Quantum engineering of photon states with entangled atomic ensembles
D. Porras and J. I. Cirac
Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany
(Dated: November 4, 2018)
We propose and analyze a new method to produce single and entangled photons which does not
require cavities. It relies on the collective enhancement of light emission as a consequence of the
presence of entanglement in atomic ensembles. Light emission is triggered by a laser pulse, and
therefore our scheme is deterministic. Furthermore, it allows one to produce a variety of photonic
entangled states by first preparing certain atomic states using simple sequences of quantum gates.
We analyze the feasibility of our scheme, and particularize it to: ions in linear traps, atoms in optical
lattices, and in cells at room temperature.
PACS numbers: PACS
The deterministic generation of collimated single and
entangled photons is of crucial importance in Quantum
Information, like in quantum cryptography [1], quantum
computation [2], quantum lithography [3] or quantum
interferometry [4, 5]. Most of the methods tested so far
require high-Q cavities, something which is very demand-
ing in practice [6, 7, 8, 9]. The engineering of quantum
states in atomic systems is now possible thanks to the
experimental progress experienced by the field of Atomic
Physics during the last years. In fact, with trapped ions
it has been already possible to create so–called W [10]
and GHZ [11] states of up to 8 ions. At the same time,
scientists have been able to produce other kinds of entan-
gled states [12] with atoms in optical lattices. Further-
more, with the advent of Rydberg techniques [13] it will
soon be possible to create W–like states in that system or
in atomic ensembles at room temperature. Apart from
their fundamental interest, some of those states may have
applications in precision spectroscopy [14, 15].
In this work we show that the ability of creating those
atomic states may have a strong impact in different sub-
fields of quantum information, as it may lead to a very
efficient way of creating certain kind of entangled pho-
tonic states which are required in various applications.
The main idea is to use a laser and an internal level con-
figuration such that we can map the atomic state onto
photonic states corresponding to modes propagating in a
well defined direction. Our scheme uses the well known
fact [16, 17] that, under certain circumstances, light scat-
tering takes place predominantly in the forward direction
due to an interference effect. In fact, this effect is the ba-
sis of one of the building blocks of the repeater scheme
proposed in [18], and has been recently demonstrated in
a series of experiments [19, 20, 21]. There, a single ex-
citation is created in an atomic ensemble by detecting a
photon emission in a certain direction. Then, the excita-
tion is released in the forward direction by using a laser.
Building on this fact, we propose to create certain kind
of excitations by using quantum gates or atomic interac-
tions, which give rise to the desired entangled states when
they are released using a laser, and which propagate in
the desired direction due to the mentioned interference
effect.
Let us consider a set of N atoms with (ground) hype-
fine levels |g〉 and |sa,b〉 (see Fig. 1 (a)). We consider
states of the form
|k(na)a ,k
b 〉 =
na! nb!
)na (
|0〉, (1)
and linear combinations thereof. Here, |0〉 = |g〉1 . . . |g〉N ,
e−ikxr
x,j , x = a, b, (2)
where σ
x,j excites an atom from |g〉j to |sx〉j , and r0j are
the equilibrium position of the atoms. In the limit nx ≪
N , Eq. (1) defines a set of orthonormal collective states
with nx atoms excited in |sx〉 and linear momentum kx.
Those states can be indeed readily created using trapped
ions or Rydberg techniques (see Appendix A1,3).
In order to release the photons, one sends a laser pulse
of wavevector kL which couples level |sx〉 to some elec-
tronically excited ones |ex〉, respectively. The large pop-
ulation of level |g〉 together with the initial entangle-
ment (coherences) between the atoms, will now stimu-
late the emission of photons from the excited states to
the level |g〉, which overall will produce the mapping be-
tween these states and the photonic states,
|k(na)a ,k
b 〉 → |na〉ka+kL,σa |nb〉kb+kL,σb ; (3)
that is, (1) is mapped to a Fock state of nx photons with
momenta kx + kL and polarization σx, where σx is the
polarization of the light in each decay channel. More-
over, due to the linearity of this process, superpositions
of states of the form (1) will be mapped onto superpo-
sitions of photonic states (3). For example, the atomic
state
|k(1),q(1)〉+ |q(1),k(1)〉
2 will emit a pair of
entangled photons in different directions. The mapping
(3) is strictly valid under ideal conditions, and in the limit
N → ∞, and the directionality in the photon emission is
directly connected to the momentum conservation which,
in turn, is a consequence of the constructive interference
in the field emitted by each atom. Thus, the crucial is-
sue in our scheme is to determine how this mapping is
http://arxiv.org/abs/0704.0641v2
modified in finite atomic ensembles under nonideal con-
ditions. In the following we analyze such questions in
detail, concentrating in the simplest case in which we
have a single excitation with momentum |k0〉 in |sa〉 (i.e.
our initial state is a W-like state) and thus we produce
a single photon. We determine a function f(Ω), which is
proportional to the probability density that the photon
is emitted in the direction Ω. In general, f = fcoh + finc;
that is, it is the sum of a coherent contribution and an
incoherent one. The later appears whenever the posi-
tions of the particles fluctuate. fcoh contains the forward
scattering contribution, which is emitted in a cone with
a width ∆Ω that decreases with the number of particles.
finc, on the contrary, describes isotropic light emission,
thus, even when the light emitted in ∆Ω is collected, the
contribution finc leads to a limitation in the efficiency of
the setup. To quantify the error probability, we define
dΩ finc(Ω)
dΩ f(Ω)
. (4)
As long as the number of excited atoms is small nx ≪ N ,
this analysis can be easily generalized to the emission of
states with many photons (1). One obtaines that the
overall error is bounded by 1− (1− E)na+nb .
The emission pattern can be obtained by studying the
Heisenberg equations of motion of the field operators.
The calculation inolves the study of the decay of the
atomic state under collective effects (see Appendix C).
To simplify our analysis we ignore the dipole pattern, in
which case we get:
f(Ω) =
i,j=1
〈e−i(kLnΩ−kL)(ri−rj)〉eik0(r
). (5)
rj are the coordinate operators of the atoms, and thus Eq.
(5) allows us to describe fluctuations in the position of the
particles during the emission of light. In the following,
we will show three different experimental set–ups where
our scheme can be implemented. In order to analyze
the performance in each of them, we first particularize
the above formula to three different situations which are
directly connected with those set–ups. We will focus on
the angular width of the forward–scattering cone, ∆Ω,
which measures the collimation of the emitted photons,
and the error probability, E , as figures of merit. Then
we will introduce the possible implementations and will
use those formulas to specify the conditions for them to
correctly operate.
(i) Fixed atomic positions. In the case of a square
lattice of particles trapped in 3D (see Fig. 1 (b)), the
emission pattern is given by
f0(Ω) =
α=x,y,z
sin2((kLn
Ω − (kL+k0)α)d0Nα/2)
sin2((kLn
Ω − (kL+k0)α)d0/2)
, (6)
with Nα the number of atoms in each direction. f0(Ω)
has a series of diffraction peaks, which are reduced to a
single one if d0 < λ/2. In this regime, the emission is cen-
tered in a cone with nΩ in the direction of kL+k0. Note
that for simultaneous energy and momentum conserva-
tion condition |kL + k0| = kL has to be fulfilled. Since
the positions of the atoms do not fluctuate, f0 has only
a coherent contribution (E = 0), and the only limitation
for the effiency of the setup is the width of the emission
cone, which scales in 3D like ∆θ3D ≈ 1/(N1/3kLd0). In
the case of a chain of atoms (1D) momentum is conserved
only along the direction of the chain. Photon emission
can be still directed efficiently along the axis of the chain,
in a cone whose width scales like ∆θ1D ≈ 1/
NkLd0.
(ii) Fluctuating atomic positions. Let us consider a
lattice of atoms at temperature T , trapped by indepen-
dent harmonic potentials. The emission pattern is now
the sum the of two contributions,
fcoh(Ω) = f0(Ω)gT (Ω), finc(Ω) = 1− gT (Ω). (7)
gT (Ω) = e
−((kLnΩ−kL)ξT )
, and ξT is the vector whose
components are the size of the position fluctuations in
each spatial direction, (ξαT )
2 = x20(1 + 2n
T ), with x
0 the
size of the ground state in harmonic potential, and nαT
the number of motion quanta at T . Light scattered into
finc represents an important fraction whenever ξ
T ≫ d0.
In this case, the emission of light is centered around kL,
since the uncertainty in the position of the particles av-
erages out the intial linear momentum k0. The scaling of
E in this regime strongly depends on the dimensionality
of the system. In particular, in the case of a chain of
atoms, E1D = d0/λ, whereas in the square 3D lattice, we
get E3D ≈ 12.6(d0/λ)2N−1/3.
(iii) Statistical distribution of particles. Consider an
ensemble of atoms (see Fig. 1 (c)), which move inside
a square box of size L, such that their motion is faster
than their radiative decay, that is, their average velocity
v is such that vL ≫ Γ, with Γ the emission rate. This
situation can be described by assuming that the atoms
are in a statistical distribution with equal probability to
be at any point in the box. The situation is thus similar
to that of a thermal state,
fcoh (Ω) = Ngbox(Ω), finc (Ω) = 1− gbox(Ω), (8)
where
gbox(Ω) =
α=x,y,z
sinc2 (L(kLn
Ω − kαL)) . (9)
Defining the average distance between particles like d0 =
LN−1/3, we find the same scalings of ∆θ as in case (i),
and of E , as in case (ii). Trapping schemes for atomic
ensembles are simpler to realize but face the difficulty
that conditions for the directionality of photon emission
are more stringent. In the case of a lattice of particles at
fixed positions, forward–scattering is ensured whenever
condition d0 < λ/2 is fulfilled. On the contrary, in the
case of atomic ensembles, the incoherent contribution finc
has to be small enough such that E ≪ 1, which implies
FIG. 1: (a) Level configuration for the release of atomic en-
tangled states in photonic channels. (b) Release of a collec-
tive state with linear momentum k0, that has been gener-
ated in a lattice of atoms. (c) Emission of photons from an
atomic ensemble, which consists of an incoherent contribu-
tion (isotropic), and a coherent one in the forward–scattering
direction.
d0 ≪ λ in 1D, or, alternatively, a number of particles
large enough in 3D.
Now we introduce three experimental set–ups where
our scheme can be implemented. In the Appendix we
show how to create the atomic states that we are consid-
ering here.
Trapped ions. This system is ideally suited to create
collective states like (1), as was demonstrated recently
in ref. [10]. Most usually ions are arranged in chains,
such that we deal with the 1D situation discussed above.
Even though trapped ions are not equally spaced, under
the condition d̄0 < λ/2, with d̄0 the average distance,
we still get light emission in the forward–scattering cone
only, see Fig. 2. Considering two different internal levels,
which can correspond to different states in an hyperfine
multiplet, states such as those defined by Eq. (1) can be
created by a number of quantum operations that scales
linearly with the number of ions N (see A1). For exam-
ple, the state 1/
2 (|0, 2kL〉+ |2kL,0〉), would emit two
photons in the forward and backward directions along the
chain axis, entangled in polarization. The main difficulty
for the implementation of this idea with ions lies on the
fact that ion–ion distances are usually in the range of a
few µm, and thus condition d0 < λ/2 is not fulfilled when
considering optical wavelengths. A way out of this prob-
lem is to use optical transitions which lie in the range of
λ & 5µm, which can be found in ions such as Hg+, Ba+,
or Yb+ [22].
Cold atoms in optical lattices. By using optical lat-
tices we fulfill the need of placing atoms at interparticle
distances comparable to optical wavelengths, since po-
tential wells in a standing–wave are indeed separated by
d0 = λsw/2, with λsw, the wavelength of the counterprop-
agating lasers. By using an optical transition such that
λ > λsw, we are in the regime in which light emission
FIG. 2: Probability of photon emission from an ion chain with
N = 30 ions initially in a W –state. The blue line corresponds
to a chain with equally spaced ions with two diffraction peaks.
Black and red lines corresponds to an ion Coulomb chain, in
which ions are in an overall trapping potential and thus are
not equally spaced. However, in the case that the average dis-
tance, d̄0 is small enough, light is also preferentially emitted
in the forward–scattering direction.
is focused into a single Bragg peak. Although one could
think of peforming quantum gates between ultracold neu-
tral atoms to generate collective atomic states [23], this
procedure faces the difficulties of quantum computation
in this system, like for example how to achieve single
atom addressability. More efficiently, one could avoid
the use of quantum gates by using the dipole–blockade
mechanism with Rydberg atoms, which allows us to gen-
erate W-states, as well as states which emit Fock states
with a number M of photons [13] (see Appendix 3).
Atomic ensembles at room temperature. The very same
techniques which can be applied to Rydberg atoms in an
optical lattice can also be used in the case of hot en-
sembles. On the one hand, this setup has the advantage
that atoms do not need to be cooled and placed in an
optical lattice. On the other hand, it can be described
by a statistical distribution of particles, and thus suffers
from the fact that high efficiency in the release of pho-
tons is achieved under more severe conditions of particle
density and atom number, as discussed above. However,
densities which are high enough to fulfill the requirement
E ≪ 1 have been recently reported in [24].
In conclusion, we have proposed to use current tech-
niques for quantum engineering to generate atomic multi-
partite entangled states which can be efficiently mapped
into photonic states. Our proposal relies on the release of
spin–wave like excitations into a given spatial direction
by means of interference effect, and can be implemented
with trapped ions, atoms in optical lattices, and atomic
ensembles at room temperature.
This work was supported by E.U. projects (SCALA
and CONQUEST), and the Deutsche Forschungsgemein-
schaft.
APPENDIX A: CREATION OF COLLECTIVE
ATOMIC STATES IN A CHAIN OF ATOMS
Entangled states of the form (1) and their linear com-
binations can be generated in a chain of particles, for
example, of trapped ions, by means of a limited number
of quantum operations. To demonstrate this, we first
show that they can be written as Matrix Product States
with a small bond dimension D, i.e. they can be written
|Ψ〉 =
i1,...,iN
〈ΦF|V iN[N ] . . . V
|ΦI〉|i1〉 . . . |iN 〉, (A1)
In (A1), the indices ij = g, sa, sb, and V
are D × D
matrices acting on an auxiliary D–dimensional Hilbert
space. D is given by the number of states which appear
in the singular value decomposition (s.v.d.) of |Ψ〉 at
any site in the chain [25]. As it is shown in [26], the state
(A1) can be prepared by performing N gates which act
on [log2 D]+1 qubits. Thus, as long as D is independent
of N , the number of gates to be applied scales linearly
with the total number of atoms.
To evaluate D, consider first the case of a state like (1)
with atoms excited in level sa only, and a partition of the
chain in two parts L and R. We get na + 1 states in the
s.v.d. with respect to this partition, which correspond to
states with a number of excited atoms in part L, ranging
from 0 to na. This result is easily generalized to a linear
combination ofM states of the form (1), in which case we
get D = M(na + 1)(nb + 1). For example, an entangled
state of the form:
|Ψ〉 = 1√
|kna=1,qnb=1〉+ |qna=1,knb=1〉
, (A2)
has D = 8.
APPENDIX B: QUANTUM STATE
ENGINEERING WITH RYDBERG BLOCKADE
Interactions between excited atomic states, like those
that take place in Rydberg atoms, can be used to the cre-
ate the states defined by Eq. (A2). This can be achieved
in a single experimental step, without the need for quan-
tum gates, if the proper configuration of atomic inter-
actions is chosen. As an example, consider the 3 level
configuration shown in Fig. 1 (a), and interactions be-
tween excited states such that atoms in levels |sa〉, |sb〉,
interact strongly only if they are in the same excited
state, that is, Uaa = Ubb = U , but Uab = 0. We ap-
ply two lasers with wavectors k1,2 and Rabi frequencies
Ω1,2, detuned with respect to the |g〉 – |sa,b〉 transition,
such that ∆1 = −∆2 = ∆. If condition ∆1,2 ≫ Ω1,2
is fulfilled, then the lasers induce a two–photon transi-
tion with Rabi frequency Ωeff = Ω1Ω2/∆. Furthermore,
if Ωeff ≪ U , states with two atoms in the same excited
state are not populated. Under these conditions there
are two possible excitation channels, depicted in Fig. 3,
which give rise to the linear combination (A2).
FIG. 3: Lasers and level configuration for the creation of
atomic entangled states which emit pairs of photons entan-
gled in polarization.
APPENDIX C: CALCULATION OF THE
PHOTON DISTRIBUTION
We consider for simplicity the lambda configuration
depicted in Fig. 1 (a), considering a single excited state
|s〉, and a single auxiliary level |e〉. The interaction of
the quantized electromagnetic field with the ensemble of
atoms, after the adiabatic elimination of level |e〉, is de-
scribed by
j,k,λ
jak,λe
i(k−kL)rj+iωLt + h.c.
gk,λ =
(ǫkλ · dge) , (C1)
σj refers to the |g〉 – |s〉 atomic transition, ΩL and kL
are the Rabi frequency and wave–vector of the classical
field, respectively, ωk is the photon energy, ǫk,λ are the
polarization vectors, and dge is the dipole matrix element
for the |g〉 – |e〉 transition.
The probability of photon emission is proportional to
the the diagonal elements of the one–photon density ma-
trix, which are obtained from the Heisenberg equation of
motion for the field operators,
ak〉 =
dτ1dτ2e
−i(ωk−ωL)(τ1−τ2)
〈e−i(k−kL)(ri−rj)〉〈σ†i (τ1)σj(τ2)〉. (C2)
Since we are interested in the conditions for momentum
conservation due to interference effects, we consider the
following atomic initial state,
|k0〉 = σ†k0 |0〉, σ
e−ik0r
j . (C3)
The emission pattern depends thus on the two–time
atomic correlation function, which in turns can be eval-
uated by means of a master equation which describes
the decay of the atomic levels. In the case of the ini-
tial atomic state k0 (C3), fixed atom positions, and ne-
glecting boundary effects, this correlation function can
be evaluated exactly,
〈σ†i (τ1)σj(τ2)〉 = e
−Γk(τ1+τ2)/2eik0(r
), (C4)
where we have neglected an energy shift due to dipole–
dipole interactions. Integrating (C2) over the absolute
value of k yields the probability of photon emission,
I(Ω) = Ī(Ω)f(Ω). (C5)
Ī(Ω) is the dipole pattern,
Ī(Ω) =
1− (negnΩ)2
, (C6)
where Γ is the single atom radiative decay rate, Γk0 is
the collective decay rate, neg is the unit vector of the
atomic transition, and nΩ is a unit vector in the direc-
tion defined by the solid angle Ω. The factor f(Ω) in I(Ω)
describes the interference between the emission from dif-
ferent atoms, and is given by Eq. (5).
Below we deduce the master equation which leads to
(C4) and we sketch its solution in the case of collective
states with a single excited atom.
APPENDIX D: MASTER EQUATION
The master equation for the reduced density matrix
of the internal levels, which describes the ratiative decay
of a set of atoms under the coupling to the quantized
radiation field given by Eq. (C1), is
∂tρ =
2 σiρσ
j − σ
i σjρ− ρσ
Gij [σ
i σj , ρ], (D1)
where the coupling constants depend on
Jij =
dτgij(τ)e
−iωLτ =
ei(ωk−ωL)τ+i(k−kL)r,(D2)
in the following way:
ℜ(Jij) =
Γij ,
ℑ(Jij) =
Gij . (D3)
The master equation (D1) can be solved for the particular
case of an initial state (C3) by noticing that the evolu-
tion of ρ is closed within the subspace spanned by the
states |k0〉, |0〉. This fact can be easily proved by direct
substitution of ρ(0) = |k0〉〈k0| in Eq. (D1), which yields
the following evolution for the atomic density matrix:
ρ(t) = e−Γk0 t|k0〉〈k0|+
1− e−Γk0 t
|0〉〈0|, (D4)
where the collective decay rate Γk0 is just the Fourier
transform of the coupling constants in the master equa-
tion,
Γk0 =
Γi,je
ik0(r
j). (D5)
A similar result holds for nondiagonal elements of ρ(t),
which together with the quantum regression theorem
yields the evolution of the atomic correlation function
(C4).
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http://arxiv.org/abs/quant-ph/0701120

  We propose and analyze a new method to produce single and entangled photons
which does not require cavities. It relies on the collective enhancement of
light emission as a consequence of the presence of entanglement in atomic
ensembles. Light emission is triggered by a laser pulse, and therefore our
scheme is deterministic. Furthermore, it allows one to produce a variety of
photonic entangled states by first preparing certain atomic states using simple
sequences of quantum gates. We analyze the feasibility of our scheme, and
particularize it to: ions in linear traps, atoms in optical lattices, and in
cells at room temperature.

Quantum engineering of photon states with entangled atomic ensembles
D. Porras and J. I. Cirac
Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany
(Dated: November 4, 2018)
We propose and analyze a new method to produce single and entangled photons which does not
require cavities. It relies on the collective enhancement of light emission as a consequence of the
presence of entanglement in atomic ensembles. Light emission is triggered by a laser pulse, and
therefore our scheme is deterministic. Furthermore, it allows one to produce a variety of photonic
entangled states by first preparing certain atomic states using simple sequences of quantum gates.
We analyze the feasibility of our scheme, and particularize it to: ions in linear traps, atoms in optical
lattices, and in cells at room temperature.
PACS numbers: PACS
The deterministic generation of collimated single and
entangled photons is of crucial importance in Quantum
Information, like in quantum cryptography [1], quantum
computation [2], quantum lithography [3] or quantum
interferometry [4, 5]. Most of the methods tested so far
require high-Q cavities, something which is very demand-
ing in practice [6, 7, 8, 9]. The engineering of quantum
states in atomic systems is now possible thanks to the
experimental progress experienced by the field of Atomic
Physics during the last years. In fact, with trapped ions
it has been already possible to create so–called W [10]
and GHZ [11] states of up to 8 ions. At the same time,
scientists have been able to produce other kinds of entan-
gled states [12] with atoms in optical lattices. Further-
more, with the advent of Rydberg techniques [13] it will
soon be possible to create W–like states in that system or
in atomic ensembles at room temperature. Apart from
their fundamental interest, some of those states may have
applications in precision spectroscopy [14, 15].
In this work we show that the ability of creating those
atomic states may have a strong impact in different sub-
fields of quantum information, as it may lead to a very
efficient way of creating certain kind of entangled pho-
tonic states which are required in various applications.
The main idea is to use a laser and an internal level con-
figuration such that we can map the atomic state onto
photonic states corresponding to modes propagating in a
well defined direction. Our scheme uses the well known
fact [16, 17] that, under certain circumstances, light scat-
tering takes place predominantly in the forward direction
due to an interference effect. In fact, this effect is the ba-
sis of one of the building blocks of the repeater scheme
proposed in [18], and has been recently demonstrated in
a series of experiments [19, 20, 21]. There, a single ex-
citation is created in an atomic ensemble by detecting a
photon emission in a certain direction. Then, the excita-
tion is released in the forward direction by using a laser.
Building on this fact, we propose to create certain kind
of excitations by using quantum gates or atomic interac-
tions, which give rise to the desired entangled states when
they are released using a laser, and which propagate in
the desired direction due to the mentioned interference
effect.
Let us consider a set of N atoms with (ground) hype-
fine levels |g〉 and |sa,b〉 (see Fig. 1 (a)). We consider
states of the form
|k(na)a ,k
b 〉 =
na! nb!
)na (
|0〉, (1)
and linear combinations thereof. Here, |0〉 = |g〉1 . . . |g〉N ,
e−ikxr
x,j , x = a, b, (2)
where σ
x,j excites an atom from |g〉j to |sx〉j , and r0j are
the equilibrium position of the atoms. In the limit nx ≪
N , Eq. (1) defines a set of orthonormal collective states
with nx atoms excited in |sx〉 and linear momentum kx.
Those states can be indeed readily created using trapped
ions or Rydberg techniques (see Appendix A1,3).
In order to release the photons, one sends a laser pulse
of wavevector kL which couples level |sx〉 to some elec-
tronically excited ones |ex〉, respectively. The large pop-
ulation of level |g〉 together with the initial entangle-
ment (coherences) between the atoms, will now stimu-
late the emission of photons from the excited states to
the level |g〉, which overall will produce the mapping be-
tween these states and the photonic states,
|k(na)a ,k
b 〉 → |na〉ka+kL,σa |nb〉kb+kL,σb ; (3)
that is, (1) is mapped to a Fock state of nx photons with
momenta kx + kL and polarization σx, where σx is the
polarization of the light in each decay channel. More-
over, due to the linearity of this process, superpositions
of states of the form (1) will be mapped onto superpo-
sitions of photonic states (3). For example, the atomic
state
|k(1),q(1)〉+ |q(1),k(1)〉
2 will emit a pair of
entangled photons in different directions. The mapping
(3) is strictly valid under ideal conditions, and in the limit
N → ∞, and the directionality in the photon emission is
directly connected to the momentum conservation which,
in turn, is a consequence of the constructive interference
in the field emitted by each atom. Thus, the crucial is-
sue in our scheme is to determine how this mapping is
http://arxiv.org/abs/0704.0641v2
modified in finite atomic ensembles under nonideal con-
ditions. In the following we analyze such questions in
detail, concentrating in the simplest case in which we
have a single excitation with momentum |k0〉 in |sa〉 (i.e.
our initial state is a W-like state) and thus we produce
a single photon. We determine a function f(Ω), which is
proportional to the probability density that the photon
is emitted in the direction Ω. In general, f = fcoh + finc;
that is, it is the sum of a coherent contribution and an
incoherent one. The later appears whenever the posi-
tions of the particles fluctuate. fcoh contains the forward
scattering contribution, which is emitted in a cone with
a width ∆Ω that decreases with the number of particles.
finc, on the contrary, describes isotropic light emission,
thus, even when the light emitted in ∆Ω is collected, the
contribution finc leads to a limitation in the efficiency of
the setup. To quantify the error probability, we define
dΩ finc(Ω)
dΩ f(Ω)
. (4)
As long as the number of excited atoms is small nx ≪ N ,
this analysis can be easily generalized to the emission of
states with many photons (1). One obtaines that the
overall error is bounded by 1− (1− E)na+nb .
The emission pattern can be obtained by studying the
Heisenberg equations of motion of the field operators.
The calculation inolves the study of the decay of the
atomic state under collective effects (see Appendix C).
To simplify our analysis we ignore the dipole pattern, in
which case we get:
f(Ω) =
i,j=1
〈e−i(kLnΩ−kL)(ri−rj)〉eik0(r
). (5)
rj are the coordinate operators of the atoms, and thus Eq.
(5) allows us to describe fluctuations in the position of the
particles during the emission of light. In the following,
we will show three different experimental set–ups where
our scheme can be implemented. In order to analyze
the performance in each of them, we first particularize
the above formula to three different situations which are
directly connected with those set–ups. We will focus on
the angular width of the forward–scattering cone, ∆Ω,
which measures the collimation of the emitted photons,
and the error probability, E , as figures of merit. Then
we will introduce the possible implementations and will
use those formulas to specify the conditions for them to
correctly operate.
(i) Fixed atomic positions. In the case of a square
lattice of particles trapped in 3D (see Fig. 1 (b)), the
emission pattern is given by
f0(Ω) =
α=x,y,z
sin2((kLn
Ω − (kL+k0)α)d0Nα/2)
sin2((kLn
Ω − (kL+k0)α)d0/2)
, (6)
with Nα the number of atoms in each direction. f0(Ω)
has a series of diffraction peaks, which are reduced to a
single one if d0 < λ/2. In this regime, the emission is cen-
tered in a cone with nΩ in the direction of kL+k0. Note
that for simultaneous energy and momentum conserva-
tion condition |kL + k0| = kL has to be fulfilled. Since
the positions of the atoms do not fluctuate, f0 has only
a coherent contribution (E = 0), and the only limitation
for the effiency of the setup is the width of the emission
cone, which scales in 3D like ∆θ3D ≈ 1/(N1/3kLd0). In
the case of a chain of atoms (1D) momentum is conserved
only along the direction of the chain. Photon emission
can be still directed efficiently along the axis of the chain,
in a cone whose width scales like ∆θ1D ≈ 1/
NkLd0.
(ii) Fluctuating atomic positions. Let us consider a
lattice of atoms at temperature T , trapped by indepen-
dent harmonic potentials. The emission pattern is now
the sum the of two contributions,
fcoh(Ω) = f0(Ω)gT (Ω), finc(Ω) = 1− gT (Ω). (7)
gT (Ω) = e
−((kLnΩ−kL)ξT )
, and ξT is the vector whose
components are the size of the position fluctuations in
each spatial direction, (ξαT )
2 = x20(1 + 2n
T ), with x
0 the
size of the ground state in harmonic potential, and nαT
the number of motion quanta at T . Light scattered into
finc represents an important fraction whenever ξ
T ≫ d0.
In this case, the emission of light is centered around kL,
since the uncertainty in the position of the particles av-
erages out the intial linear momentum k0. The scaling of
E in this regime strongly depends on the dimensionality
of the system. In particular, in the case of a chain of
atoms, E1D = d0/λ, whereas in the square 3D lattice, we
get E3D ≈ 12.6(d0/λ)2N−1/3.
(iii) Statistical distribution of particles. Consider an
ensemble of atoms (see Fig. 1 (c)), which move inside
a square box of size L, such that their motion is faster
than their radiative decay, that is, their average velocity
v is such that vL ≫ Γ, with Γ the emission rate. This
situation can be described by assuming that the atoms
are in a statistical distribution with equal probability to
be at any point in the box. The situation is thus similar
to that of a thermal state,
fcoh (Ω) = Ngbox(Ω), finc (Ω) = 1− gbox(Ω), (8)
where
gbox(Ω) =
α=x,y,z
sinc2 (L(kLn
Ω − kαL)) . (9)
Defining the average distance between particles like d0 =
LN−1/3, we find the same scalings of ∆θ as in case (i),
and of E , as in case (ii). Trapping schemes for atomic
ensembles are simpler to realize but face the difficulty
that conditions for the directionality of photon emission
are more stringent. In the case of a lattice of particles at
fixed positions, forward–scattering is ensured whenever
condition d0 < λ/2 is fulfilled. On the contrary, in the
case of atomic ensembles, the incoherent contribution finc
has to be small enough such that E ≪ 1, which implies
FIG. 1: (a) Level configuration for the release of atomic en-
tangled states in photonic channels. (b) Release of a collec-
tive state with linear momentum k0, that has been gener-
ated in a lattice of atoms. (c) Emission of photons from an
atomic ensemble, which consists of an incoherent contribu-
tion (isotropic), and a coherent one in the forward–scattering
direction.
d0 ≪ λ in 1D, or, alternatively, a number of particles
large enough in 3D.
Now we introduce three experimental set–ups where
our scheme can be implemented. In the Appendix we
show how to create the atomic states that we are consid-
ering here.
Trapped ions. This system is ideally suited to create
collective states like (1), as was demonstrated recently
in ref. [10]. Most usually ions are arranged in chains,
such that we deal with the 1D situation discussed above.
Even though trapped ions are not equally spaced, under
the condition d̄0 < λ/2, with d̄0 the average distance,
we still get light emission in the forward–scattering cone
only, see Fig. 2. Considering two different internal levels,
which can correspond to different states in an hyperfine
multiplet, states such as those defined by Eq. (1) can be
created by a number of quantum operations that scales
linearly with the number of ions N (see A1). For exam-
ple, the state 1/
2 (|0, 2kL〉+ |2kL,0〉), would emit two
photons in the forward and backward directions along the
chain axis, entangled in polarization. The main difficulty
for the implementation of this idea with ions lies on the
fact that ion–ion distances are usually in the range of a
few µm, and thus condition d0 < λ/2 is not fulfilled when
considering optical wavelengths. A way out of this prob-
lem is to use optical transitions which lie in the range of
λ & 5µm, which can be found in ions such as Hg+, Ba+,
or Yb+ [22].
Cold atoms in optical lattices. By using optical lat-
tices we fulfill the need of placing atoms at interparticle
distances comparable to optical wavelengths, since po-
tential wells in a standing–wave are indeed separated by
d0 = λsw/2, with λsw, the wavelength of the counterprop-
agating lasers. By using an optical transition such that
λ > λsw, we are in the regime in which light emission
FIG. 2: Probability of photon emission from an ion chain with
N = 30 ions initially in a W –state. The blue line corresponds
to a chain with equally spaced ions with two diffraction peaks.
Black and red lines corresponds to an ion Coulomb chain, in
which ions are in an overall trapping potential and thus are
not equally spaced. However, in the case that the average dis-
tance, d̄0 is small enough, light is also preferentially emitted
in the forward–scattering direction.
is focused into a single Bragg peak. Although one could
think of peforming quantum gates between ultracold neu-
tral atoms to generate collective atomic states [23], this
procedure faces the difficulties of quantum computation
in this system, like for example how to achieve single
atom addressability. More efficiently, one could avoid
the use of quantum gates by using the dipole–blockade
mechanism with Rydberg atoms, which allows us to gen-
erate W-states, as well as states which emit Fock states
with a number M of photons [13] (see Appendix 3).
Atomic ensembles at room temperature. The very same
techniques which can be applied to Rydberg atoms in an
optical lattice can also be used in the case of hot en-
sembles. On the one hand, this setup has the advantage
that atoms do not need to be cooled and placed in an
optical lattice. On the other hand, it can be described
by a statistical distribution of particles, and thus suffers
from the fact that high efficiency in the release of pho-
tons is achieved under more severe conditions of particle
density and atom number, as discussed above. However,
densities which are high enough to fulfill the requirement
E ≪ 1 have been recently reported in [24].
In conclusion, we have proposed to use current tech-
niques for quantum engineering to generate atomic multi-
partite entangled states which can be efficiently mapped
into photonic states. Our proposal relies on the release of
spin–wave like excitations into a given spatial direction
by means of interference effect, and can be implemented
with trapped ions, atoms in optical lattices, and atomic
ensembles at room temperature.
This work was supported by E.U. projects (SCALA
and CONQUEST), and the Deutsche Forschungsgemein-
schaft.
APPENDIX A: CREATION OF COLLECTIVE
ATOMIC STATES IN A CHAIN OF ATOMS
Entangled states of the form (1) and their linear com-
binations can be generated in a chain of particles, for
example, of trapped ions, by means of a limited number
of quantum operations. To demonstrate this, we first
show that they can be written as Matrix Product States
with a small bond dimension D, i.e. they can be written
|Ψ〉 =
i1,...,iN
〈ΦF|V iN[N ] . . . V
|ΦI〉|i1〉 . . . |iN 〉, (A1)
In (A1), the indices ij = g, sa, sb, and V
are D × D
matrices acting on an auxiliary D–dimensional Hilbert
space. D is given by the number of states which appear
in the singular value decomposition (s.v.d.) of |Ψ〉 at
any site in the chain [25]. As it is shown in [26], the state
(A1) can be prepared by performing N gates which act
on [log2 D]+1 qubits. Thus, as long as D is independent
of N , the number of gates to be applied scales linearly
with the total number of atoms.
To evaluate D, consider first the case of a state like (1)
with atoms excited in level sa only, and a partition of the
chain in two parts L and R. We get na + 1 states in the
s.v.d. with respect to this partition, which correspond to
states with a number of excited atoms in part L, ranging
from 0 to na. This result is easily generalized to a linear
combination ofM states of the form (1), in which case we
get D = M(na + 1)(nb + 1). For example, an entangled
state of the form:
|Ψ〉 = 1√
|kna=1,qnb=1〉+ |qna=1,knb=1〉
, (A2)
has D = 8.
APPENDIX B: QUANTUM STATE
ENGINEERING WITH RYDBERG BLOCKADE
Interactions between excited atomic states, like those
that take place in Rydberg atoms, can be used to the cre-
ate the states defined by Eq. (A2). This can be achieved
in a single experimental step, without the need for quan-
tum gates, if the proper configuration of atomic inter-
actions is chosen. As an example, consider the 3 level
configuration shown in Fig. 1 (a), and interactions be-
tween excited states such that atoms in levels |sa〉, |sb〉,
interact strongly only if they are in the same excited
state, that is, Uaa = Ubb = U , but Uab = 0. We ap-
ply two lasers with wavectors k1,2 and Rabi frequencies
Ω1,2, detuned with respect to the |g〉 – |sa,b〉 transition,
such that ∆1 = −∆2 = ∆. If condition ∆1,2 ≫ Ω1,2
is fulfilled, then the lasers induce a two–photon transi-
tion with Rabi frequency Ωeff = Ω1Ω2/∆. Furthermore,
if Ωeff ≪ U , states with two atoms in the same excited
state are not populated. Under these conditions there
are two possible excitation channels, depicted in Fig. 3,
which give rise to the linear combination (A2).
FIG. 3: Lasers and level configuration for the creation of
atomic entangled states which emit pairs of photons entan-
gled in polarization.
APPENDIX C: CALCULATION OF THE
PHOTON DISTRIBUTION
We consider for simplicity the lambda configuration
depicted in Fig. 1 (a), considering a single excited state
|s〉, and a single auxiliary level |e〉. The interaction of
the quantized electromagnetic field with the ensemble of
atoms, after the adiabatic elimination of level |e〉, is de-
scribed by
j,k,λ
jak,λe
i(k−kL)rj+iωLt + h.c.
gk,λ =
(ǫkλ · dge) , (C1)
σj refers to the |g〉 – |s〉 atomic transition, ΩL and kL
are the Rabi frequency and wave–vector of the classical
field, respectively, ωk is the photon energy, ǫk,λ are the
polarization vectors, and dge is the dipole matrix element
for the |g〉 – |e〉 transition.
The probability of photon emission is proportional to
the the diagonal elements of the one–photon density ma-
trix, which are obtained from the Heisenberg equation of
motion for the field operators,
ak〉 =
dτ1dτ2e
−i(ωk−ωL)(τ1−τ2)
〈e−i(k−kL)(ri−rj)〉〈σ†i (τ1)σj(τ2)〉. (C2)
Since we are interested in the conditions for momentum
conservation due to interference effects, we consider the
following atomic initial state,
|k0〉 = σ†k0 |0〉, σ
e−ik0r
j . (C3)
The emission pattern depends thus on the two–time
atomic correlation function, which in turns can be eval-
uated by means of a master equation which describes
the decay of the atomic levels. In the case of the ini-
tial atomic state k0 (C3), fixed atom positions, and ne-
glecting boundary effects, this correlation function can
be evaluated exactly,
〈σ†i (τ1)σj(τ2)〉 = e
−Γk(τ1+τ2)/2eik0(r
), (C4)
where we have neglected an energy shift due to dipole–
dipole interactions. Integrating (C2) over the absolute
value of k yields the probability of photon emission,
I(Ω) = Ī(Ω)f(Ω). (C5)
Ī(Ω) is the dipole pattern,
Ī(Ω) =
1− (negnΩ)2
, (C6)
where Γ is the single atom radiative decay rate, Γk0 is
the collective decay rate, neg is the unit vector of the
atomic transition, and nΩ is a unit vector in the direc-
tion defined by the solid angle Ω. The factor f(Ω) in I(Ω)
describes the interference between the emission from dif-
ferent atoms, and is given by Eq. (5).
Below we deduce the master equation which leads to
(C4) and we sketch its solution in the case of collective
states with a single excited atom.
APPENDIX D: MASTER EQUATION
The master equation for the reduced density matrix
of the internal levels, which describes the ratiative decay
of a set of atoms under the coupling to the quantized
radiation field given by Eq. (C1), is
∂tρ =
2 σiρσ
j − σ
i σjρ− ρσ
Gij [σ
i σj , ρ], (D1)
where the coupling constants depend on
Jij =
dτgij(τ)e
−iωLτ =
ei(ωk−ωL)τ+i(k−kL)r,(D2)
in the following way:
ℜ(Jij) =
Γij ,
ℑ(Jij) =
Gij . (D3)
The master equation (D1) can be solved for the particular
case of an initial state (C3) by noticing that the evolu-
tion of ρ is closed within the subspace spanned by the
states |k0〉, |0〉. This fact can be easily proved by direct
substitution of ρ(0) = |k0〉〈k0| in Eq. (D1), which yields
the following evolution for the atomic density matrix:
ρ(t) = e−Γk0 t|k0〉〈k0|+
1− e−Γk0 t
|0〉〈0|, (D4)
where the collective decay rate Γk0 is just the Fourier
transform of the coupling constants in the master equa-
tion,
Γk0 =
Γi,je
ik0(r
j). (D5)
A similar result holds for nondiagonal elements of ρ(t),
which together with the quantum regression theorem
yields the evolution of the atomic correlation function
(C4).
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http://arxiv.org/abs/quant-ph/0701120
