SUPPLEMENTARY INFORMATION
I. EXPERIMENTAL METHODS
In the experiment, a resonator structure is required to amplify the applied ac signal; the
need for high quality factor
Q
and field homogeneity ruled out the use of planar waveguide
and cavity resonator structures. Instead, we adopted a tunable loop-gap resonator (LGR)
design [1, 2] (Fig. 1), which can be tuned over a wide range of frequencies while maintaining
a high
Q
factor and an ac magnetic field highly concentrated on the sample volume.
LGRs are 3D lumped-element structures, with a uniform magnetic field in the loops and
an electric field largely confined to the thin gaps. For measurements above 1.5 GHz, the
4-loop 3-gap design in Fig. 1(a) allows the simultaneous probe of two resonant modes [2];
measurements between 900 MHz and 1.5 GHz used a 2-loop 1-gap design. The resonant
frequencies can be tuned by a factor
∼
4 by varying the gap capacitance, via partial or
complete filling with pieces of sapphire wafer, thereby increasing the dielectric constant.
Each resonator was fabricated from a single block of oxygen-free high-conductivity
(OFHC) copper using wire electrical-discharge machining (EDM) to eliminate potential
losses arising from seams in two-piece designs. The resonator, contained inside an OFHC
copper shield, was attached to the cold finger of a helium dilution refrigerator. Stub anten-
nas attached to stainless-steel semirigid coaxial cables provide transmit and receive ports.
The transmission coefficient,
S
21
, was measured using a vector network analyzer (Tektronix
TTR506A). The measurements presented were taken at an incident power level of approx-
imately 1
μ
W (-30 dBm) at the resonator, for which sample heating was negligible and
the sample was demonstrated—by varying power levels—to be well into the linear response
regime.
Static magnetic fields were supplied using an 8 T superconducting solenoid and a home-
built superconducting Helmholtz pair oriented parallel to the Ising axis, allowing alignment
of the applied transverse field to within 0
.
5
◦
of the crystalline
a
-axis. The LiHoF
4
sample
used was a single crystal measuring 1
.
8
×
2
.
5
×
2
.
0 mm
3
. As noted in the text, the resonator
and sample were oriented with the AC probe field along the Ising
z
-axis, a solenoid along
the transverse
x
-axis, and a split coil along the
z
-axis, allowing us to measure
χ
zz
.
1
FIG. 1: Low-Energy Electronuclear Spectroscopy in LiHoF
4
. (a) Multi-mode loop-gap resonator
for generating ac magnetic fields at frequencies ranging from 900 MHz to 4.5 GHz (3 to 19
μ
eV).
Magnetic field lines join the circular loops to the square sample chamber. The corresponding
electric fields (not shown) are largely confined to the narrow gaps, reducing dielectric heating of
the sample. The lowest-frequency magnetic mode is illustrated here. (b) Magnified view of the
resonator sample chamber, showing directions of ac and dc magnetic fields with respect to the
crystal axes of the sample. The Ising axis is the crystallographic
c
-axis.
2
II. THEORETICAL METHODS
In what follows we neglect the effect of phonons in the system; these appear to have a
negligible effect at the temperatures and power levels used in the experiment. The effective
low-energy Hamiltonian for pure LiHoF
4
, under these circumstances, has been discussed
extensively in previous work [3–5]. We use the description and parameter values given in
Ref. [4].
A. Finite Temperature RPA
To treat experiments at finite temperature we generalize the Random Phase Approxi-
mation (RPA) results in Ref. [4], to take account of (i) finite temperature
T
, and (ii) the
presence of a net longitudinal field
B
z
.
Only a very small part (a fraction
γ
=
g
n
μ
n
/gμ
B
<<
1) of the AC response signal
comes directly from the nuclear spins, so we drop this part. The longitudinal dynamic
response of the Ising spins is then
χ
zz
(
k
,ω
) =
−
βG
(
k
,iω
r
→
w
+
i
0
+
), where
β
is the
inverse temperature,
G
(
k
,iω
r
) is the connected imaginary time correlation function of the
Ising spins, written at Bose Matsubara frequency
ω
r
= 2
πr/β
, and wave vector
k
; and
J
z
=
C
zz
τ
z
relates the
Ho
spin operator
J
z
to the spin-
1
2
Ising operator
τ
z
,. In the RPA
G
(
k
,iω
r
) =
g
(
iω
r
)
1 +
βV
k
g
(
iω
r
)
(1)
where
V
k
is the Fourier transform to momentum space of the effective interaction
V
zz
ij
be-
tween Ising spin operators
τ
z
i
and
τ
z
j
. The ‘mean field’
g
(
iω
r
) are connected imaginary time
correlation functions, with
τ
the imaginary time, so that
g
(
τ
) =
−
〈
T
τ
δJ
z
(
τ
)
δJ
z
(0)
〉
0
,
(2)
in which the time ordered thermal average
〈···〉
0
is taken with respect to the mean field
Hamiltonian for a single site, and
δJ
z
=
J
z
−〈
J
z
〉
0
.
We write mean field single ion electronuclear eigenstates
{|
n
〉}
, so that
H
MF
=
∑
n
E
n
|
n
〉〈
n
|
. We also write their differences
E
nm
=
E
n
−
E
m
, and write matrix elements of
the Ising spin operator as
c
mn
=
〈
m
|
τ
z
|
n
〉
0
. Finally, we write mean field population factors
as
p
m
=
e
−
βE
m
/Z
0
and
Z
0
=
∑
m
e
−
βE
m
, with diffrences
p
mn
=
p
m
−
p
n
.
3
Then the inelastic component
̃
G
(
k
,iω
r
) of the RPA Green’s function, written in terms of
the mean field energy levels and matrix elements, is found to be
̃
G
(
k
,iω
r
) =
−
1
β
C
2
zz
∑
n>m
|
c
mn
|
2
p
mn
2
E
nm
∏
t>s
6
=
nm
(
E
2
ts
−
(
iω
r
)
2
)
∏
n>m
(
E
2
nm
−
(
iω
r
)
2
)
−
V
k
C
2
zz
∑
n>m
|
c
mn
|
2
p
mn
2
E
nm
∏
t>s
6
=
mn
(
E
2
ts
−
(
iω
r
)
2
)
,
(3)
At
T
= 0, we have
p
1
n
= 1, and the populations of all the excited states are zero. We then
recover the results of ref. [4]; the poles of the Green’s function yield the zero temperature
RPA modes of the system
{
E
p
k
}
. The modes split into 2 ‘bands’, with 8 modes in one and
7 modes in the other (see figure in main text). A mode splits off from the upper band at a
field
H
x
∼
3
T
, and drops in energy until
H
x
=
H
C
; at higher fields it rises again in energy.
In the main text we have identified this mode with the excitation seen in prior neutron
scattering experiments [6].
At finite
T
, there are 120 possible modes coming from the poles of (3), and finding them
numerically is no longer straightforward. We may simplify the task by truncating the mean
field Green’s function, considering only transitions between a subset of the mean field modes
which lie below a specified threshold. We write the inelastic component of the mean field
Green’s function as
̃
g
(
z
) =
̃
g
Λ
(
z
) +
̃
g
(
z
), where
̃
g
Λ
(
z
) =
−
C
2
zz
β
∑
n>m
∈
Λ
|
c
mn
|
2
p
mn
2
E
nm
E
2
nm
−
z
2
(4)
contains the dominant contributions to the MF Green’s function, and
̃
g
(
z
) contains terms
such that
∣
∣
∣
∣
|
c
mn
|
2
p
mn
β
2
E
nm
E
2
nm
−
z
2
∣
∣
∣
∣
< .
(5)
The inelastic part of the RPA Green’s function is then
̃
G
(
k
,z
) =
̃
g
Λ
(
z
) +
̃
g
(
z
)
1 +
βV
k
[
̃
g
Λ
(
z
) +
̃
g
(
z
)]
.
(6)
The approximate RPA modes of the system follow from the zeros of 1 +
βV
k
̃
g
Λ
(
z
), which is
a polynomial in
z
with a lower degree than in the original Green’s function. This procedure
accurately captures the relevant modes, and eliminates much of the numerical noise present
when the full degree 120 polynomial is factored.
4
In LiHoF
4
we can write [4] the inter-electronic spin interaction in the form
V
k
=
J
D
D
zz
k
−
J
nn
(
k
), where where
D
zz
k
is the usual dipole sum,
J
D
=
μ
0
(
gμ
B
)
2
/
4
π
, and
J
nn
(
k
) is the
superexchange interaction between nearest neighbour
Ho
ions. In LiHoF
4
there are four
spins per unit cell (volume
V
cell
=
a
2
c
, where
a
= 5
.
175
̊
A, and
c
= 10
.
75
̊
A. This gives a spin
density
ρ
s
= 1
.
3894
×
10
28
m
−
3
, and a magnetization
m
=
ρ
s
m
i
=
ρ
s
gμ
B
〈
J
i
〉
. We can then
write the dipolar energy in the form
J
D
ρ
s
= 13
.
5
mK
. The antiferromagnetic superexchange
term
J
nn
= 1
.
16
mK
. At zero wavevector the interaction is
V
0
=
J
D
D
zz
0
−
4
J
nn
where the
zero wavevector component of the dipole sum in SI units is
D
zz
0
ρ
s
=
4
π
3
+
λ
dip
−
4
πN
z
.
(7)
The demagnetization factor ranges between
N
z
= 0 for a needle-shaped sample and
N
z
= 1
for a flat disc. Direct summation over the LiHoF
4
lattice yields a lattice correction of
λ
dip
= 1
.
664.
0
1
2
3
4
5
6
7
8
B
x
(T)
0
5
10
15
20
25
Spectral Weight
2
2.5
3
3.5
4
0
0.05
0.1
0.15
0.2
0.25
0.3
4.8
5
5.2
5.4
5.6
5.8
6
B
x
(T)
-5
-4
-3
-2
-1
0
1
2
log[Spectral Weight]
FIG. 2: Spectral weights of the four lowest RPA modes of a long thin cylinder of LiHoF
4
at 50mK.
The spectral weights are obtained using the truncation procedure discussed in the text. The inset
to the figure on the left shows detail of the region in which the RPA modes cross. The dominant
spectral weight is transferred from the lowest (soft) mode to higher lying excitations. On the right,
we plot the logarithm of the spectral weights in the vicinity of the phase transition. We see that
the spectral weight carried by the higher lying excitations is several orders of magnitude less than
the spectral weight carried by the soft mode.
With all these results in mind, we can determine numerically the spectral weights of
the different electronuclear modes, using the truncation procedure discussed in eqtns. (4)-
(6) above. Let us write the dynamic susceptibility (ie., the electronuclear spin fluctuation
5
propagator), given above as
χ
zz
(
k
,ω
) =
−
βG
(
k
,iω
r
→
w
+
i
0
+
), in the form
χ
(
k
,z
6
= 0) =
∑
p
A
m
k
[
1
z
+
E
m
k
−
1
z
−
E
m
k
]
,
(8)
where the
{
E
m
k
}
are the electronuclear mode energies, and their spectral weights are
A
m
k
=
χ
0
(
E
m
k
)
∏
q>p
[
E
2
qp
−
(
E
m
k
)
2
]
2
E
m
k
∏
s
6
=
m
[(
E
s
k
)
2
−
(
E
m
k
)
2
]
.
(9)
The spectral weights of the lowest few modes resulting from this procedure are then shown
in Fig. 2. Note the divergence in the spectral weight of the soft mode as one approaches
the critical point.
B. Domain Structure and Susceptibility
If the LiHoF
4
crystal were uniformly magnetized along ˆ
z
, then in a single domain ellip-
soidal sample we could write a local field
h
loc
=
h
a
+
h
dip
+
h
ex
, with externally applied field
h
a
, dipolar field
h
dip
, and exchange field
h
ex
[7]. The dipolar field is
h
z
dip
=
1
4
π
[
4
π
3
+
λ
dip
]
m
z
−
N
z
m
z
(10)
in SI units; here
h
D
=
N
z
m
z
is the demagnetization field. The longitudinal component of
the exchange field is
h
z
ex
=
λ
ex
m
z
. The local field is related to the longitudinal mean field
acting on each electronic holmium spin by
B
z
0
=
gμ
B
μ
0
h
z
loc
, from which it follows that
λ
ex
=
−
4
J
nn
4
πρ
s
J
D
=
−
2
.
7317
×
10
−
2
.
(11)
The shape independent component of the dipolar field is roughly 17 times larger than the
exchange field.
In the RPA, the uniform spin susceptibility is
χ
zz
J
=
χ
zz
0
,J
1
−
V
0
χ
zz
0
,J
,
(12)
where
χ
zz
0
,J
is the mean field spin susceptibility. In a uniformly magnetized ellipsoid we have
χ
zz
=
[
[
χ
zz
0
]
−
1
−
Φ
]
−
1
,
(13)
6
where
χ
zz
0
= 4
πJ
D
ρ
s
χ
zz
0
,J
and Φ =
V
0
/
(4
πJ
D
ρ
s
) = Φ
0
−
N
z
.; the component of Φ depending
only on the lattice is
Φ
0
=
1
4
π
[
4
π
3
+
λ
dip
]
+
λ
ex
= 0
.
4385
.
(14)
Because
λ
ex
is so small, it is very easy for the system to form domain walls; we will have
many small domains, with an alternating pattern
m
z
(
r
) =
±
m
z
, unless these are removed
by a strong longitudinal field
H
z
. In general the magnetization
M
(
r
) is expected to be
non-uniform, with local field
B
(
r
) given by
B
(
r
) =
μ
0
4
π
∇
r
×
∫
∇
r
′
×
M
(
r
′
)
|
r
−
r
′
|
d
3
r
′
(15)
+
μ
0
4
π
∇
r
×
∮
S
M
(
r
′
)
×
n
′
|
r
−
r
′
|
da
′
.
Any strong field inhomogeneity in the experiment would make the lower energy electronu-
clear modes unobservable, because of their rapid dependence on
B
z
. However, as noted in
the text, a proliferation of small domains actually homogenizes the field except near the
edges, and we can again write a local relationship between
B
(
r
) and
M
(
r
). In regions where
B
z
is homogeneous, with mean magnetization
M
z
=
V m
z
, where
V
= (
V
m
+
z
−
V
m
−
z
)
/V
is
the relative difference in volume between the up and down domains, the average demag-
netization field present in the sample is
h
z
D
=
N
eff
z
m
z
, where the effective
N
eff
z
= 0 if
M
z
= 0
Thus, to account for the domain structure of the material we use the substitution
N
z
→
N
eff
z
. In mean field theory, the total longitudinal field acting on the
J
z
i
=
C
zz
τ
z
i
operators is
B
z
0
=
gμ
B
B
z
a
+
V
0
〈
J
z
〉
. We define the longitudinal mean field via
B
z
MF
=
4
πJ
D
ρ
s
(Φ
0
−
N
eff
z
)
gμ
B
〈
J
z
〉
,
(16)
and we subtract off the shape-independent part to get the average longitudinal demagnetiz-
ing field
B
z
D
=
−
4
πJ
D
ρ
s
N
eff
z
〈
J
z
〉
gμ
B
.
(17)
In a system of stripe domains, we have
B
z
D
=
−
μ
0
M
z
=
−
μ
0
V m
z
, so that
N
eff
z
=
V
. If
the domain wall energies are negligible, then the average demagnetization field is equal and
opposite the applied field,
B
z
a
=
−
B
z
D
. This result no longer holds exactly when the small
energy of the domain walls is taken into account (see main text).
7
4.60
4.55
4.50
4.45
H
x
(T)
60
40
20
0
H
z
(mT)
FIG. 3: Location of the soft-mode peak for large
H
z
Returning to the collective mode spectrum, we see that the RPA modes will now be
defined by the poles of the inelastic component of the Green’s function in eqtn. (1); where
now, in the
k
= 0 limit, we use
V
0
→
4
πJ
D
ρ
s
(Φ
0
−
N
eff
z
) in the denominator of eqtns. (1)
and (3).
Experimentally, for
H
l
comparable to or larger than the saturation field of order 40 mT,
the suppression of domain formation and the resultant demagnetization field act in opposi-
tion to the applied field, inducing a non-monotonic behavior in the location of the soft-mode
peak (Figure 3).
C. Transmission Coefficient
The interaction between the linearly polarized cavity photons and the spins is given by
H
int
=
i
~
α
(
a
−
a
†
)
∑
i
τ
z
i
→
i
~
∑
m
g
m
(
b
m
+
b
†
m
)(
a
†
−
a
)
,
(18)
where in the final line we have written the spin-photon interaction in terms of the magnon
modes and a renormalized coupling
g
2
m
∝
α
2
A
m
=
g
2
A
m
.
(19)
The renormalized coupling follows from a microscopic analysis of the spin-photon Hamilto-
nian. This will be the subject of a subsequent paper; here we note that the temperature
dependent proportionality constant in the renormalized coupling is of order one. The spec-
tral weights of the magnon modes have been incorporated into the coupling; here
A
m
is
8
the
k
→
0 limit of the spectral function
A
m
k
given in (9) above (for the zero mode, we see
that the coupling diverges at the critical point). One would expect this to lead to a strong
avoided level crossing in the resonator transmission spectrum. In the experiment, there is
no avoided level crossing when the soft mode is degenerate with the cavity mode; instead,
a weak resonance in the linewidth of the cavity mode is observed. This can be explained
by incorporating an oscillator bath environment into the model, describing, for example,
thermal photons.
We consider an oscillator bath of Caldeira-Leggett form [10, 11], where the bath compo-
nent of the Hamiltonian is given by
H
bath
=
∑
qm
~
g
qm
(
a
q
+
a
†
q
)(
b
m
+
b
†
m
) +
∑
q
~
ω
q
a
†
q
a
q
+
∑
qm
~
g
2
qm
ω
m
(
a
q
+
a
†
q
)
2
,
(20)
and the damping of the
m
th
magnon mode due to the bath is given by
γ
m
= lim
→
0
+
∑
q
g
2
qm
ω
q
−
iω
2
ω
2
q
−
ω
2
−
iω
(21)
What’s important here is that the dampings of the magnon modes are proportional to the
square of the coupling strengths; the damping of the soft mode will then diverge at the phase
transition along with the spectral weight of the mode.
To find the measured transmission coefficient
S
21
(
ω
) for the cavity, we must determine
the spectrum of electronuclear magnetopolariton modes, which themselves arise via coupling
between cavity photons and electronuclear magnons. We write the total Hamiltonian of the
coupled system as
H
=
H
s
+
H
γ
+
H
int
,
(22)
where the spectrum of the LiHoF
4
Hamiltonian
H
s
is described using the RPA above, and
the photon Hamiltonian includes the resonator mode as well as a bath
H
γ
=
ω
r
a
†
a
+
∑
q
ω
q
a
†
q
a
q
.
(23)
The operator
a
†
creates resonant cavity photons, while
a
†
q
creates a non-resonant bath mode
labelled by index
q
. This Hamiltonian describes free photons in the cavity; one can add small
damping coefficients Γ
r
,
{
Γ
q
}
to the propagator for these photons, to describe cavity surface
and other damping processes. The interaction between the magnons and the cavity photons
9
is given in eqtn. (18), and the interaction between the magnons and the bath photons takes
on the Caldeira-Leggett form in eqtn. (20).
One can now treat the coupled electronuclear magnon-photon system as a set of coupled
oscillators; this can be done in various ways [8, 9]. In the present case the result for the
resonant photon transmission coefficient can be written as
S
21
(
ω
)
∝
1
(
ω
−
ω
r
) +
i
Γ
r
/
2 +
g
2
χ
(
ω
)
,
(24)
in which Γ
r
is the photon damping coming from the cavity walls, noted above, and where
we write the
k
= 0 magnetic susceptibility in the form (compare eqtn. (8) above):
χ
(
ω
) =
∑
m
A
m
[
1
ω
+
E
m
+
iγ
m
(
ω
)
/
2
−
1
ω
−
E
m
+
iγ
m
(
ω
)
/
2
]
(25)
in which we have added electronuclear magnon damping coefficients
γ
m
(
ω
), given in eqtn.
21, coming from the magnon-photon coupling.
At frequencies relevant to the resonator, we have
χ
(
ω
)
≈ −
∑
m
[
A
m
(
ω
−
E
m
)
(
ω
−
E
m
)
2
+ (
γ
m
/
2)
2
−
i
A
m
(
γ
m
/
2)
(
ω
−
E
m
)
2
+ (
γ
m
/
2)
2
]
,
(26)
and the transmission function is given by
S
21
(
ω
)
∝
1
ω
−
ω
r
−
∑
m
̄
g
2
m
(
ω
−
E
m
)
(
ω
−
E
m
)
2
+(
γ
m
/
2)
2
+
i
[
Γ
r
/
2 +
∑
m
̄
g
2
m
γ
m
/
2
(
ω
−
E
m
)
2
+(
γ
m
/
2)
2
]
,
(27)
where the renormalized coupling between the resonant photons and the
m
-th magnon mode
is
g
2
m
=
g
2
A
m
.
We see that the effect of the divergence at the critical point in the zero mode spectral
weight
A
0
, and in the coupling ̄
g
0
between the zero mode and the resonant photons, is
cancelled in the expression for
S
21
(
ω
). Thus, as we approach the critical point, the correction
to the resonant frequency
ω
r
coming from the coupling to the
m
= 0 zero mode is
∼
|
̄
g
0
/γ
o
|
2
∼
1
/
|
̄
g
0
|
2
→
0, while the correction to the damping Γ
r
to the resonant mode coming
from the coupling to the electronuclear zero mode is
∼ |
̄
g
0
|
2
/γ
0
∼
constant. In both cases
the divergence in the coupling has been cancelled by the divergence in the damping. The
result is that for the zero mode, rather than an avoided level crossing, we see a single mode
that exhibits a resonance when the soft mode frequency is tuned to the cavity resonance.
10
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
B
x
(T)
0
2
4
6
8
10
12
Spectral Weight
B
z
=1nT
B
z
=1mT
B
z
=3mT
B
z
=7mT
B
z
=14mT
B
z
=60mT
FIG. 4: Spectral of a needle shaped sample of LiHoF
4
at 25mK for a series of longitudinal fields.
We can also show that the weak effective coupling between the magnon and photon
modes persists in the presence of a longitudinal field - this must be verified, since it is what
is seen in the experiments. In Figure 4, we show the dependence of the spectral weight of
the soft mode on a longitudinal applied field. As expected, the longitudinal field gaps the
soft mode and leads to a reduction of its spectral weight. However, in fields of up to 60mT
the spectral weight of the soft mode is still very large, and exceeds the spectral weights of
the excited states by orders of magnitude. Thus the damping of the soft mode, which goes
like the square of the coupling strength, will remain large in a longitudinal field, so that
the cancellation mechanism still applies. As a result the weak resonance observed in the
experiments, and described by eqtn. (27), will not be strongly affected by this field.
D. Relation to Electron-photon Decoupling
It is interesting to ask whether the cancellation mechanism we have just discussed has
anything to do with the ‘light-matter decoupling’ mechanism argued to exist [12] in cavities
where light is strongly coupled to matter. In both mechanisms a strong coupling to photons
leads to a strong reduction, rather than an enhancement, of a signal. In what follows we
compare and contrast these two mechanisms.
Consider first a system of mobile charged particles in an electromagnetic field. For a
single atomic dipole transition, making use of the TRK sum rule, one gets the the atomic
11
light matter Hamiltonian, or Dicke model, of form
H
LM
=
~
ω
0
b
†
b
+
~
ω
r
a
†
a
+
~
g
(
b
+
b
†
)(
a
+
a
†
) +
~
g
2
ω
0
A
2
,
(28)
where
ω
r
is the resonator frequency, and
A
= (
a
+
a
†
). The atomic dipole transition has
frequency
ω
0
, and we make use bosonic creation and annihilation operators for the matter
modes. We note the
A
2
(diamagnetic) term, which comes from squaring the canonical
momentum of the charge carriers. This term has excited some controversy [13]; it forestalls
a superradiant phase transition in the model [14].
In a metallic planar cavity enclosing a two dimensional wall of dipoles, De Liberato has
shown [12] that the diamagnetic term in the atomic light-matter Hamiltonian leads, under
strong coupling, to a decoupling of the light and matter modes. The photon field is then
localized away from the dipoles and the polaritonic mode operators have either a distinct
light or matter character.
Our resonator spin-photon system is different because there are no mobile charge carriers,
and the usual diamagnetic term is not present. Despite this, using a canonical transformation
that swaps the effective position and momentum operators of the photon field [11, 15], we
may arrive at a similar Hamiltonian in which the roles of the light and the matter are
reversed. In terms of the position and momentum operators
q
=
√
~
2
mω
(
a
†
+
a
)
p
=
i
√
~
mω
2
(
a
†
−
a
)
,
(29)
we may write the Lagrangian for the interaction as
L
int
=
−
H
int
= ̇
q
∑
m
c
m
X
m
.
(30)
The canonical momentum associated with the photons is then
P
=
p
+
∑
m
c
m
X
m
, and the
Hamiltonian in terms of the position and momentum operators is
H
=
∑
m
[
P
2
m
2
M
m
+
1
2
M
m
ω
2
m
X
2
m
]
+
[
(
p
−
∑
m
c
m
X
m
)
2
2
m
γ
+
1
2
m
γ
ω
2
r
q
2
]
.
(31)
We then swap the position and momentum coordinates of the photon field,
p
→
m
γ
ω
r
q
and
q
→
p/
(
m
γ
ω
r
). Restricting our attention to the soft mode (m=0), this may be written in
quantized form as
H
=
~
ω
r
a
†
a
+
~
ω
0
b
†
0
b
0
+
~
g
0
(
a
†
+
a
)(
b
†
0
+
b
0
) +
~
g
2
0
ω
r
(
b
†
0
+
b
0
)
2
.
(32)
12
where the final term is a Caldeira-Leggett counter term, arising here as a natural consequence
of the effective
̂
x
̂
p
coupling between the magnons and photons. Comparing with eqtn. (28),
we see that the resonator spin-photon Hamiltonian is, in a sense, the inverse of the atomic
light-matter Hamiltonian. Thus, a canonical transformation on the Hamiltonian of the
resonator spin-photon system maps to the atomic light-matter (Dicke) Hamiltonian with
the roles of the light and matter reversed. In this mapping, the Caldeira-Leggett counter
term in the resonator spin-photon Hamiltonian corresponds to the diamagnetic photon term
in the atomic light matter system.
Whether or not light-matter decoupling occurs in our system of interest is not clear.
However, even in the absence of this decoupling, we have seen in section C above that
the damping due to an oscillator bath environment will lead to a cancellation between a
divergent coupling constant and a divergent mode damping. As already discussed, this
cancellation will close the avoided level crossing associated with strong coupling, leading to
a weak resonance, as seen in the experiment.
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