Performance Analysis of Superconductor-constriction-Superconductor Transmon Qubits

The Hamiltonian of a transmon can be written as:

where $n_{g}$ is the offset charge and $E_{J}(\varphi)$ is the potential energy of the junction. The latter is defined by the CPR of the junction, $I_{J}(\varphi)$, using the following integral:

For a SIS junction having a sinusoidal CPR, $I_{J}(\varphi)=I_{c}\sin\varphi$, $E_{J}(\varphi)$ takes a cosine form and the eigenvalues of the Hamiltonian can be solved analytically [14]. In contrast, the CPR of a nanobridge ScS junction typically distorts away from the sinusoidal form, leading to a modified potential energy, and so in general the eigenvalue problem must be solved numerically.

However, we can estimate the anharmonicity of a ScS transmon using a perturbative approach. The discussion here is limited to a single-valued CPR that is distorted from the sinusoidal form, but retains its $2\pi$-periodicity and odd parity, which set $I_{J}(n\pi)=0$ for all $n\in\mathbb{Z}$.[16] Near $\varphi=0$, the CPR can be expressed by a Maclaurin expansion carrying only odd-order terms:

in which $I_{0}=I_{J}^{\prime}(0)$ and the coefficients $a_{n}=I_{J}^{(2n+1)}(0)/I_{0}$ (note this definition sets $a_{0}\equiv 1$). Note that the CPR of a SIS junction is thus a special case of Eq. 3, with $I_{0}=I_{c}$ and $a_{n}=(-1)^{n}$.

Near $\varphi=0$, the potential energy of a ScS transmon is obtained by integrating the Maclaurin series in Eq. 3, as

By comparing Eq. 4 with the potential energy of a SIS transmon,

we may define the Josephson energy of a ScS transmon as $E_{J,\textrm{ScS}}=I_{0}\Phi_{0}/2\pi$ and recognize that the anharmonicity, led by a $\varphi^{4}$ term, is approximately $-a_{1}$ times that of a SIS transmon.

The leading anharmonic term ($a_{1}\varphi^{4}/4!$) in Eq. 4 can be treated as a first-order perturbation to the quantum harmonic osscillator (QHO), so that the $m$th eigenenergy of a ScS transmon is approximated as:

in which $\hbar\omega_{p}=\sqrt{8E_{J}E_{C}}$ is the Josephson plasma energy. The transition energy between the $(m-1)$th and $m$th levels is therefore:

which recovers the familiar SIS result with $a_{1}=-1$. [14] In general, if $a_{1}<0$, the ScS transmon will have negative quartic anharmonicity, similar to the SIS transmon[19].

A concrete form of the CPR of a ScS Josephson junction can be derived from an appropriate model of superfluid transport. In this work, we limit our analysis to the approximation based on the Ginzburg-Landau model which, although limited, can provide an estimate of the relationship between the junction nonlinearity and its physical dimensions. Now consider a ScS Josephson junction comprised of a diffusive quasi-one-dimensional nanobridge that is placed in the interval $x\in[-d/2,d/2]$, with length $d$ and width $w$ (Fig. 1b, inset), connecting two large superconducting islands that respectively have uniform order parameters $\Psi=\Psi_{0}\exp(\pm i\varphi/2)$, where $|\Psi_{0}|^{2}=n_{s}$ is the bulk superfluid density. The two islands thus maintain a phase difference of $\varphi$ across the nanobridge. This neglects any phase drop within the islands, or equivalently, their kinetic inductance. Vijay et al. have noted that the rigid boundary condition is a crude approximation and is valid only when the bridge width $w$ is much shorter than the superconducting coherence length $\xi$.[20, 18, 21] The analysis and following discussion is therefore confined to this regime. With $w\ll\xi$, the spatial variation of the supercurrent density across the nanobridge width can be neglected, so that the order parameter along the nanobridge follows the one dimensional Ginzburg-Landau (GL) equation[22, 16],

In the dirty limit, $\xi$ can be approximated by the geometric mean of the Pippard coherence length $\xi_{0}$ and the electron mean free path $l$, i.e., $\xi\approx\sqrt{\xi_{0}l}$[23]. By neglecting the self-magnetic field and therefore the vector potential $A$, the simplified equation

can be solved numerically with the boundary conditions of $\Psi(\pm d/2)=\Psi_{0}\exp(\pm i\varphi/2)$. The solution is then used to compute the CPR, following

in which $S$ is the cross-sectional area of the nanobridge.

As shown in Fig. 2a, the CPR gradually distorts away from the $\sin\varphi$ form of a SIS junction with increasing constriction length ($d/\xi$). The characteristic current $I_{0}$, as defined in Eq. 3, is given by

to ensure that $a_{0}=I_{J}^{\prime}(0)/I_{0}=1$. The coefficients $a_{1}$ and $a_{2}$ are calculated from the numerical solutions using finite differences, and shown in Fig. 2b for constriction lengths from zero to 3.5$\xi$. Unlike the sinusoidal CPR that has $a_{n}=(-1)^{n}$, both $a_{1}$ and $a_{2}$ decrease in amplitude toward larger $d/\xi$. According to the plot, 50% of anharmonicity, as measured by $|a_{1}|$, is retained when $d/\xi\approx 2.25$. For short constrictions with $d/\xi\ll\sqrt{15}\approx 3.87$, Likharev and Yakobson gave an approximate CPR in closed form[24, 16], by solving Eq. 8 through a perturbation approach:

As shown in Appendix A, the approximation matches the numerical result for constrictions with length $d/\xi$ up to 1.5, but departs more significantly for longer nanobridges.

The diminishing anharmonicity is more clearly visualized by numerically calculating the potential energy $E_{J}(\varphi)$ using the integral in Eq. 2. As shown by Fig. 2c, $E_{J}(\varphi)$ gradually deviates from the cosine curve (dashed black) and approaches the parabola (dotted cyan), as $d/\xi$ increases.

Finally, we note that the GL theory was developed for $T\approx T_{c}$. Although generally GL remains valid at lower $T$, more precise descriptions of superfluid transport at arbitrary $T$ are available. For example, Kulik and Omelyanchuka have given a solution (KO-1) for short, 1D ScS junctions with $w\ll d$ and $d\ll\xi$ in the dirty limit.[15] As shown in Appendix B, the CPR of a KO-1 junction at $T=0$ K has $a_{1}=-1/2$ and $a_{2}=0$. By coincidence, this makes the KO-1 CPR numerically very similar as the GL CPR for a nanobridge with length $d/\xi=2.25$. In general, an arbitrary skewness can be introduced into the constriction junction CPR using the form first proposed by Likharev:

in which $\mathcal{L}$ parameterizes the skewness.[16] As shown in Appendix C, this “skewed sinusoidal” CPR behaves nearly identically as the GL CPR, within given range of $\mathcal{L}$. The skewness factor $\mathcal{L}$ has the physical meaning of being the kinetic inductance associated with the constriction junction, [25] such that when phase drop in the contact leads (i.e., their kinetic inductance) is non-negligible, we may capture its effect by absorbing the additional kinetic inductance into the $\mathcal{L}$ term. Similarly, we may also view GL $d/\xi$ as a pure parameter, with the value of $d$ approaching the physical length of the nanobridge only in the limit of $w\ll\xi$. For these reasons, we will limit the rest of our discussion to CPRs based on the GL solution, while recognizing that the discussion is generally applicable to CPRs of other forms or derived from a different transport model.

For a nanobridge ScS transmon with the potential energy illustrated in Fig. 2c, the wave equation can be solved numerically using the finite difference method, in which the Hamiltonian is expressed in a discretized space of phase $\varphi\in[-\pi,\pi)$, with the periodic boundary condition applied to both ends. The validity of our computation is confirmed by comparing a similar numerical solution of the wave equation for a SIS transmon with the analytical solutions presented by Koch et al [14] (see also Appendix C). Figure 3 compares the first four eigenstates of a SIS transmon and a nanobridge ScS transmon with length $d/\xi=2.25$ and anharmonicity coefficient $a_{1}=-1/2$, both with $E_{J}/E_{C}=20$ and $n_{g}=1/2$. Although the lower level eigenenergies and eigenfunctions are similar, the differences become more apparent at higher energies. This trend is more clearly observed for the transition energies $E_{0m}=E_{m}-E_{0}$ calculated for SIS transmon and the nanobridge ScS transmons at $n_{g}=1/2$, across a range of $E_{J}/E_{C}$ from 1 and 100 (Figure 4a). The calculation is made for three different nanobridge lengths, $d/\xi$ = 1.60, 2.25, and 3.20, which correspond to anharmonicity coefficients of $a_{1}=-2/3$, $-1/2$, and $-1/3$, respectively. In general, ScS transmons with shorter constrictions ($d/\xi$) behave more like SIS transmons, with longer constrictions becoming more QHO-like.

The anharmonicities of ScS transmons, defined as $\alpha=E_{12}-E_{01}$, are plotted against $E_{J}/E_{C}$ and compared with SIS transmons (Fig. 4b), and show diminishing anharmonicity as constriction length $d/\xi$ increases, closely following the perturbation theory result in Eq. 7, i.e. $\alpha=a_{1}E_{C}$. The smaller anharmonicity of a ScS transmon means that the transitions $E_{01}$ and $E_{12}$ lie closer in energy, so that a longer RF pulse is needed to correctly excite the desired transition $E_{01}$. The minimal pulse duration can be estimated as $\tau_{p}\approx 10\hbar|\alpha|^{-1}$, where the factor of 10 is included for practical reasons such as the requirement for pulse shaping.[26] The reduced anharmonicity will therefore require longer pulses to maintain the gate fidelity, decreasing gate speed as compared to conventional SIS transmons. As shown in Figure 4c, despite its lower anharmonicity, $\tau_{p}$ of the ScS transmon remains less than 10 ns even for $E_{J}/E_{C}=100$, when the qubit operates at 10 GHz. For applications where qubit pulse durations are a few to a few tens of ns, we conjecture that the negative impact of the lower anharmonicity on ScS transmon performance may be tolerable.

A primary benefit of the transmon architecture is its relative immunity to charge noise, when designed to operate in the regime of $E_{J}\gg E_{C}$. In a SIS transmon, the charge dispersion of the $m$th level decreases exponentially with $\sqrt{8E_{J}/E_{C}}$, following [14]:

Intuitively, the charge dispersion is related to the tunneling probability between neighboring potential energy valleys (Fig. 2c), e.g., when $\varphi$ makes a full $2\pi$ rotation [14]. By this reasoning, we may expect the higher barrier height of a nanobridge ScS transmon to better suppress the tunneling probability and provide lower charge dispersion, compared to a SIS transmon.

Figure 5a plots the first three eigenenergies $E_{m}$ $(m=0,1,2)$ versus the effective offset charge $n_{g}$ for both SIS (dashed) and nanobridge ScS (solid) transmons, with $E_{J}/E_{C}=10$. Clearly, in longer constrictions (i.e., larger $d/\xi$), the ScS transmon eigenenergies are more weakly perturbed by $n_{g}$. Calculations of the charge dispersion, $\epsilon_{m}=E_{m}(n_{g}=1/2)-E_{m}(n_{g}=0)$, across a wide range of $1\leq E_{J}/E_{C}\leq 100$ show that suppression of charge dispersion in the ScS transmon becomes more effective for larger $E_{J}/E_{C}$ ratios and longer contriction length, $d/\xi$ (Figure 5b). When $E_{J}/E_{C}=100$, the charge dispersion of the first excited state of a ScS transmon with length $d/\xi=2.25$ is over one order of magnitude less than the corresponding SIS transmon. It is noted that the computation for a SIS transmon matches the analytical result very well [14], again demonstrating the high numerical precision of our finite difference computation. The computational error only becomes significant as the normalized charge dispersion, $|\epsilon_{m}|/E_{01}$, approaches $10^{-11}$ and smaller (visible in the lower right corner of Figure 5b). This is due to the accumulation of floating-point error that eventually shows up when evaluating the vanishing difference between the two eigenenergies at $n_{g}=0$ and $1/2$.

In Figure 5b, the $y-$axis is presented on a logarithmic scale and the $x-$axis is scaled as $\sqrt{E_{J}/E_{C}}$, so that all curves take a linear form approaching large $E_{J}/E_{C}$ values. For the SIS transmon, the slope matches the expected $\exp(-\sqrt{8E_{J}/E_{C}})$ dependence in Eq. 14. For the ScS transmon with length $d/\xi=2.25$, the slope is larger, and is best described by:

The improved charge dispersion makes the ScS transmon less sensitive to charge noise and, in turn, gives it a longer dephasing time $T_{2}$. For dephasing caused by slow charge fluctuations of large amplitude, Koch et al. [14] has found an upper limit of $T_{2}$ given by:

Using this relation, we compare $T_{2}$ for both SIS and ScS transmons for $E_{J}/E_{C}$ between 1 and 100 (Figure 5c). The ScS transmon improves $T_{2}$ across the entire range of $E_{J}/E_{C}$ and especially at higher ratios. At $E_{J}/E_{C}=100$, the SIS transmon has a $T_{2}$ ceiling of about 3 ms, compared to about 15 ms for the ScS transmon with $d/\xi=1.60$ ($a_{1}=-2/3$), a 5-fold increase. More significantly improvement is achieved for longer constrictions (i.e., larger $d/\xi$). At present, because the $T_{1}$ lifetime of SIS transmon qubits is still beyond 1 ms and not limited by the charge noise, this advantage of the ScS transmon architecture will yield little performance benefit. However, because we expect the lifetimes of superconducting qubits to continue improving (Schoelkopf’s Law) [27], there may be a point when charge noise dephasing becomes a bottleneck, for which the ScS transmon architecture can offer effective mitigation despite the tradeoff of slower gate speed.

The operational behavior of a ScS transmon is determined by its $E_{J}$, $E_{C}$, and anharmonicity $\alpha$, which define the operating frequency $\omega_{01}$, the relative immunity to charge noise ($\epsilon_{1}$), and the minimum excitation pulse duration ($\tau_{p}$). These three quantities are not independent. We can visualize this interdependence with three sets of contour lines plotted in the plane of $E_{J}$ versus $E_{C}$ (Figure 6a), in which we set the anharmonicity to $\alpha=-E_{C}/2$, corresponding to a nanobridge ScS with length $d/\xi=2.25$. These contours represent: (1) a transmon operating frequency ($\omega_{01}/(2\pi)$) between 1 and 10 GHz (set of red, descending diagonal lines), (2) ratios of $E_{J}/E_{C}$ from 10, 100, and 1000 (set of blue, ascending diagonal lines), and (3) the minimum excitation pulse duration $\tau_{p}$ between 3.2 and 100 ns (set of dashed, predominantly vertical lines). Selecting two of these defines the third one. For example, a ScS transmon designed to operate at $\omega_{01}/(2\pi)=5$ GHz and with an excitation pulse of $\tau_{p}=32$ ns (green dot in Figure 6) must have a $E_{J}/E_{C}$ ratio of about 400. Instead, a shorter excitation pulse of $\tau_{p}=10$ ns (purple dot in Figure 6) requires a tradeoff of smaller $E_{J}/E_{C}\approx 40$, and thus less immunity against charge noise.

Importantly, $E_{J}$ and $E_{C}$ of a ScS transmon are set by the physical device dimensions and fundamental properties of the materials composing it. $E_{J}$ is given by $I_{0}\Phi_{0}/2\pi$, in which the characteristic current $I_{0}$ may be approximated by $\pi\Delta/eR_{n}$ in the dirty limit, where $\Delta$ is the superconducting energy gap and $R_{n}$ is the normal state resistance of the junction. For a BCS superconductor where $\Delta=1.76k_{B}T_{c}$, we can express $E_{J}$ in terms of the material properties $R_{n}/T_{c}=1.76k_{B}\Phi_{0}/(2eE_{J,\textrm{ScS}})$, which is shown as the second (right) $y$-axis in Figure 6a. Similarly, because $E_{C}$ is set by the total capacitance ($C_{\Sigma}=e^{2}/2E_{C,\textrm{ScS}}$) which depends on device geometry and dielectric properties, we can express $E_{C}$ as a capacitance, shown as a second (top) $x$-axis in Figure 6a.

Returning to the example, we can now see from Fig. 6a that designing a nanobridge ScS transmon with $\omega_{01}/(2\pi)=5$ GHz, $\tau_{p}=32$ ns, and $E_{J}/E_{C}$ ratio of about 400 (green dot) requires a junction with $R_{n}/T_{c}\approx 6$ $\mathrm{k\Omega\cdot K^{-1}}$ and capacitor with $C_{\Sigma}\approx 250$ fF. If one instead desires the shorter excitation pulse time of $\tau_{p}=10$ ns (purple dot), the constriction must have $R_{n}/T_{c}\approx 20$ $\mathrm{k\Omega\cdot K^{-1}}$ and $C_{\Sigma}\approx 75$ fF. Because $R_{n}$ is related to the constriction sheet resistance through $R_{n}=R_{\square}d/w$, and $d$ is fixed at $2.25\xi$ in this example, the following constraint applies to the values of material parameters ($R_{\square}$, $T_{c}$, $\xi$) and nanobridge width $w$:

to design a nanobridge to any $R_{n}/T_{c}$ value in Fig. 6a.

Equation 17 shows that to achieve the $R_{n}/T_{c}$ values of a few $\mathrm{k\Omega\cdot K^{-1}}$ required for 5 GHz transmon operation, $R_{\square}$ and $\xi$ should be large and $T_{c}$ and $w$ small. The nanobridge width $w$ is limited by available nanofabrication techniques. Setting $w=20$ nm, the constraints on $R_{\square}$, $T_{c}$, and $\xi$ applied by Eq. 17 are illustrated in Figures 6b and 6c for $R_{n}/T_{c}$ values specified by the green and purple dots in Figure 6a, respectively. The requirement for large $R_{\square}$ is eased with lower $T_{c}$. However, $T_{c}$ must be high enough to minimized quasiparticle loss at the working temperature of transmon (typically 20 mK), so we set the lowest $T_{c}$ to 0.2 K in this example.

These plots show that, in general, $R_{\square}$ must be $\sim 1-10$ $\mathrm{k\Omega/\square}$, which is difficult to achieve in conventional metallic superconductors, but is within the range of high kinetic inductance (KI) superconductors, such as granular aluminum[28, 29] or amorphous niobium-silicon alloys ($a$-Nb_xSi_1-x)[30, 31, 32]. The latter material is also promising because of its relatively long coherence length ($30-50$ nm) and suitable $T_{c}$ ($0.3-1$ K)[30, 31].

Lastly, we note that the requirement for large $R_{n}$ can be relaxed for transmons operated at higher frequency approaching the range of millimeter waves. For example, by extrapolating the plot in Figure 6a, we can estimate that a transmon operated at $\omega_{01}=2\pi\times 20$ GHz with $E_{J}/E_{C}=500$ requires $\tau_{p}=10$ ns and $R_{n}/T_{c}$ only about 1 $\mathrm{k\Omega\cdot K^{-1}}$, which can be achieved with a much wider selection of materials. In general, because ScS junctions can support large critical current densities, they have advantages for higher frequency operations that requires large $E_{J}$ (i.e., $I_{c}$).