Le-Tian Zhu(朱乐天) Tao Tu(涂涛) Ao-Lin Guo(郭奥林) and Chuan-Feng Li(李传锋)

1Key Laboratory of Quantum Information,University of Science and Technology of China,Chinese Academy of Sciences,Hefei 230026,China

2Hefei National Laboratory,University of Science and Technology of China,Chinese Academy of Sciences,Hefei 230088,China

Keywords: quantum sensing,magnons excitations,spin qubits in quantum dots

1. Introduction

Magnons in ferromagnets are promising solid-state platforms for building interfaces between different quantum systems,which play an important role in various quantum information applications.[1]Using magnon mode conduction or virtual magnon exchange in spin chains, the robust transfer of quantum information between different nodes and the longrange entanglement between qubits can be achieved.[2–6]Using the coupling between magnon modes and different quantum components, quantum hybrid architectures can be realized,which is an attractive pathway to future scalable quantum processors.[7–10]

Electron spins in semiconductor quantum dots are one of the most promising candidates for qubits.[11,12]Because they are compatible with conventional semiconductor technology, quantum dots are leading the way in terms of largescale fabrication and integration.[13,14]In purified silicon materials, electron spins exhibit long coherence times of up to 100µs because the effect of surrounding nuclear spins is suppressed. Quantum dots have excellent gate controllability,so both single-qubit and two-qubit operations can be implemented with high fidelity in an electrical manner.[15–18]Recent significant advances include demonstration of demanding and complex quantum algorithms, such as variational quantum algorithm for calculating molecular energies, on a programmable quantum dot spin processor.[18,19]

The magnetic dipole moment of a single electron spin in quantum dots is small, so it couples weakly to the external magnetic excitations.On the one hand,this is an advantage for electron spin to maintain a long coherence time even in complex solid-state environments. On the other hand, this also makes it difficult to couple and control the magnetic excitations. Therefore, how to develop quantum dots for applications in the field of magnons has been an outstanding challenge.

Among the various quantum dot structures,double quantum dot systems have received more attention.In double quantum dots, in addition to the spin degree of freedom, the electron also has the orbital degree of freedom, such as the delocalization of the electron across the two quantum dots. The presence of orbital degrees of freedom enables the electron to have a large electric dipole moment. The control of the detuning and tunneling between the two quantum dots allows the tunability of the electric dipole moment. In this way, strong coupling to external control fields or other qubits can be obtained when needed. Thus, the additional charge degrees of freedom in the double quantum dot provide an effective interface between the electron and other systems. Based on double quantum dots,a number of impressive progresses have been made recently,such as achieving 99.5%measurement fidelity,the highest value reported so far in any solid-state spin system.[20,21]

In this article,we propose a scheme to combine quantum dots and magnons by using spin–charge hybridization. We focus on a single electron in a double quantum dot. In addition to having spin degree of freedom, it also has orbital degree of freedom, so it is a spin–charge hybrid system. Since the charge state has an electric dipole moment, it gives rise to a large coupling between the spin state and the external cavity field. Further, using the cavity field as an mediator, effective interaction between single electron spin and magnetic excitations can be achieved. Therefore, quantum dots can be used for entanglement with magnetic modes and for the detection of magnon excitations. Our analysis shows that the detection efficiency is as high as 0.94 even in the presence of realistic noises and imperfections. We note that the readout efficiency is only about 0.1–0.5 in the experiments using optical methods to manipulate spin-wave excitations in solids as a quantum memory due to various noise effects.[22,23]These results provide opportunities for a wide range of applications using quantum dots to manipulate magnetic excitations.

2. The hybrid system

2.1. Spin-charge hybridization in the quantum dots

As shown in Fig.1(a),we first consider a single electron confined in two quantum dots.[24–27]The electron has two basic orbital states of|L〉and|R〉, and two basic spin states of|↑〉and|↓〉. The potential difference between the two dots isεand the tunnel coupling between the two dots istc. Both parameters can be adjusted by the voltage applied to the quantum dots. In addition to an overall external magnetic fieldB,a nearby micromagnet produces a magnetic field gradient ∆Bbetween the two dots. This double-quantum dot system can be described by the Hamiltonian

where ˜τiand ˜σiare the Pauli matrices defined in the charge and spin subspaces,respectively.

In Fig. 1(b), we show the energy levels of the quantum dots as a function of the bias parameterε. This is a fourlevel system with energy eigenvaluesEnand energy eigenstates|nqd〉. When|ε| is large, the electron is localized in one quantum dot, whose energy eigenstates can be described as|L,↓〉,|L,↑〉,|R,↓〉,|R,↑〉. In contrast, whenεis near the avoided crossing pointε ≈0, the electron is delocalized across the two quantum dots, forming the bonding and antibonding states|−〉,|+〉. Further,due to the presence of spin–orbit interactions (i.e., the last term in Eq. (1)), the bonding and anti-bonding states with opposite spins|−,↑〉,|+,↓〉are hybridized, forming the energy eigenstates|1qd〉and|2qd〉.While, the ground state is approximately unperturbed, leading to|1qd〉≈|−,↓〉. Thus,the spin states of the quantum dots depend on the charge states. The effective two-level Hamiltonian of the quantum dots can be written as

whereσirepresents the Pauli matrices defined in the basis of spin–charge hybridized states of|0qd〉and|1qd〉.

Fig.1. (a)In the system we consider, there are three components: a single electron spin in the double quantum dot, a microwave mode in the cavity,and a magnon excitation in the magnet. The single spin and the cavity mode are coupled by an electric dipole interaction,labeled as gqd,p. The magnon mode and the cavity mode are coupled by a magnetic dipole interaction,labeled as gm,p. In this way the cavity mode acts as an intermediator,allowing an effective coupling gqd,m between the single spin and the magnon mode. Here the double quantum dot is a double potential well structure. On the one hand,adjusting the voltage of the side gate can change the shape of the confinement potential,so that the electron is confined to either the left quantum dot or the right quantum dot,labeled as|L〉and|R〉. On the other hand,adjusting the voltage of the middle gate can change the barrier height between the two quantum dots,so that the electron can tunnel between the two quantum dots. (b)Energy levels of quantum dots vs bias ε. (c)Effective coupling strength between quantum dots and magnons vs bias ε. Here we use the parameters as tc =20 µeV, B=33 µeV, ∆B=2 µeV, ωm/2π =7.92 GHz,ωp/2π =8.45 GHz,and gm,p/2π =22.9 MHz.

2.2. Spin–photon coupling between the quantum dots and the cavity

Then we consider a microwave cavity with the Hamiltonian

wherea+pandapare the cavity photon operators. In the original basis of|L〉and|R〉,the interaction between the quantum dots and the cavity can be described by

where the coupling strength isgqd,p=〈0qd|˜τz|1qd〉gc,p. In this way, the combination of electric dipole moment and spin–charge hybridization can lead to a large spin–cavity couplinggqd,p.

2.3. Spin–magnon effective coupling between the quantum dots and the magnons

We now consider a hybrid system that consists of a yttrium iron garnet (YIG) ferromagnetic crystal and a double quantum dot, both coupled to the same microwave cavity, as illustrated in Fig. 1. Our system hosts three subsystems: the magnon modes in the magnetically ordered crystal can be depicted as

On the other hand,the quantum dots are coupled to the cavity through the spin–charge hybridization with a coupling strengthgqd,p,as described in Eq.(5). Thus,using the cavity as a mediator,these interactions lead to an effective coupling between the magnetic modes and the quantum dots. When the frequency of the cavity is far detuned from those of the quantum dots and the magnetic modes,the effective interaction between the quantum dots and the magnetic modes can be described as

As shown in Fig.1(c),the value of the coupling strengthgqd,mcan increase by a factor of 5 when the bias of the quantum dotsεis changed from the large detuning regime to the near avoided crossing regime. This result is due to the fact that changing the bias voltage can change the electron from being localized in one quantum dot to being delocalized across the two quantum dots. This increases the charge distribution length of the electron and thus enhances its electric dipole moment,leading to a large electric dipole interactiongqd,p.In this way, adjusting the biasεchanges the coupling strengthgqd,pof the quantum dots to the cavity, which further changes the coupling strengthgqd,mof the quantum dots to the magnons.

Furthermore, when the frequency of the quantum dots is detuning from the frequency of the magnetic modes,the interaction Hamiltonian can be reduced as

Hereχqd,m=g2qd,m/(ωqd−ωm) characterizes the shift of the energy levels of quantum dots due to the presence of magnetic modes.

3. Quantum sensing scheme of the magnons

As shown in Fig.2(a),the scheme for controlling and detecting magnetic modes using quantum dots is designed as follows:

(i)First,the state of the magnetic excitations is prepared,which is a superposition state of magnon Fock states

Meanwhile,the quantum dots are prepared in the ground state.

Fig.2. (a)Schematic diagram of the scheme for detecting the magnon states. (b) Dynamics of the protocol for detecting the magnon states.Here the red dashed line shows the expectation value of the magnons〈nm〉, and the blue solid line shows the probability of the quantum dots in the ground state P|0qd〉. Here we use the pulse parameter of magnons Ωm/2π = 0.8 MHz, the pulse parameter of quantum dotsΩqd/2π =1.25 MHz,and the other parameters as in Fig.1.

(ii) As described in Eq. (10), the frequency of the quantum dots is shifted by the magnetic modes and therefore depends on the state of the magnetic modes. This magnon-statedependent frequency shift is the key to the present scheme.Aπ-pulse is applied to the quantum dots, whose operating frequency corresponds to the frequency of the quantum dots,when the magnetic modes are in the vacuum state|0m〉.By applying such a conditionalπ-pulse, the state of the composite system becomes

(iii) The above equation suggests that the state of the quantum dots and the state of the magnetic modes are entangled after the application of theπ-pulse. The probability that the quantum dots are in the excited state indicates that the magnetic modes are in the vacuum state|0m〉,while the probability that the quantum dots are in the ground state indicates that the magnetic modes are in the magnon number state|nm〉. Therefore, using standard techniques to measure the quantum dots provides information about the magnetic modes.

4. The models and simulations of the hybrid system

To demonstrate this manipulation and detection scheme,we perform numerical simulation of dynamics of the whole system. In the rotating frame of the quantum dot and the magnon frequencies,the Hamiltonian of the composite system can be written as

Here∆qd=ωqd−ωdand∆m=ωm−ωdare the detuning of the quantum dot frequency and magnon frequency from the control pulse frequency, respectively. The last two terms in the Hamiltonian represent the control pulse fields applied to the quantum dots and the magnons,respectively. The dynamics of the system is determined by the master equation

The calculation of the master equation requires long time resources because of the large state space of the hybrid system.Therefore,we derive the equations of motion for the expectation values of the operators of the hybrid system. From the master equation, we can obtain a series of coupled equations involved high order moments. Then, we factor the higher order terms into the first order terms,e.g.,〈cσz〉≃〈c〉〈σz〉. This is a good approximation because the dynamics of the magnons in the hybrid system is dominated by the expectation value and the fluctuation is small. In this way, we obtain the following equation of motion for the hybrid system:

In the simulations, we numerically solve this set of coupled differential equations. Moreover, we find that the results of the equation of motion are consistent with those of the master equation,which indicates that our factorization approximation captures the underlying dynamical structure of the hybrid system.

5. Quantum sensing results

5.1. Detection process

We use the scheme to detect the coherent state of magnons. First, we apply a driving pulse to the ferromagnet to prepare the coherent state of magnons|Ψm〉=|α〉. Second,after a delay time oftd=200 ns,we apply a Gaussian-shapedπpulse to the quantum dots to generate an entangled state of the quantum dots and the magnons. Finally, we measure the probability of the quantum dots in the ground stateP|0qd〉.

In Fig. 2(b), we plot the time evolution of the expectation value of magnons〈nm〉=〈Ψm|c+c|Ψm〉as well as the ground state population of the quantum dotsP|0qd〉. In the first stage, the expectation value of magnons increases with time and reaches a peak,indicating the preparation of the coherent state of magnons. Here,the magnon excitation is prepared by microwave pulses with a peak value of〈nm〉=0.18. In this stage the quantum dots remain in the initial ground state and do not change. In the second stage, the state of the quantum dots starts to respond to the magnons due to the application of the conditionalπpulse, which allows the quantum dots to be entangled with the magnons. The ground state population of the quantum dots continues to decrease and reaches a steady value. In this case, the population of quantum dotsP|0qd〉significantly changes from the initial value of 1 to the final value of 0.14 due to the entanglement of quantum dot spins and magnons. This final ground state population corresponds to the probability of the magnon state|nm〉.

5.2. Detection efficiency for the magnon coherent states

Fig.3. Probability of the quantum dots in the ground state as a function of the amplitude|α|2 of the magnon coherent state. The circles are simulation results and the solid line is the fit to extract the efficiency. Here we use the same parameters as in Fig.2.

We further demonstrate the detection of various coherent states of magnons. First, we prepare the coherent states of result,i.e.,the dark count rated>0 of the detection process.Using the fit in Fig.3,we can obtain two performance metrics for the detection process in a realistic environment asη=0.94 andd=0.06,when the magnons are in the state|nm〉.

5.3. Detection efficiency with respect to various system parameters

To explore the limitation factors of the present scheme,we investigate the effect of decoherence parameters on the detection efficiency. In Fig.4(a),we find that the detection efficiencyηincreases with the increase of the relaxation timeT1and the dephasing timeTφof the quantum dots. For example,for a fixed relaxation timeT1=25 µs, when the dephasing timeTφis varied from 100 ns to 4µs,the detection efficiency increases rapidly from the lower value of 0.1 to close to the ideal value of 1.This result is expected because in this scheme,the quantum dots serve as sensors whose quantum coherence time determines the detection efficiency. Further,we find that the detection efficiencyηincreases more significantly with the dephasing timeTφthan with the relaxation timeT1. Because the key of this quantum sensing scheme is to generate the entangled states between the quantum dots and the magnons, it is more significantly affected by the dephasing process.

We also investigate the detection efficiency by varying the operation time of the conditionalπpulse. In Fig.4(b),we find that the detection efficiencyηgradually decreases with the increase of the durationtpof theπpulse.For example,when the operation time increases to 400 ns,which is comparable to the dephasing timeTφof quantum dots, the detection efficiency drops to a lower value of 0.76. This result is also anticipated because the total measurement time increases as theπpulse duration increases,the decoherence effect starts to play a significant role,leading to a decrease in detection efficiency. Ideally,if the pulse duration is very short,the detection efficiency can be close to 1. However,under practical experimental conditions, it is difficult to implement fast pulse or short pulse duration. Therefore, this result motivates the design of optimized pulse sequences to achieve higher detection efficiency in future work.

Fig.4. (a)Detection efficiency as a function of relaxation time T1 and dephasing time Tφ. (b)Detection efficiency as a function of π-pulse operating time tp. (c)Detection efficiency as a function of quantum dot bias ε. Here we use the same parameters as in Fig.2.

In Fig.3,we simulate the ground state probability of the quantum dots for different magnon coherent states. The solid line in the figure is the fitted equation:P|0qd〉=η(1−e−〈nm〉)+d. Hereηandddenote the efficiency and dark-count rate of the detection process, respectively. In the ideal case, the efficiencyη=1 and the dark count rated=0,where the detection results reduce to a simple scaling law 1−e−〈nm〉.However,the state of the quantum dots is subject to the decoherence effects from the environment, which limits the efficiencyη<1 of the detection process. In addition,decoherence from the environment causes changes in the quantum dot state even in the absence of magnons,which gives rise to an incorrect detection

In Fig.4(c),we find that the detection efficiency changes substantially as the biasεof the quantum dots changes. The detection efficiency increases significantly from a lower value of 0.3 to a higher value close to 1 when the bias voltage of the quantum dots is changed from the large detuning regime to the near avoided crossing regime. This result is consistent with the electrical tunability result of Fig. 1(c). As shown in Fig.1(c),the effective coupling strengthgqd,mbetween quantum dots and magnons can be electrically tuned over a wide range by the bias parameterε,leading to a greater strength of interaction between quantum dots and magnons, thus resulting in the higher detection efficiency shown in Fig.4(c). This result indicates that our scheme has rich electrical tunability compared to other quantum sensing schemes.

6. Conclusions

In summary,we propose a scheme for the control and detection of magnetic excitation modes in ferromagnets using quantum dots. In a realistic solid state environment, the detection efficiency can reach a high value of 0.94. This efficiency value is several times larger than the state-of-art spinwave readout efficiency value using optical methods.[22,23]Our analysis shows that decoherence effects limit the performance of the scheme,so detection efficiency can be further improved by designing optimal working points or using composite pulses.[28–30]Here we demonstrate the detection of magnetic coherent states, it would be interesting to design quantum non-demolition measurement protocols, which could enable richer detection methods.[31]In addition,the scheme can be used for the selective generation of magnon states, which is a crucial step toward the magnon-based quantum technologies. In the future,these schemes can be used to probe exotic magnetic excitations in various solid materials,and to develop quantum technology applications for composite quantum systems.

Acknowledgements

Project supported by the National Natural Science Foundation of China (Grant No. 11974336) and the National Key Research and Development Program of China (Grant No.2017YFA0304100).