Abstract
Quantum sensing techniques have been successful in pushing the sensitivity limits in numerous fields, and hold promise for scanning probes that study nanoscale devices and materials. However, forming a nanoscale qubit that is simple and robust enough to be placed on a scanning tip, and sensitive enough to detect various physical observables, is still a great challenge. Here, we demonstrate, in a carbon nanotube, an implementation of a charge qubit that achieves these requirements. Our qubit’s basis states are formed from the natural electronic wavefunctions in a single quantum dot. Different magnetic moments and charge distributions of these wavefunctions make it sensitive to magnetic and electric fields, while difference in their electrical transport allows a simple transportbased readout mechanism. We demonstrate electric field sensitivity better than that of a single electron transistor, and DC magnetic field sensitivity comparable to that of NV centers. Due to its simplicity, this qubit can be fabricated using conventional techniques. These features make this atomiclike qubit a powerful tool, enabling a variety of imaging experiments.
Introduction
Ultrasensitive nanoscale detectors of electric and magnetic fields take an increasingly central role in advancing the research of devices and materials. When used as scanning probes, these detectors provide a unique insight to electronic and spin systems on the nanoscale. To date, a large variety of scanning probe sensors have been developed, optimized to measure specific physical quantities: Magnetic fields are primarily imaged via scanning SQUIDs^{1}, Hall probes^{2} and NV centers^{3}, while electric fields are primarily probed with Kelvin probes^{4}, scanning tunneling potentiometry^{5}, and scanning single electron transistors (SET)^{6}. From all the above techniques, only NV centers utilize a quantum twolevel system (qubit) that takes full advantage of the power of quantum manipulations. This gives NV centerbased probes unprecedented sensitivity to local magnetic fields, and additionally a modest sensitivity to electric fields^{7}. NV scanning probes have also few limitations: optical readout introduces a significant challenge at cryogenic temperatures, and using the sensor at high magnetic fields requires impractical RF frequencies. A different type of a scanning qubit that can sense electric fields ultrasensitively on the nanoscale and simultaneously probe magnetic fields with modest sensitivity, will have complementary capabilities and is thus highly desirable.
Candidate solid state qubits for nanosensing applications generically divide into two groups—atomic and engineered. Atomic qubits (such as NV centers^{8} or P dopants in Si^{9,10}) utilize natural atomic wavefunctions as their basis, and hence are small and often have long coherence times. Engineered qubits, on the other hand (e.g., transmon^{11} or semiconducting double quantum dot qubits^{12,13}) provide finer control over the energy spectrum and the dipole coupling to the physics of interest, however, they are larger (μms to mms), require complex planar circuit designs, and often rely on external detectors for their readout, thus making them less suitable for nanoscale sensing applications. A qubit that can combine the simplicity of atomic qubits with the tunability and control of engineered qubits could therefore lead to a potentially powerful scanning nanosensor.
A conceptually simple qubit that may combine the above advantages can be based on the natural electronic wavefunctions of a single quantum dot. Similar to atomic orbitals, such wavefunctions have distinct spatial structure. This structure, however, occurs on much larger spatial scales and can therefore provide larger and more tunable electric moments. Carbon nanotubes present an excellent setting for realizing this concept; in their recent generations they are electronically pristine, allowing the creation of quantum dots with exceptional level of control over their wavefunctions and energy spectrum^{14,15}. So far, engineered double quantum dot qubits have been demonstrated successfully in carbon nanotubes^{16,17,18,19}, but the lithographic complexity of these devices and their frequent reliance on external readout elements such as onchip resonators, may be prohibitive for using them in scan probes. Moreover, most of these qubits were intentionally designed to be insensitive to external fields and thus are poor sensors. At the same time, single quantum dot devices in carbon nanotubes have been successfully used as ultrasensitive scanning SETs, allowing to image oxide interfaces^{20}, Wigner crystals^{21}, as well as the mapping of ballistic^{22} and hydrodynamic^{23} electron flows, demonstrating the compatibility of these devices with scanning probe applications.
In this work, we realize a qubit in a carbon nanotube, and demonstrate its application as a highlysensitive electric and magnetic fields nanosensor. Our qubit has a simple builtin transportbased readout, a highly local response to electric fields, which we image directly using capacitive techniques, and simplicity that allows placing it at the edge of a scanning probe cantilever. We determine its decay and dephasing times using timedomain and Landau–Zener–Stuckelberg^{22} interferometry experiments and show that its coherencelimited transition leads to significantly improved electric potential sensitivity as compared to the thermallybroadened Coulomb blockade peak of an SET. Furthermore, we demonstrate that the same qubit can simultaneously detect magnetic fields parallel to the nanotube axis. Although the short coherence time did not allow us to implement dynamic decoupling protocols and compete with the AC sensitivity of NV centers, we achieve DC magnetic field sensitivity that is on par with that of NV center^{24} and Hall barbased^{25} scanning probes.
Results
Forming a qubit using natural wavefunctions in a nanotube
The basis of our qubit is given by two electronic wavefunctions in a single quantum dot, formed in a suspended carbon nanotube. Within the singleparticle picture, a parabolic confinement potential along the nanotube leads to a ladder of harmonic oscillator levels, whose wavefunctions’ extent along the nanotube axis increases with increasing level number (Fig. 1a, gray illustrations). Each level is 4fold degenerate, due to the spin (↑, ↓) and valley (K, K′) degrees of freedom. In a gapped nanotube, K and K′ electrons rotate in opposite directions around the nanotube circumference, leading to opposite orbital magnetic momenta. Applying a magnetic field parallel to the tube axis, B_{}, breaks the spectrum into four independent ladders with slopes given by the orbital and spin magnetic moments, \({\textstyle{{\partial E} \over {\partial B_{}}}} = \pm \mu _{{\mathrm{spin}}} \pm \mu _{{\mathrm{orb}}}\), with μ_{orb} ≫ μ_{spin} (red and blue lines in Fig. 1a correspond to K and K′ states). Spinorbit coupling splits^{26} the 4fold degeneracy at B_{} = 0, and Coulomb repulsion changes the simple noninteracting wavefunctions into Wigner crystals with finer realspace structures^{21}, yet, since the valley remains a good quantum number, the simple picture in which tuning B_{} leads to crossing between levels with different magnetic moments and different spatial structures remains valid for the discussion below.
To make an atomiclike qubit that is sensitive to local electric and magnetic fields, we choose a crossing between a highlying K state and a lowlying K′ state with opposite spin directions, which we will denote as K_{n} and \(K^{\prime}_{\mathrm{m}}\) (n ≫ m). The charge density of the K_{n} state is spatially extended (Fig. 1b. left, red) whereas that of the \(K^{\prime}_{\mathrm{m}}\) state is spatially localized (Fig. 1b. left, blue), endowing the qubit transition a localized electric moment. Contrary to a standard charge qubit in double quantum dots, whose charge is localized on the left or right dots, separated by lithographic dimensions, in our case the electrical moment results from the difference in charge distribution of different wavefunctions within a single quantum dot. Due to symmetry of the charge distributions around the center of the dot, the dipole moment of the qubit is approximately zero, potentially reducing its sensitivity to homogenous electric fields and farfield noise, however, this qubit has quadrupole or higher moments that yield strong sensitivity to local fields, which is beneficial for high resolution imaging. The K_{n} and \(K^{\prime}_{\mathrm{m}}\) states have also opposite orbital momenta (Fig. 1b, center), endowing the qubit transition also a large magnetic moment (~20 μ_{B}, μ_{B} is the Bohr magneton). We choose opposite spins for the basis states to minimize their overlap, which leads to long decay times.
The schematic charge stability diagram for a holedoped nanotube single quantum dot is plotted in Fig. 1c as a function of gate voltage, V_{G}, and B_{}. The diagram is obtained by adding the charging energy U to singleparticle energies in Fig. 1a, inset. Transport occurs along Coulomb blockade (CB) charging lines, across which a hole is added to the system, zigzaging between K (red) and K′ (blue) character as a function of B_{}. The relevant triple point for our experiment separates the state \(\left N \right\rangle\), having N holes, from the two qubit states, \(\left B \right\rangle = \left N \right\rangle + K_{\mathrm{n}}\), and \(\left D \right\rangle = \left N \right\rangle + K^{\prime}_{\mathrm{m}}\), both having N + 1 holes. Within the Coulomb valley there should be a boundary line separating the \(\left B \right\rangle\) and \(\left D \right\rangle\) ground states (dashed green) along which the ground state changes its last occupied wavefunction while maintaining the total charge in the dot fixed. This line will however be invisible in transport, as the system is in Coulomb blockade.
Experimental realization and transport signature
We implement the atomiclike qubit in a device that has an single carbon nanotube, assembled at the edge of a cantilever (Fig. 2a). The nanotube is suspended over a distance of 1.2 μm between two Au contacts (S,D) over an array of seven individually controlled gates (Fig. 2a, inset). The height difference between the contacts and the gates (suspension height) is 60 nm. Note that this geometry is identical to our scanning nanotubebased SET cantilever geometry, which we previously used to image 1D^{19} and 2D^{20,23,27} systems, making the technique developed here directly applicable for scanning probe applications. The device is fabricated using a nanoassembly technique^{15}, in which nanotube growth and lithography are performed independently, allowing deterministic assembly of a nanotube with desired properties on a complex circuit. In this paper we will use the multiple gates to directly image the charge density of the electronic wavefunctions that form the basis for a qubit. However, we want to emphasize that the qubit that we demonstrate here does not require multiple gates and should work equally well in the simplest singlegated nanotube transistor device which can be formed by standard device fabrication techniques^{14}. Our device is cooled in a dry dilution refrigerator, with an electron temperature of T_{el} ~ 60 mK as measured by the width of CB peaks, and with a magnetic field parallel to the nanotube axis. The nanotube conductance, G, is measured at zero DC V_{sd} bias using an LC tank circuit connected to the drain contact^{28}, and with a small AC excitation on source contact (~15 μVrms) at the tank circuit resonant frequency (Supplementary Note 1).
Figure 2b shows G, measured as a function of a common gate voltage, V_{G}, applied together on all gates, and B_{}. The expected zigzag behavior of the CB peaks is clearly visible, however, while the K transitions exhibit finite conductance at the CB peak (bright), the K′ transitions have no observable conductance (dark), and are marked in the figure by dashed blue lines. Similar dark/bright behavior of the two valleys at finite B_{} was observed previously^{29,30}. The triple point used for our experiments is shown in the zoomin measurement (Fig. 2c) with the three relevant ground states, \(\left N \right\rangle\), \(\left B \right\rangle\) and \(\left D \right\rangle\), labeled. To clarify, the ‘dark’ states described here are not equivalent to the recently reported dark states due to coherent population trapping^{31}, which occur at finite V_{sd}, and do not require B_{} > 0.
At finite B_{}, the p − n junction barriers that confine the holes in the two valleys differ significantly (Fig. 1b right). Their height and spatial extent, given by the nanotube bandgap, decreases with B_{} for the K states (\(dE_{{\mathrm{gap}}}^{\mathrm{K}}/dB_\parallel =  2\mu _{{\mathrm{orb}}}\)) and increases with B_{} for the K′ states (\(dE_{{\mathrm{gap}}}^{{\mathrm{K}}^\prime }/dB_\parallel = 2\mu _{{\mathrm{orb}}}\)), leading to markedly different transport for the two states. The difference in transport visibility between the \(\left N \right\rangle\) ↔ \(\left B \right\rangle\) and \(\left N \right\rangle\) ↔ \(\left D \right\rangle\) transitions thus gives a builtin transportbased readout mechanism for the qubit state, which does not require an external charge detector.
Spatial mapping of charge distributions
In order to form a charge qubit, sensitive to its electric environment, its basis states (\(\left B \right\rangle\) and \(\left D \right\rangle\)) should differ in their charge distribution along the nanotube. We image these charge distributions directly using the array of gates, as follows; First, we tune the voltage common to all gates, V_{G}, to observe the Coulomb peak at the \(\left N \right\rangle\) ↔ \(\left B \right\rangle\) transition (illustrated in gray, Fig. 3a bottom). Then, we repeat this scan but with a voltage offset ΔV added to gate i. This will lead to a shift in the Coulomb blockade peak by δV_{i} along the V_{G} axis, proportional to the local charge density just above this gate (Colored curves in Fig. 3a bottom, Supplementary Note 2). By measuring the individual shifts with respect to all gates, δV_{i}, i = 1..7, we thus image the charge density added on the transition at seven spatial points, which is essentially the discrete version of the scanning imaging of Wigner crystals that we performed previously^{21}. Although the \(\left N \right\rangle\) ↔ \(\left D \right\rangle\) transition is dark in transport, we still know its position accurately by connecting the corners of the bright transitions (dashed blue, Fig. 2c). Thus, using the same method we can also image the spatial charge density within the \(\left D \right\rangle\) state.
Figure 3b zooms in on the triple point around \(\left N \right\rangle\), \(\left B \right\rangle\) and \(\left D \right\rangle\) (white square in Fig. 2c). Upon addition of ΔV = 0.5 mV to gate 4 the bright and dark transition shifts by independent amounts, \(\delta V_4^{\mathrm{B}}\) and \(\delta V_4^{\mathrm{D}}\) (gray arrows). Similar measurements with all gates yields the shifts \(\delta V_{\mathrm{i}}^{\mathrm{B}}\) and \(\delta V_{\mathrm{i}}^{\mathrm{D}}\), which when plotted as a function of gate position (Fig. 3c and d) trace the spatial distribution of the charge added at the \(\left N \right\rangle\) ↔ \(\left B \right\rangle\) and \(\left N \right\rangle\) ↔ \(\left D \right\rangle\) transitions, ρ_{NB}(x) and ρ_{ND}(x), where x is the spatial coordinate along the nanotube. Visibly, while ρ_{NB}(x) is homogenously spread over all gates, ρ_{ND}(x) is localized at the dot’s center.
Timedomain measurements
To study the dynamics of a \(\left D \right\rangle\), \(\left B \right\rangle\) qubit we turn to time domain experiments that use gate voltage on the central three gates, V_{G}, as a fast control axis, with the following sequence: First, the dot is initialized in the \(\left B \right\rangle\) state, on the \(\left N \right\rangle\) ↔ \(\left B \right\rangle\) Coulomb peak (V_{G} = V_{CB}, black star, Fig. 4a). Then, a fast ramp to V_{G} = V_{probe} is applied, after which the system is left to evolve for time τ_{probe}. Finally, the voltage is swept back to the initial CB point for readout, dwelling for time τ_{read,init}. If after the probing stage the system ended up in the ground state \(\left B \right\rangle\), the dot will freely conduct in the readout stage. However, if the system switched to the excited state \(\left D \right\rangle\), it will remain in the dark state during readout, blocking the conductance. The characteristic blocking time is given by the fastest of two possible decay routes, \(\left D \right\rangle\) → \(\left N \right\rangle\) or \(\left D \right\rangle\) → \(\left B \right\rangle\), both of which initialize the system to its ground state. The above sequence is repeated periodically, and we measure the conductance averaged over this sequence, which contains two terms: \(\left\langle G \right\rangle = \left( {G\left( {V_{{\mathrm{probe}}}} \right)\tau _{{\mathrm{probe}}} + G\left( {V_{{\mathrm{CB}}}} \right)\left\langle {P_{\mathrm{B}}} \right\rangle \tau _{{\mathrm{read,init}}}} \right)/\!\left( {\tau _{{\mathrm{probe}}} + \tau _{{\mathrm{read,init}}}} \right)\). The first term reflects the conductance measured during the probing stage, and is nonzero only for V_{probe} near the Coulomb peak, where the dot has a finite conductance. The second term reflects the conductance measured in the readout stage, and is directly proportional to the mean bright state probability, \(\left\langle {P_B} \right\rangle\) during this stage.
Figure 4b shows \(\left\langle G \right\rangle\) measured as a function of V_{probe} within the above sequence (blue), using τ_{probe} = 0.8 μs and τ_{read,init} = 5 μs, as well as the measured quasiDC conductance, G, (red). The CB peak in G appears also in \(\left\langle G \right\rangle\), as expected from the first term the equation above. Interestingly, however, inside the Coulomb valley \(\left\langle G \right\rangle\) shows a sharp dip at V_{probe} = V_{BD} ≈ −343 mV, not present in G. This dip is much narrower (~40 μV) than the thermallylimited CB peak (~200 μV). From \(\left\langle {P_B} \right\rangle\) extracted from \(\left\langle G \right\rangle\) and G using the equation above (Fig. 4c) we see that for most values of V_{probe} the state remains bright (P_{B} = 1), but at the dip the dark state becomes significantly occupied (\(\left\langle {P_{\mathrm{B}}} \right\rangle\) ≈ 0.7). Repeating the above measurement at various values of B_{} (Fig. 4d) shows that this dip traces a straight line terminating at the \(\left N \right\rangle\), \(\left B \right\rangle\), \(\left D \right\rangle\) ‘triple point’, as expected from the \(\left B \right\rangle\) ↔ \(\left D \right\rangle\) degeneracy line (dashed green in Fig. 1c). Its finite slope suggests that this transition is sensitive to both local magnetic and electric fields, where the latter attests to the different charge distribution within the \(\left B \right\rangle\) and \(\left D \right\rangle\) states.
Similarly to Fig. 3, we can directly image the charge density distribution change at the \(\left B \right\rangle\) ↔ \(\left D \right\rangle\) transition, by measuring the response of the transition line position, V_{BD}, to small gate perturbations. The measured density distribution, ρ_{BD}(x) (Fig. 4e, left) compares well to difference between the bright (Fig. 3c) and dark (Fig. 3d) state densities, ρ_{BD}(x) ≈ ρ_{NB}(x) − ρ_{ND}(x) (Fig. 4e, right), further establishing the narrow transition line as the boundary between the \(\left B \right\rangle\) and \(\left D \right\rangle\) ground states. From the measured ρ_{BD}(x) we see that the qubit charge redistribution is also narrow in space. The observed width (~200 nm, Fig. 4a) is limited by the resolution of the imaging method due to the size and distance to the gates. A more quantitative analysis that deconvolves the known shape of the potential distribution produced by the gates concludes that the actual width is ~100 nm (Supplementary Note 4). This width sets the spatial resolution of the qubit sensor.
To measure the transition rate (T_{1} time) and its dependence on the detuning from the degeneracy point, V_{BD} (Fig. 5a), we repeat the measurements above but with different dwell times, τ_{probe}. In Fig. 5b we plot \(\left\langle {P_{\mathrm{B}}} \right\rangle\) as a function of the voltage offset, ΔV_{G} = V_{probe} − V_{BD}, for various τ_{probe} values, and in Fig. 5c we plot it as a function of τ_{probe} for different values of ΔV_{G}. Away from the dip, the decay time is too long to reliably be extracted from this figure, whereas at the dip it becomes significantly shorter, T_{1} ~ 1 μs, indicating a fast transition from \(\left B \right\rangle\) to \(\left D \right\rangle\).
Coherent behavior and estimating system parameters
To observe quantum coherence of the \(\left D \right\rangle\), \(\left B \right\rangle\) qubit and estimate its \(T_2^ \ast\), we use LandauZenerStuckelberg (LZS) interference^{22,32}. In this case, instead of waiting at V_{probe} for τ_{probe}, the detuning is steered as V_{G}(t) = V_{BD} + ΔV_{G} + A_{LZS} sin(2πf_{LZS}t) (Fig. 5d) and we probe \(\left\langle {P_B} \right\rangle\) after time τ_{probe} by moving V_{G} to the CB peak and measuring \(\left\langle G \right\rangle\) as before. The \(\left\langle G \right\rangle\) measured as a function of ΔV_{G} and A_{LZS} at a frequency of f_{LZS} = 0.7 GHz (Fig. 5e) shows the characteristic LZS interference pattern. The peak width, δω ~ 2π × 180 MHz, indicates that the qubit maintains coherence over several oscillations, having a \(T_2^ \ast\) time of ~0.9 ns.
The observations can be quantitatively explained by a simple model, describing the evolution of the system in the \(\left D \right\rangle\), \(\left B \right\rangle\) manidfold; The unitary evolution is described by the Hamiltonian \(H = {\it{\epsilon }}\left( t \right)\sigma _{\mathrm{z}} + {\mathrm{\Delta }}\sigma _{\mathrm{x}}\), where the \(\left D \right\rangle\) and \(\left B \right\rangle\) are the eigenvectors of σ_{z}. The dominant decoherence mechanism with rate γ_{2} results from coupling to charge noise acting along the energy detuning axis, \({\it{\epsilon }}\left( t \right)\), coupling only to σ_{z}. In the fardetuned regime (\({\it{\epsilon }} \gg {\mathrm{\Delta }},\,{\upgamma}_2\)), the noise changes the phase difference between the basis states (T_{2} processes), however, close to zero detuning it translates to incoherent transition rate between the basis states (T_{1} process) (Supplementary Note 5). Consequently, the transition rate between the states depends sharply on the detuning (Fig. 5a, bottom). Using this model we quantitatively fit the results in Fig. 5b, c (solid lines) and obtain the qubit’s splitting, Δ = 2π × 2 MHz, and its decoherence rates γ_{1} = 2π × 1.5 kHz, γ_{2} = 2π × 185 MHz (details in Supplementary Note 6), where γ_{1} results from the noise coupled through σ_{x}, σ_{y}. With the same parameters we also reproduce (Fig. 5f), quantitatively well the LZS measurements in Fig. 5e (Simulation details in Supplementary Note 13).
Estimating performance of the device as a local sensor
The strong dependence of the qubit transition on electric potential and B_{} implies that it can serve as an excellent nanoscale probe of these quantities. Since SET is the most sensitive scanning electrometer to date, we benchmark the qubit sensitivity against measurements in SET modality. DC electric potential sensitivity is measured by slowly ramping the central gate voltage (V_{4}) up and down, and monitoring conductance changes in the two modalities: On the qubit \(\left B \right\rangle\) ↔ \(\left D \right\rangle\) transition line using fast gating, and on the SET \(\left N \right\rangle\) ↔ \(\left B \right\rangle\) CB line within the same triple point. Parameters are optimized separately for each modality, and the results are converted to a common potential scale (Fig. 6a). Visibly, the qubit provides a significantly improved sensitivity, primarily due to its sharper transition line as compared to the temperaturelimited Coulomb blockade peak. The sensitivity to detuning that we obtain in the qubit measurements is ~60 neV Hz^{−0.5} which translates to a potential sensitivity of ~600 nV Hz^{−0.5} (Supplementary Note 8), significantly improving over the performance of the device as an SET, and surpassing the sensitivity of our best SETs to date^{27}. DC B_{} sensitivity is measured in a similar fashion (Fig. 6b). Here, the advantage of qubit detection as compared to an SET becomes even more evident, reaching a sensitivity of ~39 μT Hz^{−0.5} (Supplementary Note 9), comparable to the DC magnetic field sensitivity of NV centers^{24}. What limits the sensitivity in the current experiment is the large contact resistance of our device (R_{c} ~ 2 MΩ) and magnetic field fluctuations inherent to a magnet power supply. Theoretical estimates predict that the performance can be improved by more than an order of magnitude by improving the contact resistance and using a persistent mode magnet (Supplementary Note 7). An additional important difference between the two modalities is their backaction on the measured system. While the SET will fluctuate between two states with different charge values on the dot, in the qubit modality, only a small redistribution of the charge along the axis of the nanotube will occur (Fig. 3d). This translates into a reduced backaction of the measurement. In principle by modifying the electrical moments of the basis states of the qubit one can continuously tradeoff sensitivity for reduced backaction. This feature is extremely important in measurements of fragile quantum states of matter^{21}.
Our sensor requires finite B_{} for its operation; however, it can operate in a wide range of magnetic fields (demonstrated at 3–8T, see Supplementary Note 12). This provides complementary capability to that of scanning SQUIDs and NV centers, which generally work only at lower fields, although achieving better magnetic field sensitivities. Importantly, the field direction required to tune the sensor is inplane for the scanned system, and will couple to electron in a probed 2D sample through a rather small Zeeman energy shift (gμ_{B}B ~ 350 μeV at 3T), which would be negligible for many of the interesting phenomena in 2D. Independently, an out of plane magnetic field component can be applied to tune the properties of the scanned 2D system. The measurement requires dilution temperatures, however, the sample under study can be thermally decoupled^{23} from the probe, and its temperature can in principle be tuned over a large temperature range while keeping the qubit cold, by using local heating only of the electron system^{33}.
The spatial resolution demonstrated here (~100 nm) was limited by the rather long (1.2 μm) device used in this study to enable the gate imaging in Fig. 3. In principle, it should be straightforward to implement the same qubit in a much shorter and simpler, singlegated suspended device and the resolution will scale in proportion, to the tens of nm range. The geometry of the current device is equivalent to the standard scanning SET cantilevers, and is thus fully compatible with scanning.
Recent work^{34} demonstrated spin qubits in carbon nanotube double quantum dots with significantly improved coherence times as compared to the results above. However, the sensitivity of these qubits to external magnetic fields is rather limited as compared to that of the atomiclike qubit demonstrated here. The two primary reasons are the exchange coupling to the feromagnetic leads, which results in weak coupling to an external field, and the fact that the magnetic moment difference between the two qubit basis states is that of a single spin, ~30 times smaller than the magnetic moment difference in the atomiclike qubit, which follows from the orbital moments.
In summary, we have demonstrated a charge qubit in carbon nanotubes that combines the advantages of atomic and engineered qubits. This qubit is conceptually simple, requires only conventional fabrication, has a small form factor allowing placing it on a scanning probe tip, has a simple builtin readout mechanism, and enables sensitive measurements of electric and magnetic fields. These features make it an enabling tool for a variety of experiments. For example, since an atomiclike qubit can be much smaller than lithographic dimensions, which constrain doubledot qubits, it can couple to high vibrational modes of suspended carbon nanotubes, which are at their quantum ground state at dilution temperatures, thus enabling quantum nanomechanical experiments in this system. As a scanning detector that measures simultaneously electric and magnetic fields, it will be instrumental in exploring phenomena that have both charge and magnetic (/electric current) signatures. Few examples include imaging current whirlpools in hydrodynamic electron flow^{35}, as well as imaging of quantum flows, including various electron optics phenomena, electron interference, electron localization, magnetic focusing of electrons and of composite fermions. The increased sensitivity and reduced backaction of the qubit will further allow to image the quasiparticles of fragile states of matters, including the observation the electrical charges of topological quasiparticles. More broadly, the addition of quantum sensing and time domain capabilities into scanning electrical field measurements opens the door for sensitivity improvements and for imaging the dynamics in quantum systems, that were so far beyond reach.
Data availability
The data that support the plots and other analysis in this work are available from the corresponding author upon request.
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Acknowledgements
We thank A. Finkler, B. Kalisky, F. Kuemmeth, and E. Zeldov for helpful suggestions. We further acknowledge support from the Minerva grant no. 712290, the Helmsley Charitable Trust grant, and the ERCCog (See1DQmatter, No. 647413).
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I.K. and S.I. conceived the experiment. I.K. built the experimental system, performed the experiments, analyzed the data, and preformed the simulations. I.K. and S.I. wrote the manuscript.
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Khivrich, I., Ilani, S. Atomiclike charge qubit in a carbon nanotube enabling electric and magnetic field nanosensing. Nat Commun 11, 2299 (2020). https://doi.org/10.1038/s41467020160015
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DOI: https://doi.org/10.1038/s41467020160015
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