No experiment could directly test the particle/antiparticle duality of Majorana fermions, so far. However, this property represents a necessary ingredient towards the realization of topological quantum computing schemes. Here, we show how to complete this task by using microwave techniques. The direct coupling between a pair of overlapping Majorana bound states and the electric field from a microwave cavity is extremely difficult to detect due to the self-adjoint character of Majorana fermions which forbids direct energy exchanges with the cavity. We show theoretically how this problem can be circumvented by using photo-assisted tunneling to fermionic reservoirs. The absence of direct microwave transition inside the Majorana pair in spite of the light-Majorana coupling would represent a smoking gun for the Majorana self-adjoint character.
Quantum simulation aims to simulate a quantum system using a controble laboratory system that underline the same mathematical model. Cavity QED lattice system is that prescribe system to simulate the relativistic quantum effect. We quantum simulate the Dirac fermion mode, Majorana fermion mode and Majorana-Weyl fermion mode and a crossover between them in cavity QED lattice. We also present the different analytical relations between the field operators for different mode excitations.
We study the spectrum of a one-dimensional Kitaev chain placed in a microwave cavity. In the off-resonant regime, the frequency shift of the cavity can be used to identify the topological phase transition of the coupled system. In the resonant regime, the topology of the system can be controlled via the microwave cavity occupation and, moreover, for a large number of photons (classical limit), the physics becomes similar to that of periodically-driven systems (Floquet insulators). We also analyze numerically a finite chain and show the existence of a degenerate subspace in the presence of the …
We present the results of low lying excitation of coupled optical cavity arrays. We derive the Dirac equation for this system and explain the existence of Majorana fermion mode in the system. We present quite a few analytical relations between the Rabi frequency oscillation and the atom-photon coupling strength to achieve the different physical situation of our study and also the condition for massless excitation in the system. We present several analytical relations between the Dirac spinor field, order and disorder operators for our systems. We also show that the Luttinger liquid physics is …
A trijunction made of three topological semiconducting wires, each supporting a Majorana bound state at its two extremities, appears as one of the simplest geometry in order to perform braiding of Majorana fermions. By embedding the trijunction into a microwave cavity allows to study the intricate dynamics of the low-energy Majorana bound states (MBSs) coupled to the cavity electric field under a braiding operation. Extending a previous work (Phys. Rev. Lett. 2019, 122, 236803), the full time evolution of the density matrix of the low-energy states, including various relaxation channels, is co…
We study the dynamical process of braiding Majorana bound states in the presence of the coupling to photons in a microwave cavity. We show theoretically that the $π/4$ phase associated with the braiding of Majoranas, as well as the parity of the ground state are imprinted into the photonic field of the cavity, which can be detected by dispersive readouts techniques. These manifestations are purely dynamical, they occur in the absence of any splitting of the MBS that are exchanged, and they disappear in the static setups studied previously. Conversely, the cavity can affect the braiding phase, …
Coupling a semiconducting nanowire to a microwave cavity provides a powerfull means to assess the presence or absence of isolated Majorana fermions in the nanowire. These exotic bound states can cause a significant cavity frequency shift but also a strong cavity nonlinearity leading for instance to light squeezing. The dependence of these effects on the nanowire gate voltages gives direct signatures of the unique properties of Majorana fermions, such as their self-adjoint character and their exponential confinement.
We found that QED in the lowest eigenstate of the microwave-cavity, capacitively coupled with Majorana populated Josephson junctions array, is described with an effective $ϕ^{4}$ model emerging in the finite domain of the system parameters. Strong anharmonicity of the cavity mode arises due to electric-dipole phase transition facilitated by the single-electron zero mode and Cooper pair tunneling between superconducting islands. An instantonic meandering of the electromagnetic field potential at finite temperatures is predicted.
We investigate the ability of selective cavity coupling to a topological chain for tailoring the connectivity of Majorana fermions. We show how topological qubits (TQs), associated with non-local Majorana fermion pairing, can be moved from the edge to the bulk of a topological chain through selective access to light-matter interaction with specific physical sites. In particular, we present a comprehensive DMRG study of ground-state features in different chain-cavity coupling geometries and validate analytical insights in the strong coupling regime. This new kind of Majorana fermion correlation…
Distinguishing Majorana bound states from accidental zero-energy modes with a microwave cavity
International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences（波兰科学院物理研究所马格托国际研究中心）；CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎理工学院综合理工学院中心）；Department of Physics, University of Basel（巴塞尔大学物理系）
We study the optical properties of a hybrid device composed by a quantum dot (QD) resonantly coupled to a photonic mode of an optical microcavity and a Majorana nanowire: a topological superconducting segment hosting Majorana bound states (MBSs) at the opposite ends. In the regime of strong light-matter coupling, it is demonstrated that the leakage of the Majorana mode into the QD opens new optical transitions between polaritonic states formed due to hybridisation of material excitation with cavity photons, which leads to the reshaping of the Jaynes-Cummings ladder and can lead to the formatio…
Poor man's Majorana bound states in quantum dot based Kitaev chain coupled to a photonic cavity
CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院）；Institute of Fundamental Physics IFF-CSIC（西班牙高等科学研究理事会基础物理研究所）；JEIP, UAR 3573 CNRS, Collège de France, PSL Research University（法兰西公学院）
We study the transport of chiral Majorana edge modes (CMEMs) in a hybrid quantum anomalous Hall insulator-topological superconductor (QAHI-TSC) system in which the TSC region contains a Josephson junction and a cavity. The Josephson junction undergoes a topological transition when the magnetic flux through the cavity passes through half-integer multiples of magnetic flux quantum. For the trivial phase, the CMEMs transmit along the QAHI-TSC interface as without magnetic flux. However, for the nontrivial phase, a zero-energy Majorana state appears in the cavity, leading that a CMEM can resonantl…
We study one-dimensional $p$-wave superconductors capacitively coupled to a microwave stripline cavity. By probing the light exiting from the cavity, one can reveal the electronic susceptibility of the $p$-wave superconductor. We analyze two superconducting systems: the prototypical Kitaev chain, and a topological semiconducting wire. For both systems, we show that the photonic measurements, via the electronic susceptibility, allows us to determine the topological phase transition point, the emergence of the Majorana fermions, and the parity of their ground state. We show that all these effect…
We investigate the quantum electrodynamics of a device based on a topological superconducting circuit embedded in a microwave resonator. The device stores its quantum information in coherent superpositions of fermion parity states originating from Majorana fermion hybridization. This generates a highly isolated qubit whose coherence time could be greatly enhanced. We extend the conventional semiclassical method and obtain analytical derivations for strong transmon-photon coupling. Using this formalism, we develop protocols to initialize, control, and measure the parity states. We show that, re…
We propose microwave-controlled rotations for qubits realized as Majorana bound states. To this end we study an inhomogeneous Kitaev chain in a microwave cavity. The chain consists of two topologically nontrivial regions separated by a topologically trivial, gapped region. The Majorana bound states at the interfaces between the left (right) regions and the central region are coupled, and their energies are split by virtual cotunneling processes. The amplitude for these cotunneling processes decreases exponentially in the number of sites of the gapped region, and the decay length diverges as th…
We construct a minimal circuit, based on the top-transmon design, to rotate a qubit formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana qubit. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi oscillations of the Majorana qubit can be monitored via oscillations in the resona…
We theoretically study a one dimensional p-wave superconducting mesoscopic ring interrupted by a weak link and coupled inductively to a microwave cavity. We establish an input-output description for the cavity field in the presence of the ring, and identify the electronic contributions to the cavity response and their dependence on various parameters, such as the magnetic flux, chemical potential, and cavity frequency. We show that the cavity response is $4π$ periodic as a function of the magnetic flux in the topological region, stemming from the so called fractional Josephson current carried …
Majorana bound states have been proposed as building blocks for qubits on which certain operations can be performed in a topologically protected way using braiding. However, the set of these protected operations is not sufficient to realize universal quantum computing. We show that the electric field in a microwave cavity can induce Rabi oscillations between adjacent Majorana bound states. These oscillations can be used to implement an additional single-qubit gate. Supplemented with one braiding operation, this gate allows to perform arbitrary single-qubit operations.
We investigate the admittance of a metallic quantum RC circuit with a spinful single-channel lead or equally with two conducting spin-polarized channels, in which Majorana fermions play a crucial role in the charge dynamics. We address how the two-channel Kondo physics and its emergent Majoranas arise. The existence of a single unscreened Majorana mode results in non-Fermi-liquid features and we determine the universal crossover function describing the Fermi-liquid to non-Fermi-liquid region. Remarkably, the same universal form emerges both at weak transmission and large transmission. We find …
Understanding the interaction between cavity photons and electronic nanocircuits is crucial for the development of Mesoscopic Quantum Electrodynamics (QED). One has to combine ingredients from atomic Cavity QED, like orbital degrees of freedom, with tunneling physics and strong cavity field inhomogeneities, specific to superconducting circuit QED. It is therefore necessary to introduce a formalism which bridges between these two domains. We develop a general method based on a photonic pseudo-potential to describe the electric coupling between electrons in a nanocircuit and cavity photons. In t…
Motivated by a recent experiment [Nadj-Perge et al., Science 346, 602 (2014)] providing evidence for Majorana zero modes in iron chains on the superconducting Pb surface, in the present work, we theoretically propose an all-optical scheme to detect Majorana fermions, which is very different from the current tunneling measurement based on electrical means. The optical detection proposal consists of a quantum dot embedded in a nanomechanical resonator with optical pump-probe technology. With the optical means, the signal in the coherent optical spectrum presents a distinct signature for the exis…
In this letter we report the first direct observation of gap-less Majorana QPs which appear as Andreev bound states on the surface of superfluid $^3$He-B. We made the precise measurements of superfluid 3He-B heat capacity at the limit of extremely low temperatures. We have separated the heat capacity of bulk Bogolyubov QPs and the surface Majorana QPs by its different temperature dependence. We have found that at 0.11 mK the Majorana fermions constitute half of the bolometer heat capacity under the conditions of our experiments.
The ability to detect single photons has become increasingly essential due to the rise of photon-based quantum computing. In this theoretical work, we propose a system consisting of a quantum dot (QD) side-coupled to a superconducting nanowire. The coupling opens a gap in both the QD mode and the Majorana zero mode (MZM) at the nanowire edge, enabling photon absorption in the system. We show that the absorbed photoelectron decays via rapid (sub-nanosecond to nanosecond) nonradiative heat transfer to the nanowire phonon modes rather than by spontaneous emission. Furthermore, we calculate the te…
The apparent splitting of zero-bias conductance peaks, apparently observed in recent experiments concerning Majorana fermions in nanowires coupled to superconductors, can be interpreted as a manifestation of a transition in the structure from (massive) Majorana to (massless) Weyl fermions. A modification of the experiments in order to test such phenomenon is proposed by making recourse to the only possible electromagnetic interaction allowed to (massive) Majorana particles and mediated by their anapole moment. In suitably designed heterostructures with toroidal symmetry, the additional anapole…
Majorana zero modes (MZMs) promise a platform for topologically protected fermionic quantum computation. However, creating multiple MZMs and generating (directly or via measurements) the requisite transformations (e.g., braids) pose significant challenges. We introduce fermion-parity-based computation (FPBC): a measurement-based scheme, modeled on Pauli-based computation, that uses efficient classical processing to virtually increase the number of available MZMs and which, given magic state inputs, operates without transformations. FPBC requires all MZM parities to be measurable, but this conf…
It is generally thought that adiabatic exchange of two identical particles is impossible in one spatial dimension. Here we describe a simple protocol that permits adiabatic exchange of two Majorana fermions in a one-dimensional topological superconductor wire. The exchange relies on the concept of "Majorana shuttle" whereby a $π$ domain wall in the superconducting order parameter which hosts a pair of ancillary Majoranas delivers one zero mode across the wire while the other one tunnels in the opposite direction. The method requires some tuning of parameters and does not, therefore, enjoy the …
In this work, we find that Majorana fermions induce selective equal spin Andreev reflections (SESARs), in which incoming electrons with certain spin polarization in the lead are reflected as counter propagating holes with the same spin. The spin polarization direction of the electrons of this Andreev reflected channel is selected by the Majorana fermions. Moreover, electrons with opposite spin polarization are always reflected as electrons with unchanged spin. As a result, the charge current in the lead is spin-polarized. Therefore, a topological superconductor which supports Majorana fermions…
We investigate the Majorana fermions in a semiconductor nanostructure with two wires connected through a ring. The nanostructure is mirror symmetric and in the proximity of a superconductor. The Rashba spin-orbit coupling and a magnetic field parallel to the wires or perpendicular to the ring are included. Moreover, a magnetic flux is applied through the center of the ring, which makes the phase difference of the superconducting order parameters in the two wires being zero or $π$ due to the fluxoid quantization and the thermodynamic equilibrium of the supercurrent in the superconducting ring. …
We propose a setup to measure the lifetime of the parity of a pair of Majorana bound states. The proposed experiment has one edge Majorana state tunnel coupled to a quantum dot, which in turn is coupled to a metallic electrode. When the Majorana Fermions overlap, even a small relaxation rate qualitatively changes the non-linear transport spectrum, and for strong overlap the lifetime can be read off directly from the height of a current peak. This is important for the usage of Majorana Fermions as a platform for topological quantum computing, where the parity relaxation is a limiting factor.
We investigate the formation of Majorana fermions in finite-size graphene strips with open boundary conditions in both directions, in the presence of an in-plane magnetic field and in the proximity of a superconducting substrate. We show that for long enough strips the Majorana states can form in the presence of a Rashba-like spin-orbit coupling, as well as in the presence of an inhomogeneous magnetic field. We find that, unlike infinite graphene ribbons in which Majorana states arise solely close to the bottom of the band and the Van Hove singularities, for finite-size systems this can happen…
We study the proximity effect in a topological nanowire tunnel coupled to an s-wave superconducting substrate. We use a general Green's function approach that allows us to study the evolution of the Andreev bound states in the wire into Majorana fermions. We show that the strength of the tunnel coupling induces a topological transition in which the Majorana fermionic states can be destroyed when the coupling is very strong. Moreover, we provide a phenomenologial study of the effects of disorder in the superconductor on the formation of Majorana fermions. We note a non-trivial effect of a quasi…
We consider an interferometer based on artificially induced topological superconductivity and chiral 1D Majorana fermions. The (non-topological) superconducting island inducing the superconducting correlations in the topological substrate is assumed to be floating. This allows probing the physics of interfering Majorana modes via microwave response, i.e., the frequency dependent impedance between the island and the earth. Namely, charging and discharging of the island is controlled by the time-delayed interference of chiral Majorana excitations in both normal and Andreev channels. We argue tha…
We study a disordered superconducting nanowire, with broken time-reversal and spin-rotational symmetry, which can be driven into a topological phase with end Majorana bound states by an externally applied magnetic field. As a function of disorder strength, it is known that the Majorana nanowire has a delocalization quantum phase transition from a topologically nontrivial phase, which supports Majorana bound states, to a nontopological insulating phase without them. On both sides of the transition, the system is localized at zero energy albeit with very different topological properties. We expl…
We show that the topological phase transition for a Kitaev chain embedded in a cavity can be identified by measuring experimentally accessible photon observables such as the Fano factor and the cavity quadrature amplitudes. Moreover, based on density matrix renormalization group numerical calculations, endorsed by an analytical Gaussian approximation for the cavity state, we propose a direct link between those observables and quantum entropy singularities. We study two bipartite entanglement measures, the von Neumann and Rényi entanglement entropies, between light and matter subsystems. Even t…
Whereas identification of Cooper pair symmetry is the first and crucial step in the investigation of unconventional superconductors, only a few have been established so far because of its own difficulties. To solve this problem, we develop a theory for identification of pairing symmetry using knowledge of topological superconductivity. Establishing the multipole theory of emergent Majorana fermions in time-reversal-invariant topological superconductors, we discover a one-to-one correspondence between the electromagnetic response of Majorana fermions and Cooper pair symmetry. The emergent Major…
Considering Rashba quantum wires with a proximity-induced superconducting gap as physical realizations of Majorana fermions and quantum dots, we calculate the overlap of the Majorana wave functions with the local wave functions on the dot. We determine the spin-dependent tunneling amplitudes between these two localized states and show that we can tune into a fully spin polarized tunneling regime by changing the distance between dot and Majorana fermion. Upon directly applying this to the tunneling model Hamiltonian, we calculate the effective magnetic field on the quantum dot flanked by two Ma…
We study a realization of a 1d chain of Majorana bound states at the interfaces between alternating ferromagnetic and superconducting regions at a quantum spin Hall insulator edge. In the limit of well separated Majoranas, the system can be mapped to the transverse field Ising model. The disordered critical point can be reached by tuning the relative magnitude or phases of the ferromagnetic and superconducting order parameters. We compute the voltage dependence of the tunneling current from a metallic tip into the Majorana chain as a direct probe of the random critical state.
We investigate the Josephson current in a Fano-Josephson junction formed by the direct coupling between two topological superconducting wires and their indirect coupling via a quantum dot. It is found that when two Majorana zero modes respectively appear in the wires, the Fano interference causes abundant Josephson phase transition processes. What is notable is that in the presence of appropriate direct and indirect inter-wire couplings, the fractional Josephson effect disappears and then such a structure transforms into a $0$-phase normal Josephson junction. On the other hand, if finite coupl…
Motivated by recent proposals for the generation of Majorana fermions in semiconducting hybrid structures, we examine possible experimental fingerprints of such excitations. Whereas previous works mainly have focused on zero-energy states in vortex cores in this context, we demonstrate analytically an alternative route to detection of Majorana excitations in semiconducting hybrid structures: interface-bound states that may be probed directly via conductance spectroscopy or STM-measurements. We estimate the necessary experimental parameters required for observation of our predictions.
We study theoretically charge transport properties of normal metal (N) / ferromagnet insulator (FI) / superconductor (S) junction and S/FI/S junction formed on the surface of three-dimensional topological insulator (TI), where chiral Majorana mode (CMM) exists at FI/S interface. We find that CMM generated in N/FI/S and S/FI/S junctions are very sensitively controlled by the direction of the magnetization ${\bm m}$ in FI region. Especially, the current-phase relation of Josephson current in S/FI/S junctions has a phase shift neither 0 nor $π$, which can be tuned continuously by the component of…
We investigate the power-dependent photoluminescence spectra from a strongly coupled quantum dot-cavity system using a quantum master equation technique that accounts for incoherent pumping, pure dephasing, and fermion or boson statistics. Analytical spectra at the one-photon correlation level and the numerically exact multi-photon spectra for fermions are presented. We compare to recent experiments on a quantum dot-micropiller cavity system and show that an excellent fit to the data can be obtained by varying only the incoherent pump rates in direct correspondence with the experiments. Our th…
We discuss the thermoelectrical properties of nanowires hosting Majorana edge states. For a Majorana nanowire directly coupled to two normal reservoirs the thermopower always vanishes regardeless of the value of the Majorana hybridization. This situation changes drastically if we insert a quantum dot. Then, the dot Majorana side coupled system exhibits a different behavior for the thermopower depending on the Majorana hybridization parameter. Thermopower reverses its sign when the half fermionic state is fully developed. As long as the hybridization becomes finite the Seebeck coefficient behav…
We show that confined Majorana fermions can exist in nanowires with proximity induced s-wave superconducting pairing if the direction of an external magnetic field rotates along the wire. The system is equivalent to nanowires with Rashba-type spin-orbit coupling, with strength proportional to the derivative of the field angle. For realistic parameters, we demonstrate that a set of permanent magnets can bring a nearby nanowire into the topologically non-trivial phase with localized Majorana modes at its ends. Without the requirement of spin-orbit coupling this opens up for a new route for demon…
Using exact diagonalization in the spherical geometry, we investigate systems of localized quasiholes at nu=5/2 for interactions interpolating between the pure Coulomb and the three-body interaction for which the Moore-Read state is the exact ground state. We show that the charge e/4 quasihole can be easily localized by means of a delta-function pinning potential. Using a tuned smooth pinning potential, the quasihole radius can be limited to approximately three magnetic length units. For systems of two quasiholes, adiabatic continuity between the Moore-Read and the Coulomb limit holds for the …
We study the formation of Majorana states in quasi-1D and 2D square lattices with open boundary conditions, with general anisotropic Rashba coupling, in the presence of an applied Zeeman field and in the proximity of a superconductor. For systems in which the length of the system is very large (quasi-1D) we calculate analytically the exact topological invariant, and we find a rich phase diagram which is strongly dependent on the width of the system. We compare our results with previous results based on a few-band approximation. We also investigate numerically open 2D systems of finite length i…
In this work, we consider a single-level quantum dot (QD) and a Majorana bound state (MBS) placed at the end of a topological superconducting nanowire (TSW). Both are coupled to the continuum and do not have a direct connection between them. We addressed the behavior of MBS leaking phenomena and its consequences into the QD physics in non-interacting and Coulomb blockade regime. By employing Green's function formalism via the equation of motion procedure, we calculate the physical quantities of interest. Our results show that the leakage of the MBS into the continuum state is achieved and can …
The Majorana representation for spin operators enables efficient application of field-theoretical methods for the analysis of spin dynamics. Moreover, a wide class of spin correlation functions can be reduced to Majorana correlations of the same order, simplifying their calculation. For the spin-boson model, direct application of this method in the lowest order allows for a straightforward computation of the transverse-spin correlations, however, for the longitudinal-spin correlations it apparently fails in the long-time limit. Here we indicate the reason and discuss, how this method can be us…
Due to their nonlocality, qubits nested in Majorana bound states may be the key to realize decoherence-free quantum computation. Majorana bound states could be achieved at the ends of a one-dimensional topological superconductor. However, when the bound states couple directly to electron reservoirs their nonlocal correlation is quenched by local Andreev reflections. Here we propose a scheme to generate nonlocal noise cross correlation between two well-separated quantum dots, mediated by a pair of Majorana bound states. Both positive and negative cross correlations can be obtained by tuning the…
Time periodic modulations of the transverse field in the closed XY spin-1/2 chain generate a very rich dynamical phase diagram, with a hierarchy of topological phases characterized by differing numbers of Floquet-Majorana modes. We show that this rich phase diagram survives when the system is coupled to dissipative end reservoirs. Circumventing the obstacle of preparing and measuring quasi energy configurations endemic to Floquet-Majorana detection schemes, we show that stroboscopic heat transport and spin density are robust observables to detect both the dynamical phase transitions and Majora…
