The quantization of resistances in the quantum Hall effect and ballistic transport through quantum point contacts is compared with the quantization of the charge relaxation resistance of a coherent mesoscopic capacitor. While the former two require the existence of a perfectly transmitting channel, the charge relaxation resistance remains quantized for arbitrary backscattering. The quantum Hall effect and the quantum point contact require only local phase coherence. In contrast quantization of the charge relaxation resistance requires global phase coherence.
We show that quantum resistance standards made of transferred graphene reach the uncertainty of semiconductor devices, the current reference system in metrology. A large graphene device (150 \times 30 \mum2), exfoliated and transferred onto GaAs, revealed a quantization with a precision of (-5.1 \pm 6.3) \times 10-9 accompanied by a vanishing longitudinal resistance at current levels exceeding 10 \muA. While such performance had previously only been achieved with epitaxially grown graphene, our experiments demonstrate that transfer steps, inevitable for exfoliated graphene or graphene grown by…
A nonlinear single-particle model is introduced, which captures the characteristic of systems in the quantum Hall regime. The model entails the magnetic Shrödinger equation with spatially variable magnetic flux density. The distribution of flux is prescribed via the postulates of the mesoscopic mechanics (MeM) introduced in my previous articles [cf. J. Phys. Chem. Solids, 65 (2004), 1507-1515; J. Geom. Phys., Vol. 55/1 (2005), 1-18]. The model is found to imply exact integer and fractional quantization of the Hall conductance. In fact, Hall resistance is found to be $R_H = \frac{h}{2e^2}\frac{…
In contrast to the case of ordinary quantum Hall effect, the resistance of ballistic helical edge channels in typical quantum spin-Hall experiments is non-vanishing, additive and poorly quantized. Here we present a simple argument connecting this qualitative difference with a spin relaxation in the current/voltage leads in an experimentally relevant multi-terminal bar geometry. Both the finite lead resistance and the spin relaxation contribute to a non-vanishing four-terminal edge resistance, explaining poor quantization quality. We show that corrections to the four-terminal and two-terminal r…
We have observed quantization of the diagonal resistance, R_xx, at the edges of several quantum Hall states. Each quantized R_xx value is close to the difference between the two adjacent Hall plateaus in the off-diagonal resistance, R_xy. Peaks in R_xx occur at different positions in positive and negative magnetic fields. Practically all R_xx features can be explained quantitatively by a ~1%/cm electron density gradient. Therefore, R_xx is determined by R_xy and unrelated to the diagonal resistivity rho_xx. Our findings throw an unexpected light on the empirical resistivity rule for 2D systems…
Time-resolved optical measurements of electron spin dynamics in modulation doped InGaAs quantum wells are used to explore electron spin coherence times and spin precession frequencies in a regime where an out of plane magnetic field quantizes the states of a two-dimensional electron gas into Landau levels. Oscillatory features in the transverse spin coherence time and effective g-factor as a function of applied magnetic field exhibit a correspondence with Shubnikov-de Haas oscillations, illustrating a coupling between spin and orbital eigenstates. We present a theoretical model in which inhomo…
We present precision measurements of the fractional quantized Hall effect where the quantized resistance $R^{[1/3]}$ in the fractional quantum Hall state at filling factor 1/3 was compared with a quantized resistance $R^{[2]}$, represented by an integer quantum Hall state at filling factor 2. A cryogenic current comparator bridge capable of currents down to the nanoampere range was used to directly compare two resistance values of two GaAs-based devices located in two cryostats. A value of $1 - (5.3 \pm 6.3) 10^{-8}$ (95% confidence level) was obtained for the ratio $(R^{[1/3]}/6R^{[2]})$. Thi…
Recent investigations of fractal conductance fluctuations (FCF) in electron billiards reveal crucial discrepancies between experimental behavior and the semiclassical Landauer-Buttiker (SLB) theory that predicted their existence. In particular, the roles played by the billiard's geometry, potential profile and the resulting electron trajectory distribution are not well understood. We present measurements on two custom-made devices - a 'disrupted' billiard device and a 'bilayer' billiard device - designed to probe directly these three characteristics. Our results demonstrate that intricate proc…
A comprehensive understanding of quantum Hall edge transmission, especially the hole-conjugate of a Laughlin state such as a $2/3$ state, is critical for advancing fundamental quantum Hall physics and enhancing the design of quantum Hall edge interferometry. In this study, we report a robust intermediate $2h/e^2$ resistance quantization in a quantum point contact (QPC) when the bulk is set at the fractional filling $2/3$ quantum Hall state. Our results suggest the occurrence of two equilibration processes. First, the co-propagating $1/3$ edges moving along a soft QPC arm confining potential fu…
The two-dimensional metal forming at the interface between an oxide insulator and SrTiO3 provides new opportunities for oxide electronics. However, the quantum Hall effect, one of the most fascinating effects of electrons confined in two dimensions, remains underexplored at these complex oxide heterointerfaces. Here, we report the experimental observation of quantized Hall resistance in a SrTiO3 heterointerface based on the modulation-doped amorphous-LaAlO$_{3}$/SrTiO$_{3}$ heterostructure, which exhibits both high electron mobility exceeding 10000 cm$^{2}$/Vs and low carrier density on the or…
The Single Cooper Pair Josephson Transistor is a device that exhibits at the same time charge quantization and phase coherence. Coherent quantum phase slip phenomenon is "dual" the Josephson phase coherence while the charge quantization is dual to the flux quantization. We present the experimental demonstration and the theoretical description of a new superconducting device - Single Flux Transistor, which is dual to the Single Cooper Pair Transistor. Our transport measurements show the periodic modulation of the critical voltage by the external magnetic field. The obtained current-voltage char…
We report on realization of 10 quantum Hall devices in series fabricated using epitaxial graphene on silicon carbide. Precision measurements with a resistance bridge indicates that the quantized Hall resistance across an array at filling factor 2 is equivalent to 5 h/e^2 within the measurement uncertainty of approximately 4 10-8. A quantum-Hall phase diagram for the array shows that a metrological quantization of 5 h/e^2 can be achieved at the magnetic field of 6 T and temperature of 4 K. This experiment demonstrates the possibility of timely unchangeable resistance reference in various ranges…
Recent experiments on quantum Hall bilayers near total filling factor 1 have demonstrated that they support an ``imperfect'' two-dimensional superfluidity, in which there is nearly dissipationless transport at non-vanishing temperature observed both in counterflow resistance and interlayer tunneling. We argue that this behavior may be understood in terms of a {\it coherence network} induced in the bilayer by disorder, in which an incompressible, coherent state exists in narrow regions separating puddles of dense vortex-antivortex pairs. A renormalization group analysis shows that it is appropr…
We explore the robust quantization of the Hall resistance in epitaxial graphene grown on Si-terminated SiC. Uniquely to this system, the dominance of quantum over classical capacitance in the charge transfer between the substrate and graphene is such that Landau levels (in particular, the one at exactly zero energy) remain completely filled over an extraordinarily broad range of magnetic fields. One important implication of this pinning of the filling factor is that the system can sustain a very high nondissipative current. This makes epitaxial graphene ideally suited for quantum resistance me…
In the quantum anomalous Hall effect, quantized Hall resistance and vanishing longitudinal resistivity are predicted to result from the presence of dissipationless, chiral edge states and an insulating 2D bulk, without requiring an external magnetic field. Here, we explore the potential of this effect in magnetic topological insulator thin films for metrological applications. Using a cryogenic current comparator system, we measure quantization of the Hall resistance to within one part per million and longitudinal resistivity under 10 m$Ω$ per square at zero magnetic field. Increasing the curre…
In some recent experiments [A. J. Bestwick, et. al., Phys. Rev. Lett. 114, 187201 (2015), Cui-Zu Chang, et. al., Nat. Materials. 14, 473-477 (2015)] it has been shown that in observations of the quantum anomalous Hall (QAH) effect the longitudinal resistance $R_L$ increases as temperature $T$ increases, while Hall resistance $R_H$ loses its quantization with increase in $T$. This behavior was explained due to increased thermal fluctuations as $T$ increases. We show that similar effects arise in QAH samples with quasi-helical edge modes as disorder increases in presence of inelastic scattering …
At total Landau level filling factor $ν_{tot}=1$ a double layer two-dimensional electron system with small interlayer separation supports a collective state possessing spontaneous interlayer phase coherence. This state exhibits the quantized Hall effect when equal electrical currents flow in parallel through the two layers. In contrast, if the currents in the two layers are equal, but oppositely directed, both the longitudinal and Hall resistances of each layer vanish in the low temperature limit. This finding supports the prediction that the ground state at $ν_{tot}=1$ is an excitonic superfl…
Effect of dc electric field on transport of highly mobile 2D electrons is studied in wide GaAs single quantum wells placed in titled magnetic fields. The study shows that in perpendicular magnetic field resistance oscillates due to electric field induced Landau-Zener transitions between quantum levels that corresponds to geometric resonances between cyclotron orbits and periodic modulation of electron density of states. Magnetic field tilt inverts these oscillations. Surprisingly the strongest inverted oscillations are observed at a tilt corresponding to nearly absent modulation of the electro…
We report a nearly ideal quantum anomalous Hall effect in a three-dimensional topological insulator thin film with ferromagnetic doping. Near zero applied magnetic field we measure exact quantization in Hall resistance to within a part per 10,000 and longitudinal resistivity under 1 ohm per square, with chiral edge transport explicitly confirmed by non-local measurements. Deviations from this behavior are found to be caused by thermally-activated carriers, which can be eliminated by taking advantage of an unexpected magnetocaloric effect.
Usually the conductivity is quantized as the inverse of the resistivity, ρ=hc/ie^2, σ=ie^2/hc and the velocity versus the electric field is linear, v=μE where μis the mobility of the electrons.However,when the applied electric field exceeds a certain value, microwaves are emitted and the relation v=μE breaks down so that the velocity actually reduces as E increases. In this region, when magnetic field is applied, the conductivity quantizes like the magnetic field, i.e., in units of hc/e which is different from the usual quantization. Because of the flux quantization, the resistivity will touch…
The effect of a DC electric field on the longitudinal resistance of highly mobile two dimensional electrons in heavily doped GaAs quantum wells is studied at different magnetic fields and temperatures. Strong suppression of the resistance by the electric field is observed in magnetic fields at which the Landau quantization of electron motion occurs. The phenomenon survives at high temperature where Shubnikov de Haas oscillations are absent. The scale of the electric fields essential for the effect is found to be proportional to temperature in the low temperature limit. We suggest that the stro…
We measure the Hall conductivity, $σ_{xy}$, on a Corbino geometry sample of a high-mobility AlGaAs/GaAs heterostructure in a pulsed magnetic field. At a bath temperature about 80 mK, we observe well expressed plateaux in $σ_{xy}$ at integer filling factors. In the pulsed magnetic field, the Laughlin condition of the phase coherence of the electron wave functions is strongly violated and, hence, is not crucial for $σ_{xy}$ quantization.
We derive electromagnetomotive force fields for charged particles moving in a rotating Hall sample, satisfying a twofold U(1) gauge invariance principle. It is then argued that the phase coherence property of quantization of the line integral of total collective particle momentum into multiples of Planck's quantum of action is solely responsible for quantization in the Hall state. As a consequence, the height of the Hall quantization steps should remain invariant in a rapidly rotating Hall probe. Quantum Hall particle conductivities do not depend on charge and mass of the electron, and are qua…
We present detailed experimental studies of the temperature dependence of the plateau conductance of GaAs quantum point contacts in the temperature range from 0.3 K to 10 K. Due to a strong lateral confinement produced by a shallow-etching technique we are able to observe the following unexpected feature: a linear temperature dependence of the measured mid-plateau conductance. We discuss an interpretation in terms of a temperature dependent, intrinsic series resistance, due to non-ballistic effects in the 2D-1D transition region. These results have been reproduced in several samples from diffe…
We show how to compute the magnetooptical functions (absorption, reflection, and transmission) when Rydberg Exciton-Polaritons appear, including the effect of the coherence between the electron-hole pair and the electromagnetic field, and the polaritonic effect. Using the Real Density Matrix Approach the analytical expressions for magnetooptical functions are obtained and numerical calculations for Cu$_2$0 crystal are performed. The influence of the strength of applied external magnetic field on the resonance displacement of excitonic spectra is discussed. We report a good agreement with recen…
We report on a comparison of four GaAs/AlGaAs-based quantum resistance standards using an original technique adapted from the well-known Wheatstone bridge. This work shows that the quantized Hall resistance at Landau level filling factor $ν=2$ can be reproducible with a relative uncertainty of $32\times 10^{-12}$ in the dissipationless limit of the quantum Hall effect regime. In the presence of a very small dissipation characterized by a mean macroscopic longitudinal resistivity $\bar{R_{xx}(B)}$ of a few $μΩ$, the discrepancy $ΔR_{\mathrm{H}}(B)$ measured on the Hall plateau between quantum H…
We have studied quantized transport in HgTe wells with inverted band structure corresponding to the two-dimensional topological insulator phase (2D TI) with locally-controlled density allowing n-p-n and n-2D TI-n junctions. The resistance reveals the fractional plateau $2h/e^{2}$ in n-p-n regime in the presence of the strong perpendicular magnetic field. We found that in n-2D TI-n regime the plateaux in resistance in not universal and results from the edge state equilibration at the interface between chiral and helical edge modes. We provided the simple model describing the resistance quantiza…
The quantum Hall effect (QHE) is a topologically protected phenomenon which has been observed in various systems. In experiments, the size of Hall bar device to realize the QHE is generally much larger than the phase coherence length, in which the quantum coherence of electrons is destroyed. Here, we theoretically study the influence of dephasing effect on the quantized Hall (QH) plateaus. We find that the QH plateau disappears in perfectly quantum coherent systems if the coupling between leads and central region is imperfect. The Hall resistance is very large and strongly oscillates instead o…
We present an analysis of the effect of dephasing on the single channel charge relaxation resistance of a mesoscopic capacitor in the linear low frequency regime. The capacitor consists of a cavity which is via a quantum point contact connected to an electron reservoir and Coulomb coupled to a gate. The capacitor is in a perpendicular high magnetic field such that only one (spin polarized) edge state is (partially) transmitted through the contact. In the coherent limit the charge relaxation resistance for a single channel contact is independent of the transmission probability of the contact an…
We report on the observation of strong backscattering of charge carriers in the quantum Hall regime of polycrystalline graphene grown by chemical vapor deposition, which alters the accuracy of the Hall resistance quantization. The temperature and magnetic field dependence of the longitudinal conductivity exhibits unexpectedly smooth power law behaviors, which are incompatible with a description in terms of variable range hopping or thermal activation, but rather suggest the existence of extended or poorly localized states at energies between Landau levels. Such states could be caused by the hi…
We have performed a metrological characterization of the quantum Hall resistance in a 1 $μ$m wide graphene Hall-bar. The longitudinal resistivity in the center of the $ν=\pm 2$ quantum Hall plateaus vanishes within the measurement noise of 20 m$Ω$ upto 2 $μ$A. Our results show that the quantization of these plateaus is within the experimental uncertainty (15 ppm for 1.5$ μ$A current) equal to that in conventional semiconductors. The principal limitation of the present experiments are the relatively high contact resistances in the quantum Hall regime, leading to a significantly increased noise …
Adiabatic quantum pumping in noninteracting, phase coherent quantum dots is elegantly described by Brouwer's formula. Interactions within the dot, while suppressing phase coherence, make Brouwer's formalism inapplicable. In this paper, we discuss the nature of the physical processes forcing a description of pumping beyond Brouwer's formula, and develop, using a controlled adiabatic expansion, a useful formalism to study the effect of interactions within a generic perturbative scheme. The pumped current consists of a first contribution, analogous to Brouwer's formula and accounting for the rema…
We report ultra-low temperature experiments on the obscure fractional quantum Hall effect (FQHE) at Landau level filling factor $ν$=5/2 in a very high mobility specimen of $μ=1.7 \times 10^7$ cm$^2$/Vs. We achieve an electron temperature as low as $\sim$ 4~mK, where we observe vanishing $R_{xx}$ and, for the first time, a quantized Hall resistance, $R_{xy}=h/(5/2e^2)$ to within 2 ppm. $R_{xy}$ at the neighboring odd-denominator states $ν$=7/3 and 8/3 is also quantized. The temperature dependences of the $R_{xx}$-minima at these fractional fillings yield activation energy gaps $Δ_{5/2}$=0.11K, …
The transport properties of a rectangular mesoscopic plaquette in the presence of a perpendicular magnetic field are studied in a tight-binding model with randomly distributed traps. The longitudinal and Hall resistances are calculted in the four-probe Landauer-Büttiker formalism which accounts automatically both for the quantum coherence and the trapping-induced localization. The localized character of eigenvectors and the specific aspect of the density of states at a given magnetic flux are correlated with the behaviour of the mentioned resistances as function of the Fermi energy. The Hall i…
We have studied the electrical conductivity of the electron gas in parallel electric and magnetic fields directed along the plane of a parabolic quantum well (across the profile of the potential). We found a general expression for the electrical conductivity applicable for any magnitudes of the magnetic field and the degree of degeneration of the electron gas. A new mechanism of generation of the negative magnetoresistance has been revealed. It has been shown that in a parabolic quantum well with a non-degenerated electron gas the negative magnetoresistance results from spin splitting of the l…
The beauty of quantum Hall (QH) effect is the metrological precision of Hall resistance quantization that originates from the topological edge states. Understanding the factors that lead to quantization breakdown not only provides important insights on the nature of the topological protection of these edge states, but is beneficial for device applications involving such quantized transport. In this work, we combine conventional transport and real space conductivity mapping to investigate whether the quantization breakdown is tied to the disappearance of edge state in the hotly studied MnBi2Te4…
We examine the concept of universal quantized resistance in the AC regime through the fully coherent quantum RC circuit comprising a cavity (dot) capacitively coupled to a gate and connected via a single spin-polarized channel to a reservoir lead. As a result of quantum effects such as the Coulomb interaction in the cavity and global phase coherence, we show that the charge relaxation resistance $R_q$ is identical for weak and large transmissions and it changes from $h/2e^2$ to $h/e^2$ when the frequency (times $\hbar$) exceeds the level spacing of the cavity; $h$ is the Planck constant and $e…
The quantum Hall effect (QHE) is a cornerstone in the new International System of Units (SI), wherein the base units are derived from seven fundamental constants such as Planck's constant h and elementary charge e. Graphene has revolutionized practical resistance metrology by enabling the realization of quantized resistance h/2e^2 = 12.9... kOhm under relaxed experimental conditions. Looking ahead, graphene also has the potential to improve realizations of the electronic kilogram using the Kibble balance, and the quantum Ampere in wide current ranges. However, these prospects require different…
The reason why the half-integer quantum Hall effect (QHE) is suppressed in graphene grown by chemical vapor deposition (CVD) is unclear. We propose that it might be connected to extended defects in the material and present results for the quantum Hall effect in graphene with [0001] tilt grain boundaries connecting opposite sides of Hall bar devices. Such grain boundaries contain 5-7 ring complexes that host defect states that hybridize to form bands with varying degree of metallicity depending on grain boundary defect density. In a magnetic field, edge states on opposite sides of the Hall bar …
We present experimental results on the quantized Hall insulator in two dimensions. This insulator, with vanishing conductivities, is characterized by the quantization (within experimental accuracy) of the Hall resistance in units of the quantum unit of resistance, h/e^2. The measurements were performed in a two dimensional hole system, confined in a Ge/SiGe quantum well, when the magnetic field is increased above the nu=1 quantum Hall state. This quantization leads to a nearly perfect semi-circle relation for the diagonal and Hall conductivities. Similar results are obtained with a higher mobi…
Physics of two-dimensional electron gases under perpendicular magnetic field often displays three distinct stages when increasing the field amplitude: a low field regime with classical magnetotransport, followed at intermediate field by a Shubnikov-de Haas phase where the transport coefficients present quantum oscillations, and, ultimately, the emergence at high field of the quantum Hall effect with perfect quantization of the Hall resistance. A rigorous demonstration of this general paradigm is still limited by the difficulty in solving models of quantum Hall bars with macroscopic lateral dim…
We consider a model for electron tunneling between a pair of ultrasmall metallic grains. Under appropriate circumstances, non-equilibrium final state effects can strongly enhance tunneling and produce electron-hole coherence between the grains. The model displays a quantum phase transition between a Coulomb blockaded state to a coherent state exhibiting subohmic tunneling conductance. The critical state of the junction exhibits a temperature independent resistance of order $h/e^2$. Finally we discuss the possible relevance to granular materials and quantum dots. In particular, similarities bet…
We study the dissipation in a system consisting of a small metallic island coupled to a gate electrode and to a massive reservoir via single tunneling junction. The dissipation of energy is caused by a slowly oscillating gate voltage. We compute it in the regimes of weak and strong Coulomb blockade. We focus on the regime of not very low temperatures when electron coherence can be neglected but quantum fluctuations of charge are strong due to Coulomb interaction. The answers assume a particularly transparent form while expressed in terms of specially chosen physical observables. We discovered …
We study the influence of the phase relaxation process on Hall resistance and spin Hall current of a mesoscopic two-dimensional (2D) four-terminal Hall cross-bar with or without Rashba spin-orbit interaction (SOI) in a perpendicular uniform magnetic field. We find that the plateaus of the Hall resistance with even number of edge states can survive for very strong phase relaxation when the system size becomes much longer than the phase coherence length. On the other hand, the odd integer Hall resistance plateaus arising from the SOI are easily destroyed by the weak phase relaxation during the c…
Robust quantum Hall resistance standard from uniform wafer-scale epitaxial graphene on SiC
Laboratoire National de Métrologie et d’Essais (LNE)（法国国家计量与测试实验室）；Université Côte d’Azur（蔚蓝海岸大学）；Centre de Recherche sur l’Hétéroépitaxie et ses applications (CRHEA)（异质外延及其应用研究中心）；CERFACS（欧洲计算流体力学与空气动力学中心）；Université Paris-Saclay（巴黎萨克雷大学）；Centre de Nanosciences et de Nanotechnologies (C2N)（纳米科学与纳米技术研究中心）
We consider the problem of finite resistance $R$ in superconducting films with geometry of a strip of width $W$ near zero temperature. The resistance is generated by vortex configurations of the phase field. In the first type of process, quantum phase slip, the vortex worldline in 2+1 dimensional space-time is space-like (i.e., the superconducting phase winds in time and space). In the second type, vortex tunneling, the worldline is time-like (i.e., the phase winds in the two spatial directions) and connects opposite edges of the film. For moderately disordered samples, processes of second typ…
We demonstrate a quantum Hall edge-channel interferometer in a high-quality graphene pn junction under a high magnetic field. The co-propagating p and n quantum Hall edge channels traveling along the pn interface functions as a built-in Aharanov-Bohm-type interferometer, the interferences in which are sensitive to both the external magnetic field and the carrier concentration. The trajectories of peak and dip in the observed resistance oscillation are well reproduced by our numerical calculation that assumes magnetic flux quantization in the area enclosed by the co-propagating edge channels. C…
In the quantum anomalous Hall effect, the edge states of a ferromagnetically doped topological insulator exhibit quantized Hall resistance and dissipationless transport at zero magnetic field. Up to now, however, the resistance was experimentally assessed with standard transport measurement techniques which are difficult to trace to the von-Klitzing constant R$_K$ with high precision. Here, we present a metrologically comprehensive measurement, including a full uncertainty budget, of the resistance quantization of V-doped (Bi,Sb)$_2$Te$_3$ devices without external magnetic field. We establishe…
We derive the quantization of magnetic helicity in the solid-state and demonstrate tunable macroscopic quantum tunneling, coherence, and oscillation for a skyrmion spin texture stabilized in frustrated magnets. We also discuss the parameter space for the experimental realization of quantum effects. Typically, for a skyrmion of 5 nm radius, quantum tunneling between two macroscopic states with distinct helicities occurs with an inverse escape rate within seconds below 100 mK, and an energy splitting in the MHz regime. Feasibility of quantum tunneling of an ensemble of magnetic spins inspires ne…
We summarize the screening theory of the integer quantized Hall effect (IQHE) and emphasize its two key mechanisms: first, the existence, in certain magnetic field intervals, of incompressible strips, with integer values of the local filling factor and quantized values of longitudinal and Hall resistivity, and second, the confinement of an imposed dissipative current to these strips, leading to the quantization of the global resistances. We demonstrate that, without any localization assumption, this theory explains the enormous experimental reproducibility of the quantized resistance values, a…
During the last decades, quantum dots within the Coulomb blockade regime of transport have been proposed as essential building blocks for a wide variety of nanomachines. This includes thermoelectric devices, quantum shuttles, quantum pumps, and even quantum motors. However, in this regime, the role of quantum mechanics is commonly limited to provide energy quantization while the working principle of the devices is ultimately the same as their classic counterparts. Here, we study quantum-dot-based nanomachines in the Coulomb blockade regime, but in a configuration where the coherent superpositi…
