Rydberg-Raman slow light: A spectrosopic tool
to measure the interaction between the
Rydberg atoms
Julius Ruseckas1
Gediminas Juzeliūnas1
Ite A. Yu2
1 Institute
of Theoretical Physics and Astronomy, Vilnius University, Lithuania
of Physics and Frontier Research Center on Fundamental and Applied
Sciences of Matters, National Tsing Hua University, Hsinchu, Taiwan
2 Department
October 18, 2014
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
1 / 38
Outline
1
Introduction
Rydberg atoms
Rydberg EIT
2
Rydberg-Raman slow light
Propagation of single photon
Storage and retrieval of two photons
Many photons: classical probe field
3
Summary
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
2 / 38
Outline
1
Introduction
Rydberg atoms
Rydberg EIT
2
Rydberg-Raman slow light
Propagation of single photon
Storage and retrieval of two photons
Many photons: classical probe field
3
Summary
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
2 / 38
Outline
1
Introduction
Rydberg atoms
Rydberg EIT
2
Rydberg-Raman slow light
Propagation of single photon
Storage and retrieval of two photons
Many photons: classical probe field
3
Summary
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
2 / 38
Rydberg atom
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
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Rydberg atom
Rydberg atom
A Rydberg atom is an excited atom with one or more electrons
that have a very high principal quantum number.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
4 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Rydberg atom
Rydberg atoms have a number of peculiar
properties:
an exaggerated response to electric
and magnetic field
long decay periods
electron wavefunctions approximate
classical orbits
excited electron experiences the
electric potential as in a hydrogen atom
radius of the orbit scales as n2
energy level spacing decreases as 1/n3
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
5 / 38
Interactions between Rydberg atoms
Atoms excited into the Rydberg states interact strongly with
each other due to the presence of resonant
long-wavelength dipole-dipole interactions
The interaction strength between atoms depends on the
principal quantum number n;
dipole moment to nearby states scales as n2 .
The strength of these interactions for n & 100 can be
comparable to the strength of the Coulomb interaction
between ions.
In a mesoscopic ensemble strong, long-range interactions
can be used for fast preparation of desired many-particle
states.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
6 / 38
Interactions between Rydberg atoms
Atoms excited into the Rydberg states interact strongly with
each other due to the presence of resonant
long-wavelength dipole-dipole interactions
The interaction strength between atoms depends on the
principal quantum number n;
dipole moment to nearby states scales as n2 .
The strength of these interactions for n & 100 can be
comparable to the strength of the Coulomb interaction
between ions.
In a mesoscopic ensemble strong, long-range interactions
can be used for fast preparation of desired many-particle
states.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
6 / 38
Interactions between Rydberg atoms
Atoms excited into the Rydberg states interact strongly with
each other due to the presence of resonant
long-wavelength dipole-dipole interactions
The interaction strength between atoms depends on the
principal quantum number n;
dipole moment to nearby states scales as n2 .
The strength of these interactions for n & 100 can be
comparable to the strength of the Coulomb interaction
between ions.
In a mesoscopic ensemble strong, long-range interactions
can be used for fast preparation of desired many-particle
states.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
6 / 38
Interactions between Rydberg atoms
Atoms excited into the Rydberg states interact strongly with
each other due to the presence of resonant
long-wavelength dipole-dipole interactions
The interaction strength between atoms depends on the
principal quantum number n;
dipole moment to nearby states scales as n2 .
The strength of these interactions for n & 100 can be
comparable to the strength of the Coulomb interaction
between ions.
In a mesoscopic ensemble strong, long-range interactions
can be used for fast preparation of desired many-particle
states.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
6 / 38
Interactions between Rydberg atoms
Long-range interactions scale as:
in the van der Waals regime
1/R 6
in the dipole-dipole regime
1/R 3
1/R 6 van der Waals potential between two neutral atoms
changes to an
1/R 7
potential for large R when the effects of retardation are
included (Casimir and Polder, 1948).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
7 / 38
Interactions between Rydberg atoms
Long-range interactions scale as:
in the van der Waals regime
1/R 6
in the dipole-dipole regime
1/R 3
1/R 6 van der Waals potential between two neutral atoms
changes to an
1/R 7
potential for large R when the effects of retardation are
included (Casimir and Polder, 1948).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
7 / 38
Interactions between Rydberg atoms
Long-range interactions scale as:
in the van der Waals regime
1/R 6
in the dipole-dipole regime
1/R 3
1/R 6 van der Waals potential between two neutral atoms
changes to an
1/R 7
potential for large R when the effects of retardation are
included (Casimir and Polder, 1948).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
7 / 38
Interactions between Rydberg atoms
Long-range interactions scale as:
in the van der Waals regime
1/R 6
in the dipole-dipole regime
1/R 3
1/R 6 van der Waals potential between two neutral atoms
changes to an
1/R 7
potential for large R when the effects of retardation are
included (Casimir and Polder, 1948).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
7 / 38
Consequences of interactions
Strong long-range interactions lead to cooperative effects
such as:
superradiance
dipole blockade.
May provide the basis for applications such as single-photon
sources and quantum gates.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
8 / 38
Consequences of interactions
Strong long-range interactions lead to cooperative effects
such as:
superradiance
dipole blockade.
May provide the basis for applications such as single-photon
sources and quantum gates.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
8 / 38
Consequences of interactions
Strong long-range interactions lead to cooperative effects
such as:
superradiance
dipole blockade.
May provide the basis for applications such as single-photon
sources and quantum gates.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
8 / 38
Consequences of interactions
Strong long-range interactions lead to cooperative effects
such as:
superradiance
dipole blockade.
May provide the basis for applications such as single-photon
sources and quantum gates.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
8 / 38
Dipole blockade
An important manifestation of Rydberg-level interactions is
excitation blockade.
If one atom is excited into the Rydberg state, it shifts the
resonance frequencies of all the surrounding atoms
suppressing their excitation.
This process has potential application in quantum
information processing
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
9 / 38
Dipole blockade
An important manifestation of Rydberg-level interactions is
excitation blockade.
If one atom is excited into the Rydberg state, it shifts the
resonance frequencies of all the surrounding atoms
suppressing their excitation.
This process has potential application in quantum
information processing
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
9 / 38
Dipole blockade
An important manifestation of Rydberg-level interactions is
excitation blockade.
If one atom is excited into the Rydberg state, it shifts the
resonance frequencies of all the surrounding atoms
suppressing their excitation.
This process has potential application in quantum
information processing
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
9 / 38
Dipole blockade
An important manifestation of Rydberg-level interactions is
excitation blockade.
If one atom is excited into the Rydberg state, it shifts the
resonance frequencies of all the surrounding atoms
suppressing their excitation.
This process has potential application in quantum
information processing
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
9 / 38
Rydberg EIT
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
10 / 38
Rydberg EIT
A quantum probe field incident onto a cold atomic gas is
coupled to high-lying atomic states (Rydberg levels) by a
second, stronger laser field (control field).
EIT allows for strong atom-light interactions without
absorption
Rydberg states provide strong long-range atom-atom
interactions
The resulting combination of EIT with Rydberg atoms can be
used for inducing strong photon-photon interactions.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
11 / 38
Rydberg EIT
A quantum probe field incident onto a cold atomic gas is
coupled to high-lying atomic states (Rydberg levels) by a
second, stronger laser field (control field).
EIT allows for strong atom-light interactions without
absorption
Rydberg states provide strong long-range atom-atom
interactions
The resulting combination of EIT with Rydberg atoms can be
used for inducing strong photon-photon interactions.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
11 / 38
Rydberg EIT
A quantum probe field incident onto a cold atomic gas is
coupled to high-lying atomic states (Rydberg levels) by a
second, stronger laser field (control field).
EIT allows for strong atom-light interactions without
absorption
Rydberg states provide strong long-range atom-atom
interactions
The resulting combination of EIT with Rydberg atoms can be
used for inducing strong photon-photon interactions.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
11 / 38
Rydberg EIT
A quantum probe field incident onto a cold atomic gas is
coupled to high-lying atomic states (Rydberg levels) by a
second, stronger laser field (control field).
EIT allows for strong atom-light interactions without
absorption
Rydberg states provide strong long-range atom-atom
interactions
The resulting combination of EIT with Rydberg atoms can be
used for inducing strong photon-photon interactions.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
11 / 38
Rydberg EIT
For a single incident probe photon
the control field induces a spectral transparency window in
the otherwise opaque medium via EIT
the probe photon travels at much reduced speed in the
form of a coupled excitation of light and matter (a Rydberg
polariton).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
12 / 38
Rydberg EIT
For a single incident probe photon
the control field induces a spectral transparency window in
the otherwise opaque medium via EIT
the probe photon travels at much reduced speed in the
form of a coupled excitation of light and matter (a Rydberg
polariton).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
12 / 38
Rydberg EIT
For a single incident probe photon
the control field induces a spectral transparency window in
the otherwise opaque medium via EIT
the probe photon travels at much reduced speed in the
form of a coupled excitation of light and matter (a Rydberg
polariton).
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
12 / 38
Rydberg EIT
If two probe photons are incident
onto the Rydberg medium
the strong interaction between
two Rydberg atoms tunes the
transition out of resonance
destroying the transparency and
leading to absorption.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
13 / 38
Rydberg EIT
If two probe photons are incident
onto the Rydberg medium
the strong interaction between
two Rydberg atoms tunes the
transition out of resonance
destroying the transparency and
leading to absorption.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
13 / 38
Rydberg EIT
If two probe photons are incident
onto the Rydberg medium
the strong interaction between
two Rydberg atoms tunes the
transition out of resonance
destroying the transparency and
leading to absorption.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
13 / 38
Rydberg EIT
Rydberg EIT can be used:
to generate non-classical light
to create strongly correlated many-body states of light
to create many-atom entanglement
to create photonic phase gates
to probe electric fields inside vapor cells and close to
surfaces
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
14 / 38
Rydberg EIT
Rydberg EIT can be used:
to generate non-classical light
to create strongly correlated many-body states of light
to create many-atom entanglement
to create photonic phase gates
to probe electric fields inside vapor cells and close to
surfaces
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
14 / 38
Rydberg EIT
Rydberg EIT can be used:
to generate non-classical light
to create strongly correlated many-body states of light
to create many-atom entanglement
to create photonic phase gates
to probe electric fields inside vapor cells and close to
surfaces
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
14 / 38
Rydberg EIT
Rydberg EIT can be used:
to generate non-classical light
to create strongly correlated many-body states of light
to create many-atom entanglement
to create photonic phase gates
to probe electric fields inside vapor cells and close to
surfaces
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
14 / 38
Rydberg EIT
Rydberg EIT can be used:
to generate non-classical light
to create strongly correlated many-body states of light
to create many-atom entanglement
to create photonic phase gates
to probe electric fields inside vapor cells and close to
surfaces
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
14 / 38
Disadvantage of Rydberg EIT
Only one photon propagates without absorption in the
Rydberg blockade region.
All additional photons are absorbed, this leads to losses.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
15 / 38
Our proposal
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
16 / 38
Possible application of proposed scheme
By storing and retrieving the Rydberg-Raman slow light, one can
reconstruct the interaction potential between the Rydberg
atoms from the measurement of the second order correlation
function of the retrieved light.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
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Second-order correlation function
The definition of the second-order correlation function is
g (2) (τ ) =
hE † (t)E † (t + τ )E(t + τ )E(t)i
hE † (t)E(t)ihE † (t + τ )E(t + τ )i
Can be measured using the
Hanbury-Brown and Twiss detection
scheme
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
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Assumptions
We assume that the incident probe field is a long flat pulse,
such that except for a short transient period, it is constant
most of the time.
We are interested in the distances for which the interaction
potential V is within the EIT window, that is in the distances
beyond the Rydberg blockade radius.
Therefore we neglect the Rydberg blockade during the
propagation in the medium and assume that dark-state
polaritons propagate independently in the atomic medium.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
19 / 38
Assumptions
We assume that the incident probe field is a long flat pulse,
such that except for a short transient period, it is constant
most of the time.
We are interested in the distances for which the interaction
potential V is within the EIT window, that is in the distances
beyond the Rydberg blockade radius.
Therefore we neglect the Rydberg blockade during the
propagation in the medium and assume that dark-state
polaritons propagate independently in the atomic medium.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
19 / 38
Assumptions
We assume that the incident probe field is a long flat pulse,
such that except for a short transient period, it is constant
most of the time.
We are interested in the distances for which the interaction
potential V is within the EIT window, that is in the distances
beyond the Rydberg blockade radius.
Therefore we neglect the Rydberg blockade during the
propagation in the medium and assume that dark-state
polaritons propagate independently in the atomic medium.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
19 / 38
Storage and retrieval of Rydberg-Raman slow
light
After switching off of the control fields the photon part of the
Rydberg-Raman polariton is absorbed
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
We can restore by switching the control fields on again.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
20 / 38
Storage and retrieval of Rydberg-Raman slow
light
After switching off of the control fields the photon part of the
Rydberg-Raman polariton is absorbed
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
We can restore by switching the control fields on again.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
20 / 38
Storage and retrieval of Rydberg-Raman slow
light
After switching off of the control fields the photon part of the
Rydberg-Raman polariton is absorbed
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
We can restore by switching the control fields on again.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
20 / 38
Storage and retrieval of Rydberg-Raman slow
light
We again neglect the Rydberg blockade during the
propagation of the restored dark-state polaritons.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
21 / 38
Storage and retrieval of Rydberg-Raman slow
light
We again neglect the Rydberg blockade during the
propagation of the restored dark-state polaritons.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
21 / 38
Coupled and uncoupled states
Coupled and uncoupled states:
1
C = (Ω1 S1 + Ω2 S2 )
Ω
1
U = (Ω∗2 S1 − Ω∗1 S2 )
Ω
where
Ω=
q
|Ω1 |2 + |Ω2 |2
is the total Rabi frequency.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
22 / 38
Coupled and uncoupled states
We can restore from the coupled state, by switching the same
control fields on again, or from the uncoupled state, by using
different control beams with Rabi frequencies
Ω01 = Ω∗2 ,
Ruseckas et al. (Vilnius and others)
Ω02 = −Ω∗1
Rydberg-Raman slow light
October 18, 2014
23 / 38
Mathematical description
Equations for slowly varying probe field
(∂t + c∂z )E = igP
and matter fields
i∂t P(z) = − iΓP(z) − Ω1 S1 (z) − Ω2 S2 (z) − gE(z)
i∂t S1 (z) =V(z)S1 (z) − Ω∗1 P(z)
i∂t S2 (z) = − Ω∗2 P(z)
where
Z
V(z) =
dz 0 V (z − z 0 )S1† (z 0 )S1 (z 0 )
describes interaction between Rydberg atoms.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
24 / 38
Propagation of single photon
The Rydberg-Raman polariton propagates with the group
velocity
Ω2
vg0 = c 2
g
Single-excitation wave functions obey the relation
Ω1 ΦS1 + Ω2 ΦS2 + gΦE = 0
Uncoupled state is not populated.
Single-excitation wave functions are related to the probe field as
ΦS1 ≈ −
Ruseckas et al. (Vilnius and others)
gΩ∗1
ΦE ,
Ω2
Φ S2 ≈ −
Rydberg-Raman slow light
gΩ∗2
ΦE
Ω2
October 18, 2014
25 / 38
Propagation of single photon
The Rydberg-Raman polariton propagates with the group
velocity
Ω2
vg0 = c 2
g
Single-excitation wave functions obey the relation
Ω1 ΦS1 + Ω2 ΦS2 + gΦE = 0
Uncoupled state is not populated.
Single-excitation wave functions are related to the probe field as
Φ S1 ≈ −
Ruseckas et al. (Vilnius and others)
gΩ∗1
ΦE ,
Ω2
Φ S2 ≈ −
Rydberg-Raman slow light
gΩ∗2
ΦE
Ω2
October 18, 2014
25 / 38
Propagation of single photon
The Rydberg-Raman polariton propagates with the group
velocity
Ω2
vg0 = c 2
g
Single-excitation wave functions obey the relation
Ω1 ΦS1 + Ω2 ΦS2 + gΦE = 0
Uncoupled state is not populated.
Single-excitation wave functions are related to the probe field as
Φ S1 ≈ −
Ruseckas et al. (Vilnius and others)
gΩ∗1
ΦE ,
Ω2
Φ S2 ≈ −
Rydberg-Raman slow light
gΩ∗2
ΦE
Ω2
October 18, 2014
25 / 38
Two photons: two-excitation wave functions
We use the two-excitation wave functions
ΦX Y (z1 , z2 , t) = hvac|X (z1 , t)Y(z2 , t)|Ψi
where operators X and Y stand for the operators E, S1 , S2 and
|Ψi is two-excitation state.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
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Storage of tow photons
(2)
The second-order correlation function gin (τ ) of the incident
probe field is related to the two-photon wave function:
(2)
gin (τ ) =
|ΦEE (0, −cτ )|2
E04
The two-photon wave function inside of the medium:
s
(2) z1 − z2
2
ΦEE (z1 , z2 , t) = E0 gin
vg0
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
27 / 38
Storage of tow photons
(2)
The second-order correlation function gin (τ ) of the incident
probe field is related to the two-photon wave function:
(2)
gin (τ ) =
|ΦEE (0, −cτ )|2
E04
The two-photon wave function inside of the medium:
s
(2) z1 − z2
2
ΦEE (z1 , z2 , t) = E0 gin
vg0
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
27 / 38
Storage of tow photons
Wave functions corresponding to one photon and one atomic
excitation:
Φ+
ES1 ≈ −
gΩ∗1
ΦEE ,
Ω2
Φ+
ES2 ≈ −
gΩ∗2
ΦEE
Ω2
Wave functions describing two atomic excitations are
ΦS1 S1 ≈ −
gΩ∗1 +
Φ
,
Ω2 ES1
Φ+
S1 S2 ≈ −
gΩ∗2 +
Φ
,
Ω2 ES1
Φ S2 S2 ≈ −
gΩ∗2 +
Φ
Ω2 ES2
After switching off of the control field the photon part is
absorbed and the only remaining non-zero two-excitation wave
functions are ΦS1 S1 , ΦS2 S2 , ΦS1 S2 .
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
28 / 38
Storage of tow photons
Wave functions corresponding to one photon and one atomic
excitation:
Φ+
ES1 ≈ −
gΩ∗1
ΦEE ,
Ω2
Φ+
ES2 ≈ −
gΩ∗2
ΦEE
Ω2
Wave functions describing two atomic excitations are
ΦS1 S1 ≈ −
gΩ∗1 +
Φ
,
Ω2 ES1
Φ+
S1 S2 ≈ −
gΩ∗2 +
Φ
,
Ω2 ES1
Φ S2 S2 ≈ −
gΩ∗2 +
Φ
Ω2 ES2
After switching off of the control field the photon part is
absorbed and the only remaining non-zero two-excitation wave
functions are ΦS1 S1 , ΦS2 S2 , ΦS1 S2 .
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
28 / 38
Free evolution during the storage
The two excitation atomic coherence picks up the phase
ΦS1 S1 (T ) = e−iV (z2 −z1 )T ΦS1 S1 (0)
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
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Restoring from the coupled state
The same control beams are switched on again.
Wave functions corresponding to one photon and one atomic
excitation:
1
(Ω1 ΦS1 S1 + Ω2 Φ+
S1 S2 )
g
1
= − (Ω2 ΦS2 S2 + Ω1 Φ+
S1 S2 )
g
Φ+
ES1 = −
Φ+
ES2
Wave function describing two photons is
1
+
ΦEE = − (Ω1 Φ+
ES1 + Ω2 ΦES2 ) .
g
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
30 / 38
Restoring from the coupled state
The same control beams are switched on again.
Wave functions corresponding to one photon and one atomic
excitation:
1
(Ω1 ΦS1 S1 + Ω2 Φ+
S1 S2 )
g
1
= − (Ω2 ΦS2 S2 + Ω1 Φ+
S1 S2 )
g
Φ+
ES1 = −
Φ+
ES2
Wave function describing two photons is
1
+
ΦEE = − (Ω1 Φ+
ES1 + Ω2 ΦES2 ) .
g
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
30 / 38
Restoring from the coupled state
(2)
The second-order correlation function gout (τ ) of the output
probe field can be related to the two-photon wave function
(2)
gout (τ ) =
|ΦEE (L, L + cτ )|2
E04
The Rydberg blockade during the free evolution modifies the
resulting second-order correlation function by a factor
!
(2)
gout (τ )
|Ω1 |4
|Ω1 |4
=
1
−
2
1
−
1
−
cos[V
(v
τ
)T
]
g0
(2)
Ω4
Ω4
g (τ )
in
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
31 / 38
Restoring from the coupled state
(2)
The second-order correlation function gout (τ ) of the output
probe field can be related to the two-photon wave function
(2)
gout (τ ) =
|ΦEE (L, L + cτ )|2
E04
The Rydberg blockade during the free evolution modifies the
resulting second-order correlation function by a factor
!
(2)
gout (τ )
|Ω1 |4
|Ω1 |4
=
1
−
2
1
−
1
−
cos[V
(v
τ
)T
]
g0
(2)
Ω4
Ω4
g (τ )
in
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
31 / 38
Restoring from the uncoupled state
We restore the light using different control beams with Rabi
frequencies
Ω02 = −Ω∗1
Ω01 = Ω∗2 ,
Nothing gets restored for the single photon input.
The second-order correlation function:
(2)
gout (τ )
(2)
gin (τ )
∼ 1 − cos[V (vg0 τ )T ]
For sufficiently short storage time T :
(2)
gout (τ )
(2)
gin (τ )
Ruseckas et al. (Vilnius and others)
∼ [V (vg0 τ )T ]2
Rydberg-Raman slow light
October 18, 2014
32 / 38
Restoring from the uncoupled state
We restore the light using different control beams with Rabi
frequencies
Ω02 = −Ω∗1
Ω01 = Ω∗2 ,
Nothing gets restored for the single photon input.
The second-order correlation function:
(2)
gout (τ )
(2)
gin (τ )
∼ 1 − cos[V (vg0 τ )T ]
For sufficiently short storage time T :
(2)
gout (τ )
(2)
gin (τ )
Ruseckas et al. (Vilnius and others)
∼ [V (vg0 τ )T ]2
Rydberg-Raman slow light
October 18, 2014
32 / 38
Restoring from the uncoupled state
We restore the light using different control beams with Rabi
frequencies
Ω02 = −Ω∗1
Ω01 = Ω∗2 ,
Nothing gets restored for the single photon input.
The second-order correlation function:
(2)
gout (τ )
(2)
gin (τ )
∼ 1 − cos[V (vg0 τ )T ]
For sufficiently short storage time T :
(2)
gout (τ )
(2)
gin (τ )
Ruseckas et al. (Vilnius and others)
∼ [V (vg0 τ )T ]2
Rydberg-Raman slow light
October 18, 2014
32 / 38
Many photons: classical probe field
The incident probe field represents the coherent (classic-like)
state with an arbitrary number of photons and an average
amplitude E0 .
We assume that the second-order correlation function
(2)
(2)
gin (τ ) of the incident light is constant: gin (τ ) = const.
The quantum state |Ψi of the atoms is the coherent state of
the C mode:
Z
gE0
†
|Ψi = exp −
dz(C (z) − C(z)) |vaci
Ω
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
33 / 38
Many photons: classical probe field
The incident probe field represents the coherent (classic-like)
state with an arbitrary number of photons and an average
amplitude E0 .
We assume that the second-order correlation function
(2)
(2)
gin (τ ) of the incident light is constant: gin (τ ) = const.
The quantum state |Ψi of the atoms is the coherent state of
the C mode:
Z
gE0
†
|Ψi = exp −
dz(C (z) − C(z)) |vaci
Ω
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
33 / 38
Many photons: classical probe field
The incident probe field represents the coherent (classic-like)
state with an arbitrary number of photons and an average
amplitude E0 .
We assume that the second-order correlation function
(2)
(2)
gin (τ ) of the incident light is constant: gin (τ ) = const.
The quantum state |Ψi of the atoms is the coherent state of
the C mode:
Z
gE0
†
|Ψi = exp −
dz(C (z) − C(z)) |vaci
Ω
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
33 / 38
Free evolution during the storage
During the free evolution the atomic field operators S1 and S2
change according to the equations
Z
∂t S1 (z, t) = − i dz 0 V (z − z 0 )S1† (z 0 , t)S1 (z 0 , t)S1 (z, t)
∂t S2 (z, t) =0
When the free evolution duration T is short enough, we use
perturbation expansion up to the second power of T .
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
34 / 38
Free evolution during the storage
During the free evolution the atomic field operators S1 and S2
change according to the equations
Z
∂t S1 (z, t) = − i dz 0 V (z − z 0 )S1† (z 0 , t)S1 (z 0 , t)S1 (z, t)
∂t S2 (z, t) =0
When the free evolution duration T is short enough, we use
perturbation expansion up to the second power of T .
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
34 / 38
Restoring from the coupled state
The restored light is related to the coupled state at time T :
g
E = − C(T )
Ω
If the electric field of the probe beam is weak, then
!
|Ω1 |4
|Ω1 |4
(2)
1−
[V (vg0 τ )T ]2
gout (τ ) ≈ 1 −
Ω4
Ω4
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
35 / 38
Restoring from the uncoupled state
The restored light is related to the uncoupled state at time T :
Ω
E = − U(T )
g
We get
(2)
gout (τ ) ∼ [V (vg0 τ )T ]2
If calculating g (2) (τ ) we divide by the intensity of the incident
light instead of the restored light, we can determine the
magnitude of interaction potential V as well:
g (2) (τ ) =
|Ω2 |4 |Ω1 |4
[V (vg0 τ )T ]2
Ω8
If the fidelity of the restoring is not 100%, one should introduce
additional fidelity factor.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
36 / 38
Restoring from the uncoupled state
The restored light is related to the uncoupled state at time T :
Ω
E = − U(T )
g
We get
(2)
gout (τ ) ∼ [V (vg0 τ )T ]2
If calculating g (2) (τ ) we divide by the intensity of the incident
light instead of the restored light, we can determine the
magnitude of interaction potential V as well:
g (2) (τ ) =
|Ω2 |4 |Ω1 |4
[V (vg0 τ )T ]2
Ω8
If the fidelity of the restoring is not 100%, one should introduce
additional fidelity factor.
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
36 / 38
Summary
By properly storing and retrieving the slow light in the
Rydberg-Raman level scheme, one can reconstruct the
interaction potential between the Rydberg atoms.
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
One can restore either from coupled or from uncoupled
state.
Restoring from the uncoupled state is more sensitive: the
second order correlation function of the retrieved light is
proportional to the square of the interaction potential
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
37 / 38
Summary
By properly storing and retrieving the slow light in the
Rydberg-Raman level scheme, one can reconstruct the
interaction potential between the Rydberg atoms.
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
One can restore either from coupled or from uncoupled
state.
Restoring from the uncoupled state is more sensitive: the
second order correlation function of the retrieved light is
proportional to the square of the interaction potential
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
37 / 38
Summary
By properly storing and retrieving the slow light in the
Rydberg-Raman level scheme, one can reconstruct the
interaction potential between the Rydberg atoms.
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
One can restore either from coupled or from uncoupled
state.
Restoring from the uncoupled state is more sensitive: the
second order correlation function of the retrieved light is
proportional to the square of the interaction potential
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
37 / 38
Summary
By properly storing and retrieving the slow light in the
Rydberg-Raman level scheme, one can reconstruct the
interaction potential between the Rydberg atoms.
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
One can restore either from coupled or from uncoupled
state.
Restoring from the uncoupled state is more sensitive: the
second order correlation function of the retrieved light is
proportional to the square of the interaction potential
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
37 / 38
Summary
By properly storing and retrieving the slow light in the
Rydberg-Raman level scheme, one can reconstruct the
interaction potential between the Rydberg atoms.
During the storage duration the atomic coherences pick up
additional phases due to interaction between atoms.
The phase of atomic coherence is transferred to the restored
light and can be detected by measuring the of the second
order correlation function.
One can restore either from coupled or from uncoupled
state.
Restoring from the uncoupled state is more sensitive: the
second order correlation function of the retrieved light is
proportional to the square of the interaction potential
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
37 / 38
Thank you for your attention!
Ruseckas et al. (Vilnius and others)
Rydberg-Raman slow light
October 18, 2014
38 / 38