{'fraction_non_alphanumeric': 0.044315839543364516, 'fraction_numerical': 0.017044553670524813, 'mean_word_length': 4.463404071026418, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A cold atom quantum simulator for pairing, condensation and pseudogaps in extended Hubbard-Holstein models

We describe a quantum simulator for the Hubbard--Holstein model (HHM), consisting of a single layer containing two dressed Rydberg atom (DRA) species held in independent painted potentials, predicting that boson-mediated fermion pairing, pseudogaps and Berezinskii-Kosterlitz-Thouless (BKT) transition temperatures are experimentally accessible. The HHM is important because it contains the essential physics of unconventional superconductors. Experimentally realizable quantum simulators for HHMs are needed: (1) since HHMs are very challenging to solve numerically and analytically (2) to understand pairing in unconventional superconductors where electron-phonon interactions (EPI) and strong Coulomb repulsion compete (3) to understand the role of boson-mediated local pairing in pseudogaps and fermion condensation. We propose and study a quantum simulator for the HHM using optical lattices, painted using zeros in the AC stark shift, to control two Rydberg atom species independently within a monolayer. We predict that interactions are sufficiently tuneable to probe: (1) both HHMs and highly unconventional phonon-mediated repulsions, (2) the competition between intermediate strength phonon and Coulomb mediated interactions, (3) how boson-mediated pairing of fermions leads to pseudogaps and BKT transition. Thus, the quantum simulator is suitable for investigating boson-mediated pairing and condensation of fermions in unconventional superconductors.

Boson-mediated pairing of fermions has not yet been observed in cold atom experiments. Cold atom quantum simulators have been very successful for simulating Hubbard models. The Mott metal-insulator and superfluid-insulator transitions have both been observed [1,2]. The Feshbach resonance can be tuned into an attractive regime, allowing local pairing in attractive Hubbard models (AHMs) to be observed directly using gas microscopy [3]. In solid state systems AHMs are the effective Hamiltonian arising from boson-mediated interactions, so it would be of significant interest to probe such interactions directly.
Probing boson-mediated pairing in a quantum simulator is technically demanding, but potentially highly rewarding as this pairing reflects the mechanism of many superconductors. The recent discovery of hydrogen based superconductivity at ambient temperatures makes bosonmediated superconductivity particularly pertinent [4,5].
Unconventional superconductors often contain significant Coulomb repulsions and boson-mediated interactions (such as EPIs) [6][7][8][9][10][11]. Furthermore, recent exact numerics provide strong upper bounds on superconductivity in the popular Hubbard model [12], identifying the need to include additional interactions alongside this model to explain superconductivity in such materials.
The HHM and its extensions [13,14] contain the essence of these interactions, but the HHM is challenging to solve numerically and analytically. A tunable quantum simulator would allow this model to be explored without the complications of competing interactions and phases in unconventional superconductors. Moreover, interesting physics, e.g. BCS to BEC crossover, could be observed directly by tuning the interaction strength with-out the limitations of stoichiometry and pressure.
The innovations within the quantum simulator for the HHM proposed here are: (1) exploitation of zeros in the AC stark shift to generate bipartite lattices within a single optical pancake to reduce experimental complexity, (2) use of Rydberg mediated interactions to tune EPI and Coulomb repulsion independently, (3) the possibility to investigate highly unconventional repulsive interactions mediated via phonons, (4) the possibility to explore boson-mediated pairing, (5) the existence of a pseudogap, (6) the potential to investigate the BKT transition.
Two atom species, a fermion representing electrons and a boson representing phonons, can be trapped in different, but coexisting, lattices by exploiting state dependence in the AC stark shift [15,16]. We consider bosonic 87 Rb and fermionic 40 K, trapped by linear polarized lasers of wavelength λ ph = 768.97 nm and λ f = 790.07 nm respectively. Wavelengths are chosen because the atom-light interaction at a wavelength between the D1 and D2 transition lines for the specified species cancels due to zeros in the AC stark shift. Thus species dependent potentials are realized. Lattice potentials are blue (red) detuned for 87 Rb ( 40 K), so bosonic atoms are trapped in an "inverse" lattice where absence of light leads to confinement. For convenience, we discuss attractive potentials for both species, but these can easily be painted from repulsive ones.
The optical lattice contains a single optical pancake with laser wavelength λ pan and width w pan , within which potentials are painted using Gaussian beams [17]. The total lattice potential is V (r) = i V spot,i (r − r i ) + V pan (z), where V pan (z) = −V 0,pan exp(−2z 2 /w 2 pan ). r is a vector that lies within the plane of the optical pancake and the z-axis is perpendicular to the pancake. The spot potential has the form, Simulator properties are fixed by the pattern of optical lattice potentials. We investigate models generated by several spot configurations, shown in Fig. 1. The fermion lattice consists of single spots (red) and the phonon lattice multiple spots (green). Rotating phonon spot patterns with respect to the fermion lattice can change model properties. Deep phonon sites with filling n = 1 in Mott insulator states are formed from multiple spots with a separation close to the Raleigh limit to make broad sites within which the atoms can oscillate. Shallow fermion sites are only partially filled.
The fermion hopping term is iσ creates a fermion on site i with spin σ.
The phonon contribution to the Hamiltonian is H ph = ν,ih ω ph,ν d † iν d iν . d † creates a phonon, and ω ph,ν is the phonon frequency of mode ν. The phonon trap is deep so that bosons are in a Mott insulating state, and confined on smaller length scales than the spot size. A broad trap for phonons can be painted using two closely positioned spots [18] with effective potential, V ph (r) = i V spot (r + D i ) where D i are the displacements of the phonon spots from the mean position r [18]. Phonon properties can be derived from the dynamical matrix, A ij = ∂ 2 V ph /∂u i ∂u j | 0 , where u i is the phonon displacement. Eigenvalues of the matrix are ω ph,ν and eigenvectors the phonon polarization ζ ν . For phonon spots separated on a single axis there is a single polarization with frequency [18] For 2D spot arrangements there are two polarizations. Phonons are not coupled between sites, so are k independent.
The Rydberg-phonon interaction (RPI) arises from coupling between DRAs of different species, Eq. 1 is calculated with van Vleck perturbation theory, and is reliable ifᾱ = Ω 2p /2∆ 2p < ∼ 0.2 [19]. Here, we use parameters for Van der Waals Rydberg-Rydberg coupling with η = 6, but η = 3 is also possible. Rabi frequency, Ω 2p characterizes the atom-laser coupling. ∆ 2p is laser detuning from the ground → Rydberg state transition.Ṽ Ryd could be repulsive or attractive without loss of generality. Eq. (1) can be Taylor expanded, . Thus, the RPI is described by φ 0 is the ground state harmonic oscillator wave function. For simplicity, we assume that φ 0 (r) = δ(r), which is valid if ( dr 3 r 2 φ 2 0 (r)) a. So, The Hubbard U can be selected via the Feshbach res- Thus the full Hamiltonian of the quantum simulator is an analogue of an extended HHM, The conditions under which this reduces to a site-local HHM will now be determined.
The form of the effective (retarded) interaction between DRAs in the (multi)polaron action [20], is highly sensitive to the spot patterns, as shown in Fig.  2. We take the limit r c a so that only near-neighbor (NN) and next-nearest-neighbor (NNN) terms are necessary in Eq. 5. Repulsive interactions between NNN sites are a feature of the simulator. These arise when f ij ∝ ζ ν,j ·r ij changes sign due to reorientation of the sense ofr ij relative to ζ ν,j . Repulsive phonon-mediated interactions might seem surprising, since the EPI is commonly identified as attractive. However, repulsive EPI is predicted in condensed matter [21].
A bipartite lattice with spot arrangements parallel to the lattice vectors (panel (B)) has an effective interaction displaying 4 fold symmetry and a small repulsive interaction on diagonal NNNs. A bipartite checkerboard pattern with no phonon site on alternate squares produces a very similar pattern (not shown).
Crossed patterns lead to effective interactions with square symmetry and the smallest off-site terms for centrally placed spots (panel (C)). Since the frequency and mass of the oscillators along the two directions are identical and the oscillators are independent, then the full Φ is the sum of the Φ for two parallel spot arrangements (panel (F)), rotated by 90 o relative to each other [20]. The effective interactions are identical for 45 o and 90 o orientations of spots. A disadvantage is that multi-spot patterns are more complicated to paint.
By translating phonon spots so they approach an individual fermion site, f ij → δ ij , leading to a better reproduction of the HHM (panels (D)-(F)). As b is decreased from 0.5a (panel (F)) to 0.4a (panel (D)), where a = a √ 2, the NNN repulsive terms reduce and for b = 0.3a , |Φ NNN |/|Φ 00 | < 5 × 10 −4 (not shown). So reproduction of the HHM, with its local coupling, depends on the lattice spacing and temperature that can be achieved (since larger systems have lower energy scales relative to their condensed matter counterparts [19]). For comparison, panel (A) shows Φ ii for the Holstein model.
The RPI can be varied through the most interesting regions of W λ ∼ U, t, where theoretical techniques often fail (W = 4t). Fig. 3 shows Hamiltonian parameters for a = 1.73µm, and various V 0 . t and U are of order 100Hz. The Rydberg-Rydberg interaction is tunable by selecting different Rydberg states and thus modifying the dipole-dipole interactions. For convenience, we consider states with n Ryd = n Rb = n K . The two-atom state is |n Rb L Rb ; n K L K where L atom is the orbital angular momentum of the atom. Suitable interactions occur for the range n Ryd ≈ 27 − 32, via the channel |n Ryd S; n Ryd S → |n Ryd S; (n Ryd − 1)P . There, C 6 ranges from 26.1 MHz µm 6 to 153 MHz µm 6 , and the energy difference between |n Ryd Sn Ryd S and |n Ryd P (n Ryd − 1)P decreases from 6303 MHz to 4800 MHz. For a = 1.73µm, a 1/r 6 potential describes the interactions well.
If repulsion is found on a single diagonal (Fig. 2(D-F)), At intermediate λ, the binding diagram is essentially unchanged from the HHM. In Fig. 4(a), U (C) is calculated for the Φ shown in Fig. 2. Attractive NN terms push U (C) → ∞ at finite λ. Repulsive terms decrease U (C) .
We predict that in experiments, phonon-mediated local pairing occurs at ∼ 20nK for λ > ∼ 1.5. Local pairs can be directly observed using gas microscopy [3]. For λ 1, local s pairs dominate below T pair ∼ (2W λ − 8t )/k B . Fig. 4(b) shows how local pairing occurs at ∼ 20nK for large λ, small V 0 . These λ values are large compared to condensed matter analogues, but are accessible in the quantum simulator at large n Ryd .
We predict that BKT temperatures of ∼ 20nK can be achieved in experiments at small V 0 (Fig. 4 (b)). BKT condensation would be identifiable via changes to the momentum distribution of the atoms, which can be measured using time of flight. No general expression exists for the BKT temperature, so we make estimates for low pair density n B = 0.01 1 where T BKT = 4πh 2 n B /a 2 k B 2m * * ln ln(4/n B ) [23,24]. For strongly coupled onsite pairs, effective pair mass m * * = h 2 √ W 2 λ 2 + 2t 2 /t 2 a 2 (see supplement). The phase diagram at 20nK shown in Fig. 4(b) has four distinct regions. If T pair < T BKT there is condensation at the BKT temperature. At T pair > T BKT , preformed pairs condense at T BKT . We predict a pseudogap for T BKT < T < T pair. The normal state is at T > T pair . So we predict it is possible to transition between normal, pseudogap and BKT phases at ∼ 20nK by selecting V 0 = 150nK, n Ryd = 34 andᾱ < 0.09961 to get λ < ∼ 5. For lower temperatures, this can be done at smaller λ. Thus the proposed simulator offers a route to the (as yet) unexplored physics of boson-mediated pairing and condensation of fermions in cold atom quantum simulators.