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Efforts to understand dark matter, a mysterious substance that constitutes about 85 % of the universe's mass, have led to major advancements in scientific technology. In this wave of innovation, quantum sensors have stood out as a promising tool for detecting and studying dark matter.

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This article explores the principles of quantum sensor technology, its application in dark matter exploration, and the latest studies highlighting its role in unraveling one of the universe's greatest mysteries.

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Quantum sensors utilize the concepts of quantum mechanics, specifically quantum entanglement and superposition, to attain remarkable sensitivity and accuracy. These concepts enable quantum sensors to measure minute changes in physical quantities such as magnetic fields, gravitational waves, and time variations with extreme precision.

Quantum Entanglement: Quantum entanglement is a phenomenon where particles become interconnected in such a way that the state of one particle instantly influences the state of another, regardless of distance. This property allows quantum sensors to measure correlations with extreme precision, far surpassing the capabilities of classical sensors. Entangled particles can provide highly sensitive measurements of magnetic fields, gravitational waves, and other physical quantities essential for detecting the subtle effects of dark matter.1

Quantum Superposition: Superposition allows quantum systems to exist in multiple states simultaneously. In quantum sensors, this means that particles can occupy multiple positions or energy levels at once, providing a richer set of data points for measurement. For instance, an atom in a superposition state can be used to measure gravitational fields with higher accuracy by comparing the different states it occupies. This principle is crucial for the high sensitivity required in dark matter detection.1

Quantum sensors employ the following techniques for sensing and measurement of dark matter:

The use of quantum sensors to detect dark matter represents a significant breakthrough in exploring the universe's mysterious components. By leveraging the unique properties of quantum mechanics, these sensors offer unmatched sensitivity and precision. This allows researchers to detect and analyze the elusive signals associated with dark matter. This section delves into the various quantum sensing techniques employed in dark matter research.2

Atomic interferometry, a technique that measures the interference of atom waves, has become a cornerstone in quantum sensing for dark matter exploration. By splitting and recombining atom waves, scientists can detect tiny perturbations caused by the presence of dark matter. The Atomic Experiment for Dark Matter and Gravity Exploration in Space (AEDGE) is one such initiative utilizing atomic interferometry. This space-based mission aims to detect fluctuations in atomic transition frequencies caused by interactions with dark matter.2

Moreover, the integration of quantum sensors with other detection technologies, such as optical and radio telescopes, enhances their capabilities. This multidisciplinary approach allows for cross-verification of dark matter signals, increasing the reliability of detections. For example, combining atomic clocks with gravitational wave detectors can help identify local variations in time caused by dark matter interactions.2

The AEDGE mission exemplifies this integration by combining atomic interferometry with gravitational wave detection. This approach not only improves the sensitivity to dark matter signals but also expands the range of detectable dark matter candidates.2

A novel approach in quantum sensing involves time reversal metrology. According to a SciTechDaily report, researchers have developed a method that enhances the sensitivity of quantum sensors using time-reversed entangled states. This method promises to significantly improve the detection capabilities of current quantum sensors.3

This technique involves evolving entangled atoms forward and then reversing their state, which amplifies quantum signals and enhances the ability to detect subtle changes caused by dark matter interactions. These entangled systems can be up to 15 times more sensitive than similar unentangled atomic systems. Such increased sensitivity is crucial for picking up the faint signals that could suggest the presence of dark matter.3

Cold atom experiments, which cool atoms to near absolute zero, offer a promising approach to dark matter detection. At these extremely low temperatures, atoms display quantum behaviors that are instrumental in identifying dark matter.

Techniques like the Bose-Einstein condensate and other cold atom laboratory experiments have shown significant potential for probing fundamental physics, including interactions involving dark matter. These advancements are vital for investigating the broad spectrum of dark matter candidates, ranging from ultra-light particles to more substantial entities like WIMPs (Weakly Interacting Massive Particles).2

A recent study published inPhysical Review Dintroduced an innovative Helium ultraLIght dark matter Optomechanical Sensor (HeLios) detector. This device utilizes Helium-3 atoms trapped in superfluid Helium-4 as quantum sensors for dark matter detection. The method leverages the changes in electrostatic energy that occur when Helium-3 atoms interact with electrons on the helium surface. These interactions can be detected with high precision, offering a novel approach to dark matter detection at extremely low temperatures.4

Quantum sensors are not only pivotal in dark matter detection but also in the study of gravitational waves, which can indirectly indicate the presence of dark matter. The use of quantum entanglement to enhance the sensitivity of gravitational wave detectors, such as those proposed by the European Space Agency (ESA), can improve the understanding of both gravitational waves and dark matter. These enhanced detectors can identify subtle changes in spacetime caused by dark matter interactions.1,2

Space-based quantum sensors are integral to missions like the Laser Interferometer Space Antenna (LISA) and AEDGE. These missions aim to place quantum sensors in space to avoid terrestrial noise and detect dark matter with higher accuracy. The cold atom experiments on the International Space Station (ISS) are precursors to these larger missions, demonstrating the feasibility and potential of quantum sensing in space.2

Quantum sensors, while celebrated for their high sensitivity, still encounter several technical hurdles. The requirements for maintaining quantum superposition and entanglementsuch as extremely low temperatures and isolation from environmental disturbancesare complex and expensive. These conditions are crucial for the precise measurements needed in dark matter detection.1

These sensors are also highly sensitive to external noise, including electromagnetic interference and vibrations. While this sensitivity is advantageous for detecting subtle signals, it complicates the task of distinguishing desired signals from background noise. To counter this, advanced shielding and noise cancellation techniques are necessary.1,2

Additionally, the large volumes of data produced by quantum sensors demand sophisticated data processing algorithms. Identifying faint signals indicative of dark matter amidst this noise presents a significant computational challenge. While quantum computing shows promise in addressing these complexities, integrating it into practical systems remains an active area of research.

Lastly, the long-term stability and calibration of quantum sensors are vital for dependable measurements. Any drift or instability in sensor performance can yield inaccurate results. Ongoing monitoring and recalibration are essential to maintain the precision of these sensors.1,3

The role of quantum sensors in dark matter exploration marks a significant advancement in human efforts to understand the cosmos. The evolution of this technology, from atomic clocks to sophisticated interferometers and space-based experiments, showcases the innovative methods developed by scientists to identify dark matter.

As quantum sensors continue to advance, their sensitivity and precision will boost dark matter research, potentially resulting in groundbreaking discoveries in the near future. The incorporation of quantum sensors with other technologies and the introduction of quantum computing will further bolster their capabilities, cementing their status as essential tools in the quest for dark matter.

Disclaimer: The views expressed here are those of the author expressed in their private capacity and do not necessarily represent the views of AZoM.com Limited T/A AZoNetwork the owner and operator of this website. This disclaimer forms part of the Terms and conditions of use of this website.

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Quantum Sensors Unveil the Mysteries of Dark Matter - AZoQuantum

Quantum circuit model

Let us consider the multimode circuit QED system in Fig.1a, namely a Josephson junction coupled to a lumped-element transmission line. As detailed in the Methods, the quantum Hamiltonian has been derived from the circuit Lagrangian depending on C and , respectively, the capacitance and inductance matrices. The system degrees of freedom are described by the vector , containing independent flux variables41, indicated in Fig.1a. To get the final form, we have performed a simultaneous diagonalization of C and via the basis change =P. Through a Legendre transform and identifying as the junction phase =kP0kk, we have obtained the classical Hamiltonian (for an analogous procedure see42). Finally, by quantizing the degrees of freedom, we get the quantum Hamiltonian in the charge gauge

$$hat{{{{{{{{mathcal{H}}}}}}}}}={hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}+ihat{N}sumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k}),$$

(1)

where ({hat{{{{{{{{mathcal{H}}}}}}}}}}_{J}=4{E}_{C}{hat{N}}^{2}-{E}_{J}cos hat{varphi }) and ({hat{a}}_{k}^{{{{dagger}}} }) is the bosonic creation operator for the kth mode and ({g}_{k}=2sqrt{hslash {omega }_{k}{E}_{C}}{P}_{0k}) its coupling to the junction. Here we used the following sum rule, due to the normalization identity PTCP=I,

$${P}_{00}^{2}+frac{1}{4{E}_{C}}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}}=1.$$

(2)

a Representation of a circuit consisting of a Josephson junction (red) of Josephson energy EJ and capacitive energy EC coupled to a transmission line resonator. The fluxes used to describe the circuit are indicated with black dots. b Energy bands of the uncoupled junction in terms of the quasi-charge for EJ=0.5EC. c Frequencies of the transmission line modes versus the mode number. d Junction-line couplings in the charge gauge as a function of the mode frequency for different values of the impedance Z. For these last two panels: plasma frequency p=5EC and free spectral range =0.2EC (see definition in the text).

The frequencies and couplings in (1) depend on the capacitive and inductive matrices. Illustrative values are shown in Fig.1c, d. A lumped-element circuit possesses an upper frequency cutoff, known as the plasma frequency p. Overall, the line has three key parameters: the impedance (Z=sqrt{L/C}), the plasma frequency ({omega }_{p}=2/sqrt{LC}), and the free spectral range ({{Delta }}=frac{pi }{sqrt{LC}}frac{1}{{N}_{m}}), where Nm is the number of modes. The phase transition is expected to emerge in the thermodynamic limit Nm, that is equivalent to 0.

It is insightful to consider the basis diagonalizing the last two terms of (1), namely (leftvert N,{{{{{{{bf{n}}}}}}}}rightrangle={leftvert Nrightrangle bigotimes }_{k=1}^{{N}_{m}}leftvert {alpha }_{N,k},{n}_{k}rightrangle), where (leftvert {alpha }_{N,k},{n}_{k}rightrangle={D}_{k}({alpha }_{N,k})leftvert {n}_{k}rightrangle), with ({D}_{k}(alpha )=exp (alpha {hat{a}}_{k}^{{{{dagger}}} }-{alpha }^{*}{hat{a}}_{k})) the displacement operator43 for the kth mode with ({alpha }_{N,k}=iNfrac{{g}_{k}}{hslash {omega }_{k}}), and n is a vector with the number of photons in each mode. The only nontrivial term in (1) is the junction Hamiltonian: the dressing of the junction by the line can be determined by inspecting its matrix elements in such a basis. As detailed in the Methods, the coupling to the line effectively produces a renormalized Josephson energy

$${widetilde{E}}_{J}=exp left(-frac{1}{2}sumlimits_{k=1}^{{N}_{m}}frac{{g}_{k}^{2}}{hslash {omega }_{k}^{2}}right){E}_{J},$$

(3)

independent of the number of photons in the modes. The remaining effect is a renormalized capacitive energy

$${widetilde{E}}_{C}={P}_{00}^{2}{E}_{C} , < ,{E}_{C}.$$

(4)

P00 vanishes in the thermodynamic limit and has been taken to be exactly zero in a recent study of the phase transition23. However, the way it approaches zero in the thermodynamic limit is important and cannot be neglected. Indeed, also the renormalized Josephson energy ({widetilde{E}}_{J}) tends to zero for 0.

The properties of the infinite system will hence be determined by the asymptotic behavior of their ratio. We computed such analytical expression by injecting the numerically evaluated values of gk, k and P00 for different values of the impedance Z and decreasing values of . The results are summarized in Fig.2. One can see that both ({widetilde{E}}_{C}) and ({widetilde{E}}_{J}) decrease for all values of Rq/Z while is decreased (Fig.2b, c). However, while ({widetilde{E}}_{C}) monotonically decreases while increasing Rq/Z, ({widetilde{E}}_{J}) first decreases and then increases. The ratio of the two quantities ({widetilde{E}}_{J}/{widetilde{E}}_{C}) instead displays a striking behavior (Fig.2d): for small enough it sharply decreases from 1 to very small values and then grows again. Remarkably, the curves for different values of all cross for Z=Rq, with the curves for larger sizes (smaller ) tending to smaller values for Rq/Z<1, and to larger values for Rq/Z>1. The inset Fig.2e shows that the slope of the curves at Z=Rq grows logarithmically with 1. We can conclude that in the thermodynamic limit the ratio ({widetilde{E}}_{J}/{widetilde{E}}_{C}) tends to 0 for Rq/Z<1 (apart from Rq/Z=0, for which no renormalization can occur) and to infinity for Rq/Z>1.

a In the thermodynamic limit the full system can be understood as a renormalized junction. b, c Renormalized capacitive and Josephson energies computed from (4) and (3) for different values of . d Ratio of the renormalized Josephson and charging energies. e Slope of the ratio at Z=Rq showing logarithmic growth in the thermodynamic limit. For these plots EJ=EC and p=2EC.

This points at the two expected phases: for Rq/Z<1 the charging energy dominates (so-called insulating behavior), for Rq/Z>1 the charging energy is negligible (so-called superconducting behavior). We emphasize that our theoretical analysis predicts a singular point for Z=Rq, independently of the bare ratio EJ/EC, and that the compact or extended nature of the junction phase does not play a role here.

To complete our investigation and fully characterize the emergence of the two phases we have applied exact diagonalization techniques44,45 to our Hamiltonian (1) for finite-size systems. To extend the reachable sizes, we have taken full advantage of Josephson potential periodicity and of Bloch theorem. Indeed, we can introduce the eigenstates of the junction Hamiltonian (leftvert nu,srightrangle={e}^{inu varphi }{u}_{nu }^{s}(varphi )), where is the quasi-charge and s is the band index. In our charge gauge representation, the full Hamiltonian is block diagonal and for each we need to diagonalize the following matrix:

$${hat{H}}_{nu }= sumlimits_{s}{varepsilon }_{nu }^{s}leftvert nu,srightrangle leftlangle nu,srightvert+sumlimits_{k=1}^{{N}_{m}}hslash {omega }_{k}{hat{a}}_{k}^{{{{dagger}}} }{hat{a}}_{k}\ +isumlimits_{k=1}^{{N}_{m}}{g}_{k}({hat{a}}_{k}^{{{{dagger}}} }-{hat{a}}_{k})sumlimits_{s,r}leftlangle nu,rrightvert hat{N}leftvert nu,srightrangle leftvert nu,rrightrangle leftlangle nu,srightvert,$$

(5)

where ({varepsilon }_{nu }^{s}) are the eigenenergies of the bare junction with [0.5,0.5] (Brillouin zone). Note that the spectrum is even with respect to , so we can restrict to positive values only.

Illustrative energy bands for the bare junction are reported in Fig.1b. An example of the obtained energy bands for the interacting system is shown in Fig.3 versus Z. Different colors correspond to different values of the free spectral range for the same p (i.e., to different numbers of modes), and the energies are rescaled by . For Rq/Z0, the junction and the modes are decoupled: the spectrum is given by the bare junction bands with replicas due to a finite number of bosons in the modes. For finite Rq/Z instead, the interaction manifests in energy anticrossings. To investigate a quantum phase transition, the behavior of the ground state and of the low-lying excited states is crucial. While increasing Rq/Z, the first energy band becomes narrower, corresponding to a reduced effective charging energy (see Fig.2b). However, the curvature of the first band cannot change, since the sum rule (2) implies that ({widetilde{E}}_{C}ge 0). This means that the ground state is always at =0. The first excited band instead changes its shape while varying Z. When the system is not coupled (Rq/Z=0), it is simply a one-photon replica of the first band. Instead, when Rq/Z is increased, the slope changes (see Fig.3f, g). The low-energy spectrum of the full system at low impedances is reminiscent of the bare junction spectrum, although at much smaller energies. This qualitative behavior of the first few bands occurs for different sizes, although at smaller energies for smaller (different colors in Fig.3). Interestingly, the rescaled bands overlap in the best way for Z=Rq, confirming the universality observed in Fig.2d. Away from this point, for both larger and smaller values of Z, the rescaled spectra do not overlap any longer. This is reminiscent of the finite-size scaling of continuous phase transitions.

ag Bands for different impedances Z, as indicated in the subplot titles. The different line colors correspond to different free spectral ranges (see legend). The differences of the energies and the ground state energy EG are plotted and the vertical axis is rescaled by . In these plots EJ=0.5EC and p=2EC.

In the exact diagonalization studies, we considered an extended phase and applied Blochs theorem. Now, since the full Hamiltonian for our finite-size system does not couple different quasi-charges , the results for a compact phase are independent of the phase being fundamentally extended or compact. Indeed, we found that the ground state always occurs for =0, which corresponds to a periodic wavefunction. This independence of the phase transition on the extended/compact nature of the junction phase was also discussed in22. For more general shunts (e.g., inductive) an extended phase is relevant27 and this may be needed to describe transport through the junction.

In Fig.4a we plot the ground state energy versus Rq/Z: surprisingly, it has a weak dependence on the system size, and no precursor of a critical behavior is apparent. Figure4b instead reports the energies of the first few excited states, separated from the ground state by an energy of the order of for all the values of Z. In the thermodynamic limit, the system is expected to be gapless for all values of Z: hence, the phase transition needs to emerge with a different mechanism. Indeed, around Z=Rq the first three excited levels exhibit an anticrossing. This corresponds to the observation we already made for Fig.3 that the first excited band passes from being a one-photon replica to a dressed Josephson band. For a compact phase, this occurs around Z=Rq, with the splitting of the resulting anticrossing (but not the position) depending on EJ, as displayed in Fig.5a. As in Fig.3, while decreasing all the rescaled spectra exhibit a universal behavior at Z=Rq. Hence, when approaching the thermodynamic limit, for a fixed EJ the anticrossing becomes narrower and narrower, meaning that the signature of the phase transition first emerges in the first excited states. At the same time, the spectrum is gapless in the 0 limit so that a singular behavior in the first excited state necessarily reflects on the ground state.

a Ground state energy for different free spectral ranges. b Corresponding dependence of the first few excited state energies for a compact phase (=0). c Junction charge fluctuations in the ground state for different free spectral ranges. d Energy difference between the first two bands at the edge of the Brillouin zone =0.5. In all the panels EJ=0.5EC and p=2EC.

a Dependence on EJ of the anticrossing involving the first three excited states for a compact phase (=0). Here =0.55EC and p=2EC. b Energy difference between the first two bands at the edge of the Brillouin zone (=0.5) showing the universality at Rq/Z=1 for a wide range of EJ. Solid lines =0.55EC, dashed lines =0.41.

While a compact phase is the only relevant quantity for the ground state of the system, the response of the system to external perturbations can depend on the full bands. To better highlight the behavior of the low-lying bands, in Fig.4d we show the behavior of the gap between the first and second bands at the edge of the Brillouin zone. The ratio of the gap and the free spectral range displays a behavior analogous to the renormalized parameters of Fig.2, with the gap closing for larger systems for Rq/Z<1. This means that the low energy spectrum is approaching a capacitive free particle" behavior. The opposite is true for Rq/Z>1. As shown in Fig.5b, this behavior extends to larger values of EJ.

Importantly, however, the ground state does not behave in the same way, as can be seen for example by computing observables for the junction degrees of freedom. As an example, we show in Fig.4c the charge fluctuations ({sigma }_{N}^{2}={{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}{hat N}^{2})-{{{{{{{rm{tr}}}}}}}}{({hat{rho }}_{J}hat N)}^{2}), with ({hat{rho }}_{J}) the reduced density matrix for the junction (analogous results are obtained, for example, for ({{{{{{{rm{tr}}}}}}}}({hat{rho }}_{J}cos hat{varphi }))). For Rq/Z>1 the charge fluctuations have a strong dependence on the system size. For Rq/Z<1 instead, the size dependence is much smaller, and the fluctuations never fall below the Cooper pair box value that is obtained for Rq/Z0 (horizontal dashed line). This, however, does not mean that the transition does not affect the ground state, as signaled by the different size dependence of the charge fluctuations on the two sides of Rq/Z=1. The emergence of the singularity in the ground state in the thermodynamic limit is, however, slow, and we can only see a precursor.

We believe that this peculiar difference of behavior between the ground state and the excited states is at the origin of much of the recent controversy over this phase transition, specifically over the characterization of the so-called insulating state. In particular, in the thermodynamic limit, the ground state is not approaching a purely capacitive behavior. However, the response of the system to a gate charge (i.e., to a change in the quasi-charge) changes sharply at the transition point.

To understand the fate of the higher frequency modes at the transition point, we focus on the compact-phase Hamiltonian, given by equation (5) for =0. For EJ=0, the dressed excited bands (charge states) energies are ({varepsilon }_{0}^{n}({E}_{J}=0)=4{widetilde{E}}_{C}{n}^{2}), with n an integer. In this limit, the effect of the transition on the photons can be understood by simply plotting the Z-dependent mode energies and the same energies shifted by the first dressed charge state ({varepsilon }_{0}^{1}({E}_{J}=0)). We show these two sets of energy branches in Fig.6. The two lowest solid and dashed curves are respectively the first photon energy 1 and ({varepsilon }_{0}^{1}({E}_{J}=0)). Note that they cross at Z=Rq. With a finite EJ the crossing is replaced by the anticrossing highlighted in Fig.5a. Above these levels, for energies small enough with respect to p, also the kth photonic mode energy k crosses the energy ({varepsilon }_{0}^{1}({E}_{J}=0)+hslash {omega }_{k-1}) at Z=Rq. This degeneracy is again lifted for EJ0. Hence, around the transition point, all the single photon states will hybridize. Note that the crossing of the first two curves,which is essentially determined by the universality highlighted in Fig.2, is robust with respect to the intrinsic ultraviolet cutoff given by the Josephson plasma frequency p. For the excited states, there is eventually an important shift of the crossing point at high energies, that would be absent in the limit p (a non-dispersive line). The same shift is also present for much smaller free spectral ranges, of the order of the ones in the experiment39, as shown by the red line in Fig.6. For modes above the third, also states involving higher dressed bands play a significant role. Moreover, states with more than one photon are also present. For energies small enough with respect to p (equidistant modes) these are also degenerate with the single-photon energies at the transition point, increasing the number of levels that participate the hybridization.

Single photon energies (solid lines) and the same energies shifted by the energy of the first excited renormalized junction band for =0 (dashed lines) with EJ0, =0.5EC and p=5EC. The nth mode energy and the nth level of the shifted branch have the same color. Their intersection point is highlighted by a black dot. The solid red line shows the intersections for a much smaller free spectral range =0.02EC.

Recent experiments have observed a striking signature of the transition from the dispersion of high-frequency modes39. With the results of our exact diagonalization, we can calculate the linear-response photonic spectral function

$$D(E)=sumlimits_{n}frac{{gamma }^{2}}{{gamma }^{2}+{(E-{E}_{n}+{E}_{G})}^{2}}| leftlangle Grightvert sumlimits_{k}({a}_{k}+{a}_{k}^{{{{dagger}}} })leftvert {E}_{n}rightrangle {| }^{2},$$

(6)

where En ((leftvert {E}_{n}rightrangle)) is the nth excited eigenenergy (eigenstate) of the full system, EG ((leftvert Grightrangle)) the ground energy (state) and is a phenomenological broadening. Here, as in the experiment, we have considered a SQUID that is equivalent to a junction with EJ that can be tuned by a magnetic field. Illustrative spectra versus the external magnetic flux are shown in Fig.7. We observe a clear change of sign of the photon frequency dispersion across the transition, as observed in39. This effect was linked to a capacitive/inductive behavior of the junction and here we see that it microscopically originates from the crossings observed in Fig.6. The broadening observed in39 for high energy modes is instead not present because of the moderate size of the system simulated here with exact diagonalization techniques. In fact, it relies on multiple resonances between single and multi-photon states37. In Fig.7, one can see the state belonging to the shifted set of levels acquiring a finite single-photon component near the crossing point for /00.5, that is for ({E}_{J} neq 0). For larger and smaller impedances, this state is dark, i.e., it does not contribute to this spectral function.

Color plot of D(E) around the bare energy of a photonic mode versus normalized flux bias (0=h/2e). The panels correspond to different values of Rq/Z with EJ=0.1EC, =0.5EC and p=5EC.

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Emergent quantum phase transition of a Josephson junction coupled to a high-impedance multimode resonator - Nature.com