Pulse shaping by coupled-cavities: Single photons and qudits

Abstract

Dynamic coupling of cavities to a quantum network is of major interest to distributed quantum information processing schemes based on cavity quantum electrodynamics. This can be achieved by active tuning a mediating atom-cavity system. In particular, we consider the dynamic coupling between two coupled cavities, each interacting with a two-level atom, realized by tuning one of the atoms. One atom-field system can be controlled to become maximally and minimally coupled with its counterpart, allowing high fidelity excitation confinement, Q-switching and reversible state transport. As an application, we first show that simple tuning can lead to emission of near-Gaussian single-photon pulses that is significantly different from the usual exponential decay in a passive cavity-based system. The influences of cavity loss and atomic spontaneous emission are studied in detailed numerical simulations, showing the practicality of these schemes within the reach of current experimental technology in solid-state environment. We then show that when the technique is employed to an extended coupled-cavity scheme involving a multi-level atom, arbitrary temporal superposition of single photons can be engineered in a deterministic way.

Full text

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arXiv:0906.2690v2 [quant-ph] 10 Sep 2009
Pulse shaping by coupled-cavities: Single photons and qudits
Chun-Hsu Su,1, ∗Andrew D. Greentree,1William J. Munro,2,3Kae Nemoto,3and Lloyd C. L. Hollenberg1
1Quantum Communications Victoria, School of Physics, University of Melbourne, VIC 3010, Australia
2Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 8QZ, United Kingdom
3National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
(Dated: September 10, 2009)
Dynamic coupling of cavities to a quantum network is of major interest to distributed quantum
information processing schemes based on cavity quantum electrodynamics. This can be achieved by
actively tuning a mediating atom-cavity system. In particular, we consider the dynamic coupling
between two coupled cavities, each interacting with a two-level atom, realized by tuning one of
the atoms. One atom-field system can be controlled to become maximally and minimally coupled
with its counterpart, allowing high fidelity excitation confinement, Q-switching and reversible state
transport. As an application, we first show that simple tuning can lead to emission of near-Gaussian
single-photon pulses that is significantly different from the usual exponential decay in a passive
cavity-based system. The influences of cavity loss and atomic spontaneous emission are studied
in detailed numerical simulations, showing the practicality of these schemes within the reach of
current experimental solid-state technology. We then show that when the technique is employed to
an extended coupled-cavity scheme involving a multi-level atom, arbitrary temporal superposition
of single photons can be engineered in a deterministic way.
PACS numbers: 42.50.Pq, 42.50.Dv, 42.60.Gd, 42.50.Ex
I.INTRODUCTION
Cavity quantum electrodynamics provides a natural
setting for distributed quantum information processing
(QIP), by bringing together matter-based quantum sys-
tems for long-term memory storage, photons for fast and
reliable transport of quantum information over long dis-
tances, and high finesse cavities for strong matter-field
interaction. State transfer [1, 2, 3, 4, 5, 6, 7], two-qubit
gates [8, 9, 10, 11, 12, 13, 14] and entanglement gen-
eration [1, 9, 10, 11, 15, 16, 17, 18] can all be realized
through controlled coupling of spatially distant atoms.
This can be achieved by connecting the cavities in which
atomic qubits reside via an optical fiber [1]. In other pro-
posals, the occupation of the fiber mode is bypassed in
the adiabatic limit [5, 11] or the mode is entirely avoided
by coupling two atoms with a common field mode within
the same cavity [8, 9]. Alternatively, coupled-cavity lat-
tices consisting of weakly-coupled optical cavities each
containing one or more atoms, have been proposed, since
each lattice site can be individually controlled and mea-
sured. Interactions in passive, coupled two atom-cavity
systems have been studied for quantum interference ef-
fects [19] and atomic state transfer [20, 21]. Large-scale
doped lattices have also been studied for photonic phase
transitions [22, 23, 24], quantum transport [25, 26], and
polaritonic cluster state preparation [27].
An all optical coupled-cavity waveguide is also of ma-
jor interest for engineering lossless guiding, slow light and
enhanced nonlinearity [28, 29]. Propagation along such a
waveguide can then be regulated by controlling an atom
∗Electronic address: chsu@ph.unimelb.edu.au
at one site [30, 31]. This type of dynamic control is also
considered in the two-cavity arrangement [14, 32] where
one atom-cavity system, sandwiched between an adjacent
waveguide and another cavity (e.g., qubit cavity), is ma-
nipulated by tuning its two-level atom. As a result, the
coupling between the waveguide and the qubit cavity can
be turned on and off on demand. Single photons can also
be prepared in this way via a reversible adiabatic process
and therefore it must also act as a receiver of single-
photon states [14]. Since the control is external to the
actual qubit cavity and the latter can be made optically
isolated from communication channels, the qubit cavity
is able to act as a separate QIP subsystem. The scheme
therefore offers flexibility in quantum network and dis-
tributed QIP and indeed, has been discussed in the con-
text of optical networks for preparing photonic 2D and
3D cluster states [33, 34, 35].
In the present paper, we examine this scheme of dy-
namic two-cavity coupling, with a particular focus on im-
proving the schemes for ultra-low loss confinement and
fast switching in Sec. II. As an application, we first show
in Sec. III that simple tuning can be used to tailor the
pulse shape of single photons for mode matching. As op-
posed the usual decay profiles from a passive cavity-based
system, the single photons prepared this way have near-
Gaussian shapes over a range of cavity parameters. Be-
ing the minimum uncertainty state, such pulses would be
most robust against mode mismatch and therefore ideal
for two-photon interference [36] and optical QIP [37]. We
note that pulse shaping has only been studied in a sin-
gle atom-cavity configuration in Refs. [38, 39, 40, 41].
In Sec. IV, the practicality of realizing our scheme in
solid-state environment (e.g., superconducting stripline
resonators with mesoscopic qubits, photonic crystals and
slot-waveguides with doped impurities such as diamond

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FIG. 1: Schematic of the two coupled atom-cavity systems
for Q-switching and pulse shaping. |gα? and |eα? denote the
ground and excited states of atom α (α = s,q). Each atom
with lifetime 1/γα is coupled to the local quasi-resonant cav-
ity mode with single-photon Rabi frequency gα. Atom s is
detuned from the local cavity mode by ?∆s whereas atom
q is detuned from cavity q by ?(∆q − δq), where ?δq is the
detuning between two cavities. The cavities are evanescently
coupled with rate κsq and the right cavity is coupled to an
adjacent waveguide with decay rate κwq.
colour centres) is discussed. We generalize the results of
Sec. II in Sec. V where the technique is applied to an
extended scheme involving a multi-level atom, and show
arbitrary temporal superposition of single photons can
be engineered in a deterministic way.
can be considered as an integrated-photonic alternative
to conventional interferometric method to preparing pho-
tonic qudits for QIP proposals such as higher-dimensional
quantum bus [42], quantum gate [43], error filtration [44]
and quantum cryptography [45, 46, 47].
Such a scheme
II.
Q-SWITCHING
An ultra-low loss (high cavity-Q) cavity represents an
ideal environment for achieving coherent atomic manip-
ulation at the single-quantum level with minimal dissi-
pation. However, the fundamental time-bandwidth re-
lation of passive low-loss cavities leads to the difficulty
of out-coupling the confined light field from the cavity
on demand. Furthermore, input fields must be of nar-
row bandwidths that limit operating speeds. This prob-
lem can be overcome using appropriate dynamic controls
that modify the effective cavity-waveguide coupling, from
high Q that enables light confinement to low Q for field
in and out-coupling, with a tuning time much shorter
than the photon lifetime of the cavity. This is termed
Q-switching and here we are interested in such a control
on the single-quantum level.
A schematic of our model coupled-cavity system is
shown in Fig. 1. It is formed by two evanescently-coupled
single-modal cavities, each contains a single two-level
atomic or atom-like system quasi-resonant with the local
cavity mode. The atom-cavity system on the right acts
as a storage (labelled with s) for a single quantum of
excitation, whereas the left atom-cavity system (q) acts
as a Q-switch to control this confinement. In turn, the
switch is coupled to an external reservoir, e.g., a waveg-
uide (w). Under the rotating-wave approximation, the
Hamiltonian of the combined system is given by
H = Hsys+ Hext+ Hint,
Hsys/? = ωsa†
+ ωqa†
(1)
?
?
(2)
sas+ νs|es??es| + gs
?
?
|es??gs|as+ h.c.
|eq??gq|aq+ h.c.
qaq+ νq|eq??eq| + gq
?
?∞
?∞
+ κsq
a†
qas+ h.c.
?
,
Hext/? =
−∞
ωb†(ω)b(ω)dω, (3)
Hint/? = i
−∞
?κwq
2π
?b†(ω)aq− a†
qb(ω)?dω (4)
where |gα? denotes atomic ground state at site α = s,q
and |eα? is the excited state. aαand b(ω) are the bosonic
annihilation operators for the excitation in cavity mode
α and the external photon mode with frequency ω, re-
spectively.
The physical meanings of the parameters are as follows:
All energies are defined with respect to the resonant en-
ergy ?ωs of cavity s, ?ωq ≡ ?(δq+ ωs) is the resonant
energy of cavity q, ?νs≡ ?(∆s+ωs) is the transition en-
ergy of atom s, and ?νq≡ ?(∆q+ωs) is the correspond-
ing energy of atom q. In other words, atom q is detuned
from cavity q by ?(∆q− δq). At site α, gα is the local
single-photon Rabi frequency. For dipole-type atom-field
interaction inside the cavity, this coupling is related to
the transition dipole moment dα of atom α and effec-
tive cavity mode volume Vαby gα= dα[ωα/(2?ǫ0Vα)]1/2.
?κsq is the (photonic) couping energy between the cav-
ity modes, and κwqis the decay rate of the cavity mode
q. The decay is not considered as a loss, but rather a
coherent out-coupling.
Here Q-switching involves modifying the transmittiv-
ity of the switch – from being reflective so that excitation
is confined at site s, to being transmissive when this en-
ergy is allowed to propagate into the adjacent waveguide.
This is achieved by tuning the transition energy of atom
q – represented by ?∆q(t) as a function of time t. In our
treatment, we assume the system is prepared in the one-
quantum manifold so that the state vector of the system
can be written as
|ψ(t)? = (Cs|gs,1s? + Ds|es,0s?)|gq,0q?|vac? +
(Cq|gq,1q?|vac? + Dq|eq,0q?)|gs,0s?|vac? +
Cout|gs,0s?|gq,0q?|φw?
(5)
where Cα,Dα and Cout are the probability amplitudes,
the last ket denotes a vacuum or photonic state in
the waveguide, and |nα? (n = 0,1) the Fock state
in cavity mode α.From the time evolution |˙ψ(t)? =
(−i/?)H|ψ(t)?, it is straightforward to obtain the set of

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differential equations below,
˙Cs = −κs/2Cs− iκsqCq− igsDs,
˙Ds = (−i∆s− γs/2)Ds− igsCs,
˙Cq = (−iδ − iκsq− κq/2 − κwq/2)Cq
−igqDq−√κwqfin,
˙Dq = (−i∆q− γq/2) − igqCq
We include the effect of atomic spontaneous emis-
sions and incoherent cavity decays with effective terms
(−γα/2)Dα and (−κα/2)Cα respectively, where γα is
the transition rate of atom α and καthe transverse de-
cay rate of cavity mode α.
the quality factor of the cavity, Qs = ωs/(2κs) and
Qq = ωq/[2(κwq+ κq)]. fin is the amplitude of the in-
put pulse and is related to the output pulse foutby the
standard input-output relation [48, 49]
fout= fin+√κCq.
(6)
(7)
(8)
(9)
The latter are related to
(10)
The pulse shape of the output field is |fout|2and the
probability of waveguide occupation is given by the inte-
grated area of the output, Pout=?∞
switching as follows.
−∞dt|fout(t)|2.
We now discuss the underlying mechanisms of Q-
A.High Q
A single quantum of excitation at site s can be stored
in the photonic |gs,1s?, the atomic mode |es,0s?, or
their superpositions |±s? ≡ A|gs,1s? ± B|es,0s?. In gen-
eral, these dressed states can be prepared using resonant
pumping and state |gs,1s? via adiabatic passage tech-
niques [50]. Here we explictly consider the case where
the system is initialized in state |−s?, and for simplicity,
we set ∆s= 0 so that A = −B = 1/√2.
To enable extended confinement, state |−s? should
be as close to one of the stationary states (i.e.
ergy eigenstates) of the Hamiltonian Hsys as possible.
For instance, we plot its eigenspectrum in Fig. 2(a) for
δq/κsq = 2,gs/κsq = 5 and gq/κsq = 20 and we iden-
tify the closest eigenstate, |Φ?, that we use in the fol-
lowing discussion.Premature photonic leakage arises
due to state mismatch between |−s? and the prescribed
eigenstate |Φ?. The overlap |?−s|Φ?|2can be maximized
in two ways. First, a very weak intercavity coupling
κsq and a large effective storage-switch detuning can
be used. However, as we will see in Eq. 13, this de-
mands a longer tuning time to out-couple this excita-
tion. The better way is to utilize quantum interference
effect. Interference can arises between two different path-
ways that populate state |gq,1q?, from states |−s? and
|eq,0q? respectively. This can be thought as a spatial
analogue of electromagnetically-induced transparency in
a bichromatically-driven Λ system [31]. Thus, by induc-
ing two-photon resonance between |−s? and atom q when
∆off
q
= E|−s?/?,
en-
(11)
FIG. 2: (Color online) (a) Eigensystem of the Hamiltonian
Hsys (Eq. 2) in one-quantum manifold as a function of atom-
cavity detuning ∆q, for parameters δq/κsq = 2,gs/κsq = 5
and gq/κsq = 20. The full view shows a closeup of the an-
ticrossing of a particular eigenstate |Φ? (blue curve) at ∆res
highlighting storage-switch resonance. Dashed lines denote
the energy of the dressed states of individual atom-cavity sys-
tem in isolation. Inset shows the energy of all four eigenstates.
(b) Overlaps showing mismatch between |Φ? and state |−s?.
Dashed, dash-dotted and solid curves corresponds to overlaps
|?gq,1q|Φ?|2, |?eq,0q|Φ?|2, and |?−s|Φ?|2, respectively.
mismatch is minimized at the dip (∆off
excitation transfers from the storage to the switch as the sys-
tem evolves along |Φ? from ∆off
q ,
The
q ). It also shows that
q
to ∆res
q .
|Φ? is considered as a dark state that is maximally de-
coupled from state |gq,1q?.
∆s = 0) is used to denote the energy of state |−s?.
This is illustrated by the dip of the curve |?|gq,1q|Φ?|2
in Fig. 2(b). On the other hand, the other overlap is
|?eq,0q|Φ?|2= κ2
pressed by using a large gq≫ κsq. For κsq/gq= 10, the
leakage probability is O(10−3). Apart from this funda-
mental limitation, the confinement is effectively lossless
when dissipation is absent.
We now examine the effect of dissipation on the con-
finement time when ∆q = ∆off
atomic and photonic decays at site α with rates γαand κα
respectively, we consider their contributions separately
in Fig. 3.For each curve, we vary one of these de-
cay rates (κs,γs,κq,γq) while keeping the others zero,
and calculate the effective decay rate r. This rate r is
obtained by fitting decay function e−rtto the overlap
|?ψ(0)|ψ(t)?|2, where |ψ(0)? = |−s?. As expected, the
confinement is most severely affected by decoherence at
E|−s? (equates −?gs for
sq/(2g2
q+ κ2
sq), which can then be sup-
q. Recall that there are

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FIG. 3: Excitation at site s dissipates at an effective rate r (in
units of κsq) when the condition ∆q = ∆off
each curve, we only vary one of these decay rates, κs(solid), γs
(solid), κq (dashed), γq (dash-dotted curve uses gq/κsq = 20
and dotted gq/κsq = 50), while keeping the others zero. The
parameters follow Fig. 2 and κwq/κsq = 5 unless stated.
q
is imposed. For
site s with r scaling with κsand γs, and is least affected
by the presence of κq since state |gq,1q? is maximally
decoupled. Similarly, for atomic emission (γq) at the
switch, we compare the curves for gs/κsq = 20 and 50
and show that increasing gq also suppresses the decay
as the storage also becomes increasing decoupled from
state |eq,0q?. Interestingly, we find that the decay rate
peaks about γq/κsq= 102−103and a larger γqimproves
confinement.
B.Low Q
When ∆q ?= ∆off
to leak out at a faster rate. In Fig. 4, we solve the effec-
tive leakage rate for different values of ∆q and δq when
the coupled-cavity system is passive. The leakage is min-
imized when the confinement condition is satisfied, con-
sistent with our earlier discussion. The leakage is most
rapid when the state |−s? is resonant with one of the
energy eigenstates of the switch, |±q?. For the setup de-
picted in Figs. 2 and 4, this corresponds to E|−s?= E|−q?
in the limit of small κsq, where the detuning condition is
q, the confined excitation is expected
∆res
q
= −gs+
g2
q
δq+ gs.(12)
We note that, although the strength of κsqis taken to be
comparable with other parameters in our analysis, Eq. 12
still proves to be useful as shown in Fig. 2.
To allow controlled out-coupling, we evolve the joint
storage-switch system along the eigenstate |Φ? adiabat-
ically by tuning atom q from ∆off
tation is transferred to the switch at ∆res
by the reduction of the overlap |?−s|Φ?|2at the anti-
crossing in Fig. 2(b). As we will see in the next section,
this approach enables us to modify the pulse shape of
the output photon. We note that the process is intrinsi-
cally reversible and therefore the switch can also couple
q
to ∆res
q. The exci-
q, represented
FIG. 4: (Color online) Light leakage rate from the storage via
the switch in the passive coupled-cavity arrangement. Cav-
ity loss is rapid at storage-switch resonance (dashed) and
is minimized at ∆off
q . We use the parameters from Fig. 2,
κwq/κsq = 5 and dissipation is ignored.
a propagating single-photon field into the storage cavity
for confinement [14].
For this adiabatic process to be successful, the rate
of change˙∆q over ∆res
q
should be slow compared to the
(photon-hopping) coupling matrix element J,
J ≡ |?−q|(κsqa†
qas)|−s?| =κsq
√2sinΘ
(13)
where Θ =
zero for large |∆q− δq|. Following this, it is easy to see
that decreasing cavity rate κsqor increasing detuning δq
implies a longer switching time is required. We can also
use the standard adiabaticity criterion [51],
1
2arctan[2gq/(∆q− δq)], which approaches
A ≡ max
?
|?Φ|˙Hsys|Φ′?|
|?Φ|Hsys|Φ? − ?Φ′|Hsys|Φ′?|2
?
≪ 1, (14)
where |Φ′? is the eigenstate closest to |Φ? in energy.
III. PULSE SHAPING
Custom shaping of single photons is of importance
for generating suitable photons for numerous QIP ap-
plications.In particular, Gaussian pulses have shown
to be optimally tolerant against mode mismatch in
interference-based optical QIP schemes [37]; also opti-
mal mode-matching leads to minimal dispersion pulses,
useful for long-range communication. However, photons
leaked from any passive cavity-based system have non-
Gaussian pulse profiles that are not robust against noise.
To illustrate, we plot in Fig. 5(a) the profile of the outgo-
ing photon from the coupled-cavity system on resonance
(∆q= ∆res
For small κwq, the output field shows oscillations indica-
tive of the strongly-coupled system. Critical damping
occurs at κwq/κsq ≈ 2 in this case. As κwq increases,
we observe the output becomes increasingly asymmet-
ric. To overcome this and produce near-Gaussian pulse
q) for different values of cavity decay rate κwq.

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FIG. 5: (Color online) (a) Pulse shapes of output field for
∆q = ∆res
q
when active switching is absent. Different values
of κwq are used. We then apply active switching – varying
∆q(t) with time t from ∆off
q
to ∆res
shapes (solid line). For different values of κwq, linear sweeps
are used in (b)–(d) whereas revised sweeps in (e)–(g). The
corresponding sweep profiles are depicted in (h)–(j). The pa-
rameters follow Fig. 2, dissipation is ignored and the system
is initialized in state |−s?. Dashed lines are Gaussian fits with
unit areas.
q
that modifies the pulse
shapes, we are interested in tuning atom q along appro-
priate trajectories while the confined excitation at site s
is out-coupled.
Before proceeding, we note that dynamic controls for
pulse shaping have only been discussed in a single atom-
cavity arrangement [38, 39, 40, 41]. In contrast, our two-
cavity approach separates the two functional elements,
interaction zone (site s) and coupling control (Q-switch)
to allow for greater flexibility and locality of control in
quantum networks.
We first ignore decoherence and consider a linear sweep
∆q(t) that varies from ∆off
q
κwqin Figs. 5(b)–(d), respectively. The switching times
T between 10/κsqand 40/κsqare chosen so that the adi-
abaticity condition is satisfied (A ∼ 10−2, J = 0.2) and
the profiles are near-Gaussian in each case. Notably, the
tuning time is much shorter than the the effective life-
time at site s. In general, we find that larger decay rates
to (∆res
q
+ ∆off
q)/2 for each
FIG. 6: Pulse shapes of the output photon using linear sweeps
for different values of gs/κsq = 0 (dashed curve), 1 (solid) and
5 (dash-dotted). When atom s is omitted from the system
(gs = 0), we initialized the system in state |1s?. In all cases,
δq/κsq = 2,gq/κsq = 20, κwq/κsq = 1, and dissipation is
ignored.
require longer T. When the sweep time is fast compared
to the effective leakage rate of the passive case (Fig 4),
the temporal widths of the output are of the same order
O(T).
The premature leakage probability is 10−3and the
successful out-coupling probability is Pout > 0.99. The
mismatch between the output pulse |fout|2and an unit
Gaussian fit |g(t)|2can be measured by the overlap in-
tegral ξ ≡ 1 −?|fout(t)||g(t)|dt = 0.014,0.033 and 0.040
prove these profiles, a more sophisticated tuning is re-
quired. Since the rate at which the switch (specifically
state |gq,1q?) is populated is related to the output pro-
file, we can modify the output by changing the rate˙∆q.
Using a map between a trial linear sweep and its cor-
responding output envelope, we reconstruct a different
sweep for each case and produce results in Fig. 5(e)–
(g), with improved ξ = 0.003,0.01 and 0.02, respectively.
The required length of the sweep is not significantly in-
creased and out-coupling probability remains > 0.99. We
also examine the effect of smaller gsin Fig. 6 and show
small improvement to switching time and pulse width.
In the limit of gs = 0 where atom s is decoupled from
the system, a pulse width of ∼ 5/κsqis possible because
the matrix element J (Eq. 13) is enhanced by a factor of
√2. This is equivalent to the case where the excitation
is initialized in the photonic mode |gs,1s?.
During Q-switching and pulse shaping, dissipation at
the switch become important. Focusing on the loss due
to spontaneous emission of atom q and decay of cavity
field at the switch, we plot in Fig 7 Poutversus the decay
rates γq,κq. The scheme is more susceptible to photon
loss than spontaneous emission and degrades as the deco-
herence rates κq/κsq> 10−2and γq> 0.1κsq. Finally, we
note that a longer sweep would not significantly reduce
Pout when the sweep is much slower than the effective
leakage rate ∼ 0.1κsq of the passive case (Fig 4). How-
ever, the excitation should be switched out at faster rates
for κwq/κsq = 1,5 and 10, respectively. To further im-

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to avoid dissipation at site s.
IV. SOLID-STATE IMPLEMENTATIONS
Our scheme provides a mechanism for controlled cou-
pling of atom-photon qubits in distributed QIP. In this
section, we consider three possible implementations in
solid-state environment and estimate their performance.
Akin to many quantum information hardware propos-
als, one requirement for suppressing loss (∝ κ2
strong atom-cavity coupling achieved with small cavity
mode volumes. As our scheme is most suited for optical
frequencies where active control of cavity-Q is difficult,
we first turn to an implementation in photonic-band-gap
(PBG) lattice and slot-waveguide structures with dia-
mond colour centres. We then consider superconducting
striplines in the microwave region.
A defect in in PBG lattice [52] constitutes a
wavelength-sized, high-Q cavity. The proposed coupled-
cavity arrangement can be formed by placing such de-
fects in close proximity [53], and doping substitutional
impurities [54] or quantum dots [55] as the suitable two-
state atomic systems. One promising platform for solid-
state quantum optics is diamond. In diamond, there are
numerous colour centres with well-defined energy levels.
In particular, the negatively-charged nitrogen-vacancy
(NV) centre has been extensively studied for quantum
electro-optical applications [56].
transition (ωs,ωq∼ 2.95 PHz, wavelength λ = 637 nm),
between excited spin triplet state (3E) to the m = 0
sublevel of the triplet ground state (3A) [57], has a
dipole moment d ∼ 10−30Cm.
ables gs,gq∼ 10 GHz coupling inside a (λ/2)3diamond-
based PBG cavity [58, 59] so that we estimate inter-
cavity coupling κsq of 1 GHz. With a long lifetime of
11.6 ns, γs,γq= 86 MHz. The transition energy of NV
can be tuned by an external control field through dc
Stark effect. Stark tuning of isolated centres has been
demonstrated [60] and the tuning range from the ear-
lier work of Redman et al. of order 1 THz [61] is more
than enough for the proposed scheme.
we estimate 10 ns operation time with Q = 106for
confinement and out-coupling probabilities ∼ 0.9. In-
deed, diamond-based PBG cavity designs in this range
are available [62, 63, 64, 65, 66, 67, 68] and a modest
quality factor of 585 has also been realized [69]. To fur-
ther improve the fidelity to 0.99, one requires either a
higher Q to reduce transverse decay or a larger gq with
smaller cavity mode volumes to reduce switching time.
Slot-waveguide geometries [70, 71], combined with mir-
rors, PBG or distributed Bragg (DBRs) reflectors in a
Fabry-Perot arrangement [72], have also shown to be
compatible with the NV and are promising for achiev-
ing ultra-small confinement [73]. A DBR-based waveg-
uide cavity with Q = 27000 has been fabricated in
silicon [74], and that using PBG structures reported
Q = 58000 [75]. The possible subwavelength confinement
sq/g2
q) is
Its zero-phonon line
In principle, this en-
Consequently,
FIG. 7: Q-switching in the presence of dissipation at the
switch. For each curve, we vary one of these decay rates,
κq (solid) and γq (dashed), while keeping the other zero, and
calculate the outcoupling probability Pout. The parameters
follow Fig. 2(b).
predicts gs,gq∼ 100 GHz [73] and nanosecond switching
time, and such implementation only requires Q = 105for
confinement and out-coupling probabilities > 0.99.
In the microwave regime, we can exploit strong cou-
pling and low loss achieved in superconducting striplines.
A segmented superconducting strip-line forms coupled
coplanar resonators and waveguides separated by capac-
itive open gaps [76]. The capacitive atom-photon cou-
pling at each site is implemented with a Cooper-pair
box, acting as a two-state artificial atom in the charge
regime [77]. Quantum superposition of its charge states
can be achieved via tuning of an applied gate voltage.
Its long coherence time (1/γq,1/γc ∼ 0.1 µs), large
dipole moment of 10−25Cm combined to make it well-
suited to our purpose. Furthermore, the transition en-
ergy (∼ 10 GHz) between its two levels can be tuned
dynamically by changing the magnetic flux through the
Josephson junction loop of the box. Since these super-
conducting qubits are coupled to strip-line cavities with a
typical strength of 0.1 GHz, we predict the time scale for
switching with κsq,κwq∼ 10 MHz to be ∼ 1µs and the
mode mismatch ξ = O(10−3). To achieve confinement
and out-coupling probabilities ∼ 0.9, modest Q = 104
should suffice. This level of coupling and photon life-
times have both been demonstrated [78, 79]. However,
for enhanced probabilities > 0.99, photonic loss must be
further suppressed with Q = 105.
V.PREPARING SINGLE PHOTONS IN
TEMPORAL SUPERPOSITION
An integrated-photonic method to preparing photonic
qudits would be a useful resource for QIP applications.
Here we generalize the results in Sec. II by examining dy-
namic coupling between two atom-cavity systems where
one of which interacts with a multi-level atom. As an-
other application, we show that, by tuning this atom,
arbitrary temporal superposition of single photons can
be engineered with high fidelity. When operated in re-

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7
FIG. 8: Schematic of the extended two coupled atom-cavity
systems for preparing two-time superposition states. The ex-
tensions to the previous schematic in Fig. 1: atom s is a three-
level atom with states |gs?,|es? and |fs? and the |es? ↔ |fs?
transition is driven by a classical field with rate Ωs and de-
tuning energy ?δs. As before, the |gs? ↔ |es? couples to the
cavity mode with rate gs and detuning ?∆s, and the cavities
are evanescently coupled.
verse, such temporally encoded qudits can be transferred
into the cavity for storage or manipulation. In particu-
lar, we consider the following extensions: the multi-level
atom replaces two-state atom s and the atomic tuning is
now performed on atom s. The switch (system q) now
acts as a passive frequency filter.
Two-time superposition:– We begin by using a ladder-
type, three-level atom with states |gs?,|es? and |fs? at site
s. The |gs? ↔ |es? transition is coupled to cavity mode s
with single-photon Rabi frequency gs, and the |es? ↔ |fs?
transition is coupled via a classical quasi-resonant field
with strength Ωsand detuning energy ?δs. With the two
atom-cavity systems evanescently coupled, the schematic
is shown in Fig. 8. Splitting the system Hamiltonian of
Eq. 2 into components, Hsys= Hs+Hq+P where Hαis
the local Hamiltonian at site α and P = ?κsq(a†
is the intercavity coupling, Hsis now
Hs/? = ∆s|es??es| + (∆s+ δs)|fs??fs|
+ gs|es??gs|as+ Ωs|fs??es| + h.c.
where the physical meanings of the other parameters are
explained following Eq. 4, and ∆s(t) represents the tun-
ing of atom s for controlling the effective coupling be-
tween system s and q. For convenience, we set δs= 0.
It is instructive to see the eigenspectrum of the system
Hamiltonian Hsys in Fig. 9. In this case, we are inter-
ested in two particular (labeled) eigenstates |Φ1? and
|Φ2?, which are close to two local eigenstates (|us? and
|vs? respectively) of system s in isolation (i.e. Hs). The
analytical solutions to the doubly-dressed system s can
be found in Ref. [80].
Both states, |Φi?, anticross with other eigenstate, and
by analogy, these anticrossings corresponds to resonance
between system s and one of the eigenstates at site q (in
this case, |+q?). In the limit of small κsq, these occur at
values of ∆sthat satisfy
qas+h.c.)
(15)
E|us?≡ −?
E|vs?≡ −?
3
?
?
−2∆s+ 2p cos
?θ
?θ
3+π
3−π
3
??
??
= E|+q?,(16)
3
−2∆s+ 2p cos
3
= E|+q?(17)
FIG. 9: (Color online) Eigensystem of the extended Hamilto-
nian Hsys (Eq. 15) with Ξ-type, three-level atom s in one-
quantum manifold, for δs = 0,δq/κsq = −15,∆q/κsq =
−8.6,gs/κsq = 1,Ωs/κsq = 4.94 and gq/κsq = 10. (a) Solid
curves are the energies of its eigenstates, and dashed curves
are the corresponding eigenenergy of system s in isolation.
The states |Φ1?,|Φ2? of the joint system are approximately
given by the eigenstates |us?,|vs? of system s at points where
their overlaps are maximum (i.e., the dips ∆off
(b) respectively). These overlaps decrease when the eigen-
states come into resonance with the state |+q? at the anti-
crossings (∆res
s;1,∆off
s;2in (c) and
s;1,∆res
s;2).
where E|us?, E|vs?are the energy of states |us? and |vs?,
p2= ∆2
Ω2
Therefore, when the system is prepared in state |us?
(|vs?), and atom s is tuned over ∆res
excitation in the dressed mode is allowed to leak out via
the switch, representing the low-Q regime similar to the
scheme previously discussed in Sec. II. Depending on the
eigenstate (|us? or |vs?) in which the system is initialized,
a high-Q regime is retrieved by inducing two-photon res-
onances between the eigenstate and atom q, as shown in
the inset of Fig. 9. The values of ∆s = ∆off
maximize the overlaps |?us|Φ1?|2and |?vs|Φ2?|2, satisfy
the respective conditions,
s+2(g2
s+Ω2
s) and cosθ = −(∆s/p3)[∆2
s+9(g2
s;1and ∆res
s/2−
s;2.
s)]. We label these resonance points ∆res
s;1(∆res
s;2), the confined
s;1,∆off
s;2that
E|us?= ?∆q, E|vs?= ?∆q.
We turn next to describe the steps for generating
single-photon superposition states.
choice of parameters, we initialize the system in a nor-
malized superposition of the eigenstates, A|us? + B|vs?
for ∆s= ∆off
laser pulse sequence engineered with the GRAPE algo-
rithm [81] or by reversing the out-coupling process. This
choice of ∆sensures confinement in mode |vs?. However,
because the optimal confinement condition for mode |us?
cannot be simultaneously imposed, we use the standard
storage-switch off-resonance to effect suitable confine-
(18)
Given the above
s;2. This can be set up, for example, using a

Page 8

8
FIG. 10: (Color online) Single photon in two-time superpo-
sition state under no dissipation. Pulse shape of the output
photon prepared with a linear sweep ∆s(t) that varies from
∆off
s;1, for different initial state (a)
(|us? + |vs?)/√2, and (b) (√2|us? + |vs?)/√3. The parame-
ters follow Fig. 9 and κwq/κsq = 4. Dashed lines are Gaussian
fits.
s;2→ ∆res
s;2→ ∆off
s;1 → ∆res
ment in this mode by ensuring ∆off
(∆res
s;1). Then at later times tB and tA (0 < tB < tA),
the resonances between state |vs? with |+q? and state
|us? with |+q?, are induced by tuning atom s from ∆off
to ∆res
out-couple with probabilities |B|2and |A|2, realizing the
output state B|tB? + A|tA? where |tA?,|tB? denote the
temporally distinguishable basis states. Since the cavity
decay (κwq) is a coherent out-coupling, the single photon
is not a mixed state but a true superposition state.
We estimate the time scale for switching by calculat-
ing the coupling matrix element Jβ≡ |?+q|(κsqa†
(β = u,v) at resonances,
s;2is far from resonance
s;2
s;2, then from ∆off
s;1to ∆res
s;1. The excitation should
qas)|βs?|
Jβ= κsqgsΩs
Nβ
cosΘ, (19)
and N2
g2
impose the condition Ju ≈ Jv so that same sweep rate
is used and the output pulse shapes at the two times are
near identical for equal superposition states. Indeed, the
parameters used in Fig. 9 are chosen for this purpose.
Applying the above prescription, we present the nu-
merical simulations for initial equal superposition state in
Fig. 10(a), and unequal superposition√2B = A =
in (b). We should expect the integrated area of each
pulse to be 0.5 in the first case, and 1/3 and 2/3 in the
latter case. For Jα∼ 0.5, a common linear sweep yields
near-Gaussian superposition components with integrated
areas of 0.502 and 0.495 for the equal superposition state,
and 0.334 and 0.658 for the unequal state. The premature
loss probability is less than O(10−2). Following curve-
fitting, the mode mismatch are ξ = 0.012 and 0.013,
respectively. Notably, the tuning time is much shorter
than both the confinement time and the lapse between
the pulses.
β= E2
sand Θ is defined previously following Eq. 13. We
|βs?Ω2
s/?2+ (E2
|βs?/?2− E|βs?∆s/? − g2
s]2+
sΩ2
?2/3
FIG. 11:
Hsys (Eq. 20) with Ξ-type, four-level atom s, for δq/κsq =
−15,∆q/κsq = −8.6,gs/κsq = 1,Ωs;i/κsq = 5 (∀i),gq/κsq =
10. Solid curves are the eigenenergies of the combined
coupled-cavity system and the dashed line denote the energy
of state |+q?. The dash-dotted line indicates the energy of
atom q (= ?∆q), intersects at points (i.e. ∆off
ing to two-photon resonance with atom q. The anticrossings
along the dashed lines (i.e. ∆res
s;i) correspond to resonances be-
tween eigenstates of system s with |+q?. (b) Corresponding
pulse shape of the output photon in three-time, unoptimized
equal superposition state.
(a) Eigensystem of the extended Hamiltonian
s;i) correspond-
Multi-time superposition:– We now generalize our results
by considering a Ξ-type, N-level atom s with states
|gs?,|es;1?,|es;2?,...,|es,N−1?.
tion is still coupled to cavity mode s with rate gswhilst
each excited-state pair (the |es;i? ↔ |es;i+1? transition)
is coupled via a quasi-resonant field with rate Ωs;i and
detuning δs;i. The Hamiltonian at site s becomes
The |gs? ↔ |es;1? transi-
Hs/? = ∆s|es;1??es;1| +
N−1
?
N−2
?
i=2
(∆s+ δs;i)|es;i??es;i|(20)
+ gs|es;1??gs|as+
i=1
Ωs;i|es;i+1??es;i| + h.c.
Resonances between N−1 eigenstates of system s and the
one of the states |±q? can be set up at different values of
∆s= ∆res
with state |+q? for N = 4. Therefore, after initializing
system s in a superposition of these eigenstates, atom s
can be tuned to induce resonances in the coupled-cavity
systems in series. Similarly, system s becomes maximally
decoupled from system q via two-photon resonance with
atom q. In principle, these mechanisms allow one to pre-
pare arbitrary temporal superposition states with N − 1
time-bins on demand. Three-time, unoptimized super-
position state is shown in Fig. 11(b). To ensure that
switching time is not too long, optimization is neces-
sary to maximize the values of the coupling matrix el-
ement J. We emphasize that a huge parameter space
{gs,Ωs;i,gq,δs;i,δq,∆q} should offer much room for this
purpose.
For the sake of definitiveness, let us consider possible
experimental realizations. Although we explicitly use Ξ-
type atoms, one could use N-type N = 4 and W-type
N = 5 atoms etc, in our discussion, other atomic con-
figurations such as Y-type N = 4, tripod N = 4 and
s;i. In Fig. 11(a), we observe three anticrossings

Page 9

9
quadrupod N = 5 can also be used. However, we will not
delve further into discussing each of these schemes, apart
from noting the abundance of systems for experimental
work. In particular, the NV can realize Λ- and tripod-
type schemes by using some or all available sublevels
(m = 0,±1) in the spin triplet (3A) ground state [57]. Al-
though unsuitable for practical solid-state applications,
rubidium [82], which has rich electronic ladder structures,
and sodium with Y -type energy levels [83], can be used
for proof-of-concept experiments.
VI.CONCLUSION
Advances in fabrication and novel cavity designs are
nearing the stage where exploration and applications of
multiple coupled atom-cavity systems are beginning to
be accessible. We have studied the dynamic coupling be-
tween two evanescently-coupled cavities realized by tun-
ing one of the intracavity atoms. Of practical interest
to distributed QIP, this mediating atom-field system can
be employed to realize high fidelity excitation confine-
ment, Q-switching and reversible state transport of single
photons. We also have shown that such control can be
used to shape single photons and engineer near-Gaussian
pulses that are distinct from the usual time-asymmetric
photons emitted from a passive cavity-based system.
The influence of cavity loss and atomic spontaneous
emissions on the performance of Q switching is quan-
titatively characterized, and we have shown that these
schemes are compatible with the state-of-the-art cavity
designs and atomic control capabilities in solid state. In
particular, when implemented with diamond centres in
PBG cavities, the effective cavity-Q of qubit cavities can
be switched in 10 ns time scale with Q = 106with a suc-
cess probability of 0.9. A shorter nanosecond switching is
expected by using slot-waveguide cavities with Q = 105
for a probability of > 0.99. In the microwave regime, 1µs
switching is possible with stripline cavities of similar Q.
By applying Q-switching to an extended coupled-
cavity system involving a multi-level atom, arbitrary
temporally encoded superposition states can be prepared
in a deterministic way. These integrated-photonic meth-
ods for generating high quality single photons and pho-
tonic qudits would be useful resources for numerous op-
tical and general QIP applications.
Acknowledgments
We thank J. H. Cole, Z. W. E. Evans, S. J. Devitt,
A. M. Stephens, S. Tomljenovic-Hanic, C. D. Hill, and
M. I. Makin for helpful discussions. W.J.M. and K.N.
acknowledge the support of QAP, HIP, MEXT, NICT
and HP. C.H.S., A.D.G. and L.C.L.H. acknowledge the
support of Quantum Communications Victoria, funded
by the Victorian Science, Technology and Innovation
(STI) initiative, the Australian Research Council (ARC),
and the International Science Linkages program. A.D.G.
and L.C.L.H. acknowledge the ARC for financial support
(Projects No. DP0880466 and No. DP0770715, respec-
tively).
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