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Particle Statistics and Lossy Dynamics of Ultracold Atoms in Optical Lattices

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Particle Statistics and Lossy Dynamics of Ultracold Atoms in Optical Lattices PHYSICAL REVIEW A 97, 053614 (2018) Particle statistics and lossy dynamics of ultracold atoms in optical lattices J. Yago Malo,1 E. P. L. van Nieuwenburg,2,3 M. H. Fischer,2 and A. J. Daley1 1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom 2Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland 3Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA (Received 12 September 2017; published 21 May 2018) Experimental control over ultracold quantum gases has made it possible to investigate low-dimensional systems of both bosonic and fermionic atoms. In closed one-dimensional systems there are many similarities in the dynamics of local quantities for spinless fermions and strongly interacting “hard-core” bosons, which on a lattice can be formalized via a Jordan-Wigner transformation. In this study, we analyze the similarities and differences for spinless fermions and hard-core bosons on a lattice in the presence of particle loss. The removal of a single fermion causes differences in local quantities compared with the bosonic case because of the different particle exchange symmetry in the two cases. We identify deterministic and probabilistic signatures of these dynamics in terms of local particle density, which could be measured in ongoing experiments with quantum gas microscopes. DOI: 10.1103/PhysRevA.97.053614 I. INTRODUCTION the eigenstates and for out-of-equilibrium dynamics induced by changing local trap quantities—will be equal as well. In the past few years, there has been rapid progress in However, single-particle loss can generate differences in local the characterization and control of dissipative dynamics for quantities due to the different exchange symmetries in the ultracold atoms in optical lattices. While these systems are many-body wave function. These differences manifest them- most known for the possibility to engineer Hamiltonians selves in local density distributions, which are accessible with for strongly interacting systems towards quantum simulation current experimental techniques in quantum gas microscopes purposes, [1,2], the same level of microscopic understanding, [20–25]. in which models can be derived from first principles under Making use of symmetries in tensor-network-based numeri- well-controlled approximations, is also available for most of cal methods, we calculate the dynamics of example systems for the dominant forms of dissipation that occur naturally in typical experimental sizes and parameter scales in the presence experiments. This applies, in particular, to our understanding of loss. The efficient simulation of such systems requires the of incoherent light scattering and the resulting dephasing of proper inclusion of symmetries in these numerical methods in the many-body state [3,4] and to our treatment of atom loss order to account for the loss process in an affordable manner. [5]. Studying these sources of dissipation is of importance We first study the loss process as a deterministic event and then well beyond gaining a better understanding of experimental employ a quantum trajectory approach [26–28] to determine imperfections: it allows for the use of dissipation (1) in probing features of bosons and fermions that survive stochastically many-body states and their dynamics [6,7], (2) in the controlled occurring loss events. preparation of interesting many-body states [6,8], and (3) in The rest of the paper is structured as follows. In Sec. II understanding how signatures of fundamental effects from we discuss the theoretical model for fermions and bosons closed systems (e.g., many-body localization) survive in the confined in one dimension subject to dissipation. In Sec. III presence of coupling to an environment [9–13]. we highlight the differences we expect to observe between the In this work, we explore how the differences between many- different types of particle statistics in the event of a loss, and in body states of hard-core bosons (HCBs) and spinless fermions Sec. IV we describe the numerical approach that allows for the confined to move in one dimension can be probed using particle computationally efficient simulation of a system subject to this loss. In one dimension, where strongly interacting bosons kind of dissipation. In Sec. V we then analyze the dynamics cannot pass each other, there are strong formal similarities following losses that occur at deterministic times and locations, between HCBs and spinless fermions [14]. These regimes identifying accessible parameter regimes where the differences have been realized in experiments with cold bosonic atoms between HCBs and spinless fermions are significant and could in strongly confined one-dimensional (1D) tubes [15], and in be engineered and observed using quantum gas microscopes lattices [16,17], and the consequences can be seen clearly, even [29–31]. In Sec. VI we study which of those features identified for just two atoms, in quantum gas microscope experiments in Sec. V survive under stochastic losses and which of them [18]. For particles moving on a lattice, this similarity can vanish when the losses occur randomly, providing local and be formalized via a Jordan-Wigner transformation to spin spatially averaged quantities that can be obtained through operators [19], where we see that for local models, the energy density measurements. Finally, in Sec. VII we discuss our eigenvalues will be identical, and local correlations—both for findings. 2469-9926/2018/97(5)/053614(9) 053614-1 ©2018 American Physical Society YAGO MALO, VAN NIEUWENBURG, FISCHER, AND DALEY PHYSICAL REVIEW A 97, 053614 (2018) II. MODEL: FERMIONS AND HARD-CORE BOSONS IN Figure 1 shows a schematic view of the density profiles after THE PRESENCE OF LOCAL PARTICLE LOSS a deterministic loss event in the middle site at t = 0, beginning from an initial product state with one atom on every lattice site. In this section, we introduce a model for particle loss in Because of the simple initial state and the single loss process, spinless fermions or hard-core bosons confined to move along the density distributions for bosons and fermions as a function one direction of an optical lattice (and tightly confined in the of time are identical, i.e., the normalized difference, other two directions). For fermions in the lowest Bloch band of the optical nb − nf n = i i , lattice, the system is well described by a tight-binding i f (4) nb + n Hamiltonian(¯h ≡ 1), i i b = f = M−1 where ni nˆ b,i and ni nˆ a,i , is zero in this case. For ˆ =− † + b + f = H J (aˆi aˆi+1 H.c.) , (1) the case of vanishing densities ni ni 0, which happens i only in certain initial states, we set the normalized difference to n = 0. In analyzing different parameter regimes and where i indicates the lattice site, with lattice length M,the i (†) identifying differences between HCBs and spinless fermions, operator aˆi annihilates (creates) a fermionic particle on we focus particularly on this quantity in the following sections. = † ∈ the site i where nˆ a,i aˆi aˆi [0,1] is the fermionic number ∈ operator of the site i [1,M], where M is the lattice system III. DIFFERENCES BETWEEN FERMIONS AND BOSONS size, and J is the tunneling amplitude in the lattice. The IN THE PRESENCE OF LOSS fermionic operators obey the usual anticommutation rules, { (†) (†)}= { †}= Bosonic and fermionic atoms will behave differently in the aˆi , aˆj 0; aˆi , aˆj δi,j . An analogous model can be considered for the case of presence of dissipation as a result of the difference in the sign hard-core bosons, for which the Hamiltonian can be seen as of the wave function under exchange of particles. One way a limiting case of the Bose-Hubbard Hamiltonian [32], and is to see this is to use a Jordan-Wigner transformation to map given by each of these cases to spin operators [19]. A single-species model for hard-core bosons can be directly rewritten as an M−1 equivalent spin-1/2 model, with the spin states associated with ˆ =− ˆ† ˆ + ˆ 2 ≡ H J (bi bi+1 H.c.), bl 0 , (2) each lattice site denoting presence (|↑) or absence (|↓)ofa i particle on that site. Because bosons commute, the mapping (†) between particle annihilation operators and spin lowering where the operator bˆ annihilates (creates) a bosonic particle − − i operatorsσ ˆ is a direct replacement, bˆ → σˆ . However, the = ˆ† ˆ ∈ l l l on the site i and nˆ b,i bi bi [0,1] is the bosonic number same mapping for fermions requires a sign determined by a operator for the site i. In contrast to the fermionic case, string operator in order to account for anticommutation of the the bosonic creation and annihilation operators obey usual annihilation operators with all other operators present in the ˆ(†) ˆ(†) = ˆ ˆ† = commutation rules, [bi , bj ] 0; [bi , bj ] δi,j . state description, We can describe the dissipative dynamics of such systems − → − i<l nˆ a,i in the presence of particle loss via a master equation for the aˆl ( 1) σˆl . (5) system density operator ρtot. The master equation arises on It is clear that a loss event can thus affect the many-body a microscopic level because in these atomic-physics systems state differently for fermions and for hard-core bosons. Our we can usually make a Born-Markov approximation, justified goal here is to identify whether there are differences that can by the existence of a single dominant frequency for each be extracted solely from the local density distribution, nˆ a/b,l, process (given by the energy of the lost atom for single-particle which translates the same way into spin operators for bosons loss, and by the photon frequency for dephasing due to light and fermions under a Jordan-Wigner transformation, nˆ a/b,l → scattering [28]).
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