27++ Best Variable Resistor Symbol Circuit – Variable Resistor Symbol Circuit

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Phase blooper in a attenuate superconducting wire occurs on the calibration of the superconducting adherence length. Appearance blooper by thermal activation1 is empiric as a arresting appendage beneath the analytical temperature. In affairs with bore beneath 10 nm and actual aerial resistance, the activity barrier is baby abundant that appearance blooper by breakthrough tunnelling can be expected2,3. Affairs of Mo–Ge deposited on abeyant carbon nanotubes accept been studied3,4,5, and the after-effects are in reasonable acceding with diminutive calculations of phase-slip rates7,8. All abstracts consisted of casual a baby d.c. current through the sample and barometer the voltage. As anniversary phase-slip accident in the attendance of a accepted I releases an activity IΦ0, breadth Φ0=h/2e is the alteration quantum, with h the Planck connected and e the electron charge, such abstracts are dissipative. Unambiguous beginning affirmation of QPS is still absent. It has been concluded5 that after-effects can be declared by thermally activated appearance blooper for affairs with beyond cross-section, and as mesoscopic deviating normal-metal conductance for the weaker wires. The abstract assay is complicated by the actuality that the behaviour of the bosonic superfluid may be overshadowed by the fermionic furnishings of localization and interaction. The superconducting activity gap in the wire may be suppressed and quasiparticles may be generated. However, we are not acquainted of any acumen that would forbid QPS to be a concrete reality. We appropriately accept that articular QPS may booty place, that it is characterized by a alteration amplitude ES/2 and that no quasiparticles are present (ES<Δ, Δ actuality the superconducting activity gap). On the base of Wentzel–Kramers–Brillouin (WKB)-type estimates3 and diminutive calculations7, we accept that the alteration amplitude for QPS in activated superconducting nanowires of 1 μm breadth can be as aerial as ES/h=100 GHz.

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Wires in which cogent QPS occurs accept ample active inductance L′ and baby capacitance C′ per assemblage length. The plasmon appearance acceleration cp=(L′C′)1/2 is of adjustment 105 m s−1 (refs 9, 10) and the appropriate impedance Zc=(L′/C′)1/2 is of adjustment 100 kΩ. Frequent QPSs are expected7 back Zc>Rq, with Rq≡h/4e2=6.45 kΩ. A abutting footfall in the compassionate of QPS was made6 by because a wire of bound breadth as a ambit element. If the blow of the ambit is modelled by a alongside attrition Rp, it was predicted that the wire will appearance a breakthrough appearance alteration at Rp=Rq, actuality superconducting at Rp<Rq and careful otherwise. As predicted11, a Josephson alliance shows qualitatively the aforementioned behaviour. The cessation was again drawn6 that the low-energy physics of QPSs reduces to that of a Josephson alliance in the aforementioned circuit.

We absolutely authenticate in this cardboard that the affiliation amid Josephson tunnelling and QPS in circuits is added intriguing. They are bifold to anniversary added with account to the barter of the canonically conjugated breakthrough variables, appearance and charge. This duality is exact, in adverse to the broadly known12 almost self-duality of Josephson alliance circuits. After-effects of ref. 6 are reproduced by appliance exact duality and almost self-duality. We use exact duality to call the dynamical acknowledgment of QPS in the absolute of able appearance slip, breadth we acknowledge actual favourable altitude for the ascertainment of Bloch-type oscillations and of a resonance at the ‘plasma’ frequency.

The duality is developed in two steps. We aboriginal analyze two simple circuits (Fig. 1). One is the Cooper-pair box13 with a Josephson alliance that is voltage-biased with voltage V through a capacitor C. In the added we acquaint the QPS junction, an aspect that represents the phase-slip action with backbone ES, in a bankrupt bend with an inductor. This is the QPS alteration qubit proposed in ref. 14, but not yet accomplished in practice. In the Cooper-pair box the charging activity depends parabolically on the induced allegation ng=CV/2e, for anniversary accumulation amount of the Cooper-pair cardinal n. The charging activity calibration is accustomed by EC=(2e)2/2C. Josephson coupling EJ mixes states with n and n 1, appropriation abasement at half-integer ethics of ng. The akin agreeable at this point equals EJ provided EJ≪EC. This is declared by the afterward hamiltonian: breadth h.c. represents the hermitian conjugate. In the QPS qubit on the appropriate of Fig. 1 the anterior activity depends parabolically on the activated alteration Φ with f=Φ/Φ0 at anniversary accumulation n that now represents the fluxoid cardinal in the loop. The calibration for the anterior activity is EL=Φ02/2L, breadth L is the inductance of the loop. QPS coupling mixes states with adjoining fluxoid numbers and lifts abasement at half-integer ethics of f. The akin agreeable at this point equals ES, provided ES≪EL. This implies the afterward hamiltonian: The diagram as fatigued for the Cooper-pair box is alone accurate back EC≫EJ, breadth allegation is the accordant breakthrough number. In analogy, the diagram for the QPS qubit is alone accurate back EL≫ES, the absolute of anemic appearance slip. Here, appearance is the accordant breakthrough number. The aloft hamiltonians are acutely agnate with account to the barter At this date we accept not yet bogus use of the actuality that allegation and appearance are canonically conjugated breakthrough variables.

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a, A Cooper-pair box and b, a QPS qubit. The diamond-shaped attribute in the QPS qubit ambit represents the breakthrough phase-slip process. The capacitive activity is E=EC(n−ng)2 and the anterior activity is E=EL(f−n)2.

We will now acquire the exact duality from this fact. We alpha by because a Josephson alliance in a beeline ambit with either a voltage or a accepted antecedent (Fig. 2). Any beeline ambit can be presented by an agnate (frequency-dependent) resistor in alternation with (for voltage bias) or alongside to (for accepted bias) the junction. A breakthrough capricious anecdotic the ambit is either the appearance beyond the alliance or the connected cardinal of Cooper pairs transferred. Those are canonically conjugated variables so that the agnate operators of allegation () and appearance () amuse . In the spirit of the Caldeira–Legett approach15, the ambit can be declared with the afterward hamiltonian: Here, represents boson-like ambiance modes, and the coupling appellation is altered for voltage or accepted bias, The operators and present accepted and voltage fluctuations, respectively, in the able resistor and are beeline combinations of the ambiance bosons. The coefficients in these beeline combinations are anchored to carbon the acknowledgment action of the environment: Here Z(ω) is the frequency-dependent impedance of a consecutive resistor, Y (ω) the admission of a alongside resistor and ħ is the bargain Planck constant.

a, Current-biased Josephson junction; b, voltage-biased Josephson junction;c, current-biased QPS junction; d, voltage-biased QPS junction. Ambit a is the exact bifold of ambit d; ambit b is the exact bifold of ambit c.

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For the hamiltonian of a QPS alliance in a circuit, we use the anterior and QPS energies from (1) to access Let us now accede the aftereffect of the approved transformation on the hamiltonian (2). Obviously, that transformation does not change the barter relations. We see that it transforms the QPS hamiltonian into the Josephson hamiltonian with the afterward parameters: Double-sided arrows beggarly that the transformation exchanges accepted and voltage bent and alternation and alongside resistors (Fig. 2). This is the capital exact aftereffect of our work. These duality relations acquiesce us to absolutely map any accepted aftereffect apropos carriage characteristics of Josephson junctions in a ambit to bifold carriage characteristics of QPS junctions in the bifold circuit.

The circuits of Fig. 1 were in the regimes EC≫EJ and EL≫ES. Now let us about-face to the adverse regimes for the aloft hamiltonians. For a Josephson junction, this absolute is accomplished at EJ≫EC, breadth the appearance φ is a categorical variable. Of accurate absorption is the dynamics of the classical Josephson alliance as declared by the resistively shunted alliance archetypal that describes a motion in a slanted sinusoidal (‘washboard’) abeyant at time t (ref. 1). Time-independent solutions accord to the zero-voltage accompaniment of the alliance breadth the appearance is trapped in one of the abeyant minima. These minima abide provided the drive accepted I does not beat the analytical accepted of the alliance Ic=2πEJ/Φ0. To access the bifold analogue of this blueprint for the voltage-biased QPS alliance we accomplish use of the transformation (3) that relates the circuits (a) and (d) from Fig. 2. The aqueduct that shunts the Josephson alliance is adapted into an alien resistor R in alternation with the QPS junction. Equation (4) becomes an blueprint for the allegation capricious q, breadth the agreement on the appropriate accord the voltage drops over QPS junction, inductance and resistor, respectively. This announcement is accurate in the absolute of ample phase-slip amplitudes ES≫EL breadth the allegation is a categorical breakthrough variable. Here we accept a washboard abeyant in the allegation variable. The allegation may be trapped in abeyant minima consistent in an careful zero-current accompaniment provided the bent voltage V does not beat the analytical voltage For the Josephson alliance in the classical regime, the acknowledgment to baby signals follows the active inductance Lkin=Φ0/(2πIccosφ). Similarly for QPS junctions the acknowledgment is according to a kinetic capacitance The geometry of Josephson adit junctions implies a blow capacitance. The aggregate of active inductance and blow capacitance leads to a resonance at the alleged claret abundance ωp=(2EJEC)1/2. Because EJ scales with the breadth and EC scales inversely with it, ωp is absolute of breadth for a connected barrier. For a QPS junction, the accustomed geometry is a wire with ample active inductance. Now, the resonance provided by active capacitance (6) and inductance has a abundance As ES is proportional and EL is inversely proportional to the wire length, the claret abundance is connected back the breadth is changed. However, in a lithographically bogus nanowire it will be accessible to lower ωp by abacus a added area of wire that lowers EL after accretion ES (Fig. 3). It is absorbing to agenda that affective from continued affairs to abbreviate wires, the arrangement ES/EL decreases in the aforementioned way as the abatement of EJ/EC back activity from large-area to small-area Josephson junctions. Whereas in the closing case a alteration is bogus from superconducting to careful response, the affairs may be careful for continued lengths and balance superconductivity at beneath lengths.

Top: The alliance allegation be anchored in a awful arresting environment. Abject capacitances are abundantly baby for the architecture given. Left: QPS junction. The ambit accord to Q=0.5,2πν=ωp,Va.c./Vc=0,1,5 for the blubbery solid, abject and attenuate solid curves respectively. With ES/h=120 GHz and EL/h=30 GHz the claret abundance is ωp/2π=85 GHz and the analytical voltage is Vc=1.56 mV; R=115 kΩ. Right: Voltage Shapiro accomplish in the bifold Josephson ambit with agnate parameters.

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The high-frequency ecology impedance of a submicrometre alliance is usually bent by the geometric capacitance and inductance of adjacent base and, after appropriate precautions, has a amount about 300 Ω≪Rq. For a QPS alliance this accustomed impedance leads to a aerial affection agency of the claret oscillations accustomed by Here βL is the agnate of the acclaimed McCumber connected βC for Josephson junctions. Appliance duality, we adumbrate a acerb nonlinear hysteretic acknowledgment to almost baby beating signals Va.c./Vc∼1/QQPS,Va.c. actuality the amplitude of the activated radiofrequency (RF) signal. The bifold aftereffect was advised for a attention breakthrough altitude with Josephson junctions16.

The best extensive appliance of QPS junctions ability be in a axiological accepted of current, bifold to the acclaimed Josephson voltage standard. Equation (5) can be acclimated to account the carriage in the alternation ambit of QPS junction, inductance and resistance. Back an RF voltage is applied, the agnate of Shapiro accomplish will action in the anatomy of plateaus at connected accepted levels breadth ν is the RF frequency. For n=1 and ν=80 GHz, as acclimated in Josephson voltage standards, the accepted akin would be 26 nA. Aerial QQPS is unfavourable for this appliance as it gives acceleration to a hysteretic current–voltage characteristic, unrealistically aerial frequencies ∼QQPS and accentuation amplitudes ∼QQPS2. To accomplish a activated device, the ambit should be bogus non-hysteretic by the use of a abundantly aerial alternation resistance. As ES/EL cannot be abate than about 4 to agreement that charge, as the accordant breakthrough variable, is able-bodied defined, the alternation attrition has to be at atomic 60 kΩ. In Fig. 3 we accord current–voltage characteristics for a ambit with affection agency Q=0.5, apprenticed at the claret frequency. Realistic ambit are accustomed in the figure. To assemble the QPS wire as able-bodied as two alternation resistors of 60 kΩ each, after shunting by abject capacitances, is arduous but seems accessible with advanced electron-beam lithography. Many accessories could be put in alongside on a chip. The accepted accepted would accept to be operated at low temperatures as the attendance of quasiparticles leads to a advice approach alongside to the QPS junction, which would abate the precision.

The QPS alliance at ES≫EL and the Josephson alliance at EC≫EJ accept in accepted that the allegation is the accordant breakthrough variable. For Josephson junctions in the charging regime, Bloch oscillations were predicted17,18 and observed19. This arrangement is generally advised as bifold to the classical Josephson junction, because of the crossover from appearance to allegation as the breakthrough variable. However, there is no austere duality as apparent for Josephson and QPS. Interestingly, the anemic QPS administration with EL≫ES and the classical Josephson alliance allotment the appearance as the breakthrough variable.

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## Variable Resistor Symbol Circuit

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