PRL 115, 123601 (2015)

PHYSICAL REVIEW LETTERS

week ending 18 SEPTEMBER 2015

Coulomb Bound States of Strongly Interacting Photons

M. F. Maghrebi,1,* M. J. Gullans,1,† P. Bienias,2 S. Choi,3 I. Martin,4 O. Firstenberg,5 M. D. Lukin,3 H. P. Büchler,2 and A. V. Gorshkov1
1Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
2Institute for Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany 3Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
4Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 5Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
(Received 14 May 2015; published 16 September 2015)
We show that two photons coupled to Rydberg states via electromagnetically induced transparency can
interact via an effective Coulomb potential. This interaction gives rise to a continuum of two-body bound
states. Within the continuum, metastable bound states are distinguished in analogy with quasibound states
tunneling through a potential barrier. We find multiple branches of metastable bound states whose energy
spectrum is governed by the Coulomb potential, thus obtaining a photonic analogue of the hydrogen atom.
Under certain conditions, the wave function resembles that of a diatomic molecule in which the two polaritons are separated by a finite “bond length.” These states propagate with a negative group velocity in
the medium, allowing for a simple preparation and detection scheme, before they slowly decay to pairs of
bound Rydberg atoms.

DOI: 10.1103/PhysRevLett.115.123601

PACS numbers: 42.50.Nn, 32.80.Ee, 34.20.Cf, 42.50.Gy

Photons are fundamental massless particles that are essentially noninteracting for optical frequencies. However, a medium that couples light to its atomic constituents can induce interactions between photons. This interaction may lead to exotic, many-body states of light [1–3], or can be used as a basis for realizing deterministic quantum gates between two photons [4–7]. A promising approach to create strongly interacting photons is to couple the light to atomic Rydberg states [3,4,6,8–36], as realized in recent experiments [37–52].
Rydberg polaritons are superpositions of Rydberg atoms and light, which propagate almost without dissipation under the conditions of electromagnetically induced transparency (EIT) [8,53–55]. EIT strongly reduces the group velocity and makes Rydberg polaritons dispersive. The large admixture of the Rydberg state can induce strong interactions between polaritons via the inherent Rydberg-Rydberg interactions. Specifically, the blockade effect prevents the formation of two Rydberg polaritons within the so-called “blockade radius” of each other [38,43,48,56–59]. When the probe photons are detuned from the atomic transition, they can form bound states. A shallow bound state of light was observed in recent experiments [45], while stronger interactions result in deep bound states of Rydberg polaritons tied together within the blockaded region [29]. One can imagine these bound states as consisting of a photon trapped by a Rydberg excitation in a deep square well.
In this Letter, we predict and explore a class of photonic states resembling diatomic molecular states in which the two bound photons can be separated by a nonzero “bond length.”

This is achieved by considering Rydberg polaritons with the quantized light red detuned from the excited atomic state. In such a system, we show the existence of metastable bound states exhibiting the Coulomb spectrum, akin to the hydrogen atom. Such states can potentially be used as building blocks for more complex quantum states of light.
To gain an intuitive understanding, consider the level structure of the Rydberg medium shown in Fig. 1(a). The probe field coupling the ground state jgi to the intermediate

(a) Ω2
Δ
|R

control Ω

|e
probe |g

Δ

(b)

r=0
r

Veff

−1

∼

1 r−rb

1 r/rb

FIG. 1 (color online). (a) The probe field couples the ground state jgi to the excited state jei and is red detuned by Δ. A control field with Rabi frequency Ω couples jei to the Rydberg state jRi and is blue detuned by Δ, thus putting the probe on an EIT
transmission resonance. The Rydberg state is thus shifted downward by Ω2=Δ. The van der Waals interaction with another reference Rydberg excitation at r ¼ 0 can bring jRi into an
absorption resonance with the two-photon transition. (b) The
effective potential of two Rydberg polaritons as a function of their separation r exhibits a singularity at jrj ¼ rb (the blockade radius) and behaves near this singularity as a Coulomb potential.

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excited state jei is red detuned by Δ > 0, and the Rabi

frequency of the control field coupling jei to the Rydberg

state jRi is Ω. For Ω ≪ Δ, the Rydberg state is shifted downward by Ω2=Δ [see Fig. 1(a)]. The van der Waals

interaction VðrÞ ¼ C6=r6 between Rydberg states modifies this picture (we assume C6 > 0 or more generally C6Δ > 0). In particular, at small separations r, the strong
interaction shifts two Rydberg states upward and out of

resonance, while at large separations, the interaction is

negligible and the energy level of each atom asymptotes to

−Ω2=Δ (we set ℏ ¼ 1). For intermediate separations on the order of the blockade radius rb, defined by VðrbÞ ¼ 2Ω2=Δ
[60], the system goes through a resonance (the factor

of 2 arises since both atoms experience the Ω2=Δ shift).

This resonance, associated with a pair (or “molecule”) of

Rydberg atoms, endows the effective interaction VeffðrÞ between two Rydberg polaritons with a singularity sepa-

rating repulsion outside the blockade region from attraction

inside; see Fig. 1(b). This effective interaction between

two Rydberg polaritons can be roughly thought of as the

difference in susceptibility of a single Rydberg polariton

with and without a Rydberg excitation at r ¼ 0 [45].

Interestingly, the effective potential near the resonant edge

is that of the Coulomb interaction that changes sign across

the blockade radius. This potential admits a continuum

of states consisting of pairs of bound Rydberg atoms

(Rydberg molecules) dressed by the photons. Within the

continuum, we identify multiple branches of metastable

states whose lifetime diverges with the strength of the

interaction. When the effective energy of the two-polariton

state lies below both Veffð∞Þ and Veffð0Þ, the bound state experiences a repulsive core and the wave function

becomes double peaked near Ærb, resembling a diatomic molecular state. We further show that the group velocity of

these states is negative, consistent with the fact that they

have a finite lifetime.

Model.—To describe a propagating polariton in a Rydberg medium, we define E†ðzÞ and S†ðzÞ as bosonic

creation operators for a photon and a Rydberg excitation,

respectively, at position z. They obey the equal-time
commutation relations ½EðzÞ; E†ðz0ފ ¼ ½SðzÞ; S†ðz0ފ ¼ δðz − z0Þ. We define g to be the collectively enhanced

atom-photon coupling [53] and assume that the decay rates 2γ of the excited state (satisfying γ ≪ Δ) and 2γ0 of the

Rydberg state can be neglected. In the regime of slow light

(g ≫ Ω) and with large single-photon detuning (Δ ≫ Ω),

one can adiabatically eliminate the excited state jei [29,45].

The two-state Hamiltonian of the Rydberg medium is then

Z H¼

 dz

E

†



−ic∂

z

þ

g2=Δ

S Ωg=Δ

Ωg=Δ  E  Ω2=Δ S

þ

1 2

Z

dzdz0Vðz − z0ÞS†ðzÞS†ðz0ÞSðz0ÞSðzÞ:

ð1Þ

In the Supplemental Material [61], we show that this treatment of the medium as a one-dimensional continuum of

stationary atoms is experimentally relevant. In the absence
of interactions, H diagonalizes into dark- and bright-state polaritons, where, at low energies, the former [∝ ðgS† − ΩE†Þ when ∂z ¼ 0] is mostly composed of jRi and travels at a reduced group velocity [53]. In the presence of interactions,
the Hamiltonian in Eq. (1) can be projected onto the sector containing two particles (at positions z and z0) described by the quantum state jΦi with two-photon amplitude EEðz; z0Þ, atom-photon amplitudes ESðz; z0Þ and SEðz; z0Þ, and twoatom amplitude SSðz; z0Þ. These are defined as EEðz; z0Þ ¼ h0jEðzÞEðz0ÞjΦi, ESðz; z0Þ ¼ h0jEðzÞSðz0ÞjΦi, SEðz; z0Þ ¼ h0jSðzÞEðz0ÞjΦi, and SSðz;z0Þ¼h0jSðzÞSðz0ÞjΦi, where j0i
is the vacuum state. The problem is simplified by noting that,
for two particles, the total energy ω and the center of mass
momentum K are good quantum numbers. In the limit g → 0, the SS component decouples
from the photonic amplitudes (ωSSðz; z0Þ ¼ ½−2Ω2=Δ þ Vðz − z0ފSSðz; z0Þ) giving rise to a continuum of (δ-
function) states of Rydberg molecules. Upon increasing
g the continuum of states is still present while the wave
function remains localized to the blockade radius. To see
this, note that the Heisenberg equations of motion for
the above amplitudes immediately lead [29,61] to the
Shrödinger-like equation



−

1 m

∂

2 r

þ

r6

−

C6 ½rbðωފ6

þ

i0þ

ψðrÞ ¼ EψðrÞ;

ð2Þ

where r is the relative coordinate of the two particles,
and ψ is the symmetrized light-Rydberg wave function ψðrÞ ≡ ½ESðrÞ þ SEðrފ=2. Notice that the van der Waals potential is replaced by an effective potential VeffðrÞ ¼ C6=(r6 − ½rbðωފ6 þ i0þ) modified within the blockaded region as in Fig. 1(b). [In contrast, for C6Δ < 0, the effective potential is a simple well VeffðrÞ ∝ −1=(r6 þ ½rbðωފ6) of width ∼2rb with no repulsive core at r ¼ 0; this potential harbors bound states of width ≳rb centered at r ¼ 0, as studied in Ref. [29] and observed in Ref. [45].] For a nonzero ω, the blockade radius rbðωÞ depends on frequency via the resonant condition C6=½rbðωފ6 ¼ 2Ω2=Δ þ ω (the dependence of rb on ω will often be implicit below). i0þ in Veff is obtained in the limit of vanishingly small γ and γ0, which is further required
by causality. In the limit of small energy and momentum,
m is the mass of a single dark-state polariton due to the
curvature of the linear susceptibility and is given by m ¼ g4=2c2Ω2Δ [62,63], while the energy is given by E ¼ ω − vgK with vg ¼ ðΩ2=g2Þc being the EIT group velocity. More generally, the parameters in Eq. (2) can be
simply derived from single-polariton physics: for two
Rydberg dark-state polaritons with momenta k1 and k2 and dispersion ω1;2 ¼ ωðk1;2Þ, the constraints ω1 þ ω2 ¼ ω and k1 þ k2 ¼ Kpffiyffiffiffiiffieffiffild an expression for the relative momentum p ¼ mE consistent with the full expressions
for m and E [29,61].

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Coulomb states.—The effective potential Veff diverges as 1=ðr Æ rbÞ near the blockade radius, like a Coulomb potential. Across the singularity, the wave function ψ
should be continuous, while its derivative does not have
to be. The full wave function Ψω;KðrÞ ¼ ðEE; ES; SE; SSÞ has various components that are related to ψ as [29,45]

EEðrÞ

¼

−

2gΩ=Δ 2g2=Δ þ ω −

cK

ψ

ðrÞ;



ESðrÞ ¼

1

−

ðg2

þ

ic Ω2Þ=Δ þ

ω

−

cK=2

∂r

ψ ðrÞ;



SSðrÞ

¼

2gΩ ΔC6

P

ψ ðrÞ r−6 − r−b 6

þ αδðr Æ rbÞ;

ð3Þ

where, for states with ψðrbÞ ≠ 0, the principal value symbol P removes the 1=½r Æ rbðωފ singularity in SS near the blockade radius. The coefficient α of the δ function is determined by the discontinuity in the derivative of ψ at the blockade radius [61].
We now notice that Eqs. (2) and (3) admit a special set of
solutions, which are a superposition of a normalizable wave function vanishing for jrj ≥ rb [ψðrbÞ ¼ 0] and a δ-function singularity in the SS component, but without the 1=½r Æ rbðωފ singularity. Such states can be interpreted in analogy to a “leaky box” where a quasibound particle tunnels through a potential barrier: for the leaky box, a true
eigenstate is a superposition of the metastable bound state
and a plane wave, which is a momentum eigenstate selected
from a continuum. Similarly, for the above special eigenstates [with ψðr ≥ rbÞ ¼ 0], the role of the continuum of eigenstates is played by the δ functions in SS, which are position eigenstates. When the δ function is removed, the other components of the wave function form a metastable
bound state. Furthermore, in the limit of an infinitely strong interaction, i.e., g → ∞, our special eigenstates lose their δ-function component [61]. This is again analogous to the leaky box, where, in the limit of an infinitely tall barrier (i.e.,
the no-leak limit), one obtains exact eigenstates confined
to the box and decoupled from the plane-wave component
sitting outside the box. Henceforth, we call the metastable bound states above (without the δ function) Coulomb states, and study their spectrum and other properties in detail.
Figure 2(a) shows the energy spectrum of the exact eigenstates (i.e., with the δ function) underlying the Coulomb states. The exact solutions are depicted as solid
lines, while the dashed lines show the energy spectrum
derived from the WKB quantization condition [applied to the case ψðÆrbÞ ¼ 0]

Z
rb
pðrÞdr ¼ nπ;

n ¼ 1; 2; …

ð4Þ

r0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with pðrÞ ¼ mðE − VeffðrÞÞ defined by Eq. (2), and r0 < rb being the classical turning point near the origin

FIG. 2 (color online). (a) Dispersion curves for the exact eigen-

states underlying the metastable Coulomb bound states (only the

first four branches n ¼ 1–4 are shown) with g2rb=cΔ ¼ 40 and
Ω=g ¼ 0.05. The solid lines give the exact solution, while the dashed lines represent the WKB results. For K → 2g2=cΔ, the

WKB results are almost exact. The dispersion curves converge to

one point with a negative slope, or group velocity. (b) Decom-

position of the metastable Coulomb bound states (Ψcn, defined

as Ψωn;Kn continuum

with the of exact

δ-function contribution removed) into the eigenstates. Here, we took g2rb=cΔ ¼ 40

and Ω=g ¼ 0.01 with a fixed center of mass momentum

cKΔ=2g2 ¼ 0.95. The width of these distributions is much less

than the energy spacing, indicating they are spectrally distin-

guishable. The inset shows the wave function components

for the n ¼ 1 Coulomb state with the parameters in (a) and cKΔ=2g2 ¼ 0.98. (The EE component is exaggerated by a factor

of 1.5 for better visibility.)

[64]. Figure 2(a) demonstrates that the WKB quantization agrees with the full solution for values of K near 2g2=cΔ.
When K is close to 2g2=cΔ, we can analytically compute
the integral in Eq. (4) to find

1 þ ωΔ=2Ω2 1 − cKΔ=2g2

¼

A

½g2rbðωÞ=cĊ2 n2

;

ð5Þ

where A ¼ ½Γð2=3Þ=Γð1=6ÞpffiπffiffiŠ2 ≈ 0.014 and Γ is the

Gamma function. If rb were independent of ω, Eq. (5) would imply that ω is quantized as 1=n2 (plus a constant),

reminiscent of the energy spectrum of the Coulomb

potential. However, due to the ω dependence of rb, the quantization changes to ωn ∼ 1=n3=2 (plus a constant) [65], which still sharply contrasts with the finite-square-well

energy quantization in Refs. [29,45].

The fact that the blockade radius, and, thus, the inter-

action strength, is sensitive to frequency is a typical feature

of nonlinear optical systems [13]. We also stress that 4g2rb=cΔ is identical to the figure of merit in the fardetuned regime ODbγ=Δ, where ODb is the optical depth
per blockade radius. The figure of merit quantifies the

strength of the interaction as two polaritons imprint a phase

∼ODbγ=Δ on each other [45]. With the dispersion in hand, we now explore the stability

of the Coulomb states. The solutions given by Eq. (3) are

a complete set of eigenstates for the two-particle Hilbert

space. To normalize these states, we take K to be fixed and use the energy normalization hΨω0;KjΨω;Ki ¼ δðω − ω0Þ.

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We can then verify the metastability of the Coulomb states {ψ½rbðωފ ¼ 0} with the δ function removed by looking at their spectral width, i.e., their decomposition into the
normalized eigenstates. Figure 2(b) shows this decomposition for several n, where we see that the Coulomb states
are sharply peaked at the expected frequencies. The width
of these distributions can be much narrower than the
spacing between states, a strong signature of spectral distinguishability [66]. Furthermore, the Coulomb states
converge to the exact eigenstates for a very strong inter-
action strength, which is analogous to the leaky box in the
limit of an infinitely deep potential [61]. A unique feature of the dispersion curves in Fig. 2(a) is
that their slope, and thus the group velocity, is negative.
While true eigenstates cannot have a negative group velocity
in the absence of left-moving modes (Supplemental Material
[61]), Coulomb states are not exact eigenstates and eventually decay into Rydberg molecules. Equation (5) gives the group velocity as v ¼ −Aðg2rb=cΔÞ2vg=n2, where vg is the EIT group velocity. Therefore, the velocity is also quantized as 1=n2 for different branches of bound states (and fixed values of ω). This quantization and the negative sign make the group velocity an ideal signature for detecting different
Coulomb states and distinguishing them from the bound states of Refs. [29,45]. We also remark that a small γ (≪ Δ) contributes to the energy a small imaginary part, proportional to γ=Δ, which thus becomes negligible for large detuning.
We now show how to prepare these states and measure
their dispersion. We assume that we have access to an additional hyperfine ground state jqi, which, as shown in Fig. 3(a), is connected to both jgi and the Rydberg state jRi through two-photon transitions via an excited state je0i. With these additional states, we can effectively turn on and off the polariton interactions by applying a fast π pulse on the two-photon transition between jqi and jRi.
The preparation procedure is as follows. First, we store
two identical photons (equivalently a weak coherent state followed by postselection) in the atomic state jqi using standard protocols [53,67]. To have a significant overlap with the Coulomb states once we map to jRi, the state has to have the correct center of mass momentum K. To achieve this, we introduce a linear energy gradient E0 along the atomic cloud for a time τ, which could be achieved with a magnetic field gradient, another optical
beam, or a microwave field. This will impart a phase e−iE0τR on the stored two-photon state. By choosing the appropriate τ and then mapping jqi to jRi, we can selectively excite different Coulomb states provided they
have a large enough spatial overlap with the initial product
state input. As the bound states travel with negative group velocity, the Coulomb state component will separate from
the rest of the wave function. To detect the state, one can then map the Rydberg state back to jqi and either measure the population of the state jqi directly or retrieve the state

(a) R e
g (b)1 |EE(z, z )|
0.8 1
0.8 0.6
0.6
0.4 0.4

e q (c)1 |EE(z, z )|
0.8 0.2

(d)1 |EE(z, z )|
0.8 0.05
0.04
0.6
0.03

0.6 0.15
0.1
0.4
0.05
0.2
0

0.4 0.02
0.01
0.2
0
t = tf /2 00 0.2

0.4 0.6
z/L

0.2 0.2
0

00
t = tf /4

0.2 0.4 0.6 0.8
z/L

1

00 0.2 0.4z/L0.6 0.8 1

t = tf
0.8 1

z /L

FIG. 3 (color online). (a) Level structure used to prepare the initial SS distribution. (b)–(d) Time evolution of a wave packet
with all components initially zero except SS, which is chosen to be a Gaussian wave packet of variational n ¼ 1 Coulomb state solutions (with the δ function removed) centered at ω ¼ 0 and having width Ω2=2Δ [61]. Specifically, jEEj, initially zero, is shown after the initial transient evolution subsides at (b) t ¼ tf=4, and at (c) t ¼ tf=2 and (d) t ¼ tf, where tf ¼ 5.5Δ=Ω2. The wave packet within the blockade radius has the expected shape of
the Coulomb state, propagates backward, and decays, while the
wave packet outside the blockade radius propagates forward with vg and disperses. We took a medium of length L ¼ 16rb, g2rb=cΔ ¼ 5, g=2π ¼ 17 GHz, Ω=2π ¼ 1.5 MHz, rb ¼ 25 μm, Δ=2π ¼ 30 MHz, γ=2π ¼ 3 MHz, and γ0=2π ¼ 5 kHz [61].

into light. In Figs. 3(b)–3(d), for realistic parameters (including γ and γ0) [41], we verify this approach by preparing a variational state that has a large overlap with the SS component of the Coulomb state [shown in Fig. 3(b)] with other components equal to zero and solve numerically for the time evolution [61]. In this case, the effective energy E of the bound state lies above Veffð0Þ and the wave function is peaked at r ¼ 0, similar to Refs. [29,45]. We have also verified that, when E < Veffð0Þ, the backward propagating state becomes double peaked (in contrast to Refs. [29,45]) and that, for smaller decay rates and larger g2rb=cΔ, the negative group velocity observed in the numerics agrees with the WKB prediction from Eq. (4) for the n ¼ 1, 2, and 3 Coulomb states within a few percent in each case [61].
Outlook.—While our proposal opens the avenue for the creation of Coulomb-like two-photon states, we expect that a wide class of both useful and exotic two-photon and multiphoton states can be created via refined engineering of photon-photon interactions, e.g., by using microwave fields [43]. The detailed understanding of the two-photon Rydberg-EIT physics provided by this work also opens up an avenue towards understanding the full—and much richer—many-body problem involving an arbitrary number of photons in any dimension.
We thank D. Chang, R. Qi, and Y. Wang for discussions. This work was supported by ARL, NSF PFC at the JQI, the NRC, NSF PIF, CUA, AFOSR, ARO, AFOSR MURI, Center for Integrated Quantum Science and Technology (IQST), the Deutsche Forschungsgemeinschaft (DFG) within SFB TRR 21, the EU Marie Curie ITN COHERENCE, DARPA QUINESS, Packard Foundation,

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and the National Science Foundation. The work of I. M. was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division.
M. F. M. and M. J. G. contributed equally to this work.
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