Communications in Theoretical Physics ›› 2020, Vol. 72 ›› Issue (8): 085501. doi: 10.1088/1572-9494/ab95fa
• Atomic, Molecular, Optical (AMO) and Plasma Physics, Chemical Physics • Previous Articles Next Articles
P Blair Blakie(),D Baillie,Sukla Pal
Received:
2020-02-21
Revised:
2020-04-21
Accepted:
2020-05-15
Published:
2020-08-01
P Blair Blakie, D Baillie, Sukla Pal, Commun. Theor. Phys. 72 (2020) 085501.
Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks
Figure 2.
(a) Comparison of the (dotted lines) analytic result (18) to (solid lines) numerically calculated ${\tilde{U}}_{\mathrm{num}}$ (obtained by numerically evaluating equation (13) using $\chi \to {\chi }_{\sigma })$ for the effective k-space kernel. Results shown for several values of η and for gs = 0. The exact result for η = 1 is also shown (dashed line). (b) The maximum absolute error of the approximation ${\tilde{U}}_{\sigma }$ compared to the ${\tilde{U}}_{\mathrm{num}}$ over the kz-range shown in (a), (in units of ${g}_{{dd}}/4\pi {l}^{2}$)."
Figure 3.
Comparison of the variational (red lines) and 3D eGPE (black lines) solutions for a uniform infinite system. The 1/e density contours of the transverse modes of the 3D eGPE χ and the variational approach χσ for (a) ${a}_{s}=120{a}_{0}$ and (b) ${a}_{s}=95{a}_{0}$. The harmonic oscillator ground state ${\chi }_{\mathrm{ho}}$ is shown for reference (blue lines). In (c) and (d) we compare the transverse mode profiles along the x (dash–dot) and y (lines) axes for the cases given in (a) and (b), respectively. (e) The effective 1D k-space interaction kernel obtained from the various transverse functions for ${a}_{s}=120{a}_{0}$ (dashed lines) and ${a}_{s}=95{a}_{0}$ (solid lines). The 3D eGPE result ${\tilde{U}}_{z}$ is obtained by evaluating equation (13) using χ. The variational ${\tilde{U}}_{\sigma }$ and the harmonic oscillator ${\tilde{U}}_{\mathrm{ho}}$ results are obtained from equation (18). Results for 164Dy using ${a}_{{dd}}=130.8\,{a}_{0}$, with ${\omega }_{x,y}=2\pi \times 150\,\mathrm{Hz}$, ${\omega }_{z}=0$, and $n=2.5\times {10}^{3}\ \mu {{\rm{m}}}^{-1}$."
Figure 4.
Comparison of line density profiles $n(z)=\int {\rm{d}}{\boldsymbol{\rho }}| {\rm{\Psi }}({\boldsymbol{x}}){| }^{2}$ and energies for a 164Dy condensate in a trap with ${\omega }_{x,y}=2\pi \,\times 150\,\mathrm{Hz}$, ${\omega }_{z}=2\pi \times 20\,\mathrm{Hz}$, for ${a}_{s}=100{a}_{0}$ and various atom numbers N. (a) The line density along the z-axis calculated using the 3D eGPE (black), variational (red) and the quasi-1D (blue) theories. Inset shows the 1/e density contours (relative to the density at ${\boldsymbol{\rho }}={\bf{0}}$) in the ${\boldsymbol{\rho }}$-plane for the various theories for N = 5 × 104. The contour of the 3D eGPE solution is evaluated at z = 0 (black) and z = 15 μm (grey). (b) Energy per particle of the three theories and (c) error in the energy of the variational and the quasi-1D theories relative to the 3D eGPE results."
Figure 5.
Excitation dispersion relations obtained from the (a) variational (dark blue) and (b) 3D (light blue) BdG theories for various values of as as labelled. In (a) the black circle indicates the roton coordinates (${k}_{\mathrm{rot}},{\epsilon }_{\mathrm{rot}}$) for ${a}_{s}=95{a}_{0}$. In (b) the higher excitations bands for ${a}_{s}=150{a}_{0}$ are shown (black dotted lines). The (c) roton energy and (d) roton wavevector as as changes for the variational (dark blue line) and 3D (light blue dots) theories. In (c) the critical value ${a}_{s}^{* }$ at which the roton energy goes to zero is indicated for each theory with an arrow and the fit function $\alpha \sqrt{{a}_{s}-{a}_{s}^{* }}$ (with α a fitting parameter) is also shown (dashed lines). In (d) the value of the critical roton wavevector (${k}_{\mathrm{rot}}^{* }$) when ${\epsilon }_{\mathrm{rot}}=0$ for ${a}_{s}={a}_{s}^{* }$ is indicated by an arrow for each theory. The roton critical values (e) ${a}_{s}^{* }$ and (f) ${k}_{\mathrm{rot}}^{* }$ as the system density changes. We compare the variational (lines) and 3D (symbols) theories both including (dark blue line for variational, light blue circles for 3D) and neglecting (red line for variational, magenta crosses for 3D) the quantum fluctuation term. Results for 164Dy with ${\omega }_{x,y}=2\pi \times 150\,\mathrm{Hz}$, and in (a)–(d) the density is n = 2.5 × 103 μm−1."
1 |
Griesmaier A Werner J Hensler S Stuhler J Pfau T 2005 Bose–Einstein condensation of chromium Phys. Rev. Lett. 94 160401
doi: 10.1103/PhysRevLett.94.160401 |
Pasquiou B Bismut G Maréchal E Pedri P Vernac L Gorceix O Laburthe-Tolra B 2011 Spin relaxation and band excitation of a dipolar Bose–Einstein condensate in 2D optical lattices Phys. Rev. Lett. 106 015301
doi: 10.1103/PhysRevLett.94.160401 |
|
2 |
Lu M Burdick N Q Youn S H Lev B L 2011 Strongly dipolar Bose–Einstein condensate of dysprosium Phys. Rev. Lett. 107 190401
doi: 10.1103/PhysRevLett.107.190401 |
Lu M Burdick N Q Lev B L 2012 Quantum degenerate dipolar Fermi gas Phys. Rev. Lett. 108 215301
doi: 10.1103/PhysRevLett.107.190401 |
|
3 |
Aikawa K Frisch A Mark M Baier S Rietzler A Grimm R Ferlaino F 2012 Bose–Einstein condensation of erbium Phys. Rev. Lett. 108 210401
doi: 10.1103/PhysRevLett.108.210401 |
4 |
Chomaz L van Bijnen R M W Petter D Faraoni G Baier S Becher J H Mark M J Wächtler F Santos L Ferlaino F 2018 Observation of roton mode population in a dipolar quantum gas Nat. Phys. 14 442
doi: 10.1038/s41567-018-0054-7 |
5 |
Petter D Natale G van Bijnen R M W Patscheider A Mark M J Chomaz L Ferlaino F 2019 Probing the roton excitation spectrum of a stable dipolar Bose gas Phys. Rev. Lett. 122 183401
doi: 10.1103/PhysRevLett.122.183401 |
6 |
Landau L D 1941 The theory of superfluidity of helium II J. Phys. 5 71
doi: 10.1103/PhysRev.60.356 |
7 |
Santos L Shlyapnikov G V Lewenstein M 2003 Roton-maxon spectrum and stability of trapped dipolar Bose–Einstein condensates Phys. Rev. Lett. 90 250403
doi: 10.1103/PhysRevLett.90.250403 |
8 |
Ronen S Bortolotti D C E Bohn J L 2007 Radial and angular rotons in trapped dipolar gases Phys. Rev. Lett. 98 030406
doi: 10.1103/PhysRevLett.98.030406 |
9 |
Blakie P B Baillie D Bisset R N 2012 Roton spectroscopy in a harmonically trapped dipolar Bose–Einstein condensate Phys. Rev. A 86 021604
doi: 10.1103/PhysRevA.86.021604 |
10 |
Corson J P Wilson R M Bohn J L 2013 Stability spectroscopy of rotons in a dipolar Bose gas Phys. Rev. A 87 051605
doi: 10.1103/PhysRevA.87.051605 |
11 |
Corson J P Wilson R M Bohn J L 2013 Geometric stability spectra of dipolar Bose gases in tunable optical lattices Phys. Rev. A 88 013614
doi: 10.1103/PhysRevA.88.013614 |
12 |
Jona-Lasinio M Łakomy K Santos L 2013 Time-of-flight roton spectroscopy in dipolar Bose–Einstein condensates Phys. Rev. A 88 025603
doi: 10.1103/PhysRevA.88.025603 |
13 |
Bisset R N Blakie P B 2013 Fingerprinting rotons in a dipolar condensate: super-poissonian peak in the atom-number fluctuations Phys. Rev. Lett. 110 265302
doi: 10.1103/PhysRevLett.110.265302 |
14 |
Baillie D Blakie P B 2015 A general theory of flattened dipolar condensates New J. Phys. 17 033028
doi: 10.1088/1367-2630/17/3/033028 |
15 |
Roccuzzo S M Ancilotto F 2019 Supersolid behavior of a dipolar Bose–Einstein condensate confined in a tube Phys. Rev. A 99 041601
doi: 10.1103/PhysRevA.99.041601 |
16 |
Kadau H Schmitt M Wenzel M Wink C Maier T Ferrier-Barbut I Pfau T 2016 Observing the Rosensweig instability of a quantum ferrofluid Nature 530 194 197
doi: 10.1038/nature16485 |
17 |
Ferrier-Barbut I Kadau H Schmitt M Wenzel M Pfau T 2016 Observation of quantum droplets in a strongly dipolar Bose gas Phys. Rev. Lett. 116 215301
doi: 10.1103/PhysRevLett.116.215301 |
18 |
Bisset R N Wilson R M Baillie D Blakie P B 2016 Ground-state phase diagram of a dipolar condensate with quantum fluctuations Phys. Rev. A 94 033619
doi: 10.1103/PhysRevA.94.033619 |
19 |
Chomaz L Baier S Petter D Mark M J Wächtler F Santos L Ferlaino F 2016 Quantum-fluctuation-driven crossover from a dilute Bose–Einstein condensate to a macrodroplet in a dipolar quantum fluid Phys. Rev. X 6 041039
doi: 10.1103/PhysRevX.6.041039 |
20 |
Schmitt M Wenzel M Böttcher F Ferrier-Barbut I Pfau T 2016 Self-bound droplets of a dilute magnetic quantum liquid Nature 539 259
doi: 10.1038/nature20126 |
21 |
Böttcher F 2019 Dilute dipolar quantum droplets beyond the extended Gross–Pitaevskii equation Phys. Rev. Res. 1 033088
doi: 10.1103/PhysRevResearch.1.033088 |
22 |
Böttcher F Schmidt J-N Wenzel M Hertkorn J Guo M Langen T Pfau T 2019 Transient supersolid properties in an array of dipolar quantum droplets Phys. Rev. X 9 011051
doi: 10.1103/PhysRevX.9.011051 |
23 |
Tanzi L Lucioni E Famà F Catani J Fioretti A Gabbanini C Bisset R N Santos L Modugno G 2019 Observation of a dipolar quantum gas with metastable supersolid properties Phys. Rev. Lett. 122 130405
doi: 10.1103/PhysRevLett.122.130405 |
24 |
Wächtler F Santos L 2016 Quantum filaments in dipolar Bose–Einstein condensates Phys. Rev. A 93 061603(R)
doi: 10.1103/PhysRevA.93.061603 |
25 |
Baillie D Wilson R M Blakie P B 2017 Collective excitations of self-bound droplets of a dipolar quantum fluid Phys. Rev. Lett. 119 255302
doi: 10.1103/PhysRevLett.119.255302 |
26 |
Lee A-C Baillie D Bisset R N Blakie P B 2018 Excitations of a vortex line in an elongated dipolar condensate Phys. Rev. A 98 063620
doi: 10.1103/PhysRevA.98.063620 |
27 |
Salasnich L Parola A Reatto L 2002 Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates Phys. Rev. A 65 043614
doi: 10.1103/PhysRevA.65.043614 |
28 |
Edler D Mishra C Wächtler F Nath R Sinha S Santos L 2017 Quantum fluctuations in quasi-one-dimensional dipolar Bose–Einstein condensates Phys. Rev. Lett. 119 050403
doi: 10.1103/PhysRevLett.119.050403 |
29 |
Lima A R P Pelster A 2011 Quantum fluctuations in dipolar Bose gases Phys. Rev. A 84 041604
doi: 10.1103/PhysRevA.84.041604 |
30 |
Sinha S Santos L 2007 Cold dipolar gases in quasi-one-dimensional geometries Phys. Rev. Lett. 99 140406
doi: 10.1103/PhysRevLett.99.140406 |
31 |
Deuretzbacher F Cremon J C Reimann S M 2010 Ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap Phys. Rev. A 81 063616
doi: 10.1103/PhysRevA.81.063616 |
32 |
Deuretzbacher F Cremon J C Reimann S M 2013 Erratum: ground-state properties of few dipolar bosons in a quasi-one-dimensional harmonic trap Phys. Rev. A 87 039903
doi: 10.1103/PhysRevA.87.039903 |
33 |
Giovanazzi S O'Dell D H J 2004 Instabilities and the roton spectrum of a quasi-1D Bose–Einstein condensed gas with dipole–dipole interactions Eur. Phys. J. D 31 439 445
doi: 10.1140/epjd/e2004-00146-7 |
34 |
Ronen S Bortolotti D C E Bohn J L 2006 Bogoliubov modes of a dipolar condensate in a cylindrical trap Phys. Rev. A 74 013623
doi: 10.1103/PhysRevA.74.013623 |
35 |
Giovanazzi S Görlitz A Pfau T 2002 Tuning the dipolar interaction in quantum gases Phys. Rev. Lett. 89 130401
doi: 10.1103/PhysRevLett.89.130401 |
36 |
Tang Y Kao W Li K-Y Lev B L 2018 Tuning the dipole–dipole interaction in a quantum gas with a rotating magnetic field Phys. Rev. Lett. 120 230401
doi: 10.1103/PhysRevLett.120.230401 |
37 |
Lu H-Y Lu H Zhang J-N Qiu R-Z Pu H Yi S 2010 Spatial density oscillations in trapped dipolar condensates Phys. Rev. A 82 023622
doi: 10.1103/PhysRevA.82.023622 |
38 |
Bao W Cai Y Wang H 2010 Efficient numerical methods for computing ground states and dynamics of dipolar Bose–Einstein condensates J. Comput. Phys. 229 7874
doi: 10.1016/j.jcp.2010.07.001 |
39 |
Antoine X Levitt A Tang Q 2017 Efficient spectral computation of the stationary states of rotating Bose–Einstein condensates by preconditioned nonlinear conjugate gradient methods J. Comput. Phys. 343 92
doi: 10.1016/j.jcp.2017.04.040 |
40 | Knight M J Bland T Parker N G Martin A M 2019 Improved low-dimensional wave equations for cigar-shaped and disk-shaped dipolar Bose–Einstein condensatesarXiv:1908.02395 [cond-mat.quant-gas] |
41 |
Chomaz L 2019 Long-lived and transient supersolid behaviors in dipolar quantum gases Phys. Rev. X 9 021012
doi: 10.1103/PhysRevX.9.021012 |
42 |
Tanzi L Roccuzzo S M Lucioni E Famà F Fioretti A Gabbanini C Modugno G Recati A Stringari S 2019 Supersolid symmetry breaking from compressional oscillations in a dipolar quantum gas Nature 574 382
doi: 10.1038/s41586-019-1568-6 |
43 |
Guo M Böttcher F Hertkorn J Schmidt J-N Wenzel M Büchler H P Langen T Pfau T 2019 The low-energy goldstone mode in a trapped dipolar supersolid Nature 574 386
doi: 10.1038/s41586-019-1569-5 |
44 |
Natale G van Bijnen R M W Patscheider A Petter D Mark M J Chomaz L Ferlaino F 2019 Excitation spectrum of a trapped dipolar supersolid and its experimental evidence Phys. Rev. Lett. 123 050402
doi: 10.1103/PhysRevLett.123.050402 |
|
Copyright © 2009-2019 Editorial Office of Communications in Theoretical Physics
Support by Beijing Magtech Co. Ltd. Tel: 86-010-62662699 E-mail: support@magtech.com.cn