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.
(Dated: January 13, 2020)
A structured electromagnetic reservoir can result in novel dynamics of quantum emitters. In
particular, the reservoir can be tailored to have a memory of past interactions with emitters, in
contrast to memory-less Markovian dynamics of typical open systems. In this Article, we investigate
the non-Markovian dynamics of a superconducting qubit strongly coupled to a superconducting
slow-light waveguide reservoir. Tuning the qubit into the spectral vicinity of the passband of this
waveguide, we find non-exponential energy relaxation as well as substantial changes to the qubit
emission rate. Further, 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 tuning of the
qubit-waveguide interaction strength, we probe a crossover between Markovian and non-Markovian
qubit emission dynamics. These attributes allow for future studies of multi-qubit circuits coupled
to structured reservoirs, in addition to constituting the necessary resources for generation of multi-
photon highly entangled states.
I. INTRODUCTION
Spontaneous emission by a quantum emitter into the
fluctuating electromagnetic vacuum is an emblematic ex-
ample of Markovian dynamics of an open quantum sys-
tem [1]. However, modification of the electromagnetic
reservoir can drastically alter this dynamic, introducing
“non-Markovian” memory effects to the emission process,
a consequence of information back-flow from the reser-
voir to the emitter [2–5]. There have been several stud-
ies investigating non-Markovian effects on the preserva-
tion of quantum information and multipartite entangle-
ment [6, 7]. These studies have generated interest in
leveraging long-lived environmental correlations for sta-
bilization and synthesis of many-body, arbitrary quan-
tum states of a quantum system [8–12].
Studies of non-Markovian physics are readily achieved
by strongly coupling an emitter to a single-mode waveg-
uide – a one-dimensional (1D) reservoir with a contin-
uum of states. Waveguides which break translational
symmetry, or which host resonant elements within the
waveguide, are of particular interest in this regard ow-
ing to the structure in their spectrum [13–15]. For ex-
ample, rich phenomena emerge upon constriction of the
1D continuum of guided modes to a transmission band
of finite bandwidth, with sharp transitions in the pho-
tonic density of states (DOS) occurring at the band-
edges. These phenomena include non-exponential radia-
tive decay, finite light trapping close to the bandedge, and
∗
These two authors contributed equally
†
opainter@caltech.edu; http://copilot.caltech.edu
the formation of protected atom-photon bound states far
from the continuum [16–21]. Spectral constriction of the
continuum, and the concomitant frequency dispersion,
can result in the slowing of light propagation which en-
ables observation of additional non-Markovian phenom-
ena. For instance, by placing a reflective boundary on
one end of a slow-light waveguide, i.e. a mirror, a frac-
tion of the emitter’s radiation is reflected back from the
mirror, thus inducing energy back-flow from the waveg-
uide reservoir at significantly delayed timescales [22–24].
Surprisingly, this deceptively simple mechanism of non-
Markovian time-delayed feedback can allow for genera-
tion of multi-dimensional photonic cluster states by a
single emitter, provided that
τ
d
Γ
1D
1, where Γ
1D
is
the emitter’s emission rate into the waveguide and
τ
d
is
the round-trip travel time of an emitted photon [12].
Superconducting microwave circuits incorporating
Josephson-Junction-based qubits [25, 26] represent a
near-ideal test bed for studying the quantum dynamics
of emitters interacting with a 1D continuum [27, 28]. In
comparison to solid-state and atomic optical systems [29–
32], superconducting microwave circuits can be created at
a deep-sub-wavelength scale, giving rise to strong qubit-
waveguide coupling far exceeding other qubit dissipa-
tive channels. This has enabled a variety of pioneer-
ing experiments probing qubit-waveguide radiative dy-
namics, employing waveguide spectroscopy [24, 33–35],
time-dependent qubit measurements [36–39] and anal-
ysis of higher-order field correlations [40, 41]. Recent
experiments have also explored the coupling of super-
conducting qubits to acoustic wave devices, demonstrat-
ing the capability of these systems to produce signifi-
cant time-delayed feedback [35, 39], albeit with other
challenges such as acoustic wave back-scattering, limited
arXiv:2001.03240v1 [quant-ph] 9 Jan 2020
2
acousto-electric coupling, and quantum-limited detection
of acoustic fields.
In this work we develop an all-electrical slow-light
waveguide consisting of a superconducting metamate-
rial waveguide with a highly structured 1D continuum,
resulting in significant retardation of propagating mi-
crowave fields over a broad bandwidth. By strongly
coupling Xmon-style superconducting qubits [42, 43] to
the slow-light waveguide, we explore through both spec-
troscopic field measurements and time-dependent qubit
measurements, the properties of this system deep within
the non-Markovian limit. By terminating one-end of the
slow-light waveguide with a reflective boundary, we also
study the time-delayed feedback of emitted light pulses
from the qubit (achieving
τ
d
Γ
1D
≈
30), providing insight
into the attainable fidelity and scale of the aforemen-
tioned multipartite entanglement proposal [12] in such a
physical system.
II. SLOW-LIGHT METAMATERIAL
WAVEGUIDE
In prior work studying superconducting qubit emission
into a photonic bandgap waveguide [36], 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 trans-
mission line with resonator loading in parallel to the line,
one obtains high efficiency transmission with a charac-
teristic impedance approximately that of the standard
CPW away from the resonance frequency of the loading
resonators, and a transmission stopband near resonance
of the resonators. In contrast, here we seek a waveguide
with high transmission efficiency and slow-light propaga-
tion within a transmission passband. In addition to a
metamaterial design that optimizes the slow-light delay
for a given chip area, secondary considerations include
a modular design that can be reliably replicated at the
unit cell level without introducing spurious cell-to-cell
couplings, and a method for matching to external input
and output lines that avoids unintended reflections and
resonances within the transmission passband.
Large delay per unit area can be obtained by employ-
ing a network of sub-wavelength resonators, with light
propagation corresponding to hopping from resonator-
to-resonator at a rate set by near-field inter-resonator
coupling. This area-efficient approach to achieving large
delays is well-suited to applications where only limited
bandwidths are necessary. In optical photonics applica-
tions, this sort of scheme has been realized in what are
called coupled-resonator optical waveguides, or CROW
waveguides [44, 45]. Here we employ a periodic array
of capacitively coupled, lumped-element microwave res-
onators to form the waveguide. Such a resonator-based
waveguide supports a photonic channel through which
light can propagate, henceforth referred to as the pass-
band, with bandwidth approximately equal to four times
the coupling between the resonators,
J
. The limited
bandwidth directly translates into large propagation de-
lays; as can be shown (see App. 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 reso-
nance frequency of the resonators.
An optical and scanning electron microscope (SEM)
image of the unit cell of the metamaterial slow-light
waveguide used in this work are shown in Fig. 1a. The
cell consists of a tightly meandered wire inductor sec-
tion (
L
0
; false color blue) and a top shunting capaci-
tor (
C
0
; false color green), forming the lumped-element
microwave resonator. 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 have included
a long superconducting island (
C
q
; false color green);
this is used in the next section as the shunting capac-
itance for Xmon qubits. Similar lumped-element res-
onators have been realized with internal quality factors
of
Q
i
∼
10
5
and small resonator frequency disorder [36],
enabling propagation of light with low extinction from
losses or disorder-induced scattering [46]. The waveguide
resonators shown in Fig. 1a have a bare resonance fre-
quency of
ω
0
/
2
π
≈
4
.
8 GHz, unit cell length
d
= 290
μ
m,
and transverse unit cell width
w
= 540
μ
m, achieving a
compact planar form factor of
̄
d/λ
= (
√
dw
)
/
(2
πv/ω
0
)
≈
1
/
60, where
v
is the speed of light in a CPW on a in-
finitely thick silicon substrate.
The unit cell is to a good approximation given by the
electrical circuit shown in Fig. 1b, in which the pho-
ton hopping rate is
J
∝
C
g
/C
0
[47]. We chose a
ratio of
C
g
/C
0
≈
1
/
70, which yields a delay per res-
onator of roughly 2 ns. Note that we have achieved
this compact form factor and large delay per resonator
while separating different lumped-element components
by large amounts of ground plane, which minimizes spu-
rious crosstalk between different unit cells. Analysis of
the periodic circuit’s Hamiltonian and dispersion can be
found in App. B, where the dispersion is shown to be
ω
k
=
ω
0
/
√
1 + 4
C
g
C
0
sin
2
(
kd/
2). Figure 1c shows a plot of
the theoretical waveguide dispersion for an infinitely pe-
riodic waveguide, where the frequency of the bandedges
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. 1d, dashed blue curve, for example).
To avoid these reflections, we taper the impedance of the
waveguide by slowly shifting the capacitance of the res-
onators 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
3
φ
x
L
0
C
0
...
...
C
g
b
c
100 μm
20 μ
m
d
4.65
4.7
4.75
Frequency (GHz)
0
100
200
Group Delay (ns)
-60
-
40
-20
0
Transmission (dB)
a
(
L
0
C
0
)
-1/2
k
Γ
X
e
≈
4.6
4.8
≈
0
∞
(
L
0
(
C
0
+4
C
g
))
-1/2
≈
≈
100 μm
FIG. 1.
Microwave Coupled Resonator Array Slow-
light Waveguide. a
, Optical image of a fabricated mi-
crowave resonator unit cell. The capacitive elements of the
resonator are false colored in green, while the inductive me-
ander is false colored in blue. The inset shows a false col-
ored 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
, Theoret-
ical dispersion relation of the periodic resonator array. See
App. B for derivation.
d
, Transmission through a metama-
terial slow-light waveguide spanning 26 resonators and con-
nected 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 res-
onators at each boundary. Red line: measured transmission
for a fabricated finite resonator array with boundary match-
ing 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
= 5 ns in the middle of the passband.
matches the Bloch impedance of the periodic structure
in the passband to the characteristic impedance of the
input-output waveguides [48]. In essence, this tapering
achieves strong coupling of all normal modes of the fi-
nite structure to the input-output waveguides by adia-
batically transforming guided resonator array modes into
guided input-output waveguide modes. This loading of
the normal modes lowers their
Q
such that they spec-
trally overlap and become indistinguishable, changing the
DOS of a finite array from that of a multi-mode res-
onator to that of finite-bandwidth continuum with sin-
gular bandedges. Further details of the design of the unit
cell and boundary resonators can be found in App. C.
Using the above design principles, we fabricated a
capacitively coupled 26-resonator array metamaterial
waveguide. The waveguide was fabricated using electron-
beam deposited aluminum (Al) on a silicon substrate
and was measured in a dilution refrigerator; transmis-
sion measurements are shown in Fig. 1d,e, and further
details of our fabrication methods and measurement set-
up can be found in App. A. We find less than 0
.
5 dB
ripple in transmitted power and less than 10% variation
in the group delay (
τ
g
≡ −
dφ
dω
,
φ
= arg(
t
(
ω
)), where
t
is transmission) across 80 MHz of bandwidth in the cen-
ter of the passband, ensuring low distortion of propagat-
ing signals. Qualitatively, this small ripple demonstrates
that we have realized a resonator array with small disor-
der and precise modification of the boundary resonators.
More quantitatively, from the transmitted power mea-
surements we extract a standard deviation in the reso-
nance frequencies of 3
×
10
−
4
×
ω
0
(see App. D). Fur-
thermore, 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 bandedges 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 dynam-
ics of a quantum emitter, a second sample was fabricated
with a metamaterial waveguide similar to that in the
previous section, this time including three flux-tunable
Xmon qubits [43] coupled at different points along the
waveguide (see Fig. 2a-c). Each of the qubits is cou-
pled to its own XY control line for excitation of the
qubit, a Z control line for flux tuning of the qubit tran-
sition frequency, and a readout resonator (R) with sep-
arate readout waveguide (RO) for dispersive read-out of
the qubit state. The qubits are designed to be in the
transmon-limit [42] with large tunneling to charging en-
ergy ratio (see Refs. [36, 49] for further qubit design and
fabrication details). As in the test waveguide of Fig. 1,
the qubit-loaded metamaterial waveguide 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 metamate-
rial waveguides together using a CPW bend and internal
impedance matching sections. The Xmon qubit capaci-
tors were designed to have capacitive coupling to a single
unit cell of the metamaterial waveguide, yielding a qubit-
unit cell coupling of
g
uc
≈
0
.
9
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 were detuned from
Q
1
by approximately 1 GHz for all of the measurements
that follow. At zero flux bias (i.e., maximum qubit fre-
quency), the measured parameters of Q
1
are:
ω
ge
/
2
π
=