Circuit QED with a Flux Qubit Strongly Coupled to a Coplanar
  Transmission Line Resonator
Circuit QED with a Flux Qubit Strongly Coupled
to a Coplanar Transmission Line Resonator
T.Lindström1, C.H. Webster1, J.E. Healey2, M. S. Colclough2 C.M.Muirhead2, A.Ya.Tzalenchuk1
1National Physical Laboratory, Hampton Road, Teddington, TW11 0LW, UK and
2 University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
(Dated: October 22, 2018)
We propose a scheme for circuit quantum electrodynamics with a superconducting flux-qubit
coupled to a high-Q coplanar resonator. Assuming realistic circuit parameters we predict that it
is possible to reach the strong coupling regime. Routes to metrological applications, such as single
photon generation and quantum non-demolition measurements are discussed.
I. INTRODUCTION
Until a few years ago it was an open question whether
true quantum effects such as quantum entanglement
would ever be observed in a man-made macroscopic elec-
tronic device. However, over the past decade, quantum
coherence has been demonstrated in a variety of macro-
scopic systems, including superconducting circuits1,2,3,4.
Many of these experiments drew inspiration from the pi-
oneering work on atomic qubits that took place a decade
earlier5. As the fields of atomic physics and quantum
optics continue to advance, it makes sense to continue to
look to them for guidance.
The universal nature of quantum mechanics is greatly
to our advantage, in that the terminology and method-
ology apply as well to macroscopic as to microscopic
systems. This allows well-known results from atomic
physics and quantum optics to be used to plan and pre-
dict the outcome of experiments on solid state devices.
Recently, such techniques have been applied with great
success to implement a number of ideas such as quan-
tum state tomography6, Mach-Zehnder interferometry7
and sideband cooling8.
This approach has also been very successful in the field
of cavity quantum electrodynamics (CQED) [see Ref.9
for an introduction]. In CQED an atomic 2-level system
(i.e. a qubit) is made to interact with a high-finesse op-
tical cavity with a coupling energy h̄g. Provided that
the relaxation rates γ of the qubit and κ of the cavity
field are smaller than g (known as the strong coupling
criterion), it is possible to observe a coherent exchange
of energy between the qubit and the cavity field. The
resulting entangled states can be detected spectroscopi-
cally. Recently, Schoelkopf and co-workers10,11 achieved
strong coupling in a macroscopic circuit comprising a su-
perconducting charge qubit and a coplanar transmission
line resonator. This new field is known as circuit-QED,
and has many potential applications such as the gener-
ation and detection of single microwave photons. More
recently strong coupling was observed in experiments on
photonic crystals12 and quantum dots13.
Much of the work to date on qubit-cavity sys-
tems has been focused on methods for making quan-
tum non-demolition (QND) measurements of the qubit
state. QND schemes have been used to read-out single
qubits10,14 and also to measure the photon number in
the cavity15. Several ways of reading out qubits using
low-Q cavities/resonators have also been reported16,17,
but these do not allow the strong coupling regime to be
reached.
The benefits of superconducting circuit-QED systems
over atomic systems are twofold. Firstly, the qubit en-
ergy can be tuned by varying the external magnetic field,
enabling control over the qubit-cavity interaction. Sec-
ondly, qubit parameters such as the level separation at
the degeneracy point can be engineered through appro-
priate circuit design. The main advantage of flux qubits
over other types of superconducting qubits is that they
are less susceptible to fluctuations in the background
charge and the associated noise. This makes them less
prone to decoherence, and therefore easier to manipulate
in a deterministic way.
In section II we summarise the established background
theory in the context of our proposed system; in III we
show that it is possible to achieve strong coupling be-
tween a superconducting flux qubit18 and a high-Q copla-
nar transmission line resonator; in IV we present com-
puter simulations of the resonator response; and in V we
propose a scheme for producing single photons on de-
mand.
II. THEORY OF CIRCUIT QED WITH A FLUX
QUBIT
In this section we analyze the processes which occur
when a superconducting qubit is coupled to a supercon-
ducting coplanar transmission line resonator, as shown in
Fig. 1. The effect of coupling a flux qubit to a resonator
has previously been experimentally demonstrated19 but
the quality factor of the resonator was too small to fulfill
the strong coupling criteria.
Two types of flux qubits will be discussed - an RF
SQUID, which consists of a single Josephson junction in
a superconducting loop, and a persistent current qubit
(PCQ), which has three junctions in the loop, one of
which is smaller than the others by a factor α18. Both
need to be biased by an external magnetic flux Φx, which
tunes the energy level separation through an anticross-
ing at Φx = 0.5Φ0. The resonator is a coplanar trans-
http://arxiv.org/abs/0704.0727v3
FIG. 1: a) Sketch of a typical coplanar waveguide resonator
of length l=λ/2 ≈ 11 mm. Shown is also how the qubit can be
placed in between the centre conductor and the ground plane
of the waveguide. b) Schematic diagram of a superconducting
qubit coupled to a coplanar transmission line resonator. (A)
Persistent current qubit. (B) RF SQUID. M is the mutual
inductance between the qubit and resonator, and ΦX is the
magnetic flux threading the qubit loop.
mission line with inductance L and capacitance C, which
is weakly coupled to external transmission lines via cou-
pling capacitors CC . The Hamiltonian H of the complete
system is
H = Hq +Hr +Hg +HI +HE . (1)
Hq describes the qubit, Hr the resonator, and Hg the
interaction between them. HI denotes the interaction
of the system with a periodic drive field. Finally, HE
describes the interaction of the resonator with its envi-
ronment, resulting in the loss of photons to the external
transmission lines and the interaction of the qubit with
its environment, resulting in spontaneous decay from the
excited state to the ground state.
The qubit Hamiltonian Hq is given by the expression
(h̄/2)(−ǫσz−δσx) where σz and σx are Pauli spin matri-
ces, δ is the the level repulsion, ǫ = (2Ip/h̄)(Φx −Φ0/2),
and Ip is the persistent current. In the case of an RF
SQUID suitable for operation as a qubit, Ip is roughly
equal to half the critical current of the single junction,
whereas, for the persistent current qubit, Ip is approxi-
mately equal to the critical current of the smallest of the
three junctions. If the qubit is operated at or near the
degeneracy point, Hq can be expressed more simply by
transforming to the basis in which the ground state | ↓〉
and excited state | ↑〉 correspond to symmetric and anti-
symmetric superpositions of clockwise and anti-clockwise
persistent currents. This yields
(| ↑〉〈↑ | − | ↓〉〈↓ |) = h̄ω0
σz, (2)
where the level separation is h̄ω0, ω0 being the qubit
Larmor frequency
ǫ2 + δ2.
Assuming the resonator supports only a single mode
of the electromagnetic field, its Hamiltonian is given by
Hr = h̄ωr
, (3)
where a†(a) is the creation(annihilation) operator which
creates (destroys) a single photon in the cavity. The
eigenstates of the resonator described by this Hamilto-
nian are Fock states |0〉 . . . |n〉 . . . with n photons. The
single mode condition corresponds to a harmonic oscilla-
tor with the energy levels at h̄ωr(n+ 1/2) and the zero-
point energy h̄ωr/2.
The interaction between the radiation and the qubit
is described as the dipole interaction Hg = −µ̂ · B̂ be-
tween a magnetic moment µ of the persistent current
circulating in the qubit loop and the magnetic field B in
the resonator. Introducing quantization, this term in the
Hamiltonian can be written in the form
Hg = h̄g(a
†σ− + σ+a), (4)
where σ+(σ−) is the raising(lowering) operator for the
qubit. This expression is valid in the so-called non-
dispersive regime where ωr ≈ ω035. The constant g char-
acterizes the qubit-photon interaction strength and the
expression in the brackets describes the process whereby
the qubit can be excited by absorbing a photon, or a
photon can be generated at the expense of de-exciting
the qubit into its ground state. In the next section we
shall explicitly calculate the dipole coupling strength g
for specific designs of the qubit and the cavity.
The interaction of the coupled system with an external
classical drive field can be seen as a periodic exchange of
photons between the resonator and the driving field:
HI = ξ(e
−iωta† + eiωta), (5)
where ξ is the drive amplitude. One can also drive the
qubit directly using a separate control line leading to
terms ξ′(e−iωtσ+ + eiωtσ−), but we shall not consider
this case any further.
Adding the terms (2)-(5) together we arrive at the
driven Jaynes-Cummings Hamiltonian20
σz + h̄ωr
+ h̄g(a†σ− + σ+a) (6)
+ ξ(e−iωta† + eiωta).
This Hamiltonian can be used to write down a mas-
ter equation which completely describes the dynamics of
the system. All the parameters which define the system
can be conveniently written in units of angular frequency
– a convention which we will follow here. If the cav-
ity is weakly coupled to an already weak drive field one
can achieve a regime when only two lower Fock states of
the resonator are relevant. Within the picture described
above we make one two-level system, the qubit, interact
with another two-level system, the resonator. When the
qubit is detuned from the resonator the eigenstates of
the coupled system can be written: |0 ↓〉, |0 ↑〉, |1 ↓〉
and |1 ↑〉, where the number represents a Fock state of
the resonator and the arrow represents the qubit state.
However, when the qubit is brought into resonance, Hg
couples the states |0 ↑〉 and |1 ↓〉 and lifts their degen-
eracy. The system will oscillate between the states |0 ↑〉
and |1 ↓〉 at a frequency Ω = 2g , known as the vac-
uum Rabi frequency21, giving rise to a splitting22 of the
central peak as shown in Fig. 2. One can visualize this
as a cycle in which the resonator and qubit continuously
exchange an amount of energy equal to one photon. As
the drive amplitude increases it will start to perturb the
system. This leads to a set of states that are shifted by
En = ±
nh̄g[1− (2ξ/g)2]3/4. (7)
Hence, the effect of the drive is to reduce the Rabi fre-
quency.
The coupling strength g can be determined experimen-
tally by making a spectroscopic measurement of the split-
ting ∆E1. The drive amplitude should be reduced until
the splitting reaches its maximum value, where it is equal
to 2gh̄.
|0 |1↑〉 ± ↓〉
Energy
hω0hωr
|1↓〉|0 |1↑〉 ± ↓〉
FIG. 2: Energy levels of the coupled qubit-cavity system. On
the left of the diagram the qubit is far detuned from the cavity.
As we move from left to right, the magnetic flux threading the
qubit loop is increased, tuning the qubit transition frequency
into resonance with the cavity. On the right of the diagram,
the qubit is once again far detuned.
The above discussion covers the resonant regime,
where the qubit is tuned into resonance with the cav-
ity. In contrast, when the detuning ∆ = ω0−ωr is large,
such that g/∆ ≪ 1, a dispersive Stark shift pulls the cav-
ity frequency by ±g2/∆. This so-called dispersive regime
can be used to perform quantum non-demolition mea-
surements of the qubit state11.
The effects of the environment on the system are taken
into account by the term HE . There are three types of
damping that need to be considered:
• Photons leak out of the cavity at a rate κ = ωr/Q,
where Q is the cavity quality factor.
• The qubit relaxes at a rate γ = 1/T1, where T1 is
the energy relaxation time.
• Pure dephasing of the qubit at a rate γφ = 1/Tφ =
1/T2 − 2/T1, where T2 is the dephasing time.
Dephasing plays a larger role in solid state systems than
atomic systems, due to the stronger interaction of solid
state qubits with their environments. In the absence of
pure dephasing we would have T2 = 2T1 but in real sys-
tems T2 is frequently much shorter than that, indicating
the need to take pure dephasing into account.
III. STRONG COUPLING WITH A FLUX
QUBIT
Below we estimate the coupling strength g using a
semi-classical approach. We treat the flux qubit as
a magnetic dipole and assume that it is placed at a
magnetic field antinode of the resonator. The coupling
strength is given by g = µB0rms/h̄, where B0rms is the
zero-point root mean square magnetic field generated by
the current fluctuations at the antinode of the resonator.
The magnetic dipole moment of the qubit is given by
µ = IpA, where Ip is the persistent current flowing
around the loop and A is the loop area. We can esti-
mate B0rms by considering the zero point energy of the
resonator, h̄ωr/2. This energy cycles continuously be-
tween inductive and capacitive components. The mag-
netic field is determined by the inductive component,
LI(t)2/2, where L is the total equivalent inductance of
the resonator near resonance and I(t) is the instanta-
neous current. At the moment when the energy is purely
inductive, we have (1/2)LI2max = (1/2)h̄ωr. Since I(t)
undergoes sinusoidal oscillation we therefore have that
Irms =
. (8)
Assuming that current flows in thin strips (whose width
is determined by the superconducting penetration depth)
at the edges of the centre conductor and ground plane,
the field at the antinode of the fundamental mode is ap-
proximately given by
B0rms ≈
µ0Irms
, (9)
where µ0 is the permeability of free space and r is half the
width of the gap between the centre conductor and the
ground plane (we assume the qubit is placed at the centre
of the gap). Therefore, the coupling strength between the
qubit and resonator is given approximately by
g ≈ IpAµ0
. (10)
By inserting realistic values for the parameters in the
above equation we can obtain an estimate of g.
First, we choose the fundamental frequency of the res-
onator. It is convenient to choose a value that lies within
the range 4–8 GHz, as this is well within the design scope
of both the qubit and resonator, and can be accessed
with commercial microwave sources and components. We
choose ωr/2π = 6 GHz. With a centre conductor of width
∼ 10 µm and a gap of width ∼ 5 µm, it is possible to
achieve a total inductance L ∼ 2 nH for a resonator op-
erated at its fundamental frequency..
Next, we choose Ip such that the transition frequency
ω0 of the qubit at the degeneracy point is slightly less
than that of the resonator. This will enable us to tune
the qubit in and out of resonance with the resonator by
changing the external flux Φx threading the qubit loop.
For the 3-junction persistent current qubit having two
junctions of critical current Ic = 800 nA and junction
capacitance C = 4 fF, and one junction of critical cur-
rent αIc, where α = 0.72, we get ω0/2π = 4.9 GHz at
the degeneracy point and Ip ≈ 580 nA. These parameters
were obtained by solving the Schroedinger equation nu-
merically. The transition frequency ω0 does not depend
on the area A of the qubit loop, provided that the loop
inductance remains small compared with the Josephson
inductance. However, we note that the larger the loop
area, the greater the (undesired) coupling to the environ-
ment. Here we choose a value A ≈ 8 µm2.
With the above parameters we obtain g/2π ≈ 35 MHz.
We now compare this with the rate of photon loss from
the resonator κ and the relaxation rate of the qubit γ.
When g > κ, γ, the coupled system is able to undergo
many cycles (≈ 2g/(κ+γ)) of vacuum Rabi oscillation be-
fore losing coherence. This is important for applications
such as single microwave photon generation. The photon
loss rate from the resonator is given by κ/2π = ωr/Q,
where Q is the loaded quality factor of the resonator. It
is possible to design a resonator with Q = 105, yield-
ing κ/2π ≈ 0.1 MHz. The relaxation rate of the qubit
is given by γ = 2π/T1. Taking T1 ∼ 1 µs3, we obtain
γ/2π ≈ 1 MHz. Naturally, the values for T1 and T2 for
a real system can not be predicted with any accuracy
and will depend on the experimental conditions. How-
ever, based on the data available in the literature23,24 we
believe that the aforementioned values are reasonable.
Hence, it is clear that our estimated value of g for the
persistent current qubit should satisfy the strong cou-
pling criterion.
For an RF SQUID with critical current Ic = 10 µA,
area A = 64 µm2, loop inductance LSQUID = 35 pH and
junction capacitance C = 50 fF, we obtain a transition
frequency ω0/2π = 4.6 GHz. In contrast to the persis-
tent current qubit, the area of the RF SQUID does affect
the transition frequency, via the loop inductance. It is
difficult to reduce the area further than the value we have
chosen, as this necessitates increasing the critical current
and decreasing the junction capacitance, which becomes
increasingly difficult to achieve in practice. Close to the
degeneracy point, the persistent current Ip in the above
SQUID is expected to be about 5 µA. Combined with the
increased loop area, this is likely to lead to an even larger
coupling strength g than predicted for the persistent cur-
rent qubit (unless the SQUID is displaced significantly
from the antinode of the resonator) but at the same time
make qubit more susceptible to noise. The fact that the
loop area of the 3-junction persistent current qubit can
be made small enough to render it relatively insensitive
to flux noise is one reason why it has been so successfully
used by e.g Mooij and co-workers3,18.
IV. NUMERICAL SIMULATION OF THE
SPECTRUM UNDER MICROWAVE
EXCITATION
Having shown that it is possible to reach the strong
coupling regime with a flux qubit coupled to a coplanar
resonator, we now simulate the results of a spectroscopic
experiment to measure g. The experiment would involve
driving the coupled system with an external microwave
field whose frequency ωl would be swept through the res-
onance of the coupled system. The most straightforward
way to probe the response of the system would be to
use what effectively amounts to a standard microwave
transmission ( S12 ) measurement of the cavity. The ex-
periment would be done by starting with the qubit far
detuned from the resonator, then stepping the external
magnetic flux to tune the qubit through the cavity reso-
nance. This type of experiment would allow us to record
the output power of the resonator as a function of qubit
Larmor frequency.
These simulations were performed by solving the mas-
ter equation using a Liouvillian with the Hamiltonian (7)
and three collapse (Lindblad) operators which account
for the decay and dephasing of the qubit and the cavity
at the aforementioned rates κ,γ and γφ (i.e. the effects of
HE). A brief description of the formalism can be found
in appendix B. All simulations were performed using the
“Quantum Optics Toolbox” developed by Tan25. Note
that all the figures show the spectrum of the intracavity
field (see appendix C for the definition and a description
of how it is calculated).
In the regime where the qubit Larmor frequency ω0 is
very far detuned from the cavity (i.e. when even disper-
sive effects are negligible) the qubit and the cavity are
effectively decoupled and no exchange of energy can take
place. If the system is probed by measuring the response
of the cavity, a single peak located at the bare resonance
−100 −80 −60 −40 −20 0 20 40 60 80 100
Frequency shift(MHz)
FIG. 3: The spectrum far from resonance (where the only
effect of the qubit is to broaden the resonance) and at zero
de-tuning for small drive (mean steady-state photon number
〈n〉 < 10−3), the latter giving rise to a Rabi splitting 2g.
Shown is also the spectrum at large (〈n〉 ≈ 8) drive amplitudes
(dashed line).
frequency ωr will be seen.
If instead the qubit is tuned exactly on resonance (∆ =
0) the effects of the coupling become clearly visible. Now,
there are two peaks located at ωr ±Ω/2 = ωr ± g as can
be seen in fig. 3. In between these two extremes there
is a gradual change from a single- to a double-peaked
spectrum where the splitting is approximately g2/∆ as
can be seen in the left picture of fig. 4. In this case the
pure dephasing rate γφ was set to zero. The result is a
”diamond-shaped” picture with a maximum splitting of
2g at zero detuning.
Far from resonance we can identify the four branches as
being associated with the states |0 ↑〉 and |1 ↓〉 above and
below the resonance. Exactly on resonance the system
is in a superposition of states |0 ↑〉 ± |1 ↓〉. This is in
agreement with the diagram shown in fig. 2. Note that
e.g. an interferometric measurement method26 must be
used in order to be able to observe the whole spectrum,
a simpler transmission experiments would only see one
peak off-resonance since such a measurement only records
transitions between photon states where the final state
can emit a photon; in this case |0 ↑〉 ↔ |1 ↓〉.
The right picture in fig. 4 shows the spectrum when
pure dephasing is introduced to the model. The result is
somewhat more complicated in that the spectral weights
are asymmetric with respect to wr − wl = 0 when the
qubit is detuned from resonance. The reason for this
is that dephasing leads to a loss of coherence, meaning
that the qubit tends to stay in its ground state and the
coupling between the states |0 ↑〉 and |1 ↓〉 is reduced,
the system is therefore no longer in a superposition of
those states. Whence, states with zero photons in the
cavity are effectively ”decoupled” from the one-photon
states. Since only states which allow for (at least) one
photon in the cavity can be measured it follows that only
the two branches (one above and one below resonance)
that (approximately) correspond to ±|1 ↓〉 will be clearly
visible. Note, however, that despite the loss of coherence
FIG. 4: (Color online) Power spectrum of the coupled qubit-
resonator system as a function of qubit detuning in the strong
coupling limit. The frequency of the drive field ωl is held at
the resonance frequency ωr of the bare resonator, while the
Larmor frequency ω0 of the qubit is tuned by changing the
external magnetic field. Left : Spectrum in the absence of
pure dephasing. Right: Adding a pure dephasing channel to
the dissipation results in an asymmetric spectrum, here the
pure dephasing rate γφ is 9.5 MHz. Shown is also one branch
of the expression ±g2/∆ (white dashed line). The following
parameters were used in the simulations (in GHz): ωr = ωl =
6, g = 0.035, κ = 0.004, γ = 0.001 and ξ/h̄ = 0.25κ.
an on-resonance measurement would still show two peaks
separated by 2g, i.e. exactly on resonance this situation is
effectively indistinguishable from the more coherent case
even when the full spectrum is measured.
While there are no bound states in the limit of strong
driving ξ > g/2 a continuum of states still exists giv-
ing rise to complex spectra (dashed line in fig. 3). The
structure is reminiscent to the so-called ”Mollow” peaks,
well known in atomic physics from e.g. fluorescence spec-
troscopy. However, in the latter case the peaks are the
result of strong driving of the atom (qubit) whereas in
this simulation the cavity is being driven so the similarity
is somewhat superficial. When as in this case the cavity
is driven so strongly that there are on average several
photons in the cavity, we see both a drive induced shift
of the position of the sidebands and a reappearance of
the central peak. Note that since it is difficult to directly
relate the parameter ξ to the power output from a mi-
crowave generator, care must be taken not to drive the
system inadvertently into this regime.
For the persistent current qubit considered here, a de-
tuning of ±0.4 GHz corresponds to an external magnetic
flux in the range ±10−4Φ0 which is a useful value for a
real experiment. However, for the RF SQUID, the flux
required is rather less: ±3 · 10−5Φ0.
One effect which is not taken into account in our sim-
ulations is the presence of thermal photons in the cavity.
Thermal effects can, in general, be neglected when work-
ing with optical cavities due to the very small average
number of photons at those frequencies. In experiment on
solid state qubits this is, however, not generally true since
they are operated in the microwave range. Also, the rele-
vant temperature scale is set not only by the the phonon
temperature (e.g. the temperature of the mixing cham-
ber of a dilution refrigerator) but also by the amount of
noise (essentially ”hot” photons) which reaches the sys-
tem via the leads. However, in a well-filtered system a
total temperature of 50 mK is attainable. The resonator
will then nearly be in its ground state with an average
thermal occupancy n̄ of 0.009 and thermal fluctuations in
the photon number of the order of 0.1. This justifies ig-
noring thermal effects in our simulations for now. That
said, even a moderate increase in temperature can sig-
nificantly change the outcome of an experiment27. One
further simplifying assumption in the model is that T1
and T2 do not change as the qubit is detuned from the
optimal bias point Φ0/2. While this is clearly unrealistic,
it has been experimentally shown24 that neither param-
eter should change dramatically in the parameter range
considered here, giving some justification to this approx-
imation.
V. APPLICATIONS OF CIRCUIT-QED
One of the most important applications of circuit QED
is the generation of single microwave photons on demand.
Single photon sources in the optical regime have been re-
alized using e.g. cavity QED with atoms and high finesse
optical cavities26. Design and fabrication of deterministic
sources that operate in the microwave regime have proved
to be more difficult, but a source based on superconduct-
ing circuit-QED was recently demonstrated28. This kind
of source could be used for quantum radiometry, as well
as for quantum information applications such as quan-
tum key distribution.
Various schemes can be envisaged29,30,31,32, by which
single photons can be generated with a circuit QED de-
vice. Below, we describe a straightforward technique,
based on manipulation of the qubit state with microwave
pulses and rapid changes in the DC magnetic field to tune
the qubit in and out of resonance with the cavity.
Our technique begins with the qubit far detuned from
the cavity and the combined system in its ground state
|0 ↓〉. A microwave π-pulse is applied to the qubit to
excite it to the state |0 ↑〉 [Fig. 5(a)], and this is fol-
lowed by a step in the magnetic field to bring the qubit
into resonance with the cavity [Fig. 5(b)]. The state of
the system immediately after the step is still |0 ↑〉, but
due to the qubit-cavity interaction on resonance, this is
no longer an eigenstate, so the system begins to precess
|0 + |1↑ ↓〉 〉|0 |1↑〉 − ↓〉
large
detuning
|0 + |1↑ ↓〉 〉
|0 |1↑〉 − ↓〉
|0↑〉|1↓〉
|0 + |1↑ ↓〉 〉
|0 |1↑〉 − ↓〉
|0↑〉|1↓〉
|0 + |1↑ ↓〉 〉|0 |1↑〉 − ↓〉
(a) (b)
(c) (d)
detuning
detuning
large
detuning
FIG. 5: (Color online). Bloch sphere diagrams showing the
state of the qubit-cavity system at successive stages of the
single photon generation process.
around the equator of the Bloch sphere at the vacuum
Rabi frequency 2g. After a time 2π/4g, the state of the
system will be |1 ↓〉 [Fig. 5(c)]. This means that coherent
energy exchange has taken place between the qubit and
the cavity, creating a photon-like state. Another step in
magnetic field detunes the qubit from the cavity so that
the state |1 ↓〉 is once more an eigenstate of the system
[Fig. 5(d)]. The system remains in this state until the
photon decays out of the cavity into one of the external
waveguides in a time of order 2π/κ. By repeating this
sequence many times, photons can be generated on de-
mand, provided that the time window within which they
are required is much longer than 2π/κ.
If a scheme similar to the one above is implemented, it
is important to prove that it generates single photons de-
terministically, rather than stochastically. At optical fre-
quencies this is done by studying photon-counting statis-
tics using interferometric measurements. Such measure-
ments require the use of a beamsplitter. An analogous
experiment can be envisaged in the microwave regime,
provided that a microwave beamsplitter can be realised.
Such a device has been proposed recently29, and could
lead to a microwave analogue of the Hanbury-Brown and
Twiss interferometer26.
VI. CONCLUSIONS
We have shown that it should be possible to reach the
strong coupling regime using a flux qubit coupled to a
coplanar waveguide resonator. If realized, it would open
the door to potential applications in metrology, quantum
communication and experimental tests of quantum me-
chanics. The fact that conventional lithography can be
used to fabricate the samples and that the experimen-
tal parameters can be chosen freely can in some cases
a be significant advantage compared to CQED implen-
tantaions utilizing e.g. atoms of ions. Since the relevant
frequencies are in the microwave regime it is also possible
to use well established methods to manipulate the sys-
tem. The main drawback compared to experiments done
at optical frequencies is the short coherence time of the
qubit and the fact that the system must be operated at
very low temperatures.
VII. ACKNOWLEDGMENTS
The authors would like to thank Mark Oxborrow,
Alexandre Zagoskin, Alexander Blais and Vladimir
Antonov for helpful discussions and comments. We
would also like to thank Scott Parkins for his help with
the Quantum Optics Toolbox. This work was funded
by the UK Department of Trade and Industry Quan-
tum Metrology Programme, project QM04.3.4. and the
Swedish Research Council.
APPENDIX A: DERIVATION OF THE
HAMILTONIAN
It is useful to compare the informal procedure used
in the introduction of this paper to derive the Hamil-
tonian with a more formal approach. Starting with
the bare qubit Hamiltonian − 1
(ǫσz + ∆σx) where ǫ =
2Ip(Φx −Φ0/2) we proceed just as before by noting that
the flux threading the qubit loop will be modulated via
the mutual inductance M that couple the fluctuations in
the cavity to the qubit. Writing the total external flux
as Φx = Φ
x + δΦ, where δΦ = M
(a†+a), adding
the Hamiltonian for the oscillator mode and the external
field and finally transforming into the eigenbasis of the
qubit we get
H = h̄ωr
σz + h̄g(a
†σ− + σ+a) sin θ
+ξ(e−iωta† + eiωta)− h̄g(a† + a)σz cos θ (A1)
in the RWA. Here we have introduced the mixing angle
θ = arctan∆/ǫ. This Hamiltonian is identical to the
J-C Hamiltonian (7) except that we now have an effec-
tive coupling g sin θ and an extra term h̄g(a†+ a)σz cos θ
which is zero when the qubit is operated at the degener-
acy point θ = π/2. By moving to an interaction frame
rotating at the drive frequency ω we see that all terms in
the Hamiltonian (A1) are time-independent except the
last term which picks up a factor exp(−iwω), meaning
it can be neglected in the rotating wave approxiamtion.
Note, however, that this additional term can potentially
play a role in the dispersive regime.
APPENDIX B: DISSIPATION
The effects of the environment on a quantum system is
in general very difficult to model but is nevertheless cru-
cial to understand since it is the cause of decoherence.
However, assuming the interaction with the environment
is Markovian the evolution of the (reduced) density ma-
trix of the system can be described by a master equation
ρ̇ = Lρ of Lindblad form33
= − i
[H, ρ] +
kCkρ+ ρC
where Ck are Lindblad operators. In the case considered
here we have 3 Lindblad operators. Firstly, the relaxation
from the excited state to the ground state at a rate γ1 =
1/T1 represented by a Lindblad operator proportional
to the lowering operator σ̂−, i.e. C1 =
−. The
cavity is loosing energy at a rate κ = ωr/Q which leads
to the ”destruction” of photons in the system, C2 =
Finally, we also need to consider pure dephasing of the
qubit at a rate γφ = 1/Tφ = 1/T2 − 1/2T1 where T2 is
the usual total dephasing time of the qubit. This process
is represented by the operator C3 =
σ̂z .
APPENDIX C: CALCULATION OF THE
SPECTRUM
Our aim is to calculate the steady-state power spec-
trum S(ω) of the intracavity field, formally this is de-
fined in terms of the photocount output from the cavity
as seen by a monochromatic detector20. The spectrum
can be calculated from the 2-time correlation function34
〈a†(t+ τ)a(τ)〉
S(ω) =
e−iωτ 〈a†(τ + t)a(t)〉dτ (C1)
which can be evaluated using the quantum-regression
theorem
< a†(τ + t)a(t) >= Tr{a†eLτaρ} (C2)
where the Liovillian (which includes the three Lindblad
operators defined in Appendix B) is given by the right
hand side of equation B1 and ρ is the steady-state den-
sity matrix which is the solution to Lρ = 0 These calcu-
lations are straightforward using the built-in routines of
the ”Quantum Optics Toolbox”.
1 Y. Nakamura, Yu.A. Pashkin, and J.S. Tsai. Coherent
control of macroscopic quantum states in a single-cooper-
pair box. Nature, 398:786–8, 1999.
2 C.H. Van Der Wal, A.C.J. Ter Haar, F.K. Wilhelm, R.N.
Schouten, C.J.P.M. Harmans, T.P. Orlando, S. Lloyd,
and J.E. Mooij. Quantum superposition of macroscopic
persistent-current states. Science, 290:773–7, 2000.
3 I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, and J.E.
Mooij. Coherent quantum dynamics of a superconducting
flux qubit. Science, 299:1869–71, 2003.
4 J.M. Martinis, S. Nam, J. Aumentado, and C. Urbina.
Rabi oscillations in a large Josephson-junction qubit. Phys.
Rev. Lett., 89(11):117901–1, 2003.
5 A. Zeilinger. Experiment and the foundations of quantum
physics. Reviews of Modern Physics, 71:288, 1999.
6 Matthias Steffen, M. Ansmann, Radoslaw C. A ND Bial-
czak, N. Katz, Erik Lucero, R. McDermott, Matthew Nee-
ley, E. M. Weig, A. N. Cleland, and John M. Martinis.
Measurement of the entanglement of two superconducting
qubits vis quantum state tomography. Science, 313:1423–
1425, 2006.
7 William D. Oliver, Yang Yu, Janice C. Lee, Karl K.
Berggren, Leonid S. Levitov, and Terry P. Orlando. Mach-
Zehnder interferometry in a strongly driven superconduct-
ing qubit. Science, 310:1653–1657, 2005.
8 S.A Valenzuela, W. Oliver, D.M. Berns, K.K. Berggren,
L.S. Levitov, and T.P. Orlando. Microwave-induced cool-
ing of a superconducting qubit. Science, 314:1589–1591,
2006.
9 C.S. Gerry and P.L. Knight. Introductory Quantum Optics.
Cambridge University Press, 2005.
10 A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.
Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.
Schoelkopf. Strong coupling of a single photon to a super-
conducting qubit using circuit quantum electrodynamics.
Nature, 431:162–167, 2004.
11 A Blais, R Huang, A Wallraff, S.M. Girvin, and R.J.
Shoelkopf. Cavity quantum electrodynamics for supercon-
ducting electrical circuits: An architecture for quantum
computation. Phys. Rev. A, 69:062320, 2004.
12 T Yoshie, A. Sherer, A. Hendrickson, G. Khitrova, H.M.
Gibbs, G. Rupper, C. Ell, O.B. Shchekin, and D.G. Deppe.
Vacuum Rabi splitting with a single quantum dot in a pho-
tonic crystal nanocavity. Nature, 432:200–203, 2004.
13 J. P. Reithmaier, A. Lffler, G. Sk, C. Hofmann, S. Kuhn,
S. Reitzenstein, L.V. Keldysh, V.D. Kulakovskii, Rei-
necke T. L., and A. Forchel. Strong coupling in a single
quantum dot-semiconductor microcavity system. Nature,
432:197–200, 2004.
14 A. Lupaşcu, S. Saito, T. Pictot, P.C. de Groot, C.J.M.
Harmans, and J.E. Mooij. Quantum non-demolition mea-
surement of superconducting two-level system. arXiv:cond-
Mat, (0611505), 2006.
15 D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff,
J. M. Gambetta, A. Blais, L. Frunzio, B. Johnson, M. H.
Devoret, S. M. Girvin, and Schoelkopf R. J. Resolving
photon number states in a superconducting circuit. Nature,
445:515, 2007.
16 G. Johansson, L. Tornberg, and C. M. Wilson. Fast quan-
tum limited readout of a superconducting qubit using a
slow oscillator. Phys. Rev. B, 74:100504R, 2006.
17 Grajcar et al. Four-qubit device with mixed couplings.
Phys. Rev. Lett., 96:047006, 2006.
18 J. E. Mooij, T. P. Orlando, L. Levitov, L. Tian, C. H.
van der Wal, and S. Lloyd. Josephson persistent-current
qubit. Science, 285:1036, 1999.
19 I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C.J.P.M.
Harmans, and J.E. Mooij. Coherent dynamics of a flux
qubit coupled to a harmonic oscillator. Nature, 431:159–
162, 2004.
20 D.F. Walls and G.J. Milburn. Quantum Optics. Springer-
Verlag, 1994.
21 J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda,
K. Semba, and H. Takayanagi. Vacuum Rabi oscillations in
a macroscopic superconducting qubit LC oscillator system.
Physical Review Letters, 96:127006, 2007.
22 P. Alsing, D.S Guo, and H.J. Carmichael. Dynamic Stark
effect for the Jaynes-Cummings system. Phys. Rev. A,
47(7):5135–5143, 1994.
23 P. Bertet, I. Chiorescu, G. Burkard, K. Semba, C. J. P. M.
Harmans, D.P. DiVincenzo, and J. E. Mooij. Relaxation
and dephasing in a flux qubit. arXiv:cond-mat, (0412485),
2004.
24 K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba,
H. Takayanagi, F. Deppe, and A. Shnirman. Dephasing
of a superconducting flux qubit. Physical Review Letters,
98:47004, 2007.
25 S. Tan. A computational toolbox for quantum and atomic
optics. Journal of Optics B, 1:424–432, 1999.
26 M. Oxborrow and A. Sinclair. Single photon sources. Con-
temporary Physics, 46(3):173–206, 2005.
27 I Rau, G. Johansson, and A. Shnirman. Cavity quantum
electrodynamics in superconducting circuits: Susceptibil-
ity at elevated temperatures. Phys. Rev. B, 70:054521,
2004.
28 A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A.
Schreier, B. R. Johnson, J. M. Chow, J. Majer, L. Frunzio,
M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Gen-
erating single microwave photons in a circuit. arXiv:cond-
mat, (0702648v1), 2007.
29 M. Mariantoni, M.J. Storcz, F.K. Wilhelm, W.D. Oliver,
A. Emmert, A. Marx, R. Gross, H. Christ, and E. Solano.
On-chip microwave Fock states and quantum homodyne
measurements. arXiv:Cond-Mat, (0509737), 2006.
30 F. Marquardt. Efficient on-chip source of microwave pho-
ton pairs in superconducting circuit qed. arXiv:cond-Mat,
(0605232), 2006.
31 A.M. Zagoskin, M. Grajcar, and A.N. Omelyanchouk. Se-
lective amplification of a quantum state. Phys. Rev. B,
70:060301(R), 2004.
32 K. Saito, M. Wubs, S. Kohler, P. Hänggi, and
Y. Kayanuma. Quantum state preparation in circuit QED
via Landau-Zener tunneling. Europhysics Letters (EPL),
76(1):22–28, 2006.
33 H. P. Breuer and F. Petruccione. The theory of open quan-
tum systems. Oxford University Press, 2004.
34 R. J. Glauber. Coherent and incoherent states of radiation
field. Physical Review, 131:2766–2788, 1963.
35 In the dispersive regime the effective Hamiltonian becomes
H = g2/∆(σ̂+σ̂− + â†âσ̂z)
