Numerical study of Faraday waves in binary non-miscible Bose-Einstein - - PowerPoint PPT Presentation

▶
numerical study of faraday waves in binary
SMART_READER_LITE
LIVE PREVIEW

Numerical study of Faraday waves in binary non-miscible Bose-Einstein - - PowerPoint PPT Presentation

Sep 09, 2023 241 likes •459 views

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates z 1 and Alexandru Nicolin 2 Antun Bala 1 Scientific

slide-1
SLIDE 1

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook

Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates∗

Antun Balaˇ z1 and Alexandru Nicolin2

1Scientific Computing Laboratory, Institute of Physics Belgrade,

University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia

2“Horia Hulubei” National Institute for Physics and Nuclear Engineering,

P.O.B. MG-6, 077125 Bucharest, Romania

∗Supported by: Serbian Ministry of Education and Science under Grant ON171017,

CNCS-UEFISCDI under Grant PD122, Contract No. 35/28.07.2010, and European Commission through projects EGI-InSPIRE, PRACE-2IP, PRACE-3IP, and HP-SEE. LENCOS’12, Sevilla 11 July 2012

slide-2
SLIDE 2

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook

Why ultracold quantum gases are interesting?

Intensive progress in the field of ultracold atoms has been recognized by Nobel prize for physics in 2001 for experimental realization of Bose-Einstein condensation Cold alkali atoms: Rb, Na, Li, K . . . T ∼ 1 nK, ρ ∼ 1014cm−3 Cold bosons, cold fermions Harmonic trap, optical lattice Short-range interactions, long-range dipolar interactions Spin-orbit-coupled BECs Tunable quantum systems concerning dimensionality, type and strength of interactions

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-3
SLIDE 3

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook

Experimental observation of Faraday waves

  • P. Engels, C. Atherton, M. A. Hoefer, PRL 98, 095301 (2007)

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-4
SLIDE 4

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Two-component BEC systems

Experimentally realized with a broad variety of types of atoms and parameters of a system

heterogeneous systems: different types of atoms homogeneous systems: same type of atoms, different internal (usually spin) states

Rich dynamics and interplay of the parameters Several possible ground states A variety of possible dynamical evolutions We focus on the study of Faraday waves and patterns in cigar-shaped two-component 87Rb BECs, with strong radial confinement, which is harmonically modulated We also study resonant waves, which appear for specific values of the frequency of radial modulation

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-5
SLIDE 5

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Mean-field description of 2C BEC

The system is described by a coupled system of GP equations:

i∂Ψ1( r, t) ∂t = » − 2 2m1 △ + V ( r, t) + g11|Ψ1( r, t)|2 + g12|Ψ2( r, t)|2 – Ψ1( r, t) i∂Ψ2( r, t) ∂t = » − 2 2m2 △ + V ( r, t) + g21|Ψ1( r, t)|2 + g22|Ψ2( r, t)|2 – Ψ2( r, t)

where the couplings are given by:

g11 = 4π2a1

m1

, g22 = 4π2a2

m2

, g12 = g21 = 2π2aint

meff

Typical experimental values we consider for two hyperfine states of 87Rb:

N1 = 2.5 · 105 , N2 = 1.25 · 105 a1 = 100.4 a0 , a2 = 98.98 a0 , aint = 100.4 a0 ωρ(t) = ωρ,0(1 + ǫ sin ωmt) , ωρ,0 = 160 · 2π Hz ωm = 250 · 2π Hz , ǫ = 0.1 , ωz = 7 · 2π Hz

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-6
SLIDE 6

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Ground state - imaginary-time propagation

0.0 × 100 5.0 × 108 1.0 × 109 1.5 × 109 2.0 × 109 2.5 × 109 3.0 × 109 3.5 × 109 4.0 × 109 4.5 × 109

  • 80
  • 60
  • 40
  • 20

20 40 60 80 longitudinal density (m-1) z (µm) n1(z) n2(z)

Density profile of the converged eigenstate obtained by propagation in the imaginary time. Discretization parameters: Nρ = Nz = 2000, ε = 10−4/ωz.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-7
SLIDE 7

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Ground state - experimental realization

  • K. M. Mertes, J. W. Merrill, R. Carretero-Gonz´

alez, D. J. Frantzeskakis,

  • P. G. Kevrekidis, D. S. Hall, PRL 99, 190402 (2007)

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-8
SLIDE 8

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Segregated state - imaginary-time propagation

0.0 ×10 0 5.0 ×10 8 1.0 ×10 9 1.5 ×10 9 2.0 ×10 9 2.5 ×10 9 3.0 ×10 9 3.5 ×10 9 4.0 ×10 9 4.5 ×10 9

  • 80
  • 60
  • 40
  • 20

20 40 60 80 longitudinal density (m-1) z (µm) n1(z) n2(z)

First excited eigenstate obtained by imaginary-time propagation. Discretization parameters: Nρ = Nz = 2000, ε = 10−4/ωz.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-9
SLIDE 9

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

First excited state - experimental realization

  • C. Hamner, J. J. Chang, P. Engels, M. A. Hoefer, PRL 106, 065302 (2011)

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-10
SLIDE 10

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Two-component BEC systems Mean-field description of 2C BEC Ground state - symbiotic pair First excited - segregated state

Energy: time dependence

308 309 310 311 312 313 314 315 316 50 100 150 200 250 300 350 400 450 E (units of - hωz) imaginary time (s) segregated → symbiotic symbiotic 314.0 315.0 316.0 10 20 30 40 50 308.258 308.260 308.262 10 20 30 40 50

Convergence of the total energy of the system during the imaginary-time propagation for the symbiotic pair and for the segregated state.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-11
SLIDE 11

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Symbiotic pair ground state - Faraday waves

Emergence of Faraday waves as a result of real-time propagation. The radial frequency of the trap is modulated at the non-resonant frequency ωm = 250 · 2π Hz, ǫ = 0.1.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-12
SLIDE 12

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Segregated state - Faraday waves

Emergence of Faraday waves as a result of real-time propagation. The radial frequency of the trap is modulated at the non-resonant frequency ωm = 250 · 2π Hz, ǫ = 0.1.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-13
SLIDE 13

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Periods of Faraday waves

0.01 0.1 1 10 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FFT amplitude k (µm-1) k1 k2 k3,1 k3,2 FFT[n1] FFT[n2] 0.0001 0.001 0.01 0.1 1 10 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 FFT amplitude k (µm-1) k1,1 k1,2 k2 k3,1 k3,2 FFT[n1] FFT[n2]

FFT of density profiles for the two condensates at t = 200 ms. For the symbiotic pair, the periods of waves are found to be 13.0 µm and 12.5 µm, while for the segregated state the periods are 11.6 µm and 13.0 µm.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-14
SLIDE 14

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Variational approach - symbiotic pair (1)

Variational ans¨ atze for the symbiotic pair wave functions:

ψ1(ρ, z, t) = N1 exp „ − ρ2 2w2

ρ(t) + iρ2α2(t)

« » 1 − exp „ − z2 2w2

z

«– ψ2(ρ, z, t) = N2 exp „ − ρ2 2w2

ρ(t) − z2

2w2

z

+ iρ2α2(t) « ˆ 1+(u(t)+iv(t)) cos kz ˜

ψ1 is considered to be unperturbed, acting as an additional confinement for ψ2 Variational analysis leads to a Mathieu-type equation:

¨ u(τ) + u(τ) [a(k, ω) + ǫb (k, ω) sin 2τ] = 0 a(k, ω) = k4 ω2 + k2 ω2 Λsym , b(k, ω) = k2 ω2 Λsym , ωt = 2τ

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-15
SLIDE 15

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Variational approach - symbiotic pair (2)

For small positive ǫ and positive b(k, ω), the Mathieu equation has solutions of the form exp(±iµτ) sin √aτ and exp(±iµτ) cos √aτ, where Im[µ] consists of a series of lobes positioned around the solution of the equation a(k, ω) = n2 The lobe centered around a(k, ω) = 1 is the largest, and yields the most unstable solutions, determined by:

kF,sym = s −Λsym 2 + r Λ2

sym

4 + ω2

This dispersion relation yields a period of 12.0 µm for the Faraday waves, which is in excellent agreement with the numerical results

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-16
SLIDE 16

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Self-resonance - symbiotic pair

Emergence of Faraday waves as a result of real-time propagation. The radial frequency of the trap is modulated at the resonant frequency ωm = 160 · 2π Hz, ǫ = 0.1.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-17
SLIDE 17

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Second resonance - symbiotic pair (1)

Emergence of resonant waves in a two-component BEC system radially modulated at ωm = 300 · 2π Hz.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-18
SLIDE 18

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Second resonance - symbiotic pair (2)

Emergence of resonant waves in a two-component BEC system radially modulated at ωm = 320 · 2π Hz.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-19
SLIDE 19

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Non-resonant Faraday waves Periods of Faraday waves Resonant waves

Second resonance - symbiotic pair (3)

Emergence of resonant waves in a two-component BEC system radially modulated at ωm = 340 · 2π Hz.

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-20
SLIDE 20

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook

Conclusions

We have studied the emergence of surface waves in two-component BEC systems First, we have calculated two initial states of interest: ground state or symbiotic pair, and the first excited state

  • r segregated state

For non-resonant modulation of the radial confinement, the usual Faraday waves are observed, with the similar period in both components For the self-resonant modulation of the radial confinement, the expected resonant waves are observed For the second-harmonic resonance, much stronger and faster-emerging resonant waves are observed, turning the system from the non-miscible to the miscible state

  • A. Balaˇ

z and A. Nicolin, PRA 85, 023613 (2012)

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates

slide-21
SLIDE 21

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook

Outlook

Study of resonances using the Mathieu-type analysis Study of miscible two-component systems Pancake-shaped two-component systems Faraday waves for two-component BEC loaded into an

  • ptical lattice

LENCOS’12, Sevilla 11 July 2012

  • A. Balaˇ

z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates