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Noise-resilient solid host for electron qubits above 100 mK

Xinhao Li Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA    Christopher S. Wang James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA    Brennan Dizdar James Franck Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA    Yizhong Huang Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA    Yutian Wen Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA    Wei Guo National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University, Tallahassee, Florida 32310, USA    Xufeng Zhang Department of Electrical and Computer Engineering, Northeastern University, Boston, Massachusetts 02115, USA    Xu Han [email protected] Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA    Xianjing Zhou [email protected] National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University, Tallahassee, Florida 32310, USA    Dafei Jin [email protected] Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA Department of Physics and Astronomy, University of Notre Dame, Notre Dame, Indiana 46556, USA

Cryogenic solid neon has recently emerged as a pristine solid host for single electron qubits. At ∼\sim10 mK temperatures, electron-on-solid-neon (eNe) charge qubits have exhibited exceptionally long coherence times and high operation fidelities. To advance this platform towards a scalable quantum information architecture, systematic characterization of its noise feature is imperative. Here, we show the remarkable resilience of solid neon against charge and thermal noises when eNe qubits are operated away from the charge-insensitive sweet-spot and at elevated temperatures. Without optimizing neon growth, the measured charge (voltage) noise on solid neon is already orders of magnitude lower than that in most stringently grown semiconductors, rivaling the best records to date. Up to 400 mK, the eNe charge qubits operated at ∼\sim 5 GHz can maintain their echo coherence times over 1 microsecond. These observations highlight solid neon as an ideal host for quantum information processing at higher temperatures and larger scales.

Solid-state electron qubits are inherently affected by decoherence mechanisms in their host materials. Spectral characterization has revealed a typical 1/f1/f noise distribution for both charge and spin qubits, often attributed to individual charge fluctuators at the surfaces and interfaces of semiconductor or superconducting materials [1, 2, 3]. Extensive effort has been made to extend electron qubit coherence times by reducing the noise density in the surrounding environment and minimizing the sensitivity of qubits to noise  [4, 2, 5]. Recently, we demonstrated solid neon as a novel host material that traps electrons at the neon-vacuum interface [6, 7]. Using a circuit quantum electrodynamics architecture, we address the charge states of eNe by coupling them to superconducting resonators. At the charge sweet-spot, eNe qubits are first-order insensitive to charge noise, showing long coherence times T2∗T_{2}^{*} of up to ∼\sim50 µs (ref [7]), which is nearly four orders of magnitude longer than that of reported semiconductor charge qubits [8]. This behavior can be favorably translated to eNe spin qubits as well with even better predicted performance [9, 10, 11].

There remains, however, more to be understood about the nature of eNe charge qubits (which we will refer to as “eNe qubits” for the remainder of the manuscript) and their environment. The specific electron trapping mechanism on solid neon is unclear, though it is likely that disorder in the neon surface plays a key role [12]. Moreover, the environmental coherence-limiting factors in current devices remain elusive. Investigating the performance of eNe qubits away from the sweet-spot and at elevated temperatures, when the qubit is subject to charge and thermal noise, can provide valuable insight into the coupling between eNe and the environment, revealing the role of the neon host and paving the way for improved qubit performance and stability. This is crucial for scaling up the eNe qubit platform, as environmental noise is a major obstacle to the precise creation and consistent retention of entangled multi-qubit states [1, 13]. It is also essential for understanding the limitations of spin-state control [2, 14, 9] of eNe using electrically sensitive mechanisms such as synthetic spin-orbit coupling [15, 16, 17, 18, 19] or exchange interactions [20, 21, 22, 23]. Furthermore, operating qubits at elevated temperatures can mitigate many engineering constraints due to limited cooling power at base temperature, advantageous for scaling [24, 25, 26].

Refer to caption

Fig. 1. : eNe charge qubit coupled to a TiN high-impedance superconducting resonator. a, Illustration of the high impedance TiN superconducting resonator with two identical electron traps. Microwave (MW) signals to control and read out the electron qubit’s charge state are sent through the resonator via the input and output ports. Gate electrodes are patterned with on-chip low-pass filters to apply DC bias voltage to the qubits while maintaining the resonator’s quality factor. The metal plane between the resonator pins is connected to the ground plane via aluminum wire bonds. b, False-color scanning electron micrograph image of the electron trapping area on the right side of the resonator with trap gates (RTG). c, Cross-section schematic of the electron trapping area.

In this work, we utilize individual eNe qubits as probes for evaluating solid neon as a robust electron qubit host. We study the coherence and noise behavior of eNe qubits under both on- and off-sweet-spot conditions, as well as their temperature-dependent coherence up to 0.5 K. When the qubit is biased to be sensitive to charge noise, dynamical decoupling (DD) effectively extends eNe’s coherence toward the relaxation limit (2T1T_{1}). Notably, the extracted high-frequency charge noise density via DD measurements, projected as voltage fluctuation on the nearby electrode of eNe, can be orders of magnitude smaller than what electron qubits experience in semiconductor materials [20, 16, 15, 21, 17, 18, 19], approaching some of the best performance [23]. Meanwhile, the varying qubit and noise properties across different eNe qubits reflect the complexity of the local charge environment of individual electrons, likely due to the disordered neon surface and adjacent excess electrons. Furthermore, we found that the thermal resilience of eNe qubits with a frequency of ∼\sim 5 GHz supports echo coherence times exceeding 1 µs up to 400 mK, primarily limited by thermally induced increases in energy relaxation and dephasing rates. These results highlight the superior noise isolation the neon host provides and the importance of engineering the local charge environment to enhance performance uniformity.

Refer to caption

Fig. 2. : Spectroscopic and coherence properties of an eNe charge qubit Q1. a, Normalized microwave transmission amplitude (A/A0A/A_{0}) centered around the resonator frequency versus the relative resonator bias voltage Δ​Vres\Delta V_{\text{res}} as described in b. Two avoided crossings appear when the eNe qubit comes into resonance with the resonator. b, Two-tone qubit spectroscopy measurement displaying the transmission phase response at the resonator frequency ωr\omega_{\text{r}} versus Δ​Vres\Delta V_{\text{res}}, for weak probe tones concurrently sent in at fdrivef_{\text{drive}}. c, Ramsey fringes at the charge sweet-spot, marked with the red arrow in b, with fitted T2∗T_{2}^{*} of 8.2 µs. PeP_{e} is the qubit’s excited-state population. d, Relaxation and Hahn echo measurements showing T1T_{1} = 11.6 µs and T2echoT_{2}^{\text{echo}} = 21.6 µs at the charge sweet-spot.

Device structure

Our device consists of a split superconducting resonator made of a 30 nm thick TiN film grown on a ⟨111⟩\langle 111\rangle-oriented intrinsic silicon (Si) substrate by atomic layer deposition [27], as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKa. An electron trap is positioned at each end of the resonator, with the broader goal of coupling two distant eNe qubits via the resonator bus [28]. Each electron trap features a simplified rectangular structure compared to the previously used oval-shaped design [6, 7]. Around the traps, the Si substrate is etched down by approximately 250 nm to host the thin neon layer, as illustrated in Fig. Noise-resilient solid host for electron qubits above 100 mKb. In this work, we utilize the differential mode of the resonator to couple to motional states of electrons trapped on solid neon (see Supplementary I), with the microwave electrical field pointing from one resonator pin to the other [29], as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKc. Considering the first two charge states of eNe, the coupled system can be described by the Jaynes-Cummings Hamiltonian [14]:

H=ℏ​ωr​(a†​a+12)+12​ℏ​ωq​σz+g​(a†​σ−+a​σ+),H=\hbar\omega_{\text{r}}\left(a^{\dagger}a+\frac{1}{2}\right)+\frac{1}{2}\hbar\omega_{\text{q}}\sigma_{z}+g(a^{{\dagger}}\sigma_{-}+a\sigma_{+}), (1)

where ωr/2​π=5.668\omega_{\text{r}}/2\pi=5.668 GHz is the resonator frequency after neon deposition, ωq/2​π\omega_{\text{q}}/2\pi is the qubit transition frequency, gg is the electron-photon coupling strength, a†a^{{\dagger}} and aa are the photon creation and annihilation operators, respectively, σz\sigma_{z} and σ±≡σx±i​σy\sigma_{\pm}\equiv\sigma_{x}\pm i\sigma_{y} are the standard Pauli operators on a two-level system.

As an improvement from previous work, we leverage the high kinetic inductance (∼\sim20 pH/□\Box) of the thin TiN film to enhance the qubit-resonator coupling strength [30]. The estimated equivalent lumped element impedance for the differential mode is Zr∼600​ΩZ_{r}\sim 600\,\Omega (see Supplementary I), approximately ten times that of the previous niobium (Nb) device [6, 29]. Since g∝ωr​Zrg\propto\omega_{r}\sqrt{Z_{r}}, we expect the high impedance resonator to support a coupling strength achieving g/2​π∼10​MHzg/2\pi\sim 10\,\text{MHz} level [30]. Meanwhile, several DC gates are placed around the electron trapping area to tune the qubit transition frequency, as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKa-b. In order to minimize microwave leakage of the resonator mode through the DC gates, all gates are equipped with on-chip low-pass filters with a ∼\sim0.5 GHz cutoff frequency providing over 60 dB attenuation at resonator frequency. This is particularly important in our device given that for high-impedance resonators, the parasitic gate capacitance can be comparable to the resonator capacitance [31, 32]. With this design, the resonator maintains a narrow linewidth κ/2​π\kappa/2\pi of 0.38 MHz (see Supplementary I). Before performing electron experiments, a thin layer of neon was grown on the surface of the device following the same procedure as in our previous work [7], resulting in less than 1 MHz redshift of the resonator frequency. Electrons are emitted from a tungsten filament and bound to the surface of the neon film. Only when both the neon film is present on the sample surface, and electrons are emitted do we see signatures of trapped eNe qubits strongly coupled to the resonator (see Supplementary II).

Refer to caption

Fig. 3. : Decoherence of eNe qubit Q1. a, Calculated 1/T2CPMG1/T_{2}^{\rm{CPMG}} for various qubit bias points and refocusing pulse numbers NN. With the increase of NN, the 1/T2CPMG1/T_{2}^{\rm{CPMG}} of the qubit biased near the charge sweet-spot approaches the limit of 2​T12T_{1}. b, The pure dephasing time TϕT_{\phi} increases as a function of NN when the qubit biased away from charge sweet-spot, with a power-law fit of Tϕ∝N0.61T_{\phi}\propto N^{0.61}. c, Repeated Ramsey fringes measured near qubit sweet-spot for 128 iterations, with each record taking 33 s. d, Detuning Δd​q\Delta_{dq} between drive tone and qubit frequency during the Ramsey measurements, revealing stochastic frequency shifts.

Qubit performance

As illustrated in Fig. Noise-resilient solid host for electron qubits above 100 mKc, the local neon profile and the trap structure define the potential energy landscape seen by the electrons. Recent theoretical work emphasized the neon surface’s important role in defining the qubit’s Hamiltonian [12], whose exact form remains unclear. Here, we approximate each eNe qubit’s transition frequency with a generalized hyperbolic model, capturing the measured qubit spectroscopic features:

ℏ​ωq=h2​(ωss+δ​ωss)2+(ϵ+δ​ϵ+2​de​(ℰ+δ​ℰ))2,\hbar\omega_{\text{q}}\,=\,\sqrt{h^{2}(\omega_{\text{ss}}+\delta\omega_{\text{ss}})^{2}+(\epsilon+\delta\epsilon+2d_{e}(\mathcal{E}+\delta\mathcal{E}))^{2}}, (2)

where ωss\omega_{\text{ss}} represents the the charge sweet-spot frequency, and ϵ\epsilon describes the energy off-set defining the corresponding bias voltage. This model has been widely applied to describe two-level quantum systems [33], with an energy landscape consistent with our bounds of large anharmonicity in eNe charge qubits [7]. The electrical tunability of the qubit’s transition frequency is described by the term 2​de​ℰ2d_{\rm{e}}\mathcal{E}, where ded_{\rm{e}} is the electron dipole moment and ℰ\mathcal{E} is the applied field [34, 35, 36]. Meanwhile, noise terms δ​ℰ\delta\mathcal{E}, δ​ϵ\delta\epsilon and δ​ωss\delta\omega_{\text{ss}} caused by DC bias or adjacent charge fluctuations lead to qubit decoherence.

Bringing eNe qubit’s transition frequency ωq\omega_{\rm{q}} onto resonance with ωr\omega_{\rm{r}} results in the vacuum Rabi splitting in the resonator’s transmission spectrum. An example of this is shown in Fig. Noise-resilient solid host for electron qubits above 100 mKa for Q1, one of the three qubits we characterized, with the electron-photon coupling strength g/2​π=6.43g/2\pi=6.43 MHz and on-resonance qubit linewidth γ/2​π=3.81\gamma/2\pi=3.81 MHz (see Supplementary III-VI for details of characterizations on the three qubits Q1-3). Compared to the previous Nb resonator [6, 7], the higher impedance of the TiN resonator enhances the qubit-resonator coupling strength, with a maximum observed g/2​πg/2\pi of approximately 1616 MHz (see Q3 in Supplementary VI). We further mapped the qubit spectrum by applying a second drive tone and probing at the resonator frequency at low power, as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKb. The extracted qubit frequency follows a hyperbolic dependence on the DC voltage VresV_{\text{res}} applied to the resonator with a charge sweet-spot at 5.065 GHz (see Supplementary IV). The sweet-spot frequency, coupling strength, and sensitivity to DC biases (∂fq/∂Vres\partial f_{\text{q}}/\partial V_{\text{res}}) vary between qubits (see Supplementary III). This variation suggests some randomness in the local trapping potential, which determines the qubit’s minimum transition frequency and the projection of the electron’s dipole moment along the resonator mode and the applied DC field.

Next, we characterize the coherence performance of the qubit biased at its charge sweet-spot. Figure Noise-resilient solid host for electron qubits above 100 mKc-d show the measured T1T_{1}, T2∗T_{2}^{*}, and T2echoT_{2}^{\text{echo}} of 11.6 µs, 8.2 µs, and 22.6 µs, respectively. Unlike the qubit reported in our previous works [7], the relaxation time (T1T_{1}) of eNe qubits observed on the new TiN device is generally not Purcell-limited at their charge sweet-spots, with over 100 MHz detuning from the resonator (Δrq/2​π\Delta_{\rm{rq}}/2\pi). The Purcell rate of Q1 to the resonator mode is Γr=κ​g2/Δrq2= 1/3.9​ms−1\Gamma_{\rm{r}}\,=\,\kappa g^{2}/\Delta_{\rm{rq}}^{2}\,=\,1/3.9\,\text{ms}^{-1}, which indicates that non-radiative decay channels dominate the energy relaxation of Q1. Measurements of the Q1’s T1T_{1} at various bias frequencies (see Noise Spectroscopy section) further confirm that the non-radiative decay dominates the energy relaxation unless the qubit frequency is tuned much closer to the resonator frequency. Additionally, the fact that the T2echoT_{2}^{\rm{echo}} approaches 2​T12T_{1} indicates that the quasi-static noise is the dominating dephasing factor for Q1 at its charge sweet-spot [37, 38].

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Fig. 4. : Noise spectroscopy of eNe qubit Q1. Main plot between 0.01 ∼\sim 1 MHz: Total noise density (colored dots) derived from dynamical decoupling data at different qubit bias points. Main plot between 10−310^{-3} to 10−110^{-1} Hz: Calculated total noise density (blue dots) from long-term Ramsey measurements when biased at charge sweet-spot. Main plot near 5 GHz: Transverse noise of the eNe qubit (green triangles). Inset plot i: Equivalent charge (voltage) noise on the resonator electrode. Inset plot ii: Zoom-in of transverse noise. Gray dashed lines: Power-law fits of frequency-dependent noise.

Noise spectroscopy

Fluctuations in the charge environment can cause stochastic frequency shifts of the eNe qubit, which leads to qubit dephasing. Armed with high fidelity control and a known voltage (charge) lever arm, we first characterize the high-frequency noise spectral density of Q1 using the Carr-Purcell-Meiboom-Gill (CPMG) sequence with N=0,1,2,4,6,8,12N=0,1,2,4,6,8,12, and 16 refocusing pulses. All bias points had positive Δ​Vres\Delta V_{\rm{res}} as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKb, with a maximum sensitivity of ∂fq/∂Vres=180.7\partial f_{\rm{q}}/\partial V_{\rm{res}}=180.7 MHz/mV, corresponding to a qubit frequency detune of 15.9 MHz from the charge sweet-spot. The measured average energy relaxation time at all bias points is 11.94 ±\pm 0.3 µs, indicating that the same non-radiative decay channel dominates. We fit the measured coherence decay to extract the decoherence time T2CPMGT_{2}^{\text{CPMG}} and pure dephasing time TϕT_{\phi} (See Methods). As shown in Fig. Noise-resilient solid host for electron qubits above 100 mKa, in the absence of refocusing pulses (i.e., the Ramsey measurement), the decoherence rate 1/T2CPMG1/T_{2}^{\text{CPMG}} increases with sensitivity to charge noise. At ∂fq/∂Vres=180.7\partial f_{\rm{q}}/\partial V_{\rm{res}}=180.7 MHz/mV, T2∗T_{2}^{*} decreases to 1.93 µs. Introducing refocusing pulses enhances T2CPMGT^{\text{CPMG}}_{2} at every biasing point, approaching two times of T1T_{1} as NN increased. At the charge sweet-spot, the rapid saturation of T2CPMGT_{2}^{\text{CPMG}} with increasing NN indicates a low noise power density in the relatively high-frequency range (100 kHz). When the qubit was biased at a point more sensitive to charge noise, more pulses were required to extend the T2CPMGT_{2}^{\text{CPMG}} towards the energy relaxation limit. In Supplementary V and VI, we show that for the other two eNe qubits, a single refocusing pulse is insufficient to mitigate the majority of noises at the qubits’ sweet-spot, suggesting a higher noise density at the high-frequency range. This variation in noise behavior suggests some complexities of the local noise environment experienced by individual electrons on the neon surface.

Figure Noise-resilient solid host for electron qubits above 100 mKb plots the fitted pure dephasing time (TϕT_{\phi}) at bias points away from the sweet-spot as a function of the number of refocusing pulses (NN). The relation between TϕT_{\phi} and NN reflects the frequency-dependent noise power distribution. For a noise spectrum following S​(f)∝ 1/fαS(f)\,\propto\,1/f^{\alpha}, TϕT_{\phi} should scale with NβN^{\beta}, where β=α/(1+α)\beta=\alpha/(1+\alpha) (ref. [39, 20, 21, 17]). The fitted β\beta via this scaling relation provides a more accurate noise distribution compared to individual fittings of the decay curve using χN​(τ)=(τ/Tϕ)α+1\chi_{N}(\tau)=(\tau/T_{\phi})^{\alpha+1} (ref. [39]). For Q1, the average fitting gives β\beta = 0.61, corresponding to α\alpha = 1.56.

In the 100 kHz frequency range of Fig. Noise-resilient solid host for electron qubits above 100 mK, we plot the calculated total noise in the unit of Hz2/Hz\text{Hz}^{2}/\text{Hz}, for the four different bias points away from the charge sweet-spot. (See Methods for details of the CPMG sequence and noise calculation.) The extracted total noise scales from 10610^{6} to 10410^{4} Hz2/Hz\text{Hz}^{2}/\text{Hz} as the frequency increases, with an averaged power-law fitting of S∝1/f1.55S\propto 1/f^{1.55}, matching the fitting result in Fig. Noise-resilient solid host for electron qubits above 100 mKb. The spectral noise behavior observed away from the sweet-spot suggests that charge noise dominates the total noise when the qubit is more sensitive to electrical fluctuation. Given the qubit’s sensitivity to VresV_{\rm{res}} and neglecting other potential noise sources, we can extract the equivalent voltage (charge) noise as fluctuations of the DC voltages applied on the resonator electrodes. The inset in Fig. Noise-resilient solid host for electron qubits above 100 mK shows the calculated charge noise of Sv∼10−5S_{v}\sim 10^{-5}\,µV2/Hz\text{V}^{2}/\text{Hz} at the 100 kHz frequency range. A separate measurement on Q2 gives a similar charge noise density with Sv∝ 1/f1.14S_{v}\,\propto\,1/f^{1.14} (see Supplementary V and Discussion and Outlook).

Material platform Qubit type SvS_{v} (μV2/\mu\text{V}^{2}/Hz) Noise frequency (kHz) ∂fq/∂V\partial f_{q}/\partial V (MHz/mV) Reference Neon single-electron charge 10−5∼10−710^{-5}\sim 10^{-7} 10∼100010\sim 1000 10∼10\sim100 this work Si/SiO2 singlet-triplet spin 10−1∼10−210^{-1}\sim 10^{-2} 100∼1000100\sim 1000 ∼\sim 10 ref. [20] Si/SiO2 single-electron spin 10−1∼10−210^{-1}\sim 10^{-2} 10∼10010\sim 100 0.20.2 ref. [15] 28Si/28SiO2 single-electron spin 100∼10−110^{0}\sim 10^{-1} 1∼101\sim 10 0.010.01 ref. [16] Si/SiGe singlet-triplet spin 10−4∼10−510^{-4}\sim 10^{-5} 100∼1000100\sim 1000 ∼10\sim 10 ref. [21] Si/SiGe single-electron spin 10−1∼10−310^{-1}\sim 10^{-3} 10∼100010\sim 1000 0.550.55 ref. [19] 28Si/SiGe single-electron spin 10−2∼10−310^{-2}\sim 10^{-3} 10∼10010\sim 100 0.090.09 ref. [17] 28Si/SiGe single-electron spin 10−110^{-1} 1010 0.360.36 ref. [18] 28Si/SiGe singlet-triplet spin 10−2∼10−410^{-2}\sim 10^{-4} 10∼100010\sim 1000 4.54.5 ref. [22] GaAs/AlGaAs singlet-triplet spin 10−7∼10−810^{-7}\sim 10^{-8} 100∼1000100\sim 1000 10∼10010\sim 100 ref. [23]

Tab. 1. : Comparison between voltage noises experienced by electron qubits on solid neon and in semiconductor material platforms.

To benchmark different electron qubit material platforms, we compare the environmental charge noise measured as voltage fluctuations on adjacent gate electrodes, situated at a distance on the scale of one hundred nanometers from the qubits. Specifically, we examine electrons trapped on solid neon and those confined in semiconductor materials, which are the leading platforms for hosting electron qubits. Table 1 summarizes our results on eNe and compares them with data from the literature on semiconductor qubits. Because of the clean qubit environment provided by the neon interface, the charge noise experienced by eNe is orders of magnitude lower than that in typical semiconductor materials, approaching some of the best performance. Given the smaller susceptibility of electron spin qubits to charge noise through effects including spin-orbit coupling and exchange interactions, as shown in the “∂fq/∂V\partial f_{q}/\partial V” column of Tab. Noise-resilient solid host for electron qubits above 100 mK, we foresee that electron spin qubits on solid neon could approach a pure dephasing time of millisecond-scale [9]. The development of the electron trap and the suppression of redundant electrons on neon will further improve the qubit performance.

After investigating the high-frequency noise, we turn to the low-frequency noise at the charge sweet-spot. Figure Noise-resilient solid host for electron qubits above 100 mKc-d show variations in Q1’s frequency revealed by performing repeated Ramsey measurements [21, 20] for approximately one hour since initially biased near its charge sweet-spot. The qubit undergoes a discrete frequency transition of varying magnitudes, as seen near the 25 minutes of the one-hour tracing. This is revealed by a Fast Fourier transform (FFT) of the Ramsey signals (See Methods and Supplementary IV). It suggests slow mechanisms driving qubit decoherence, akin to low-frequency dynamics in semiconductor and superconducting qubits [21, 20, 40], which can be mitigated with feedback control [41, 42]. Using a periodogram method, we convert this one-hour qubit frequency measurement into a frequency noise spectrum over 10−310^{-3} to 10−110^{-1} Hz, as shown in Fig. Noise-resilient solid host for electron qubits above 100 mK. A power-law fit of the data gives the relation of S∝f−1.11S\propto f^{-1.11}. Despite the qubit being first-order insensitive to small charge (voltage) fluctuation at the sweet-spot, it can still experience slow frequency drifts due to nearby charge fluctuations, as observed in both semiconductor and superconducting qubits via second-order effects [43, 44]. Other detection methods, such as single-electron charge sensing techniques [21], could complete the noise spectrum ranging from 1 Hz to 10410^{4} Hz.

To complete the discussion, we calculate the transverse noise at the qubit frequencies contributing to energy relaxation with 1/T1=π/2×S​(2​π​fq)1/T_{1}=\pi/2\times S(2\pi f_{\rm{q}}) (ref. [37]). The fitted total transverse noise versus frequency follows S∝f2.22S\propto f^{2.22} between 5.065 to 5.498 GHz, with a mean of 3.6×1053.6\times 10^{5} Hz2/Hz\text{Hz}^{2}/\text{Hz}, as shown in the inset of Fig. Noise-resilient solid host for electron qubits above 100 mK.

Temperature dependence

To obtain more information about the noise environment of eNe qubits, we measured the temperature dependence of Q1’s coherence (T1T_{1}, T2∗T_{2}^{*}, and T2echoT_{2}^{\rm{echo}}) at its charge sweet-spot from 10 mK to 500 mK. The energy relaxation T1T_{1} is well described by a model that only accounts for the coupling of a two-level quantum system to a bosonic thermal bath within the experiment’s temperature range, T1∝tanh​(ℏ​ωq/2​kB​T)T_{1}\propto\text{tanh}(\hbar\omega_{\rm{q}}/2k_{\text{B}}T) (Fig. Noise-resilient solid host for electron qubits above 100 mKa) (ref. [45, 46]). Similarly, the thermal population follows that of a Maxwell-Boltzmann distribution [47] (See Supplementary IV). The corresponding single electron temperature closely tracks the mixing-chamber (MXC) temperature between 100 to 500 mK and saturates around 40 mK, as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKa. This effective cooling of electron qubits below 100 mK is difficult to achieve with electrons in semiconductors, due to the suppressed electron-environment interaction at low temperatures [48].

A separate measurement on a different qubit Q2, presented in Supplementary V, also reveals strikingly thermal behavior in T1T_{1} and thermal population but with a different T1T_{1} in the low-temperature limit. It suggests a picture in which individual eNe qubits couple differently to their environment (perhaps through phonons [14, 9]), which sets the low temperature T1T_{1}. This motivates further studies of the microscopic limitations of T1T_{1}. At these qubit frequencies, T1T_{1} drops to only about half the low-temperature value at 200 mK, facilitating operation at higher temperatures. In contrast, electron qubits in semiconductor materials are susceptible to phonon effects [25, 49], which can significantly degrade T1T_{1} at elevated temperatures. Given these favorable scalings, we can also anticipate further improvement of the high-temperature performance of eNe qubits by engineering the electron trapping and detection structure to support charge- or spin-qubit operations at higher frequencies [24].

The temperature dependence of the coherence data, however, remains complex. T2∗T_{2}^{*} and T2echoT_{2}^{\rm{echo}} show that quasi-static and high-frequency noise components behave differently with temperature, as shown in Fig. Noise-resilient solid host for electron qubits above 100 mKb. With increasing temperature, T2∗∝T−0.74T_{2}^{*}\propto T^{-0.74}, indicating the quasi-static noise power scales almost linearly against temperature. Below 50 mK, T2echoT_{2}^{\rm{echo}} approaches 2​T12T_{1}, suggesting that the single echo pulse still effectively mitigates the quasi-static noise that contributes to most of the dephasing. However, starting from 75 mK, T2echoT_{2}^{\rm{echo}} scales with temperature as ∝T−1.1\propto T^{-1.1}, and quickly degrades from 2​T12T_{1}, indicating the rise of non-quasistatic dephasing noises at higher temperatures [23]. Despite that, Q1 maintains >> 1 µs T2echoT_{2}^{\rm{echo}} at elevated temperatures up to 400 mK with qubit frequency only at ∼\,\sim 5 GHz, showing eNe’s thermal-resilient coherence performance and potential for high-temperature operation.

Refer to caption

Fig. 5. : Temperature-dependent coherence of eNe qubit Q1 at charge sweet-spot. a, T1T_{1} (blue dots, data) versus mixing chamber (MXC) temperature. The solid curve represents the predicted decay rate T1​(T)=T1​(T=0)⋅tanh​(ℏ​ωq/2​kB​T)T_{1}(T)=T_{1}(T=0)\cdot\text{tanh}(\hbar\omega_{\rm{q}}/2k_{B}T) where we use the measured value at 10 mK for T1​(T=0)T_{1}(T=0). Extracted electron temperature (gray diamonds, data) versus MXC temperature based on the measured thermal population in Supplementary IV. b, Decoherence time, T2∗T_{2}^{*} (orange dots) and T2echoT_{2}^{\rm{echo}} (green dots) versus the MXC temperature, with power-law fitting. Gray dots and curves show 2​T12T_{1} for comparison. c, Extracted pure dephasing time TϕT_{\phi} as a function of the MXC temperature. Red dashed curve: Parameter-free calculation of the resonator-induced dephasing based on Q1 properties.

We then extract the relation between the pure dephasing time (TϕT_{\phi}) and MXC temperature, following the coherence decay formula described in the Methods. As shown in Fig. Noise-resilient solid host for electron qubits above 100 mKc, TϕT_{\phi} begins to decrease significantly when the MXC temperature exceeds 100 mK. In this range, the measured data matches well with a parameter-free model accounting for the effect of resonator thermal photon on the qubit dephasing [24, 50]:

Tϕ−1=κ2​Re​[(1+2​i​χκ)2+8​i​χκ​nth−1]T_{\phi}^{-1}=\frac{\kappa}{2}\text{Re}\left[\sqrt{\left(1+\frac{2i\chi}{\kappa}\right)^{2}+\frac{8i\chi}{\kappa}n_{\text{th}}}-1\right] (3)

where κ\kappa is the resonator decay rate, and χ\chi is the resonator’s dispersive shift (see Supplementary IV). nthn_{\text{th}} represents the resonator thermal photon population, equal to 1/(eh​fr/kB​T−1)1/(e^{hf_{\rm{r}}/k_{\text{B}}T}-1). At higher temperatures, qubit coherence is degraded primarily due to thermal effects. Nevertheless, the data’s deviation from the model below 100 mK suggests the presence of other low-temperature dephasing mechanisms. Separate measurements on Q2 (Supplementary V) give similar results.

Discussion and outlook

In this work, we have quantitatively evaluated, for the first time, the environmental noise isolation provided by the thin neon layer for eNe qubits, demonstrating a number of distinct advantages compared to semiconductor host materials for single electron qubits. Additionally, eNe qubits exhibit thermally resilient coherence, making them promising for high-temperature operation. These systematic coherence and noise characterization highlight the following directions for future research.

The inconsistent spectral properties of eNe qubits indicate that the local neon profile is crucial in electron-trapping mechanisms. Variations in the neon surface profile, caused by liquid-phase solidification and substrate roughness, introduce randomness in the energy potential landscape that hosts the electron qubits. Additionally, the relative position of trapped electrons to nearby electrodes determines the frequency lever-arm and the qubit’s sensitivity to voltage fluctuations. To enhance consistency and control over individual qubit properties, refined neon growth methods, and gate-defined electron trapping mechanisms should be developed to mitigate the effects of neon surface randomness.

On the other hand, diverse noise behavior observed in eNe qubits reflects the complexity of the local charge environment. The inconsistent spectral noise distribution suggests locally non-uniform noise sources. Theoretical studies have shown the high sensitivity of qubit’s coherence to fluctuators’ density and distribution [51]. Excess electrons emitted by the tungsten filament could result in artificial charge fluctuators with high mobility on the neon surface. This charge rearrangement can further contribute to the low-frequency qubit drift and occasional electron escape events [52]. Strategies include developing electron loading procedures [29] and protection gates to separate trapped qubits from adjacent charge reservoirs [25] could be implemented to improve the qubit’s stability.

Finally, the limiting factor for the non-radiative energy relaxation rate of eNe qubits needs further investigation. The non-monotonic variations in T1T_{1} of qubits with different charge sweet-spot frequencies point to localized non-radiative decay channels. Nearby charges may create a sparse bath of two-level fluctuators weakly coupled to the eNe qubit, whose density determines the transverse noise intensity [53, 51]. Electrical manipulations could be applied to bias or repeal those weakly coupled charges to reduce the relaxation rate [54]. With the development of improved electron loading and trapping methods, we foresee the improvements in both the coherence and long-term stability of eNe qubits.

Methods

Device and setup

The resonator, electrode, and on-chip filter were patterned with electron beam lithography followed by reactive-ion etching. See details of the TiN film, resonator, and on-chip filter characterization in Supplementary I. The experiment setup is similar to the ones in our previous works [6, 7]. The chip was mounted on a customized printed circuit board within a vacuum-sealed copper cell, which provides direct current (DC) and microwave (MW) interfaces. On the top lid of the cell, a gas filling line was attached to deposit neon at cryogenic temperature, and a tungsten filament was used as the electron source. The cell is mounted on the mixing chamber (MXC) plate of a dilution refrigerator. A total attenuation of 60 dB was applied on the cryogenic segment of the MW input line with infrared filters (QMC-CRYOIRF-004). The MW output line was equipped with cryogenic isolators (LNF-ISISC4_12A) at the MXC plate, followed by a high electron mobility transistor (LNF-LNC4_8C) at 4K plate and room temperature amplification. All DC connections were filtered with thermocoaxes cable, LC filters (Mini-Circuits RLP-30+), and homemade low-pass filters with 150 Hz cutoff. Qubit spectroscopic measurements were conducted with a vector network analyzer (Keysight N5222B) and a signal generator (Anritsu MG3692C). Time-domain pulse measurements were conducted with Quantum Machine OPX+ and Octave. The DC gates were applied with QDevil QDAC-II.

Neon growth

Neon is filled with the following procedure. The fridge is warmed up from its base temperature with a heater mounted on the 4 K plate. At this moment, the helium mixture circulation is turned off, and all the mixture has been collected while the pulse tube is still on. The heater power is set to a value such that it creates a temperature gradient from 27 K at the 4 K plate to about 25 K at the MXC plate. Under such conditions, the neon gas is filled and deposited onto the device chip in its liquid phase. After filling, the heater is turned off, and the whole fridge is cooled down again to let the liquid neon freeze into solid. During the cool down, we further anneal the neon film at 10 K for 1 hour.

Electron deposition

Electrons are ejected from the tungsten filament mounted on the lid of a hermetic copper cell. When the dilution fridge is cooled down to the base temperature, the total resistance on the filament loop is 2 Ω\Omega. A current pulse train applied by a pulse generator with −0.6-0.6 V voltage output, pulse width of 0.1 ms, repetition frequency of 1 kHz, and duration of 0.3 s was used to fire electrons. We noticed that applying higher voltage or longer duration would cause too many electrons to land on the top of the chip, reducing the stability of the trapped electrons.

Noise characterization

We use the dynamical decoupling technique with Carr-Purcell-Meiboom-Gill (CPMG) pulse sequences to study the spectral distribution of high-frequency noise affecting the eNe qubits [37, 38]. In the sequence, NN refocusing YπY_{\pi} pulses are applied between two Xπ/2X_{\pi/2} pulses with identical separation τ/N\tau/N between two pulses. Under such sequences, qubits’ coherence decaying follows Pe​(N,τ)=P0+a⋅exp​(−τ/2​T1)​exp​(−χP)​exp​(−χN​(τ))P_{e}(N,\tau)=P_{0}+a\cdot\text{exp}\left(-\tau/2T_{1}\right)\text{exp}\left(-\chi_{P}\right)\text{exp}\left(-\chi_{N}(\tau)\right), accounting for pure dephasing −χN-\chi_{N}, energy relaxation −τ/2​T1-\tau/2T_{1}, and decay during driving pulses −χP-\chi_{P} (ref. [37]). We define T2CPMGT_{2}^{\text{CPMG}} as the time when the qubit’s coherence decays by a factor of 1/e1/e due to energy relaxation and pure dephasing. Increasing the number of YπY_{\pi} pulses reduces the time the qubit is exposed to noise before each refocusing, thereby extending the coherence time. Under Gaussian noise assumption [37, 38], a CPMG sequence with NN refocusing pulses and total delay time of τ\tau imposes a filter function gN​(ω,τ)g_{N}(\omega,\tau) to the qubit noise caused by source λ\lambda with spectral distribution Sλ​(ω)S_{\lambda}(\omega), which determines the qubit’s dephasing χN​(τ)\chi_{N}(\tau):

χN​(τ)=τ2​∑λ(∂ωq∂λ)2​∫0∞Sλ​(ω)​gN​(ω,τ)​𝑑ω\chi_{N}(\tau)=\tau^{2}\sum_{\lambda}\left(\frac{\partial\omega_{q}}{\partial\lambda}\right)^{2}\int_{0}^{\infty}S_{\lambda}(\omega)g_{N}(\omega,\tau)\,d\omega (4)

where ∂ωq/∂λ\partial\omega_{\rm{q}}/\partial\lambda is the qubit transition frequency’s sensitivity to noise source λ\lambda, the filter function gN​(ω,τ)=|yN​(ω,τ)|2/(ω​τ)2g_{N}(\omega,\tau)=|y_{N}(\omega,\tau)|^{2}/(\omega\tau)^{2}, and yN​(ω,τ)=1+(−1)1+N​exp​(i​ω​τ)y_{N}(\omega,\tau)=1+(-1)^{1+N}\text{exp}(i\omega\tau) +2​∑j=1N(−1)j​exp​(i​ω​τ​(j−0.5)/N)​cos​(ω​τπ/2)+2\sum_{j=1}^{N}(-1)^{j}\text{exp}(i\omega\tau(j-0.5)/N)\text{cos}(\omega\tau_{\pi}/2).

With the noise filter imposed by the CPMG sequence, we could approximate the total noise power spectrum density SS at frequency fNf_{N} as (ref. [21, 20]):

S​(2​π​fN)=1(Tϕ)2​gN​(2​π​fN,Tϕ)​Δ​ωNS(2\pi f_{N})=\frac{1}{(T_{\phi})^{2}g_{N}(2\pi f_{N},T_{\phi})\Delta\omega_{N}} (5)

where 2​π​fN2\pi f_{N} is the peak angular frequency of the first harmonic of gN​(ω,Tϕ)g_{N}(\omega,T_{\phi}), and Δ​ωN\Delta\omega_{N} is the full width at half maximum of the peak.

To investigate the low-frequency dynamics of qubit decoherence near the charge sweet-spot, we repeatedly conducted Ramsey measurements for 128 iterations, with each recording taking 33 seconds [21, 20]. As shown in Fig. Noise-resilient solid host for electron qubits above 100 mKd, we extracted the qubit frequency drift through a fast Fourier transform (FFT) analysis of the measured Ramsey fringes. During the measurement, we also observed more drastic qubit frequency fluctuations on the scale of 10 – 100 MHz, corresponding to 0.14 – 0.44 mV in Δ​Vres\Delta V_{\text{res}}. Sometimes, we even observed the disappearance of qubit signatures in the usual gate scanning range. These qubit frequency ”jumps” usually occur less than 0.5 times per hour. We attribute these large fluctuations to charge rearrangements of nearby weakly trapped electrons on neon [52]. We ensured that the qubit remained relatively stable and free from such significant fluctuations during data collection for all the presented measurements at various bias points.

Data availability

The data that support the findings of this study are available from the corresponding authors upon request. Source data are provided with this paper.

Code availability

The codes used to perform the experiments and to analyse the data in this work are available from the corresponding authors upon request.

Acknowledgements.
Work performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, was supported by the U.S. DOE, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. D. J., X. H., and X. L. acknowledge support from Argonne National Laboratory Directed Research and Development (LDRD) program. D. J. and Y. W. acknowledge support from the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-23-1-0636. D. J. and W. G. acknowledge support from the National Science Foundation (NSF) under Award No. OSI-2426768. D. J., X. Zhou, and Y. H. acknowledge support from the Julian Schwinger Foundation for Physics Research. D. J. acknowledges support from the Department of Energy (DOE) under Award No. DE-SC0025542. B. D. acknowledges support from the NSF under Award No. DMR-1906003. C. S. W. and X. L. acknowledge support from Q-NEXT, one of the US DOE Office of Science National Quantum Information Science Research Centers. W. G. acknowledges support from the Gordon and Betty Moore Foundation through Grant DOI 10.37807/gbmf11567 and the National High Magnetic Field Laboratory at Florida State University, which is supported by the NSF Cooperative Agreement No. DMR-2128556 and the state of Florida. X. Zhang and Y. H. acknowledge support from the Office of Naval Research (ONR) Young Investigator Program (YIP) program under Award No. N00014-23-1-2144. X. H. acknowledges support from France and Chicago Collaborating in the Sciences (FACCTS) program. This work was partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the NSF under award number DMR-2011854. This work made use of the Pritzker Nanofabrication Facility of the Institute for Molecular Engineering at the University of Chicago, which receives support from SHyNE, a node of the NSF National Nanotechnology Coordinated Infrastructure (NSF NNCI-1542205). The authors thank David I. Schuster and Amir Yacoby for their helpful discussions.
Author contributions X. L., X. Zhou, and D. J. devised the experiment. X. L. and C. S. W. conducted the experiment. X. L. designed and fabricated the device. X. L., C. S. W., and B. D. analyzed the data. Y. H., X. Zhang ,and X. H. supported the experimental measurement. Y. W. and W. G. supported the theoretical understanding. D. J. conceived the idea and led the project. X. L., C. S. W., X. Zhou, and D. J. wrote the original manuscript. All authors contributed to the work.

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