Collapse and Revival of an Artificial Atom Coupled to a Structured Photonic Reservoir
Vinicius S. Ferreira ,
1,2
,*
Jash Banker ,
1,2
,*
Alp Sipahigil,
1,2
Matthew H. Matheny,
1,2
Andrew J. Keller,
1,2
Eunjong Kim ,
1,2
Mohammad Mirhosseini,
1,2
and Oskar Painter
1,2
,
†
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics, California
Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology,
Pasadena, California 91125, USA
(Received 9 March 2020; revised 12 September 2021; accepted 17 September 2021; published 2 December 2021)
Quantum emitters in the presence of an electromagnetic reservoir with varying density of states, or
structure, can undergo a rich set of dynamical behavior. In particular, the reservoir can be tailored to have a
memory of past interactions with emitters, in contrast to memoryless Markovian dynamics of typical open
systems. In this article, we investigate the non-Markovian dynamics of a superconducting qubit strongly
coupled to a superconducting waveguide engineered to have both a sharp spectral variation in its
transmission properties and a slowing of light by a factor of 650. Tuning the qubit into the spectral vicinity
of the passband of this slow-light waveguide reservoir, we observe a 400-fold change in the emission rate of
the qubit, along with oscillatory energy relaxation of the qubit resulting from the beating of bound and
radiative dressed qubit-photon states. Furthermore, upon addition of a reflective boundary to one end of the
waveguide, we observe revivals in the qubit population on a timescale 30 times longer than the inverse of
the qubit
’
s emission rate, corresponding to the round-trip travel time of an emitted photon. By
in situ
tuning
of the qubit-waveguide interaction strength, we also probe a crossover between Markovian and non-
Markovian qubit emission dynamics in the presence of feedback from waveguide reflections. With this
superconducting circuit platform, future studies of multiqubit interactions via highly structured reservoirs
and the generation of multiphoton highly entangled states are possible.
DOI:
10.1103/PhysRevX.11.041043
Subject Areas: Metamaterials, Quantum Physics
I. INTRODUCTION
Spontaneous emission by a quantum emitter into the
fluctuating electromagnetic vacuum, and the corresponding
exponential decay of the emitter excited state, is an
emblematic example of Markovian dynamics of an open
quantum system
[1]
. However, modification of the electro-
magnetic reservoir can drastically alter this dynamic,
introducing
“
non-Markovian
”
memory effects to the emis-
sion process, a consequence of information backflow from
the reservoir to the emitter
[2
–
5]
. A canonical example of
this, considered in early theoretical work
[6
–
8]
, is the
behavior of a quantum emitter whose natural emission
frequency lies close to the gap edge of a photonic band-gap
material
[9,10]
where a sharp transition of the photonic
density of states (DOS) occurs. Inside the band gap, the
emitter sees a reservoir devoid of electromagnetic states,
while just outside of the band gap lies a continuum of
states. This structure of the photonic band-gap reservoir
leads to a strong dressing of the emitter and a resulting
emission dynamics modified by the interplay between
bound and radiative emitter-photon resonant states
[11
–
15]
.
More recently, theoretical studies have explored how a
structured reservoir with non-Markovian memory alters the
entanglement within a quantum system coupled to such a
reservoir
[16
–
18]
. This has led to the paradigm of reservoir
engineering, where non-Markovianity is a quantifiable
resource for quantum information processing and commu-
nication. Theory work from this quantum information
perspective shows that long-lived reservoir correlations
can be used for the generation and preservation of entan-
glement
[19,20]
and quantum control
[21]
of a quantum
system, enhancement of the capacity of quantum channels
[22]
, and the synthesis of exotic many-body quantum states
of light from single emitters
[23]
.
In practice, observation of non-Markovian emission
phenomena can be achieved by strongly coupling an
emitter to a single-mode waveguide
—
a one-dimensional
(1D) reservoir with a continuum of states. Waveguides
which break continuous translational symmetry, or which
*
These authors contributed equally to this work.
†
opainter@caltech.edu;
http://copilot.caltech.edu
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI.
PHYSICAL REVIEW X
11,
041043 (2021)
2160-3308
=
21
=
11(4)
=
041043(27)
041043-1
Published by the American Physical Society
host resonant elements within the waveguide, are of
particular interest in this regard owing to the structure in
their spectrum
[24
–
26]
. For example, an array of coupled
resonant elements leads to a constriction of the 1D
continuum of guided modes to a transmission band of
finite bandwidth, with sharp transitions in the photonic
DOS occurring at the band edges as in a photonic band-gap
material.
Spectral constriction of the waveguide continuum, and
the concomitant frequency dispersion, can also result in the
slowing of light propagation which enables observation of
additional non-Markovian phenomena. For instance, by
placing a reflective boundary (mirror) on one end of a slow-
light waveguide, a fraction of the emitter
’
s radiation can be
fed back from the waveguide reservoir to the emitter at
significantly delayed timescales
[27
–
29]
. The non-
Markovian regime is reached when
τ
d
Γ
1
D
>
1
, where
Γ
1
D
is the emitter
’
s emission rate into the waveguide and
τ
d
is the round-trip travel time of an emitted photon.
Theoretical studies show that such non-Markovian time-
delayed feedback in a 1D waveguide reservoir can lead to
revivals in the excited-state population of an emitter as it
undergoes spontaneous emission decay
[27,30
–
35]
, reali-
zation of stable bound states in a continuum
[36,37]
, and
enhanced collective effects including multipartite entangle-
ment and superradiant emission from emitters interacting
via a common waveguide channel
[18,38
–
42]
. This decep-
tively simple mechanism of time-delayed feedback can also
be used for the generation of multidimensional photonic
cluster states by a single emitter and has been proposed as a
means for generating the universal resource states neces-
sary for measurement-based quantum computation
[23]
.
Superconducting microwave circuits incorporating
Josephson-junction-based qubits
[43,44]
represent a near-
ideal test bed for studying the quantum dynamics of
emitters interacting with a 1D continuum
[45,46]
.In
comparison to solid-state and atomic optical systems
[47
–
50]
, superconducting microwave circuits can be cre-
ated at a deep-subwavelength scale, giving rise to strong
qubit-waveguide coupling far exceeding other qubit dis-
sipative channels. This has enabled a variety of pioneering
experiments probing qubit-waveguide radiative dynamics,
employing waveguide spectroscopy
[29,51
–
53]
, time-
dependent qubit measurements
[54
–
57]
, and analysis of
higher-order field correlations
[58,59]
. Recent experiments
also explore the coupling of superconducting qubits to
acoustic wave devices, demonstrating the capability of
these systems to produce significant time-delayed feedback
and remote entanglement of qubits
[53,57]
.
In this work, we present the design and characterization
of an all-electrical slow-light waveguide consisting of a
chain of coupled lumped-element superconducting reso-
nators patterned on a silicon microchip. We demonstrate
that this compact, low-loss microwave waveguide has sharp
band edges and a passband with group delay of 55 ns per
centimeter over an 80-MHz bandwidth. Through the
addition of strongly coupled Xmon-style superconducting
qubits
[60,61]
to the slow-light waveguide, we are able to
realize a quantum emitter-reservoir system operating deep
within the non-Markovian limit. Spectroscopic measure-
ment of the coupled system shows the emergence of
dressed qubit-photon resonant states near the band edges
of the constricted passband of the waveguide
[7,8,52]
.
Using nonadiabatic tuning of the qubit emission frequency,
we also measure the time-dependent dynamics of the qubit
excited-state population when it is resonant at different
points across the band gap and passband of the waveguide.
We directly observe nonexponential, oscillatory radiative
decay of the qubit, which modeling indicates is a result of
the interference of the pair of bound and radiative dressed
qubit-photon states that exist on either side of the band edge
of the slow-light waveguide
[11]
. Furthermore, by termi-
nating one end of the slow-light waveguide with a reflective
boundary, we explore the effects of time-delayed feedback
on the qubit emission as it emits into the passband of the
slow-light waveguide. In this regime, we observe multiple,
well-resolved revivals in the qubit excited-state population
and explore the crossover between Markovian and non-
Markovian emission dynamics through
in situ
tuning of the
qubit coupling to the waveguide. From this series of
measurements, we estimate the achievable fidelity of
entangling a number of photon pulses via qubit emission
and subsequent time-delayed feedback and find that the
demonstrated qubit-waveguide system is a promising plat-
form for the sequential generation of multidimensional
photonic cluster states as described in the theoretical
proposals of Refs.
[23,62
–
64]
.
II. SLOW-LIGHT METAMATERIAL WAVEGUIDE
In prior work studying superconducting qubit emission
into a photonic band-gap waveguide
[54]
, we employed a
metamaterial consisting of a coplanar waveguide (CPW)
periodically loaded by lumped-element resonators. In that
geometry, whose circuit model simplifies to a transmission
line with resonator loading in parallel to the line, one
obtains high-efficiency transmission with a characteristic
impedance approximately that of the standard CPW away
from the resonance frequency of the loading resonators and
a transmission stop band near resonance of the resonators.
The spectral characteristics of the metamaterial in Ref.
[54]
were studied via spontaneous emission lifetime and Lamb-
shift measurements of a weakly coupled superconducting
qubit, which revealed information about the local DOS at
the qubit frequency that was consistent with the metama-
terial engineered dispersion. In contrast, here we seek a
waveguide with high transmission efficiency, slow-light
propagation within a transmission passband, and consid-
erably stronger qubit coupling to the waveguide-guided
modes. The stronger coupling renders the Born approxi-
mation inapplicable in such a system, where the effect of
VINICIUS S. FERREIRA
et al.
PHYS. REV. X
11,
041043 (2021)
041043-2
the qubit interaction with the photonic reservoir takes on
significantly more complexity than simply a decay rate
dependent solely on the DOS at the qubit frequency.
Furthermore, the increased propagation delay gives rise
to non-Markovian memory effects in the waveguide-medi-
ated interactions between qubits, for which the waveguide
degrees of freedom can no longer be traced out, as in
Ref.
[55]
, for instance.
Large delay per unit area can be obtained by employing a
network of subwavelength resonators, with light propaga-
tion corresponding to hopping from resonator to resonator
at a rate set by near-field interresonator coupling. This area-
efficient approach to achieving large delays is well suited to
applications where only limited bandwidths are necessary.
However, realizing such a waveguide system in a compact
chip-scale form factor requires a modular implementation
that can be reliably replicated at the unit-cell level without
introducing spurious cell-to-cell couplings. In optical
photonics applications, this sort of scheme is realized in
what are called coupled-resonator optical waveguides, or
CROW waveguides
[65,66]
. Here, we employ a periodic
array of capacitively coupled, lumped-element microwave
resonators to form the waveguide. Such a resonator-based
waveguide supports a photonic channel through which light
can propagate, henceforth referred to as the passband, with
bandwidth approximately equal to 4 times the coupling
between the resonators,
J
. The limited bandwidth directly
translates into large propagation delays; as can be shown
(see the Appendix
B
), the delay in the resonator array is
roughly
ω
0
=J
longer than that of a conventional CPW of
similar area, where
ω
0
is the resonance frequency of the
resonators.
Optical and scanning electron microscope (SEM) images
of the unit cell of the metamaterial slow-light waveguide
used in this work are shown in Fig.
1(a)
. The cell consists of
a tightly meandered wire inductor section (
L
0
; false color
blue) and a top shunting capacitor (
C
0
; false color green),
forming the lumped-element microwave resonator. Note
that these delineations between inductor and capacitor
are not strict and that the meandered wire inductor (top
shunting capacitor) has a small parasitic capacitance (para-
sitic inductance). The resonator is surrounded by a large
ground plane (gray) which shields the meander wire section.
Laterally extended
“
wings
”
of the top shunting capacitor
also provide coupling between the cells (
C
g
; false color
green). Note that at the top of the optical image, above
each shunting capacitor, we include a long superconducting
island (
C
q
; false color green); this is used in the next section
as the shunting capacitance for Xmon qubits. Similar
lumped-element resonators have been realized with internal
quality factors of
Q
i
∼
10
5
and small resonator frequency
disorder
[54]
, enabling propagation of light with low
extinction from losses or disorder-induced scattering
[67]
.
The waveguide resonators shown in Fig.
1(a)
have a bare
resonance frequency of
ω
0
=
2
π
≈
4
.
8
GHz, unit-cell length
d
¼
290
μ
m, and transverse unit-cell width
w
¼
540
μ
m,
achieving a compact planar form factor of
̄
d=
λ
¼ð
ffiffiffiffiffiffi
dw
p
Þ
=
ð
2
π
v=
ω
0
Þ
≈
1
=
60
, where
v
is the speed of light in a CPWon
an infinitely thick silicon substrate.
The unit cell is to a good approximation given by the
electrical circuit shown in Fig.
1(b)
, in which the photon
hopping rate is
J
∝
C
g
=C
0
[68]
. We choose a ratio of
C
g
=C
0
≈
1
=
70
, which yields a delay per resonator of
roughly 2 ns. Note that we achieve this compact form
factor and large delay per resonator while separating
different lumped-element components by large amounts
of ground plane, which minimizes spurious cross talk
between different unit cells. Analysis of the periodic
circuit
’
s Hamiltonian and dispersion can be found in
Appendix
B
, where the dispersion is shown to be
(b)
(c)
(d)
(a)
FIG. 1. Microwave coupled resonator array slow-light wave-
guide. (a) Optical image of a fabricated microwave resonator unit
cell. The capacitive elements of the resonator are false colored in
green, while the inductive meander is false colored in blue. The
inset shows a false-colored SEM image of the bottom of the
meander inductor, where it is shorted to ground. (b) Circuit
diagram of the unit cell of the periodic resonator array waveguide.
(c) Theoretical dispersion relation of the periodic resonator array.
See Appendix
B
for the derivation. (d) Transmission through a
metamaterial slow-light waveguide spanning 26 resonators and
connected to
50
-
Ω
input-output ports. Dashed blue line: theo-
retical transmission of finite array without matching to
50
-
Ω
boundaries. Black line: theoretical transmission of finite array
matched to
50
-
Ω
boundaries through two modified resonators at
each boundary. Red line: measured transmission for a fabricated
finite resonator array with boundary matching to input-output
50
-
Ω
coplanar waveguides. The measured ripple in transmission
is less than 0.5 dB in the middle of the passband. (e) Measured
group delay
τ
g
. Ripples in
τ
g
are less than
δτ
g
¼
5
ns in the
middle of the passband.
COLLAPSE AND REVIVAL OF AN ARTIFICIAL ATOM
...
PHYS. REV. X
11,
041043 (2021)
041043-3
ω
k
¼
ω
0
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
4
ð
C
g
=C
0
Þ
sin
2
ð
kd=
2
Þ
q
. Figure
1(c)
shows a
plot of the theoretical waveguide dispersion for an infinitely
periodic waveguide, where the frequency of the band edges
of the passband are denoted with the circuit parameters of
the unit cell.
For finite resonator arrays, care must be taken to avoid
reflections at the boundaries that would result in spurious
resonances [see Fig.
1(d)
, dashed blue curve, for example].
To avoid these reflections, we taper the impedance of the
waveguide by slowly shifting the capacitance of the
resonators at the boundaries. In particular, we modify
the first two unit cells at each boundary, but, in principle,
more resonators could have been modified for a more
gradual taper. Increasing
C
g
to increase the coupling
between resonators, and decreasing
C
0
to compensate
for resonance frequency changes, effectively impedance
matches the Bloch impedance of the periodic structure in
the passband to the characteristic impedance of the input-
output waveguides
[69]
. In essence, this tapering achieves
strong coupling of all normal modes of the finite structure
to the input-output waveguides by adiabatically transform-
ing guided resonator array modes into guided input-output
waveguide modes. This loading of the normal modes
lowers their
Q
such that they spectrally overlap and become
indistinguishable, changing the DOS of a finite array from
that of a multimode resonator to that of finite-bandwidth
continuum with singular band edges. Further details of the
design of the unit cell and boundary resonators can be
found in Appendix
C
.
Using the above design principles, we fabricated a
capacitively coupled 26-resonator array metamaterial
waveguide. The waveguide is fabricated using electron-
beam deposited aluminum (Al) on a silicon substrate and
is measured in a dilution refrigerator; transmission mea-
surements are shown in Figs.
1(d)
and
1(e)
, and further
details of our fabrication methods and measurement setup
can be found in Appendix
A
.Wefindlessthan0.5dB
ripple in transmitted power and less than 10% variation in
the group delay (
τ
g
≡
−
ð
d
φ
=d
ω
Þ
,
φ
¼
arg
½
t
ð
ω
Þ
, where
t
is transmission) across 80 MHz of bandwidth in the center
of the passband, ensuring low distortion of propagating
signals. Qualitatively, this small ripple demonstrates that
we have realized a resonator array with small disorder and
precise modification of the boundary resonators. More
quantitatively, from the transmitted power measurements,
we extract a standard deviation in the resonance frequen-
cies of
3
×
10
−
4
×
ω
0
(see Appendix
D
). Furthermore, we
achieve
τ
d
≈
55
ns of delay across the 1-cm metamaterial
waveguide, corresponding to a slow-down factor given by
the group index of
n
g
≈
650
. We stress that this group
delay is obtained across the center of the passband rather
than near the band edges where large (and undesirable)
higher-order dispersion occurs concomitantly with large
delays.
III. NON-MARKOVIAN RADIATIVE DYNAMICS
In order to study the non-Markovian radiative dynamics
of a quantum emitter, a second sample is fabricated with a
metamaterial waveguide similar to that in the previous
section, this time including three flux-tunable Xmon qubits
[61]
coupled at different points along the waveguide [see
Figs.
2(a)
–
2(c)
]. Each of the qubits is coupled to its own
XY
control line for excitation of the qubit, a
Z
control line
for flux tuning of the qubit transition frequency, and a
readout resonator (
R
) with separate readout waveguide
(RO) for dispersive readout of the qubit state. The qubits
are designed to be in the transmon limit
[60]
with large
tunneling to charging energy ratio (see Refs.
[54,70]
for
further qubit design and fabrication details). As in the test
waveguide in Fig.
1
, the qubit-loaded metamaterial wave-
guide is impedance matched to input-output
50
-
Ω
CPWs.
In order to extend the waveguide delay further, however,
this new waveguide is realized by concatenating two of the
test metamaterial waveguides together using a CPW bend
and internal impedance-matching sections. The Xmon
qubit capacitors are designed to have capacitive coupling
to a single unit cell of the metamaterial waveguide, yielding
a qubit
–
unit-cell coupling of
g
uc
≈
0
.
8
J
.
In this work, only one of the qubits,
Q
1
, is used
to probe the non-Markovian emission dynamics of the
qubit-waveguide system. The other two qubits are to be
used in a separate experiment and are detuned from
Q
1
by approximately 1 GHz for all of the measurements
that follow. At zero flux bias (i.e., maximum qubit
frequency), the measured parameters of
Q
1
are
ω
ge
=
2
π
¼
5
.
411
GHz,
η
=
2
π
¼ð
ω
ef
−
ω
ge
Þ
=
2
π
¼
−
235
MHz,
ω
r
=
2
π
¼
5
.
871
GHz, and
g
r
=
2
π
¼
88
MHz. Here,
j
g
i
,
j
e
i
, and
j
f
i
are the vacuum, first excited, and second excited states of
the Xmon qubit, respectively, with
ω
ge
the funda-
mental qubit transition frequency,
ω
ef
the first excited-
state transition frequency, and
η
the anharmonicity.
ω
r
is
the readout resonator frequency, and
g
r
is the bare coupling
rate between the qubit and the readout resonator.
As an initial probe of qubit radiative dynamics, we
spectroscopically probe the interaction of
Q
1
with the
structured 1D continuum of the metamaterial waveguide.
These measurements are performed by tuning
ω
ge
into the
vicinity of the passband and measuring the waveguide
transmission spectrum at low power (such that the effects of
qubit saturation can be neglected). A color-intensity plot of
the measured transmission spectrum versus flux bias used
to tune the qubit frequency is displayed in Fig.
2(d)
. These
spectra show a clear anticrossing as the qubit is tuned
toward either band edge of the passband [an enlargement
near the upper band edge of the passband is shown in
Fig.
2(e)
]. As shown theoretically
[11,12]
, in the single-
excitation manifold, the interaction of the qubit with the
waveguide results in a pair of qubit-photon dressed states
of the hybridized system, with one state in the passband
(a delocalized
“
continuum
”
state) and one state in the band
VINICIUS S. FERREIRA
et al.
PHYS. REV. X
11,
041043 (2021)
041043-4
gap (a localized
“
bound
”
state). This arises due to the
large peak in the photonic DOS at the band edge (in the
lossless case, a van Hove singularity), the modes of which
strongly couple to the qubit with a coherent interaction rate
of
Ω
WG
≈
ð
g
4
uc
=
4
J
Þ
1
=
3
, resulting in a dressed-state splitting
of
2
Ω
WG
. This splitting has been experimentally shown to
be a spectroscopic signature of a non-Markovian interac-
tion between an emitter and a photonic crystal reservoir
[51,52]
. Further details and discussion can be found in
Appendixes
B
and
E
.
The dressed state with frequency in the passband is a
radiative state which is responsible for decay of the qubit
into the continuum
[8]
. On the other hand, the state with
frequency in the gap is a qubit-photon bound state, where
the qubit is self-dressed by virtual photons that are emitted
and reabsorbed due to the lack of propagating modes in the
waveguide for the radiation to escape. This bound state
assumes an exponentially shaped photonic wave function
of the form
P
x
e
−
j
x
j
=
λ
ˆ
a
†
x
j
vac
i
, where
j
vac
i
is the state with
no photons in the waveguide,
ˆ
a
†
x
is the creation operator of a
photon in unit cell at position
x
(with the qubit located at
x
¼
0
), and
λ
≈
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
J=
ð
E
b
−
ω
0
Þ
p
is the state
’
s localization
length. In the theoretical limit of an infinite array, and in
the absence of intrinsic resonator and qubit losses, the
qubit component of the bound state does not decay even
though it is hybridized with the waveguide continuum,
a behavior distinct from conventional open quantum
systems. Practically, however, intrinsic losses and the
overlap between the bound state
’
s photonic wave function
and the input-output waveguides results in decay of the
qubit-photon bound state.
In complement to spectroscopic probing of the qubit-
reservoir system and in order to directly study the pop-
ulation dynamics of the qubit-photon dressed states, we
also perform time-domain measurements as shown in
Fig.
3
. In this protocol [illustrated in Fig.
3(a)
], we excite
the qubit to state
j
e
i
with a resonant
π
pulse on the
XY
control line and then rapidly tune the qubit transition
frequency using a fast current pulse on the
Z
control line
to a frequency (
ω
0
ge
) within, or in the vicinity of, the slow-
light waveguide passband. After an interaction time
τ
, the
qubit is then rapidly tuned away from the passband, and the
remaining qubit population in
j
e
i
is measured using a
microwave probe pulse (RO) of the readout resonator
which is dispersively coupled to the qubit. The excitation
of the qubit is performed far from the passband, permitting
initialization of the transmon qubit while it is negligibly
hybridized with the guided modes of the waveguide.
Dispersive readout of the qubit population is performed
outside of the passband in order to minimize the loss
of population during readout. Note that, as illustrated in
Fig.
3(a)
, the qubit is excited and measured at different
frequencies on opposite sides of the passband; this is
necessary to avoid Landau-Zener interference
[71]
.
Results of measurements of the time-domain dynamics
of the qubit population as a function of
ω
0
ge
(the estimated
(b)
(a)
(d)
(c)
(e)
b
FIG. 2. Artificial atom coupled to a structured photonic reservoir. (a) False-colored optical image of a fabricated sample consisting of
three transmon qubits (
Q
1
,
Q
2
, and
Q
3
) coupled to a slow-light metamaterial waveguide composed of a coupled microwave resonator
array. Each qubit is capacitively coupled to a readout resonator (false-color dark blue) and an
XY
control line (false-color red) and
inductively coupled to a
Z
flux line for frequency tuning (false-color light blue). The readout resonators are probed through feedlines
(false-color lilac). The metamaterial waveguide path is highlighted in false-color dark purple. (b) SEM image of the
Q
1
qubit, showing
the long, thin shunt capacitor (false-color green),
XY
control line, the
Z
flux line, and coupling capacitor to the readout resonator (false-
color dark blue). (c) SEM enlarged image of the
Z
flux line and superconducting quantum interference device (SQUID) loop of the
Q
1
qubit, with Josephson junctions and its pads false colored in crimson. (d) Transmission through the metamaterial waveguide as a
function of the flux. The solid magenta line indicates the expected bare qubit frequency in the absence of coupling to the metamaterial
waveguide, calculated based on the measured qubit minimum and maximum frequencies and the extracted anharmonicity. The dashed
black lines are numerically calculated bound-state energies from a model Hamiltonian of the system; see Appendix
E
for further details.
(e) Enlargement of transmission near the upper band edge, showing the hybridization of the qubit with the band edge, and its
decomposition into a bound state in the upper band gap and a radiative state in the continuum of the passband.
COLLAPSE AND REVIVAL OF AN ARTIFICIAL ATOM
...
PHYS. REV. X
11,
041043 (2021)
041043-5
bare qubit frequency during interaction with the wave-
guide) are shown as a color-intensity plot in Fig.
3(b)
.In
this plot, we observe a 400-fold decrease in the
1
=e
excited-
state lifetime of the qubit as it is tuned from well outside the
passband to the middle of the slow-light waveguide
passband, reaching a lifetime as short as 7.5 ns. Beyond
the large change in qubit lifetime within the passband,
several other more subtle features can be seen in the qubit
population dynamics near the band edges and within the
passband. These more subtle features in the measured
dynamics show nonexponential decay, with significant
oscillations in the excited-state population that is a
hallmark of strong non-Markovianity in quantum systems
coupled to amplitude damping channels
[72,73]
.
The observed qubit emission dynamics in this non-
Markovian limit are best understood in terms of the
qubit-waveguide dressed states. Fast (i.e., nonadiabatic)
tuning of the qubit in state
j
e
i
into the proximity of the
passband effectively initializes it into a superposition of the
bound and continuum dressed states. The observed early-
time interaction dynamics of the qubit with the waveguide
then originate from interference of the dressed states, which
leads to oscillatory behavior in the qubit population
analogous to vacuum-Rabi oscillations
[74]
. The frequency
of these oscillations is thus set by the difference in energy
between the dressed states. The amplitude of the oscil-
lations, on the other hand, quickly decay away as the
energy in the radiative continuum dressed state is lost into
the waveguide.
All of these features can be seen in Fig.
3(c)
, which
shows plots of the measured time-domain curves of the
qubit excited-state population for bare qubit frequencies
near the top, middle, and bottom of the passband. Near the
upper band-edge frequency, we observe an initial oscil-
lation period as expected due to dressed-state interference.
Once the continuum dressed state decays away, a slower
decay region free of oscillations can be observed (this is
due to the much slower decay of the remaining qubit-
photon bound state). Finally, around
τ
≈
115
ns, there is an
onset of further small-amplitude oscillations in the qubit
population. These late-time oscillations can be attributed to
interference of the remaining bound state at the site of the
qubit with weak reflections occurring within the slow-light
waveguide of the initially emitted continuum dressed state.
The 115-ns timescale corresponds to the round-trip time
between the qubit and the CPW bend that connects the two
slow-light waveguide sections.
In the middle of the passband, we see an extended region
of initial oscillation and rapid decay, albeit of smaller
oscillation amplitude. This is a result of the much smaller
initial qubit-photon bound-state population when tuned to
the middle of the passband. Near the bottom of the pass-
band, we see rapid decay and a single period of a much
slower oscillation. This is curious, as the dispersion near the
upper and lower band-edge frequencies of the slow-light
waveguide is nominally equivalent. Further modeling
shows that this is a result of weak nonlocal coupling of
the Xmon qubit to a few of the nearest-neighbor unit cells
of the waveguide. Referring to Fig.
1(c)
, the modes near the
lower band edge occur at the
X
point of the Brillouin zone
edge, where the modes have alternating phases across each
unit cell; thus, extended coupling of the Xmon qubit causes
cancellation effects which reduce the qubit-waveguide
coupling at the lower-frequency band edge. Further detailed
numerical model simulations of our qubit-waveguide sys-
tem via a tight-binding model and a circuit model, as well
as the correspondence between the observed dynamics and
(b)
(a)
(c)
Lower band edge
Upper band edge
FIG. 3. Non-Markovian radiative dynamics in a structured
photonic reservoir. (a) Pulse sequence for the time-resolved
measurement protocol. The qubit is excited while its frequency
is 250 MHz above the upper band edge, and then it is quickly
tuned to the desired frequency (
ω
0
ge
) for a interaction time
τ
with
the reservoir. After interaction, the qubit is quickly tuned below
the lower band edge for dispersive readout. (b) Intensity plot
showing the excited-state population of the qubit versus inter-
action time with the metamaterial waveguide reservoir as a
function of the bare qubit frequency. (c) Line cuts of the intensity
plot shown in (b), where the color of the plotted curve matches
the corresponding horizontal dot-dashed curve in the intensity
plot. Solid black lines are numerical predictions of a model with
experimentally fitted device parameters and an assumed 0.8%
thermal qubit population (see Appendix
E
for further details).
VINICIUS S. FERREIRA
et al.
PHYS. REV. X
11,
041043 (2021)
041043-6
the theory of spontaneous emission by a two-level system
near a photonic band edge
[11]
, are given in Appendix
E
.
IV. TIME-DELAYED FEEDBACK
In order to further study the late-time, non-Markovian
memory effects of the qubit-waveguide dynamics, we
also perform measurements in which the end of the
waveguide furthest from qubit
Q
1
is terminated with an
open circuit, effectively creating a
“
mirror
”
for photon
pulses stored in the slow-light waveguide reservoir. As
illustrated in Fig.
4(a)
, we achieve this
in situ
by connecting
the input microwave cables of the dilution refrigerator to
the waveguide via a microwave switch. The position of the
switch, electrically closed or open, allows us to study a
truly open environment for the qubit or one in which
delayed feedback is present, respectively (see Appendix
A
for further details).
Performing time-domain measurements with the mirror
in place and with the qubit frequency in the passband, we
observe recurrences in the qubit population at 1 and 2
times the round-trip time of the
slow-light waveguide that
do not appear in the absence of the mirror [see Fig.
4(b)
].
The separation of timescales between full population
decay of the qubit and its time-delayed reexcitation
demonstrates an exceptionally long memory of the res-
ervoir due to its slow-light nature and places this experi-
ment in the deep non-Markovian regime
[27]
. The small
recurrence levels as they appear in Fig.
4(b)
arenotdueto
inefficient mirror reflection but rather can be explained
as follows. Because the qubit emits toward both ends of
the waveguide, half of the emission is lost to the
unterminated end, while the other half is reflected by
the mirror and returns to the qubit. In addition, the
exponentially decaying temporal profile of the emission
leads to inefficient reabsorption by the qubit and further
limits the recurrence (see, for instance, Refs.
[75,76]
for
details). These two effects can be observed in simulations
of a qubit coupled to a dispersionless and lossless wave-
guide (pink dotted line; for more details, see Ref.
[31]
and
Appendix
G
). The remaining differences between the
simulation and the measured population recurrence (blue
solid line) can be explained by the effects of propagation
loss and pulse distortion due to the slow-light waveguide
’
s
dispersion.
We also further probe the dependence of this phenome-
non on the strength of coupling to the waveguide con-
tinuum by parametric flux modulation of the qubit
transition frequency
[77]
when it is far detuned from the
passband. This modulation creates sidebands of the qubit
excited state, which are detuned from
ω
ge
by the frequency
of the flux tone
ω
mod
. By choosing the modulation
frequency such that a first-order sideband overlaps with
the passband, the effective coupling rate of the qubit with
the waveguide at the sideband frequency is reduced
approximately by a factor of
J
1
2
½
ε
=
ω
mod
, where
ε
is
the modulation amplitude and
J
1
is a Bessel function of
the first kind (
ε
=
ω
mod
is the modulation index). Keeping a
fixed
ω
mod
, we observe purely exponential decay at small
modulation amplitudes. However, above a modulation
amplitude threshold, we again observe recurrences in the
qubit population at the round-trip time of the metamaterial
waveguide, demonstrating a continuous transition from
Markovian to non-Markovian dynamics (see Appendix
G
for further comparisons between these data and the theo-
retical model in Ref.
[31]
).
(a)
(b)
(c)
FIG. 4. Time-delayed feedback from a slow-light reservoir with
a reflective boundary. (a) Illustration of the experiment, showing
the qubit coupled to the metamaterial waveguide which is
terminated on one end with a reflective boundary via a microwave
switch. (b) Measured population dynamics of the excited state of
the qubit when coupled to the metamaterial waveguide terminated
in a reflective boundary. Here, the bare qubit is tuned into the
middle of the passband. The onset of the population revival occurs
at
τ
¼
227
ns, consistent with round-trip group delay
ð
τ
d
Þ
measurements at that frequency, while the emission lifetime of
the qubit is
ð
Γ
1
D
Þ
−
1
¼
7
.
5
ns. The magenta curve is a theoretical
prediction for emission of a qubit into a dispersionless, lossless
semi-infinite waveguide with equivalent
τ
d
and
Γ
1
D
(see Appen-
dix
G
for details). (c) Population dynamics under parametric flux
modulation of the qubit, for varying modulation amplitudes,
demonstrating a Markovian to non-Markovian transition. When
the modulation index (
ε
=
ω
mod
) is approximately 0.4, we have
Γ
1
D
ð
ε
Þ¼
1
=
τ
d
; the corresponding dynamical trace is colored
in blue.
COLLAPSE AND REVIVAL OF AN ARTIFICIAL ATOM
...
PHYS. REV. X
11,
041043 (2021)
041043-7
V. CONCLUSION
In conclusion, by strongly coupling Xmon qubits to a 1D
structured photonic reservoir consisting of a metamaterial
slow-light waveguide, we are able to probe the non-
Markovian dynamical regime of waveguide quantum
electrodynamics. In this regime, we observe nonexponen-
tial qubit spontaneous decay near the band edges of
the slow-light waveguide, attributable to interference result-
ing from the splitting of the qubit state into a radiative
state in the passband and a bound state in the band-gap
region of the metamaterial waveguide. Moreover, by
placing a reflective boundary on one end of the wave-
guide, we observe recurrences in the qubit population
at the round-trip time of an emitted photon, as well as a
Markovian to non-Markovian transition when varying the
qubit-waveguide interaction strength.
The demonstrated ability to achieve a true finite-
bandwidth continuum with time-delayed feedback opens
up several new research avenues for exploration
[28,30
–
42,78]
. As a straightforward extension of the current work,
one may probe the qubit-waveguide-mirror system in a
continuous, strongly driven fashion and use tomography to
study photon correlations in the output radiation field
[28]
.
This output field, with an expected photon stream of high
entanglement dimensionality, has a direct mapping to
continuous matrix product states which can used for analog
simulations of higher-dimension interacting quantum fields
[78,79]
. With technical advancements in the tomography of
microwave fields
[59,80]
and realization of single-micro-
wave-photon qubit detectors
[81
–
83]
, the basic tools for
characterization of these entangled photonic states and their
quantum many-body-system analogues are now available.
Looking forward even further, the use of the multilevel
structure of the transmon qubit, in conjunction with a
second distant qubit side coupled to the waveguide as a
switchable mirror, can be used to generate 2D cluster states
[23]
. This system is capable of entangling consecutively
emitted photons as well as photons separated in time by the
round-trip waveguide delay
τ
d
, thus achieving an
N
×
M
2D cluster state, where
N
is limited by the number
of nonoverlapping photons that can fit in the slow-light
waveguide and
N
·
M
is limited by the coherence time
of the emitter. With our achieved device parameters, we
estimate that a
3
×
3
2D cluster state could be generated
with fidelity greater than 50% (see Ref.
[23]
and
Appendix
F
for further details). Realistic improvements
in
τ
d
and
T
2
could increase the size of the state by at least
an order of magnitude, with even further improvement
possible via incorporation of compact high kinetic induct-
ance superconducting thin-film resonators or acoustic delay
lines
[57,84]
. Additionally, by controlling the number
of reflections a photon undergoes before exiting the
metamaterial waveguide, cluster states of 3D or higher
entanglement dimensionality can be generated, enabling
the realization of fault-tolerant measurement-based quan-
tum computation schemes
[23,64,85]
.
The essential paradigm of our experiment, consisting of a
single artificial atom coupled to a waveguide with a long
propagation delay and sharp spectral cutoffs, could, in
principle, be achieved in other solid-state and atomic optical
systems, such as trapped atoms coupled to a nanofiber
or defect centers coupled to photonic crystal waveguides
[47
–
50]
. The challenge with such modalities, however, is
achieving a large coupling of the emitter to the guided modes
of the waveguide relative to its decay rate as well as the
propagation delay of the waveguide. From an application
standpoint, however, the optical domain is of great interest
due to the mature technology in single-photon detectors,
photonic integrated circuits for linear and nonlinear optics,
and optical fibers for long-range communication.
ACKNOWLEDGMENTS
We thank Hannes Pichler for fruitful discussions regard-
ing the mirror measurements, MIT Lincoln Laboratories
for the provision of a traveling-wave parametric amplifier
[86]
used for both spectroscopic and time-domain mea-
surements in this work, Jen-Hao Yeh and Ben Palmer for
the use of one of their cryogenic attenuators
[87]
for
reducing thermal noise in the metamaterial waveguide, and
Hengjiang Ren and Xueyue Zhang for help during mea-
surements, fabrication, and writing. This work was sup-
ported by the AFOSR MURI Quantum Photonic Matter
(Grant No. 16RT0696), the Institute for Quantum Infor-
mation and Matter (IQIM), an NSF Physics Frontiers
Center (Grant No. PHY-1125565) with support of the
Gordon and Betty Moore Foundation, and the Kavli
Nanoscience Institute (KNI) at Caltech. V. S. F. gratefully
acknowledges support from NSF GFRP fellowship, and
M. M. (A. S.) gratefully acknowledges support from a
KNI (IQIM) postdoctoral fellowship.
APPENDIX A: FABRICATION AND
MEASUREMENT SETUP
1. Device fabrication
The devices used in this work are fabricated on
10
mm ×
10
mm silicon substrates [float zone grown,
525
μ
m thick-
ness,
>
10
k
Ω
-cm resistivity], following similar techniques
as in Ref.
[70]
. After standard solvent cleaning of the
substrate, our first aluminum (Al) layer consisting of the
ground plane, CPWs, metamaterial waveguide, and qubit
capacitor is patterned by electron-beam lithography of our
resist followed by electron-beam evaporation of 120 nm
aluminum at a rate of
1
nm
=
s. A liftoff process performed
in
n
-methyl-2-pyrrolidone at
80
°C for 2.5 h (with 10 min of
ultrasonication at the end) then yields the aforementioned
metal structures.
In our qubit device, the Josephson junctions are fabri-
cated using double-angle electron beam evaporation of
VINICIUS S. FERREIRA
et al.
PHYS. REV. X
11,
041043 (2021)
041043-8