Quantum Simulation with Fluxonium Qutrit Arrays

TL;DR

The paper proposes a realistic route to quantum simulate strongly correlated bosonic matter using arrays of fluxonium qutrits. By mapping circuit Hamiltonians to a truncated three-level per-site model, it reveals a rich extended Bose-Hubbard–like framework with correlated hopping, pair hopping, and nonlocal density interactions, tunable via external flux. Through Gutzwiller mean-field theory and exact diagonalization, it uncovers a diverse ground-state phase diagram featuring SF, MI, PSF, PSF*, PCB, and clustering phases, and discusses dynamical protocols to identify regimes without full ground-state cooling. The work argues for experimental feasibility with current fluxonium technology and points to avenues for exploring lattice gauge theories, Pfaffian states, and non-Abelian topological order using qutrit-based quantum simulators.

Abstract

Fluxonium superconducting circuits were originally proposed to realize highly coherent qubits. In this work, we explore how these circuits can be used to implement and harness qutrits, by tuning their energy levels and matrix elements via an external flux bias. In particular, we investigate the distinctive features of arrays of fluxonium qutrits, and their potential for the quantum simulation of exotic quantum matter. We identify four different operational regimes, classified according to the plasmon-like versus fluxon-like nature of the qutrit excitations. Highly tunable on-site interactions are complemented by correlated single-particle hopping, pair hopping and non-local interactions, which naturally emerge and have different weights in the four regimes. Dispersive corrections and decoherence are also analyzed. We investigate the rich ground-state phase diagram of qutrit arrays and propose practical dynamical experiments to probe the different regimes. Altogether, fluxonium qutrit arrays emerge as a versatile and experimentally accessible platform to explore strongly correlated bosonic matter beyond the Bose-Hubbard paradigm, and with a potential toward simulating lattice gauge theories and non-Abelian topological states.

Figures (18)

• Figure 1: Sketch of two fluxonium circuits $i$ and $j$ capacitatively coupled. The operators $\hat{n}_j, \hat{\phi}_j$ describe the number and superconducting phase of the Cooper pairs with respect to the upper island of the circuit. The main control parameter is provided by the external magnetic flux $\Phi$.
• Figure 2: Qutrits arising in four different regimes of $H_{\rm at}$ are reported. The black solid line depicts the Josephson potential $V(\phi)$. The red dashed lines represent the energy levels of the qutrit, and the solid red curves sketch the wavefunction of the states (not normalized). The blue dashed lines indicate the other levels of $H_{\rm at}$, which are off resonant with respect to the transitions at energy $\omega_{10}=\omega_{21}$ (in particular, the red dotted line provides a visual hint of the detuning). The qutrit levels are labeled by their photon number $a=0,1,2$, while we used barred numbers to index the full spectrum of the circuit. Panels (a-d) correspond to the $\Pi\Pi$, $\Phi\Phi$, $\Pi\Phi$ and $\Phi\Pi$ qutrits. The names illustrate the nature of the $0\leftrightarrow1$ and $1\leftrightarrow 2$ excitations of the qutrit. Notice that state $|2\rangle=|\bar{3}\rangle$ in the $\Pi\Phi$ qutrit and $\Phi\Pi$ qutrit, since the second excited state of the circuit $|\bar{2}\rangle$ is off-resonant.
• Figure 3: Qutrit parameters for different inductive energies $E_L$ and Josepshon junction energies $E_J$. In this plot, the qutrit levels 0,1,2 correspond to the three lowest eigenstates of $H_{\rm at}$. In panel (a), we plot the detuning $\delta$ of the closest off-resonant level. The black dotted line indicates where $\delta$ crosses zero. Panel (b) reports the degree of correlation of single-particle hopping $\alpha$. Panel (c) illustrates the relevance of pair-hopping processes through $P/J$ (in log scale). For reference, the black dashes indicate the line where $P/J=1$. Panel (d) reports the dephasing time due to noise in the external flux bias. In all four panels, the light green pentagon corresponds to the $\Pi\Pi$ qutrit of Fig. \ref{['fig:inductance_qutrits']}.(a), and the cross to the $\Phi\Phi$ qutrit of Fig. \ref{['fig:inductance_qutrits']}.(b).
• Figure 4: Difference between the single-particle coherence $g^{(1)}= \langle b^\dagger_i \ b_j \rangle$ and the pair coherence $g_{\rm pair}^{(1)}= \langle (b^\dagger_i)^2 \ b_j^2 \rangle$, calculated for the ground-state of $H_{\alpha\Delta}$ at unit filling $n=1$, within the Gutzwiller approximation. We can distinguish a standard superfluid (SF), a pair superfluid with small stiffness (PSF$^*$) and a Mott insulator phase (MI) phase. Below the cyan dashed line we have $\psi_1=0$, while $\psi_0=\psi_2=0$ below the orange dots. The red hatched area is defined by the thermodynamic instability condition $\frac{d^2 e}{dn^2} <0$.
• Figure 5: Difference between single-particle coherence $g^{(1)}$ and pair coherence $g_{\rm pair}^{(1)}$ in the ground-state of $H$, as a function of the hopping correlation $\alpha$ and of the pair-hopping strength $P/J$. Here, we take $\Delta=2.5zJ$ and unit density $n=1$, and perform the calculations within the Gutzwiller approximation. We can distinguish a standard superfluid phase (SF, green region), a pair superfluid (PSF, pink area) and a Mott insulator (MI, dark region). The cyan dashed line separates the SF from the PSF, the orange dots the SF from the MI, and the solid light green line the MI from the PSF. Analytical expressions for these transition lines are reported in the text.