Higher-order spin and charge dynamics in a quantum dot-lead hybrid system

Understanding the dynamics of open quantum systems is important and challenging in basic physics and applications for quantum devices and quantum computing. Semiconductor quantum dots offer a good platform to explore the physics of open quantum systems because we can tune parameters including the coupling to the environment or leads. Here, we apply the fast single-shot measurement techniques from spin qubit experiments to explore the spin and charge dynamics due to tunnel coupling to a lead in a quantum dot-lead hybrid system. We experimentally observe both spin and charge time evolution via first- and second-order tunneling processes, and reveal the dynamics of the spin-flip through the intermediate state. These results enable and stimulate the exploration of spin dynamics in dot-lead hybrid systems, and may offer useful resources for spin manipulation and simulation of open quantum systems.


CHARGE AND SPIN RELAXATION SIGNALS
We describe the dynamics of the charge and spin on the QD by considering the rate equation for the probability P σ that the dot is occupied by a single electron with spin σ ∈ {↑, ↓} (we alternatively use σ = ±1), where P e is the probability that the dot is empty. We do not consider any other states, which gives the normalization condition P ↑ + P ↓ + P e = 1. Eq. (1) includes the process of an electron with a given spin leaving the dot into the lead where an empty state exists with the probability (1 − f σ ) and entering an empty dot from the lead state occupied with probability f σ . This Fermi factor is given by being the energy cost to add an electron into the dot, which includes the electrostatic potential energy eV g , and the orbital (quantization) energy ϵ 1 . The Zeeman energy is E z = gµ B B, and the Fermi-Dirac distribution depends on the temperature T , and the lead Fermi energy µ F . Apart from the Fermi factors f σ , the tunneling rates for hopping on and off the dot are identical, Γ σ . We, however, allow for a spin dependence of the tunneling rate which has been found to be an appreciable effect (the asymmetry of the rates can be of the order of the rates themselves), most probably due to the exchange interaction in the lead [1-3].
To expose the spin and charge dynamics, we introduce new variables, the probability of charge occupation, P o = P ↑ + P ↓ and the spin polarization, s = P ↑ − P ↓ , and new parameters, for the average, Γ, and the dimensionless asymmetry α, in the tunneling rates, by writing Γ σ = Γ(1+σα), and similarly for the Fermi factors, f σ = f + σf δ . Equation (1) can be now cast into the matrix form for the vector of unknowns, v = (P o , s) T , namely with the matrix defining the system propagator and the steady state solution The steady state is independent of the tunneling rates, and depends only on the lead Fermi factors for the two spins, as it should be, while the propagator matrix depends on all parameters of the problem. Even though it is straightforward to solve the problem in the most general case, it is useful to consider M for α = 0 (spin independent tunneling rates), which gives To demonstrate this difference, seen also experimentally, we plot the charge and spin signals in

CO-TUNNELING RATE
To derive the formula for the spin relaxation by cotunneling, which was used in the main text to fit the data on Fig. 4(a), we consider the Hamiltonian of a QD coupled to a lead, H = Here the dot Hamiltonian is where the index α labels the states of the dot |α⟩ with energies ϵ α , and |0⟩ denotes an empty dot, |σ⟩ = d † σ |0⟩ a dot with a single electron with spin σ, and |S⟩ = d † ↑ d † ↓ |0⟩ a dot with a two electron singlet state, and d † σ is the creation operator of a dot electron with spin σ. The lead is described by where k is a wave-vector (for simplicity, we consider a one dimensional lead, so that k is a scalar).

Finally, the dot-lead coupling is
which desctribes a spin-preserving lead-dot tunneling with, in general complex and spin and energy dependent, tunneling amplitudes t kσ .
We now repeat the standard calculation [5][6][7][8][9] with minor adjustments to arrive at the inelastic spin decay rather than the co-tunneling current. To this end, we define the transition rate by the Fermi's Golden rule formula where i and f are the initial and final states with energies E i and E f , respectively, considered to be separable (to the lead and dot components) eigenstates of the unperturbed system described by As we are not conditioning the transitions on the states of the lead, the rate is summed over all possible initial lead states, with the corresponding probabilities p i lead , and all lead final states. The former gives the prescription for a replacement ∑ i lead p i lead |i lead ⟩⟨i lead | → ρ thermal lead , with the latter the equilibrium density matrix corresponding to a system with Hamiltonian H L , at a temperature T . Finally, G is the transition operator which can be expanded in powers of the tunneling term with E = E i = E f . The two terms describe, respectively, the direct tunneling and the cotunneling, and γ is a regularization factor [10].
Simple results can be derived in the well justified case of a negligible dependence of the tunneling amplitudes on the wave vector, t kσ ≈ t σ . Using the first term in Eq. (11) gives in this limit the following expression for the direct tunneling rates defined in Eq. (1) where we denoted Γ σ ≡ Γ σ→0 = Γ 0→σ and g F is the density of states in the lead at the Fermi energy. Similarly, keeping only the second term in Eq. (11) gives the co-tunneling rate for a spin-flip (from σ to the opposite value σ) of a single electron occupying the dot, where µ(2) = ϵ S − ϵ 1 is (the spin independent part of) the energy cost to add a second electron into the dot. The expression can be further simplified if the dot is deep in the Coulomb blockade, so that the charge excitation energies are much larger than the temperature, namely µ(1) ≪ µ F ≪ µ (2) are well fulfilled on the energy scale of the temperature, k B T . The energy dependence of the last term in Eq. (13) can be then neglected, replacing ϵ → µ F , and the remaining integral can be evaluated resulting in where we also neglected the regularization factors. In the large temperature limit, k B T ≫ E z , the temperature dependent factor becomes k B T , while in the opposite limit, k B T ≪ E z , it gives 2E z for σ =↓, and 0 for σ =↑. However, Eq. (14) is already in the form which was used to fit the data and is thus the final result of this section. * tomohiro.otsuka@riken.jp † tarucha@ap.t.u-tokyo.ac.jp