Realization of a Carbon-Nanotube-Based Superconducting Qubit

TL;DR

This paper demonstrates a hybrid superconducting qubit utilizing a carbon nanotube Josephson junction, with voltage tunability and strong coupling to a microwave resonator, advancing solid-state quantum computing components.

Contribution

It introduces a novel hybrid superconducting qubit with a carbon nanotube junction and demonstrates its tunability and strong coupling in a circuit QED setup.

Findings

Achieved voltage tunability of qubit frequency.

Observed strong dispersive coupling to a microwave resonator.

Measured qubit relaxation and coherence times of 10-200 ns.

Abstract

Hybrid circuit quantum electrodynamics (QED) involves the study of coherent quantum physics in solid state systems via their interactions with superconducting microwave circuits. Here we present an implementation of a hybrid superconducting qubit that employs a carbon nanotube as a Josephson junction. We realize the junction by contacting a carbon nanotube with a superconducting Pd/Al bi-layer, and implement voltage tunability of the qubit frequency using a local electrostatic gate. We demonstrate strong dispersive coupling to a coplanar waveguide resonator via observation of a resonator frequency shift dependent on applied gate voltage. We extract qubit parameters from spectroscopy using dispersive readout and find qubit relaxation and coherence times in the range of $10-200~\rm{ns}$.

Tables

Table 1. Table 1: Parameters extracted from resonator and qubit spectroscopy measurements for devices QA and QB.

	$f_{\text{0}}$ [GHz]	$f_{\text{Q}}$ [GHz]	$g_{\text{max}}$ [MHz]	${\chi }_{\text{max}}$ [MHz]
Qubit A	5.572	$2.8-4.2$	113	10
Qubit B	4.595	$2.4-3.5$	85	7

Equations

$$h\alpha =E_{\text{C}}\left(1-\frac{3E_{\text{J}}}{\Delta N}\right),$$

$$h{f}_{01}=\sqrt{8E_{\text{J}}{E}_{\text{C}}}-E_{\text{C}}\left(1-\frac{3E_{\text{J}}}{\Delta N}\right),$$

$$S(\tau )=\frac{1}{2}+\frac{T_1(1-e^{-\tau /(2T_1)})}{\tau }$$

$$2\pi \delta {\nu }_{\text{HWHM}}=\frac{1}{T_2^{\prime }}={\left(\frac{1}{T_2^2}+n_{\text{s}}{\omega }_{\text{vac}}^2\frac{T_1}{T_2}\right)}^{1/2}$$

Full text

Circuit quantum electrodynamics (cQED) with superconducting circuits Wallraff et al. (2004) has been a remarkably successful platform for on-chip quantum optics and quantum information processing research Wendin (2017). Hybrid superconducting circuits provide the platform to access coherent quantum properties of other systems via their interactions with microwave photons or artificial atoms Xiang et al. (2013); Kurizki et al. (2015); Cottet et al. (2017). In recent years, a variety of hybrid superconducting qubits have been realised by replacing the conventional aluminum (SIS) Josephson junctions (JJ) with normal- or semiconductor-based (SNS) JJs, such as InAs nanowires de Lange et al. (2015); Larsen et al. (2015), InGaAs heterostructures Casparis et al. (2018) and graphene Kroll et al. (2018); Wang et al. (2019). For these SNS JJs the normal- or semiconductor is contacted with a superconducting material enabling a supercurrent to flow due to the superconducting proximity effect Meissner (1960). Cooper pair transport in such devices is described by Andreev reflections Andreev (1965); de Gennes and Saint-James (1963); Saint-James (1964). The conductance of semiconductors can be adjusted by applying a voltage to a nearby gate-electrode, which for an SNS JJ tunes the Cooper-pair transport and hence the Josephson energy of the junction.

A strong technical motivation for these new semiconductor-superconductor hybrid JJ qubits is to realize gate voltage tunable qubits and hence eliminating decoherence due to magnetic flux noise. Further, electric fields are much easier to localise compared to magnetic fields, which makes complex multi-qubit devices simpler to engineer. Additionally, qubit operation in moderate magnetic fields, for example to explore interactions with different spin systems, can be made possible due to the robustness of these hybrid JJs to magnetic field Luthi et al. (2018).

A variety of novel physical investigations could be envisaged with carbon nanotube (CNT)-based superconducting qubits. Using a CNT as the junction allows to make use of its exceptional mechanical properties, which could offer a potential platform for creating quantum interference between a qubit and mechanical motion Khosla et al. (2018). Further, ultra-clean CNTs offer ballistic transport characteristics and could be combined with growth from a nuclear spin free C12 precursor. This would allow for perfect, defect free JJs as opposed to conventional Al JJs with an amorphous tunnel barrier. Further, this could have a positive impact on qubit coherence via elimination of two-level fluctuator defects in the amorphous oxide Shnirman et al. (2005); Oh et al. (2006); Müller et al. (2017). Furthermore, hybrid devices incorporating proximitized CNTs would allow to study Andreev levels Gramich et al. (2017); Tosi et al. (2019); Janvier et al. (2015a) and they are also predicted to carry Majorana fermions Klinovaja et al. (2012); Sau and Tewari (2013); Marganska et al. (2018), which could be beneficial for topological quantum computing Kitaev (2003).

In the work presented here proximitized CNTs are used as the JJ in a common planar 2D superconducting qubit architecture and their performance as a qubit is analyzed via a coupled microwave resonator. Resonator and qubit spectroscopy are performed as a function of applied gate voltage and strong dispersive coupling on the order of 100 MHz is observed. Power dependent qubit spectroscopy is used to extract the Josephson energy $E_{\text{J}}$ and charging energy $E_{\text{C}}$ of the qubits. Further, the coherence is investigated and $T_{1}$ and $T_{2}$ times in the range of 10-200 ns are observed.

Fig. 1 shows images and a circuit diagram of the device studied. The chip consists of 10 $\lambda/4$ resonators at different frequencies, multiplexed through a single coplanar microwave transmission line (Fig. 1 (a)). Close to each resonator’s electric field anti-node, qubits are fabricated and a dedicated DC electrostatic gate is used for control of the chemical potential of the CNT (Fig. 1 (b-c)).

The system is cooled below 20 mK in a dilution refrigerator and measured using standard cQED measurement techniques, see Fig. 1 (d). First, the transmission spectrum of a device is measured via the feedline to identify the individual resonance frequencies of the 10 resonators, at which a narrow ($\sim 1$ MHz) absorption dip is observed. Subsequently, $S_{21}$ spectroscopy as a function of gate voltage $V_{\text{G}}$ of each individual resonance is performed and resonators which exhibit a clear gate-dependent resonance frequency are selected for further investigations as these potentially correspond to working CNT-qubits. Usually 20-50% of all devices on one chip show this dependence. From these, two devices showing similar gate-dependent behaviour are carefully characterized, hereafter labeled as device QA and QB.

Fig. 2 (a) shows resonator spectroscopy as a function of DC voltage applied to the gate electrode ($V_{\textrm{G}}$) on device QA. At a single gate-voltage $V_{\text{G}}$, an absorption line corresponding to the resonator is observed (inset Fig. 2 (d)). Tuning $V_{\text{G}}$ the absorption line moves in frequency, exhibiting a broad gate region ($V_{\text{G}}<-30$ V) with an approximately constant resonator frequency of $f_{0}=5.572$ GHz (bare resonator) and quasi-periodic excursion to higher frequencies for $V_{\text{G}}>-20$ V. In cQED this is indicative of the resonator being dispersively coupled to a qubit with tunable transition frequency $f_{\textrm{Q}}<f_{0}$. Here, the resonator frequency is observed to vary by $\sim 10$ MHz, which is consistent with the dispersive regime of cQED Blais et al. (2004).

In a second measurement, we carry out spectroscopy to identify the potential qubit’s frequency. Here the cavity drive is tracking the particular resonance frequency $f_{r}$ at each $V_{\textrm{G}}$ and simultaneously a spectroscopic probe tone is fed onto the input line (qubit drive). While the qubit drive is swept in frequency the amplitude response at $f_{r}$ is measured. This measurement performed on device QA is presented in Fig. 2 (b). A spectroscopic response is observed for $V_{\textrm{G}}>-30$ V. The values of $V_{\textrm{G}}$ exhibiting a spectroscopic response coincide exactly with the values exhibiting shifts in $f_{r}$, cf. Fig. 2 (a). This is consistent with the presence of a qubit, frequency tunable between $2.8$ GHz $<f_{\textrm{Q}}<4.2$ GHz. The data in Fig. 2 (b) likely shows the $f_{01}=f_{\text{Q}}$, i.e. ground to first excited state transition of the qubit as a function of gate voltage.

Similar measurements were also performed on device QB and are presented in Fig. 2 (c) and (d). The resonator of this device exhibits a bare resonance frequency of $f_{0}=4.595$ GHz for $V_{\text{G}}>20$ V. When $V_{\text{G}}$ is tuned below 20 V $f_{r}$ exhibits upwards shifts, see Fig. 2 (c). The strongest shift in $f_{r}$ is observed to be $\sim 7$ MHz. For $V_{\text{G}}<20$ V a spectroscopic response is visible at the same gate voltages as the shifts in $f_{r}$ (Fig. 2 (d)), consistent with the qubit frequency in the range $2.4$ GHz$\leavevmode\nobreak\ <f_{\textrm{Q}}<3.5$ GHz, cf. Table 1.

The coupling strength $g$ between the resonator and the qubit can be estimated using the dispersive shift $\chi$ of the resonator. To first order, due to the coupling to the $f_{01}$ transition, $\chi=\frac{g^{2}}{\Delta}$, where $\Delta=f_{r}-f_{\text{Q}}$ is the detuning between the qubit and the resonator. Using this expression and the data of QA (Fig. 2 (a) and (b)) yields an estimation of the coupling strength that shows to increase with observed $f_{\text{Q}}$, see Supplementary Information. In particular, we extract a minimum value of $g=48$ MHz and a maximum value of $g=113$ MHz. Similar values are observed with QB, summarized in Table 1.

Mapping out $f_{01}$ is the first step of characterizing the voltage tunable qubit. However, to fully describe the qubit Hamiltonian, $E_{\textrm{J}}$ and $E_{\textrm{C}}$ are needed. These parameters cannot be deduced from the qubit’s fundamental frequency alone. Here, we show that they can nevertheless be extracted by measuring the qubit at different drive powers. For this purpose, we use the same qubit spectroscopy technique as in Fig. 2 (b), holding $V_{\text{G}}$ fixed and varying the qubit drive power. Such a measurement is presented in Fig. 3 (a). At a low drive power, $P=-45$ dBm at the output of the microwave generator, only a single peak is observed in the qubit spectroscopy (Fig. 3 (a) bottom trace). If the power of the qubit drive is increased to $P=-30$ dBm, a more complicated multi-peak response is observed, which exhibits a relatively clear second peak at frequencies just lower than $f_{01}$, see Fig. 3 (a). This is indicative of a weakly anharmonic circuit, such as a transmon qubit, where the lower peak corresponds to the two photon transition from the ground state to the second excited state, i.e. $f_{02}/2$. Note that this second spectral peak is not always clearly present, and a broader spectral feature is consistently seen at higher drive powers, within which it is not possible to resolve clear individual peaks. This may be due to significant charge dispersion of higher qubit energy levels, or other unknown sources of decoherence.

We measure the frequency of the two clearest spectral lines over a range of $V_{\text{G}}$ from data similar to that seen in Fig. 3 (a). Interpreting them as the $f_{01}$ and $f_{02}/2$ transitions of a transmon, we can extract the anharmonicity $\alpha=2\left(f_{01}-f_{02}/2\right)$. By using a first order perturbation theory approach to a modified Cooper pair box Hamiltonian presented by Kringhøj et al. Kringhøj et al. (2018) $\alpha$ and $f_{01}$ are given by

[TABLE]

[TABLE]

where $\Delta$ is the induced superconducting gap of the CNT JJ (we estimate $\Delta=90\leavevmode\nobreak\ \mu$eV using DC bias spectroscopy measurements on CNT devices contacted with identical processing) and $N$ the number of conduction channels contributing to Cooper-pair transport. Using Eq. 1 and Eq. 2 together with the values extracted for $f_{01}$ and $\alpha$, the qubit parameters $E_{\text{J}}$ and $E_{\text{C}}$ can be calculated by solving the resulting set of equations.

If all channels are assumed to exhibit equal transmission $\mathcal{T}$, Eq. 1 can be rewritten using $E_{\text{J}}=\frac{\Delta}{4}N\mathcal{T}$ such that $\mathcal{T}=\frac{4}{3}(1-\alpha/E_{\text{C}})$ Kringhøj et al. (2018). The measured $\alpha$ can now be used to calculate $\mathcal{T}$ at each $V_{\text{G}}$ (or $f_{01}$) resulting in pairs of $\left[\mathcal{T},f_{01}\right]$. Comparing these with $hf_{01}(\mathcal{T})=\sqrt{2\Delta E_{\text{C}}\mathcal{T}N}-E_{\text{C}}\left(1-\frac{3}{4}\mathcal{T}\right)$ (resulting from rearranging Eq. 2) for $N\in\{1,2,3,4\}$ and $E_{\text{C}}$ taken from electrostatic modelling for the exact qubit design ($E_{\text{C}}^{QA}=508$ MHz, $E_{\text{C}}^{QB}=391$ MHz), good agreement was found for $N=1$, see Supplementary Information.

In Fig. 3 (b) the Josephson energy $E_{\text{J}}$, extracted with the method mentioned above, is presented as a function of qubit frequency $f_{01}=f_{\text{Q}}$. The dashed lines correspond to expected values for $E_{\text{J}}$ according to Eq. 2 (with $E_{\text{C}}$ extracted from simulations). The spread in values is attributed to the error in extracting $f_{02}/2$ from the qubit power spectroscopy data as these can be noisy and sometimes exhibit a complicated multi-peak structure, making the peak distinction difficult. The Josephson energy can also be used to calculate the critical current $I_{\text{c}}$ of the qubit’s JJ using $E_{\text{J}}=\hbar I_{\text{c}}/(2e)$, see Fig. 3 (b). We find $5$ nA $<I_{\text{c}}<13$ nA, with an average value $\langle I_{\textrm{c}}\rangle=9.6\pm 1.6$ nA across both qubit devices. These values of $I_{\text{c}}$ are comparable to $1$ nA $<I_{\text{c}}<16$ nA that we independently observed in DC measurements of CNT JJs. The normalized histogram for the values of $E_{\text{C}}$, extracted simultaneously with the values of $E_{\text{J}}$, is presented in Fig. 3 (c). Dashed vertical lines correspond to simulated values of $E_{\text{C}}$, as mentioned above, and show good agreement with the data.

The obtained values of $E_{\text{J}}$ and $E_{\text{C}}$ can be used to calculate the ratio $E_{\text{J}}/E_{\text{C}}$. We find a mean $E_{\text{J}}/E_{\text{C}}\approx 11$ for QA and $E_{\text{J}}/E_{\text{C}}\approx 13$ for QB. Again, these values agree well with electrostatic simulations of the device design, yielding $E_{\text{J}}/E_{\text{C}}\sim 12$, giving confidence in the extraction method of $E_{\text{J}}$ and $E_{\text{C}}$. Note that the estimated average $E_{\text{J}}/E_{\text{C}}$ ratio places the qubit between the transmon regime $(E_{\text{J}}/E_{\text{C}}>20)$ and the Cooper pair box regime $(E_{\text{J}}/E_{\text{C}}<1)$, where a low $E_{\text{J}}/E_{\text{C}}$ leads to a complex energy level structure with charge dispersion making the qubit susceptible to charge noise Koch et al. (2007). However, charge dispersion is predicted to vanish in channels where the transmission is approaching unity Averin (1999), which is the case for JJs made from CNTs.

We finally report on investigations of the coherence of the CNT-qubit devices. Conclusive measurements of Rabi oscillations were not observed, likely due to poor coherence (see Supplementary Information). We therefore resort to an alternative pulsed technique for measuring the relaxation time, previously used in quantum dot charge qubits Petta et al. (2004); Penfold-Fitch et al. (2017). The method was tested on standard superconducting qubits to confirm that it yields the same result as standard techniques (see Supplementary Information).

To measure $T_{1}$, a pulse chopping method Petta et al. (2004); Penfold-Fitch et al. (2017) is used. The resonator is continuously measured with a weak cavity drive at $f_{0}$ for a time of 100 $\mu$s. Simultaneously, the qubit is driven on resonance with a pulse train exhibiting a 50% duty cycle and for each measurement the pulse period $\tau$ is varied, see inset Fig. 4 (a). For very short $\tau$, i.e. $\tau\ll T_{1}$, the qubit drive randomizes the qubit between the ground and first excited state and it has no time to relax. In the case of very long $\tau$, i.e. $\tau\gg T_{1}$, the qubit has time to relax to the ground state in between drive pulses. Therefore, in the latter case, the measured signal is the time average of the qubit being in the ground and excited state, giving a signal of half the value found in the limit $\tau\rightarrow 0$. A measurement following this procedure is presented in Fig. 4 (a). The data is normalized with respect to a measurement with the qubit drive turned off and fitted to

[TABLE]

where $T_{1}$ is the only free parameter. In the measurement shown in Fig. 3 (a) the fit yields $T_{1}=117.3\pm 5.8$ ns.

The relaxation time $T_{1}$ was measured across a range of gate voltages, i.e. $f_{\textrm{Q}}$, for both devices, see Fig. 4 (b). QB on average exhibits a longer $\langle T_{1,QB}\rangle=151\pm 71$ ns than QA where $\langle T_{1,QA}\rangle=74\pm 30$ ns. Largest $T_{1}$ values of 250 ns and 150 ns correspondingly for QA and QB were observed

A lower bound, $T_{2}^{\prime}$, on qubit coherence time, $T_{2}$ can be found by measuring the linewidth of a low power qubit spectroscopy trace Schuster et al. (2005); Abragam (1961), see Fig. 4 (c). Fitting a Lorentzian with linewidth $2\delta\nu_{\textrm{HWHM}}$ to the qubit transition peak, $T_{2}^{\prime}$ can be calculated via

[TABLE]

where $n_{\textrm{s}}\omega^{2}_{\textrm{vac}}$ is proportional to the microwave input power Schuster et al. (2005); Abragam (1961). Hence, at low qubit drive powers the linewidth should be the least broadened. This results in $T_{2}^{\prime}=19.0\pm 1.5$ ns for the data presented in Fig. 4 (c).

The measurement and its analysis was repeated for different qubit frequencies $f_{\textrm{Q}}$, see Fig. 4 (d). While $T_{2}^{\prime}$ seems to increase slightly with increasing $f_{\textrm{Q}}$ for QA, this is not true for QB. Coherence is highest at around $f_{\textrm{Q}}=3$ GHz, with $T_{2}^{\prime}=25$ ns for QB, but significantly reduced at $f_{\textrm{Q}}=3.5$ GHz. On average QA exhibits a longer $\langle T_{2,QA}^{\prime}\rangle=10\pm 5$ ns compared to QB $\langle T_{2,QB}^{\prime}\rangle=6\pm 6$ ns. Although these coherence times seem rather short, these only represent a lower bound for $T_{2}$. However, the low coherence times could be attributed to dissipation due to dirty, disordered CNTs, Purcell decay into the gate line and residual resistance to the superconducting leads. Dissipation in nanoscale weak link JJ oscillators, made from aluminium, was previously mentioned as a possible source of decoherence Vijay et al. (2009, 2010); Levenson-Falk et al. (2011). Additionally, the short $T_{2}^{\prime}$ could also stem from Andreev levels in the junction interacting with acoustic phonons Zazunov et al. (2005); Janvier et al. (2015b); Gramich et al. (2015).

The experiments described here demonstrate a first implementation of a voltage tunable superconducting qubit based on a CNT JJ. The device is of similar geometry and exhibits similar gate voltage behaviour to previously reported voltage tunable superconducting qubit devices de Lange et al. (2015); Larsen et al. (2015); Casparis et al. (2018); Kroll et al. (2018); Wang et al. (2019). Simultaneous resonator and qubit spectroscopy showed clear evidence of qubit-resonator coupling with coupling strength on the order of $g\sim 100$ MHz, comparable to state of the art cQED experiments with conventional transmon qubits. Qubit spectroscopy at high drive powers was used to extract the qubit parameters $E_{\text{J}}$, $E_{\text{C}}$, their ratio $E_{\text{J}}/Ec$ and $I_{\text{c}}$ of the JJ. Further, qubit relaxation and coherence times in the range $10-200$ ns were observed.

Although the coherence of these first devices is not competitive with state-of-the-art superconducting qubits, advances in fabrication, e.g. using suspended ultra-clean CNT JJs could lead to significant improvements. Such JJs have already been individually realized Schneider et al. (2012). Additionally, growing CNTs from purified C12 methane would offer a platform for ballistic and defect free JJs, potentially having a huge impact on coherence of the qubits. Fabrication improvements in other hybrid qubit designs, such as those based on InAs nanowires resulted in $T_{1}=5-20\leavevmode\nobreak\ \mu$s Casparis et al. (2016); Luthi et al. (2018), similar to state-of-the-art, aluminum-based flux-tunable transmon qubits Kelly et al. (2015). The implementation of a superconducting qubit with a CNT presented here offers potential for unique experiments in order to create quantum interference between a qubit and mechanical motion Khosla et al. (2018). Additionally, CNT-based qubits could be used as ultrasensitive force sensors Moser et al. (2013) and if arranged in a SQUID geometry as a detector for magnetic moments Cleuziou et al. (2006). Furthermore, these qubits based on proximitized CNTs could be utilized to study Andreev physics Gramich et al. (2017); Tosi et al. (2019); Janvier et al. (2015a) and investigate the prediction of carrying Majorana fermions Klinovaja et al. (2012); Sau and Tewari (2013); Marganska et al. (2018), which could be valuable for topological quantum computing.

I Methods

Device fabrication begins with CNTs grown via chemical vapour deposition on a Si/SiO2 (450 $\mu$m/300 nm) substrate. The microwave resonators are then patterned via electron beam lithography (EBL). Prior to metal deposition an oxygen plasma etch is carried out to remove any CNTs that might electrically short the microwave circuits. Afterwards, 100 nm of Al is deposited via electron beam evaporation. Following lift-off, SEM imaging is used to locate and select CNTs for the qubits. The contacts to the CNT and island of the qubit are then patterned with EBL, post-development cleaned using UV ozone and metalized with a Pd/Al (4/80 nm) bi-layer. Before sample mounting, the room-temperature resistances of the CNT JJs are measured to check the fabrication yield. Roughly 80% of the fabricated devices conduct at room-temperature and exhibit resistances between 7 k$\Omega<R_{n}<100$ k$\Omega$. A series of 3 chips were fabricated, each consisting of 10 potential qubits. Each chip contains 10 $\lambda/4$ resonators with different frequencies, multiplexed via capacitive coupling through a single microwave transmission line, see Fig. 1 (a). All chips are based on the same architecture, except that on the chip of QA the capacitive coupling of the resonator to the transmission line was increased, compared to the chip of QB.

II Acknowledgements

Acknowledgements.

We acknowledge support from the Royal Academy of Engineering, EPSRC (EP/R029229/1), Swiss Nanoscience Institute (SNI) and the Swiss National Science Foundation. M. M. acknowledges support from the Stiftung der Deutschen Wirtschaft (sdw).

III Author contributions

A.B. grew the CNTs at facilities of C.S., A.N. developed the Pd/Al bi-layer and M.M. fabricated the devices. A.P., M.E. assisted in the experimental setup, M.M. performed the measurements and M.M. and P.J.L analyzed the measurements. The Manuscript was prepared by M.M. with P.J.L. and E.A.L, A.N., A.P., M.E. and A.B. providing input. G.A.D.B., E.A.L. and P.J.L supervised the project.