Decoherence of a spin-valley qubit in a MoS2 quantum dot

Transition metal dichalcogenide (TMD)-based quantum dots (QDs) have proven to be a successful and promising device for physically implementing electron spin-valley based qubits. Although the electron spin in a TMDs monolayer semiconductor QD can be isolated and controlled with high precision, decoherence occurs due to unavoidable coupling with the surrounding environment, such as nuclear spin environments. In this paper, using an exact master equation (ME) of spin qubit dynamics coupled to a nuclear spin bath in terms of hyperfine interaction (HI), we have investigated the controllability of dynamics processes with varying degrees of non-Markovianity. In large magnetic fields, we show that pure spin or valley qubits can be created. We calculate the loss of fidelity due to the Overhauser field of HI in a wide range of nuclear spin N . In this context, we prove that this field restricts the decoherence process of the central electron spin, which can regain its coherence. Finally, we discuss how the coherence of the spin qubit remains robust for large N .


Introduction
Modern life is fueled by data, which drives the demand for computational power to the point where new generations of semiconductor technology are constantly being deployed. Several publications since then have focused on improving and developing valleytronic devices [1,2], and transition metal dichalcogenide (TMD) qubits, including valley qubits, spin qubits, spin-valley qubits, and even impurity-based qubits [3][4][5][6][7][8][9]. TMDbased quantum dots (QDs) have attracted significant attention due to their potential applications in optoelectronics [10][11][12][13] including light sensors [14], logic circuits [15,16] and valleytronics [17][18][19][20]. Recently, MoS 2 -QDs have gained attention as a novel qubit pattern with potential for future quantum devices. However, the decoherence on this system is seen as a significant barrier to the deployment of quantum devices. The major obstacles addressed pose a challenge to researchers because noise hinders the transmission of quantum signals. One of the essential tools in the study of spin-valley qubits in QDs is stability, which is sensitive to disturbances of a noisy environment. Indeed, the inevitable coupling with their environment quickly destroys the phase relations (superposition of incompatible states) between the quantum states until they become classical states. The electron spin in a QD has two main decoherence channels: (i) (Markovian) phonon-assisted relaxation channel, due to the presence of spin-orbit interaction, (ii) (non-Markovian) spin bath constituted by the spins of the nuclei in the QD that interact with the electron spin via the hyperfine interaction (HI) [6,[21][22][23][24][25][26] where the number of nuclear spins varies between ∼10 2 and 10 6 . In our case, the calculations are carried out at zero temperature, so that phonon absorption and multi-phonon processes are negligible [6,27]. Full polarized baths are employed to facilitate qubit operations and extend coherence times [28]. Theoretically, it has been suggested as the storage of an electron spin state in a paper by Kurucz et al [29]. In straightforward experimental setups, a spin bath is polarized by manipulating a single qubit [23,[30][31][32], whose polarization can be reset repeatedly.
The central spin model, which entails a central spin interacting with many surrounding spins, can be used to describe realistic quantum many-spin systems, including the semiconductor quantum dots [33,34], the nitrogen-vacancy center in the diamond [35,36], and others. These solid-state central spin systems have recently received a lot of attention from researchers working in quantum information, quantum computation [34,37], quantum metrology, and quantum sensing [38][39][40]. In this context, a dynamic framework describes how to suppress HI-induced decoherence using the second-order time-convolutionless (TCL) master equation (ME) method [41][42][43][44][45][46][47][48][49][50], which is an effective method for dealing with such systems, was proposed [51]. This work inspired us to seek a dynamic scheme of spin qubit system in order to achieve high fidelity. We apply our theory to the prototypical model system for the spin-valley qubit hosted by MoS 2 -QD proposed by Kormányos et al [3].
In this paper, we use the TCL ME to solve the central spin problem. This equation enables us to study the dynamics of the central spin, and more interestingly to show the fidelity of the spin qubit to investigate the non-Markovian signature in the dynamical decoherence of open quantum systems. In response to this idea, we examine a precisely solvable decoherence phenomenon at a MoS 2 -QD qubit system in a fluctuating environment with fully polarized nuclear spins.
The paper is organized as follows. In section 2 we derive an effective Hamiltonian describing this singleelectron spin and  spin nuclei in interaction. Our approach is based on deriving an appropriate exact non-Markovian TCL ME describing the evolution of the qubit system. We further apply this result to quantify the fidelity loss that noise induces. In the final section, we discuss and conclude our findings.

Master equation of an electron spin in MoS 2 -QD
Our major aim is to study the HI-induced decoherence in a prototype model system for the spin-valley qubit hosted by MoS 2 -QD for the states , . In appendix A, the proper choice of the spinvalley qubit is discussed in detail. We assume that the initial energy of this problem is the total energy of the desired spin-valley qubits ¢  , which may be set as origin. In this case, the electron wavefunction is localized within a QD with R = 26 nm. The effective Hamiltonian for single-electron spin with a bath of a  spin-I 0 nuclei, through the contact HI, in a magnetic field b z along the z-axis, given by (setting ÿ = 1), x y z is the electron spin operator.
is the electron (nuclear) Zeeman splitting in a magnetic field b z , with an effective g-factor ( ) g g sp I for the electron (nuclei) and Bohr  (1), we have neglected the anisotropic HI, electron-electron interaction, dipole-dipole interaction between nuclear spins, and nuclear quadrupolar splitting [52,53]. Appendix C contains detailed calculations of this approximation. The overall activity of a  nuclear spin environment can be understood as a nuclear magnetic field. This is later called the Overhauser field [33,[54][55][56][57] z is the nuclear spin operator at the lattice site k at position r k . = S I I z k k z is the total z component of nuclear spin and A k is the associated hyperfine coupling constant. We also introduced the raising and lowering operators S ± = S x ± iS y and =   I I iI k k x k y respectively, and the nuclear magnetic field operators h ± = h x ± ih y . Therefore, we can rewrite equation (1) as follows: The third and the fourth terms in equation (2) are the hyperfine contact interaction between the spin electron and the nuclei in the QD, which describe the flip-flop interaction and (longitudinal) Overhauserʼs field, giving rise to the inhomogeneous broadening and dephasing, indeed the last term, å A S I k k z z k , produces an effective magnetic field for the electron is determined by the electron density at the nuclei site [58], which corresponds to the HI of an electron with nuclear spin at the site k with position r k . Here, is the two-dimensional volume of a unit cell containing one nucleus, and a is the lattice  nuclei within the dot. However, only the isotopes that allow nonzero nuclear spin will be part of these nuclei, 95 Mo, 97 Mo and 33 S. Furthermore, the concentration of 33 S is negligible compared to that of Mo isotopes, and the decoherence of the electron spin mainly originates from the presence of 95 Mo and 97 Mo nuclear spins [59]. This leads to the number of nuclear spins within the QD, is the envelope wavefunction of the localized electron, and is the total hyperfine coupling constant to a nuclear spin of species i k at site k [55]. μ 0 is the vacuum permeability. u i k is the amplitude of the periodic part of the Bloch function at the position of the nucleus of i k species. γ S is the gyromagnetic ratio of free electrons, and its value is always negative. However, the nuclear gyromagnetic ratio g i k can take either sign. As a result, the hyperfine coupling constant A i k might be positive or negative. For convenience, in a material containing several nuclear isotopic species, i k , we define an average hyperfine coupling constant. Here, we take the root-mean-square (RMS) average [52,60],  Γ i is the contribution of flip-flops between the nuclei of common species i. Figure 1(a) displays the decay rates for a MoS 2 -QD; the Mo isotopes have the greatest influence, whilst the S isotopes are negligible. The quadratic dependence on isotopic abundance δ i , shown in equation (3), is particularly an important factor in the decoherence rate. Due to this dependence, electron spins in MoS 2 will show decay mainly due to flip-flops between Mo spins, notably 95 Mo. Significantly, we can notice relatively high flip-flop rates for the Mo isotopes, as a result of a large nuclear spin 5/2 and isotopic natural abundance, respectively. Here, we considered isotopically purified Mo. It is possible to raise the T 2 of a single layer MoS 2 by performing isotopic purification [59]. Reduction of this noise is possible by fabricating devices using isotopically purified silicon [59,62]. Moreover, the decoherence time T 2 in MoS 2 -QD is several hundreds of ns, which is serval orders larger than the operation time in the optical quantum control of spin-valley qubits. Indeed, the coherence time increases with the number of nuclear spin N within the dot size R. For instance, in the MoS 2 crystal's conduction electron, T 2 = 307 ns for N = 1000 and T 2 = 97 ns for N = 100 [6]. We also notice that T 2 rises with fewer layers, although for MoS 2 , the enhancement for a single layer is only two times that of the bulk [59]. Additionally, according to the author J Pawłowski [63], the spin-valley qubit MoS 2 SWAP operations have the lowest fidelity, with a maximum time of 2 ns for 99%-fidelity. Compared to coherence time, operations should last substantially shorter. According to the hyperfine interaction decoherence from Mo nuclear spins, the electron spin-valley degrees of freedom in the MoS 2 monolayer have an estimated coherence time of 100 ns. Now, we consider a localized electron in its orbital ground state. For a MoS 2 -QD with parabolic confinement, the wavefunction for the ground state of a single electron isolated in the CB near the valley ¢ K under a magnetic field reads as follows  is the effective length scale equal to the magnetic length ℓ , therefore, the ground state wavefunction that will be considered for this work is ( ) Adding and subtracting å A I 2 k k k z to the total Hamiltonian (5) can be split into two main parts, an unperturbed part (longitudinal) represented by the first two terms, which consist of all Zeeman terms, and a perturbation part (transverse) containing the virtual flip-flop processes of the HI. In the interaction picture with respect to the unperturbed part, we can write is the down (upper) state of the central spin. In fact, . The transversal hyperfine term results in off-resonant transitions between the system and environmental spins, whereas the longitudinal hyperfine term provides additional contributions to the energy splitting.
The time evolution of the combined system, consisting of the electron spin and  nuclear spins, and given by the action of the total Hamiltonian tot I  in equation (6), is described in the following. (i) when t < 0 we assume that the electron spin and the nuclear system are decoupled, and both of them prepared independently in the states described by the density operators ρ S (0) and ρ E (0), respectively. (ii) At t = 0, the electron and nuclear spin system are brought into contact on a switching timescale is the largest energy scale in this problem. The state of the entire system,  (4), which in turn leads to a non-uniform HI between the electron spin and the nuclear spins. An external magnetic field is applied perpendicular (B z ) to the plane of the dot (x, y). described by the total density operator ρ(t), where it's given at t = 0, , this is called the reduced density matrix of the subsystem S and E, respectively. (iii) The evolution of the density operator ρ(t) for t 0 is governed by the Hamiltonian ( ) t tot I  for an electron spin coupled to an environment of nuclear spins.
This model is comparable to previous research investigations of an electron spin confined to a QD of Gallium arsenide (GaAs) [21] and graphene QD [22], but there are MoS 2 specific properties. Given that the natural abundance δ i of spin-carrying isotopes is small for molybdenum Mo and sulfur S, hence only  of all atoms tot  within the MoS 2 QD carry spin. However, in semiconducting materials such as GaAs, all isotopes possess a spin. To highlight these differences, we compare the most important characteristics of MoS 2 , graphene, and GaAs, which are given in table 2. Indeed, the HI coupling constant A MoS 2 , is about approximately the same magnitude as A Graphene and it's about two orders of magnitude smaller than the constant A GaAs in GaAs, which even further reduces the nuclear magnetic field by the same amount. Furthermore, the relatively small hyperfine energy t =  eV), in MoS 2 increases the timescale τ HF . It also relies on both the hyperfine strength A and the isotopic abundance δ i , of this interaction significantly compared to GaAs and graphene. The switching timescale τ sw for various materials is also listed in table 2 as well.
Here, we drive into the method that used alongside with this work. We suppose at t = 0, that the total system, electron and nuclear, describe by where |ψ S (0)〉, is the initial state for the central spin and |ψ E (0)〉 is the initial state for the nuclear spin bath where we start with a perfectly polarized nuclear ensemble, as shown in figure 1(b). Equation (7) can be written in the subspace spanned by the bases denotes the vacuum state of the bath. Noting that, c k (0) = 0 ∀ k. This means that the environment is initially in the vacuum state. Detailed calculations of the time evolution of the total system and the definition of the correlation function are provided in the appendix B. The correlation function in the continuum limit assumes the form is the spectral density of the bath given by the sum of (coupling strength) 2 (density of modes), which is therefore simply the Fourier transform of the correlation function  . Here, the parameter λ defines the width of the Lorentzian spectral density; in fact, it is the measure of memory capacity or non-Markovianity of the environment [51] and is related to the bath correlation time τ E = 1/λ. On the other hand, γ 0 measures the coupling strength between the qubit and its environment, and hence the characteristic time of the system τ S = 1/γ 0 denotes the relaxation time. Taking a Lorentzian spectral density in resonance with the qubit's transition frequency, we obtain an exponential two-point correlation function, denoted as , which defined as the solution of the integro-differential equation,˜( where ( ) c g l = -1 2 0 . To get an ME in differential form with a generator local in time, we first provide the precise time evolution mapping [50], which transforms the initial states into the states at time t  where ρ ij (t) = 〈i|ρ(t)|j〉 for i, j = 0,1. We can construct the exact TCL equation by introduce a time-local generator We can obtain an exact TCL ME to second order in the interaction picture The dynamics of the exact TCL ME, parameterized by , behave as a time-dependent Lamb shift caused by coupling with noisy surroundings, and , behave as a time-dependent decay rate (decoherence rate). Furthermore, the decay rate γ(t) might have negative values, indicating a significant non-Markovian tendency in the dynamics of the system [64]. We use the fidelity [51,66] ( ) ( )| ( )| ( ) y r y = á ñ t t 0 0  to measure the decoherence dynamics of the central spin. Indeed, the coherence of spin qubits is highly affected by the nuclear spins of the host material and their hyperfine coupling to the electron spin. When the system is prepared in the initial state |ψ(0)〉 = |1〉, the fidelity reads The fidelity depicted in figure 2, may be divided into two main regimes. In the weak-coupling case, which reflects that the decoherence dynamics of the quantum system is Markovian, λ > 2γ 0 , one has c   (c g l = -1 2 0 ) so that G(t) is always positive. When there is no longer any exchange between the qubit and his surroundings, the coupling strength γ 0 → 0 the fidelity ( )g  t lim 1 0 0  . However, in the case of strong coupling, λ < 2γ 0 one has c i  , so G(t) oscillates between positive and negative values reaching zero. This means the fidelity ( ) t  will then decay with an oscillating. Indeed, for λ/γ 0 < 1, ( ) t  decays non-exponentially and displays a clear beating pattern as shown in figure 2, indicating that the quantum information flow bounces back from the spin bath to the qubit system as the fidelity lifetime lowers. This is the signature of non-Markovian behavior.
We analyze the scenario in which the central spin of a MoS 2 -QD qubit overlaps with around  nuclear spins and interacts through HI. This may lead to entanglement between the qubit and the nuclear bath and to back-action effects from the qubit to the nuclei and vice versa. In this situation, we assume that the HI strength  follows, is usually understood as a measure of memory capacity or non-Markovianity of the environment. Then, the effective correlation function of the spin bath can be evaluated as The fidelity ( ) t  can be obtained numerically by inserting equation (17) into equation (10). To perform the numerical simulation, we selected the following MoS 2 parameters: The hyperfine coupling's strength has been estimated to be A = 0.29 μ eV. This estimate is based on an average of the hyperfine coupling constants for the two nuclear isotopes 95 Mo and 97 Mo, weighted by their relative abundance. The naturally occurring isotopes carry spin with ( ) Mo Mo 95 97 . In this model, we have the Overhauser field » » h I A A 0 5 2 . Here, we study the case of a localized electron spin trapped in MoS 2 -QD that interacts with  polarized nuclear spin environments via HI, where we employ the same TCL method as in some previous work [51].  Figure 3, for = 10 2  , shows that the oscillation of the real and imaginary components of the amplitude c 1 (t) decays non-exponentially and displays a clear beating pattern. This oscillation eventually decreases with time to an equilibrium value. |c 1 (t)| shows nonmonotonic oscillatory decay with zero coherence revivals, which occurs by the electron spin-flip transition. Remarkably, the dynamics describes the initial oscillations of c 1 (t) appearing in the non-Markovian description of open quantum systems. However, figure 3(b) depicts the oscillation for = 10 4  , where the decay rate of amplitude c 1 (t) increases significantly in comparison to figure 3(a) which depicts the transition from nonmonotonic oscillatory decay to monotonic decay. This indicates that increasing the number of nuclear spins  can decrease the quantum fluctuations caused by nuclear dynamics and boost the qubit's coherence.
To better understand how the environment of nuclear spins through HI affects the coherence of the MoS 2 qubit system, we analyze the temporal evolution of the density matrix ρ(t). Figure 4 illustrate the evolution of populations (diagonal elements of the density matrix) and coherence (non-diagonal elements of the density matrix) as a function of dimensionless time νt. Figure 4(a) illustrates that the oscillation decays nonexponentially for non-diagonal elements and exhibits a distinct beating. These beating patterns clearly originate from the peaked nature of the environmental spectrum. The ρ 11 (t) shows nonmonotonic oscillatory decay with zero coherence revivals, which reflects that the decoherence dynamics of the quantum system is non-Markovian. However, by increasing the number of nuclear spins  and examining the figure 4(b), we can see that the non-diagonal elements exhibit a damped oscillatory behavior with the disappearance of the beat pattern. We can attribute this effect to the fact that when we increase  , we go from a strong coupling regime between the qubit and the noisy environment described by the non-Markovian character to a weak coupling regime described by the Markovian character. Figure 5 show the fidelity ( ) t  as a function of dimensionless time νt induced by the non-Markovian HI for different values of the number of nuclear spins  within the bath. The value of ( ) t  exactly reflects the decoherence of the central spin. The closer the value of ( ) t  to 1, the smaller the difference between the current state and the initial state of the central spin. As shown in figure 5 we can clearly see that when = 10 2  , the fidelity shows nonmonotonic oscillatory decay with zero coherence revivals, meaning that the exchange of the quantum information and energy between the system and bath spins having a noticeable or major effect, the quantum information flow bounces from the spin bath back to the system. On the contrary, when  increases, the decay rate of the central spin gradually decreases, as shown by the transition from nonmonotonic oscillatory decay to monotonic decay. In addition, the coherence of the central spin remains robust for large  . These results indicate that the fidelity improves as the effect of environmental memory increases. The spin bath with  up to 10 6 (R = 230 nm) works as it provides natural protection for central spin coherence. For our qubit  scenario, the fidelity is represented in figure 5 with a solid purple line for = 10 4  (R = 26 nm). The varying of the nuclear spins  value depends on QD radius R as shown in appendix A, which mean by increasing R we increase  . It is also important to know that selecting R is necessary for implementing the MoS 2 spin-valley qubit.

Conclusion and outlook
In this paper, we have proposed an exact ME for a central spin coupled to a spin bath via HI. We focus on the TCL ME for the full polarized environment bath. By studying an ML-MoS 2 spin valley QD with a perpendicular external magnetic field that can be used to tune the energy splitting between these two states. We analyze the decoherence dynamics of the induced noise determined by the correlation function corresponding to this spinbath model, the uniform HI strength, and the Gaussian distribution in terms of the bath-spin frequency, respectively. Described by this noise, the effect of the spin bath on the central spin gives rise to a reduced dynamics. Furthermore, we have found that the Overhauser field in a QD system helps to restrict the decoherence process of the central electron spin, which can regain its coherence and retain its initial state in an environment with a larger number of bath spins. An obvious extension of this work is to use the fidelity to explicitly show the signature of non-Markovian behavior. As a consequence of this, the environmental non-Markovian feature can increase the coherence in the single-qubit dynamics. This model is qualitatively valid for other systems that satisfy the requirements of this later, where the most important demands are a Gaussian-like envelope function, slow dynamics of the nuclear bath, and sufficiently large Zeeman splitting with respect to the HI energy scale. The study of this qubit system in an unpolarized spin bath will be very interesting for future research, as it will allow us to see how the inhomogeneity of the hyperfine interaction affects the spin coherence and relaxation time.
Several methods and architectural designs have been proposed and tested experimentally to isolate a QD from TMD [67]. These include electrostatic gating [3,9,[68][69][70], strain bubbles [71][72][73], nanoflakes [8,63], lattice defects [74][75][76][77][78] or by forming potential wells with TMD lateral heterostructures [79]. Following the selection of a QD type, the next option for implementing a quantum processor in a TMD is a qubit space. The development of semiconductor nanostructures as low-noise hosts for qubits is a major undertaking. Understanding the band structure and external field replies is required for TMD monolayer (ML) to achieve qubits with carriers [3,80]. However, the strong intrinsic spin-orbit coupling (SOC) [81,82] within this TMD material has the opposite effect, making this a difficult task. Indeed, TMD is characterized by a significant splitting of the spin states ∼150 meV in the valence band (VB) [83,84] and up to a few tens of meV in the conduction band (CB) [85][86][87] within the same valley ( ¢ K K ), meaning that a single electron within a QD in TMD will not explain the required degeneracies wanted for a spin qubit. Favorably, the band crossing seen in the spin-resolved CB structures in ML-MoS 2 submit that is reasonable to accomplish spin degeneracy localized within a given valley ( ) ¢ K K , see figure 6(a). Consequently, such spin-degenerate regimes allow the possibility of realizing the desired qubits in the MoS 2 -QD [3].
In particular, the band structure shown in figure 6(a) is calculated based on a self-consistent scheme, the present first principle study consists in solving the Kohn-Sham (KS) equations by using the all electron Full-Potential Linearized Augmented Plane Wave as embedded in a WIEN2k [88,89] simulation package. The generalized gradient approximation framework with a Perdew-Burke-Ernzerhof functional is used for the exchange correlation potential [90].
The eigenenergies of a single electron confined in a MoS 2 -QD by the parabolic potential in a perpendicular magnetic field B = (0, 0, B z ), B z > 0, about the valley K and ¢ K can be obtained by solving the effective lowenergy Hamiltonian [3]   , denote respectively the parabolic confinement and the cyclotron frequency. Here, R is the QD radius. Thus, the QD eigenvalues as a function of the out-ofplane magnetic field B z and the QD radius R are given as (c) Part of the spectrum shown in (b) represent the ground state (n = 0, ℓ = 0) energy spectra experience a perpendicular magnetic field B z . Inset: region about which the twofold degeneracy localized to valley ¢ K is observed in the spectrum of the two state | ¢ñ K and | ¢ñ K .