Subradiant states of quantum bits coupled to a one-dimensional waveguide
Andreas Albrecht,
1,
∗
Lo ̈ıc Henriet,
1,
∗
Ana Asenjo-Garcia,
2, 3
Paul B. Dieterle,
3, 4,
†
Oskar Painter,
3, 4
and Darrick E. Chang
1, 5
1
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain
2
Norman Bridge Laboratory of Physics MC12-33,
California Institute of Technology, Pasadena, CA 91125, USA
3
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
4
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125, USA
5
ICREA-Instituci ́o Catalana de Recerca i Estudis Avan ̧cats, 08015 Barcelona, Spain
(Dated: March 7, 2018)
The properties of coupled emitters can differ dramatically from those of their individual con-
stituents. Canonical examples include sub- and super-radiance, wherein the decay rate of a collective
excitation is reduced or enhanced due to correlated interactions with the environment. Here, we
systematically study the properties of collective excitations for regularly spaced arrays of quantum
emitters coupled to a one-dimensional (1D) waveguide. We find that, for low excitation numbers,
the modal properties are well-characterized by spin waves with a definite wavevector. Moreover,
the decay rate of the most subradiant modes obeys a universal scaling with a cubic suppression in
the number of emitters. Multi-excitation subradiant eigenstates can be built from fermionic com-
binations of single excitation eigenstates; such “fermionization” results in multiple excitations that
spatially repel one another. We put forward a method to efficiently create and measure such sub-
radiant states, which can be realized with superconducting qubits. These measurement protocols
probe both real-space correlations (using on-site dispersive readout) and temporal correlations in
the emitted field (using photon correlation techniques).
Superconducting qubits coupled to photons propagat-
ing in open transmission lines [1–3] offer a platform to
realize and investigate the fascinating world of quan-
tum light-matter interactions in one dimension – so-
called “waveguide quantum electrodynamics (QED)”[4–
10]. Such systems enable a number of exotic phenomena
that are difficult to observe or have no obvious analogue
in other settings, such as near-perfect reflection of light
from a single resonant qubit [1, 2, 4, 11–13] or the dy-
namical Casimir effect [14]. One particularly interesting
feature of these systems is that the interaction between
multiple qubits, mediated by photon absorption and re-
emission, is of infinite range, as the mediating photon is
unable to diffract energy into other directions in a one-
dimensional channel. This can give rise to strong col-
lective effects in multi-qubit systems [8, 15]. For exam-
ple, it has been observed that two qubits separated by a
substantial distance can exhibit super- or sub-radiance,
wherein a single collective excitation can decay at a rate
faster or slower than that of a single qubit alone [3].
The physics associated with collective effects in waveg-
uide QED has attracted growing interest, and there have
been a number of proposals that implicitly exploit sub-
and super-radiant emission to realize atomic mirrors [16],
photon Fock state synthesis [17], or quantum computa-
tion [18, 19]. However, the fundamental properties of the
∗
A. Albrecht and L. Henriet contributed equally to this work
†
Current address: Department of Physics, Harvard University,
Cambridge, MA 02138, USA
collective qubit excitations themselves, such as their spa-
tial character and decay spectrum, have been less sys-
tematically studied [20].
Here, we aim to provide a systematic description of
subradiant states in ordered arrays by using a spin-model
formalism, wherein emission and re-absorption of pho-
tons by qubits is exactly accounted for. Our study re-
veals a number of interesting characteristics. In particu-
lar, as the number of qubits
N
increases, we show that
the Liouvillian “gap” closes, i.e., there exists a smooth
distribution of decay rates associated with subradiant
states whose value approaches zero. Furthermore, we find
that the most subradiant multi-excitation states exhibit
“fermionic” correlations in that the excitations obey an
effective Pauli exclusion principle. We propose a realis-
tic experimental protocol to measure these exotic spatial
properties, and finally investigate the correlations in the
corresponding emitted field. Interestingly, similar behav-
ior can also be found in other settings, such as in arrays
of atoms in three-dimensional space [21], which suggests
a certain degree of “universality” to the phenomenon of
subradiance. Taken together, these results show that the
physics of subradiance is itself a rich many-body problem.
Spin model -
We consider
N
regularly-spaced two-level
transmon qubits [22] with ground and excited states
|
g
〉
,
|
e
〉
and resonance frequency
ω
eg
. The qubits are dipole
coupled to an open transmission line supporting a contin-
uum of left- and right-propagating modes with linear dis-
persion and velocity
v
, thus realizing a waveguide QED
setup [see Fig. 1 (a)]. Integrating out the quantum elec-
arXiv:1803.02115v1 [quant-ph] 6 Mar 2018
2
XY
Z
¡
1
D
j
e
i
j
g
i
d
(a)
(a)
-
1
-
0
.
5
0
0
.
5
1
kd/π
-
15
-
10
-
5
0
5
ln
(
Γ
k
/
Γ
1
D
)
(c)
-
1
-
0
.
5
0
0
.
5
1
kd/π
-
3
0
3
J
k
/
Γ
1
D
(b)
3
4
ln
N
-
15
-
12
-
9
ln (Γ
/
Γ
1
D
)
(b)
(c)
FIG. 1.
(a)
Schematic of planar transmon qubits capacitively
coupled to a coplanar waveguide with rate Γ
1D
. Photon-
mediated interactions couple the qubits together with an am-
plitude determined by the single-qubit emission rate Γ
1D
into
the waveguide, and a phase determined by the phase veloc-
ity of the transmission line and the distance between qubits.
Collective frequency shifts
(b)
and decay rates
(c)
for qubits
coupled through a waveguide with
k
1D
d/π
= 0
.
2. Blue cir-
cles correspond to the results for a finite system with
N
=
30 qubits. Black dotted lines correspond to
k
=
±
k
1D
. The
frequency shift for the infinite chain is denoted by the solid
line (see Appendix A). The inset in
(c)
shows the scaling of
the decay rate with qubit number Γ
/
Γ
1D
∼
N
−
3
for the 4
most subradiant states.
tromagnetic environment in the Markovian regime, one
finds that emission of photons into the waveguide leads
to cooperative emission and exchange-type interactions
between the qubits [16, 18, 21, 23, 24]. The dynamics of
the qubit density matrix
ρ
can be described by a mas-
ter equation of the form ̇
ρ
=
−
(
i/
̄
h
)[
H
eff
ρ
−
ρ
H
†
eff
] +
∑
m,n
Γ
m,n
σ
m
ge
ρσ
n
eg
, where the effective (non-Hermitian)
Hamiltonian reads [16, 23, 24]
H
eff
= ̄
h
N
∑
m,n
=1
(
J
m,n
−
i
Γ
m,n
2
)
σ
n
eg
σ
m
ge
,
(1)
with
J
m,n
= Γ
1D
sin(
k
1D
|
z
m
−
z
n
|
)
/
2 and Γ
m,n
=
Γ
1D
cos(
k
1D
|
z
m
−
z
n
|
) denoting the coherent and dis-
sipative interaction rates, respectively.
Here, Γ
1D
is
the single qubit emission rate into the transmission
line,
k
1D
=
ω
eg
/v
is the resonant wavevector,
σ
m
αβ
=
|
α
m
〉〈
β
m
|
denotes an operator acting on the internal
states
{
α,β
} ∈ {
g,e
}
of qubit
m
at position
z
m
=
md
, with
d
the inter-qubit distance.
The photonic
degrees of freedom can be recovered after solving the
qubit dynamics [16, 24].
In particular, the positive-
frequency component of the left- and right-going field
emitted by the qubits reads
ˆ
E
+
L/R
(
t
) =
ˆ
E
in
L/R
(
z
±
vt
) +
i(Γ
1D
/
2)
∑
N
n
=1
e
i
k
1D
|
z
L/R
−
z
n
|
ˆ
σ
n
ge
(
t
), where the field
ˆ
E
+
L
(
ˆ
E
+
R
) is measured directly beyond the first (last) qubit,
at position
z
L
=
d
(
z
R
=
Nd
). Here,
ˆ
E
in
L/R
denotes the
(quantum mechanical) input field. For a given number
of excitations, the effective Hamiltonian
H
eff
defines a
complex symmetric matrix that can be diagonalized to
find collective qubit modes with corresponding complex
eigenvalues defining their resonance frequencies (relative
to
ω
eg
) and decay rates.
In the simple case of
k
1D
d
= 2
nπ
[(2
n
+ 1)
π
], with
n
an integer, the coherent qubit-qubit interactions
J
m,n
vanish and the effective Hamiltonian is purely dissipa-
tive,
H
eff
=
−
i
( ̄
h
Γ
1D
N/
2)
S
†
k
=0[
k
=
π/d
]
S
k
=0[
k
=
π/d
]
, where
S
†
k
= 1
/
√
N
∑
n
e
ikz
n
σ
n
eg
. The
k
= 0 [
k
=
π/d
] col-
lective mode defined by
S
†
k
is coupled superradiantly to
the waveguide at a rate
N
Γ
1D
, while all other modes are
dark, with decay rate Γ = 0. This realizes the ideal Dicke
model of superradiance [25]. Within the setting of a 1D
waveguide, it has also been shown that this configura-
tion has interesting quantum optical functionality. For
example, the qubits act as a nearly perfect mirror for
near-resonant photons [16, 26, 27] and can generate ar-
bitrary photon Fock states on demand [28]. Away from
this spacing, the system becomes multimode [20]; below,
we provide the first comprehensive look at these modal
properties.
Single-excitation modes -
Numerical diagonalization
of
H
eff
in the single-excitation sector gives
N
distinct
eigenstates
|
ψ
(1)
ξ
〉
=
S
†
ξ
|
g
〉
⊗
N
=
∑
n
c
ξ
n
|
e
n
〉
that obey
H
eff
|
ψ
(1)
ξ
〉
= ̄
h
(
J
ξ
−
i
Γ
ξ
/
2)
|
ψ
(1)
ξ
〉
, where
J
ξ
and Γ
ξ
repre-
sent the frequency shift and decay rate associated with
|
ψ
(1)
ξ
〉
.
Here,
|
e
n
〉
=
σ
n
eg
|
g
〉
⊗
N
corresponds to hav-
ing atom
n
excited, with all other atoms being in the
ground state. We obtain a broad distribution of de-
cay rates defining superradiant (Γ
ξ
>
Γ
1D
) and subra-
diant (Γ
ξ
<
Γ
1D
) states. Ordering the eigenstates by
increasing decay rates, i.e., from
ξ
= 1 for the most sub-
radiant to
ξ
=
N
for the most radiant, we find that
strongly subradiant modes are characterized by a decay
rate Γ
ξ
Γ
1D
that is suppressed with qubit number as
Γ
ξ
/
Γ
1D
∝
ξ
2
/N
3
, similar to the case of a 1D chain of
atoms in free space [21]. Thus, in the thermodynamic
limit, the spectrum of decay rates becomes smooth and
the “gap” of minimum decay rate closes. For an infi-
nite chain, the eigenstates of
H
eff
take the form of Bloch
spin waves
|
ψ
(1)
k
〉
=
S
†
k
|
g
〉
⊗
N
, with
k
being a quantized
wavevector within the first Brillouin zone (
|
k
|≤
π/d
). In
such states, the single excitation is delocalized and coher-
ently shared among all the qubits. For finite chains, the
eigenstates are instead described in momentum space by
a wavepacket with a narrow distribution of wavevectors
around a dominant wavevector
k
, which can thus serve
as an unambiguous label of states. In Fig. 1 (b) and (c),
we show the distribution of frequency shifts
J
k
and de-
cay rates Γ
k
with
k
for
N
=30 qubits and
k
1D
d/π
= 0
.
2.
We find large decay rates and frequency shifts for eigen-
states with wavevectors
k
close to the resonant wavevec-
3
tors
±
k
1D
. Conversely, we obtain decay rate minima and
small frequency shifts around
kd
= 0 and
|
k
|
d
=
π
. For
k
1D
d >
0
.
5
π
(
k
1D
d <
0
.
5
π
), wavevectors
k d
= 0 form
the global (local) and
|
k
|
d
=
π
the local (global) de-
cay rate minimum, respectively. The
k
-dependence can
be understood by considering the infinite lattice limit,
where the qubits and waveguide generally hybridize to
form two lossless polariton bands (see Appendix A). The
dispersion relation is plotted in Fig. 1 (b) as a solid line,
and matches well with the frequency shifts obtained for
a finite system. For a finite system, the polaritons with
wavevector around
k
∼
k
1D
(
k
= 0
,π/d
) are most (least)
impedance-matched at their boundaries to the dispersion
relation of propagating photons in the bare waveguide,
thus giving rise to super-radiant (sub-radiant) emission.
Multi-excitation modes -
A quadratic bosonic Hamilto-
nian would enable us to easily find the multi-excitation
eigenstates of
H
eff
from the single-excitation sector re-
sults. Here, however, the spin nature prevents multiple
excitations of the same qubit. Specifically, two-excitation
states
|
φ
(2)
ξ
〉
=
N
2
(
S
†
ξ
)
2
|
g
〉
⊗
N
, with
N
2
a normaliza-
tion factor, are not eigenstates of the effective Hamil-
tonian (1). Moreover, in the case of an index
ξ
corre-
sponding to a subradiant single-excitation mode, the ini-
tial decay rate of
|
φ
(2)
ξ
〉
is significantly greater than twice
the single-excitation decay rate Γ
ξ
. This discrepancy can
be explained by noting that the spatial profile of
|
φ
(2)
ξ
〉
,
i.e. the probability
p
m,n
=
|〈
e
m
,e
n
|
φ
(2)
ξ
〉|
2
for qubits
m
and
n
to be excited, contains a sharp cut along the di-
agonal
m
=
n
(
p
m,m
≡
0). In reciprocal space, this
corresponds to a broad distribution of wavevector com-
ponents, including radiant contributions responsible for
an increased decay rate. From this qualitative discussion,
one expects the excitations forming a multi-excitation
subradiant eigenstate to be smoothly repelled from one
another.
We numerically find the existence of two-excitation
subradiant eigenstates
|
ψ
(2)
ξ
〉
, with a decay rate scaling as
Γ
(2)
ξ
/
Γ
1D
∼
N
−
3
– as in the single-excitation sector – for
the most subradiant eigenstates. These eigenstates reveal
interesting properties in real and momentum space. One
example is illustrated in the top of Fig. 2 (a), where we
consider the most subradiant two-excitation wavefunc-
tion
|
ψ
(2)
ξ
=1
〉
=
∑
m<n
c
mn
|
e
m
,e
n
〉
for
k
1D
d/π
= 0
.
2 and
N
= 20 qubits, and plot both the probability amplitude
|
c
mn
|
2
in real space (left) and
|
c
k
1
,k
2
|
2
in reciprocal space
(right). Here,
c
k
1
,k
2
refers to the two-dimensional dis-
crete Fourier transform of
c
m,n
. In real space, one sees
that the maximum in
|
c
mn
|
2
occurs for
m
≈
15
,n
≈
6, re-
vealing a tendency for the excitations to both repel each
other, and avoid the system boundaries where they can
be radiated. At the same time, in momentum space, a
peak occurs around
k
1
,
2
d/π
=
±
1, coinciding with the
dominant wavevectors
kd/π
≈ ±
1 of the most subradi-
ant single-excitation states [Fig. 1 (c)]. A natural two-
excitation wavefunction ansatz that realizes both the
real- and momentum-space properties consists of taking
5
10
15
20
5
10
15
20
0
0.005
0.01
0.015
0.02
-1
-0.5
0
0.5
-1
-0.5
0
0.5
0
0.5
1
-1
-0.5
0
0.5
-1
-0.5
0
0.5
0
0.05
0.1
0.15
5
10
15
20
5
10
15
20
0
0.01
0.02
0.03
0.04
n
n
n
m
m
m
(b)
(a)
k
1
d/
⇡
k
1
d/
⇡
k
1
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
k
1
d/
⇡
k
1
d/
⇡
k
1
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
0
1
a.u.
n
n
n
m
m
m
F
F
F
k
1
d/
⇡
k
1
d/
⇡
k
1
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
k
2
d/
⇡
0
1
a.u.
FIG. 2.
(a)
Probability amplitude
|
c
mn
|
2
in real space (left)
and in reciprocal space
|
c
k
1
,k
2
|
2
of the wavefunction profile of
the most subradiant two-excitation eigenstate for
k
1D
d/π
=
0
.
2 (top) and
k
1D
d/π
= 0
.
5 (bottom), for
N
= 20 qubits.
c
mn
is only defined for
m < n
, but for visual appeal here we
symmetrize the plot by taking
c
mn
=
c
nm
. Dotted dashed
circles are a guide to the eye to highlight the positions of
the maximum momentum components.
(b)
Fidelity between
the exact two-excitation eigenstate of the Hamiltonian with
quasi-momentum values (
k
1
,
k
2
) and the fermionized ansatz
for
N
=50 qubits and
k
1D
d/π
= 0
.
2.
an anti-symmetric combination of single-excitation eigen-
states, which enforces a Pauli-like exclusion (“fermion-
ization”). In particular, starting from the wavefunctions
of the two most subradiant single-excitation eigenstates,
we find that we can construct an accurate approxima-
tion of the most subradiant two-excitation eigenstate,
|
ψ
(
F
)
ξ
=1
〉
=
N
∑
m<n
(
c
ξ
=1
m
c
ξ
=2
n
−
c
ξ
=2
m
c
ξ
=1
n
)
|
e
m
e
n
〉
, with
N
a normalization factor. For
k
1D
d
mod
π
6
= 0 and
k
1D
d/π
away from 0
.
5, the
ξ
= 1
,
2 single-excitation
states have dominant wavevectors (
k
1
,k
2
) near the global
decay rate minimum, e.g., at
k
=
π/d
for
k
1D
d/π
= 0
.
2.
For
k
1D
d/π
= 0
.
5, the fermionic ansatz also works well
to describe the most subradiant two-excitation eigenstate
(bottom of Fig. 2 (a)). In this case, it is built from the
most subradiant single-excitation eigenstates
k
1
=
π/d
and
k
2
= 0 (degenerate in decay rate), and results in
the checkerboard pattern seen in the plot. To more gen-
erally examine the accuracy of the ansatz, we take the
two-dimensional Fourier transform of each two-excitation
eigenstate, and unambiguously assign a label of quasi-
momentum indices (
k
1
,k
2
) to each state
|
ψ
(2)
(
k
1
,k
2
)
〉
based
upon where the Fourier transform is peaked. We then
compute the overlap fidelity
F
=
|〈
ψ
(
F
)
(
k
1
,k
2
)
|
ψ
(2)
(
k
1
,k
2
)
〉|
2
between the exact state and the fermionic ansatz com-
posed of the single-excitation eigenstates (
k
1
,k
2
). As il-
lustrated in Fig. 2 (b), the ansatz works well when the
two single-excitation states composing the eigenstate are
strongly subradiant. In this case we find that the infi-
delity 1
−F
scales with the qubit number as 1
/N
2
(see
Appendix B 1). In the thermodynamic limit
N
→∞
, we
find that the decay rate of such subradiant “fermionized”
eigenstates approaches the sum of the decay rates of the
single-excitation states they are composed of (see Ap-
pendix B 2). The conclusions made about the subradiant
decay rate scaling and their fermionic nature – exem-
4
plified here for two-excitations – are found to extend to
higher excitation numbers provided that the density of
excitations is dilute:
m
ex
N
.
0
10
20
30
0.4
0.6
0.8
1
(b)
a.u.
n
n
n
m
m
m
n
n
n
n
n
n
(a)
0
1
(i)
(ii)
(iii)
(iii)
(ii)
(i)
}
(2)
(
t
)
}
(2)
(
t
)
}
(2)
(
t
)