Three-dimensional transmon coherence measurement method based on network analyser

Wu Xiao-Yu; Zhao Hu; Li Zhi

doi:10.7498/aps.69.20200252

Abstract

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1. 引　言
2. 3D Transmon量子比特
3. 色散法测量3D Transmon
4. 基于网络分析仪的3D Transmon相干测量
5. 结　论

References

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Abstract

Three-dimensional(3D) transmon is a kind of superconducting qubit with long decoherence time, which has important applications in superconducting quantum computation, quantum optics, cavity quantum electrodynamics, et al. Rabi oscillation is a vital method to characterize the decoherence time of quantum system, and it is also a basic experiment to demonstrate the energy level evolution of quantum system. In order to test the Rabi oscillation of 3D transmon, strict timing control is necessary, and the process of testing and debugging is complicated. In this paper, 3D transmon samples are fabricated and their basic parameters E_C = 348.74 MHz and E_J = 11.556 GHz are tested and characterized. The coupling coefficient g²/Δ between qubit and the 3D cavity is 43 MHz, which is located in the dispersive regime. The qubit’s first transition frequency f₀₁ = 9.2709 GHz, and the second transition frequency f₁₂ = 9.0100 GHz. The 3D resonator is fabricated by the material 6061T6 aluminum, the loaded quality factor is 4.8 × 10⁵, and the bare frequency of the resonator is 8.108 GHz.Through comparison, it is found that the Rabi oscillation time obtained by the proposed method is shorter than by the Jaynes-Cummings method. The main reasons are as follows. First, the measurement of network analyzer is a continuous measurement, and the test signal always affects the decoherence process of 3D transmon. Second, the quantum bit is in the ground state after decoherence, and the ground state measured by the network analyzer accounts for a relatively high proportion, which causes the curve measured by the network analyzer to be one-sided attenuation oscillation. Third, the dispersive readout method is related to the quality factor of the superconducting cavity. The storage time of microwave photons in the superconducting cavity is longer than the decoherence time of 3D transmon, so the quantum information is partially decohered before leaving the superconducting cavity, which will shorten the Rabi oscillation time.An innovative Rabi oscillation test method based on network analyzer is presented. The test system based on this method is simple to build and can be used as a new way to quickly verify whether 3D transmon has quantum characteristics. This method can also be extended to other quantum systems for preliminarily verifying the time domain characteristics.

Keywords:

PACS:

03.67.Lx	(Quantum computation architectures and implementations)
42.50.Pq	(Cavity quantum electrodynamics; micromasers)
85.25.-j	(Superconducting devices)

Authors and contacts

1.
College of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China
2.
AVIC Xi’an Flignt Automatic Control Research Institute, Xi’an 710065, China

Corresponding author: Li Zhi, lizhi@scu.edu.cn

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1. 引　言

基于约瑟夫森结的超导量子计算^[1-4]由于能够与经典微电子工艺兼容、退相干时间长、制备基态简单、量子比特之间的耦合较为容易以及可规模化等优点, 被认为是目前最有可能实现量子计算机的方案之一^[5-8]. 国际上有许多团队都致力于用超导的方式来实现量子计算机^[9-14]. 超导量子计算的核心单元为超导量子比特, 目前已实现的超导量子比特有charge qubit^[10], flux qubit^[15], phase qubit^[12], quantronium^[16], Xmon^[17], C-shunt flux qubit^[18]等, 除此之外, 还有结合超导谐振腔实现的量子比特, 包括基于共面波导谐振腔实现的量子比特^[13], 以及基于三维超导谐振腔实现的量子比特, 即3D Transmon^[14]. 3D Transmon中两个电极板(pad)和微波腔一起构成并联在Josephson结的大电容, 用于压制电荷噪音对量子比特相干性的影响. 这种方法(即用大电容并联在Josephson结来压制电荷噪音对超导量子比特相干性的影响的方法)最早是由You 等^[19]提出, 对应的超导量子比特叫capacitively shunted flux qubit (也叫C-shunt flux qubit), 即“电容并联磁通量子比特”.

本文通过双角度蒸发^[15]的方法在氧化硅衬底上制备了样品, 并将样品放置在三维超导谐振腔中心处形成3D Transmon样品. 采用基于腔量子电动力学的色散读出方法^[20], 对量子比特的基本性能进行了测试表征, 用网络分析仪、微波源以及任意波形发生器对3D Transmon进行了相干测量操作, 当改变激励微波信号的功率时, 相干振荡的频率与微波信号的幅度呈现出较好的线性. 对比数据采集卡Jaynes-Cummings^[21]方法进行拉比振荡的测试结果, 可以看出网络分析仪色散法得到的退相干时间小于数据采集卡Jaynes-Cummings得到的退相干时间, 由于网络分析仪色散测试法对于测量时序的要求较为简单, 测试系统搭建较为容易, 可作为验证样品具备量子特性的一种方法补充.

2. 3D Transmon量子比特

Transmon^[22]量子比特实际上是由charge qubit演变而来, 设charge qubit的栅电容为C_g, 栅电压为V_g, 约瑟夫森结的结电容为C_J, 则栅电压吸引的库伯对数目n_g = C_gV_g/2 e, 库伯盒的总充电能为4E_C = (2 e)²/2(C_J+C_g), 则charge qubit的哈密顿量可以写为

$\begin{split} H =\; & 4{E_C}\sum\limits_n {{{\left( {n - {n_g}} \right)}^2}} \left| n \right\rangle\left\langle n \right| \\ & - \frac{{{E_J}}}{2}\sum\limits_n \left[ {\left| n \right\rangle \left\langle {n + 1} \right| + \left| {n + 1} \right\rangle \left\langle n \right|} \right]. \end{split}$

(1)

其中E_J为库伯盒的约瑟夫森能, n为库伯对的数目, $\left| {\rm{n}} \right\rangle$ 为粒子数算符. 选取不同的E_J和E_C, 对(1)式进行求解, 就可以得到不同E_J/E_C下的能带图. 当E_J/E_C较小时, 量子比特的能级随n_g变化较大; 当E_J/E_C较大时, 量子比特的能级变得较为扁平, 此时构成的量子比特为Transmon.

将Transmon与超导谐振腔耦合起来, 可通过谐振腔的状态对Transmon的状态进行读取, 腔量子电动力学的JC哈密顿量为^[7]

$\begin{split} {H_{JC}} ={}& \hbar {\omega _r}\Big( {{a^ + }a + \frac{1}{2}} \Big) + \hbar \frac{{{\omega _q}}}{2}{\sigma _Z} \\ & + \hbar g( {{a^ + }{\sigma ^ - } + a{\sigma ^ + }}).\end{split}$

(2)

其中 $\hbar$ 是约化普朗克常数, a⁺与a为光子的产生和湮灭算符, σ⁺算符可将量子比特从基态 $\left| {\rm{0}} \right\rangle$ 变换到激发态 $\left| {\rm{1}} \right\rangle$ , σ^–算符可将量子比特从激发态 $\left| {\rm{1}} \right\rangle$ 变换到基态 $\left| {\rm{0}} \right\rangle$ . 哈密顿量的第一项为电磁场的能量, 谐振腔的角频率为 ${\omega _r}$ , 第二项为二能级的量子比特, ${\omega _q}$ 是量子比特的跃迁角频率, 第三项为量子比特与谐振腔的相互作用, g是耦合强度, a⁺σ^–与aσ⁺分别表示量子比特向腔中发射一个光子和从腔中吸收一个光子. 设 $\Delta = {\omega _q} - {\omega _r}$ 为量子比特与谐振腔的角频率差, 当g $\ll$ Δ时, 量子比特与谐振腔的耦合进入色散区间, 量子比特的装态可通过测量谐振腔的状态得到.

通过双角度蒸发的方法得到的Transmon样品的f₀₁ = 9.2709 GHz, f₁₂ = 9.0100 GHz, 充电能E_C = 348.74 MHz, 约瑟夫森能E_J = 11.5560 GHz. 将该Transmon放置在三维超导谐振腔^[23]的第一模态处, 得到3D Transmon样品. 三维超导谐振腔的第一模态频率f_C = 8.108 GHz, 有载品质因数为4.8 × 10⁵. 对3D Transmon的S₂₁参数进行变功率扫描得到如图1所示的结果.

$图 1 3D Transmon的S21变功率扫描结果　(a) S21强度图; (b)部分S21曲线\r\nFig. 1. The power change scan S21 of 3D Transmon: (a) Intensity graph of S21; (b) partial S21 curve.$

图 1 3D Transmon的S₂₁变功率扫描结果　(a) S₂₁强度图; (b)部分S₂₁曲线

Fig. 1. The power change scan S₂₁ of 3D Transmon: (a) Intensity graph of S₂₁; (b) partial S₂₁ curve.

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从图1可看出, 当输入功率逐渐下降时, 谐振腔内的微波光子数逐渐下降, 即量子比特与光子的相互作用对谐振腔的影响越来越明显, 谐振腔的第一模态频率从8.108 GHz逐渐变到8.065 GHz, 这反映出谐振腔与量子比特的耦合g²/Δ=43 MHz.

3. 色散法测量3D Transmon

为了对3D Transmon进行测试, 搭建了基于稀释制冷机的超低温测试系统, 整个测试系统的搭建思路是输入信号逐级衰减, 输出信号逐级放大, 为了有更好的信噪比. 如图2所示, 任意波形发生器与微波源通过一套混频电路可提供脉冲长度可调的量子比特激励信号, 网络分析仪通道1输出信号, 通道2接收信号, 为了使不同的设备频率基准完全相同, 三台设备都用同一个10 MHz的基准源.

$图 2 3D Transmon测试系统\r\nFig. 2. 3D Transmon measurement system.$

图 2 3D Transmon测试系统

Fig. 2. 3D Transmon measurement system.

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3D Transmon安装在稀释制冷机的10 mK温区, 输入信号经过了5个衰减器, 共计–49 dB的衰减, 输出级有两个隔离器, 为了尽量少的将放大器以及输出线路的噪声引入样品, 提高整个测试系统的信噪比. 输出线路从10 mK到4 K之间采用超导微波线缆, 输出线路在4 K级进行一次放大, 在室温区进行了二次放大.

给量子比特施加f₀₁ = 9.2709 GHz的连续微波激励信号, 微波功率为–110 dBm, 网络分析仪的扫描功率设定为–140 dBm, 扫描S₂₁参数, 得到如图3所示的结果.

$图 3 微波信号连续激励量子比特时的S21曲线\r\nFig. 3. The S21 curve for the qubits excited by continuously microwave.$

图 3 微波信号连续激励量子比特时的S₂₁曲线

Fig. 3. The S₂₁ curve for the qubits excited by continuously microwave.

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从图3可以看出, S₂₁曲线上有两个谐振峰, 8.065与8.045 GHz分别代表量子比特处于 $\left| {\rm{0}} \right\rangle$ 和 $\left| {\rm{1}} \right\rangle$ 的两个状态, 对8.065 GHz处的谐振峰幅度和相位进行测量, 可判断量子比特的状态, 这种量子比特的读出方法为色散读出法.

4. 基于网络分析仪的3D Transmon相干测量

受限于目前的工艺以及测试系统等原因, 目前制备的3D Transmon样品的退相干时间普遍在1 µs的水平, 因此激励量子比特的周期选为5 µs可以满足量子比特激励后充分冷却到基态 $\left| {\rm{0}} \right\rangle$ 的要求.

网络分析仪的采样周期取决于中频带宽(IF Bandwidth), 网络分析仪Agilent N5323 A的中频带宽最大为15 MHz, 其对应采样时间约为67 ns一个点, 这样的速度对于测量退相干时间为1 µs的3D Transmon太短, 为了得到3D Transmon的相干振荡, 只能尽量拉长网络分析仪的测试时间窗口, 通过取平均的方法得到量子比特的相干振荡. 将中频带宽设定为5 Hz, 对应的一个测试点时间为0.2 s, 测试的时序图如图4所示, 网络分析仪的一次读取包含0.2 s/5 µs=40000次激励过程, 通过改变微波激励时间t, 就可以得到3D Transmon的相干振荡曲线. 当微波激励功率为–100 dBm, 网络分析仪读取功率为–138 dBm时得到的相干振荡幅度与相位曲线如图5所示.

$图 4 网络分析仪测量相干振荡时序图\r\nFig. 4. Time sequence diagram of coherent oscillation measured by network analyzer.$

图 4 网络分析仪测量相干振荡时序图

Fig. 4. Time sequence diagram of coherent oscillation measured by network analyzer.

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$图 5 网络分析仪测量3D Transmon相干振荡幅度与相位图\r\nFig. 5. Amplitude and phase diagram of 3D transmon coherent oscillation measured by network analyzer$

图 5 网络分析仪测量3D Transmon相干振荡幅度与相位图

Fig. 5. Amplitude and phase diagram of 3D transmon coherent oscillation measured by network analyzer

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图5可以明显的看出3D Transmon的相干振荡, 与传统意义的拉比振荡所不同的是, 该曲线只能看到振荡的峰值, 却看不到振荡的谷值, 这是由于超导谐振腔的品质因数为4.8 × 10⁵, 对应的光子存储时间约为2 µs, 大于量子比特的退相干时间. 测试时, 用网络分析仪持续监测8.065 GHz处的谐振峰, 幅度曲线上的峰值代表量子比特处于基态 $\left| {\rm{0}} \right\rangle$ , 当量子比特处于激发态 $\left| {\rm{1}} \right\rangle$ 时, 量子信息在还没有离开超导谐振腔之间前已经进行了部分衰减, 因此看不到振荡曲线的波谷. 从图5的振荡曲线来看, 该3D Transmon的在微波激励信号长度为1 µs时, 曲线还在振荡. 改变微波激励信号的功率进行测试, 得到不同功率下的拉比振荡结果如图6(a)所示, 画出拉比振荡与激励微波幅度的关系图(图6(b)). 从拟合结果来看, 拉比振荡频率与微波激励幅度呈现出比较好的线性关系. 这也说明了通过网络分析仪测量相干振荡为拉比振荡.

$图 6 (a) 不同微波功率下的相干振荡强度图; (b) 相干振荡与微波激励幅度关系\r\nFig. 6. (a) Intensity diagram of coherent oscillation at different microwave power; (b) relationship between coherent oscillation and microwave excitation amplitude.$

图 6 (a) 不同微波功率下的相干振荡强度图; (b) 相干振荡与微波激励幅度关系

Fig. 6. (a) Intensity diagram of coherent oscillation at different microwave power; (b) relationship between coherent oscillation and microwave excitation amplitude.

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为了对比该方法得到的振荡曲线与经典拉比振荡曲线的关系, 我们采用了数据采集卡Jaynes-Cummings方法测试拉比振荡^[24], 如图7所示, 得到的拉比振荡时间为1151.8 ns.

$图 7 数据采集卡Jaynes-Cummings方法测试拉比振荡\r\nFig. 7. Rabi oscillation by Jaynes-Cummings method based on data acquisition card.$

图 7 数据采集卡Jaynes-Cummings方法测试拉比振荡

Fig. 7. Rabi oscillation by Jaynes-Cummings method based on data acquisition card.

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通过对比发现网络分析仪色散法得到的拉比振荡时间较短, 其主要原因是: 1)网络分析仪测试拉比振荡是持续性测量, 因此测试信号时刻在影响3D Transmon的退相干过程, 而数据采集卡方法通过严格的时序控制, 可在需要测试时, 精准地介入; 2)网络分析仪测试方法, 激励信号的周期为5 µs, 该时间段内包含量子比特有激励的过程和没有激励的过程, 随着激励时间的增长, 无微波激励的占比越来越低, 而在这些没有微波激励的时间段内, 量子比特退相干后将处于基态 $\left| {\rm{0}} \right\rangle$ , 也就是说没有微波激励的区域所占比例越高, 网络分析仪测量得到的基态 $\left| {\rm{0}} \right\rangle$ 所占比例越高, 这就造成了网络分析仪测量拉比振荡得到的曲线是单边衰减振荡的, 与标准拉比振荡曲线不同; 3)色散读出法与超导谐振腔的品质因数非常相关, 选用的超导谐振腔对微波光子的存储时间大于3D Transmon的退相干时间, 因此量子信息在还没有离开超导谐振腔之前已部分退相干, 这对于测试拉比振荡是极为不利的. 而基于数据采集卡的Jaynes-Cummings方法, 量子比特的读出功率非常高, 三维超导谐振腔可以“继承”量子比特在读出时的状态, 因此得到的退相干时间较长.

5. 结　论

本文创新性地提出了一种基于网络分析仪色散法的3D Trasnmon相干振荡测试方法, 该方法的测试系统搭建较为简单, 但由于受到谐振腔品质因数、微波激励信号占激励总时间比例以及持续测量的影响, 网络分析仪色散法得到的拉比振荡时间小于数据采集卡Jaynes-Cummings方法得到时间. 该方法可作为一种快速验证量子系统是否具备时域特性的一种辅助手段, 可推广至其他量子系统.

感谢日本NEC基础创新实验室Tsai J S提供的样品加工支持.

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图 1 3D Transmon的S₂₁变功率扫描结果　(a) S₂₁强度图; (b)部分S₂₁曲线

Figure 1. The power change scan S₂₁ of 3D Transmon: (a) Intensity graph of S₂₁; (b) partial S₂₁ curve.

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图 2 3D Transmon测试系统

Figure 2. 3D Transmon measurement system.

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图 3 微波信号连续激励量子比特时的S₂₁曲线

Figure 3. The S₂₁ curve for the qubits excited by continuously microwave.

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图 4 网络分析仪测量相干振荡时序图

Figure 4. Time sequence diagram of coherent oscillation measured by network analyzer.

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图 5 网络分析仪测量3D Transmon相干振荡幅度与相位图

Figure 5. Amplitude and phase diagram of 3D transmon coherent oscillation measured by network analyzer

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图 6 (a) 不同微波功率下的相干振荡强度图; (b) 相干振荡与微波激励幅度关系

Figure 6. (a) Intensity diagram of coherent oscillation at different microwave power; (b) relationship between coherent oscillation and microwave excitation amplitude.

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图 7 数据采集卡Jaynes-Cummings方法测试拉比振荡

Figure 7. Rabi oscillation by Jaynes-Cummings method based on data acquisition card.

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