Circuit Quantum Electrodynamics - Yale School of Engineering ...

CHAPTER 3. CAVITY QED WITH SUPERCONDUCTING CIRCUITS 47
1/Z(ω). For a parallel LCR oscillator the impedance is given by
ZLCR(ω) =
�
jωC + 1
�−1 1
+
jωL R
The imaginary 1 part of this impedance will be zero at the resonance frequency ω0 = 1/ √ LC. Near
resonance, this expression can be expanded to first order in δω = ω −ω0, allowing the approximation
ω 2 − ω 2 0 ∼ 2ω(ω − ω0), and yielding
R
ZLCR(ω) =
1 + 2jQδω/ω0
where the quality factor is Q = ω0RC = ω0/γ. The quality factor is dimensionless and can be
thought of as the number of oscillations in a cavity decay time 2/κ = RC, before the system comes
to equilibrium. Large resistance results in high Q, with the limit as R → ∞ giving the lossless
impedance
ZLC(ω) =
1
2jC(ω − ω0)
which now approaches infinity at δω = 0. As with the differential equation one can use the lossless
expression with a complex frequency ω ′ 0 = ω0(1 + j/2Q) to represent the decay.
3.1.2 Transmission Line as Series of LC Circuits
The microwave frequency regime is a particularly interesting slice of the electromagnetic spectrum
because its characteristic wavelength, λ = c/f, occurs at human size scales of 100 µmto 1 m. Because
the waves interfere constructively and destructively on these scales, the electrical response of a
structure becomes highly dependent on its geometry. In a microwave circuit both the topology and
the geometry are critical, which presents a challenge when engineering microwave circuits. It is well
worth facing this challenge, as the geometric dependence brings the opportunity to make fantastically
sharp resonances, and even to make a short act as an open! Here, a section of transmission line is
used to create an effective LCR resonator.
A quick review of transmission line resonators following [Pozar1990] chapters 2 and 3 is given. A
transmission line can be modeled as consisting of many small lumped elements that have the same
impedances (admittances) per unit length as the transmission line (see Fig. 3.2). The impedance of
one small section is
Z0 =
�
Rℓ + jωLℓ
Gℓ + jωCℓ
1 In electrical engineering it is conventional to use exp(jωt) whereas quantum mechanics uses exp(iωt) and j = −i.
(3.3)
(3.4)
(3.5)
(3.6)