Strongly Interacting One-Dimensional Bose SystemsQuantum Magnetism - From Few To Many
arXiv.org/abs/1409.4224
Amin S. Dehkharghani1 †, A. G. Volosniev1, E. J. Lindgren2, J. Rotureau2, C. Forssen2,D. V. Fedorov1, A. S. Jensen1 & N. T. Zinner1
1Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark2Department of Fundamental Physics, Chalmers University of Technology, SE-412 96 Goteborg, Sweden
† [email protected]/abs/1409.4224
IntroductionA general and complete description of unequal intra- and inter-species interactions in cold atomic systems such as Bose mix-tures is difficult. Although, experimentally these systems havebeen recently realized and studied [1, 2], the crossover from fewto many-body particles in such multi-components subsystems isalmost impossible to describe theoretically.
Here we provide a particularly clean realization of a ’ferromag-netic’ system confined in a harmonic trap. Using numerical andnewly developed analytical techniques we obtain and analyzethe exact wave function.
Main Objectives1. Explore the crossover between few- and many-body behavior.2. Demonstrate that the strongly interacting regime in Bose mix-
tures realizes a perfect ferromagnet in the ground state.3. And excited states will form perfect antiferromagnetic order.4. Demonstrate the behavior in extremely imbalanced system,
with one strongly interacting ’impurity’
Methods•Our two-component bosonic system has N particles:
N = NA +NB (1)
split into two different kinds, NA and NB identical bosons.•All N particles have mass m and move in the same external
harmonic trapping potential with single-particle Hamiltonian:
H0 =p2
2m+1
2mω2x2 (2)
Units for length, b =√
~/mω, and energy, ~ω are used.• Short-range interspecies interactions between A and B parti-
cles with interaction strength, g, i.e.
HI = g
NA∑i=1
NB∑j=1
δ(xi − yj), (3)
where x denote the coordinates of A and y for B.• The intraspecies interaction strengths are assumed� g.• To access the quantum mechanical properties of our system
we must solve theN -body Schrodinger equation. This is doneby adapting an effective interaction approach that has recentlybeen successfully applied to fermions in harmonic traps [3].The analytical and numerical methods allow us to address upto ten particles in balanced systems and up to 20 in imbal-anced systems, which is larger than most previous studies notbased on stochastic or Monte Carlo techniques.
Results
2+1 system• In the limit 1/g → 0, the ground state becomes doubly degen-
erate and has half-integer energy: 2.5~ω - shown in Figure 1.•Notice how the two lowest states merge at 1/g = 0 and be-
come two excited states branches on the attractive side of theresonance but the even parity state remains the lower one.
Figure 1: The energy spectrum of a 2+1 system as a function of interac-tion strength, g, obtained by numerical calculations. The contribution fromcenter-of-mass motion has been removed.
Balanced systems• 2+2 system has also the two-fold degenerate ground state for1/g → 0 that has a non-integer energy - shown in figure 2a).• The system realizes a perfect spatially ferromagnetic quantum
state. Shown by numerical and also analytical results.•Analytically, we transform into Jacobian coordinates, remove
center-of-mass, and end up with figure 2b)• Red regions indicate AABB structure and occupy the largest
part of the space - most favored structure.
Figure 2: a) 2+2 energy spectrum for g > 0. The 1/g → 0+ limits are ana-lytically known and indicated by triangles. Blue is for even states and red forodd. b) Three-dimensional representation of the coordinate space, with theshown Jacobi coordinates at the bottom.
• 3+3, 4+4 and 5+5 show the same ferromagnetic behavior inground state, as illustrated in figure 3.• odd-even superposition is considered in order to prove perfect
ferromagnetic behavior, shown in figure 3b)-c)• The system does behave as two ideal Bose gases - figure 3d)
Figure 3: a) Total ground state density for 2+2 and different values of g.Dotted (red) line corresponds to 1/g = 100 while the solid (black) line is for1/g = 0.01. Blue dots show the analytical solution for 1/g = 0. b)-c) Densi-ties for a superposition of the two-fold degenerate ground state at g = 100 forNA = NB = 2, 3, 4, and 5. d) Rescaled plot of the total density at 1/g = 0.01.
Imbalanced systems• 1 +NB systems = strongly interacting Bose polaron in 1D.• Cases such as 1+3 and 1+8 systems show that impurity sits
at the edge of the system, illustrated in figure 4. The samestructure is also true for NA 6= NB with NA > 1
•Analytical results are obtained by using a method similar tothe Born-Oppenheimer method.
Figure 4: a) Impurity ground state density in 1+3 and 1+8 system. Theanalytical results for 1/g = 0 are shown as triangles. b) and c) shown theodd-even mixture. d) The density of the majority component (NB).
• Figure 5 shows that the energy per particle tends to saturatefor large systems and happens faster the more imbalanced thesystem is.• Balanced systems have almost linear energy dependence.
Figure 5: The filled circles show the ground state energy for a NA+NB sys-tem at 1/g = 0.01 relative to the zero-point energy. The diamonds representour analytical results. The dashed lines are quadratic interpolations, whilethe solid line is an interpolation of the energy for the balanced systems.
Conclusions•A mixture of two ideal Bose systems in one dimension have
non-integer multiples of ~ω• The separation of components in the ground state is intrinsic
to both balanced and imbalanced mixtures.•Other spatial configurations are true in specific excited states.• The effect is not connected to the harmonic confinement.• Physical picture can be given in terms of domain walls, which
the system tends to minimize.
Upcoming ResearchIn the upcoming months we aim to demonstrate the behaviorin extremely imbalanced system, with one and two weakly in-teracting impurity with arbitrary number of majority particles.In addition, we will generalize our model so it also takes careof different particle masses, non-identical harmonic frequenciesand displaced harmonic oscillator potentials. The intra-speciesinteraction among majority particles will also be considered andinvestigated. Figure 6 shows one of the newest results for g=1.
−4 −3 −2 −1 0 1 2 3 4
x [b]0.0
0.4
0.8
Dens
ity [b
−1]
1 particle subsystem
-1/g = -1.0numericalideal
−4 −3 −2 −1 0 1 2 3 4
x [b]0
2
4
Dens
ity [b
−1] 8 particle subsystem
−4 −3 −2 −1 0 1 2 3 4
x [b]0
2
4
Dens
ity [b
−1] 1+8 total system
Figure 6: Density distribution in a 1+8 polaron system with finite g=1. Theagreement with numerically calculated data is within few percent and getsbetter as the number of majority particles increases.
AcknowledgementsThis work was funded by the Danish Council for Indepen-dent Research DFF Natural Sciences and the DFF Sapere Audeprogram and the European Research Council under the Euro-pean Community’s Seventh Framework Programme - ERC grantagreement no. 240603.
References[1] G. Zurn et al., Phys. Rev. Lett. 108, 075303 (2012).[2] A. N. Wenz et al., Science 342, 457 (2013).[3] E. J. Lindgren, et al. New J. Phys. 16, 063003
