Transfer of a Polaritonic Qubit through a Coupled Cavity Array
Transfer of a Polaritonic Qubit through a Coupled Cavity Array
Sougato Bose 1, Dimitris G. Angelakis2,∗ and Daniel Burgarth1,3
1Department of Physics and Astronomy,
University College London, Gower St., London WC1E 6BT, UK
2Centre for Quantum Computation,
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Wilberforce Road, CB3 0WA, UK and
3Computer Science Departement, ETH Zürich,
CH-8092 Zürich, Switzerland
Abstract
We demonstrate a scheme for quantum communication between the ends of an array of coupled
cavities. Each cavity is doped with a single two level system (atoms or quantum dots) and the
detuning of the atomic level spacing and photonic frequency is appropriately tuned to achieve
photon blockade in the array. We show that in such a regime, the array can simulate a dual rail
quantum state transfer protocol where the arrival of quantum information at the receiving cavity
is heralded through a fluorescence measurement. Communication is also possible between any pair
of cavities of a network of connected cavities.
∗Electronic address: dimitris.angelakis@qubit.org
http://arxiv.org/abs/0704.0984v1
mailto:dimitris.angelakis@qubit.org
I. INTRODUCTION
Recently, the exciting possibility of coupling high Q cavities directly with each other
has materialized in a variety of settings, namely fiber coupled micro-toroidal cavities [1],
arrays of defects in photonic band gap materials (PBGs) [2, 3] and microwave stripline
resonators joined to each other [4]. A further exciting development has been the ability to
couple each such cavity to a quantum two-level system which could be atoms for micro-
toroid cavities, quantum dots for defects in PBGs or superconducting qubits for microwave
stripline resonators[5]. Possibilities with such systems are enormous and include the the
implementation optical quantum computing [6], the production of entangled photons [7],
the realization of Mott insulating and superfluid phases and spin chain systems [8, 9, 10] .
Such settings can also be used to verify the possibilities of distributed quantum computation
involving atoms coupled to distinct cavities [11] also to generate cluster states for efficient
measurement based quantum computing schemes[12].
When the coupling between the cavity field and the two-level system (which we will just
call atom henceforth, noting that they need not necessarily be only atoms) is very strong (in
the so called strong coupling regime), each cavity-atom unit behaves as a quantum system
whose excitations are combined atom-field excitations called polaritons. The nonlinearity
induced by this coupling or as it is otherwise known, the photon blockade effect[13], forces the
system to a state where maximum one excitation (polariton) per site is allowed. However, a
superposition of two different polaritons, which is equivalent to a superposition of two energy
levels of the cavity-atom system, is indeed allowed and naturally the question arises as to
whether that can be used as a qubit. Purely atomic qubits (formed from purely atomic energy
levels) in cavities have long been discussed in the literature (see references cited in [11], for
example), but such qubits in distinct cavities do not directly interact with each other unless
mediated through light. On the other hand, a purely photonic field in a cavity is not easy to
manipulate in the sense of one being able to create arbitrary superpositions of its states by an
external laser. Being a mixed excitation, polaritons interact with each other as well as permit
easy manipulations with external lasers in much the same manner as one would manipulate
and superpose atomic energy levels. Is there any interesting form of quantum information
processing that can be performed by encoding the quantum information in a superposition of
polaritonic states? While an ultimate aim might be to accomplish full quantum computation
|e, 1〉; |g, 2〉
|e, 0〉; |g, 1〉
|g, 0〉
ω0 − g P (−)†
|e, 1〉; |g, 2〉
|e, 0〉; |g, 1〉
|g, 0〉
ω0 − g
FIG. 1: A series of coupled cavities coupled through light and the polaritonic energy levels for
two neighbouring cavities. These polaritons involve an equal mixture of photonic and atomic
excitations and are defined by creation operators P
(±,n)†
= (|g, n〉k〈g, 0|k ± |e, n − 1〉k〈g, 0|k)/
where |n〉k, |n − 1〉k and |0〉k denote n, n − 1 and 0 photon Fock states in the kth cavity. The
polaritons of the kth atom-cavity system are denoted as |n±〉k and given by |n±〉k = (|g, n〉k ±
|e, n − 1〉k)/
2 with energies E±n = nωd ± g
with polaritonic qubits (it has been recently shown this to possible using the cluster state
approach [12]), we concentrate here on a more modest aim of transferring the state of a
qubit encoded in polaritonic states (a polaritonic qubit) from one end of the coupled cavity
array to another.
Assume a chain of N coupled cavities. We will describe the system dynamics using
the operators corresponding to the localized eigenmodes (Wannier functions), a
k(ak). The
Hamiltonian is given by
kak +
kak+1 +H.C.). (1)
and corresponds to a series quantum harmonic oscillators coupled through hopping photons.
The photon frequency and hopping rate is ωd and A respectively and no nonlinearity is
present yet. Assume now that the cavities are doped with two level systems (atoms/ quantum
dots/superconducting qubits) and |g〉k and |e〉k their ground and excited states at site k.
The Hamiltonian describing the system is the sum of three terms. Hfree the Hamiltonian
for the free light and dopant parts, H int the Hamiltonian describing the internal coupling of
the photon and dopant in a specific cavity and Hhop for the light hopping between cavities.
Hfree = ωd
kak + ω0
|e〉k〈e|k (2)
H int = g
k|g〉k〈e|k +H.C.) (3)
Hhop = A
kak+1 +H.C) (4)
where g is the light atom coupling strength. The Hfree +H int part of the Hamiltonian can
be diagonalized in a basis of mixed photonic and atomic excitations, called polaritons (Fig.
1). While |g, 0〉k is the ground state of each atom cavity system, the excited eigenstates of
the kth cavity-atom system are given by |n±〉k = (|g, n〉k ± |e, n − 1〉k)/
2 with energies
E±n = nωd ± g
n. One can then define polariton creation operators P
(±,n)†
k by the action
(±,n)†
k |g, 0〉k = |n±〉k. As we have proved elsewhere, due to the blockade effect, once a site
is excited to |1−〉 or |1+〉, no further excitation is possible[8]. In simplified terms, this is
because it costs more energy to add another excitation in already filled site so the system
prefers to deposit it if possible to an a nearby empty site. This effect has recently lead to
the prediction of a Mott phase for polaritons in coupled cavity systems[8]. If we restrict to
the low energy dynamics of the system such that states with n ≥ 1 are not occupied, which
can be ensured through appropriate initial conditions, the Hamiltonian in becomes (in the
interaction picture):
HI = A
k+1 + A
k+1 +H.C. (5)
where P
k = P
(±,1)†
k is the polaritonic operator creating excitations to the first polaritonic
manifold (Fig. 1). In deriving the above, the logic is that the terms of the type P
which inter-convert between polaritons, are fast rotating and they vanish[8].
We are now in a position to outline the basic idea behind the protocol. A qubit is encoded
as a superposition of the polaritonic states |1+〉 and |1−〉 in the first cavity. The multi-
cavity system is then allowed to evolve according to HI . At the receiving cavity at the other
end we then do a measurement inspired by a dual rail quantum state transfer protocol [14]
which heralds the perfect reception of the qubit for one outcome of the measurement, while
for the other outcome of the measurement the process is simply to be repeated once more
after a time delay. Before presenting the scheme in detail, let us first present a special set of
initial conditions under which HI describes the dynamics of two identical parallel uncoupled
spin chains.
Suppose we are restricting our attention to a dynamics in which the initial state is ob-
tained by the action of only one of the operators among P
k and P
k on the state
k |g, 0〉k which has all the sites in the state |g, 0〉. As P
k does not act after P
k has
acted and vice versa, under the above restricted initial conditions, the system is going to
evolve only according to one of the terms in Eq.(5) i.e., only according to the first or the
second term. To be more precise, if we start with a state P
k |g, 0〉k only the term
k=1 P
k+1 is going to be active and cause the time evolution, while if we start with
the state P
k |g, 0〉k only the term A
k=1 P
k+1 will be responsible for the time
evolution. Each of the operators P
k and P
k individually have the same algebra as the
Pauli operator σ+k = σ
k + iσ
k , which makes both the parts of the Hamiltonian individually
equivalent to a XY spin chain with a Hamiltonian HXY = A
k+1 + σ
k+1). The
restricted set of initial states mentioned above can be mapped on to those of two parallel
chains of spins labeled as chain I and chain II respectively. Let |0〉 and |1〉 be spin-up and
spin-down states of a spin along the z direction, |0〉(I)|0〉(II) be a state with all spins of
both chains being in the state |0〉, |k〉(I)|0〉(II) represent the state obtained from |0〉(I)|0〉(II)
by flipping only the kth spin of chain I and |0〉(I)|k〉(II) represents the state obtained from
|0〉(I)|0〉(II) by flipping only the kth spin of chain II. Then, the restricted class of initial
conditions for polaritonic states can be mapped on to states of the parallel spin chains as
|g, 0〉1|g, 0〉2....|g, 0〉N → |0〉I |0〉II , (6)
|g, 0〉1..|g, 0〉k−1|1+〉k|g, 0〉k+1..|g, 0〉N → |k〉(I)|0〉II , (7)
|g, 0〉1..|g, 0〉k−1|1−〉k|g, 0〉k+1..|g, 0〉N → |0〉I |k〉(II) (8)
Under the above mapping and under the above restrictions on state space, HI becomes
equivalent to the Hamiltonian of two identical parallel XY spin chains completely decoupled
from each other. Precisely such a Hamiltonian is known to permit a heralded perfect quan-
tum state transfer from one end of a pair of parallel spin chains to the other [14], and we
discuss that below.
Spin chains are capable to transmitting quantum states by natural time evolution [15].
However it is well known that due to the disperion on the chain [16] the fidelity of transfer
is quite low except for specific engineered couplings in the spin chains [17, 18] or when the
receiver has access to a significant memory [19]. The advantage of the polariton system is
that we have two parallel and identical chains. We have recently shown how this can be
made use of in a dual rail protocol [14]. The main idea of this protocol is to encode the
state in a symmetric way on both chains. The sender Alice encodes a qubit α|0〉+ β|1〉 to
be transmitted as
|Φ(0)〉 = α|0〉(I)|1〉(II) + β|1〉(I)|0〉(II), (9)
which evolves with time as
|Φ(t)〉 =
f1j(t)(α|0〉(I)|j〉(II) + β|j〉(I)|0〉(II)), (10)
where f1j is the transition amplitude of a spin flip from the 1st to the jth site of a chain.
Clearly, if after waiting a while Bob performs a joint parity measurement on the two spins
at his (receiving) end of the chain and the parity is found to be “odd”, then the state of the
whole system will be projected to α|0〉(I)|N〉(II) + β|N〉(I)|0〉(II), which implies the perfect
reception of Alice’s state (albeit encoded in two qubits now). The protocol presented in
Ref.[14] in fact suggested the use of a two qubit quantum gate at Bob’s end which measured
both the parity as well as mapped the state to a single qubit state. However, here the
presentation as above suffices for what follows. Physically, this protocol, which is called
the dual rail protocol, allows one to perform measurements on the chain that monitor the
location of the quantum information without perturbing it. As such it can also be used for
arbitrary graphs of spins (as long as there are two identical parallel graphs) with the receiver
at any node of the graph. Furthermore, for the Hamiltonian at hand (XY spin model) it is
known [20] that the probability of success converges exponentially fast to one if the receiver
performs regular measurements. The time it takes to reach a transfer fidelity F scales as
t = 0.33A−1N5/3| ln(1− F )|. (11)
The difference between our current coupled cavity system and the spin chain system
considered in [14] is that in our case, the two chains are effectively realized in one sys-
tem. Therefore, it is not necessary to perform a two-qubit measurement such as a parity
measurement at the receiving ends of the chain. The qubit to be transferred is encoded as
′|1+〉1 + β
′|1−〉1 ≡ α|e, 0〉1 + β|g, 1〉1. This state can be created by the sender Alice using
a resonant Jaynes-Cummings interaction between the atom and the cavity field. Then the
whole evolution will exactly be as in Eq.(10) with the spin chain states have to be replaced
by polaritonic states according to the mapping given in Eqs.(6)-(8). The measurement to
herald the arrival of the state at the receiving end is accomplished by a exciting (shelving)
|g, 0〉 repeatedly to a metastable state by an appropriate laser (which does not do anything
if the atom is either in |1±〉). The fluorescence emitted on decay of the atom from this
metastable state to |g, 0〉 implies that another measurement has to be done after waiting
a while. No fluorescence implies success and completion of the perfect transfer of the po-
laritonic qubit. Interestingly enough, the measurement at the receiving cavity need not
be snapshot measurements at regular time intervals, but can also be continuous measure-
ments under which the scheme can have very similar behavior to the case with snap-shot
measurements for appropriate strength of the continuous measurement process [21].
We now briefly discuss the parameter regime needed for the scheme of this paper. In order
to achieve the required limit of no more than one excitation per site, the parameters should
have the following values[8]. The ratio between the internal atom-photon coupling and the
hopping of photons down the chain should be g/A = 102. We should be on resonance,
∆ = 0, and the cavity/atomic frequencies ωd, ω0 ∼ 104g which means we should be well in
the strong coupling regime. The losses should also be small, g/max(κ, γ) ∼ 103, where κ
and γ are cavity and atom/other qubit decay rates. These values are expected to be feasible
in both toroidal microcavity systems with atoms and stripline microwave resonators coupled
to superconducting qubits [5], so that the above states are essentially unaffected by decay
for a time 10/A (10ns for the toroidal case and 100ns for microwave stripline resonators type
of implementations).
We conclude with a brief discussion about the positive features of the scheme and sit-
uations in which the scheme might be practically relevant. The scheme combines the best
aspects of both atomic and photonic qubits as far as communication is concerned. The
atomic content of the polaritonic state enables the manipulation to create the initial state
and measure the received state of the cavity-atom systems with external laser fields, while
the photonic component enables its hopping from cavity to cavity thereby enabling transfer.
Unlike quantum communication schemes where an atomic qubit first has to be mapped to
the photonic state in the transmitting cavity and be mapped back to an atomic state in
the receiving cavity by external lasers, here the polaritonic qubit simply has to be created.
Once created, it will hop by itself though the array of cavities without the need of further
external control or manipulation.
In what situations might such a scheme have some practical utility? One case is when
Alice “knows” the quantum state she has to transmit to Bob. She can easily prepare it as
a polaritonic state in her cavity and then let Bob receive it through the natural hopping
of the polaritons. Another situation is when a multiple number of cavities are connected
with each other through an arbitrary graph. The protocol of Ref.[14] still works fine in
this situation with Alice’s qubit being receivable in any of the cavities simply by doing the
receiving fluorescence measurements in that cavity.
We acknowledge the hospitality of Quantum Information group in NUS Singapore, and
the Kavli Institute for Theoretical Physics where discussions between DA and SB took place
during joint visits. This work was supported in part by the QIP IRC (GR/S82176/01),
the European Union through the Integrated Projects QAP (IST-3-015848), SCALA (CT-
015714) and SECOQC., and an Advanced Research Fellowship from EPSRC.
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http://arxiv.org/abs/quant-ph/0606033
http://arxiv.org/abs/quantum-ph/0410189
http://arxiv.org/abs/quant-ph/0606157
http://arxiv.org/abs/quant-ph/0702133
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	Introduction
	References
