Quantum dark solitons in the one- dimensional Bose gas Joachim Brand Humboldt Kolleg – Controlling quantum matter: from ultracold atoms to solids– July 2018
Sophie S. Shamailov
Solitons Water Optics BEC Coupled Pedula Tikhonenko et al. (1996) 3 Sengstock group (2008) credit: Alex Kasman
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 Solitons in trapped BEC oscillate more slowly than COM Theory: ✓ T s • Busch, Anglin PRL (2000) ◆ 2 M in • Konotop, Pitaevskii, PRL (2004) = = 2 M ph T trap Experiment: • Becker et al. Nat. Phys. (2008) • Weller et al. PRL (2008) Movie credits: Nick Parker Hamburg Experiment: Becker et al. (2008)
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Low-lying excitation spectrum (yrast states) Experimental probe of dynamic structure Here for Tonks-Girardeau gas, γ = ∞ , N=15 Energy factor Fabbri et al. (2015) Lieb’s type II elementary excitations Umklapp excitation (ring current) P Momentum Eigenstates are translationally invariant! 2 π ~ n Where are the solitons?
Comparison of dispersion relations Takayama, Ishikawa, JPSP (1980): Asymptotically (weak interaction, thermodynamic limit) is GP dark soliton congruent with yrast dispersion of Lieb-Liniger model 0 ( E − E g ) 5 Circles: finite system (ring), 4 N = 10, γ =1 3 2 1 Thermodynamic limit, h 2 n 2 2 m 0 N = ∞ , γ =1 ¯ π / 2 π 0 3 π / 2 2 π P gm hn 0 γ = ¯ n 0 ~ 2 − 1
How to get over the translational invariance of the eigenstates? • Syrwid, Sacha, PRA (2015): Soliton emerges during particle measurement. • Sato et al. NJP (2012, 2016): Localised density dip by superposition of all yrast eigenstates • Our proposal: Gaussian wave packet of yrast states Soliton velocity: 5 0 ( E − E g ) 4 v s = dE s 3 dp c 2 Fialko, Delattre, JB, Kolovsky (2012) 1 h 2 n 2 2 m Shamailov, Brand, arXiv:1805.07856 0 ¯ π / 2 π 3 π / 2 2 π 0 P hn 0 ¯ − 1
Simulating time evolution n ( x, t ) = h P 0 ( t ) | ˆ ρ ( x ) | P 0 ( t ) i X C P 0 ∗ C P 0 = p h q, yr | ˆ ρ (0) | p, yr i q p,q ⇥ exp[ i ( p � q ) x/ ~ � i ( E p � E q ) t/ ~ ] , The form factor is calculated by determinantal formula from the rapidities. Formula derived from algebraic Bethe ansatz: Slavnov (1989), Korepin (1982), Caux (2007)
Time evolution of Gaussian wave packet (exact) N = 100 γ = 1 1.2 0.8 n s /n 0 30 0.4 20 hn 2 ¯ 0 2 m t 10 0 40 50 60 70 0 80 90 n 0 x Shamailov, Brand, arXiv:1805.07856
Time evolution of quantum dark soliton Use the following ansatz, in analogy to quantum bright solitons: 2 + σ 2 ∆ x 2 ( t ) = σ fs CoM ( t ) , " ◆ 2 # ✓ ~ t σ 2 CoM ( t ) = σ 2 1 + 0 2 M σ 2 0 where Z ( x � h x i ) 2 [ n ( x ) � n 0 ] dx ∆ x 2 = N − 1 d Z N d = [ n ( x ) � n 0 ] dx ~ 2 σ 2 0 = 4 ∆ P 2 Ballistic spreading of the CoM – fit two parameters: σ 2 fs , M Shamailov, Brand, arXiv:1805.07856
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