PDF (double-sided) - Physics Department, UCSB - University of ...

Figure 2.3: Modified Inductor-Capacitor Oscillator – a) Circuit modification: The
oscillator’s capacitor is replaced by a josephson tunnel junction with its intrinsic
capacitance C J designed to match the original capacitance C. b) Relevant currents
and voltages: The voltage V (t) across the capacitor drives the current I J (t)
through the junction and I(t) around the oscillator loop through the inductor.
is set equal to the current flowing through the junction and its capacitor given a
voltage V (t) across the elements:
I(t) = − 1 L
∫
V (t) dt = I J (t) + C d V (t) (2.11)
dt
Plugging in the Josephson relations gives:
I(t) = − 1 L
∫
Φ0
2π
This can be rewritten as:
( ) (
2 Φ0 d 2
C
2π dt δ + ∂ − Φ 0
2 ∂δ 2π I c cos δ + 1
2L
d
dt δ(t) dt = I c sin δ(t) + C d Φ 0
dt 2π
d
δ(t) (2.12)
dt
( ) )
2 Φ0
δ 2 − Φ 0
2π 2π I dc δ = 0 (2.13)
Here, δ is a function of t, and I dc is the constant of integration which captures an
initial flux bias applied to the inductor.
This equation can be interpreted as an equation of motion of a particle of mass
m = C ( )
Φ 0 2
2π at position δ in the potential V (δ) = −
Φ 0
I 2π c cos δ + 1
2L
(
Φ0
2π
) 2
δ 2 −
26