MQT Makroskopsko kvantno tuneliranjeMacroscopic quantum tunnelling

Josephsonov spoj · nagnjeni periodični potencial · pobeg · kvantizirani nivojiJosephson junction · tilted washboard · escape · quantized levels

Računa brskalnikBrowser compute

Nobelovo nagrado za fiziko 2025 so prejeli John Clarke, Michel Devoret in John Martinis za odkritje makroskopskega kvantnega tuneliranja in kvantizacije energije v električnem vezju. Fazna razlika ϕ na Josephsonovem spoju se obnaša kot kroglica v nagnjenem periodičnem potencialu. Spreminjajte tok in parametre spoja ter opazujte nivoje v jami, pobeg s termično aktivacijo ali s tuneliranjem ter količine, ki so jih leta 1985 izmerili v Berkeleyju. Stran dopolnjuje poljudne članke v Obzorniku za matematiko in fiziko, Proteusu in Gei. The 2025 Nobel Prize in Physics was awarded to John Clarke, Michel Devoret and John Martinis for the discovery of macroscopic quantum mechanical tunnelling and energy quantization in an electric circuit. The phase difference ϕ across a Josephson junction behaves like a particle in a tilted washboard potential. Change the bias current and the junction parameters, and follow the levels in the well, escape by thermal activation or by tunnelling, and the quantities measured at Berkeley in 1985. The page accompanies popular articles (in Slovene) in Obzornik za matematiko in fiziko, Proteus and Gea.

Nagnjeni periodični potencialTilted washboard

U(ϕ) = −EJ(cos ϕ + ϕ I/Ic), EJ = ħIc/2e

RCSJ
Pralna deska: kroglica v modelu RCSJWashboard: phase particle in the RCSJ model
Jama od blizu: kvantizirani nivojiWell close-up: quantized levels

Kroglica sledi klasični enačbi RCSJ z dušenjem Q0 = ωp0RC. Pobeg iz jame je naključen dogodek s pogostostjo Γ = Γt + Γq; čakalni čas teče pospešeno. Pri tunelskem pobegu se kroglica pojavi na izhodni točki za pregrado. Pri velikem Q ostane po pobegu v stanju s končno napetostjo (histereza); vrne se šele pod I ≈ 4Ic/πQ0. The particle follows the classical RCSJ equation with damping Q0 = ωp0RC. Escape from the well is a random event with rate Γ = Γt + Γq; the waiting time is shown in time-lapse. After escape by tunnelling the particle appears at the exit point beyond the barrier. For large Q it stays in the finite-voltage state after escape (hysteresis) and is retrapped only below I ≈ 4Ic/πQ0.

Ubežna temperaturaEscape temperature

Γ = (ωp/2π) exp(−ΔU/kBTesc)

DMC 1985

Pri visokih temperaturah sledi Tesc temperaturi (termična aktivacija), pod Tcr se ustali pri vrednosti, ki jo določa tuneliranje. Tako so Devoret, Martinis in Clarke leta 1985 prikazali svoje meritve (sl. 2 v članku v Obzorniku). Krivulja iz vsote po nivojih je neodvisen približek brez dušenja. At high temperature Tesc follows T (thermal activation); below Tcr it saturates at the value set by tunnelling. This is how Devoret, Martinis and Clarke presented their 1985 data. The level-sum curve is an independent, undamped approximation.

Spektroskopija nivojevLevel spectroscopy

Prehodi En,n+1/h v odvisnosti od tokaTransitions En,n+1/h vs bias current

MDC 1985

Mikrovalovi s frekvenco f povečajo pogostost pobega, ko je hf enak razmiku med sosednjima nivojema. Zaradi kubičnega člena v potencialu se razmiki z n manjšajo; klasično bi pričakovali resonanco pri ωp/2π. Nivoji so lastne vrednosti 4ECn̂² + U(ϕ), izračunane numerično. Microwaves at frequency f enhance the escape rate when hf matches the spacing of adjacent levels. Because of the cubic term in the potential, the spacing decreases with n; classically one would expect a resonance at ωp/2π. The levels are eigenvalues of 4ECn̂² + U(ϕ), computed numerically.

Preklopni tokSwitching current

P(I) = (Γ/İ) exp(−∫Γ dI/İ)

P(I)

Tok linearno povečujemo od nič, dokler spoj ne preklopi v stanje s končno napetostjo. Pri termični aktivaciji se porazdelitev z nižanjem temperature oži in pomika proti Ic; v kvantnem režimu postane neodvisna od temperature. The current is ramped linearly from zero until the junction switches to the finite-voltage state. With thermal activation the distribution narrows and moves towards Ic as the temperature is lowered; in the quantum regime it becomes temperature independent.

OzadjeBackground

Josephsonov spoj kot kroglica

Skozi Josephsonov spoj teče supertok I = Ic sin ϕ, kjer je ϕ razlika faz parametrov reda na obeh superprevodnih kontaktih in Ic kritični tok. Napetost na spoju je V = (Φ₀/2π) dϕ/dt, Φ₀ = h/2e. Spoj s kapaciteto C in vzporedno upornostjo R, skozi katerega teče tok I, opisuje enačba modela RCSJ

C (Φ₀/2π)² d²ϕ/dt² + (Φ₀/2π)² (dϕ/dt)/R + ∂U/∂ϕ = 0,   U(ϕ) = −EJ(cos ϕ + ϕ I/Ic),   EJ = ħIc/2e.

To je enačba gibanja kroglice z maso, sorazmerno s C, s trenjem, sorazmernim z 1/R, v nagnjenem periodičnem potencialu (»pralni deski«). Pri I < Ic je kroglica ujeta v minimumu in na spoju ni napetosti. Ko pobegne, se zakotali po deski in na spoju se pojavi napetost.

Plitva jama blizu kritičnega toka

Kroglica niha na dnu jame s plazemsko frekvenco ωp = ωp0[1 − (I/Ic)²]1/4, ωp0 = (2πIc/Φ₀C)1/2 = √(8EJEC)/ħ, kjer je EC = e²/2C. Pregrada je visoka ΔU = 2EJ[√(1 − s²) − s arccos s] ≈ (4√2/3) EJ(1 − s)3/2, s = I/Ic; približek velja, ko je jama blizu Ic dobro opisana s kvadratnim in kubičnim členom. Razmerje ΔU/ħωp je približno število nivojev v jami. Faktor kakovosti je Q = ωpRC.

Termična aktivacija in tuneliranje

V klasičnem režimu, kBT ≫ ħωp, kroglica pobegne čez pregrado zaradi termičnih fluktuacij (Kramers 1940):

Γt = at (ωp/2π) exp(−ΔU/kBT),   at = 4/[(1 + QkBT/1,8ΔU)1/2 + 1]².

V kvantnem režimu, kBT ≪ ħωp, pobegne s tuneliranjem skozi pregrado. Za kubični potencial z ohmskim dušenjem sta Caldeira in Leggett dobila

Γq = aq (ωp/2π) exp[−7,2 (ΔU/ħωp)(1 + 0,87/Q)],   aq = [120π · 7,2 ΔU/ħωp]1/2.

Disipacija zmanjša tuneliranje eksponentno, termično aktivacijo pa le prek predfaktorja. Režima se zamenjata pri Tcr = ħωR/2πkB, ωR = ωp[(1 + 1/4Q²)1/2 − 1/2Q]. Stran uporablja Γ = Γt + Γq; daleč od Tcr en prispevek prevladuje za več redov velikosti, blizu Tcr pa je vsota le interpolacija (natančno teorijo pri končni temperaturi so razvili Grabert, Olschowski in Weiss). Ubežno temperaturo Tesc definira zveza Γ = (ωp/2π) exp(−ΔU/kBTesc).

Kvantizirani nivoji

Hamiltonova funkcija spoja je H = 4ECn̂² + U(ϕ), kjer je n̂ = −i∂/∂ϕ število Cooperjevih parov, ki so prešli spoj. Stran lastne vrednosti izračuna numerično s končnimi diferencami (4000 točk). Nad vrhom pregrade je potencial konstanten, zato so stanja kvazivezana; nivoji, ki so blizu vrha, so le približni. Pogostost tuneliranja iz n-tega nivoja je ocenjena po WKB: Γn = (ωn/2π)/(1 + e2Sn), Sn = ∫|n| dϕ pod pregrado, ωn iz razmika med nivoji. Vsota Σ pnΓn z Boltzmannovimi zasedenostmi (brez dušenja) je na grafu ubežne temperature prikazana kot neodvisna primerjava; pri T = 0 se Γ0 od Γq(Q → ∞) razlikuje za 10–25 %.

Poskusi iz leta 1985

Devoret, Martinis in Clarke so vse parametre spoja (Ic, C, R) izmerili v klasičnem režimu, med drugim z resonančno aktivacijo z mikrovalovi, in nato pri nizkih temperaturah brez prilagajanja parametrov potrdili napoved za MQT. Martinis, Devoret in Clarke so z mikrovalovno spektroskopijo izmerili razmike med nivoji v jami. Prednastavitvi uporabita Ic in C iz slik 2 in 3 članka Phys. Rev. Lett. 55, 1543 (1985): Ic = 30,572 µA, C = 47,0 pF (sl. 2, Q ≈ 80, T = 28 mK, f = 2,0 GHz) in Ic = 9,489 µA, C = 6,35 pF (sl. 3, Q = 30 ± 15, T = 18 mK). Upornost R je izbrana tako, da da navedeni Q pri ΔU/ħωp ≈ 6 oziroma ≈ 2, kot ga navaja članek.

Animacija

Kroglica sledi klasični enačbi RCSJ v brezdimenzijskem času τ = ωp0t; šuma ni. Pobeg iz jame je naključen dogodek s pogostostjo Γ: čakalni čas je izžreban iz eksponentne porazdelitve in prikazan pospešeno, vrsta pobega pa je izbrana z verjetnostma Γt/Γ in Γq/Γ.

Poljudni članki

Viri

The Josephson junction as a particle

A supercurrent I = Ic sin ϕ flows through a Josephson junction, where ϕ is the difference between the phases of the order parameters of the two superconducting electrodes and Ic is the critical current. The voltage across the junction is V = (Φ₀/2π) dϕ/dt, Φ₀ = h/2e. A junction with capacitance C and shunt resistance R, biased with current I, obeys the RCSJ equation

C (Φ₀/2π)² d²ϕ/dt² + (Φ₀/2π)² (dϕ/dt)/R + ∂U/∂ϕ = 0,   U(ϕ) = −EJ(cos ϕ + ϕ I/Ic),   EJ = ħIc/2e.

This is the equation of motion of a particle with mass proportional to C and friction proportional to 1/R in a tilted washboard potential. For I < Ic the particle is trapped in a minimum and there is no voltage. Once it escapes, it runs down the washboard and a voltage appears.

A shallow well near the critical current

The particle oscillates at the bottom of the well at the plasma frequency ωp = ωp0[1 − (I/Ic)²]1/4, ωp0 = (2πIc/Φ₀C)1/2 = √(8EJEC)/ħ, with EC = e²/2C. The barrier height is ΔU = 2EJ[√(1 − s²) − s arccos s] ≈ (4√2/3) EJ(1 − s)3/2, s = I/Ic; the approximation holds when the well close to Ic is well described by a quadratic plus a cubic term. The ratio ΔU/ħωp is approximately the number of levels in the well. The quality factor is Q = ωpRC.

Thermal activation and tunnelling

In the classical regime, kBT ≫ ħωp, the particle escapes over the barrier through thermal fluctuations (Kramers 1940):

Γt = at (ωp/2π) exp(−ΔU/kBT),   at = 4/[(1 + QkBT/1.8ΔU)1/2 + 1]².

In the quantum regime, kBT ≪ ħωp, it escapes by tunnelling through the barrier. For a cubic potential with ohmic damping, Caldeira and Leggett obtained

Γq = aq (ωp/2π) exp[−7.2 (ΔU/ħωp)(1 + 0.87/Q)],   aq = [120π · 7.2 ΔU/ħωp]1/2.

Dissipation suppresses tunnelling exponentially but affects thermal activation only through the prefactor. The crossover between the regimes is at Tcr = ħωR/2πkB, ωR = ωp[(1 + 1/4Q²)1/2 − 1/2Q]. The page uses Γ = Γt + Γq. Far from Tcr one term dominates by orders of magnitude; near Tcr the sum is only an interpolation (the finite-temperature theory is due to Grabert, Olschowski and Weiss). The escape temperature Tesc is defined by Γ = (ωp/2π) exp(−ΔU/kBTesc).

Quantized levels

The junction Hamiltonian is H = 4ECn̂² + U(ϕ), where n̂ = −i∂/∂ϕ is the number of Cooper pairs transferred across the junction. The page computes its eigenvalues numerically by finite differences (4000 points). Above the barrier top the potential is held constant, so the states are quasi-bound; levels close to the top are only approximate. The tunnelling rate from level n is estimated in the WKB approximation, Γn = (ωn/2π)/(1 + e2Sn), with Sn = ∫|n| dϕ under the barrier and ωn from the level spacing. The sum Σ pnΓn with Boltzmann populations (no damping) is shown in the escape-temperature plot as an independent comparison; at T = 0, Γ0 differs from Γq(Q → ∞) by 10–25 %.

The 1985 experiments

Devoret, Martinis and Clarke measured all junction parameters (Ic, C, R) in the classical regime, among other ways by microwave resonant activation, and then confirmed the MQT prediction at low temperature with no adjustable parameters. Martinis, Devoret and Clarke measured the level spacings in the well by microwave spectroscopy. The presets use Ic and C from Figs. 2 and 3 of Phys. Rev. Lett. 55, 1543 (1985): Ic = 30.572 µA, C = 47.0 pF (Fig. 2, Q ≈ 80, T = 28 mK, f = 2.0 GHz) and Ic = 9.489 µA, C = 6.35 pF (Fig. 3, Q = 30 ± 15, T = 18 mK). R is chosen to give the quoted Q at ΔU/ħωp ≈ 6 and ≈ 2, respectively, as stated in the paper.

Animation

The particle follows the classical RCSJ equation in dimensionless time τ = ωp0t, without noise. Escape from the well is a random event with rate Γ. The waiting time is drawn from the exponential distribution and shown in time-lapse. The escape channel is chosen with probabilities Γt/Γ and Γq/Γ.

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