Introduction
Polaron Systems
Summary
Strongly Interacting One-Dimensional Bose Systems and
Bose Polarons
Arbitrary inter-species interaction strength, confinement and different masses.
Amin S. Dehkharghani1 , Artem Volosniev1 , Nikolaj Zinner1 .
1 Aarhus
University, Denmark, [email protected]
Quantum Technologies Conference VI, Warsaw, June 24, 2015
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Outline
1
Introduction
One-Dimensional Physics
Experimental Techniques
2
Polaron Systems
System
General 1+N system
Semi-analytical Solution
Results
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
One-Dimensional Physics
Why One-Dimensional Physics?
1931: Bethe solves the Heisenberg model of Ferromagnetism.
1960’s: Bethe’s method: used in different many-body models.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
One-Dimensional Physics
Why One-Dimensional Physics?
1931: Bethe solves the Heisenberg model of Ferromagnetism.
1960’s: Bethe’s method: used in different many-body models.
Recently: One-dimensional (1D) quantum systems realized
experimentally.
1D quantum nature: different and interesting
Verification of Tonks-Girardeau model
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Summary
One-Dimensional Physics
Motivation
Problem: Only few analytical models.
Motivation: Many regimes and multi-component systems not
explored yet.
Solution: Come up with analytical solutions to describe the 1D
systems (as general as possible).
Most interesting systems:
N+M systems [Scientific Reports 5, 10675 (2015)]
N+1 polaron systems [arXiv:1503.03725]
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Experimental Techniques
Experiments
Bose-Einstein-Condensate (BEC): T ≈ µK
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Experimental Techniques
Experiments
Bose-Einstein-Condensate (BEC): T ≈ µK
Cooling techniques:
Doppler-Cooling.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Summary
Experimental Techniques
Experiments
Bose-Einstein-Condensate (BEC): T ≈ µK
Cooling techniques:
Doppler-Cooling.
Magneto-Optical Trap (MOT).
Credit: Wikipedia
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Experimental Techniques
Experiments
Bose-Einstein-Condensate (BEC): T ≈ µK
Cooling techniques:
Doppler-Cooling.
Magneto-Optical Trap (MOT).
Cooling by Evaporating.
Credit: Wikipedia
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Experimental Techniques
Experiments
3D BEC → 1D BEC
3D BEC ≈ 10.000 atoms.
2D optical lattice to freeze out transversal motion: tubes (each
with 8-25 atoms).
Interaction strength, g3D
Through Feshbach resonance
Scattering resonance in 1D: Confinement-induced resonances
(CIR): g3D → g1D
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Summary
Experimental Techniques
Experiments
Credit: E. Haller: Science Vol.325 no. 5945 pp.1224-1227
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
System
Defining the System
1+N Systems
Two components: M = 1 and N.
Two species: A and B for bosons.
Intra-species interaction: gBB .
Inter-species interaction: gAB (or just g).
Masses: mA , mB and trapping frequencies: ωA , ωB .
Figure : 8+1 system.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Summary
General 1+N system
1+N Particles
Formalism
Mass: mA for impurity and mB for majority.
p
Measure length in b = ~/mB ωB .
Hamiltonian
HA (x) =
N
X
HB (yi ) =
i
2 x2
p2x + m2AB ωAB
,
2mAB
N
X
p2y + y2i
i
y
2
where mAB = mA /mB , ωAB = ωA /ωB , x for impurity and yi for
majority particle’s position.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
General 1+N system
1+N Particles
Total Hamiltonian
P B
P B
P
H = HA (x) + Ni=1
HB (yi ) + Ni=1
gδ(x − yi ) + i<k gBB δ(yi − yk )
Wave Function
Adiabatic decomposition
Ψ(x, y1 , . . . , yNB ) =
X
φj (x)Φj (y1 , . . . , yNB |x),
j=1
where
PNB Φj is the jth normalized eigenstate of the eigenvalue problem
i=1 HB (yi )Φj = Ej (x)Φj
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
General 1+N system
1+N Particles
Ideal gas, gBB = 0
Φj (y1 , . . . , yNB |x) = Ŝ
NB
Y
fkj (yi |x)
i
i=1
with a symmetrization operator Ŝ (acting on the yi coordinates) and fkj (yi |x) being the
kij th normalized eigenstate of HB (yi ) for a given x.
i
Gross-Pitaevskii Equation (GPE)
For gBB 6= 0, but small, use 1D GPE.
1 ∂2
1 2
2
µ(x)f̃kj = −
+
y
+
N
·
g
|
f̃
f̃kj
j|
B
BB
i
ki
i
i
2 ∂y2i
2
where µ(x) is a chemical potential and gBB is determined through the 3D
boson-boson scattering length.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Semi-analytical Solution
Solution
Coupled system of equations
[HA (x) + Ei (x)] φi =
∂φj
1 X
Qij (x)φj + Pij (x)
mAB
∂x
j=1
2
∂
∂
where Pij (x) = hΦi | ∂x
|Φj iy and Qij (x) = 12 hΦi | ∂x
2 |Φj iy .
Considering only Qii gives us an upper bound for the exact
energy of the ground state.
This approximation gets better as NB grows.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Results
Energies
1+8
9
8.5
8
E n erg y [h̄ω ]
7.5
7
6.5
6
5.5
5
analytical
4.5
numerical (even)
numerical (odd)
4
−10
−9
−8
−7
−6
−5
−4
- 1 / g [bh̄ω ]
−1
Amin S. Dehkharghani
−3
−2
−1
0
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Results
Density
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Results
Pair-Correlation
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Results
Momentum
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Summary
Summary
Developed a new and fast method to solve the Bose 1+N Polaron
problem.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Summary
Developed a new and fast method to solve the Bose 1+N Polaron
problem.
The interaction strength between the impurity and majority
particles is arbitrary.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Summary
Developed a new and fast method to solve the Bose 1+N Polaron
problem.
The interaction strength between the impurity and majority
particles is arbitrary.
The method can implement any confining potential, trapping
frequency and mass-difference.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Summary
Summary
Developed a new and fast method to solve the Bose 1+N Polaron
problem.
The interaction strength between the impurity and majority
particles is arbitrary.
The method can implement any confining potential, trapping
frequency and mass-difference.
In addition, we can include interactions among the majority
particles under the assumption that these may be described by
the 1D Gross-Pitaevskii equation.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Introduction
Polaron Systems
Last Words
Thanks to
Nikolaj Zinner
Artem Volosniev
Aksel Jensen
Dmitri Fedorov
Christian Forssén.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Last Words
Thanks to
Nikolaj Zinner
Artem Volosniev
Aksel Jensen
Dmitri Fedorov
Christian Forssén.
Thanks to the organizers of the Quantum Technologies
Conference VI, Warszawa.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary
Introduction
Polaron Systems
Last Words
Thanks to
Nikolaj Zinner
Artem Volosniev
Aksel Jensen
Dmitri Fedorov
Christian Forssén.
Thanks to the organizers of the Quantum Technologies
Conference VI, Warszawa.
Thank you for your attention.
Amin S. Dehkharghani
Quantum Systems in One-Dimensional Traps
Summary