## Abstract

Motivated by the progress on shortcuts to adiabaticity, we propose three schemes for speeding up (fractional) stimulated Raman adiabatic passage, and achieving rapid and non-adiabatic creation and transfer of maximal coherence in a triple-quantum-dot system. These different but relevant protocols, designed from counter-diabatic driving, dress-state method, and resonant technique, require their own pumping fields, applied gate voltages and varying tunneling couplings between two spatially separated dots. Such fast and reliable shortcuts not only allow for feasibly experimental realization in solid-state architectures but also may have potential applications in quantum information processing and quantum control.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Precise preparation and coherent control of quantum states have emerged as an essential physical basis for quantum information processing and quantum computation [1]. Adiabatic passages are robust against the fluctuation of control parameters, compared with the resonant *π* pulse and its variations, when the long operation time is requested to fulfill the adiabatic criteria. Among them, the conventional rapid adiabatic passage, stimulated Raman adiabatic passage (STIRAP) [2] and fractional STIRAP (F-STIRAP) [3], invented in quantum optics over two decades ago, provide versatile tools for controlling two or three-level systems in cold atoms and other systems, including charged ions, artificial atoms, coupled waveguide, and electron charge and spins, also see recent reviews [4–6].

Quantum dots (QDs), regarded as confined artificial atoms in three dimension, provide an ideal platform to implement qubits for quantum computing and quantum information processing. For instance, the coherent generation and manipulation can be achieved in a double QD by adjusting the tunneling between the neighbouring dots through the tunable electric gates [7]. Also such coherent control can be extended to triangular triple quantum dots (TQDs), in which Kondo effect [8,9] and many-body effects such as many body interference [10,11] and channel blockade [12] are demonstrated due to its symmetric geometry. Adiabatic passage has been used to realize electron transport in linear-arranged TQD [13–17], triangle-arranged TQD [18,19], and a QD donor chain [20,21]. Nevertheless, the demanding on shorter time never ends not only for pursuing more efficient operation but also avoiding the decoherence. Therefore, the optimal control theory [22] and non-adiabatic passage [23] has been proposed in solid-state devices for fast electron shuttling and quantum teleportation.

Motivated by the techniques of “shortcuts to adiabaticity” (STA) [27], we shall investigate the fast non-adiabatic but high-fidelity preparation and manipulation of quantum states in a TQD. In analogy with the previous treatment of the three-level system [18, 19], the pumping laser fields and tunneling couplings are designed to speed up the conventional F-STIRAP and STIRAP schemes for creation and transfer of coherence. Here we begin with the counter-diabatic driving (equivalent to transitionless quantum algorithm) [28–32], since such protocols have been experimentally demonstrated in the systems of cold atoms [33, 34] and spins of a single NV center in a diamond [35, 36]. Theoretically, QDs in the presence of external field and spin-orbit coupling also provide the flexility to implement the complementary interaction by unitary transformation [37], for achieving fast non-adiabatic spin control [38] and electron transport [39]. In addition, other shortcuts designed from dressed states [40] and resonant technique [23] are discussed and compared, to show their relevance and features.

## 2. Model and hamiltonian

Quantum state preparation and transfer of charge in a triangular TQD have been widely studied in the literature [24–26]. Here we consider the TQD system, composed by three QDs arranged triangularly, as shown in Fig. 1(a), fabricated on a GaAs/AlGaAs two-dimensional electron gas wafer. The TQD is defined by 15/30 nm thick Ti/Au metallic gates, which can be patterned by electron-beam lithography [25]. An exciton in QD1 is produced by shining a pumping laser with the Rabi frequency Ω_{1}(*t*). With the applied voltage controlled by gate electrodes, the energy levels of the conduction band are in resonance so that the electron can tunnel through the barrier between QD1 and QD2 or between QD1 and QD3. Meanwhile, the valence-energy levels are off-resonant so that the hole remains in the first dot and tunneling to other two dots is neglected. The tunneling pulses Ω_{2}(*t*) (Ω_{3}(*t*)) between QD1 and QD2 (QD3) can be manipulated by varying the bias voltage of external electrodes. This four-level system, indicated in Fig. 1(b), is comprised of: the ground state |0〉 without any excitations in any dots; the first excited state |1〉 where the exciton, including both the electron and the hole, is in the first dot; the states |2〉 and |3〉, where the electron is in the second and the third dot, respectively. Therefore, this semiconductor architecture allows for the creation and transfer of coherence, and total Hamiltonian for the four-level system with the pumping laser field and the tunneling pulses is thus expressed as [19]

*ω*corresponds to the energy difference between the states |

_{ij}*i*〉 and |

*j*〉, the detuning is

*δ*=

_{p}*ω*

_{10}−

*ω*

_{1}, and

*ω*

_{1}is the frequency of pumping pulse. In the case of resonance,

*ω*

_{12}=

*ω*

_{13}= 0,

*ω*

_{10}coincides with

*ω*

_{1}, so that

*δ*are set to zero. The solution of the time-dependent Shrödinger equation of

_{p}*H*

_{0}in the basis of on-site state |

*j*〉 (

*j*= 0, 1, 2, 3) is written as where

*c*(

_{j}*t*) is the probability amplitude of each state. Correspondingly, the populations on each state is

*P*(

_{j}*t*) = |

*c*(

_{j}*t*)|

^{2}. Since the initial state is |0〉, populations at the initial time are

*P*

_{0}(0) = 1 and the others are zero.

Under the resonant coupling of a pump laser field with QD1, an electron is excited in QD1. Then, the electron can be transferred to QD2 and QD3 with the tunneling. The coherent control of quantum states is naturally divided into two steps of creation and transfer. Step I is to generate $|{\mathrm{\Psi}}_{1}\u3009=\left(|0\u3009-|2\u3009\right)/\sqrt{2}$ with the pumping laser pulse Ω_{1} and the tunneling pulse Ω_{2}. Step II is to finally achieve $|{\mathrm{\Psi}}_{2}\u3009=\left(|0\u3009-|3\u3009\right)/\sqrt{2}$ with the tunneling pulses Ω_{2} and Ω_{3}. As each step evolves as a three-level system, we consider the following Hamiltonian:

_{0}

*t*≫ 1 is satisfied, with the maximum value of pulse intensity Ω

_{f}_{0}[2]. Here we choose such two-step scheme to create the coherence between ground state |0〉 and state |3〉, in order to demonstrate that our shortcut protocol works for both F-STIRAP and STIRAP. Of course, there are alternative ways to create the coherence among multiple states by using single F-STIRAP or STIRAP.

Using the strategy of F-STIRAP and STIRAP respectively, we can achieve the two steps with the fidelity above 0.9999 by choosing the pulses of hyperbolic functions, as plotted in Fig. 2(a),

*P*

_{0}= 1 initially to

*P*

_{2}= 1/2 at step II, as described by Fig. 2(b). With the maximal amplitude ${\mathrm{\Omega}}_{0}^{I}=5$ and ${\mathrm{\Omega}}_{0}^{\mathit{II}}=10$, the operation times are ${t}_{f}^{I}=100$ at step I and ${t}_{f}^{\mathit{II}}=100$ at step II to guarantee the adiabatic condition. Here, we should note that the Rabi frequency, describing the pulse intensity, is scaled by 5 × 2

*π*MHz. As a consequence, the maximal pulse intensity in the STIRAP protocol is 10, which corresponds to 50 × 2

*π*MHz. And the operation times ${t}_{f}^{I}={t}_{f}^{\mathit{II}}=100$ are in units of 2

*π*/Ω

_{0}(= 0.01

*μs*), corresponding to the timescale of 1

*μs*. Shortening the operation time to ${t}_{f}^{I}=1$, the adiabatic condition is broken down, and the target state cannot be reached at the final time at all. Therefore, the techniques of STA are strongly demanding, especially when the dephasing effect is considered.

## 3. Shortcuts to adiabaticity

In this section, we shall design the shortcuts to adiabatic creation and transfer of coherence in a TQD. Based on the adiabatic basis, three protocols in the TQD system are presented to achieve the maximal coherence between |0〉 and |3〉 from the initial state |0〉. The first protocol is counter-diabatic driving along the unitary transformation [37]. The second protocol is to accelerate the adiabatic passage by using dressed-state method [40]. The third protocol is to use a resonant technique, in analogy to the precession of spin 1 [23]. Each STA protocol requires its own modified laser field and tunneling coupling, but the relevance and feature are clarified by comparison.

#### 3.1. Protocol a

The Hamiltonian (3), corresponding to the first step, has instantaneous eigenstates

*E*

_{0}= 0 and

*E*

_{±}(

*t*) = ±Ω(

*t*), where the mixing angle tan

*θ*= Ω

_{1}(

*t*)/Ω

_{2}(

*t*) and $\mathrm{\Omega}(t)=\sqrt{{\mathrm{\Omega}}_{1}^{2}(t)+{\mathrm{\Omega}}_{2}^{2}(t)}$. The state evolution in STIRAP can be along the dark state of the instantaneous eigenstates, when the adiabatic criteria mentioned above is fulfilled. In the following sections, we take such instantaneous eigenstate as reference to speed up the adiabatic passage and design adiabatic-like evolution but within shorter time, at least having the same initial and final states coinciding with the dark state (9).

By introducing the unitary transformation,

_{+}〉, |D〉, |B

_{−}〉}, ${\mathscr{H}}_{\mathit{ad}}^{I}=A(t)H(t){A}^{\u2020}(t)-i\hslash A\frac{\partial}{\partial t}{A}^{\u2020}(t)$, that is,

*λ*

_{1,5,6}are the Gell-Mann matrices, represented as,

The combined Hamiltonian can take the state along the instantaneous eigenstate of ${H}_{0}^{I}$, within shorter time, which means the adiabatic transfer can be accelerated. However, the additional counter-diabatic terms becomes very complicated, hard to implement or even unphysical. Following [41], we choose

*H̃*=

*ħ*[Ω̃

_{1}(

*t*)

*λ*

_{1}+ Ω̃

_{2}(

*t*)

*λ*

_{6}− Ω̃

_{a}λ_{5}], where the pumping field and tunneling pulse are modified as

*= 0, yielding tan*

_{a}*φ*=

*θ̇*/Ω

_{1}, so that the effective pumping and tunneling pulses have the following forms: The boundary conditions $U(0)=U\left({t}_{f}^{I}\right)=1$, that is $\varphi (0)=\varphi \left({t}_{f}^{I}\right)=0$, should be satisfied to guarantee that the populations at initial and final times are the same before and after the transformation. Looking back to the three-level system, we solve the time-dependent Schrödinger equation of Hamiltonian Eq. (3) with the modified pulses Ω̃

_{1}and Ω̃

_{2}in Fig. 3(a). By using such accelerated F-STIRAP, the maximal coherence between |0〉 and |2〉 is achieved at ${t}_{f}^{I}=1$, see Fig. 3(b). There exists intermediate excitation of population at the state |1〉, which implies that the dynamics is not the one before transformation.

Similarly, we derive the modified pulses,

for step II, corresponding to the Hamiltonian ${H}_{0}^{\mathit{II}}$, where*φ*

_{2}= arctan(

*θ̇*/Ω

_{2}). With modified pulses in Fig. 3(a), the population of level |2〉 is completely transferred to |3〉 by accelerated STIRAP, and eventually the maximal coherence between |0〉 and |3〉 is achieved at ${t}_{f}^{\mathit{II}}=1$, see Fig. 3(b).

#### 3.2. Protocol b

Next, we shall construct STA for fast non-adiabatic creation and transfer of coherence by using dress-state method [40]. In step I, the Rabi frequency of pumping laser Ω_{1} and the tunneling pulse Ω_{2} can be parameterized by the time-dependent frequency Ω and the angle *θ*,

*J*,

_{x}*J*and

_{y}*J*are spin 1 generator matrices, satisfying SU(2) algebra, [

_{z}*J*,

_{μ}*J*] =

_{ν}*iJ*, with the structure constants

_{γ}ε_{μ}_{νγ}*ε*and

_{μ}_{νγ}*ν*=

*x*,

*y*,

*z*[42]. Since the term

*J*is responsible for the non-adiabatic coupling, the protocol A is applicable. One can add the additional term to cancel the non-adiabatic coupling and further apply the unitary transformation to modify the two original pulses of pumping laser and tunneling couplings.

_{y}Here we choose an alternative correct Hamiltonian [40],

in the basis of dressed state, which is different from typical counter-diabatic term, −*θ̇J*. However, in order to cancel the non-adiabatic coupling but provide more flexibility, the time-dependent unitary transformation

_{y}*V*(

*t*), parameterized as a rotation of the spin with Euler angles,

*ξ*(

*t*),

*μ*(

*t*) and

*η*(

*t*), is introduced, as

*μ*(0) =

*μ*(

*t*) = 0 and the other two angles having any values. In addition, the boundary conditions,

_{f}*V*(0) =

*V*(

*t*) = 1, must be satisfied so that the system dynamics is the same one before and after the transformation. After the unitary transformation, the total Hamiltonian in the adiabatic basis becomes

_{f}_{1}= Ω̃ sin

*θ̃*and Ω̃

_{2}= Ω̃ cos

*θ̃*. In addition, the control parameters are also obtained as

*P*

_{1}= sin

^{2}

*μ*cos

^{2}

*ξ*. For simplicity,

*ξ*is set to zero. Of course, this protocol B is the same as protocol A, when

We solve the time-dependent Schrödinger equation of Hamiltonian Eq. (3) with the modified pulses Ω̃_{1} and Ω̃_{2} in Fig. 4(a) and obtain the maximal coherence between |0〉 and |2〉 at ${t}_{f}^{I}=1$, indicated in Fig. 4(b), where *P*_{1} = sin^{2} *μ*.

For step II, the tunneling pulses are parameterized as

*μ*= − arctan(

*θ̇*

_{2}/Ω) and obtain Fig. 4, with the modified Ω̃

_{2}and Ω̃

_{3}, in which the populations transfer from ${P}_{2}\left({t}_{f}^{I}\right)=1/2$, to ${P}_{3}\left({t}_{f}^{\mathit{II}}\right)=1/2$, keeping

*P*

_{0}(

*t*) = 1/2.

Moreover, the dressed-state approach provides additional freedom to suppress the population excitations in the intermediate state by setting

*A*> 0 is a tunable parameter to lower the value of

*μ*and

*τ*is the operation time in each step. Consequently, the control parameters are derived as

*μ*, in order to decrease

*P*

_{1}= sin

^{2}

*μ*. As a matter of fact, when the operation time is shorter and more intermediate excitation is suppressed, the pulse intensity will be dramatically increased, bounded by time-energy uncertainty relation [38]. Here we set that the pulse intensity is scaled by 5 × 2

*π*MHz and the maximal amplitude of the designed pulses is about 1 ∼ 10 GHz, when the operation times are ${t}_{f}^{I}={t}_{f}^{\mathit{II}}=1$, corresponding to 10 ns. This is completely feasible in state-of-the-art experiments, see Ref. [43]. Fig. 5 illustrates an example to create the desired states and to keep the intermediate states lower than 1% simultaneously, where

*A*is chosen 15 and 55 at Step I and Step II, respectively.

#### 3.3. Protocol c

Next, we turn to simple resonant technique, inferred from the dressed state. In the adiabatic basis, the Hamiltonian is

which is in analogy to spin 1 in a magnetic field. The spin precession is used to get the adiabatic-like result of coherence transfer. In detail, when the initial state begins with the eigenstates of*J*, and the following resonant condition holds [23], with

_{z}*k*= 1, 2, 3, ..., the final state at

*t*=

*t*will coincide with dark state, which is exactly the eginestate of

_{f}*J*. This resembles a modified resonant pulse, and provides an alternative shortcuts to adiabatic passage. In step I, we choose the pumping laser and the tunneling pulse in the form of trigonometric functions,

_{z}*n*= 1, $\omega =\pi /\left(4{t}_{f}^{I}\right)$, as a result, $\theta \left({t}_{f}^{I}\right)=\pi /4$ due to

*θ*=

*ωt*. Under these conditions, the state evolves from (1, 0, 0)

*to ${\left(1/\sqrt{2},0,-1/\sqrt{2}\right)}^{T}$ at $t={t}_{f}^{I}$.*

^{T}Again, we have similar expression of operating time for step II,

But the frequency of the pulses, $\omega =\pi /\left(2{t}_{f}^{\mathit{II}}\right)$, is required to adapt the condition, $\theta \left({t}_{f}^{\mathit{II}}\right)=\pi /2$, for the complete population transfer.Figure 6 demonstrates the creation and transfer of coherence by using trigonometric pulses, where the maximum values of pulses are Ω = 5, and Ω = 10, respectively, for two steps, and the corresponding operation times, ${t}_{f}^{I}=1.24$ and ${t}_{f}^{\mathit{II}}=0.61$, are shown in Fig. 7. The resonant condition and oscillation of fidelity, relevant to spin-1 precession, can be explained by an SU(2) group method, see Ref. [44]. Also the resonant technique is similar to the multi-mode driving, based on Lewis-Riesenfeld invariants and inverse engineering [45].

## 4. Discussion and conclusion

Regarding the techniques of STA, we have developed the method for speeding up the conventional F-STIRAP and STIRAP, which is extremely useful for creating and transferring the maximum coherence between two states in fast and reliable way. In general, the fidelity can be improved by shortening the operation time, when the decoherence effect is considered [46]. In addition, we focus on three protocols, including counter-diabatic driving, dressed-state method, and resonant technique. In the adiabatic frame, we clarify their relevance, feature and difference. Clearly, counter-diabatic driving and dressed-state method requires supplementary interaction to cancel the diabatic transition, while the resonant protocol does not. In detail, counter-diabatic driving provides the shortcut in which the state evolves exactly along the dark state of reference Hamiltonian. But after unitary transformation, the intermediate state will be excited and only the initial and final state coincide with the dark state, due to the boundary conditions. In the dressed state approach, more freedoms in corrected Hamiltonian and unitary transformation are introduced, which can be useful to suppress the excitation of intermediate state [40]. Slightly different, the physics of resonant technique is analogous to spin precession, which is straightforward but irrelevant to adiabatic reference. Here we emphasize that there are other methods for STA, i.e. inverse engineering and fast-forward protocols [27], though some techniques are mathematically equivalent. But each system requires its own appropriate protocol based on the criteria, constraints or/and optimization, so that the physical implementation could be totally different and unique as well.

To summarize, we have investigated the fast non-adiabatic creation and transfer of quantum states in a TQD by using the techniques of STA. We speed up the F-STIRAP in the first step to create the coherence between the ground state and one indirect exciton state, that is, achieve the maximum coherence between the state |0〉 and |2〉. Then in the second step, we design the shortcuts to speed up STIRAP to obtain the complete transfer of coherence between the ground state and the other indirect exciton, which means the state transfer from |2〉 to |3〉, but keeping the same population of state |0〉. Here we focus on the example of coherence between the ground state and one indirect exciton state with both F-STIRAP and STIRAP involved. Of course, the accelerated F-STIRAP can be directly applied as well. Moreover, the value of coherence distribution among the multiple states can reach 1/3 by using accelerated STIRAP and its variation. All the pumping and tunneling pulses designed from various protocols allow for feasibly experimental realization in current semiconductor structures. The fast control and manipulation of coherence may also have potential applications in quantum information processing based on the atomic coherence effect, such as slow-light storage and quantum logical gates.

## Funding

National Natural Science Foundation of China (NSFC) (61404079, 11474193, 11775139); Shuguang Program (14SG35); STCSM (18010500400 and 18ZR1415500); Juan de la Cierva Fellowship.

## Acknowledgment

Y. B. and X.C. appreciate the warm hospitality of ICMM-CSIC in Madrid, where the work is finished.

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