Detecting One-Dimensional Dipolar Bosonic Crystal Orders via Full Distribution Functions

Budhaditya Chatterjee, Camille Lévêque, Jörg Schmiedmayer, and Axel U. J. Lode

Phys. Rev. Lett. 125, 093602 – Published 26 August 2020

Abstract

We explore the ground states of a few dipolar bosons in optical lattices with incommensurate filling. The competition of kinetic, potential, and interaction energies leads to the emergence of a variety of crystal state orders with characteristic one- and two-body densities. We probe the transitions between these orders and construct the emergent state diagram as a function of the dipolar interaction strength and the lattice depth. We show that the crystal state orders can be observed using the full distribution functions of the particle number extracted from simulated single-shot images.

Received 8 May 2019
Accepted 8 July 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.093602

Physics Subject Headings (PhySH)

Bose-Einstein condensates Cold gases in optical lattices Dipolar gases

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Budhaditya Chatterjee^1,*, Camille Lévêque^2,3,†, Jörg Schmiedmayer ^2,‡, and Axel U. J. Lode ^4,2,3,§

¹Department of Physics, Indian Institute of Technology-Kanpur, Kanpur 208016, India
²Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria
³Wolfgang Pauli Institute c/o Faculty of Mathematics, University of Vienna, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria
⁴Institute of Physics, Albert-Ludwig University of Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany

^*bchat@iitk.ac.in
^†camille.leveque@tuwien.ac.at
^‡Schmiedmayer@atomchip.org
^§auj.lode@gmail.com

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Vol. 125, Iss. 9 — 28 August 2020

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Images

Figure 1
(a) One-body density $ρ (x)$ as a function of the dipolar interaction strength $g_{d}$ for a lattice depth $V = 8$ . For the SMI, $g_{d} ≲ 0.8$ , the density exhibits a fivefold structure. As $g_{d}$ increases, the density develops a twofold splitting in the doubly-occupied central three wells displaying the onset of $SM I \to KCS$ transition. For even larger $g_{d}$ , the density transitions to a pattern with alternating single and double occupations signifying the DWCS transition. See the Supplemental Material [83], Section S2, for $ρ (k)$ in momentum space. (b) Natural occupations (plotted cumulatively) as a function of $g_{d}$ . For small $g_{d}$ , the SF fraction results in the dominance of $λ_{1}$ . With increasing $g_{d}$ , the fragmentation increases as several $λ_{k > 1}$ become significant. Beyond the $SM I \to KCS$ transition, the system is maximally fragmented with eight almost equally contributing eigenvalues.
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Figure 2
(a)–(d) Two-body density $ρ^{(2)} (x_{1}, x_{2})$ . (a) $g_{d} = 0.0005$ : the pure SF shows a nearly square-lattice-like equispaced distribution. (b) $g_{d} = 0.01$ : the localization of bosons in the SMI results in a diagonal depletion. (c) $g_{d} = 1.5$ : for KCS, a correlation hole develops; the probability to find two bosons at the same position vanishes. (d) $g_{d} = 6.0$ : the DWCS shows a completely split but nonuniform density with density-wave order. (e)–(h): second-order spatial correlation function [ $| g^{(2)} (x_{1}, x_{2}) |$ ]. (e) $g_{d} = 0.0005$ : the bosons are significantly coherent for the SF. (f) $g_{d} = 0.01$ : reduction of off-diagonal coherence for the SMI. (g) $g_{d} = 1.5$ : diagonal coherent blocks split as the bosons crystalize at the KCS. (h) $g_{d} = 6$ : completely split coherent blocks centered at each boson position in the DWCS.
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Figure 3
Characterization and detection of the KCS and DWCS. (a) Crystal order parameter $Δ$ as a function of the interaction strength $g_{d}$ and lattice depth $V$ . The maximum, $Δ = 1$ , corresponds to the condensed SF. The SMI is revealed by intermediate $Δ < 1$ . The minimum value, $Δ = 0.125$ , corresponds to a CS thereby characterizing its formation. (b) Imbalance parameter $Θ$ as a function of $g_{d}$ and $V$ . Both SMI and KCS correspond to low values of $Θ$ , while the maximum value indicates the $KC S \to DWCS$ transition. (c)–(e) Full distribution functions $P_{n} (x)$ as a function of $g_{d}$ for $V = 8$ evaluated from 10000 single shots. The plots of the probability to detect zero [ $P_{0} (x)$ ] or one [ $P_{1} (x)$ ] particle are reminiscent of the one-body density [compare Fig. 1]. The probability of detecting two particles [ $P_{2} (x)$ ] vanishes clearly when the CS transition occurs. At large interactions, $g_{d} ≳ 6$ , $P_{0} (x)$ and $P_{1} (x)$ exhibit a density-wave pattern. This pattern along with vanishing $P_{2} (x)$ reveals the DWCS. See [83], Section S6, for the FDF in momentum space.
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