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Superradiance-like Electron Transport through a Quantum Dot

We theoretically show that intriguing features of coherent many-body physics can be observed in electron transport through a quantum dot (QD). We first derive a master equation based framework for electron transport in the Coulomb-blockade regime which includes hyperfine (HF) interaction with the nuclear spin ensemble in the QD. This general tool is then used to study the leakage current through a single QD in a transport setting. We find that, for an initially polarized nuclear system, the proposed setup leads to a strong current peak, in close analogy with superradiant emission of photons from atomic ensembles. This effect could be observed with realistic experimental parameters and would provide clear evidence of coherent HF dynamics of nuclear spin ensembles in QDs.


I. INTRODUCTION
Quantum coherence is at the very heart of many intriguing phenomena in today's nanostructures 1,2 . For example, it is the essential ingredient to the understanding of the famous Aharonov-Bohm like interference oscillations of the conductance of metallic rings 3 or the well-known conductance steps in quasi one-dimensional wires 4,5 . In particular, nonequilibrium electronic transport has emerged as a versatile tool to gain deep insights into the coherent quantum properties of mesoscopic solidstate devices 6,7 . Here, with the prospect of spintronics and applications in quantum computing, a great deal of research has been directed towards the interplay and feedback mechanisms between electron and nuclear spins in gate-based semiconductor quantum dots [8][9][10][11][12][13][14] . Current fluctuations have been assigned to the random dynamics of the ambient nuclear spins 15 and/or hysteresis effects due to dynamic nuclear polarization [15][16][17][18] . Spinflip mediated transport, realized in few-electron quantum dots in the so-called spin-blockade regime 19 , has been shown to exhibit long time scale oscillations and bistability as a result of a buildup and relaxation of nuclear polarization 15,16 . The nuclear spins are known to act collectively on the electron spin via hyperfine interaction. In principle, this opens up an exciting testbed for the observation of collective effects which play a remarkable role in a wide range of many-body physics [20][21][22] .
In Quantum Optics, the concept of superradiance, describing the cooperative emission of photons, is a paradigm example for a cooperative quantum effect 1,23,24 . Here, initially excited atoms emit photons collectively as a result of the buildup and reinforcement of strong interatomic correlations. Its most prominent feature is an emission intensity burst in which the system radiates much faster than an otherwise identical system of independent emitters. This phenomenon is of fundamental importance in quantum optics and has been studied extensively since its first prediction by Dicke in 1954 23 . Yet, in its original form the observation of optical superradiance has turned out to be difficult due to dephasing dipole-dipole van der Waals interactions, which suppress a coherence buildup in atomic ensembles. This paper is built upon analogies between mesoscopic solid-state physics and Quantum Optics: the nuclear spins surrounding a QD are identified with an atomic ensemble, individual nuclear spins corresponding to the internal levels of a single atom and the electrons are associated with photons. Despite some fundamental differences -for example, electrons are fermions, whereas photons are bosonic particles -this analogy stimulates conjectures about the potential occurence of related phenomena in these two fields of physics. Led by this line of thought, we will address the question if superradiant behaviour might also be observed in a solid-state environment where the role of photons is played by electrons.
To this end, we analyze a gate-based semiconductor QD in the Coulomb blockade regime, obtaining two main results, of both experimental and theoretical relevance: First, in analogy to superradiant emission of photons, we show how to observe superradiant emission of electrons in a transport setting through a QD. We demonstrate that the proposed setup, when tuned into the spin-blockade regime, carries clear fingerprints of cooperative emission, with no van der Waals dephasing mechanism on relevant timescales. The spin-blockade is lifted by the HF coupling which becomes increasingly more efficient as correlations among the nuclear spins build up. This markedly enhances the spin-flip rate and hence the leakage current running through the QD. Second, we develop a general theoretical master equation framework that describes the nuclear spin mediated transport through a single QD. Apart from the collective effects due to the HF interaction, the electronic tunneling current is shown to depend on the internal state of the ambient nuclear spins through the effective magnetic field (Overhauser field) produced by the hyperfine interaction.
The paper is structured as follows: In Sec. II, we highlight our key findings and provide an intuitive picture of our basic ideas, allowing the reader to grasp our main results on a qualitative level. By defining the underlying Hamiltonian, Sec. III then describes the system in a more rigorous fashion. Next, we present the first main result of this paper in Sec. IV: a general master equation for electron transport through a single QD which is coherently enhanced by the HF interaction with the ambient . Schematic illustration of the transport system: An electrically defined QD is tunnel-coupled to two electron reservoirs, the left and right lead respectively. A bias voltage eV = µL − µR is applied between the two leads in order to induce a current through the QD. An external magnetic field is used to tune the system into the sequentialtunneling regime and the QD effectively acts as an spin-filter. The resulting spin-blockade can be lifted by the HF interaction between the QD electron and the nuclear spins in the surrounding host environment.
nuclear spins in the QD. It features both collective effects and feedback mechanisms between the electronic and the nuclear subsystem of the QD. Based on this theoretical framework, Sec. V puts forward the second main result, namely the observation of superradiant behavior in the leakage current through a QD. The qualitative explanations provided in Sec. II should allow to read this part independently of the derivation given in Sec. IV. Sec. VI backs up our analytical predictions with numerical simulations. When starting from an initially polarized nuclear spin ensemble, the leakage current through the QD is shown to exhibit a strong peak whose relative height scales linearly with the number of nuclear spins, which we identify as the characteristic feature of superradiant behaviour. In Sec. VII we draw conclusions and give an outlook on future directions of research.

II. MAIN RESULTS
In this section we provide an intuitive exposition of our key ideas and summarize our main findings. We study a single electrically-defined QD in the Coulomb-blockade regime which is attached to two leads, as schematically depicted in Fig. 1. Formally, the Hamiltonian for the total system is given by Here, H Z describes the Zeeman splitting of the electronic levels inside the QD in presence of an external magnetic field. H B refers to two independent reservoirs of noninteracting electrons, the left and right lead respectively. The coupling between these and the QD is described in terms of a tunneling Hamiltonian H T and H HF models the collective interaction between an electron confined inside the QD and an ensemble of N proximal nuclear spins surrounding the QD. Note that the specific form of H will be given later on in Sec. III. Our analysis is built upon a Quantum master equation approach, a technique originally rooted in the field of Quantum Optics. By tracing out the unobserved degrees of freedom of the leads we derive an effective equation of motion for the density matrix of the QD system ρ S -describing the electron spin inside the QD as well as the nuclear spin ensemble -irreversibly coupled to source and drain electron reservoirs. In addition to the standard assumptions of a weak system-reservoir coupling (Born approximation), a flat reservoir spectral density, and a short reservoir correlation time (Markov approximation), we demand the hyperfine flip-flops to be strongly detuned with respect to the effective magnetic field seen by the electron throughout the dynamics. Under these conditions, the central master equation can be written aṡ where the tunneling rates α σ (t) and β σ (t) describe dissipative processes by which an electron of spin σ tunnels from one of the leads into or out of the QD, respectively. Here, the fermionic operator d † σ creates an electron of spin σ inside the QD. While a detailed derivation of Eqn. (2) along with the precise form of the tunneling rates is presented in Sec. IV, here we focus on a qualitative discussion of the theoretical and experimental implications thereof.
Our central master equation exhibits two core features: First, dissipation only acts on the electronic subsystem with rates α σ (t) and β σ (t) that depend dynamically on the state of the nuclear subsystem. This non-linear behavior potentially results in hysteretic behavior and feedback mechanisms between the two subsystems as already suggested theoretically 11,14,20,21 and observed in experiments in the context of double QDs in the Pauli-blockade regime; see e.g. Refs. 12,13,18 . Second, the collective nature of the HF interaction H HF allows for the observation of coherent many-body effects.
The effect of the hyperfine interaction between an electron inside the QD and the ambient nuclear spin ensemble is two-fold giving rise to the two main results outlined above: First, the nuclear spins provide an effective magnetic field for the electron spin, the Overhauser field, whose strength is proportional to the polarization of the nuclear spin ensemble. Thus, a changing nuclear polarization can either dynamically tune or detune the position of the electron levels inside the QD. This, in turn, can have a marked effect on the transport properties of the QD as they crucially depend on the position of these resonances with respect to the chemical potentials of the leads. In our model, this effect is directly captured by the tunneling rates dynamically depending on the state of the nuclei. Second, to show that this system supports the observation of intriguing, purely collective effects we refer to the following example: Consider a setting in which the bias voltage and an external magnetic field are tuned such that only one of the two electronic spin-components, say the level |↑ , lies inside the transport window. In this spin-blockade regime the electrons tunneling into the right lead are spin-polarized, i.e., the QD acts as an spin filter 32,34 . If the HF coupling is sufficiently small compared to the external Zeeman splitting, the electron is predominantly in its |↓ spin state allowing to adiabatically eliminate the electronic QD coordinates. In this way we obtain an effective equation of motion for the nuclear density operator µ only. It readṡ where A µ = N i=1 g i σ µ i with µ = +, −, z are collective nuclear spin operators, composed of all N individual nuclear spin operators σ µ i , with g i being proportional to the probability of the electron being at the location of the nucleus of site i. Again, we will highlight the core implications of Eqn.(3) and for a full derivation thereof, including the definition of the effective rates c r and c i , we refer to Sec. V. Most notably, Eqn.(3) closely resembles the superradiance master equation which has been discussed extensively in the context of atomic physics 24 and therefore similar effects might be expected.
Superradiance is known as a macroscopic collective phenomenon which generalizes spontaneous emission from a single emitter to a many-body system of N atoms 1 . Starting from a fully polarized initial state the system evolves within a totally symmetric subspace under permutation and experiences a strong correlation build-up. As a consequence, the emission intensity is not of the usual exponentially decaying form, but conversely features a sudden peak occuring on a very rapid timescale ∼ 1/N with a maximum ∼ N 2 .
In this paper, we show that the same type of cooperative emission can occur from an ensemble of nuclear spins surrounding an electrically-defined QD, a phenomenon we term as electronic superradiance: The spin-blockade can be lifted by the HF interaction as the nuclei pump excitations into the electron. When starting from a highly polarized, weakly correlated nuclear state, due to the collective nature of the HF interaction, this process becomes increasingly more efficient as correlations among the nuclei build up, directly giving rise to an increased leakage current. Therefore, the current is collectively enhanced   2. (color online). Normalized leakage current through a QD in the spin-blockade regime for N nuclear spins, initial nuclear polarization p and external Zeeman splitting ω0 in units of the total HF coupling constant AHF ≈ 100 µeV, summarized as (N, p, ω0/AHF). For homogeneous HF coupling the dynamics can be solved exactly (black dotted line). Compared to this idealized benchmark, the effects are reduced for realistic inhomogeneous HF coupling, but still present: The relative peak height becomes more pronounced for smaller detuning ω0 or higher polarization p (solid red line compared to the blue dashed and green dash-dotted line, respectively). Even under realistic conditions, the relative peak height is found to scale linearly with N , corresponding to a strong enhancement for typically N ≈ 10 5 − 10 6 . by the electron's HF interaction with the ambient nuclear spin ensemble giving rise to the pure many-body effect of electronic superradiance.
Compared to its conventional atomic counterpart, our system incorporates two major differences: First, our setup describes superradiant behaviour from a single emitter, since in the strong Coulomb-blockade regime the electrons are emitted antibunched. As described above, the superradiant character is due to the nuclear spins acting collectively on the electron spin leading to an increased leakage current on timescales longer than single electron tunneling events. The second crucial difference is the inhomogeneous nature (g i = const.) of the collective operators A µ . Accordingly, the collective spin is not conserved, leading to dephasing between the nuclei which in principle could prevent the observation of superradiant behavior. However, as exemplified in Fig. 2, we show that under realistic conditions -taking into account a finite initial polarization of nuclear spins p and dephasing processes due to the inhomogeneous nature of the HF coupling -the leakage current through the QD still exhibits the characteristic peak whose relative height will be shown to scale linealry with the number of nuclear spins. Even though the effect is reduced compared to the ideal atomic case, for an experimentally realistic number of nuclei N ≈ 10 5 −10 6 a strong increase is still predicted. The experimental key signature of this effect, the relative peak height of the leakage current, can be varied by ei-ther tuning the external Zeeman splitting or the initial polarization of the nuclear spins.
In the remainder of the paper, Eqn. (2) and Eqn.(3) will be derived from first principles; in particular, the underlying assumptions and approximations will be listed. Based on this general theoretical framework, more results along with detailed discussions will be presented. For both the idealized case of homogeneous HF coupling -in which an exact solution is feasible even for relatively large N -and the more realistic inhomogeneous case, further numerical simulations will prove the existence of a strong superradiant peaking in the leakage current of single QD in the spin-blockade regime.

III. THE SYSTEM
This section gives an in-depth description of the Hamiltonian under study, formally introduced in Eqn.(1). The system we consider consists of a single electrically-defined QD in a transport setting as schematically depicted in Fig. 1. Due to strong confinement only a single orbital level is relevant. Moreover, the QD is assumed to be in the strong Coulomb-blockade regime so that at maximum one electron resides inside the QD. Therefore, the effective Hilbert-space of the QD electron is span {|↑ , |↓ , |0 } where the lowest energy states for an additional electron in the QD with spin σ =↑, ↓ are split by an external magnetic field. The Hamiltonian for the total system is given in Eqn. (1).
Here, the first term H Z = σ σ d † σ d σ describes the electronic levels of the QD. The Zeeman splitting between the two spin components is ω 0 = ↑ − ↓ (we set = 1) and the QD electron operators are d † σ = |σ 0|, describing transitions from the state |0 with no electron inside the QD to a state |σ with one electron of spin σ inside the QD.
Electron transport through the QD is induced by attaching the QD to two electron leads (labeled as L and R) which are in thermal equilibrium at chemical potentials µ L and µ R , respectively. The leads themselves constitute reservoirs of non-interacting electrons where c † αkσ (c αkσ ) creates (annihilates) an electron in lead α = L, R with wavevector k and spin σ. The operators c † αkσ (c αkσ ) fulfill the usual Fermi commutation relations: {c † αkσ , c † α k σ } = {c αkσ , c α k σ } = 0 and {c † αkσ , c α k σ } = δ α,α δ k,k δ σ,σ . The effect of the Coulomb interaction in the leads can be taken into account by renormalized effective quasi-particle masses. A positive source-drain voltage eV = µ L − µ R leads to a dominant tunneling of electrons from left to right. Microscopically, the coupling of the QD system to the electron reservoirs is described in terms of the tunneling Hamil- with the tunnel matrix element T (α) k,σ specifying the transfer coupling between the lead α = L, R and the system. There is no direct coupling between the leads and electron transfer is only possible by charging and discharging the QD.
The cooperative effects are based on the the collective hyperfine interaction of the electronic spin of the QD with N initially polarized nuclear spins in the host environment of the QD 25 . It is dominated by the isotropic contact term 26 given by Here S µ and A µ = N i=1 g i σ µ i with µ = +, −, z denote electron and collective nuclear spin operators, respectively. The coupling coefficients are normalized such that i g 2 i = 1 and individual nuclear spin operators σ µ i are assumed to be spin 1/2 for simplicity; g is related to the total HF coupling strength A HF via g = A HF / i g i . We neglect the typically very small nuclear Zeeman and nuclear dipole-dipole terms 26 . For simplicity, we also restrict our analysis to one nuclear species only. These simplifications will be addressed in more detail in Sec. VI. The effect of the HF interaction with the nuclear spin ensemble is two-fold: The first part of the above Hamiltonian H ff = g 2 (A + S − + A − S + ) is a Jaynes-Cummingstype interaction which exchanges excitations between the QD electron and the nuclei. The second term H OH = gA z S z constitutes a quantum magnetic field, the Overhauser field, for the electron spin generated by the nuclei. If the Overhauser field is not negligible compared to the external Zeeman splitting, it can have a marked effect on the current by (de)tuning the hyperfine flip-flops.

IV. GENERALIZED QUANTUM MASTER EQUATION
Electron transport through a QD can be viewed as a tool to reveal the QD's nonequilibrium properties in terms of the current-voltage I/V characteristics. From a theoretical perspective, a great variety of methods such as the scattering matrix formalism 49 and non-equilibrium Green's functions 7,46 have been used to explore the I/V characteristics of quantum systems that are attached to two metal leads. Our analysis is built upon the master equation formalism, a tool widely used in quantum optics for studying the irreversible dynamics of quantum systems coupled to a macroscopic environment.
In what follows, we will employ a projection operator based technique to derive an effective master equation for the QD system -comprising the QD electron spin as well as the nuclear spins -which experiences dissipation via the electron's coupling to the leads. This dissipation is shown to dynamically depend on the state of the nuclear system potentially resulting in feedback mechanisms between the two subsystems. We will derive conditions which allow for a Markovian treatment of the problem and list the assumptions our master equation based framework is based on.
A. Superoperator Formalism -Nakajima-Zwanzig Equation The state of the global system that comprises the QD as well as the environment is represented by the full density matrix ρ (t). However, the actual states of interest are the states of the QD which are described by the reduced density matrix ρ S = Tr B [ρ], where Tr B . . . averages over the unobserved degrees of freedom of the Fermi leads. We will derive a master equation that governs the dynamics of the reduced density matrix ρ S using the superoperator formalism. We start out from the von Neumann equation for the full density matriẋ where H (t) can be decomposed into the following form which turns out to be convenient later on Here, H 0 (t) = H Z + H B + g A z t S z comprises the Zeeman splitting caused by the external magnetic field via H Z and the Hamiltonian of the non-interacting electrons in the leads H B ; moreover, the time-dependent expectation value of the Overhauser field has been absorbed into the definition of H 0 (t). The HF interaction between the QD electron and the ensemble of nuclear spins has been split up into the flip-flop term H ff and the Overhauser field H OH , that is H HF = H OH + H ff . The term H 1 (t) = H ∆OH (t) + H ff comprises the Jaynes-Cummings-type dynamics H ff and fluctuations due to deviations of the Overhauser field from its expectation value, i.e., H ∆OH (t) = gδA z S z , where δA z = A z − A z t . The introduction of superoperators -operators acting on the space of linear operators on the Hilbert space -allows for a compact notation. The von Neumann Next, we define the superoperator P as a projector onto the relevant subspace where ρ 0 B describes separate thermal equilibria of the two leads whose chemical potentials are different due to the bias voltage eV = µ L −µ R . Essentially, P maps a density operator onto one of product form with the environment in equilibrium but still retains the relevant information on the system state. The complement of P is Q = 1 − P.
By inserting P and Q in front of both sides of the von Neumann equation one can derive a closed equation for the projection Pρ (t), which for factorized initial condition, where Qρ (0) = 0, can be rewritten in the form of the generalized Nakajima-Zwanzig master equation (10) which is non-local in time and contains all orders of the system-leads coupling 41 . Here,T denotes the chronological time-ordering operator. Since P and Q are projectors onto orthogonal subspaces that are only connected by L T , this simplifies to (11) Starting out from this exact integro-differential equation, we introduce some approximations: In the weak coupling limit we neglect all powers of L T higher than two (Born approximation). Consequently, we replace L (τ ) by L (τ ) − L T in the exponential of Eqn. (11). Moreover, we make use of the fact that the nuclear spins evolve on a time-scale that is very slow compared to all electronic processes: In other words, the Overhauser field is quasi-static on the timescale of single electronic tunneling events 22,27 . That is, we replace A z τ by A z t in the exponential of Eqn. (11) which removes the explicit time dependence in the kernel. By taking the trace over the reservoir and using Here, we also used the relations PL T P = 0 and L B P = 0 and switched the integration variable to τ = t − t . Note that, for notational convenience, we will suppress the explicit time-dependence of L 0(1) (t) in the following.
In the next step, we iterate the Schwinger-Dyson identity In what follows, we will keep only the first term of this infinite series (note that the next two leading terms are explicitly calculated in Appendix A). In quantum optics, this simplification is well known as approximation of independent rates of variation 47 . In our setting it is valid, if L 1 (t) is small compared to L 0 (t) and if the bath correlation time τ c is short compared to the HF dynamics, A HF 1/τ c . Pictorially, this means that during the correlation time τ c of a tunneling event, there is not sufficient time for the Rabi oscillation with frequency g A HF to occur. For typical materials 48 , the relaxation time τ c is in the range of ∼ 10 −15 s corresponding to a relaxation rate Γ c = τ −1 c ≈ 10 5 µeV. Indeed, this is much faster than all other relevant processes. In this limit, the equation of motion for the reduced density matrix of the system simplifies tȯ Note, however, that this master equation is not Marko-vian as the rate of change of ρ S (t) still depends on its past. Conditions which allow for a Markovian treatment of the problem will be addressed in the following.

B. Markov Approximation
Using the general relation e −iL0τ O = e −iH0τ Oe iH0τ for any operator O, we rewrite Eqn. (14) aṡ In accordance with the previous approximations, we will replace e −iH0τ ρ S (t − τ ) e iH0τ by ρ S (t) which is approximately the same since any correction to H 0 would be of higher order in perturbation theory 43,44 . In other words, the evolution of ρ S (t − τ ) is approximated by its unperturbed evolution which is legitimate provided that the relevant timescale for this evolution τ c is very short (Markov approximation). This step is motivated by the typically rapid decay of the lead correlations functions 43 ; the precise validity of this approximation is elaborated below. In particular, this simplification disregards dissipative effects induced by H T which is valid selfconsistently provided that the tunneling rates are small compared to the dynamics generated by H 0 . Moreover, in Eqn. (15) we introduced the tunneling Hamiltonian in the interaction picture asH T (τ ) = e −iH0τ H T e iH0τ . For simplicity, we will only consider one lead for now and add the terms referring to the second lead later on. Therefore, we can disregard an additional index specifying the left or right reservoir and write explicitlỹ Here, the resonances σ (t) are explicitly time-dependent as they dynamically depend on the polarization of the nuclear spins The quantity can be interpreted as an effective Zeeman splitting which incorporates the external magnetic field as well as the mean magnetic field generated by the nuclei.
Since the leads are assumed to be at equilibrium, their correlation functions are given by where the Fermi function with inverse temperature β = 1/k B T gives the thermal occupation number of the respective lead in equilibrium. Note that all terms comprising two lead creation c † kσ or annihilation operators c kσ vanish since ρ 0 B contains states with definite electron number only 43 . The correlation functions are diagonal in spin space and the tunneling Hamiltonian preserves the spin projection; therefore only co-rotating terms prevail. If we evaluate all dissipative terms appearing in Eqn. (15), due to the conservation of momentum and spin in Eqn. (19) and Eqn. (20), only a single sum over k, σ survives. Here, we single out one term explicitly, but all other terms follow analogously. We obtaiṅ where the correlation time of the bath τ c is determined by the decay of the noise correlations Here, we made use of the fact that the leads are macroscopic and therefore exhibit a continuous density of states per spin n ( ). On top of that, we have introduced the spectral density of the bath as where D σ ( ) = n ( ) |T σ ( )| 2 is the effective density of states. The Markovian treatment manifests itself in a self-consistency argument: We assume that the spectral density of the bath J σ ( ) is flat around the (timedependent) resonance σ (t) over a range set by the characteristic width Γ d . Typically, both the tunneling matrix elements T σ ( ) as well as the density of states n ( ) are slowly varying functions of energy. In the so-called wide-band limit the effective density of states D σ ( ) is assumed to be constant so that the self-consistency argument will exclusively concern the behaviour of the Fermi function f ( ) which is intimately related to the temperature of the bath T . Under the condition, that J σ ( ) behaves flat on the scale Γ d , it can be replaced by its value at σ (t), and the noise correlation simplifies to Using the relation with P denoting Cauchy's principal value, we find that the Markov approximation Re [C σ (τ )] ∝ δ (τ ) is fulfilled provided that the self-consistency argument holds. This corresponds to the white-noise limit where the correlation-time of the bath is τ c = 0. Pictorially, the reservoir has no memory and instantaneously relaxes to equilibrium. We can then indeed replace e −iH0τ ρ S (t − τ ) e iH0τ by ρ S (t) and extend the integration in Eqn. (15) to infinity, with neglibile contributions due to the rapid decay of the memory kernel. In the following, we will derive an explicit condition for the selfconsistency argument to be satisfied. Let us first consider the limit T = 0: As schematically depicted in Fig. 3, in this case f ( ) behaves perfectly flat except for = µ where the self-consisteny argument is violated. Therefore, the Markovian approximation is valid at T = 0 given that the condition | σ (t) − µ| Γ d is fulfilled. In this limit, all tunneling rates are constant over time and effectively decoupled from the nuclear dynamics. Note that for the observation of electronic superradiance it will be sufficient to restrict oneself to this case.
For a more general analysis, we now turn to the case of finite temperature T > 0. We require the absolute value of the relative change of the Fermi function around the resonance σ (t) over a range of the characteristic width Γ d to be much less than unity, that is An upper bound for the first factor can easily be obtained as this quantity is maximized at the chemical potential µ, for all temperatures. Evaluating the derivative at σ (t) = µ results in the compact condition Thus, finite temperature T > 0 washes out the rapid character of f ( ) at the chemical potential µ and, provided that Eqn. (27) is fulfilled, allows for an Markovian treatment. Two distinct mechanisms contribute to the width Γ d : dissipation due to coupling to the leads and the effect of H 1 (t), because both of them have been neglected self-consistently in the memory kernel when going from Eqn. (11) to Eqn. (14). Typically, the tunneling rates are of the order of ∼ 5 − 20µeV, depending on the transparency of the tunnel-barrier. Regarding the contribution due to H 1 (t), we first consider two limits of particular importance: For a completely mixed state the fluctuation of the nuclear field around its zero expectation value is of the order of ∼ A HF / √ N ≈ 0.1 µeV. In contrast, for a fully polarized state these fluctuations can be neglected whereas the effective strength of the flip-flop dynamics is ∼ A HF / √ N as well. Therefore, in both limits considered here, the dominant contribution to Γ d is due to the coupling to the leads and the self-consistency condition could still be met with cryostatic temperatures k B T 10 µeV, well below the orbital level spacing. However, we note that in the course of a superradiant evolution, where strong correlations among the nuclei build up, the dominant contribution to Γ d may come from the flip-flop dynamics, which are A HF /4 ≈ 25 µeV at maximum for homogeneous coupling. For realisitic conditions, though, this effect is significantly reduced, as will be demonstrated in our simulations in Sec. VI.

C. General Master Equation for Nuclear Spin Assisted Transport
Assuming that the self-consistency argument for a Markovian treatment is satisfied, we now apply the following modifications to Eqn.(15): First, we neglect level shifts due to the coupling to the continuum states which can be incorporated by replacing the bare frequencies σ (t) with renormalized frequencies. Second, one adds the second electron reservoir that has been omitted in the derivation above. Lastly, one performs a suitable transformation into a frame rotating at the frequencȳ = ( ↑ + ↓ ) /2 leaving all terms invariant but changing After these manipulations one arrives at the central master equation as stated in Eqn. (2) where the tunneling rates govern the dissipative processes in which the QD system exchanges single electrons with the leads. The tunneling rates, as presented here, are widely used in nanostructure quantum transport problems 42,43,45 . However, in our setting they are evaluated at the resonances σ (t) which dynamically depend on the polarization of the nuclear spins; see Eqn. (17). Note that Eqn. (2) incorporates finite temperature effects via the Fermi functions of the leads. This potentially gives rise to feedback mechansims between the electronic and the nuclear dynamics, since the purely electronic diffusion markedly depends on the nuclear dynamics. Since Eqn.
(2) marks our first main result, at this point we quickly reiterate the assumptions our master equation treatment is based on: • The system-lead coupling is assumed to be weak and therefore treated perturbatively up to second order (Born-approximation).
• In particular, the tunneling rates are small compared to the effective Zeeman splitting ω.
• Level shifts arising from the coupling to the continuum states in the leads are merely incorporated into a redefinition of the QD energy levels σ (t).
• There is a separation of timescales between electron-spin dynamics and nuclear-spin dynamics.
In particular, the Overhauser field g A z t evolves on a timescale that is slow compared to single electron tunneling events.
• We have applied the approximation of independent rates of variation: If the HF dynamics generated by H 1 (t) = H ff + H ∆OH (t) is (i) sufficiently weak compared to H 0 and (ii) slow compared to the correlation time of the bath τ c , their effect in the memory kernel of the master equation can be neglected. The latter holds for A HF τ c 1. Note that the flipflop dynamics can become very fast as correlations among the nuclei build up culminating in a maximum coupling strength of A HF /4 for homogeneous coupling. This potentially drives the system into the strong coupling regime where condition (i), that is ω ||H 1 (t) ||, might be violated. However, under realistic conditions of inhomogeneous coupling this effect is significantly reduced.
• The effective density of states D σ ( ) = n ( ) |T σ ( )| 2 is weakly energy-dependent (wideband limit). In particular, it is flat on a scale of the characteristic widths of the resonances.
• The Markovian description is valid provided that either the resonances are far away from the chemical potentials of the leads on a scale set by the characteristic widths of the resonances or the temperature is sufficiently high to smooth out the rapid character of the Fermi functions of the leads. This condition is quantified in Eqn. (27).
In summary, we have derived a Quantum master equation describing electronic transport through a single QD which is collectively enhanced due to the interaction with a large ancilla system, namely the nuclear spin ensemble in the host environment. Eqn.(2) incorporates two major intriguing features both of theoretical and experimental relevance: Due to a separation of timescales, only the electronic subsystem experiences dissipation with rates that depend dynamically on the state of the ancilla system. This non-linearity gives rise to feedback mechanisms between the two subsystems as well as hysteretic behavior. Moreover, the collective nature of the HF interaction offers the possibility to observe intriguing coherent many-body effects. Here, one particular outcome is the occurence of electronic superradiance, as will be shown in the remainder of this paper. Note that in the absence of HF interaction between the QD electron and the proximal nuclear spins, i.e., in the limit g → 0, our results agree with previous theoretical studies 44 .

V. ELECTRONIC SUPERRADIANCE
Proceeding from our general theory derived above, this section is devoted to the prediction and analysis of superradiant behavior of electrons tunneling through a single QD in the Coulomb-blockade regime; see Fig. 1 for the scheme of the setup.
We note that, in principle, an enhancement seen in the leakage current could simply arise from the Overhauser field dynamically tuning the hyperfine flip-flops. However, we can still ensure that the measured change in the leakage current through the QD is due to cooperative emission only by dynamically compensating the Overhauser field. This can be achieved by applying a time dependent magnetic or spin-dependent AC Stark field such that H comp (t) = −g A z t S z which will be done in most of our simulations below to clearly prove the existence of superradiant behaviour in this setting. Consequently, in our previous analysis H 0 (t) is replaced by H 0 = H 0 (t) − g A z t S z = H Z + H B so that the polarization dependence of the tunneling rates is removed and we can drop the explicit time-dependence of the resonances σ (t) → σ . Under this condition, the master equation for the reduced system density operator can be written aṡ In accordance with our previous considerations, in this specific setting the Markovian treatment is valid provided that the spectral density of the reservoirs varies smoothly around the (time-independent) resonances σ on a scale set by the natural widths of the level and the fluctuations of the dynamically compensated Overhauser field. More specifically, throughout the whole evolution the levels are assumed to be far away from the chemical potentials of the reservoirs 39,40 ; for an illustration see Fig. 3. In this wide band limit, the tunneling rates α σ , β σ are independent of the state of the nuclear spins. The master equation is of Lindblad form which guarantees to preserve the complete positivity and the hermiticity of the density matrix. Eqn. (28) agrees with previous theoretical results 44 except for the appearance of the collective HF interaction between the QD electron and the ancilla system in the Hamiltonian dynamics of Eqn. (28).
To some extent, Eqn.(28) bears some similarity with the quantum theory of the laser. While in the latter the atoms interact with bosonic reservoirs, in our transport setting the QD is pumped by the nuclear spin ensemble and emits fermionic particles 45,46 .
If the HF dynamics are the slowest timescale in the problem, Eqn. (28) can be recast into a form which makes its superradiant character more apparent. In this case, the system is subject to the slaving principle 46 : The dy-namics of the whole system follow that of the subsystem with the slowest time constant allowing to adiabatically eliminate the electronic QD coordinates and to obtain an effective equation of motion for the nuclear spins. In this limit, the Overhauser field is much smaller than the Zeeman splitting so that a dynamic compensation of the OH can be disregarded for the moment. For simplicity we consider a transport setting in which only four tunneling rates are different from zero, see Fig. 1. The QD can be recharged from the left and the right lead, but only electrons with spin projection σ =↑ can tunnel out of the QD into the right lead. We define the total recharging rate β = β ↓ + β ↑ = β  4. (color online). The electronic QD system in the local moment regime after the adiabatic elimination of the |0 level including the relevant dissipative processes. Within the effective system (box) we encounter an effective decay term and an effective pure dephasing term, given by the rates γ and Γ respectively. This simplification is possible for fast recharging of the QD, i.e., β α.
electronic levels and the coherences in spin space according to ρ mn = m| ρ S |n , where m, n = 0, ↑, ↓. This yieldṡ We can retrieve an effective master equation for the regime in which on relevant timescales the QD is always populated by an electron. This holds for a suffi-ciently strong recharging rate, that is in the limit β α, which can be implemented experimentally by making the left tunnel barrier more transparent than the right one.
Then, the state |0 is populated negligibly throughout the dynamics and can be eliminated adiabatically according to ρ 00 ≈ α β ρ ↑↑ . In analogy to the Anderson impurity model, in the following this limit will be referred to as local moment regime. The resulting effective master equation readsρ where γ = β ↓ β α is an effective decay rate and Γ = β ↑ β α an effective dephasing rate. This situation is schematized in Fig. 4. The effective decay (dephasing) describes processes in which the QD is recharged with a spin down (up) electron after a spin up electron has tunneled out of the QD.
In the next step we aim for an effective description that contains only the nuclear spins: Starting from a fully polarized state, SR is due to the increase in the operative HF matrix element A + A − . The scale of the coupling is set by the total HF coupling constant A HF = g i g i . For a sufficiently small relative coupling strength 25 where the electron is predominantly in its |↓ spin state and we can project Eqn. (33) to the respective subspace. As shown in detail in Appendix B, in this limit the reduced master equation for the nuclear density operator µ = Tr el [ρ S ] is given by Eqn. (3), where the effective coefficients read This master equation is our second main result. In an optical setting, it has previously been predicted theoretically to exhibit strong SR signatures 25 . Conceptually, its superradiant character can be understood immediately in the ideal case of homogeneous coupling in which the collective state of all nuclear spins can be described in terms of Dicke states |J, m : The enhancement of the HF interaction is directly associated with the transition through nuclear Dicke states |J, m , m J. In this idealized setting, the angular momentum operator I = √ N A of the nuclear spin ensemble obeys the SU(2) Lie algebra, from which one can deduce the ladder operator relation I − |J, m = J(J + 1) − m (m − 1) |J, m − 1 . This means that, starting from an initially fully polarized state |J = N/2, m = N/2 , the system cascades down the Dicke-ladder with an effective rate since, according to the first term in Eqn. (3), the populations of the Dicke states evolve aṡ While the effective rate is Γ N/2→N/2−1 = c r at the very top of the the ladder it increases up to Γ |m| N/2 ≈ cr 4 N at the center of the Dicke ladder. This implies the characteristic intensity peaking as compared to the limit of independent classical emitters the emission rate of which would be Γ cl = cr N N ↑ = cr N (N/2 + m). However, there is also a major difference compared to the superradiant emission of photons from atomic ensembles: In contrast to its atomic cousin, the prefactor c r /N ∝ 1/N 2 is N -dependent, resulting in an overall time of the SR evolution t D which increases with N . By linearizing Eqn. (38) for the beginning of the superradiant evolution 24 as Γ m→m−1 ≈ c r (s + 1), where s = N/2 − m gives the number of nuclear flips, one finds that the first flip takes place in an average time c −1 r , the second one in a time (2c r ) −1 and so on. The summation of all these elementary time intervals gives an upper bound estimate for the process duration till the SR peaking as which, indeed, increases with the number of emitters as ∼ N ln(N ), whereas one obtains t D ∼ ln(N )/N for ordinary superradiance 24 . Accordingly, in our solid-state system the characteristic SR peak appears at later times for higher N . The underlying reason for this difference is that in the atomic setting each new emitter adds to the overall coupling strength, whereas in the central spin setting a fixed overall coupling strength A HF is distributed over an increasing number of particles. Note that in an actual experimental setting N is not a parameter, of course. For our theoretical discussion, though, it is convenient to fix the total HF coupling strength A HF and to extrapolate from our findings to an experimentally relevant number of nulear spins N . For large relative coupling strength 1 the QD electron saturates and superradiant emission is capped by the decay rate α/2, prohibiting the observation of a strong intensity peak. In order to circumvent this bottleneck regime, one has to choose a detuning ω 0 such that 0 < ≤ 1. However, to realize the spin-blockade regime, where the upper spin manifold is energetically well separated from the lower spin manifold, the Zeeman splitting has to be of the order of ω 0 ∼ A HF which guarantees < 1. In this parameter range, the early stage of the evolution -in which the correlation buildup necessary for SR takes place 24 -is well described by Eqn. (3).
In reality, the inhomogeneous nature (g i = const) of the collective operators A µ leads to dephasing between the nuclei, possibly preventing the phased emission necessary for the observation of SR 24,25,36,37 . The inhomogeneous part of last term in Eqn.(3) -the electron's Knight field -causes dephasing 38 ∝ g Var (g i )/2, possibly leading to symmetry reducing transitions J → J − 1. Still, it has been shown that SR is also present in realistic inhomogeneous systems 25 , since the system evolves in a many-body protected manifold (MPM): The second term in Eqn.(3) energetically separates different total nuclear spin-J manifolds protecting the correlation build-up for large enough .
The superradiant character of Eqn.(3) suggests the observation of its prominent intensity peak in the leakage current through the QD in the spin-blockade regime. We have employed the method of Full-Counting-Statistics (FCS) 51,52 in order to obtain an expression for the current and find (setting the electron's charge e = 1) This result is in agreement with previous theoretical findings: The current through the device is completely determined by the occupation of the levels adjacent to one of the leads 39,42,49 . The first term describes the accumulation of electrons with spin σ =↑ in the right lead, whereas the second term describes electrons with σ =↓ tunneling from the right lead into the QD. As done before 25 , we take the ratio of the maximum current to the initial current (the maximum for independent emitters) I coop /I ind as our figure of merit: a relative intensity peak height I coop /I ind > 1 indicates cooperative effects. One of the characteristic features of SR is that this quantity scales linearly with the number of spins N . In the local-moment regime, described by Eqn.(33), the expression for the current simplifies to showing that it is directly proportional to the electron inversion. This, in turn, increases as the nuclear system pumps excitations into the electronic system. A compact expression for the relation between the current and the dynamics of the nuclear system can be obtained immediately in the case of homogeneous coupling Since the nuclear dynamics are in general much slower than the electron's dynamics, the approximate solution of this equation is S + S − t ≈ − d dt I z t /γ. As a consequence, the current I (t) is proportional to the timederivative of the nuclear polarization Still, no matter how strong the cooperative effects are, on a timescale of single electron tunneling events, the electrons will always be emitted antibunched, since in the strong Coulomb-blockade regime the QD acts as a singleelectron emitter 53 . Typically, the rate for single-electron emission events is even below the single tunneling rate α due to the spin-blockade. On electronic timescales ∼ 1/α, the SR mechanism manifests in lifting this blockade; as argued above, the efficiency of this process is significantly enhanced by collective effects.
Before we proceed with an in-depth analysis of the current I(t), we note that an intriguing extension of the present work would be the study of fluctuations thereof. Insights into the statistics of the current could be obtained by analyzing two-time correlation functions such as n ↑ (t + τ )n ↑ (t) , where n ↑ = d † ↑ d ↑ . This can conveniently be done via the Quantum Regression Theorem 54 which yields the formal result n ↑ (t + τ )n ↑ (t) = Tr S n ↑ e Wτ (n ↑ ρ S (t)) . Here, W denotes the Liouvillian governing the system's dynamics according toρ S = Wρ S and Tr S [. . . ] refers to the trace over the system's degree of freedoms. This procedure can be generalized to higher order correlation functions and full evaluation of the current statistics might reveal potential connections between current fluctuations and cooperative nuclear dynamics.

A. Experimental Realization
The proposed setup described here may be realized with state-of-the-art experimental techniques. First, the Markovian regime, valid for sufficiently large bias eV , is realized if the Fermi functions of the leads are smooth on a scale set by the natural widths of the levels and residual fluctuations due to the dynamically compensated Overhauser field. Since for typical materials the hyperfine coupling constant is A HF = 1 − 100µeV 8 and tunneling rates are typically of the order of ∼ 10 µeV 9 , this does not put a severe restriction on the bias voltage which are routinely in the range of hundreds of µV or mV 18,19 . Second, in order to tune the system into the spin-blockade regime, a sufficiently large external magnetic field has to be applied. More precisely, the corresponding Zeeman splitting ω 0 energetically separates the upper and lower manifolds in such a way that the Fermi function of the right lead drops from one at the lower manifold to zero at the upper manifold. Temperature smeares out the Fermi function around the chemical potential by approximately ∼ k B T . Accordingly, with cryostatic temperatures of k B T ∼ 10 µeV being routinely realized in the lab 10 , this condition can be met by applying an external magnetic field of ∼ 5 − 10T which is equivalent to ω 0 ≈ 100 − 200µeV in GaAs 8,35 . Lastly, the charging energy U , typically ∼ 1 − 4meV 9,19 , sets the largest energy scale in the problem justifying the Coulomb-blockade regime with negligible double occupancy of the QD provided that the chemical potential of the left lead is well below the doubly occupied level. Lastly, we note that similar setups to the one proposed here have previously been realized experimentally by Hanson et al. 34,35 .
Proceeding from these considerations, we will now show by numerical simulation that an SR peaking of several orders of magnitude can be observed for experimentally relevant parameters in the leakage current through a quantum dot in the spin-blockade regime. We will first consider the idealized case of homogeneous coupling for which an exact numerical treatment is feasible even for a larger number of coupled nuclei. Then, we will continue with the more realistic case of inhomogeneous coupling for which an approximative scheme is applied. Here, we will also study scenarios in which the nuclear spins are not fully polarized initially. Finally, we will discuss intrinsic nuclear dephasing effects and undesired cotunneling processes which have been omitted in our simulations.

Idealized Setting
The homogeneous case allows for an exact treatment even for a relatively large number of nuclei as the system evolves within the totally symmetric low-dimensional subspace {|J, m , m = −J, . . . , J}. Starting from a fullypolarized state, a strong intensity enhancement is observed; typical results obtained from numerical simulations of Eqn. (28) are depicted in Fig. 5 for N = 60 and N = 100 nuclear spins. The corresponding relative peak heights display a linear dependence with N , cf. Fig. 6, which we identify as the characteristic feature of superradiance. Here, we have used the numerical parameters A HF = 1, ω 0 = 1 and α = β Before we proceed, some further remarks on the dynamic compensation of the Overhauser field seem appropriate: We have merely introduced it in our analysis in order to provide a clear criterion for the presence of purely collective effects, given by I coop /I ind > 1. In other words, dynamic compensation of the Overhauser field is not a necessary requirement for the observation of collective effects, but it is rather an adequate tool to display them clearly. From an experimental point of view, the dynamic compensation of the Overhauser field might be challenging as it requires accurate knowledge about the evolution of the nuclear spins. Therefore, we also present results for the case in which the external magnetic field is constant and no compensation is applied. This markedly relaxes the experimental challenges. Here, we can distinguish two cases: Depending on the sign of the HF coupling constant A HF , the time-dependence of the effective Zeeman-splitting ω can either give rise to an additional enhancement of the leakage current (A HF > 0) or it can counteract the collective effects (A HF < 0). As shown in Fig. 6, this sets lower and upper bounds for the observed enhancement of the leakage current.
In Fig. 6 we also compare the results obtained for dynamic compensation of the Overhauser field to the idealized case of perfect compensation in which the effect of the Overhauser field is set to zero, i.e., H OH = gA z S z = 0. Both approaches display the same features justifying our approximation of neglecting residual (de)tuning effects of the dynamically compensated Overhauser field w.r.t. the external Zeeman splitting ω 0 . This will also be discussed in greater detail below.

Beyond the Idealized Setting
We now turn to the more realistic case of inhomogeneous coupling which in principle could prevent the phasing necessary for SR. However, as shown below, SR is still present in realistic inhomogeneous systems. In contrast to the idealized case of homogeneous coupling, the dynamics cannot be restricted to a low-dimensional subspace so that an exact numerical treatment is not feasible due to the large number of nuclei. We therefore use an approximate approach which has previously been shown to capture the effect of nuclear spin coherences while allowing for a numerical treatment of hundreds of spins 22,25 . For simplicity, we restrict ourselves to the local moment regime in which the current can be obtained directly from the electron inversion I (t) ∝ S + S − t . By Eqn. (33), this expectation value is related to a hierachy of correlation terms involving both the electron and nuclear spins. Based on a Wick type factorization scheme, higher order expressions are factorized in terms of the covariance matrix γ + ij = σ + i σ − j and the "mediated covariance matrix" γ − ij = σ + i S z σ − j . For further details, see Refs. 22,25 . Moreover, the coupling constants g j have been obtained from the assumption of a two-dimensional Gaussian spatial electron wavefunction of width √ N /2. Specifically, we will present results for two sets of numerical parameters, corresponding to a relative coupling strength of = 0.5, where A HF = 1, ω 0 = 1, γ = 0.1 and Γ = 0.08, and = 0.55 with A HF = 1, ω 0 = 0.9, γ = 0.1 and Γ = 0.067.
As shown in Fig. 7 and 8, the results obtained with these methods demonstrate clear SR signatures. In comparison to the ideal case of homogeneous coupling, the relative height is reduced, but for a fully polarized initial state we still find a linear enhancement I coop /I ind ≈ 0.043N ( = 0.5); therefore, as long as this linear dependence is valid, for typically N ≈ 10 5 − 10 6 a strong intensity enhancement of several orders of magnitude is predicted ∼ 10 3 − 10 4 .
If the initial state is not fully polarized, SR effects are reduced: However, when starting from a mixture of symmetric Dicke states |J, J with polarization p = 80(60)%, we find that the linear N dependence is still present: I coop /I ind ≈ 0.0075(0.0028)N ( = 0.5), i.e., the scaling is about a factor of ∼ 5(15) weaker than for full polarization 55 . Still, provided the linear scaling holds up to an experimentally realistic number of nuclei N ≈ 10 5 − 10 6 , this amounts to a relative enhancement of the order of I coop /I ind ∼ 10 2 − 10 3 . In our simulations we have self-consistently verified that the fluctuations of the Overhauser field, defined via  are indeed small compared to the external Zeeman splitting ω 0 throughout the entire evolution. This ensures the validity of our perturbative approach and the realization of the spin-blockade regime. From atomic superradiance it is known that in the limit of homogeneous coupling large fluctuations can build up, since in the middle of the emission process the density matrix becomes a broad distribution over the Dicke states 24 . Accordingly, in the idealized, exactly solvable case of homogeneous coupling we numerically find rather large fluctuations of the Overhauser field; as demonstrated in Fig. 9, this holds independently of N . In particular, for a relative coupling strength = 0.5 the fluctuations culminate in max [∆ OH ] /ω 0 ≈ 0.35. However, in the case of inhomogeneneous HF coupling the Overhauser field fluctuations are found to be smaller as the build-up of these fluctuations is hindered by the Knight term causing dephasing among the nuclear spins. As another limiting case, we also estimate the fluctuations for completely independent homogeneously coupled nuclear spins via the Binominal distribution as max [∆ OH ] ∼ 0.5A HF / √ N 56 . Moreover, we have also ensured self-consistently the validity of the perturbative treatment of the flip-flop dynamics; that is, throughout the entire evolution, even for maximum operative matrix elements A + A − t , the strength of the flipflop dynamics H ff was still at least five times smaller than ω 0 .
Initially, the HF mediated superradiance dynamics is rather slow, with its characteristic time scale set by c −1 r ; for experimentally realistic parameters -in what follows we use the parameter set = 0.5, α ≈ 10 µeV, N ≈ 10 5 for numerical estimates -this corresponds to c −1 r ≈ 10 µs. Based on fits as shown in Fig. 10, we then estimate for the SR process duration t D ≈ 50c −1 r ≈ 500 µs which is still smaller than recently reported 33 nuclear decoherence times of ∼ 1 ms. Therefore, it should be possible to observe the characteristic enhancement of the leakage current before the nuclear spins decohere.
Accordingly, in the initial phasing stage, the HF mediated lifting of the spin-blockade is rather weak resulting in a low leakage current, approximatively given by I (t = 0) ≈ 2 α/N . Therefore, the initial current due to HF processes is inversely proportional to the number of nuclear spins N . However, as correlations among the nuclei build up, the HF mediated lifting becomes more efficient culminating in a maximum current of I max ≈ 2 α, independent of N . For realistic experimental valuesalso taking into account the effects of inhomogeneous HF coupling and finite initial polarization p ≈ 0.6we estimate the initial (maximum) leakage current to be of the order of I (t = 0) ≈ 6 fA (I max ≈ 10 pA). Leakage currents in this range of magnitudes have already been detected in single QD spin-filter experiments 34 as well as double QD Pauli-blockade experiments 15,16,18,19 ; here, leakage currents below 10 fA and 150 fA, respectively, have been attributed explicitly to other spurious processes 18,34 . Among others, these will be addressed in greater detail in the following.
In our simulations we have disregarded species inhomogeneities in the nuclear Zeeman energies. In principle, these are large enough to cause additional dephasing between the nuclear spins, similar to the inhomogeneous Knight field. However, this dephasing mechanism only applies to nuclei of different Zeeman energies, that is nuclei which belong to different species 22 . This leads to two or three mutually decohered subsystems each of which is described by our theory. Moreover, we have neglected the dipolar and quadrupolar interactions among the nuclear spins. First, the latter is absent for nuclear spin I = 1/2 (CdSe QDs) or strain-free QDs 28 . Second, the nuclear dipole-dipole interaction can cause diffusion and dephasing processes. Diffusion processes that can change A z are strongly detuned and therefore of minor importance, as corroborated by experimentally measured spin diffusion rates 29,30 . Resonant processes such as ∝ I z i I z j can lead to dephasing similar to the inhomogenous Knight shift. This competes with the phasing necessary for the observation of SR as expressed by the first term in Eqn. (3). The SR process is the weakest at the very beginning of the evolution where we estimate its strength as c min r ≈ 10 µeV/N . An upper bound for the dipole-dipole interaction in GaAs has been given in Ref. 26 as ∼ 10 −5 µeV, in agreement with values given in Refs. 27,31 . Therefore, the nuclear dipole-dipole interaction can safely be neglected for N 10 5 . In particular, its effect should be further reduced for highly polarized ensembles. Moreover, as argued above due to the presence of the MPM-term in Eqn. (3) and demonstrated by our simulations, the observation of SR is even robust against dephasing caused by the much stronger Knight field.
Our transport setting is tuned into the sequential tunneling regime and therefore we have disregarded cotunneling processes which are fourth order in H T . In principle, cotunneling processes could lift the spin-blockade and add an extra contribution to the leakage current that is independent of the HF dynamics. However, note that cotunneling current scales as I ct ∝ α 2 , whereas sequential tunneling current I ∝ α; accordingly, cotunneling current can always be suppressed by making the tunnel barriers less transparent 34 . Moreover, inelastic cotunneling processes exciting the QD spin can be ruled out for eV, k B T < ω 0 due to energy conservation 32 . The effectiveness of a single quantum dot to act as an electrically tunable spin filter has also been demonstrated experimentally 34 : The spin-filter efficiency was measured to be nearly 100%, with I ct being smaller than the noise floor ∼ 10 fA. Its actual value has been calculated as ∼ 10 −4 fA, from which we roughly estimate I ct ∼ 10 −2 fA in our setting. This is smaller than the initial HF mediated current I (t = 0) and considerably smaller than I max , even for an initially not fully polarized nuclear spin ensemble. Still, if one is to explore the regime where cotunneling cannot be neglected, phenomenological dissipative terms -effectively describing the corresponding spin-flip and pure dephasing mechanisms for inelastic and elastic processes respectively -should be added to Eqn. (28).

VII. CONCLUSION AND OUTLOOK
In summary, we have developed a master equation based theoretical framework for nuclear spin assisted transport through a QD. Due to the collective nature of the HF interaction, it incorporates intriguing many-body effects as well as feedback mechanisms between the electron spin and nuclear spin dynamics. As a prominent application, we have shown that the current through a single electrically defined QD in the spin-blockade regime naturally exhibits superradiant behavior. This effect stems from the collective hyperfine interaction between the QD electron and the nuclear spin ensemble in the QD. Its most striking feature is a lifting of the spin-blockade and a sudden peak in the leakage current. The experi-mental observation of this effect would provide clear evidence of coherent HF dynamics of nuclear spin ensembles in QDs.
Finally, we highlight possible directions of research going beyond our present work: Apart from electronic superradiance, the setup proposed here is inherently well suited for other experimental applications like dynamic polarization of nuclear spins (DNP): In analogy to optical pumping, Eqn.(3) describes electronic pumping of the nuclear spins. Its steady states are eigenstates of A z , which lie in the kernel of the collective jump-operator A − . In particular, for a completely inhomogeneous system the only steady state is the fully polarized one, the ideal initial state required for the observation of SR effects. When starting from a completely unpolarized nuclear state, the uni-directionality of Eqn.(3) -electrons with one spin orientation exchange excitations with the nuclear spins, while electrons of opposite spin primarily do not -implies that the rather warm electronic reservoir can still extract entropy out of the nuclear system. More generally, the transport setting studied here possibly opens up the route towards the (feedback-based) electronic preparation of particular nuclear states in single QDs. This is in line with similar ideas previously developed in double QD settings, see e.g. Refs. 12,15,18,20,33 .
In this work we have specialized on a single QD. However, our theory could be extended to a double QD (DQD) setting which is likely to offer even more possibilities. DQDs are routinely operated in the Pauli-blockade regime where despite the presence of an applied sourcedrain voltage the current through the device is blocked whenever the electron tunneling into the DQD has the same spin orientation as the one already present. The DQD parameters and the external magnetic field can be tuned such that the role of the states |σ , σ =↓, ↑, in our model is played by a pair of singlet and triplet states, while all other states are off-resonant. Then, along the lines of our study, non-linearities appear due to dependencies between the electronic and nuclear subsystems and collective effects enter via the HF-mediated lifting of the spin-blockade.
While we have focused on the Markovian regime and the precise conditions for the validity thereof, Eqn. (14) offers a starting point for studies of non-Markovian effects in the proposed transport setting. All terms appearing in the memory kernel of Eqn. (14) are quadratic in the fermionic creation and annihilation operators allowing for an efficient numerical simulation, without having to explicitly invoke the flatness of the spectral density of the leads. This should then shed light on possibly abrupt changes in the QD transport properties due to feedback mechanism between the nuclear spin ensemble and the electron spin.
Lastly, our work also opens the door towards studies of dissipative phase transitions in the transport setting: when combined with driving, the SR dynamics can lead to a variety of strong-correlation effects, non-equilibrium and dissipative phase transitions 1,57-59 , which could now be studied in a mesoscopic solid state system, complementing other approaches to dissipative phase transitions in QDs 60-63 .

ACKNOWLEDGMENTS
We acknowledge support by the DFG within SFB 631 and the Cluster of Excellence NIM.

Appendix A: Microscopic Derivation of the Master Equation
In this Appendix we provide some details regarding the derivation of the master equations as stated in Eqn. (2) and Eqn. (28). It comprises the effect of the HF dynamics in the memory kernel of Eqn. (12) and the subsequent approximation of independent rates of variation.
In the following, we will show that it is self-consistent to neglect the effect of the HF dynamics L 1 (t) in the memorykernel of Eqn. (12) provided that the bath correlation time τ c is short compared to the Rabi flips produced by the HF dynamics. This needs to be addressed as cooperative effects potentially drive the system from a weakly coupled into a strongly coupled regime. First, we reiterate the Schwinger-Dyson identity in Eqn. (13) as an infinite sum over time-ordered nested commutators where for any operator XL More explicitly, up to second order Eqn.(A1) is equivalent to Note that the time-dependence ofH 1 (τ ) is simply given bỹ where the effective Zeeman splitting ω = ω 0 + g A z t is time-dependent. Accordingly, we defineL 1 (τ ) =L + (τ ) + L − (τ ) +L ∆OH (τ ) = e iωτ L + + e −iωτ L − + L ∆OH , where L x · = [H x , ·] for x = ±, ∆OH. In the next steps, we will explicitly evaluate the first two contributions to the memory kernel that go beyond n = 0 and then generalize our findings to any order n of the Schwinger-Dyson series.

First order correction
The first order contribution n = 1 in Eqn.(12) is given by Performing the integration in τ 1 leads to where, for notational convenience, we introduced the operators In accordance with previous approximations, we have replaced e −iH0τ ρ S (t − τ ) e iH0τ by ρ S (t) since any additional term besides H 0 would be of higher order in perturbation theory 43,44 . In particular, this disregards dissipative effects: In our case, this approximation is valid self-consistently provided that the tunneling rates are small compared to effective Zeeman splitting ω. The integrand decays on the leads-correlation timescale τ c which is typically much faster than the timescale set by the effective Zeeman splitting, ωτ c 1. This separation of timescales allows for an expansion in the small parameter ωτ , e.g. g ω e iωτ − 1 ≈ igτ . We see that the first order correction can be neglected if the the bath correlation time τ c is sufficiently short compared to the timescale of the HF dynamics, that is gτ c 1. The latter is bounded by the total hyperfine coupling constant A HF (since ||gA x || ≤ A HF ) so that the requirement for disregarding the first order term reads A HF τ c 1.

Second order correction
The contribution of the second term n = 2 in the Schwinger-Dyson expansion can be decomposed into fz . (A7) The first term Ξ (2) zz contains contributions from H ∆OH only Similarly, Ξ ff which comprises contributions from H ff only is found to be Here, we have used the following simplification: The time-ordered products which include flip-flop terms only can be simplified to two possible sequences in which L + is followed by L − and vice versa. This holds since L ± L ± X = [H ± , [H ± , X]] = H ± H ± X + XH ± H ± − 2H ± XH ± = 0.
Here, the first two terms drop out immediately since the electronic jump-operators S ± fulfill the relation S ± S ± = 0.
In the problem at hand, also the last term gives zero because of particle number superselection rules: In Eqn.(12) the time-ordered product of superoperators acts on X = H T , ρ S (t − τ ) ρ 0 B . Thus, for the term H ± XH ± to be nonzero, coherences in Fock space would be required which are consistently neglected; compare Ref. 44 . This is equivalent to ignoring coherences between the system and the leads. Note that the same argument holds for any combination H µ XH ν with µ, ν = ±.
Similar results can be obtained for Ξ (2) fz which comprises H ± as well as H ∆OH in all possible orderings. Again, using that the integrand decays on a timescale τ c and expanding in the small parameter ωτ shows that the second order contribution scales as ∼ (gτ c ) 2 . Our findings for the first and second order correction suggest that the n-th order correction scales as ∼ (gτ c ) n . This will be proven in the following by induction.

n-th order correction
The scaling of the n-th term in the Dyson series is governed by the quantities of the form where the index suggests the order in which H ± (giving an exponential factor) and H ∆OH (resulting in a factor of 1) appear. Led by our findings for n = 1, 2, we claim that the expansion of ξ Since we have already verified this result for n = 1, 2, the general result follows by induction. This completes the proof.

Appendix B: Adiabatic Elimination of the QD Electron
For a sufficiently small relative coupling strength the nuclear dynamics are slow compared to the electronic QD dynamics. This allows for an adiabatic elimination of the electronic degrees of freedom yielding an effective master equation for the nuclear spins of the QD.
Our analysis starts out from Eqn.(33) which we write aṡ where Note that the superoperator W 0 only acts on the electronic degrees of freedom. It describes an electron in an external magnetic field that experiences a decay as well as a pure dephasing mechanism. In zeroth order of the coupling parameter the electronic and nuclear dynamics of the QD are decoupled and SR effects cannot be expected. These are contained in the interaction term W 1 . Formally, the adiabatic elimination of the electronic degrees of freedom can be achieved as follows 50 : To zeroth order in the eigenvectors of W 0 with zero eigenvector λ 0 = 0 are where ρ SS = |↓ ↓| is the stationary solution for the electronic dynamics and µ describes some arbitrary state of the nuclear system. The zero-order Liouville eigenstates corresponding to λ 0 = 0 are coupled to the subspaces of "excited" nonzero (complex) eigenvalues λ k = 0 of W 0 by the action of W 1 . Physically, this corresponds to a coupling between electronic and nuclear degrees of freedom. In the limit where the HF dynamics are slow compared to the electronic frequencies, i.e. the Zeeman splitting ω 0 , the decay rate γ and the dephasing rate Γ, the coupling between these blocks of eigenvalues and Liouville subspaces of W 0 is weak justifying a perturbative treatment. This motivates the definition of a projection operator P onto the subspace with zero eigenvalue λ 0 = 0 of W 0 according to where µ = Tr el [ρ] is a density operator for the nuclear spins, Tr el . . . denotes the trace over the electronic subspace and by definition W 0 ρ SS = 0. The complement of P is Q = 1 − P . By projecting the master equation on the P subspace and tracing over the electronic degrees of freedom we obtain an effective master equation for the nuclear spins in second order perturbation theorẏ µ = Tr el P W 1 P ρ − P W 1 QW −1 0 QW 1 P ρ .
Using Tr el [S z ρ SS ] = −1/2, the first term is readily evaluated and yields the Knight shift seen by the nuclear spins The derivation of the second term is more involved. It can be rewritten as This leads to Similarly, one finds Analogously, one can show that terms containing two flip or two flop terms give zero. The same holds for mixed terms that comprise one flip-flop and one Overhauser term with ∼ A z S z . The term consisting of two Overhauser contributions gives Performing the integration and separating real from imaginary terms yields where c r = g 2 / 4ω 2 0 + α 2 α and c i = g 2 / 4ω 2 0 + α 2 ω 0 . Combining Eqn.(B7) with Eqn.(B20) directly gives the effective master equation for the nuclear spins given in Eqn. (3) in the main text.