t=9 s - University of Virginia€¦ · Long-range Rydberg-Rydberg molecules bound by the...

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Long-range Rydberg-Rydberg molecules bound by the dipole-dipole interaction Hyunwook Park 1 , Martin Kiffner 2 , Wenhui Li 2 , and T. F. Gallagher 1,2 1 Department of Physics, University ofVirginia 2 Center for Quantum Technologies, National University of Singapore II. Existence of the wells: θ=0 well III. Robustness of wells IV. Dynamics Fig. 1 Rb ground-Rydberg diatomic potential (5s+30f) resembles a trilobite [1]. I. Introduction Rydberg atoms, which are highly excited atoms with one or more electrons with a large principal quantum number n, are a fascinating system to study a broad range of quantum mechanical problems due to their exaggerated properties, such as, huge geometrical cross sections, long radiative life times, and large polarizabilities, etc [1]. As one of the most fundamental problems in atoms and molecules, the dipole- dipole interaction has been intensively studied and observed in diverse disciplines, physics, chemistry, and bio-science, etc [2-4]. Recently, it has garnered much attention due to its potential application to quantum gates through the notion of dipole blockade [5]. For the quantum manipulation of atoms, an intriguing question was raised about the possibility of forming Rydberg molecules by Greene et al. [6]. It was claimed that when two closely spaced atoms make a pair, one of the atoms in the pair can be excited to Rydberg state while the other one remains in the ground state, forming a trilobite-like potential, as shown in Fig. 1. Similarly, the existence of Rydberg-Rydberg (RR) dimers was also predicted [7]. However, such Rydberg molecules have not been observed, although RR pairs were produced in an optical lattice potential[8, 9]. Here, we present a new approach to produce a RR molecule bound in a dipole-dipole potential. Specifically, we discuss the Rb ns 1/2 np 3/2 state which generates long range dipole-dipole potential wells in the presence of a small electric field. This new method should be able to produce a larger RR dimer than the trilobite- like potential and provide a new way for the quantum information physics. VI. Concluding Remarks A. It is found that a very large diatomic pot well exists in the ns 1/2 np 3/2 dipole-dipole potential when an electric field is applied. B. For a given n, R 0 can be adjusted to match, for example, lattice spacing by adjusting the electric field. C. We have observed the suppressed attractive dipole-dipole potentials as an evidence of the dipole-coupled anti-crossings induced by the electric field. D. In the future work, the microwave field can be used to engineer the large Rydberg-Rydberg dimers. * This research was supported by the Air Force Office of Scientific Research Basis states M=0 : 4 states ns 1/2 np -1/2 np -1/2 ns 1/2 ns -1/2 np 1/2 np 1/2 ns -1/2 M=1 : 4 states ns 1/2 np 1/2 np 1/2 ns 1/2 ns -1/2 np 3/2 np 3/2 ns -1/2 M=-1 : 4 states ns -1/2 np -1/2 np -1/2 ns -1/2 ns 1/2 np -3/2 np -3/2 ns 1/2 M=2 : 2 states ns 1/2 np 3/2 np 3/2 ns 1/2 M=-2 : 2 states ns -1/2 np -3/2 np -3/2 ns -1/2 2 4 6 8 10 12 -40 -20 0 20 40 n=40, =0, =0 Energy [MHz] R [m] ns 1/2 np 3/2 levels θ R atom1 (μ 1 ) atom 2 (μ 2 ) z ˆ A simple example of V dd in E: when two atoms are aligned along the field E, R // E (θ=0) 3 2 1 2 1 ) ˆ )( ˆ ( 3 R R R V dd The dipole-dipole interaction in ns 1/2 np 3/2 pair of atoms [10] where μ 1 = μ 2 = μ sp in this work θ R atom1 (μ 1 ) atom 2 (μ 2 ) z ˆ E ˆ The well exists at θ=0. How wide is the well in θ? In a realistic problem, two atoms are randomly oriented. 2 4 6 8 10 12 -10 0 10 n=40 =-10 MHz = o Energy [MHz] R [m] The potential well is plotted in a 3-dimensional space. The pot well has two minima along the z- axis (θ=0) and becomes shallow at other θ. The deep wells are in -20 o < θ < 20 o . 2 4 6 8 10 12 -10 0 10 n=40 =-10 MHz =90 o Energy [MHz] R [m] 5.3 5.4 5.5 5.6 5.7 5.8 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 (c) =0.08 rad R [m] (b) =0.02 rad Energy [MHz] |M|=1 |M|=2 (a) =0 2 fold degenerate |M|=0 |M|=2 |M|=1 |M|=0 around R=5.5 m (a) At θ=0, ΔM=0 and |M|=0, 1, and 2 levels cross. If θ 0, M is not conserved. (b) and (c) at θ 0, |M|=0, 1, and 2 levels are coupled, resulting in anti-crossings. 2 4 6 8 10 12 -20 -10 0 10 20 30 m j =3/2 Dash M=0 Solid |M|=1 Dot |M|=2 : Stark shift due to E n=40, =-10 MHz=0 Energy [MHz] R [m] m j =1/2 ns 1/2 np 3/2 levels =-10 MHz Potential well Question: What happens to the dipole-dipole interaction in the presence of an electric field, E ? Degenerate potential wells with equilibrium at R 0 =5.5 μm (≈ 10 5 Bohr radius a 0 ) Huge Rydberg-Rydberg molecule R 0 can be adjusted independent of n by changing δ, R 0 ≈δ -1/3 2 cos ) ( 9 3 ) ( 3 ) ( 28 ) ( 4 3 6 1 ) ( 2 R V R V R V R V R V dd dd dd dd An analytic formula for the bound state of the potential is given by, The vibrational frequency around the equilibrium position ,R 0 =5.5 μm, is computed to be 15 kHz, which is so slow that the atom’s motion remains on the adiabatic potential curves. Accordingly, this potential well keeps the atoms from collisional ionization, maintaining the Rydberg-Rydberg molecular state. References [6]Chris H. Greene et al. Phys. Rev. Lett. 85, 2458 (2000) [7] Christophe Boisseau et al. Phys. Rev. Lett. 88, 133004 (2002) [8] B. Vaucher et al. Phys. Rev. A 78 043415 (2008) [9] S. E. Anderson et al. Phys. Rev. Lett. 107, 263001 (2011) [10] Hyunwook Park et al. Phys. Rev. A 84, 022704 (2011) [1] T. F. Gallagher. Rydberg Atoms. Cambridge University Press (1994) [2] R. Vincent et al. Phys. Rev. B 83, 165426 (2011) [3] David Beljonne et al. J. Phys. Chem. B 113 19 (2009) [4] K. Autumn et al. Nature (London) 405, 681 (2000) [5] M. D. Lukin et al. Phys. Rev. Lett. 87, 037901 (2001) [11] Hyunwook Park et al. Phys. Rev. A 84, 052708 (2011) The degenerate ns 1/2 np 3/2 levels split and shift up and down due toV dd as the atoms approach close each other V. Preliminary evidence of the well: probing attractive potentials Experimental approach [11]: 1. Transfer atoms to an attractive potential by a microwave field 2. Allow a time for atoms to move and collide in the attractive potential between the microwave and detection 3. Detect ions produced by the atom-atom collision 2 4 6 8 10 12 -30 -20 -10 0 10 20 30 n=40 =45 o Energy [MHz] R [m] m j =1/2 =+10 MHz m j =3/2 12375 12380 12385 12390 12395 12400 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.006 0.008 0.010 0.012 0.014 I 2 ion Microwave frequency [X6 MHz] I 1 t=0 t=9 s ion 38p 3/2 [arb. units] allowed time t=0 m j =1/2 3/2 Two resonances occur due to Stark shift. The ionization from m j =1/2, I 1 , is much stronger than that of m j =3/2, I 2 , because seven attractive potentials exist in m j =1/2, contrast to only one attractive potential in m j =3/2. Due to the dipole-dipole coupling in the presence of the electric field, only one of m j =3/2 levels (red solid line) shifts down to form an attractive potential, which leads to ionizing collisions which can be detected. The dipole-dipole potential with δ>0 θ=90 o well with δ>0 (m j =3/2 above m j =1/2 in this case) We obtain a ring shaped potential about the z-axis 2 / 3 p 2 / 3 p 2 / 1 p 2 / 1 p 2 / 1 s 2 / 1 s δ Internal level structure of each Rydberg atom. The Stark shift of the |p ±3/2 > levels is denoted by δ = W(|p ±3/2 >)- W(|p ±3/2 >) (δ < 0 for the above configuration). The dipole transitions indicated by solid, dashed, and dotted lines couple to π, σ-, and σ+ polarized light, respectively.

Transcript of t=9 s - University of Virginia€¦ · Long-range Rydberg-Rydberg molecules bound by the...

Page 1: t=9 s - University of Virginia€¦ · Long-range Rydberg-Rydberg molecules bound by the dipole-dipole interaction Hyunwook Park1, Martin Kiffner2, Wenhui Li2, and T. F. Gallagher1,2

Long-range Rydberg-Rydberg molecules bound by the dipole-dipole interaction

Hyunwook Park1, Martin Kiffner2, Wenhui Li2, and T. F. Gallagher1,2

1Department of Physics, University of Virginia

2Center for Quantum Technologies, National University of Singapore

II. Existence of the wells: θ=0 well

III. Robustness of wells

IV. Dynamics

Fig. 1 Rb ground-Rydberg diatomic potential (5s+30f) resembles a trilobite [1].

I. Introduction

Rydberg atoms, which are highly excited atoms with one or more electrons with a large principal quantum number n, are

a fascinating system to study a broad range of quantum mechanical problems due to their exaggerated properties, such as, huge

geometrical cross sections, long radiative life times, and large polarizabilities, etc [1]. As one of the most fundamental problems in atoms and molecules, the dipole-

dipole interaction has been intensively studied and observed in diverse disciplines, physics, chemistry, and bio-science, etc [2-4]. Recently, it has garnered much

attention due to its potential application to quantum gates through the notion of dipole blockade [5].

For the quantum manipulation of atoms, an intriguing question was raised about the possibility of forming Rydberg molecules by Greene et al. [6]. It was claimed

that when two closely spaced atoms make a pair, one of the atoms in the pair can be excited to Rydberg state while the other one remains in the ground state,

forming a trilobite-like potential, as shown in Fig. 1. Similarly, the existence of Rydberg-Rydberg (RR) dimers was also predicted [7]. However, such Rydberg

molecules have not been observed, although RR pairs were produced in an optical lattice potential[8, 9].

Here, we present a new approach to produce a RR molecule bound in a dipole-dipole potential. Specifically, we discuss the Rb ns1/2np3/2 state which generates

long range dipole-dipole potential wells in the presence of a small electric field. This new method should be able to produce a larger RR dimer than the trilobite-

like potential and provide a new way for the quantum information physics.

VI. Concluding Remarks

A. It is found that a very large diatomic pot well exists in the ns1/2np3/2 dipole-dipole potential when an electric field is applied.

B. For a given n, R0 can be adjusted to match, for example, lattice spacing by adjusting the electric field.

C. We have observed the suppressed attractive dipole-dipole potentials as an evidence of the dipole-coupled anti-crossings induced by the electric field.

D. In the future work, the microwave field can be used to engineer the large Rydberg-Rydberg dimers.

* This research was supported by the Air Force Office of Scientific Research

Basis states M=0 : 4 states

ns1/2np-1/2 np-1/2ns1/2 ns-1/2np1/2 np1/2ns-1/2

M=1 : 4 states

ns1/2np1/2 np1/2ns1/2 ns-1/2np3/2 np3/2ns-1/2

M=-1 : 4 states

ns-1/2np-1/2 np-1/2ns-1/2 ns1/2np-3/2 np-3/2ns1/2

M=2 : 2 states

ns1/2np3/2 np3/2ns1/2

M=-2 : 2 states

ns-1/2np-3/2 np-3/2ns-1/2 2 4 6 8 10 12

-40

-20

0

20

40

n=40,=0,=0

En

erg

y [

MH

z]

R [m]

ns1/2

np3/2

levels

θ

R atom1

(μ1)

atom2

(μ2)

z

A simple example of Vdd in E:

when two atoms are aligned along the field E, R // E (θ=0)

3

2121 )ˆ)(ˆ(3

R

RRVdd

The dipole-dipole interaction in ns1/2np3/2 pair of atoms [10]

where μ1= μ2 = μsp in this work

θ

R

atom1

(μ1)

atom2

(μ2)

z

E

The well exists at θ=0. How wide is the well in θ?

In a realistic problem, two atoms are randomly oriented.

2 4 6 8 10 12

-10

0

10 n=40

=-10 MHz

=o

En

erg

y [

MH

z]

R [m]

The potential well is plotted in a 3-dimensional

space. The pot well has two minima along the z-

axis (θ=0) and becomes shallow at other θ. The

deep wells are in -20o < θ < 20o.

2 4 6 8 10 12

-10

0

10n=40

=-10 MHz

=90o

En

erg

y [

MH

z]

R [m]

5.3 5.4 5.5 5.6 5.7 5.8

-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

(c) =0.08 rad

R [m]

(b) =0.02 rad

En

erg

y [

MH

z]

|M|=1 |M|=2

(a) =0

2 fold degenerate

|M|=0

|M|=2

|M|=1

|M|=0

around R=5.5 m (a) At θ=0, ΔM=0 and |M|=0, 1, and 2 levels cross. If θ ≠ 0, M is not

conserved. (b) and (c) at θ ≠ 0, |M|=0, 1, and 2 levels are coupled, resulting in anti-crossings.

2 4 6 8 10 12

-20

-10

0

10

20

30

mj=3/2

Dash M=0

Solid |M|=1

Dot |M|=2

: Stark shift due to E

n=40, =-10 MHz=0

En

erg

y [

MH

z]

R [m]

mj=1/2

ns1/2

np3/2

levels

=-10 MHz

Potential well

Question: What happens to the dipole-dipole interaction

in the presence of an electric field, E ?

Degenerate potential wells

with equilibrium at R0=5.5 μm

(≈ 105 Bohr radius a0) Huge Rydberg-Rydberg molecule R0 can be adjusted independent of n by changing δ, R0≈δ-1/3

2cos)(93)(3)(28)(436

1)( 2 RVRVRVRVRV dddddddd

An analytic formula for the bound state of the potential is given by, The vibrational frequency around the equilibrium position ,R0=5.5 μm, is computed to be 15 kHz, which is so slow

that the atom’s motion remains on the adiabatic potential curves. Accordingly, this potential well keeps the atoms

from collisional ionization, maintaining the Rydberg-Rydberg molecular state.

References [6]Chris H. Greene et al. Phys. Rev. Lett. 85, 2458 (2000)

[7] Christophe Boisseau et al. Phys. Rev. Lett. 88, 133004 (2002)

[8] B. Vaucher et al. Phys. Rev. A 78 043415 (2008)

[9] S. E. Anderson et al. Phys. Rev. Lett. 107, 263001 (2011)

[10] Hyunwook Park et al. Phys. Rev. A 84, 022704 (2011)

[1] T. F. Gallagher. Rydberg Atoms. Cambridge University Press (1994)

[2] R. Vincent et al. Phys. Rev. B 83, 165426 (2011)

[3] David Beljonne et al. J. Phys. Chem. B 113 19 (2009)

[4] K. Autumn et al. Nature (London) 405, 681 (2000)

[5] M. D. Lukin et al. Phys. Rev. Lett. 87, 037901 (2001)

[11] Hyunwook Park et al. Phys. Rev. A 84, 052708 (2011)

The degenerate ns1/2np3/2 levels split and shift up and down

due to Vdd as the atoms approach close each other

V. Preliminary evidence of the well: probing attractive potentials

Experimental approach [11]:

1. Transfer atoms to an attractive

potential by a microwave field

2. Allow a time for atoms to

move and collide in the

attractive potential between

the microwave and detection

3. Detect ions produced by the

atom-atom collision 2 4 6 8 10 12

-30

-20

-10

0

10

20

30

n=40

=45o

En

erg

y [

MH

z]

R [m]

mj=1/2

=+10 MHz

mj=3/2

12375 12380 12385 12390 12395 12400

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.006

0.008

0.010

0.012

0.014

I2io

n

Microwave frequency [X6 MHz]

I1

t=0

t=9 s

ion

38

p3

/2 [a

rb. u

nits]

allowed time

t=0

mj=1/2 3/2

Two resonances occur due

to Stark shift.

The ionization from mj=1/2,

I1, is much stronger than

that of mj=3/2, I2, because

seven attractive potentials

exist in mj=1/2, contrast to

only one attractive potential

in mj=3/2.

Due to the dipole-dipole coupling in the presence of the electric

field, only one of mj=3/2 levels (red solid line) shifts down to form

an attractive potential, which leads to ionizing collisions which can

be detected.

The dipole-dipole potential with δ>0 θ=90o well with δ>0 (mj=3/2 above mj=1/2 in this case) We obtain a ring shaped potential about the z-axis

2/3p 2/3p

2/1p 2/1p

2/1s 2/1s

δ

Internal level structure of each Rydberg atom.

The Stark shift of the |p±3/2 > levels is denoted by

δ = W(|p±3/2>)- W(|p±3/2>) (δ < 0 for the above configuration). The dipole transitions indicated by solid, dashed, and dotted lines couple to π, σ-, and σ+ polarized light, respectively.