Phase-Slip Based SQUID Used as a Photon Switch in Superconducting Quantum Computation Architectures

Abstract

The photon storage time in a superconducting coplanar waveguide (CPW) resonator is contingent on the loaded quality factor, primarily dictated by the input and output capacitance of the resonator. The phase-slip based superconducting quantum interference device (PS-SQUID) comprises two phase-slip (PS) junctions connected in series with a superconducting island in between. The PS-SQUID can manifest nonlinear capacitance behavior, with the capacitance finetuned by the gate voltage to minimize the impact of magnetic field noise as much as possible. By substituting the coupling capacitance of the CPW resonator with the PS-SQUID, the loaded quality factor of the resonator can be changed by three orders, thus, we get a microwave photon switch in superconducting quantum computation architectures. Furthermore, by regulating the loaded quality factors, the coupling strength between the CPW and superconducting quantum circuits can be controlled, enabling the ability to manipulate stationary qubits and flying qubits.

1. Introduction

Superconducting quantum circuits [1,2] are one of the most likely ways to achieve quantum networks [3,4] and quantum computing [5,6]. In actual quantum networks and quantum computing, in order to meet the processing of quantum information at different quantum device connections, such as between adjacent atoms or cavities, between the atom and the cavity, or between the atom (cavity) and the transmission line, the coupling at this connections need to be adjusted in real time. For example, when an atom or cavity interacts with a traveling wave field (such as an electromagnetic field), due to the high fidelity transmission of quantum information between atoms and fields, i.e., the conversion between stationary qubits (the state of the atom or cavity at rest relative to the traveling wave field) and flying qubits (the state of the traveling wave field), appropriate coupling conditions need to be met. So, the controlled coupling in real time applies not only to the adjustment of stationary qubits [1], but also to the adjustment of flying qubits [7,8]. Usually, this coupling can be achieved through inductance or capacitance, which are respectively called coupling inductance or coupling capacitance. Therefore, it is very important to seek coupling inductors or coupling capacitors that can vary over time.

In superconductiong quantum circuits, the superconducting coplanar waveguide (CPW) resonator stands as a pivotal component. This versatile device serves multiple functions: it acts as a quantum bus for photon transmission, functions as quantum memory, operates as a single-photon generator, and facilitates the coupling of superconducting qubits with various atomic systems. The loaded quality factor ($Q_L$) of the CPW resonator can decisively influence the coupling strength $g\propto 1/Q_L$ with external inputs or outputs [9], when the CPW resonator is connected to other circuit components. For example, resonators with low $Q_L$ are suitable for performing fast measurement, while resonators with high $Q_L$ are ideal for storing photons as a quantum memory.Furthermore, to make the CPW resonator more tunable for different applications, arrays of direct current superconducting quantum interference device (DC-SQUID) are often inserted into the CPW central lines. This allows the resonant frequency of the CPW to be tuned through the external magnetic field [10,11].

However, once a CPW resonator is fabricated in a traditional way, its $Q_L$ will be fixed. This greatly limits the ability to control the coupling strength between the cavity and other objects in real-time, such as the capacitive coupling case between a two-level atom and a CPW resonator in Figure 1a, making it impossible to achieve the required photon (the black wavy line) entry (the arrow to the right) and exit (the arrow to the left). This type of problem can be understood from the perspective of flying qubits below, that is, the relationship between stationary qubits and flying qubits can usually be divided into four categories. Category 1: The conversion of a stationary qubit into a flying qubit, that is, the generation of a flying qubit. Category 2: The conversion of a flying qubit into a stationary qubit, i.e., the reception of a flying qubit. Category 3: A flying qubit is first converted into a stationary qubit, followed by another flying qubit, which is the conversion of a flying qubit. Category 4: A stationary qubit is first converted into a flying qubit, and then converted into another stationary qubit, that is, the transmission or indirect coupling of two stationary qubits of a flying qubit. At present, relevant studies [8,12] have shown that the control process of the four types of flying qubits mentioned above requires coupling matching between stationary qubits and flying qubits. That is, for controlling flying qubits with specific time-domain shapes, such as the generation, reception, conversion, or transmission of a microwave single photon, coupling needs to be regulated.

Fortunately, the related studies indicate that a class of CPW resonators with tunable $Q_L$ can be fulfilled by using an inductance transformer and DC-SQUID as the input-output coupling inductance of a CPW resonator [13,14,15]. This method used a magnetic field to tune the equivalent input-output coupling inductance of the CPW resonator. However, introducing magnetic field into the circuit has many disadvantages, which will bring new difficulties to the analysis and regulation of the desired process. First, the designed $Q_L$ of the CPW resonator will be suppressed by magnetic field. Second, magnetic vortexes and magnetic noises are also introduced into the circuit. Third, most of superconducting qubits are sensitive to magnetic field, and the DC-SQUID flux bias will alter the qubit energy band, making it hard to tune the qubit independently.

Meanwhile, the PS phenomenon was mainly concentrated near the transition temperature of superconductors in the early stages. PS junctions can be made of superconducting nanowires [16,17], and can be used to form qubits [18], which correspond to superconducting Josephson junction qubits. In the past decade or so, by applying the principle of PS junctions, different functional devices were proposed, including the multilevel memory element [19], quantum current standard [20,21], the charge control [22], the charge-based superconducting digital logic [23], the single-charge transistor [24], the optical control of states [25], the single photon detection [26]. In related research [27,28,29,30], microwave photon switches are mainly used to represent the ability of microwave single photons to enter different quantum channels. This includes two types: one is to generate microwave single photons to different quantum channels, and the other is to route the incident microwave single photon to other quantum channels. At the same time, there is currently no microwave photon switch that can change the coupling strength over time and regulate the coupling channel. Therefore, such microwave photon switches are crucial in regulating the conversion process between stationary qubits and flying qubits in quantum networks and quantum computing.

2. Materials and Methods

In this paper, we proposed a more directly way to implement the $Q_L$ tunable resonator using a device called the phase-slip based superconducting quantum interference device (PS-SQUID) [31] (Figure 1b). Through this, we obtain a type of microwave photon switchs on using voltage regulation PS junctions to act as an adjustable coupling capacitor to control the entry and exit of microwave photons. In the method, we employed two PS-SQUID, which are formed by two series PS junctions connected by a superconducting island, as the input or output capacitors of the resonator. The PS-SQUID is a nonlinear capacitor, and can be tuned by the gate voltage. By applying voltage pulses, we can control the $Q_L$ of the CPW resonator in time. This approach allowed us to realize a microwave photon switch in superconducting quantum computation architectures.

The basic element in a PS-SQUID is the PS junction. The voltage and charge relation in a PS junction can be written as:

$$\begin{array}{c}V=V_c\sin \left(2\pi q\right),\end{array}$$

(1)

where $V_c$ is the critical voltage of the PS junction. The current and charge relation in a PS junction is

$$\begin{array}{c}I=2e\dot{q}.\end{array}$$

(2)

The nonlinear capacitance in a PS junction can be deduced by considering the time derivative of Equation (1), to yield $\dot{V}=V_{c}2\pi \cos \left(2\pi q\right)I/2e$ by combining Equation (2). According to the capacitance voltage relation $C=I/\dot{V}$, the equivalent capacitance of a PS junction can be obtained as:

$$\begin{array}{c}C=\frac{2e}{2\pi V_c\cos \left(2\pi q\right)}.\end{array}$$

(3)

The PS junction has an intrinsic kinetic inductance $L_k$ [18], as

$$L_k=a\frac{\hslash R_n}{k_B{T}_c},$$

(4)

here, a is a numerical constant ($a\approx 0.14$), ℏ is the reduced Planck constant, $R_n$ is the normal state resistance at temperature zero, $k_B$ is the Boltzmann constant and $T_c$ is the critical temperature. In usual, the kinetic inductance of a PS junction is about two orders of magnitude larger than its geometric inductance. If we make a combination of the kinetic inductance and its capacitance, the PS junction can also exhibit a feature of a nonlinear oscillator with a resonant angular frequency ${\omega }_p=1/\sqrt{L_{k}C}=\sqrt{2\pi k_B{T}_c{V}_c\cos \left(2\pi q\right)/2ea\hslash R_n}$.

Assuming the two PS junctions in a PS-SQUID are identical, the critical voltage $2V_C\cos (\pi C_g{V}_g/2e)$ of a PS-SQUID can be tuned by the gate voltage $V_g$, thus, the equivalent capacitance of a PS-SQUID can be written as:

$$\begin{array}{c}C\left(V_g\right)=\frac{2e}{2\pi 2V_c\cos (\pi C_g{V}_g/2e)\cos \left(2\pi q\right)}.\end{array}$$

(5)

From the above equation, it is clear that the kinetic capacitance of a PS-SQUID can be tuned by the gate voltage.

It is known that the Josephson junction in the circuit can be considered as a combination composed of a nonlinear inductance parallel with a capacitance, as the Josephson junction has supercurrent. Correspondingly, the equivalent circuit of a PS junction should be considered as a combination composed of an inductance in series with a nonlinear capacitance, as shown in Figure 1c, so the PS junction is exactly dual to the Josephson junction.

The half-wave CPW resonator is a distributed device with voltages and currents varying in magnitude and phase over its length. If we substitute the input and output capacitance with two PS-SQUID, as in Figure 2a, the coupling capacitances ($C_{in}\left(V_g\right),C_{out}\left(V_g\right)$) can be tuned by the gate voltage, for simplicity, we assume the two PS-SQUID are identical and its equivalent capacitance can be abbreviated as $C\left(V_g\right)$. In some sense, the behavior of the PS-SQUID is just like a photon switch, where the on/off is controlled by the gate voltage $V_g$. To understand the function of the circuit (Figure 2a) clearly and intuitively, an equivalent circuit is illustrated in Figure 2b, where the CPW resonator is approximated by a lumped parallelled $LCR$ oscillator. The PS-SQUID is represented by a series of inductance ($L_k$) and capacitance ($C\left(V_g\right)$) as mentioned in Figure 1b, the $R_L$ is the loaded resistance of the feedin lines, which is usually $50\Omega$. To simplify the analysis of the influence of coupling capacitance on CPW resonators, Figure 2b can be transformed to Figure 2c by Norton equivalent parallel connection of resistors $R_{in}^{\ast }$, $R_{out}^{\ast }$ and capacitors $C_{in}^{\ast }$, $C_{out}^{\ast }$, where

$$\begin{array}{c}{R}^{\ast }=R_{in}^{\ast }=R_{out}^{\ast }=\frac{1+{\omega }_n^2{C}_e^2{R}_L^2}{{\omega }_n^2{C}_e^2{R}_L},\\ \\ C^{\ast }=C_{in}^{\ast }=C_{out}^{\ast }=\frac{C_e}{1+{\omega }_n^2{C}_e^2{R}_L^2}.\end{array}$$

Here, $C_e$ is equal to $C\left(V_g\right)/(1-{\omega }_n^2{L}_k{C}_{out}\left(V_g\right))$, ${\omega }_n=n{\omega }_0=1/\sqrt{L_{n}C}$ is the angular frequency of the nth mode, where n is the mode number. The $L_n$ (relevant to nth mode) and C is the equivalent inductance and capacitance of the CPW resonator. The above equivalence is only valid for $\omega \approx {\omega }_n$. In practice, the kinetic inductance ($L_k$) of a PS junction is in the order of $nH$, and the input/output capacitance of a CPW resonator is in the order of fF, thus, the $C\left(V_g\right)$ should be worked in the order of fF, only in this order, the quality factor of a CPW resonator can be tuned effectively. The loaded quality factor $Q_L$ in Figure 2c is,

$$\begin{array}{c}{Q}_L={\omega }_n^{\ast }\frac{C+2C\left(V_g\right)}{1/R+2/R^{\ast }}.\end{array}$$

(6)

where ${\omega }_n^{\ast }=1/\sqrt{L_n(C+2C^{\ast })}$ is the nth mode resonant frequency after considering the input and output capacitance, R is the CPW resonator equivalent resistance, C is the CPW resonator equivalent capacitance.

The charge conservation relation in a PS-SQUID can be written as,

$$\begin{array}{c}{q}_1-q_2=\frac{C_g{V}_g}{2e},\end{array}$$

(7)

the voltage relation is,

$$\begin{array}{c}V=V_c\sin \left(2\pi q_1\right)+V_c\sin \left(2\pi q_2\right).\end{array}$$

(8)

Combining Equations (7) and (8) together, and setting $q^{-}=q_1-q_2$ and $q^{+}=q_1+q_2$, and we will get,

$$\begin{array}{c}{q}^{-}=\frac{C_g{V}_g}{2e},\end{array}$$

(9)

$$\begin{array}{c}V=2V_c\sin \left(\pi q^{+}\right)\cos \left(\pi q^{-}\right).\end{array}$$

(10)

According to Equations (5), (9) and (10), the equivalent kinetic capacitance of a PS-SQUID can also be represented by the voltage V and the gate voltage $V_g$,

$$\begin{array}{c}{C}_k=\frac{2e}{2\pi V_c}\frac{1}{\cos (\frac{\pi V_g{C}_g}{2e})\cos (\arcsin (\frac{V}{2V_c\cos (\frac{\pi V_g{C}_g}{2e})}))}.\end{array}$$

(11)

To demonstrate the predicted effect, the appropriate values for experiments should be found according to Equation (11). From Equation (11), it is easy to known the kinetic capacitance of a PS-SQUID is periodic with the gate voltage. Considering one period, the $V_g{C}_g$ should be in the order of an electron charge, we set gate capacitance $C_g$ to $2\times {10}^{-14}$ F that can make the gate voltage $V_g$ located in the order of $\mathsf{\mu }$V, which is easy to be implemented in practical. Meanwhile, the critical voltage of a PS junction is equal to $h\Gamma /2e$, where h is the Plank constant, $\Gamma$ is the PS tunnelling rate, and $\Gamma /2\pi$ can be changed from the order $0.1$ GHz to 100 GHz by different fabrication parameters. So the critical voltage can be changed from 1.3 $\mathsf{\mu }$V∼1.3 mV.

First, we want to know what kind of $V_C$ that we need to tune the quality factor of a resonator effectively. The cavity quantum electrodynamics experiments are often performed in the low power regime, where just contain several photons, so the voltage between the PS-SQUID is in the order of 10 nV. For simplicity, we can set the low power condition as $V\approx 0$ nV. For the initial attempt, we set $V_g$ to zero, then based on Equation (11), the PS-SQUID kinetic capacitance $C_k$ versus the critical voltage of a PS junction $V_C$ in the low power regime can be illustrated in Figure 3.

From Figure 3, we can see the the kinetic capacitance can be changed by three orders with different critical voltage. Initially, the kinetic capacitance drops drastically as the critical voltage increases, and then decreases slowly as the critical voltage approaches 1.3 mV. Without a gate voltage, the kinetic capacitance of the PS-SQUID can range from ${10}^{-13}$ F to ${10}^{-17}$ F, which includes the typical range of the input and output capacitance of the CPW resonator.

3. Results

To observe clearly how the kinetic capacitance of the PS-SQUID and the loaded quality factor of the CPW resonator change with gate voltage, an example that can be practically implemented is introduced below. First, the kinetic capacitance of the PS-SQUID is assumed to be 0.5 fF when the gate voltage is zero. Thus, the critical voltage $V_C$ of the PS junction is $1.0186\times {10}^{-4}$ V which can be calculated by Equation (11). Second, we assumed the CPW resonator is working in the low power regime, where the voltage is assumed to be ${10}^{-4}{V}_C$. Based on these assumptions, we can determine how the kinetic capacitance changed with the gate voltage from Equation (11) (Figure 4).

Figure 4 clearly demonstrate the kinetic capacitance periodically modulated by the gate voltage. The kinetic capacitance of the PS-SQUID is set to 0.5 fF when the gate voltage is zero. By changing the gate voltage in the $\mathsf{\mu }$V range, the kinetic capacitance can be changed by three orders. Then we can use Equation (6) to simulate the loaded quality factor changed with input/output capacitance (Figure 5). The CPW equivalent resistance and capacitance can be calculated by $Z_0/\alpha l$ and $C_{l}l/2$, respectively. The $Z_0$ is the resonator’s characteristic resistance, $\alpha$ is the attenuation constant, l is the length of the resonator, $C_l$ is the capacitance per unit length. In this example, we use the experiment results [32], where $Z_0=50\Omega$, $\alpha =2.4\times {10}^{-4}$ m⁻¹, $l=1.1\times {10}^{-2}$ m, $C_l=1.2894\times {10}^{-10}$ F/m, and only the first mode is to be considered. Besides, the kinetic inductance $L_k$ of the wires that exhibit quantum phase-slip is of the order nH, and the geometric inductance $L_g$ is of order 10 pH which can be neglected in the calculation. Based on the thoery [16,18], the kinetic inductance of the PS-SQUID $L_k$ can be set to 100 nH.

Figure 5 demonstrates the loaded quality factor $Q_L$ of the CPW resonator changing with the gate voltage. The $Q_L$ can be altered by three orders when the gate voltage changes in the $\mathsf{\mu }$V range, making it easily implementable in experiments. The $Q_L$ represents the photon store capability of a CPW resonator. A high $Q_L$ implies that photons are less likely to escape from the resonator, while a low $Q_L$ indicates easier photon escape. Consequently, this configuration provides a voltage-controlled photon switch in the microwave domain.

4. Discussion

Meanwhile, when a CPW resonator connected to a superconducting quantum circuit as the stationary qubit with a coupling coefficient $\gamma$ [33,34], which is corresponds to the coupling strength g mentioned at the beginning of the article, the quality factor $Q_L$ of this CPW resonator has an inverse relationship with $\gamma$, such as $\gamma \propto 1/Q_L$. Here, $\gamma$ is an important parameter for the quantum information transmission between stationary qubits and flying qubits [35], the latter is the state of the quantum traveling wave field in the CPW resonator. Therefore, $\gamma$ can be regulated by adjusting the gate voltage, and it is easy to find that its curve with the gate voltage is opposite to Figure 5.

Furthermore, assuming that the stationary qubit initially stay at the excited state $|e\rangle$ (for simplicity, treating the stationary qubit as a two-level atom), when it starts releasing $|1_{\xi }\rangle$ into the waveguide, related studies [8,12] indicate that $\gamma$ changes over time and satisfies the relationship

$$\gamma \left(t\right)=\frac{{\left|\xi \left(t\right)\right|}^2}{{\int }_t^{t_1}{\left|\xi \left(\tau \right)\right|}^2\mathrm{d}\tau }.$$

(12)

Here,

$$|1_{\xi }\rangle ={\int }_{t_0}^{t_1}\xi \left(t\right)b^{†}\left(t\right)|vac\rangle \mathrm{d}t,$$

(13)

$b^{†}\left(t\right)$ is the creation operator at the moment t, $|vac\rangle$ indicates that the quantum channel coupled with the stationary qubit is in the vacuum state, $\xi \left(t\right)$ is the time-domain shape of the single photon, which corresponds to the frequency-domain shape $\xi \left(\omega \right)$, while the latter is the Fourier transform of the former and has a representation similar to Equation (13). $\xi \left(t\right)$ denotes the probability density amplitude of releasing a photon at the moment t, and complies with the normalized relation ${\int }_{-\infty }^{\infty }{\left|\xi \left(t\right)\right|}^2\mathrm{d}t=1$. $t_0$ and $t_1$ represent the initial and final moments of the interaction between $|1_{\xi }\rangle$ and the stationary qubit, respectively. After the single photon are completely released, the the stationary qubit stay at the excited state $|e\rangle$.

On the contrary, assuming that the stationary qubit initially remains at the ground state $|g\rangle$, when it starts receiving $|1_{\xi }\rangle$, $\gamma$ meet the relationship [8,12]

$$\gamma \left(t\right)=\frac{{\left|\xi \left(t\right)\right|}^2}{{\int }_{t_0}^t{\left|\xi \left(\tau \right)\right|}^2\mathrm{d}\tau }.$$

(14)

After the single photon is completely received, the stationary qubit stay at the ground state $|e\rangle$.

Comparing Equations (12) and (14), it is easy to notice that the numerators on the right side of the equations represent the probability of generating or receiving photons at moment t, while the denominators represent the probability of the stationary qubit being the excited state $|e\rangle$.

Combining the relationship between $\gamma$, Q and $Q_L$ mentioned earlier, the relationship between coupling strength and gate voltage can be further obtained. Furthermore, regarding the regulation of the generation or reception of single photons (during which the photon switch is required to be in an open state, and vice versa, it must be in a closed state at other times), the relationship between the waveform of the required single photon and the gate voltage can be obtained. Therefore, further exploration on how to regulate $\gamma$ by the gate voltage has a great research value for regulating the generation or reception process of single photons with the desired shape.

5. Conclusions

In summary, the PS junction can be judged as a kinetic capacitance in circuit. Two series PS junction with a gate voltage controlled island can form a device call PS-SQUID. Similar to DC-SQUID, the kinetic capacitance of the PS-SQUID can be tuned by the gate voltage. Then, a voltage controlled kinetic capacitance are realized in the superconducting circuits. By substituting the input and output capacitance of the CPW resonator with the appropriate PS-SQUID, the loaded quality factor of the CPW resonator can be changed. Thus, we get a photon switch in the microwave domain.