Application of the Landau-Zener-Stückelberg-Majorana dynamics in an electrically driven flip of a hole spin
An idea of employing the Landau-Zener-Stückelberg-Majorana (LZSM) dynamics to flip a spin of a single ground state hole is introduced and explored by a time-dependent simulation. This configuration interaction study considers a hole confined in a quantum molecule formed in InSb quantum wire by application of an electrostatic potential. An up-down spin-mixing avoided crossing is formed by non-axial terms in the Kohn-Luttinger Hamiltonian and the Dresselhaus spin-orbit one. Manipulation of the system is possible by dynamic change of external vertical electric field, which enables the consecutive driving of the hole through two anticrossings. Moreover, a simple model of the power-law type noise that impedes the precise electric control of the system is included in the form of random telegraph noise to estimate the limitations of the working conditions. We show that in principle the process is possible, but it requires a precise control of parameters of the driving impulse.
The electric dipole spin resonance (EDSR) is a process in which the spin state of a quantum system is manipulated by the means of an ac electric field.edsr1 (); edsr2 (); edsr3 (); edsr4 (); edsr5 (); edsr6 (); edsr7 (); edsr8 (); edsr9 (); edsr10 () This process can utilize a few different mechanisms of electric-spin coupling: the spin-orbit interaction,edsr1 (); edsr3 (); edsr6 (); edsr7 (); edsr8 (); edsr10 () the spatial inhomogeneity of the applied magnetic fieldedsr2 (); edsr5 () or of the hyperfine interaction.edsr4 () If the frequency of the electric signal is resonant to the relevant energy difference of two levels with different spin, then a transition between the levels may be induced, depending on a certain set of selection rules.
A typical EDSR transition is done between two uncoupled spin states. However, when two levels are involved in an avoided crossing, then driving the system through this anticrossing is described by the Landau-Zener dynamics instead. When the driving is periodic, the system accumulates the Stückelberg phase between the transitions and this leads to a constructive or destructive interference, depending on the specific parameters of a given system. The theory related to systems of this kind is described in a review article of Ref. LZ_theory, .
Multiple harmonic generation in EDSR in an InAs nanowire double quantum dot was recently observed for conduction band electrons in double quantum dots.LZSM_elec_experiment () The harmonics display a remarkable detuning dependence: near the interdot charge transition as many as eight harmonics were observed, while at large detunings only the fundamental spin resonance condition was detected. In following theoretical studies the transport dynamics of a periodically driven system, modeling the level structure of a two-electron double quantum dot, was studied.LZSM_elec_theory (); rudner () It was shown that the observed multiphoton resonances, which are dominant near interdot charge transitions, are due to multilevel Landau-Zener-Stückelberg-Majorana interference. The main features observed in the experiments of Ref. LZSM_elec_experiment, were replicated: multiphoton resonances up to eight photons, a robust odd-even dependence, and oscillations in the electric dipole spin-resonance signal as a function of energy-level detuning.
The Landau-Zener dynamic was used to study the possibility of manipulating the S-T+ avoided crossing that arises due to the hyperfine interactions in a system of two electrons in a double quantum dot in GaAs.burkard-1 (); burkard-2 () The results concern a two electron Landau-Zener system with the spin-mixing singlet-triplet avoided crossing, resulting from the hyperfine interaction. In the both works, the necessity of going beyond the simplest infinite time Landau-Zener model is stressed out, and the finite-time Landau-Zener theory is employed. Moreover, the formulated master-equation formalism allowed to study the impact of phonon-mediated hyperfine relaxation and charge-noise-induced dephasing on the evolution of the system.burkard-2 () In the corresponding experimental work, Ref. Petta, , an all-electrical method for quantum control was presented that relies on electron-nuclear spin coupling and drives and drives spin rotations on nanosecond time scales. Interference patterns were observedPetta () in singlet-state occupation as a function of waiting time between consecutive sweeping the system back and forth through a singlet-triplet avoided crossing, due to phase accumulation, with agreement in the Landau-Zener theory.
In a recent work, a p-channel silicon metal-oxide-semiconductor field-effect transistor with a double dot in the channel, formed by a pair of defects or impurities, was studied.Ono () A two-spin EDSR was realized experimentally, with the main line as well as additional few-photon lines visible. A supression of the spin resonance was found in the viccinity of a singlet-triplet avoided crossing.
The EDSR manipulation scheme was realized for valence band holes in a quantum molecule created in a gated InSb nanowire.holes_exp () The mentioned work employed the strong spin-orbit coupling of this material for the spin flipping and measured the transport through the system as a result of lifting the Pauli spin blockade.
In this work we suggest a scheme for reversing the spin state of a single hole confined in two quantum dots with tunneling coupling, whose energy level structure is presented schematically in Fig. 1. We have taken advantage of an avoided crossing involving the states localized in the same dot but with an opposing spin characteristics. We propose an electrical control signal that leads to the transition from the ground state, of the heavy hole spin-down type, to the first excited state, of the heavy hole spin-up type. Instead of using a long periodic signal, as typically done in EDSR processes, the system is driven through two anticrossings only a few times by a short cosine impulse. High transition efficiency of about is obtained by the detailed balancing of the impulse parameters. The process is one order of magnitude faster than the alternative EDSR realized in the same system, which is especially important in the context of limited spin coherence time when performing spin operations.
iii.1 Geometry of the system
We consider a system of a single hole in a quantum molecule consisting of two quantum dots coupled vertically. The dots are made by applying an external electrostatic potential to an InSb quantum wire of a zincblende crystal structure. The nanowire is assumed to have circular shape in cross-section and a radius of nm. The geometry of the system is similar to the one considered in Ref. holes_exp, .
We model a confinement potential of two vertically stacked quantum dots in the form of an infinite circular quantum well in the plane (corresponding to the cross-section of the wire) and two finite quantum wells along the axis (the growth axis of the wire). The zero of the energy scale is set to the degenerated top of heavy and light hole bands outside the dots. The total potential is , where is the unity matrix,
In the equation above, the part corresponds to the shape of the confinement of one of the dots and part corresponds to the shape the confinement of the other one. The meV is the depth of the confinement. The nm parameter describes the separation of the dots and the nm describes the width of the dots. The potential is presented in Fig. 2(a) and the shape of the dots in Fig. 2(b).
The adopted potential defining the system is symmetric in respect to reversing the nanowire axis. In any experimental realization, the potential for each dot would be slightly different. We have studied the impact of the asymmetry of the dots in Appendix D.
iii.2 Kohn-Luttinger Hamiltonian
We work in the effective mass approximation. The kinetic energy of holes is calculated using the 4-band Kohn-Luttinger Hamiltonian.chuang () The Hamiltonian for the crystal orientation is given (in atomic units) by:
where are the spin matrices for spin , , are the Luttinger parameters, and .KL111-1 (); KL111-2 () To obtain the expression of the same Hamiltonian for orientation, one should express the and vectors of the orientation in the terms of and vectors of the orientation, respectively (see Appendix A).
If written in the basis the Hamiltonian for the orientation has the following form
where the operators used in this definition are listed in Table 1.
The Kohn-Luttinger Hamiltonian can be divided into two parts:
The part is axially symmetric and hence its eigenstates have defined -components of total angular momentum , which is the sum of Bloch and envelope -components. For this reason, the computation of the system described by is much easier as the process for each one of subspaces can be done separately. The envelope eigenfunctions of are 4-dimensional vector functions:
Moreover, the non-axially-symmetric part is relatively small: the constant in is about 14 times greater than the one in , and the constant in about 54 times greater than the one in .
The diagonal terms of the Hamiltonian:
have the corresponding envelope eigenfunctions:
where is a Bessel function of the first kind of -th order and is the -th zero of that function. The quantum numbers order the functions by ascending energy and (separately) order the functions in the same way.
Finally, the Hamiltonian for the axially-symmetric static system is
where and are the magnetic field and electric field Hamiltonians (as defined below), respectively. As the last two terms do not mix states with different quantum numbers, the eigenfunctions of Eq. (9) also have the form presented in Eq. (6).
iii.3 Electric and magnetic fields
The Hamiltonian of the external electric field, applied along the growth -axis, in atomic units, has the form of
where is the electric field amplitude and is the shape function. The has such character that it simulates an electric field that: I) is homogeneous in the area of the dots, II) is continuous everywhere up to second derivative and III) decreases as in area far from the system. The is the simplest polynomial that meets the continuity assumptions.annotation1 ()
The Hamiltonian of the homogeneous magnetic field , in atomic units, is given by:
where is the -factor for heavy and light holes in the system. This is a model that was used in Ref. climente_magnetic, , but with two changes: I) the inverted effective mass values for heavy holes in plane and the light holes one for the orientation were substituted by analogous values for the system (i.e. and , respectively) and II) we omit the terms proportional to as they are very small for the range of magnetic field that was considered.annotation2 () The -factor for bulk InSb is equal to , but in a system of this type the value is significantly quenched, i.e. .holes_exp () The Lande value, which does not take into account the influence of remote bands, is . We decided to adopt a middle value of .
iii.4 Dresselhaus Hamiltonian
In order to account for the mixing of the states with different spins, the Dresselhaus Hamiltonian was included. In the case of the crystal orientation it has the form of
where , , , , are material parameters, and c.p. stands for cyclic permutations of the preceding terms.climente_dresselhaus (); winkler () The procedure for obtaining the Hamiltonian for the orientation is the same as for the Kohn-Luttinger Hamiltonian (see Appendix A). It leads to the following result:
where the element operators are defined as follows:
where constants and to are defined in the terms of , , , (see Table 2). Please note, that if the angular dependencies of and states are and , respectively, then the matrix element is nonzero only for , the matrix element is nonzero only for , the and the matrix elements are nonzero only for . This leads to a significant simplification of the Hamiltonian, see Appendix B.
iii.5 Computational method
Our computational method consists of several separable steps. At the beginning, the one-band hole Hamiltonian eigenequations Eq. (7) are solved. The and functions in Eq. (8) are determined by direct diagonalisation on one-dimensional mesh with mesh spacing nm and computation box of . Afterwards, the eigenfunctions of the axially-symmetric Hamiltonian Eq. (9) are obtained in a base constructed by taking functions of type as in formula Eq. (8) with , , and . As it was mentioned before, each defines a separable subspace and the quantum number sorts the eigenfunctions of each subspace in the order of ascending energy.
The next step is to include the non axial part into the calculation. This part consists of the small non-axial terms in Kohn-Luttinger Hamiltonian and of the Dresselhaus Hamiltonian:
This operation is made in a basis consisting of selected set of lowest-lying eigenstates:
where is the basis for the non-axially-symmetric calculation and is the projection of the -th non-axial state onto individual basis state.
The evolution simulation of the system is done with the Runge-Kutta method of the -th order with the time step of fs. The basis for the evolution is the set of states obtained in the non-axially-symmetric calculation, without the external electric field:
where is the evolution basis and is the projection of the time-dependent state onto individual -th basis state (in order of ascending energy). The specific algorithm for the evolution of projections is given below:
Please note that all results for the evolution are presented in local basis, and not in the basis, for ease of interpretation. The projections , obtained as have been explained above, are recalculated to represent the projections as if the levels at each given were the basis states instead of the states. In this way, one can refer to these projections as corresponding to the hole energy spectrum in each point of the axis.
iv.1 Time-independent system
We begin by studying the energy spectrum with a static electric field applied. The energy spectrum of the hole system for the axially symmetric Hamiltonian [see Eq. (9)] is presented in Fig. 3(a). The set of levels with the lowest energies has the following elements: (,), (,), (,), and (,). This set of levels is separated energetically from the next ones for any in the considered range by about meV. The characteristics of the four lowest-lying eigenstates are given in Table 3. In every case, the dominating valence band is the one with the lowest value (that equals for the first four levels), and, in each case, it is one of the heavy hole bands.
|set of values|
The main two features of this spectrum are the two avoided crossings: the one of the two levels [marked as A in Fig. 3(a)] and the one of the two levels [marked as B in Fig. 3(a)]. The mentioned avoided crossings occur due to the tunneling coupling between the dots. At the confinement potential of the system is symmetric in respect to , hence the eigenfunctions are equally distributed between both dots. For , away from the crossing, the hole is localized in the energetically preferable dot in the ground level of each subspace (i.e. levels with ) and in the energetically impreferable dot in the excited levels (i.e. levels with ). The situation is reversed for .
In the case of the non-axially-symmetric calculation, due to computational constraints, we are interested in the lowest-lying states only. Because of the energy separation of about meV, the first four energy levels, shown in Fig. 3(a), create a natural basis for this calculation. Thus we define the basis in Eq. (16) as the set of levels listed in Table 3.
The hole energy spectrum for the total Hamiltonian is presented in Fig. 3(b). In comparison to the axial one [see Fig. 3(a)] the only important difference is the formation of two additional smaller avoided crossings, marked as C and D in Fig. 3(b). These anticrossings correspond to the mixing of the (,), that is the spin-down state, and (,) state, that is the spin-up one. Apart from that, the energy shifts are very small, and the Fig. 3(b) spectrum is nearly the same as the Fig. 3(a) one.
The one-band probability densities, integrated over the coordinate:
are shown for kV/m in Fig. 4(a) for the ground state, and in Fig. 4(b) for the first excited level, respectively. The ground state of the hole is localized in the dot and is strongly dominated by the band. The first excited state is localized in the same dot and is strongly dominated by the band. This corresponds to our idea to flip the spin of a hole by transferring it from the ground state to the first excited level. This would result in reversing the state from being dominated to being dominated, while remaining in the same dot. In order to do so, we plan to employ the Landau-Zener transitions of A and D avoided crossings, see Fig. 3(b).
The idea of the spin flip is presented in Fig. 3(b). The initial state of the simulation is the time-independent ground state at kV/m and the intended final state is the time-independent first excited state at the same electric field. The transfer is planned to be made in five steps. It should start with tuning the electric field to the side of the larger avoided crossing in such a way that the time-dependent state would remain equal to the time-independent ground level [arrow in Fig. 3(b)]. The second step consists of using this anticrossing to transfer the time-dependent hole state to the first excited level at the same electric field [arrow in Fig. 3(b)]. The third stage is the drive of the hole from the tunneling-generated anticrossing to the lower side of the smaller spin-mixing one [arrow in the inset of Fig. 3(b)]. After that, the hole state should be transferred across the smaller avoided crossing without the leak of the time-dependent state to another time-independent levels [arrow in Fig. 3(b)]. It is here that the spin flip takes place. The final step is the drive of the system to kV/m [arrow in Fig. 3(b)].
Each of stages was optimized separately for transfer efficiency in terms of relevant parameters (see Appendix C) and the total driving impulse was constructed by joining all the parts together. The evolution of the total transfer is presented in Fig. 5. The initial state is equal to the time-independent ground state , that is . After the evolution, the evolving state ends in the first excited state with . This means that the system started and ended evolution in the same electric field. Both the initial and the final states are localized in the same dot. However, the system started evolution in a state dominated by the spin state and it ended it in a state dominated by the opposite spin state. This proposition of the process that reverses the spin state of the hole is the main result of this work.
iv.3 Comparison with the EDSR of uncoupled levels
The Landau-Zener type of spin-flip is an alternative for the EDSR one. In case of the latter the levels involved in transition are not engaged in an avoided crossing. In order to make the comparison between these two mechanisms, the transition in Fig. 3(b) was calculated. The frequency of the driving signal
is tuned to resonance with the energy difference between the final and the initial level: GHz for the offset kV/m. The amplitude of EDSR signal is kV/m. The results are presented in Fig. 6. The evolving state starts in the ground state of the time-independent system and then typical Rabi oscillations begin. The evolving state occupies the first excited state after ns, which corresponds to about periods of the signal function. Please note that this time is about one order of magnitude larger that the time of the total evolution for the scheme.
v.1 The models of the noise
The evolution simulation assumes total control of the driving electric field . In an experiment, such a precise control is impossible. The impact of the power-law noise on the effectiveness of the transfer is studied by implementation of a simple random telegraph noise model (RNT), as described in Ref. RTN, . According to the model, the actual time dependence of the electric field is given by:
where is the non-distorted electric field drive, is the jump amplitude of RTN,
is the electric distortion of the RTN, stands for the Heaviside step function and the time of the -th jump is defined as follows:
In Eqs. (22) and (23) the variable is the current iteration of random generation of a set of numbers from a uniform distribution over range and the sign of the first jump is determined randomly. The characteristic frequency is related to the average number of jumps occurring during the evolution time :
It was shown that this model simulates the power-law noise well for a sufficiently high amount of jumps per one evolution ( GHz).RTN () On the opposite end of the scale, where the frequency of the noise change is low, e.g. if of cases have no jumps ( MHz), a different model was adopted. The little variability of the noise signal during the evolution can be simulated by adopting a static shift in electric field:
where the sign of the shift is determined randomly.
v.2 The noise simulation
The results for the RTN noise model are presented in Fig. 7. The final efficiency was averaged over simulations for each pair of and values. Keeping transfer efficiency equal, an increase in leads to an increase in . The mechanism of the observed effect is very similar to the one responsible for the motional narrowing effect in magnetic resonance (see e.g. Ref. motional, pp. 212-213 and Appendix E). If the noise changes very quickly, then the system does not adapt to each individual shift value. The mean value of the noise shift is equal to zero, and so the overall effect of the noise is diminished in comparison to the noise with lower .
The results for the static offset are presented in Fig. 8. The value of the final projection was averaged over both possible signs of . The effectiveness of the operation is nearly one for and it drops to nearly-zero as increases. The condition for high fidelity is in approximation kV/m and if kV/m, then the probability of a successful operation is less than a half. These relations may be seen as estimates for the necessary conditions of an electric field control in any experimental realization of the presented scheme.
The imperfections in the control process can be intuitively divided into two categories. The first one can be thought of, in a simplified way, as a systematic error type: instead of the desired value, the system is tuned to at a given time and the lifetime of the error value is large. The second type consists of errors constantly oscillating around the correct value (i.e. RTN model). We have shown that the first type of error, the static one, is more destructive to the described process than the second one. Thus the experimental setup used for realization of the proposed scheme should especially minimize the systematic kind of control error.
The values of the parameters of any given realization of the quantum dot system are not perfectly known a priori. This relates to the exact size and geometry of the dots, the confining potential, the precise value of the -factor, among others. Fortunately, the presented scheme is – on the general level – adaptive to the specifics of a given system. The only necessary condition is that the four lowest-lying eigenstates of a system need to be qualitatively similar to the ones presented in Fig. 3(b) for some magnetic field and some electric field range. That is, the two avoided crossings used for the transitions need to be present in the energy spectrum. Unfortunately, the efficiency of the whole process is strongly dependent on the specifics of the driving impulse, and these specifics depend in turn on the details of the previously mentioned parameters of the system. Thus, for a practical realization of this idea, it is necessary to study a given system experimentally in order to establish reliable estimates of the parameters. This is especially true in the case of the characteristic energy of the avoided crossing C in Fig. 3(b). Any imperfections in the axial symmetry of the nanowire shape as well as the piezoelectric effects (also breaking this symmetry) will contribute to the mixing of the states with different values. In practice, it would be most efficient to take the approach of Ref. LZSM_elec_theory, , i.e. to treat the anticrossing energy as a fittable parameter and try to deduce its value from experimental data. After that, one can employ the presented scheme and optimize each of the steps and join them together as has been presented above.
Vii Summary and Conclusions
In the presented work a system of double quantum dots, created in an InSb nanowire by application of an external potential, was investigated. The energy spectrum of the system in a static electric field, applied in the direction of the wire, was obtained. The presence of the non-axially symmetric terms in the overall Hamiltonian leads to the formation of an avoided crossing in the spectrum, which involves two states of opposite spin states, in addition to the tunnel-coupling one. A scheme for reversing the spin state of a hole by manipulating the evolution with electric field was proposed, based on driving the hole state through two anticrossings. The results provided show that a perfect realization of the process, with an exact control over the electric field, is possible and the total process time is one order of magnitude smaller than the realization time of an alternative classic EDSR approach. The impact of an imperfect control of the driving factor was studied with two simple models that correspond to two different kinds of error, respectively.
Appendix A The transformation of Kohn-Luttinger and Dresselhaus Hamiltonians from to direction
The 4-band Kohn-Luttinger Hamiltonian for the crystal orientation is given by Eq. (3) and the Dresselhaus spin-orbit Hamiltonian in this orientation is given by Eq. (12). Please note, that both are defined in the terms of and vectors. To transform the Hamiltonians to the crystal orientation, one needs to express the coordinates of these vectors for in terms of their coordinates for the orientation. Within the scope of this appendix, the