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

TL;DR

This paper investigates various crystal order states of dipolar bosons in optical lattices, analyzing their transitions and how to observe these states through full distribution functions derived from simulated imaging.

Contribution

It introduces a method to identify crystal orders in dipolar bosonic systems using full distribution functions from simulated single-shot images.

Findings

Multiple crystal states emerge depending on interaction strength and lattice depth.

Transitions between different crystal orders are characterized and mapped.

Full distribution functions effectively reveal the underlying crystal structures.

Abstract

We explore the groundstates 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.

Figures

Equations

$$H=\sum _{i=1}^N\left[T_i+V_{ol}(x_i)\right]+\sum _{i<j}{V}_{int}(x_i-x_j).$$

$$\Delta =\sum _k{\left(\frac{{\lambda }_k}{N}\right)}^2.$$

$${\rho }^{(1)}(x,x^{\prime })=⟨\Psi |{\hat{\Psi }}^{†}(x)\hat{\Psi }(x^{\prime })|\Psi ⟩=\sum _i{\lambda }_i{\phi }_i^{\ast }(x){\phi }_i(x^{\prime }),$$

$$\Theta =\frac{1}{N}\sum _{e,o}⟨n_o⟩-⟨n_e⟩,$$

Full text

The realization of Bose-Einstein condensates (BECs) of dipolar atoms griesmaier05 ; beaufils08 ; lu11 ; aikawa12 ; ni08 ; zwierlein15 and molecules ni08 ; zwierlein15 provides new perspectives to study phase transitions in correlated quantum systems baranov08 ; lahaye09 . The anisotropic and long-ranged dipole-dipole interactions lead to a plethora of new phenomena absent in conventional BEC, e.g., directional elongation yi01 ; santos00 ; goral00 and geometric stabilization yi01 ; santos00 ; eberlein05 ; goral02a ; koch08 . A lower dimensionality results in additional physical features: $p$-wave superfluidity bruun08 ; cooper09 , Luttinger-liquid-like behavior arkhipov05 ; citro07 ; depalo08 ; pedri08 , and anisotropy in curved geometries zollner11 ; zollner11pra ; maik11 .

Atoms in optical lattices are experimentally very tunable and serve as quantum simulators for condensed matter systems bruder ; greiner02 ; Liberto16 ; Kock16 ; Kock15 ; landig15 ; jotzu14 ; Landini18 ; Langen15 ; Schweigler17 ; Engelsen17 ; lahaye10 ; anna13 ; fischer13 ; gallemi13 ; gallemi16 . Few-particle systems especially provide an experimental bottom-up access to many-body physics selim1 ; selim2 ; selim3 . The interplay between anisotropic long-range and contact interactions results in the emergence of new phases beyond the usual superfluid ($SF$) and Mott-Insulator ($MI$). A density-wave phase ($DW$) goral02 ; dalla06 ; biedron18 ; Maluckov12 , characterized by an alternate filling of lattice sites and a supersolid phase Kovrizhin2005 ; Yi2007 ; Grimmer2014 ; Cinti2016 ; Danshita2009 ; biedron18 , with coexistent $DW$ and superfluidity were observed. Exotic phases such as Haldane insulators dalla06 ; deng11 , checkerboards goral02 ; menotti07 and Mott solids zoller10 were predicted. Remarkably, a crystal state ($CS$) emerges for dominant dipolar interactions arkhipov05 ; astrakharchik08b ; deuretzbacher10 ; Schachenmayer10 ; bera18 ; zollner11 ; zollner11pra ; chatterjee12 ; chatterjee17a ; chatterjee17b .

In this Letter, we establish protocols to detect the remarkable plethora of crystal orders emerging in lattices incommensurately filled with dipolar bosons. The physics of the crystal state cannot be addressed using the Hubbard model Note_Hubbard and requires an ab-initio many-body description chatterjee17a ; chatterjee17b . We study groundstates of a few dipolar atoms in one-dimensional lattices by numerically solving the full Schrödinger equation with the multiconfigurational time-dependent Hartree method for bosons alon08 ; streltsov07 ; ultracold ; axel1 ; axel2 ; lode:19 .

As Refs. chatterjee15 ; cao17 ; Mistakidis17 ; Koutentakis17 ; Mistakidis18a ; Streltsov13 ; Streltsova14 ; Klaiman18 ; Marchukov19 , we explore the full range of interaction strengths to observe emergent distinct crystal orders, due to the competition between kinetic, potential, and interaction energies. Our main finding is that all these crystal orderings and thereby the phase diagram can be unequivocally characterized using the full distribution functions of the position-dependent particle number operator extracted from simulated experimental absorption or single-shot images sakmann16 ; lode17 ; tsatsos17 ; chatterjee17a . We emphasize that our results are experimentally feasible as single-shot images with single-atom sensitivity for few-atom systems has been observed Bakr:09 ; Sherson:10 . With our present finite-size considerations, we cannot make claims about the true quantum phases in the thermodynamic limit. However, we demonstrate that the crystal orders are valid for the state diagram, the finite-size analog of the thermodynamic phase diagram for different lattice sizes, particle numbers, and different types of boundary conditions SI .

Consider polarized, dipolar bosons in a quasi-one-dimensional lattice potential having tight transversal confinement of characteristic length $a_{\perp}$. The $N$-body Hamiltonian reads

[TABLE]

Here $T_{i}$ is the kinetic energy of the $i$-th boson, $V_{ol}=V\sin^{2}(\kappa x)$ is the lattice potential with a depth $V$ and a wave-vector $\kappa$. Hard wall boundaries at $x=\pm S\pi/2\kappa$ restrict the lattice to $S$ sites. $V_{int}(x_{i}-x_{j})=\frac{g_{d}}{|x_{i}-x_{j}|^{3}+\alpha}$ is the (purely) dipolar interaction of strength $g_{d}$  Note1 . The transversal confinement introduces the short-scale cutoff $\alpha\approx{a_{\perp}}^{3}$ sinha07 ; deuretzbacher10 ; cai10 ; Note2 . All quantities are given in terms of recoil energy Note_recoil . We consider a cutoff $\alpha=0.05$, $N=8$ bosons, and $S=5$ lattice sites Note_parameter .

We now discuss possible crystal-state orderings using the density $\rho(x)=\langle\Psi|\hat{\Psi}^{\dagger}(x)\hat{\Psi}(x)|\Psi\rangle$ as a function of the interaction strength $g_{d}$ [Fig. 1(a)]. For $g_{d}=0$, we obtain a pure $SF$. The incommensurate setup implies the absence of a pure $MI$; as $g_{d}$ increases we find an $MI$ which coexists with an $SF$ ($SMI$) brouzos10 . Observed from $\rho(x)$: the two outer wells have a smaller population than the central wells; $N=5$ atoms in the $MI$ coexist with $N=3$ bosons in the $SF$ fraction. The $SF$ localizes in the central wells to minimize the kinetic energy with hard-wall boundary conditions.

The crystal transition occurs at $g_{d}\approx 0.8$: due to their repulsive dipolar interactions, the bosons avoid each other and save interaction energy minimizing their overlap. The density [Fig. 1(a)] splits in the doubly occupied central wells – a signature of the crystal transition. The crystal state’s site occupations are $1,[11],[11],[11],1$, where $[11]$ denotes the two-hump density in doubly-occupied lattice sites. Since the double occupation of the three central wells results from the kinetic energy in the Hamiltonian, we term this state “kinetic crystal state” ($KCS$).

A further increase of $g_{d}$ makes the interaction energy overcome the kinetic energy; the double occupation of adjacent sites is energetically unfavorable [Fig. 1(a)]: instead of nearest neighbors, next-nearest neighbors are now doubly occupied in a density-wave-ordered structure at $g_{d}\gtrsim 3$. The “standard” density-wave-order for our system would have the occupations $2,1,2,1,2$ (Ref. Maluckov12 ). Due to the strong dipolar interactions, the bosons’ density in doubly occupied sites is spatially split and a density-wave crystal state ($DWCS$) with occupations $[11],1,[11],1,[11]$ results. The $DWCS$ represents a completely new crystal order, featuring the coexistence of $DW$ arrangement and crystallization. The $KCS$ and the $DWCS$ are possible crystal orders and hence a subset of a general crystal state $CS$. Here, and henceforth, we use the label $CS$ for a crystal state in either the $KCS$ or the $DWCS$ arrangement. See Ref. SI , Secs. S4, S5, and S9, for finite-size and boundary-condition effects on the crystal orders; importantly, the $KCS$ fades away for a larger system size while the $DWCS$ and $CS$ prevail.

Various crystal arrangements of the atoms with respect to the lattice can be obtained from a purely classical model (Sec. S4 SI ). However, many-body calculations are necessary to capture the quantum properties of the crystal states: many modes contribute to the quantum field of the crystal states [Fig. 1(b)] – their properties are only accessible via a realistic model of the many-body wavefunction. Moreover, the $SF$ and $MI$ are purely quantum states and the transition from such a non-crystal to a crystal order cannot be obtained from a classical model.

To explain the emergent many-body properties and the localization in the observed states, we discuss the two-body density $\rho^{(2)}(x_{1},x_{2})=\langle\Psi|\hat{\Psi}^{\dagger}(x_{1})\hat{\Psi}^{\dagger}(x_{2})\hat{\Psi}(x_{1})\hat{\Psi}(x_{2})|\Psi\rangle$ for characteristic values of $g_{d}$ [Fig. 2(a–d)]. Unlike the particle arrangement corresponding [one-body density, Fig. 1(a)], the analysis of the two-body density exhibits the many-bodyness of the crystal state: the two-body densities in Fig. 2(b)–(d)  – unlike for classical, semi-classical, and mean-field models – is not a product of one-body densities.

For small interaction strength ($g_{d}=0.0005$) in the $SF$ the maxima of $\rho^{(2)}$ are nearly uniformly distributed at positions $(x_{1},x_{2})$ in the vicinity of the minima of the lattice [Fig. 2(a)]. For larger interaction, $g_{d}=0.01$, a partial depletion along the diagonal ($x_{1},x_{2}\approx x_{1}$) of $\rho^{(2)}$ occurs. This depletion results from the formation of an $MI$ that coexists with an $SF$ [Fig. 2(b)]. At stronger interactions, $g_{d}=1.5$, the bosons in the doubly-occupied central wells crystallize forming the $KCS$. The diagonal of $\rho^{(2)}(x_{1},x_{1})\approx 0$ is completely depleted: a correlation hole is formed, the probability of detecting two bosons at the same position vanishes. The split maxima in the three central wells result from the on-site interaction-driven splitting $2\rightarrow[11]$. For $g_{d}=6$, in the $DWCS$ [Fig. 2(d)], a split inter-site structure of the two-body density $\rho^{(2)}$ is present for every odd site of the lattice potential. The diagonal depletion is wider compared to the $KCS$ [Fig. 2(c),(d) for $x_{1}\approx x_{2}$]. The non-uniform distribution of the maxima of $\rho^{(2)}$ heralds the $DWCS$.

For extremely strong interactions $g_{d}>50$ (not shown), the $DWCS$ transforms into a crystal state $CS$. To minimize the interaction that overwhelms the potential the distribution of the maxima of both $\rho(x)$ and $\rho^{(2)}(x_{1},x_{2})$ becomes equidistant [ SI , Sec. S3]. This $CS$ differs from the $SMI$, $KCS$, and the $DWCS$, because the distribution of maxima of the densities is not dictated by the minima of the lattice potential ( SI ,Sec. S4,S5,S9) for other possible crystal orders at different particle numbers, lattice sizes and boundary conditions.

To explore the coherence properties of the emergent states, we analyze the $2^{nd}$ order spatial (Glauber) correlation functions sakmann08 ; Glauber63 , defined as $g^{(2)}(x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime})=\frac{\rho^{(2)}(x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime})}{\sqrt{\rho(x_{1})\rho(x_{1}^{\prime})\rho(x_{2})\rho(x_{2}^{\prime})}}$ and quantifies the $2^{nd}$ order coherence in the system. Fig. 2(e)–(h) display the diagonal $|g^{(2)}|\equiv|g^{(2)}(x_{1},x_{2})|=|g^{(2)}(x_{1},x_{2},x_{1}^{\prime}=x_{1},x_{2}^{\prime}=x_{2})|$.

In the $SF$ ($g_{d}=0.0005$) the system is delocalized and shows substantial coherence [$|g^{(2)}|\approx 1$ almost throughout Fig. 2(e)]. For the $SMI$, $g_{d}=0.01$, a partial localization in the lattice sites reduces the off-diagonal coherence, reflected in the diagonal anticorrelation blocks $|g^{(2)}|<1$, Fig. 2(f). For $g_{d}=1.5$ ($KCS$), the diagonal coherent blocks split as the bosons crystallize. The (anti) correlation regions are now centered on each boson, [Fig. 2(g)]; this persists also for the $DWCS$ [$g_{d}=6$, Fig. 2(h)], showing the complete localization and decoherence of atoms in crystal states. These coherence properties are inherently a quantum many-body effect absent in mean-field approaches ( SI , Sec. S8). The two-body correlation functions and densities in Fig. 2, as expected for multiple significant eigenvalues of the reduced one-body density matrix [Fig. 1(b) and below] demonstrate the appealing many-body structure of crystallized bosons.

The $SMI\rightarrow KCS$ and $KCS\rightarrow DWCS$ transitions depend on $g_{d}$ and $V$; we construct the state diagram for these parameters. The $SMI\rightarrow CS$ transition can be determined from a crystal state order parameter $\Delta$ chatterjee17a ,

[TABLE]

The $\lambda_{k}$ are the $k^{th}$ eigenvalues (natural occupations) obtained from diagonalizing the one-body reduced density matrix,

[TABLE]

together with the eigenfunctions $\varphi_{i}(x)$ (natural orbitals).

The values $\lambda_{k}$ determine if the system is condensed: one natural occupation is macroscopic ($\lambda_{1}\approx N$)  penrose56 or fragmented: multiple $\lambda_{k}$’s are macroscopic spekkens ; mueller . Fig. 1(b) shows $\lambda_{k}$ as a function of $g_{d}$. For $0<g_{d}<0.002$, the system is condensed $SF$ and $\lambda_{1}\approx N$. For $0.002<g_{d}<0.8$, the system fragments and multiple $\lambda_{k>1}$ gradually increase for increasing interactions. At $g_{d}\gtrsim 0.8$, the $CS$ is reached with $N$ natural orbitals (almost) equally populated [Fig. 1(b)]. It is a hallmark of the many-body properties of the $CS$: the high-order density matrices, $\rho^{(p)}(p>1)$ are not a product of densities $\rho^{(1)}(x,x^{\prime})$ [Fig. 2]. Thus observables such as the correlation functions and the full distribution functions (see below) require a many-body model while classical models fail, see Sec. S5 SI .

The maximal value of the crystal order parameter, Eq. (2), is obtained for the $SF$ with $\Delta=1$, while the $CS$ is identified by the minimum value, $\Delta=\frac{1}{N}=0.125$, hence, characterizing the $CS$ transition. Fig. 3(a) shows the value of $\Delta$ as a function of $g_{d}$ and $V$, which clearly displays the transition from the $SMI$ to the $CS$.

For small $g_{d}$ and $V$, the bosons are condensed into a $SF$ and $\Delta\approx 1$. With increasing $g_{d}$ and/or $V$, the $SMI$ forms; fragmentation and – consequently – a diminution of $\Delta$ is seen. A further increase of $g_{d}$ ($>0.8$) decreases $\Delta$ gradually towards its minimum value ($\Delta=0.125$) for all values of $V$ marking the onset of the $KCS$. When $\Delta$ reaches its minimum, the maximally (eightfold) fragmented $CS$ is reached; the orderings of the $CS$, analyzed in the following, are the $KCS$ or the $DWCS$. Importantly, by analyzing $\Delta$ alone, the $KCS\rightarrow DWCS$ transition cannot be identified.

To identify the $KCS\rightarrow DWCS$, we use the population imbalance of even (e) and odd (o) sites, defined as

[TABLE]

with $\langle n_{e}\rangle$ and $\langle n_{o}\rangle$ their respective population. $\Theta$ is maximal for the density modulation corresponding to the $DWCS$. Fig. 3(b) shows $\Theta$ as a function of $g_{d}$ and $V$. For small values of $g_{d}$, the localization of the atoms in the central wells leads to small values of $\Theta$. For larger $g_{d}$, $\Theta$ decreases because of the uniform density of the $SMI$. As $g_{d}$ increases further, $\Theta$ reaches its maximum value for the $KCS\rightarrow DWCS$ transition, which shows a stronger dependence on $V$ compared to $SMI\rightarrow KCS$ transition. A shallower lattice potential favors the $KCS\rightarrow DWCS$ transition at lower $g_{d}$ while deeper lattices require larger $g_{d}$, Fig. 3(b).  The order parameters $\Delta$ and $\Theta$ quantify state diagram of any finite-size system as they are applicable for any finite $N$ and $S$.

We now propose a general experimental protocol to detect all emergent states and thereby the state diagram of crystal orders using standard imaging Bakr:09 ; Sherson:10 . These single-shot measurements correspond to a projective measurement of the wavefunction. Ideally, the images contain an instantaneous sample of the position of all $N$ particles distributed according to the $N$-particle probability distribution $|\Psi|^{2}$.  Here, we compute a set of single-shot measurements from our MCTDH-B groundstate wavefunctions sakmann16 ; lode17 ; tsatsos17 ; Mistakidis18b and evaluate the full distribution functions of the position-dependent particle number operator, i.e., we quantify the probability $P_{n}(x)$ to detect $n$ particles at positions $x$ [Fig. 3(c–e)].

In the delocalized $SMI$ several particles can be detected in the same site with a significant probability: $P_{n}(x)$ are nonzero for $n\leq 2$. When the $KCS$ is reached at $g_{d}\approx 0.8$, the bosons become completely localized resulting in $P_{n\geq 2}(x)\approx 0$.  The transition from the $SMI$ via the $KCS$ to the $DWCS$ with increasing interaction strength $g_{d}$ is characterized unequivocally through the analysis of the full distribution functions $P_{0}(x)$, $P_{1}(x)$ and $P_{2}(x)$. While $P_{0}$ and $P_{1}$ exhibit the distribution patterns of the $KCS$ and the $DWCS$, $P_{2}\rightarrow 0$ signals the bosons’ complete isolation in crystal states – independent of the crystal ordering. The results of Fig. 3(c–e) show a good agreement with the ones of Fig. 1(a) and Fig. 3(a–b), demonstrating the $KCS$ and $DWCS$ transition for the same values of $g_{d}$. The simultaneous presence of density-wave order in $P_{0/1}$ and isolation, $P_{n\geq 2}\rightarrow 0$, can thus experimentally identify the $KCS$ and the $DWCS$ (See SI Sec. S5, for $P_{n\geq 3}\approx 0$). The FDF as shown in Fig. 3(c–e) is thus sufficient for detecting all the crystal orders and emergent states and therby the order-parameter and state-diagram. Since the full distribution function has been experimentally implemented betz:11 , our protocol thus provides a general and experimentally feasible way to explore the plethora of crystal orders of dipolar bosons in a lattice.

We emphasize that our crystal-order detection protocol using full distribution functions makes no reference to $N$ or $S$: it thus represents a viable method for any finite-size system.

We note that finite-size cold-atom systems necessarily exhibit pinning, thus precluding difficulties arising due to sliding phases that may be seen for periodic boundary conditions (S9  SI ). Moreover, the required accurate single-shot imaging techniques with high detection efficiency and close-to-single-atom detection are available for ultracold atoms Bakr:09 .

An alternative experimental protocol using the variance of single-shot measurements to quantify the order parameter $\Delta$ lode17 ; chatterjee17a intertwined with a binning of them to quantify the even-odd-imbalance $\Theta$ is described in  SI Sec. S7.

Beyond the static properties, the exploration of collective excitations like roton modes which have been theoretically predicted Maluckov12 ; Maluckov13 ; Santos03 ; Ronen07 ; Lasinio13 and experimentally observed Mottl12 ; Chomaz18 ; Petter19 would be of further interest.

We expect low lying phonon-like excitations in our many-body system, similarly to  Maluckov13 . In contrast to Maluckov13 , the dependence of the crystal spacing on the strength of the dipole-dipole interactions alters the dispersion relation and, therewith, the phonon modes in our setup.