The fundamental structure of a Josephson junction consists of a sandwich of two su-perconductors separated by an insulating layer, typically fabricated from oxidation of the superconductors, and thin enough to allow tunneling of discrete charges through the barrier. That is why a Josephson junction is also called a superconducting tunnel junction or a Josephson tunnel junction.
For the purposes of creating a two-level system which is isolated from and not by external excitation resonant with other energy levels, the harmonic system is not suitable, in which all of energy gaps are the same. A nonlinear system is required.
A Josephson junction [15, 16] is the electronic circuit element that has nonlinear and non-dissipative properties at arbitrarily low temperatures. Because of the properties of nonlinearity, when the driving frequency ω is detuned from the natural oscillation frequency ω0, the system is very sensitive between two possible oscillation states that differ in amplitude and phase. So, a Josephson junction is an important element not only of creating a qubit but also of quantum readout measurement.
2.1.1 The Josephson effect
As stated above, a junction consists two strongly superconducting electrodes con-nected by a weak link. The weak link can be an insulating layer as Josephson origi-nally proposed, or a normal metal layer made weakly superconductive by the so-called proximity effect, or simply a short, narrow constriction in otherwise continuous su-perconducting material [16]. According to quantum mechanics, the electrons would tunnel through the weak link or barrier layer. There are two effects of pair tunneling, DC and AC Josephson effects [15].
DC Josephson effect: a dc current flows across the junction in the absence of any electric or magnetic field. The relationship between the phase difference δ and the current I of superconducting pairs across the junction is
IJ = Icsin δ . (2.1)
The critical current Icis the maximum zero-voltage superconducting current that can pass through the junction above which the superconducting state will become normal state. It is proportional to the transfer interaction. Because no voltage apply, the phase difference δ is a constant. For finite voltage situations involving the ac Joseph-son effect, a more complete description is required.
AC Josephson effect: when a dc voltage is applied across the junction, an ac current flows across the junction. The phase difference δ is no longer a constant. The relationship between voltage and phase difference is
˙δ = −2eV/~ (2.2)
or
δ(t) = −2e
~ Z t
0
V dt + δ(0) . (2.3)
and the superconducting current is
IJ = Icsin (δ(0) − 2eV t/~) . (2.4)
Furthermore, considering more general cases, we can apply a time-dependent voltage, and write down the function in some significant symbols,
IJ(t) = IcsinΦJ(t)
ϕo = Icsin δ(t) , (2.5)
where the generalized flux is defined by ΦJ =Rt
−∞V (t0)dt and ϕ0 = ~/2e is the re-duced flux quantum, or ϕ0 = Φ0/2π, where Φ0, h/2e, is the magnetic flux quantum.
Actually, phase difference is not a gauge-invariant quantity; for a given physical situation, there is not only one unique value of phase difference. Hence it cannot in general determine the current IJ, which is a well-defined gauge-invariant physical quantity. The phase difference mentioned before is not the real phase difference between two superconductor [16], defined by
δ ≡ δ0 −2π Φ0
Z
A · dl . (2.6)
where δ0 is the real phase difference and the integration over the vector potential A is from one electrode of the weak link to the other. Thus, the difficulty is cured. In addition to curing the conceptual problem, the introduction of the gauge-invariant phase difference is the key to working out the effects in a magnetic field, which cannot be treated without introducing the vector potential A.
2.1.2 A Josephson junction with a nonlinear inductance
At first, let’s take a short review of a conventional inductance.
L = Φ/I or I = φ/L ,
where L is the inductance, Φ is the magnetic flux, and I is the current.
We thus expand Eq. (2.5) IJ(t) = 1
LΦJ(t) − 1
6LJϕ20Φ3J(t) + O[Φ5J(t)] (2.7) or simplely
IJ(t) = Icsin δ(t) = Ic
δ(t) − δ3(t) 3! + ...
. (2.8)
I
J C R
Figure 2.1 The current-biased josephson junciton and its equivlent circuit.
By comparing the functions of a Josephson junction and a conventional inductance, it is very easy to find that besides the linear term in the relation of current and magnetic flux, there are additional nonlinear high-order terms in a Josephson junction. A Josephson junction, therefore, can be considered having a nonlinear inductance.
2.1.3 The current-biased Josephson junction
A Josephson junction schematically shown in Fig. 2.1 as a sandwich structure can be modeled as a parallel circuit which consists of a nonlinear inductance, a resistance, and a capacitance.
According to Kirchhoff’s rule and some relationships, I = C ˙V = C ¨δ, δ = 2e
~Φ, and Ij = Icsin δ, the equation of the circuit is
~
2eC ¨δ + ~
2eR˙δ + Icsin δ = Ie, (2.9) where C is the capacitance, R is the resistance, and V is the voltage across the capacitance. Then, it is useful to define some meaningful parameters, EC ≡ (2e)2C2 and
δ
U
Figure 2.2 The ”tilted-washboard” effective potential versus phase differ-ence of a current-biased Josephson junction.
EJ ≡ 2e~Ic. The kinetic energy of the quasi-partical of phase δ is
K( ˙δ) = ~2δ˙2
4EC , (2.10)
the potential energy of it is
U (δ) = EJ(1 − cos δ) − ~
2eIeδ , (2.11)
and the Hamiltonian has the form
H = ECn2− EJcos δ − ~
2eIeδ . (2.12)
The relationship of potential versus phase is shown in Fig. (2.2). It is obvious that nonlinear inductance, cos δ, makes potential oscillate and bias current makes it slope.
When current bias is applied, the pendulum potential becomes tilted. By the way, a current-biased Josephson junction can be considered as a qubit, because the potential is cosine function, making energy gaps different.
V g
C g
Figure 2.3 The single Cooper pair box. One side of a small superconducting island is connected via a Josephson tunnel junction to a large superconduct-ing reservoir, and another side is coupled capacitively to a voltage source.