Probing the many-body localization phase transition with superconducting circuits

Abstract

Chains of superconducting circuit devices provide a natural platform for studies of synthetic bosonic quantum matter. Motivated by the recent experimental progress in realizing disordered and interacting chains of superconducting transmon devices, we study the bosonic many-body localization phase transition using the methods of exact diagonalization as well as matrix product state dynamics. We estimate the location of transition separating the ergodic and the many-body localized phases as a function of the disorder strength and the many-body on-site interaction strength. The main difference between the bosonic model realized by superconducting circuits and similar fermionic model is that the effect of the on-site interaction is stronger due to the possibility of multiple excitations occupying the same site. The phase transition is found to be robust upon including longer-range hopping and interaction terms present in the experiments. Furthermore, we calculate experimentally relevant local observables and show that their temporal fluctuations can be used to distinguish between the dynamics of Anderson insulator, many-body localization, and delocalized phases. While we consider unitary dynamics, neglecting the effects of dissipation, decoherence and measurement back action, the timescales on which the dynamics is unitary are sufficient for observation of characteristic dynamics in the many-body localized phase. Moreover, the experimentally available disorder strength and interactions allow for tuning the many-body localization phase transition, thus making the arrays of superconducting circuit devices a promising platform for exploring localization physics and phase transition.

Phases of the disordered Bose-Hubbard model with attractive interactions

“…The disordered attractive Bose-Hubbard model with L sites is defined in the basis of the local bosonic annihilationâ , creationâ † , and occupation numbern =â † â operators by the Hamiltonian [15,16,33,34]…”

“…where |n 0 = N denotes the state where N bosons occupy the site 0 and other sites are empty. We refer to this as the localized state, not to be confused with Anderson localization of the noninteracting situation or the many-body localization of the highly excited states [15][16][17]. When we take into account the hopping up to first order in nondegenerate perturbation theory, the localized state |ψ 0 0 gets a correction of the form…”

“…The interaction term originates from an approximation of the cosinusoidal potential of the Josephson junctions of the transmons [26], with typical values of U/2π between 200 MHz and 300 MHz. The disorder ω models the small unintentional variations in manufactured devices, but it can also be artificially magnified or reduced by applying magnetic flux through the loop formed by the two parallel Josephson junctions of the transmons [15,16,26,33,34,36], yielding D/2π in the range from 100 kHz to 2 GHz. The experimentally realistic range of parameters belongs roughly to the interval J/U ∈ [0.03, 0.5] and D/U ∈ [10 −4 , 10 2 ], allowing the realization of all the ground-state phases of the disordered attractive Bose-Hubbard model.…”

“…The attractive Bose-Hubbard model has remained much less studied than its repulsive counterpart. At strong disorder, many-body localization emerges [15][16][17] for highly excited states. The quantum ground-state behavior changes dramatically when switching from repulsive to attractive interaction [18][19][20][21][22].…”

“…The size of experimentally demonstrated transmon arrays has grown rapidly from a few to over 50 sites [30][31][32][33][34][35][36][37][38]. Thus, an array of coupled transmons realizes the disordered attractive Bose-Hubbard model in a natural manner [16,33]. Furthermore, the attractive Bose-Hubbard model is immediately applicable also for cold atoms in optical lattices, where the interaction can be tuned from repulsive to attractive via the Feshbach resonance [4,39].…”