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arXiv:2408.08559

Stability of Majorana zero modes with quantum optical lattices

Santiago F. Caballero-Benítez

Published Aug 16, 2024• 0 citations
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Abstract

I analyze the emergence of Majorana zero modes (MZM) in a one dimensional ultracold fermionic system confined by an optical lattice inside a high-Q cavity. This forms a quantum optical lattice due to the cavity backaction, with emergent long range interactions controlled by the light pumped into the system and thus long range pairing. I investigate the possibility of formation and emergence of edge modes using exact diagonalization and singular value decomposition of the Hamiltonian in the Majorana representation, while computing the mass gap of the MZM and its spectral properties. The interplay of cavity induced interactions and effective $p$-wave short range interactions controls the emergence of the signatures of MZM. I find that under certain conditions MZM distinctively emerge leading to a superradiant topological phase of matter. These MZM have potential applications for quantum information as they are topologically protected analogous to the behaviour of the Kitaev chain. The techniques employed lead to a scheme based on the Majorana representation and exact diagonalization with self consistent conditions that can be applied to other fermionic and analog systems termed Light-Matter-Majorana.

DOI
10.48550/arxiv.2408.08559

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I analyze the emergence of Majorana zero modes (MZM) in a one dimensional ultracold fermionic system confined by an optical lattice inside a high-Q cavity. This forms a quantum optical lattice due to the cavity backaction, with emergent long range interactions controlled by the light pumped into the system and thus long range pairing. I investigate the possibility of formation and emergence of edge modes using exact diagonalization and singular value decomposition of the Hamiltonian in the Majorana representation, while computing the mass gap of the MZM and its spectral properties. The interplay of cavity induced interactions and effective ppp-wave short range interactions controls the emergence of the signatures of MZM. I find that under certain conditions MZM distinctively emerge leading to a superradiant topological phase of matter. These MZM have potential applications for quantum information as they are topologically protected analogous to the behaviour of the Kitaev chain. The techniques employed lead to a scheme based on the Majorana representation and exact diagonalization with self consistent conditions that can be applied to other fermionic and analog systems termed Light-Matter-Majorana.
Abstract I analyze the emergence of Majorana zero modes (MZM) in a one dimensional ultracold fermionic system confined by an optical lattice inside a high-Q cavity. This forms a quantum optical lattice due to the cavity backaction, with emergent long range interactions controlled by the light pumped into the system and thus long range pairing. I investigate the possibility of formation and emergence of edge modes using exact diagonalization with self-consistent order parameters and singular value decomposition of the Hamiltonian in the Majorana representation. I analyze the spectral properties of the system, the mass gap and amplitude of the MZM with a unified framework. The interplay of cavity induced interactions and effective p -wave short range interactions controls the emergence of the signatures of MZM. I find that under certain conditions MZM distinctively emerge leading to superradiant topological phases of matter. These MZM have potential applications for quantum information as they are topologically protected analogous to the behavior of the Kitaev chain. The techniques employed lead to a scheme based on the Majorana representation and exact diagonalization with self consistent conditions using Hartree–Fock–Bogoliubov theory in real space for finite systems that can be applied to other fermionic and analog systems termed Light-Matter-Majorana. A promising landscape for the exploration of short range and long range topological order with superradiance is found.
Abstract Topological excitations, such as Majorana zero modes, are a promising route for encoding quantum information. Topologically protected gates of Majorana qubits, based on their braiding, will require some form of network. Here, we propose to build such a network by entangling Majorana matter with light in a microwave cavity QED set-up. Our scheme exploits a light-induced interaction which is universal to all the Majorana nanoscale circuit platforms. This effect stems from a parametric drive of the light-matter coupling in a one-dimensional chain of physical Majorana modes. Our set-up enables all the basic operations needed in a Majorana quantum computing platform such as fusing, braiding, the crucial T-gate, the read-out, and importantly, the stabilization or correction of the physical Majorana modes.
Quantum matter at ultralow temperatures offers a test bed for analyzing and controlling desired properties in strongly correlated systems. Under typical conditions the nature of the atoms fixes the magnetic character of the system. Beyond classical light potentials leading to optical lattices and short-range interactions, high-Q cavities introduce novel dynamics into the system via the quantumness of light. Here we propose a theoretical model and we analyze it using exact diagonalization and density matrix renormalization group simulations. We explore the effects of cavity mediated long-range magnetic interactions and optical lattices in ultracold matter. We find that global interactions modify the underlying magnetic character of the system while introducing competition scenarios. Antiferromagnetic correlated bosonic matter emerges in conditions beyond what nature typically provides. These allow new alternatives toward the design of robust mechanisms for quantum information purposes, exploiting the properties of magnetic phases of strongly correlated quantum matter.
Long-range interactions can fundamentally alter properties in gapped topological phases such as emergent massive edge modes. However, recent research has shifted attention to topological nontrivial critical points or phases, and it is natural to explore how long-range interactions influence them. In this work, we investigate the topological behavior at the quantum critical point of extended Kitaev chains with long-range interactions, which can be derived from the critical Ising model via the Jordan-Wigner transformation in the short-range limit. Specifically, we analytically find that the critical edge modes at the critical point remain stable against long-range interactions. More importantly, we observe that these critical edge modes remain massless even when long- range interactions become very strong. As a by-product, we numerically find that the critical behavior of the long-range model belongs to the free-Majorana-fermion universality class, which is entirely different from the long-range universality class in the usual long-range spin models. Our work could shed new light on the interplay between long-range interactions (frustrated) and the gapless topological phases of matter.
Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.
Abstract We investigate the effects of coupling a local electromagnetic cavity to a segment of a topological Kitaev chain (KC), with particular emphasis on the interplay between photons and Majorana zero modes (MZMs). In addition to the well-known scissor effect-which effectively partitions the chain and isolates free MZMs in the bulk-we provide evidence of non-trivial fusion rules and braiding operations, hallmark signatures of non-Abelian anyons, enabled by spatially selective ultrastrong KC-cavity coupling. We propose that these distinctive MZM properties can be experimentally probed via fermionic parity measurements and photon-induced Berry phases. Furthermore, we demonstrate that, in the so-called sweet-spot regime, the coupled system can be mapped onto a Rabi-like model with a homodyne-rotated quadrature, offering a simplified yet powerful theoretical description. Exploiting the symmetry of fermionic modes within a two-site cavity configuration, we also show the feasibility of generating hybrid MZM-polariton Schrödinger cat states. Our findings offer a novel approach to manipulating topological quantum matter through local light–matter interactions and provide theoretical tools for future experimental realizations in platforms such as quantum materials or superconducting circuits.