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Subradiant states of quantum bits coupled to a one-dimensional
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2019
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New J. Phys.
21
(
2019
)
025003
https:
//
doi.org
/
10.1088
/
1367-2630
/
ab0134
PAPER
Subradiant states of quantum bits coupled to a one-dimensional
waveguide
Andreas Albrecht
1
,
6
, Loïc Henriet
1
,
6
, Ana Asenjo-Garcia
1
,
2
,
3
,
8
, Paul B Dieterle
3
,
4
,
7
, Oskar Painter
3
,
4
and
Darrick E Chang
1
,
5
,
9
1
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, E-08860, Castelldefels
(
Barcelona
)
, Spain
2
Norman Bridge Laboratory of Physics MC12-33, California Institute of Technology, Pasadena, CA 91125, United States of America
3
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, United States of America
4
Kavli Nanoscience Institute and Thomas J Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA
91125, United States of America
5
ICREA-Institució Catalana de Recerca i Estudis Avançats, E-08015, Barcelona, Spain
6
A Albrecht and L Henriet contributed equally to this work.
7
Current address: Department of Physics, Harvard University, Cambridge, MA 02138, United States of America.
8
Current address: Department of Physics, Columbia University, New York, NY 10027, United States of America.
9
Author to whom any correspondence should be addressed.
E-mail:
darrick.chang@icfo.eu
Keywords:
waveguide QED, subradiance, superradiance, superconducting qubits
Abstract
The properties of coupled emitters can differ dramatically from those of their individual constituents.
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 waveguide. We
fi
nd that, for low excitation numbers, the modal properties are
well-characterized by spin waves with a de
fi
nite 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 combinations of single excitation
eigenstates; such
fermionization
results in multiple excitations that spatially repel one another. We
put forward a method to ef
fi
ciently create and measure such subradiant 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
fi
eld
(
using photon correlation
techniques
)
.
1. Introduction
Superconducting qubits coupled to photons propagating in open transmission lines
[
1
3
]
offer a platform to
realize and investigate the fascinating world of quantum light
matter interactions in one dimension
so-called
waveguide quantum electrodynamics
(
QED
)
[
4
11
]
. Such systems enable a number of exotic phenomena that
are dif
fi
cult to observe or have no obvious analog in other settings, such as near-perfect re
fl
ection of light from a
single resonant qubit
[
1
,
2
,
4
,
12
14
]
, or the dynamical Casimir effect
[
12
]
, and allow for the measurement of
quantum vacuum
fl
uctuations
[
13
]
. One particularly interesting feature of these systems is that the interaction
between multiple qubits, mediated by photon absorption and re-emission, is of in
fi
nite range. This can give rise
to strong collective effects in multi-qubit systems
[
8
,
17
,
18
]
. For example, 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 waveguide 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
[
14
]
, photon Fock state synthesis
[
15
]
, or quantum computation
[
16
,
17
]
. The fundamental properties of
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)
. Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft
the qubit modes themselves, such as their spatial character and decay spectrum, have been studied recently in the
classical single-excitation regime
[
18
]
.
Here, we aim to provide a systematic description of single- and multi-excitation subradiant states in ordered
arrays by using a spin-model formalism, wherein emission and re-absorption of photons by qubits is exactly
accounted for. Our study reveals a number of interesting characteristics. In particular, 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
fi
nd that the most subradiant
multi-excitation states exhibit
fermionic
correlations in that the excitations obey an effective Pauli exclusion
principle. These calculations parallel a similar investigation involving subradiant states of an ordered chain of
atoms in three-dimensional space
[
19
]
. The
fi
nding of similar properties suggests a certain degree of
universality
to the phenomenon of subradiance. Next, we propose a realistic experimental protocol to measure
these exotic spatial properties, and
fi
nally investigate the correlations in the corresponding emitted
fi
eld. Taken
together, these results show that the physics of subradiance is itself a rich many-body problem.
2. Eigenmodes of the atom-waveguide system and collective emission properties
2.1. Setup and spin model description
We consider
N
regularly-spaced two-level transmon qubits
[
20
]
with ground and excited states
ñ
g
,
ñ
e
and
resonance frequency
ω
eg
. The qubits are dipole coupled to an open transmission line supporting a continuum of
left- and right-propagating modes with linear dispersion and velocity
v
(
see
fi
gure
1
(
a
))
. Integrating out the
quantum electromagnetic environment in the Markovian regime, one
fi
nds that emission of photons into the
waveguide leads to cooperative emission and exchange-type interactions between the qubits
[
14
,
16
,
19
,
21
,
22
]
.
The dynamics of the qubit density matrix
ρ
can be described by a master equation of the form
å
rrrsrs
=-
- + G
̇
()[
]
()
/
HH
i,1
mn
mn
ge
m
eg
n
eff
eff
,
,
where the effective
(
non-Hermitian
)
Hamiltonian reads
[
14
,
21
,
22
]
å
ss
=-
G
=
⎜⎟
()
J
i
2
,2
mn
N
mn
mn
eg
n
ge
m
eff
,1
,
,
with
=G
-
(∣
∣)
J
kz z
sin
2
mn
m
n
,1D1D
and
G
=G
-
(∣
∣)
kz z
cos
mn
m
n
,1D1D
denoting the coherent and dissipative
interaction rates, respectively. Here,
Γ
1D
is the single qubit emission rate into the transmission line,
w
=
kv
eg
1D
Figure 1.
(
a
)
Schematic of planar transmon qubits capacitively coupled to a coplanar waveguide. Photon-mediated interactions couple
the qubits together with an amplitude determined by the single-qubit emission rate
Γ
1D
into the waveguide, and a phase determined
by the phase velocity of the transmission line and the distance between qubits. Collective frequency shifts
(
b
)
and decay rates
(
c
)
for
qubits coupled through a waveguide with
p
=
kd
0.2
1D
. Blue circles correspond to the results for a
fi
nite system with
N
=
30 qubits.
Black dashed lines correspond to
=
kk
1D
. The frequency shift for the in
fi
nite chain is denoted by the solid line. The inset in
(
c
)
shows the scaling of the decay rate with qubit number
G
G~
-
N
1D
3
for the 4 most subradiant states.
2
New J. Phys.
21
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is the resonant wavevector,
s
ab
=ñá
ab
∣∣
m
mm
acts on the internal states
ab
Î
{
}{ }
ge
,,
of qubit
m
at position
=
zmd
m
, with
d
the inter-qubit distance. The photonic degrees of freedom can be recovered after solving the
qubit dynamics
[
14
,
22
]
. In particular, the positive-frequency component of the left- and right-going
fi
eld
emitted by the qubits reads
å
s
=+
G
+
=
-
ˆ
()
ˆ
()
ˆ()
()
∣∣
Et E zvt
t
i
2
e,
3
LR
LR
n
N
kz z
ge
n
in
1D
1
i
LR
n
1D
where the
fi
eld
+
ˆ
E
L
(
+
ˆ
E
R
)
is measured directly beyond the
fi
rst
(
last
)
qubit, at position
=
zd
L
(
=
zNd
R
)
. Here,
ˆ
E
L
R
in
denotes the quantized input
fi
eld.
The Markov approximation holds when retardation effects are negligible, that is, when the timescale
L
/
v
for
a photon to travel within the qubit chain of length
L
is small as compared to the timescale
G
-
1D
1
of qubit-photon
interactions. This condition amounts to
L
10
m, for typical values of
v
10
8
ms
1
and
G
10 Hz
1D
7
.
For a given number of excitations, the effective Hamiltonian
eff
de
fi
nes a complex symmetric matrix that
can be diagonalized to
fi
nd collective qubit modes with complex eigenvalues de
fi
ning their resonance
frequencies
(
relative to
ω
eg
)
and decay rates.
2.2. The Dicke limit
In the simple case of
k
1D
d
=
2
n
π
[
(
2
n
+
1
)
π
]
, with
n
an integer, the coherent qubit
qubit interactions
J
m
n
,
vanish and the effective Hamiltonian is purely dissipative,
=-
G
p
p
==
==
()
[]
[]
N
SS
i
2
,4
kk d
kk d
eff
1D
0
0
where
s
=
å
SN
1e
k
n
kz
n
i
eg
n
. The
p
==
[
]
kk d
0
collective mode emits 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
[
23
]
. Within the setting of a one-dimensional
(
1D
)
waveguide, it has also been shown that this
con
fi
guration has interesting quantum optical functionality. For example, the qubits act as a nearly perfect
mirror for near-resonant photons
[
14
,
24
,
25
]
and can generate arbitrary photon Fock states on demand
[
26
]
.
Away from this spacing, the system becomes multimode
[
18
]
, and results in interesting properties for the
single- and multi-excitation eigenstates.
2.3. Single-excitation modes
Numerical diagonalization of
eff
in the single-excitation sector gives
N
distinct eigenstates
y
ñ= ñ =
å
ñ
xx
x
Ä
∣∣
()
Sg
ce
N
n
n
n
1
that obey
yy
ñ= -G ñ
x
xx
x
∣()∣
()
()
()
J
i2 .
5
eff
11
Here,
s
ñ= ñ
Ä
eg
n
n
N
eg
corresponds to having atom
n
excited, and
J
ξ
and
Γ
ξ
represent the frequency shift and
decay rate associated with
y
ñ
x
()
1
. Their interpretation as shifts and decay rates can be understood from the
equivalent quantum jump interpretation
[
27
]
of the master equation
(
1
)
. In particular, within the jump
formalism, a wave function evolves under the Schrödinger equation governed by
eff
, and thus, an eigenstate
y
ñ
x
()
1
evolves in time as
y
--G ñ
xx
x
[(
) ] ∣
()
Jt
exp i
2
1
. The loss of amplitude at a rate
Γ
ξ
during evolution is
supplemented by quantum jump operators stochastically applied to the wave function
(
corresponding to the last
term
srs
å
G
mn
mn
ge
m
e
g
n
,
,
in equation
(
1
))
, which physically describes the new state following the decay of an
excitation.
For our particular system of interest, we obtain a broad distribution of decay rates de
fi
ning superradiant
(
G
>G
x
1D
)
and subradiant
(
G
<G
x
1D
)
states. Ordering the eigenstates by increasing decay rates, i.e. from
ξ
=
1
for the most subradiant to
x
=
N
for the most radiant, we
fi
nd that strongly subradiant modes exhibit a decay
rate
G
G
x
1
D
that is suppressed with qubit number as
x
G
x
N
1D
23
[
28
,
29
]
. This decay rate scaling is similar
to the case of a 1D chain of atoms in 3D free space with lattice spacing smaller than half of the transition
wavelength
[
19
]
, while in the present case there is no restriction on the lattice constant other than not being in
the Dicke limit discussed earlier. Such a cubic scaling is rather generic to so-called 1D
boundary dissipation
models
[
19
,
30
,
31
]
, where losses occur solely at the ends of the physical system. In our system, the periodic chain
of qubits guides light perfectly in the form of polaritons, which are then dissipated into the waveguide when they
hit the ends of the chain.
An interesting consequence of the scaling of
Γ
ξ
with
N
for the most subradiant states is that, in the
thermodynamic limit, the spectrum of decay rates becomes smooth and the
gap
of minimum decay rate closes.
For an in
fi
nite chain, the eigenstates of
eff
take the form of Bloch spin waves
y
ñ= ñ
Ä
()
Sg
kk
N
1
, with
k
being a
quantized wavevector within the
fi
rst Brillouin zone
(
p
k
d
)
. For
fi
nite 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
fi
gures
1
(
b
)
and
(
c
)
, we show the
distribution of frequency shifts
J
k
and decay rates
Γ
k
with
k
for
N
=
30 qubits and
p
=
kd
0.2
1D
.We
fi
nd large
3
New J. Phys.
21
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2019
)
025003
A Albrecht
et al
decay rates and frequency shifts for eigenstates with wavevectors
k
close to the resonant wavevectors
k
1
D
.
Conversely, we obtain decay rate minima and small frequency shifts around
kd
=
0 and
p
=
kd
. For
p
>
kd
0.5
1D
(
p
<
kd
0.5
1D
)
, wavevectors
kd
=
0 form the global
(
local
)
and
p
=
kd
the local
(
global
)
decay
rate minimum, respectively. Such a behavior differs from what is found in a free-space atomic chain
[
19
]
, where
subradiant states are located in a region
w
>
k
c
eg
.
The
k
-dependence can be understood by considering the in
fi
nite lattice limit, where the qubits and
waveguide generally hybridize to form two lossless polariton bands. For an in
fi
nite system, the total Hamiltonian
describing both the qubits and photonic degrees of freedom is given by

å
ww
=+++
{[]}()
††
SS
aa
g aS
h.c. .
6
k
eg
k
kk
k
k
k
k
k
tot
Here,
S
k
creates a collective spin excitation with
k
a quantized wavevector in the
fi
rst Brillouin zone, and
a
k
is the
creation operator of a propagating excitation with wave-vector
k
and frequency
w
=
vk
k
in the transmission
line. The third term of equation
(
6
)
describes the interaction between the qubits and the electromagnetic
fi
eld,
where the parameter
g
k
quanti
fi
es the strength of the interaction. We take a light
matter coupling of the form
[
21
]
dw w wqw w
å
-= -
() ()
gg
k
k
kf
2
2
, where
w
w
>
feg
is a high-frequency cutoff and
q
()
.
is the Heaviside
step function.
For each wavevector, and within the single-excitation sector, the Hamiltonian
(
6
)
represents a 2
×
2 matrix
that can be diagonalized to yield frequencies
W
k
, as shown in
fi
gure
2
. Physically, the two distinct solutions
correspond to a qubit branch and a waveguide branch, with signi
fi
cant hybridization of the two around their
intersection at
=
kk
1D
. For a
fi
nite system, this implies that a collective excitation of qubits with wavevector
close to
k
1
D
ef
fi
ciently radiates into the waveguide, as con
fi
rmed in
fi
gure
1
(
c
)
. Polaritons with wavevector
around
~
kk
1
D
(
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.
In the regions where
w
J
1
k
eg
, with
w
=W -

J
kk
eg
the frequency shift, we recover a good agreement
with the expression obtained from the direct Bloch diagonalization of the effective spin-model Hamiltonian
(
2
)
,
which predicts
~G
+ + -
[(())(())]
///
J
kkd
kkd
cot
2 cot
2 4
k
1D
1D
1D
for
¹
kk
1D
and
d
G
~G
N
2
kkk
1D ,
1D
,
with
pw
G
=
g
2
eg
1D
2
(
see
fi
gure
2
)
. That dispersion relation is plotted in
fi
gure
1
(
b
)
as a solid line, and matches
well with the frequency shifts obtained for a
fi
nite system. While the single-excitation limit is readily solvable
either within the spin model or the full qubit-
fi
eld Hamiltonian of equation
(
6
)
, the spin model is a powerful
simplifying tool to understand the properties of multiple excited qubits interacting via common photonic
modes.
2.4. Multi-excitation modes
A quadratic bosonic Hamiltonian would enable us to easily
fi
nd the multi-excitation eigenstates of
eff
from the
single-excitation sector results. Here, however, the spin nature prevents multiple excitations of the same qubit.
Speci
fi
cally, two-excitation states
j
ñ= ñ
x
x
Ä
()∣
()
Sg
N
2
2
2
, with
2
a normalization factor, are not eigenstates of
the effective Hamiltonian
(
2
)
. Moreover, for an index
ξ
corresponding to a subradiant single-excitation mode,
the initial decay rate of
j
ñ
x
()
2
is signi
fi
cantly greater than twice the single-excitation decay rate
Γ
ξ
.
This discrepancy can be explained by noting that the spatial pro
fi
le of
j
ñ
x
()
2
, i.e. the probability
j
=á ñ
x
∣∣∣
()
pee
,
mn
mn
,
2
2
for qubits
m
and
n
to be excited, contains a sharp cut along the diagonal
m
=
n
Figure 2.
The dots show the two eigenvalue solutions of equation
(
6
)
, which are plotted in black
(
red
)
when the qubit
(
photon
)
component of the polariton is the largest in absolute value. The solid blue line corresponds to the result obtained from the direct Bloch
diagonalization of
eff
, and the dashed black lines show the bare dispersion relations of the isolated qubits and photons. Here,
p
=
kd
0.32
1D
and
g
=
0.01.
4
New J. Phys.
21
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2019
)
025003
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et al