# Rydberg Atoms In the Vicinity of an Optical Nanofiber

Erwan Stourm, Maxence Lepers, Jacques Robert, Sîle Nic Chormaic, Klaus

Mølmer, Etienne Brion

► To cite this version:

Erwan Stourm, Maxence Lepers, Jacques Robert, Sîle Nic Chormaic, Klaus Mølmer, et al., Rydberg Atoms In the Vicinity of an Optical Nanofiber. Optical Manipulation and Its Applications, Apr 2023, Vancouver, Canada. pp.AW1D.1, 10.1364/OMA.2023.AW1D.1. [hal-05527359]

HAL Id: hal-05527359

https://hal.science/hal-05527359v1

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# Interaction of two RYDBERG rubidium atoms in the vicinity of an optical nanofiber

E. STOURM<fnref n="1" />*, M. LEPERS<fnref n="2" />, J. ROBERT<fnref n="1" />, S. NIC CHORMAIC<fnref n="3" />, K. MØLMER<fnref n="4" />, **E. BRION**<fnref n="5" />

<footnote n="1">¹Laboratoire de Physique des gaz et des plasmas, Université PARIS-SACLAY, CNRS, ORSAY, FRANCE</footnote>

<footnote n="2">²Laboratoire Interdisciplinaire CARNOT de BOURGOGNE, CNRS, Université de BOURGOGNE FRANCHE-COMTÉ, DIJON, FRANCE</footnote>

<footnote n="3">³Light-Matter Interactions for Quantum Technologies Unit, OKINAWA Institute of Science and Technology Graduate University, OKINAWA, JAPAN</footnote>

<footnote n="4">⁴Niels BOHR Institute, University of COPENHAGEN, Blegdamsvej 17, 2100 COPENHAGEN, DENMARK</footnote>

<footnote n="5">⁵Laboratoire Collisions Agrégats Réactivité, CNRS, Université de TOULOUSE III Paul SABATIER, TOULOUSE, FRANCE
* now teacher in classes préparatoires aux grandes écoles.</footnote>
# RYDBERG atoms and blockade

## RYDBERG atoms

Rydberg state – $n \ge 30$

radiative lifetime – $n^5$

dipole matrix elements – $n^2$

<figure><img src="image_2.png" /></figure>

rubidium

$$n \approx 50$$

$$\tau_r \approx 50\mu s$$

$$a \approx 1\mu m$$

## RYDBERG blockade

$$V_{dd} = \frac{e^2}{4\pi\epsilon_0 R^3} \left[ \vec{d}_A \cdot \vec{d}_B - \frac{3}{R^2} \left( \vec{d}_A \cdot \vec{R} \right) \left( \vec{d}_B \cdot \vec{R} \right) \right]$$

<figure><img src="image_3.png" /></figure>

rubidium : for $n \approx 60$ and $R \approx 4\mu m$, $\Delta_{dd} \approx 10$MHz

<figure><img src="image_4.png" /></figure>

VAN DER WAALS
(nonresonant
configuration)
# RYDBERG atoms and blockade

## RYDBERG atoms

Rydberg state – $n \ge 30$

radiative lifetime – $n^5$

dipole matrix elements – $n^2$

<figure><img src="image_5.png" /></figure>

rubidium

$$n \approx 50$$

$$\tau_r \approx 50\mu s$$

$$a \approx 1\mu m$$

## RYDBERG blockade

$$V_{dd} = \frac{e^2}{4\pi\varepsilon_0 R^3} \left[ \vec{d}_A \cdot \vec{d}_B - \frac{3}{R^2} \left( \vec{d}_A \cdot \vec{R} \right) \left( \vec{d}_B \cdot \vec{R} \right) \right]$$

<figure><img src="image_6.png" /></figure>

$$\Delta_{dd} \approx \frac{C_6}{R^6}$$

$$C_6 \sim n^{11}$$

blockade conditions

$$\Omega \ll \Delta_{dd} \Leftrightarrow R \ll R_b \equiv \left(\frac{C_6}{\hbar\Omega}\right)^{\frac{1}{6}}$$

<figure><img src="image_7.png" /></figure>
# RYDBERG atoms and blockade

## RYDBERG atoms

Rydberg state – $n \ge 30$

radiative lifetime – $n^5$

dipole matrix elements – $n^2$

<figure><img src="image_8.png" /></figure>

rubidium

$$n \approx 50$$

$$\tau_r \approx 50 \mu s$$

$$a \approx 1 \mu m$$

## RYDBERG blockade in an atomic sample

<figure><img src="image_9.png" /></figure>

blockade conditions

$N$ atoms

<figure><img src="image_10.png" /></figure>

$$\Omega \ll \Delta_{dd} \Leftrightarrow R \ll R_b \equiv \left(\frac{C_6}{\Omega}\right)^{\frac{1}{6}}$$
### Fast Quantum Gates for Neutral Atoms

D. Jaksch, J. I. Cirac, and P. Zoller

Institut für Theoretische Physik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria

S. L. Rolston

National Institute of Standards and Technology, Gaithersburg, Maryland 20899

R. Côté¹ and M. D. Lukin²

¹Physics Department, University of Connecticut, 2152 Hillside Road, Storrs, Connecticut 06269-3046

²ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138

(Received 7 April 2000)

We propose several schemes for implementing a fast two-qubit quantum gate for neutral atoms with

PRL 104, 010502 (2010)

PHYSICAL REVIEW LETTERS

<figure><img src="image_11.png" /></figure>

### Entanglement of Two Individual Neutral Atoms Using Rydberg Blockade

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys

Laboratoire Charles Fabry, Institut d'Optique, CNRS, Univ Paris-Sud, Campus Polytechnique, RD 128, 91127 Palaiseau cedex, France

(Received 5 August 2009; published 8 January 2010)

week ending
8 JANUARY 2010

## RYDBERG blockade in an atom

PRL 99, 260501 (2007)

PHYSICAL REVIEW LETTERS

week ending
31 DECEMBER 2007

### Quantum Computing with Collective Ensembles of Multilevel Systems

E. Brion and K. Mølmer

Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark

M. Saffman

Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706, USA
(Received 10 August 2007; published 26 December 2007)

PRL 113, 053601 (2014)

PHYSICAL REVIEW LETTERS

week ending
1 AUGUST 2014

<figure><img src="image_12.png" /></figure>

### Single-Photon Transistor Mediated by Interstate Rydberg Interactions

H. Gorniaczyk,¹,² C. Tresp,¹ J. Schmidt,¹ H. Fedder,² and S. Hofferberth¹,¹

¹5. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

²3. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
(Received 27 March 2014; published 28 July 2014)

$$ \Omega \ll \Delta_{dd} \Leftrightarrow R \ll R_b \equiv $$

### Dipole Blockade and Quantum Information Processing in Mesoscopic Atomic Ensembles

M. D. Lukin,¹ M. Fleischhauer,¹,² and R. Cote³

¹ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138

²Fachbereich Physik, Universität Kaiserslautern, D-67663 Kaiserslautern, Germany

³Physics Department, University of Connecticut, Storrs, Connecticut 06269

L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller

Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria
(Received 7 November 2000; published 26 June 2001)

We describe a technique for manipulating quantum information stored in collective states of mesoscopic ensembles. Quantum processing is accomplished by optical excitation into states with strong dipole-dipole interactions. The resulting "dipole blockade" can be used to inhibit transitions into all but singly excited collective states. This can be employed for a controlled generation of collective atomic spin states as well as nonclassical photonic states and for scalable quantum logic gates. An example involving a cold Rydberg gas is analyzed.

<figure><img src="image_13.png" /></figure>

$$ \begin{aligned} n &\approx 50 \\ \tau_r &\approx 50\mu s \\ a &\approx 1\mu m \end{aligned} $$

PHYSICAL REVIEW LETTERS

week ending
8 JANUARY 2010

<figure><img src="image_14.png" /></figure>

### Demonstration of a Neutral Atom Controlled-NOT Quantum Gate

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson,* T. G. Walker, and M. Saffman

Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706 USA
(Received 5 August 2009; published 8 January 2010)

## LETTER

A

<figure><img src="image_15.png" /></figure>

doi:10.1038/nature11361

### Quantum nonlinear optics with single photons enabled by strongly interacting atoms

Thibault Peyronel¹, Ofer Firstenberg¹,², Qi-Yu Liang¹, Sebastian Hofferberth¹,², Alexey V. Gorshkov³, Thomas Pohl⁴, Mikhail D. Lukin² & Vladan Vuletić³

REVIEWS OF MODERN PHYSICS, VOLUME 82, JULY–SEPTEMBER 2010

### Quantum information with Rydberg atoms

M. Saffman and T. G. Walker

Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, Wisconsin 53706, USA

K. Mølmer

Lundbeck Foundation Theoretical Center for Quantum System Research, Department of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark

(Published 18 August 2010)
<figure><img src="image_16.png" /></figure>

Tapered silica
optical fibers

$r < \lambda$

<figure><img src="image_17.png" /></figure>

(J. Raskop's thesis – LKB – Paris)

→ in the waist region, strong coupling of nearby atoms to guided mode **evanescent** component

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

**Atom trapping**

Le Kien et al, Phys. Rev. A 70 063403 (2004),

E. Vetsch et al, Phys. Rev. Lett. 104 203603 (2010)

**EIT-based quantum memory**

B. Gouraud et al, Phys. Rev. Lett. 114 180503 (2015)
# RYDBERG atom – nanofiber interface

<figure><img src="image_23.png" /></figure>

Rydberg-atom-nanofiber interface
for quantum technologies ?
# RYDBERG atom – nanofiber interface

<figure><img src="image_24.png" /></figure>

## Previous work

Spontaneous emission rates towards the guided modes

<figure><img src="image_25.png" /></figure>

## CASIMIR-POLDER potential

<figure><img src="image_26.png" /></figure>
# Spontaneous emission and energy shifts of a Rydberg rubidium atom close to an optical nanofiber

E. Stourm,¹ M. Lepers,² J. Robert,¹ S. Nic Chormaic,³ K. Mølmer,⁴ and E. Brion⁵,*

¹Université Paris-Saclay, CNRS, Laboratoire de physique des gaz et des plasmas, 91405, Orsay, France

²Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne Franche-Comté, 21078 Dijon, France

³Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa, 904-0495, Japan

⁴Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark

⁵Laboratoire Collisions Agrégats Réactivité, IRSAMC and UMR5589 du CNRS, Université de Toulouse III Paul Sabatier, F-31062 Toulouse Cedex 09, France

<figure><img src="image_27.png" /></figure>

(Received 11 February 2020; accepted 15 April 2020; published 14 May 2020)

2a

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

optical
nanofiber

<figure><img src="image_30.png" /></figure>

$n_2 = 1$

Previous work

Spontaneous
emission rates
towards the
guided modes

<figure><img src="image_31.png" /></figure>

CASIMIR-POLDER potential

<figure><img src="image_32.png" /></figure>
THEORY

# Spontaneous emission and energy shifts of a Rydberg rubidium atom close to an optical nanofiber

E. Stourm,¹ M. Lepers,² J. Robert,¹ S. Nic Chormaic,³ K. Mølmer,⁴ and E. Brion,⁵,*

¹Université Paris-Saclay, CNRS, Laboratoire de physique des gaz et des plasmas, 91405, Orsay, France

²Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Université de Bourgogne Franche-Comté, 21078 Dijon, France

³Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa, 904-0495, Japan

⁴Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark

⁵Laboratoire Collisions Agrégats Réactivité, IRSAMC and UMR5589 du CNRS, Université de Toulouse III Paul Sabatier, F-31062 Toulouse Cedex 09, France

<figure><img src="image_33.png" /></figure>

(Received 11 February 2020; accepted 15 April 2020; published 14 May 2020)

2a

↓

↓

↓

optical
nanofiber

<figure><img src="image_34.png" /></figure>

$n_2 = 1$

EXPERIMENT

PHYSICAL REVIEW RESEARCH 2, 012038(R) (2020)

Rapid Communications

# Generation of cold Rydberg atoms at submicron distances from an optical nanofiber

Krishnapriya Subramonian Rajasree,¹ Tridib Ray,¹,* Kristoffer Karlsson,¹ Jesse L. Everett,¹ and Síle Nic Chormaic¹,²,†

¹Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

²Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

<figure><img src="image_35.png" /></figure>

(Received 31 July 2019; accepted 20 January 2020; published 14 February 2020)

$R$ (nm)

$R$ (nm)
# RYDBERG atom – nanofiber interface

<figure><img src="image_36.png" /></figure>

New J. Phys. 25 (2023) 023022

**New Journal of Physics**

The open access journal at the forefront of physics

## PAPER

Interaction of two Rydberg atoms in the vicinity of an optical nanofibre

E Stourm¹, M Lepers², J Robert¹, S Nic Chormaic³, K Mølmer¹ and E Brion⁵,\*

## Here

VAN DER WAALS interaction of two RYDBERG $^{87}$Rb atoms → RYDBERG blockade?
Effective Hamiltonian of the system

$$H_{\text{eff}} = \frac{1}{\epsilon_0} \hat{\mathbf{d}}_A \cdot \overline{\mathbf{T}}(\mathbf{r}_A, \mathbf{r}_B) \cdot \hat{\mathbf{d}}_B + \text{quadrupolar terms}$$

J. Block and S. Scheel, Phys. Rev. A **96**, 062509 (2017).

<figure><img src="image_37.png" /></figure>

Non-retarded GREEN's function

$$\overline{\mathbf{T}}(\mathbf{r}_A, \mathbf{r}_B) \equiv - \lim_{\omega \to 0^+} \left(\frac{\omega}{c}\right)^2 \overline{\mathbf{G}}(\mathbf{r}_A, \mathbf{r}_B, \omega)$$

Total GREEN function

$$\overline{\mathbf{T}} = \overline{\mathbf{T}}_0 + \overline{\mathbf{T}}_{\text{fib}}$$

$$\overline{\mathbf{T}}_0 = -\frac{1}{4\pi r_{AB}^3} (\overline{\mathbf{I}} - 3\mathbf{u}_{AB} \otimes \mathbf{u}_{AB})$$
S states

<figure><img src="image_38.png" /><figcaption>⁸⁷Rb, |30S₁/₂⟩, a = 200nm</figcaption></figure>

• A, B prepared in |30S₁/₂⟩

• R<sub>A</sub> = R<sub>B</sub> = R

• a = 200 nm

<figure><img src="image_39.png" /></figure>
S states

<figure><img src="image_40.png" /></figure>

⁸⁷Rb, |30S₁/₂⟩, a = 200nm

• $\frac{U_{AB}}{U_{AB}^{(0)}} < 1$ at short distance

• $\frac{U_{AB}}{U_{AB}^{(0)}} \to 1$ for $\Delta z \to 0$

<figure><img src="image_41.png" /></figure>
S states

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>
S states

<figure><img src="image_44.png" /></figure>

asymptote for $\Delta z \gg R$ ?

$^{87}$Rb, $|30S_{1/2}\rangle$, $a = 200$nm

<figure><img src="image_45.png" /></figure>
S states

<figure><img src="image_46.png" /></figure>

* Same qualitative behaviour for all *n*'s

* RYDBERG blockade seemingly favoured by the nanofiber for RYDBERG S states

<figure><img src="image_47.png" /></figure>
# P states

<figure><img src="image_48.png" /></figure>

A, B prepared in $|30P_{3/2}, M = 3/2\rangle$

$R = 250\text{nm}$

$a = 200\text{nm}$

quantization axis along (Oz)

<figure><img src="image_49.png" /></figure>
VAN DER WAALS Hamiltonian
couples transitions in *A* and *B*

In free-space, certain transitions
cannot be coupled when quantization
and interatomic axes coincide,
because of a rotation symmetry.

<figure><img src="image_50.png" /></figure>
VAN DER WAALS Hamiltonian
couples transitions in *A* and *B*

The presence of the nanofiber
breaks the symmetry and
makes these couplings allowed

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>
Interpretation

A pair of $\sigma^+ - \sigma^+$ transitions
activated by the fiber becomes
dominant

<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /></figure>

Near resonant process
for $n \approx 37$

Strong enhancement
of the potential
Interpretation

A pair of $\sigma^+ - \sigma^+$ transitions
activated by the fiber becomes
dominant

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

Energy mismatch becomes
negative for $n > 37$
Consequences on the potential

<figure><img src="image_59.png" /></figure>

A, B prepared in $|nP_{3/2}, M = 3/2\rangle$

$R = 250\text{nm}$

$a = 200\text{nm}$

quantization axis along (Oz)

Enhancement for $n = 30$

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>
Consequences on the potential

<figure><img src="image_62.png" /></figure>

A, B prepared in $|nP_{3/2}, M = 3/2\rangle$

$R = 250\text{nm}$

$a = 200\text{nm}$

quantization axis along (Oz)

<figure><img src="image_63.png" /></figure>
# Consequences on the potential

<figure><img src="image_64.png" /></figure>

A, B prepared in $|nP_{3/2}, M = 3/2\rangle$

$R = 250\text{nm}$

$a = 200\text{nm}$

quantization axis along (Oz)

<figure><img src="image_65.png" /></figure>
# Consequences on the potential

<figure><img src="image_66.png" /></figure>

A, B prepared in $|nP_{3/2}, M = 3/2\rangle$

$R = 250\text{nm}$

$a = 200\text{nm}$

quantization axis along (Oz)

<figure><img src="image_67.png" /></figure>

New behaviour for $n = 45$

<figure><img src="image_68.png" /></figure>

$n = 45$, free-space

$\Delta z (\mu\text{m})$
Anisotropy of interaction

<figure><img src="image_69.png" /></figure>

A, B in $|30P_{3/2}, M_J = 3/2\rangle$

$R = 350$ nm

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>

fiber
Anisotropy of interaction

<figure><img src="image_72.png" /></figure>

A, B in $|30P_{3/2}, M_J = 3/2\rangle$
$R = 250$ nm

<figure><img src="image_73.png" /></figure>

<figure><img src="image_74.png" /></figure>

fiber
Anisotropy of interaction

<figure><img src="image_75.png" /></figure>
Anisotropy of interaction

<figure><img src="image_76.png" /></figure>
Conclusion

The presence of an optical nanofiber substantially modifies the interaction of
two RYDBERG-excited $^{87}$Rb atoms in its vicinity

* for S (isotropic) RYDBERG states, blockade seems to be enhanced by the presence of the nanofiber,

* for P (anisotropic) RYDBERG states, the interaction may change in nature (repulsive → attractive, formation of a potential well) and the anisotropy profile is modified with respect to free-space.

For more details

New J. Phys. 25 (2023) 023022

https://doi.org/10.1088/1367-2630/ab83f

New Journal of Physics

The open access journal at the forefront of physics

<figure><img src="image_77.png" /></figure>

IOP Institute of Physics

<figure><img src="image_78.png" /></figure>

Published in partnership
with: Deutsche Physikalische
Gesellschaft and the Institute
of Physics

PAPER

Interaction of two Rydberg atoms in the vicinity of an optical nanofibre

E Stourm¹, M Lepers², J Robert³, S Nic Chormaic⁰, K Mølmer⁰ and E Brion⁰

Perspectives

* More systematic studies of possible geometric configurations and states

* sub-, super- and selective radiance effects with RYDBERG atoms coupled to a nanofiber

* chiral effects