Quantum dark solitons in the one- dimensional Bose gas Joachim - - PowerPoint PPT Presentation

Quantum dark solitons in the one- dimensional Bose gas

Humboldt Kolleg – Controlling quantum matter: from ultracold atoms to solids– July 2018

Joachim Brand

Sophie S. Shamailov

Solitons

3 credit: Alex Kasman Tikhonenko et al. (1996)

Optics

Sengstock group (2008)

Water Coupled Pedula BEC

Solitons are solutions of nonlinear partial differential equations. Can we find solitons in strongly-correlated quantum fluids? What are their properties?

Let’s find them in the one-dimensional Bose gas.

Dark soliton oscillations in BEC experiment

Hamburg Experiment: Becker et al. (2008)

Solitons in trapped BEC oscillate more slowly than COM Min Mph = ✓ Ts Ttrap ◆2 = 2

Theory:

• Busch, Anglin PRL (2000)
• Konotop, Pitaevskii, PRL (2004)

Experiment:

• Becker et al. Nat. Phys. (2008)
• Weller et al. PRL (2008)

Movie credits: Nick Parker

Dark solitons

• Mean field (classical) theory:

Defocussing nonlinear Schrödinger (Gross-Pitaevskii) equation

Dark and grey soliton solution (g>0): Tsuzuki, JLTP (1971)

vs = 0 Velocity vs 6= 0

Kivshar, Luther-Davis (1998)

Superfluid phase step ∆φ

Number density

Number of depleted particles Nd = Z [n(x) − nbg]dx

Effective mass, length scale

Phase

The one-dimensional Bose gas

Ptot = ~

N

X

j=1

kj, Etot = ~2 2m

N

X

j=1

k2

j.

kj -

Rapidities/quasimomenta

• Integer quantum numbers
• number of bosons

N

x k a k x exp ,

i i S N i i i

⟨{ } { }⟩ ( )

( ) ( )

⎛ ⎝ ⎜ ⎞ ⎠ ⎟

∑ ∑

| =

∈

P

P P

Bethe ansatz wave function

Ij

kj + 1 L X

l

2 arctan kj − kl mgh−2 = 2π L Ij

H = − ~2 2m

N

X

j=1

d2 dx2

j

+ g X

hi,ji

δ (xi − xj)

Lieb-Liniger model: Bosons with contact interactions in one dimension

Low-lying excitation spectrum (yrast states)

Momentum Energy Here for Tonks-Girardeau gas, γ = ∞, N=15 Lieb’s type II elementary excitations Umklapp excitation (ring current)

P 2π~n

Eigenstates are translationally invariant! Where are the solitons?

Experimental probe

• f dynamic structure

factor Fabbri et al. (2015)

Comparison of dispersion relations

−1 1 2 3 4 5

P ¯ hn0 2m ¯ h2n2

0(E − Eg)

π/2 π 3π/2 2π

Takayama, Ishikawa, JPSP (1980): Asymptotically (weak interaction, thermodynamic limit) is GP dark soliton congruent with yrast dispersion of Lieb-Liniger model Circles: finite system (ring), N = 10, γ=1 Thermodynamic limit, N = ∞, γ=1

γ = gm n0~2

How to get over the translational invariance

• f the eigenstates?
• Syrwid, Sacha, PRA (2015): Soliton emerges during particle

measurement.

• Sato et al. NJP (2012, 2016): Localised density dip by superposition
• f all yrast eigenstates
• Our proposal: Gaussian wave packet of yrast states

−1 1 2 3 4 5

P ¯ hn0 2m ¯ h2n2

0(E − Eg)

π/2 π 3π/2 2π

Soliton velocity:

vs = dEs dpc

Fialko, Delattre, JB, Kolovsky (2012) Shamailov, Brand, arXiv:1805.07856

Simulating time evolution

n(x, t) =hP0(t)|ˆ ρ(x)|P0(t)i = X

p,q

CP0∗

q

CP0

p hq, yr|ˆ

ρ(0)|p, yri ⇥ exp[i(p q)x/~ i(Ep Eq)t/~], The form factor is calculated by determinantal formula from the rapidities. Formula derived from algebraic Bethe ansatz: Slavnov (1989), Korepin (1982), Caux (2007)

40 50 60 70 80 90 10 20 30 0.4 0.8 1.2

ns/n0

¯ hn2 2m t

n0x

Time evolution of Gaussian wave packet (exact)

γ = 1 N = 100

Shamailov, Brand, arXiv:1805.07856

Time evolution of quantum dark soliton

Use the following ansatz, in analogy to quantum bright solitons:

∆x2(t) = σfs

2 + σ2 CoM(t),

σ2

CoM(t) = σ2

" 1 + ✓ ~t 2Mσ2 ◆2#

Ballistic spreading of the CoM – fit two parameters: where

∆x2 =N −1

d

Z (x hxi)2 [n(x) n0] dx Nd = Z [n(x) n0] dx σ2

0 =

~2 4∆P 2

σ2

fs, M

Shamailov, Brand, arXiv:1805.07856

Fits to the numerical time evolution

50 100 150 200 20 40 60

n2

0¯

h2 ∆p2

∆x2n2

5 10 15 20 25 50 100 150

∆x2n2

5 10 15 20 25 50 100 150

∆x2n2

5 10 15 20 25 50 100 150

• ¯

hn2 2m

2 t2 ∆x2n2

50 100 150 2 4 6 8 10 12

• ¯

hn2 2m

• t

∆xn0

x103 x102 x104

Shamailov, Brand, arXiv:1805.07856

γ = 0.1

γ = 10

γ = 1

The fundamental soliton width

Soliton width from GP theory

σ2

GP =

π2 3γ2n2

0N 2 d

Shamailov, Brand, arXiv:1805.07856

γ

10-2 10-1 100 101 102 103 104

σ/ξ

0.5 1 1.5 2 2.5 σ0,min/ξ σfs/ξ

Nd

• 10
• 9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

σ2

fsc2N 2 d

5 10 15

P0 = π¯ hn0 P0 = 2.5¯ hn0 P0 = 2¯ hn0 P0 = 1.5¯ hn0 P0 = 1¯ hn0 π2/3

(b) (a)

Particle number depletion and phase step

hn0 P ¯ hn0

∆φ

−7 −5 −3 −1

P ¯ hn0

Nd

hn0

π/2 π 3π/2 2π π/2 π 3π/2 2π 3π/2 2π π/2 π

Nd = ✓ 1 v2

s

c2 ◆−1 ✓∂Es ∂µ + vsP mc2 ◆

P = mvsNd + 1 2~n0∆φ

So does the phase step mean anything here?

Shamailov, Brand, arXiv:1805.07856

Yes, the phase step is very important.

EN

s (P) ≈ E∞ s (P) + Psvcf + 1

2Nmv2

cf + N 2 d

2L dµ dn0 , Ps = mvsNd vcf = ~∆φ mL

P = mvsNd + 1 2~n0∆φ

In ring geometry (periodic box), the phase step demands backflow current. Energy and momentum have corrections from Galilean boost.

Phase uncertainty is inherited from momentum uncertainty.

Comparison of dispersion relations

−1 1 2 3 4 5

P ¯ hn0 2m ¯ h2n2

0(E − Eg)

π/2 π 3π/2 2π

Takayama, Ishikawa, JPSP (1980): Asymptotically (weak interaction, thermodynamic limit) is GP dark soliton congruent with yrast dispersion of Lieb-Liniger model Circles: finite system (ring), N = 10, γ=1 Thermodynamic limit, N = ∞, γ=1

γ = gm n0~2

Energy corrections from phase step provide an excellent approximation of finite size dispersions.

Quantum soliton collisions

What have we learned…

• Even in the absence of true long-range order, solitons persist
• Solitons behave like quantum-mechanical bound states of (a

noninteger number of) holes

• The phase step is relevant for the backflow current – on a global

scale

• Yrast excitation spectrum may hold the key to soliton-like

excitations even in non-integrable models: many properties can be

• btained as derivatives
• Spreading of density is controlled by effective mass
• Generalisations to fermions, long-range interactions, …

…beyond the 1D Bose gas?

Other projects / future:

• Stochastic exact diagonalisation in Fock space for ultra-

cold atoms: Collaboration with Ali Alavi (Stuttgart) on FCIQMC

• Accelerating Fock space expansion of one-dimensional

quantum gas with contact interaction with the transcorrelated method. Improvement from to

• P Jeszenszki, HJ Luo, A Alavi, JB, arXiv:1806.032888

δE ∼ M −1

δE ∼ M −3

as =1L, kc =4π L as =2L, kc =4π L as =1L, kc =8π L

• 0.4
• 0.2

0.0 0.2 0.4 x L 1.00 1.02 1.04 1.06 exp (u0 (x))

DODD

ODD-WALLS ALLS CENTRE ENTRE

for Photonic and Quantum Technologies

Thank you!