
Dark Soliton Dynamics in Atomic Bose-Einstein
Condensates Are Extremely Sensitive to
Soliton-Sound Interactions
Interaction Controlled Via
EXTERNAL DRIVING
SYSTEM GEOMETRY
Create Artificial Sound Field To Pump Energy Into
Soliton
? Is Sound Trapped Within System ?
PRL 90, 220401 (2003)
PRL (In Press) cond-mat/0403566 (2004)
? Is Emitted Sound Field Modified?
Sensitive on Relative Phase Between Drive
Soliton Oscillations
J. Phys. B 37, S175 (2004)
J. Opt. B 6, S380 (2004)
Can Stabilize Soliton Against Dissipation
Similar Studies for Vortices
PRL 92, 160403 (2004) JLTP (In Press, 2004)
cond-mat/0405635 (2004)
See Also PRL 86, 1410 (2001) PRA 67, 015601
(2003)
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SOLITON STABILIZATION IN DISSIPATIVE SYSTEMS
NO DRIVING
DRIVING (WITH APPROPRIATE REPHASING)
DRIVE POTENTIAL