Page 196 - Principles and Applications of NanoMEMS Physics

184 Chapter 4

state is embodied, not in the wavefunction of elemental particles, but on the
coherent collective behavior of many particles, e.g., a superfluid. Thus, the
qubit states are defined by macroscopically observed quantities, such as the
charge or the current of particle condensates.
The key to superconducting qubits is the nonlinear nature of the resonant
LC circuit embodied in the Josephson junction [206]. The quantum
mechanical behavior of a linear LC circuit is captured by the flux Φ
through the inductor, which plays the role of position coordinate, and the
charge Q on the capacitor, which plays the role of conjugate momentum,
=
=
thus enabling the commutation relation [ Qĭ, ] i . With the Hamiltonian
given by, H = Φ 2 2 L + Q 2 2 C , the usual eigenenergy states are given by
E = = Ȧ 0 (n + 1 ) 2 , where ω 0 = 1 LC is the resonance frequency.
Reflecting the quadratic nature of the potential, the energy states are equally
spaced. Thus, it is difficult to define the two lowest states as the qubit states,
since transitions between higher-lying states are as equally likely [206].
The LC resonator may be made useful as a qubit if its energy spectrum is
caused to exhibit two lowest-lying states separated from the higher-lying
states. This is accomplished if a nonlinear inductance is introduced [206]. In
I
particular, the nonlinear Josephson inductance, L = Φ / 2 π cos δ ,
J 0 0
where =δ φ − φ , φ is the phase of the wavefunction on either side of
L R R , L
the junction, and I is the critical current, introduces a nonlinear potential in
0
which the two lowest-lying states are well separated from the higher-lying
states. These variables afford characterization of the Josephson junction in
terms of its energy, (ΦE ) Φ= I cos δ 2 π = E cos δ . In this context,
/
J ext 0 0 J
the conjugate variables of the quantum mechanical description of the LC
resonator become the flux, now given by Φ = ϕ θ , where ϕ = Φ 2 π,
0 0 0
and θ = δ mod 2 π represents a point in the unit circle (an angle module
2 π), and the charge, now given by Q = 2 eN , which represents the charge
that has tunneled through the junction, and N an operator with integer
eigenvalues capturing the number of Cooper pairs that have tunneled. The
commutation relation now is given by [ N,θ ] i= [206]. The Hamiltonian is

given by,
2
H = E ( − QN e 2 / ) − E cos θ, (42 )
CJ r J
where E = () 2e2 2 C embodies the Coulomb energy for adding one
CJ J
Cooper pair worth of charge to the junction capacitance C , and Q
J r
embodies a residual random charge capturing an initial charge existing on