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This paper introduces a novel framework to study default dependence and systemic risk in a financial network that evolves over time. We analyse several indicators of risk, and develop a new latent space model to assess the health of key European banks before, during, and after the recent financial crises. First, we adopt the measure of CoRisk to determine the impact of such crises on the financial network. Then, we use minimum spanning trees to analyse the correlation structure and the centrality of the various banks. Finally, we propose a new statistical model that permits a latent space visualisation of the financial system. This provides a clear and interpretable model-based summary of the interaction data, and it gives a new perspective on the topology structure of the network. Crucially, the methodology provides a new approach to assess and understand the systemic risk associated with a financial system, and to study how debt may spread between institutions. Our dynamic framework provides an interpretable map that illustrates the default dependencies between institutions, highlighting the possible patterns of contagion and the institutions that may pose systemic threats., Comment: 40 pages

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot., Comment: 47 pages, 2 figures; v2: references added, conventions fixed; v3: published version

We present a detailed investigation of different excitonic states weakly confined in single GaAs/AlGaAs quantum dots obtained by the Al droplet-etching method. For our analysis we make use of temperature-, polarization- and magnetic field-dependent $\mu$-photoluminescence measurements, which allow us to identify different excited states of the quantum dot system. Besides that, we present a comprehensive analysis of g-factors and diamagnetic coefficients of charged and neutral excitonic states in Voigt and Faraday configuration. Supported by theoretical calculations by the Configuration interaction method, we show that the widely used single-particle Zeeman Hamiltonian cannot be used to extract reliable values of the g-factors of the constituent particles from excitonic transition measurements.

The appearance of single photon sources in atomically thin semiconductors holds great promises for the development of a flexible and ultra-compact quantum technology, in which elastic strain engineering can be used to tailor their emission properties. Here, we show a compact and hybrid 2D-semiconductor-piezoelectric device that allows for controlling the energy of single photons emitted by quantum emitters localized in wrinkled WSe2 monolayers. We demonstrate that strain fields exerted by the piezoelectric device can be used to tune the energy of localized excitons in WSe2 up to 18 meV in a reversible manner, while leaving the single photon purity unaffected over a wide range. Interestingly, we find that the magnitude and in particular the sign of the energy shift as a function of stress is emitter dependent. With the help of finite element simulations we suggest a simple model that explains our experimental observations and, furthermore, discloses that the type of strain (tensile or compressive) experienced by the quantum emitters strongly depends on their localization across the wrinkles. Our findings are of strong relevance for the practical implementation of single photon devices based on two-dimensional materials as well as for understanding the effects of strain on their emission properties.

Scalable quantum photonic architectures demand highly efficient, high-purity single-photon sources, which can be frequency matched via external tuning. We demonstrate a single-photon source based on an InAs quantum dot embedded in a micropillar resonator, which is frequency tunable via externally-applied stress. Our platform combines the advantages of a Bragg micropillar cavity and the piezo-strain-tuning technique enabling single photon spontaneous emission enhancement via the Purcell effect and quantum dot (QD) with tunable wavelength. Our optomechanical platform has been implemented by integration of semiconductor-based QD-micropillars on a piezoelectric substrate. The fabricated device exhibits spontaneous emission enhancement with a Purcell factor of 4.4$\pm$0.7 and allows for a pure triggered single-photon generation with $g^{(2)}(0)$ < 0.07 under resonant excitation. A quantum dot emission energy tuning range of 0.75 meV for 27 kV/cm applied to the piezo substrate has been achieved. Our results pave the way towards the scalable implementation of single-photon quantum photonic technologies using optoelectronic devices., Comment: 16 pages, 4 figures

The optomechanical coupling of quantum dots and flexural mechanical modes is studied in suspended nanophononic strings. The investigated devices are designed and monolithically fabricated on an (Al)GaAs heterostructure. Radio frequency elastic waves with frequencies ranging between $f$=250 MHz to 400 MHz are generated as Rayleigh surface acoustic waves on the unpatterned substrate and injected as Lamb waves in the nanophononic string. Quantum dots inside the nanophononic string exhibit a 15-fold enhanced optomechanical modulation compared to those dynamically strained by the Rayleigh surface acoustic wave. Detailed finite element simulations of the phononic mode spectrum of the nanophononic string confirm, that the observed modulation arises from valence band deformation potential coupling via shear strain. The corresponding optomechanical coupling parameter is quantified to $0.15 \mathrm{meV nm^{-1}}$. This value exceeds that reported for vibrating nanorods by approximately one order of magnitude at 100 times higher frequencies. Using this value, a derive vertical displacements in the range of 10 nm is deduced from the experimentally observed modulation. The results represent an important step towards the creation of large scale optomechanical circuits interfacing single optically active quantum dots with optical and mechanical waves., Comment: Submitted manuscript

Semiconductor devices are strong competitors in the race for the development of quantum com-putational systems. In this work, we interface two semiconductor building blocks of different di-mensionality and with complementary properties: (1) a quantum dot hosting a single exciton andacting as a nearly ideal single-photon emitter and (2) a quantum well in a 2D microcavity sustain-ing polaritons, which are known for their strong interactions and unique hydrodynamics propertiesincluding ultrafast real-time monitoring of their propagation and phase-mapping. In the presentexperiment we can thus observe how the injected single particles propagate and evolve inside themicrocavity, giving rise to hydrodynamics features typical of macroscopic systems despite their in-trinsic genuine quantum nature. In the presence of a structural defect, we observe the celebratedquantum interference of a single particle that produces fringes reminiscent of a wave propagation.While this behaviour could be theoretically expected, our imaging of such an interference pattern,together with a measurement of antibunching, constitutes the first demonstration of spatial mappingof the self-interference of a single quantum particle hitting an obstacle., Comment: 9 pages, 4 figures, supplementary 4 pages, 6 figures

We investigate nonlinear dispersive mode coupling between the flexural in- and out-of-plane modes of two doubly clamped, nanomechanical silicon nitride string resonators. As the amplitude of one mode transitions from the linear response regime into the nonlinear regime, we find a frequency shift of two other modes. The resonators are strongly elastically coupled via a shared clamping point and can be tuned in and out of resonance dielectrically, giving rise to multimode avoided crossings. When the modes start hybridizing, their polarization changes. This affects the nonlinear dispersive coupling in a non-trivial way. We propose a theoretical model to describe the dependence of the dispersive coupling on the mode hybridization., Comment: 6 pages, 6 figures

We discuss two theoretical proposals for controlling the nonequilibrium steady state of nanomechanical resonators using quantum electronic transport. Specifically?, we analyse two approaches to achieve the ground-state cooling of the mechanical vibration coupled to a quantum dot embedded between (i) spin-polarised contacts or (ii) a normal metal and a superconducting contact. Assuming a suitable coupling between the vibrational modes and the charge or spin of the electrons in the quantum dot, we show that ground-state cooling of the mechanical oscillator is within the state of the art for suspended carbon nanotube quantum dots operating as electromechanical devices., Comment: 11 pages, 2 figures

We analyze the N=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. We find fourth-order linear modular differential equations for the vacuum characters/Schur indices of these theories, which are again uniform across the full family of theories and parameterized by the dual Coxeter number. We comment briefly on expectations for the still higher-rank cases., Comment: 30 pages; v2: minor typos corrected, published version

Hybrid quantum dot-oscillator systems have become attractive platforms to inspect quantum coherence effects at the nanoscale. Here, we investigate a Cooper-pair splitter setup consisting of two quantum dots, each linearly coupled to a local resonator. The latter can be realized either by a microwave cavity or a nanomechanical resonator. Focusing on the subgap regime, we demonstrate that cross-Andreev reflection, through which Cooper pairs are split into both dots, can efficiently cool down simultaneously both resonators into their ground state. Moreover, we show that a nonlocal heat transfer between the two resonators is activated when opportune resonance conditions are matched. The proposed scheme can act as a heat-pump device with potential applications in heat control and cooling of mesoscopic quantum resonators., Comment: 9 pages (including Appendix), 3 figures; added Journal ref

We revisit the calculation of holographic correlators in $AdS_3$. We develop new methods to evaluate exchange Witten diagrams, resolving some technical difficulties that prevent a straightforward application of the methods used in higher dimensions. We perform detailed calculations in the $AdS_3 \times S^3 \times K3$ background. We find strong evidence that four-point tree-level correlators of KK modes of the tensor multiplets enjoy a hidden 6d conformal symmetry. The correlators can all be packaged into a single generating function, related to the 6d flat space superamplitude. This generalizes an analogous structure found in $AdS_5 \times S^5$ supergravity., Comment: 29 pages

Hybrid systems consisting of a quantum emitter coupled to a mechanical oscillator are receiving increasing attention for fundamental science and potential applications in quantum technologies. In contrast to most of the presented works, in which the oscillator eigenfrequencies are irreversibly determined by the fabrication process, we present here a simple approach to obtain frequency-tunable mechanical resonators based on suspended nanomembranes. The method relies on a micromachined piezoelectric actuator, which we use both to drive resonant oscillations of a suspended Ga(Al)As membrane with embedded quantum dots and to fine tune their mechanical eigenfrequencies. Specifically, we excite oscillations with frequencies of at least 60 MHz by applying an AC voltage to the actuator and tune the eigenfrequencies by at least 25 times their linewidth by continuously varying the elastic stress state in the membranes through a DC voltage. The light emitted by optically excited quantum dots is used as sensitive local strain gauge to monitor the oscillation frequency and amplitude. We expect that our method has the potential to be applicable to other optomechanical systems based on dielectric and semiconductor membranes possibly operating in the quantum regime., Comment: 17 pages, 4 figures

Photon emission by tunneling electrons can be encouraged by locating a resonator close to the tunnel junction and applying an appropriate voltage-bias. However, studies of normal metals show that the resonator also affects how the charges flow, facilitating processes in which correlated tunneling of two charges produces one photon. We develop a theory to analyze this kind of behavior in Josephson junctions by deriving an effective Hamiltonian describing processes where two Cooper-pairs generate a single photon. We determine the conditions under which the transport is dominated by incoherent tunneling of two Cooper-pairs, whilst also uncovering a regime of coherent double Cooper-pair tunneling. We show that the system can also display an unusual form of photon-blockade and hence could serve as a single-photon source.

We establish a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra $U(1)^c$ maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in $d=2c$ dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For $c=4$ and $c=12$, these functionals exactly reproduce the "magic functions" used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. The same functionals are also applied to general 2D CFTs, with only Virasoro symmetry. In the limit of large central charge, we relate sphere packing to bounds on the black hole spectrum in 3D quantum gravity, and prove analytically that any such theory must have a nontrivial primary state of dimension $\Delta_0 \lesssim c/8.503$., Comment: 70 pages, 7 figures

We study a single-level quantum dot strongly coupled to a superconducting lead and tunnel-coupled to a normal electrode which can exchange energy with a single-mode resonator. We show that a such system can sustain lasing characterized by a sub-Poissonian Fock-state distribution of the resonator. The lasing regime is clearly identifiable in the subgap transport regime: in the resonant case, the current is pinned to the maximum value achievable in this hybrid nanostructure., Comment: 5 pages, 4 figures

Hybrid interfaces between distinct quantum systems play a major role in the implementation of quantum networks. Quantum states have to be stored in memories to synchronize the photon arrival times for entanglement swapping by projective measurements in quantum repeaters or for entanglement purification. Here, we analyze the distortion of a single photon wave packet propagating through a dispersive and absorptive medium with high spectral resolution. Single photons are generated from a single In(Ga)As quantum dot with its excitonic transition precisely set relative to the Cesium D1 transition. The delay of spectral components of the single photon wave packet with almost Fourier-limited width is investigated in detail with a 200 MHz narrow-band monolithic Fabry-P\'erot resonator. Reflecting the excited state hyperfine structure of Cesium, "slow light" and "fast light" behavior is observed. As a step towards room-temperature alkali vapor memories, quantum dot photons are delayed for 5 ns by strong dispersion between the two 1.17 GHz hyperfine-split excited state transitions. Based on optical pumping on the hyperfine-split ground states, we propose a simple, all-optically controllable delay for synchronization of heralded narrow-band photons in a quantum network.

Employing the concept of two-mode squeezed states from quantum optics, we demonstrate a revealing physical picture for the antiferromagnetic ground state and excitations. Superimposed on a N{\'e}el ordered configuration, a spin-flip restricted to one of the sublattices is called a sublattice-magnon. We show that an antiferromagnetic spin-up magnon is comprised by a quantum superposition of states with $n+1$ spin-up and $n$ spin-down sublattice-magnons, and is thus an enormous excitation despite its unit net spin. Consequently, its large sublattice-spin can amplify its coupling to other excitations. Employing von Neumann entropy as a measure, we show that the antiferromagnetic eigenmodes manifest a high degree of entanglement between the two sublattices, thereby establishing antiferromagnets as reservoirs for strong quantum correlations. Based on these novel insights, we outline strategies for exploiting the strong quantum character of antiferromagetic (squeezed-)magnons and give an intuitive explanation for recent experimental and theoretical findings in antiferromagnetic magnon spintronics.

Light emission from solid-state quantum emitters is inherently prone to environmental decoherence, which results in an inhomogeneous line broadening and in the deterioration of photon indistinguishability. Here we employ photon correlation Fourier spectroscopy (PCFS) to study the temporal evolution of such a broadening for the biexciton and exciton emission in resonantly driven GaAs quantum dots. Differently from previous experiments, the time scales we probe range from a few nanoseconds to milliseconds and, simultaneously, the spectral resolution we achieve can be as small as 2 $\mu$eV. We find pronounced differences in the temporal evolution of the two lines, which we attribute to differences in their homogeneous linewidth and sensitivity to charge noise. We then analyze the effect of irradiation with additional white light, which reduces blinking at the cost of enhanced charge noise. Due to its robustness against experimental imperfections and its high temporal resolution and bandwidth, PCFS outperforms established spectroscopy techniques, such as Michelson interferometry. We discuss its practical implementation, its limitations, and the possibility to use it to estimate the indistinguishability of consecutively emitted single photons for applications in quantum communication and photonic-based quantum information processing.

We propose a new mechanism of friction in resonantly driven vibrational systems. The form of the friction force follows from the time- and spatial-symmetry arguments. We consider a microscopic mechanism of this resonant force in nanomechanical systems. The friction can be negative, leading to an instability of forced vibrations of a nanoresonator and the onset of self-sustained oscillations in the rotating frame.

Because of the broken time-translation symmetry, in periodically driven vibrational systems fluctuations of different vibration components have different intensities. Fluctuations of one of the components are often squeezed, whereas fluctuations of the other component, which is shifted in phase by {\pi}/2, are increased. Squeezing is a multifaceted phenomenon; it attracts much attention from the perspective of high-precision measurements. Here we demonstrate a new and hitherto unappreciated side of squeezing: its direct manifestation in the spectra of driven vibrational systems. With a weakly damped nanomechanical resonator, we study the spectrum of thermal fluctuations of a resonantly driven nonlinear mode. In the attained sideband-resolved regime, we show that the asymmetry of the spectrum directly characterizes the squeezing. This opens a way to deduce squeezing of thermal fluctuations in strongly underdamped resonators, for which a direct determination by a standard homodyne measurement is impeded by frequency fluctuations. The experimental and theoretical results are in excellent agreement. We further extend the theory to also describe the spectral manifestation of squeezing of quantum fluctuations.

We present free field realizations for the associated vertex operator algebras of a number of four-dimensional $\mathcal{N}=2$ superconformal field theories. Our constructions utilize an exceptionally small set of chiral bosons whose number matches the complex dimensionality of the Higgs branch of the superconformal field theory. In the case of theories whose Higgs branches support additional degrees of freedom (free vector multiplets or decoupled interacting SCFTs), the corresponding "free field realizations" include additional ingredients: symplectic fermions in the case of vector multiplets and a $C_2$ co-finite VOA in the case of a residual interacting SCFT. The resulting picture is that the associated VOA can be constructed from the Higgs branch effective theory via free field realization. Our constructions also provide a natural realization of the $R$-filtration of the associated VOA., Comment: 31 pages. v2: minor typos fixed, references added

The generation of high-quality entangled photon pairs has been being a long-sought goal in modern quantum communication and computation. To date, the most widely-used entangled photon pairs are generated from spontaneous parametric downconversion, a process that is intrinsically probabilistic and thus relegated to a regime of low pair-generation rates. In contrast, semiconductor quantum dots can generate triggered entangled photon pairs via a cascaded radiative decay process, and do not suffer from any fundamental trade-off between source brightness and multi-pair generation. However, a source featuring simultaneously high photon-extraction efficiency, high-degree of entanglement fidelity and photon indistinguishability has not yet been reported. Here, we present an entangled photon pair source with high brightness and indistinguishability by deterministically embedding GaAs quantum dots in broadband photonic nanostructures that enable Purcell-enhanced emission. Our source produces entangled photon pairs with a record pair collection probability of up to 0.65(4) (single-photon extraction efficiency of 0.85(3)), entanglement fidelity of 0.88(2), and indistinguishabilities of 0.901(3) and 0.903(3), which immediately creates opportunities for advancing quantum photonic technologies., Comment: Submitted to Nature Nanotechnology on 9th March 2018 and accepted for publication on 1st March 2019. The formal acceptance of the manuscript was delayed because of the unfounded message (from a self-declared reviewer without a full/authorized access to the manuscript) addressed to the editorial board of the recipient journal after the completion of the peer review process

A semiconductor quantum dot (QD) can generate highly indistinguishable single-photons at a high rate. For application in quantum communication and integration in hybrid systems, control of the QD optical properties is essential. Understanding the connection between the optical properties of a QD and the growth process is therefore important. Here, we show for GaAs QDs, grown by infilling droplet-etched nano-holes, that the emission wavelength, the neutral-to-charged exciton splitting, and the diamagnetic shift are strongly correlated with the capture zone-area, an important concept from nucleation theory. We show that the capture-zone model applies to the growth of this system even in the limit of a low QD-density in which atoms diffuse over $\mu$m-distances. The strong correlations between the various QD parameters facilitate preselection of QDs for applications with specific requirements on the QD properties; they also suggest that a spectrally narrowed QD distribution will result if QD growth on a regular lattice can be achieved.

We study the vibrational motion of membrane resonators upon strong drive in the strongly nonlinear regime. By imaging the vibrational state of rectangular siliconnitride membrane resonators and by analyzing the frequency response using optical interferometry, we show that upon increasing the driving strength, the membrane adopts a peculiar deflection pattern formed by concentric rings superimposed onto the drum head shape of the fundamental mode. Such a circular symmetry cannot be described as a superposition of a small number of excited linear eigenmodes. Furthermore, the different parts of the membrane oscillate at different multiples of the drive frequency, an observation that we denominate as 'localization of overtones'. We introduce a phenomenological model that is based on the coupling of a very small number of effective nonlinear oscillators, representing the different parts of the membrane, and that describes the experimental observations.

Semiconductor quantum dots are converging towards the demanding requirements of photonic quantum technologies. Among different systems, quantum dots with dimensions exceeding the free-exciton Bohr radius are appealing because of their high oscillator strengths. While this property has received much attention in the context of cavity quantum electrodynamics, little is known about the degree of indistinguishability of single photons consecutively emitted by such dots and on the proper excitation schemes to achieve high indistinguishability. A prominent example is represented by GaAs quantum dots obtained by local droplet etching, which recently outperformed other systems as triggered sources of entangled photon pairs. On these dots, we compare different single-photon excitation mechanisms, and we find (i) a "phonon bottleneck" and poor indistinguishability for conventional excitation via excited states and (ii) photon indistinguishablilities above 90% for both strictly resonant and for incoherent acoustic- and optical-phonon-assisted excitation. Among the excitation schemes, optical phonon-assisted excitation enables straightforward laser rejection without a compromise on the source brightness together with a high photon indistinguishability.

We propose a new dynamic stochastic blockmodel that focuses on the analysis of interaction lengths in networks. The model does not rely on a discretization of the time dimension and may be used to analyze networks that evolve continuously over time. The framework relies on a clustering structure on the nodes, whereby two nodes belonging to the same latent group tend to create interactions and non-interactions of similar lengths. We introduce a fast variational expectation-maximization algorithm to perform inference, and adapt a widely used clustering criterion to perform model choice. Finally, we test our methodology on artificial data, and propose a demonstration on a dataset concerning face-to-face interactions between students in a high-school., Comment: 23 pages, 5 figures, 3 tables

Photonic quantum technologies call for scalable quantum light sources that can be integrated, while providing the end user with single and entangled photons on-demand. One promising candidate are strain free GaAs/AlGaAs quantum dots obtained by droplet etching. Such quantum dots exhibit ultra low multi-photon probability and an unprecedented degree of photon pair entanglement. However, different to commonly studied InGaAs/GaAs quantum dots obtained by the Stranski-Krastanow mode, photons with a near-unity indistinguishability from these quantum emitters have proven to be elusive so far. Here, we show on-demand generation of near-unity indistinguishable photons from these quantum emitters by exploring pulsed resonance fluorescence. Given the short intrinsic lifetime of excitons confined in the GaAs quantum dots, we show single photon indistinguishability with a raw visibility of $V_{raw}=(94.2\pm5.2)\,\%$, without the need for Purcell enhancement. Our results represent a milestone in the advance of GaAs quantum dots by demonstrating the final missing property standing in the way of using these emitters as a key component in quantum communication applications, e.g. as an entangled source for quantum repeater architectures.

Photonic entanglement swapping, the procedure of entangling photons without any direct interaction, is a fundamental test of quantum mechanics and an essential resource to the realization of quantum networks. Probabilistic sources of non-classical light can be used for entanglement swapping, but quantum communication technologies with device-independent functionalities demand for push-button operation that, in principle, can be implemented using single quantum emitters. This, however, turned out to be an extraordinary challenge due to the stringent requirements on the efficiency and purity of generation of entangled states. Here we tackle this challenge and show that pairs of polarization-entangled photons generated on-demand by a GaAs quantum dot can be used to successfully demonstrate all-photonic entanglement swapping. Moreover, we develop a theoretical model that provides quantitative insight on the critical figures of merit for the performance of the swapping procedure. This work shows that solid-state quantum emitters are mature for quantum networking and indicates a path for scaling up., Comment: The first four authors contributed equally to this work. 17 pages, 3 figures

We develop an analytic approach to Boundary Conformal Field Theory (BCFT), focussing on the two-point function of a general pair of scalar primary operators. The resulting crossing equation can be thought of as a vector equation in an infinite-dimensional space ${\cal V}$ of analytic functions of a single complex variable. We argue that in a unitary theory, functions in ${\cal V}$ satisfy a boundedness condition in the Regge limit. We identify a useful basis for ${\cal V}$, consisting of bulk and boundary conformal blocks with scaling dimensions which appear in OPEs of the mean field theory correlator. Our main achievement is an explicit expression for the action of the dual basis (the basis of liner functionals on ${\cal V}$) on an arbitrary conformal block. The practical merit of our basis is that it trivializes the study of perturbations around mean field theory. Our results are equivalent to a BCFT version of the Polyakov bootstrap. Our derivation of the expressions for the functionals relies on the identification of the Polyakov blocks with (suitably improved) boundary and bulk Witten exchange diagrams in $AdS_{d+1}$. We also provide another conceptual perspective on the Polyakov block expansion and the associated functionals, by deriving a new Lorentzian OPE inversion formula for BCFT., Comment: 67 pages, 8 figures

We study the decoherence and the relaxation dynamics of topological states in an extended class of quantum Ising chains which can present a manyfold ground state subspace. The leading interaction of the spins with the environment is assumed to be the local fluctuations of the transverse magnetic field. By deriving the Lindblad equation using the many-body states, we investigate the relation between decoherence, energy relaxation and topology. In particular, in the topological phase and at low temperature, we analyze the dephasing rates between the different degenerate ground states.