Probing weakly bound molecules with nonresonant light Mikhail Lemeshko
Bretislav Friedrich
Fritz Haber Institute of the Max Planck Society, Berlin
DPG Jahrestagung March 8 – 12, 2010
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
2 / 32
Structure of weakly bound molecules
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
3 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
r
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
n = 1: Coulomb potential (H atom) n = 2: ion/electron + polar molecule
r
n = 3: two polar molecules n = 4: ion/electron + atom n = 6: two atoms (molecular potential)
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
n = 1: Coulomb potential (H atom) n = 2: ion/electron + polar molecule
r
n = 3: two polar molecules n = 4: ion/electron + atom n = 6: two atoms (molecular potential)
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
n = 1: Coulomb potential (H atom) n = 2: ion/electron + polar molecule
r
n = 3: two polar molecules n = 4: ion/electron + atom n = 6: two atoms (molecular potential)
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
n = 1: Coulomb potential (H atom) n = 2: ion/electron + polar molecule
r
n = 3: two polar molecules n = 4: ion/electron + atom n = 6: two atoms (molecular potential)
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
Molecular potentials
In molecular physics, most potentials have an asymptotic form V (r) ∼ −Cn /rn
V(r)
n = 1: Coulomb potential (H atom) n = 2: ion/electron + polar molecule
r
n = 3: two polar molecules n = 4: ion/electron + atom n = 6: two atoms (molecular potential)
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
4 / 32
Structure of weakly bound molecules
What happens near the threshold? Near-threshold states have large quantum numbers We like to think that this justifies the semiclassical (WKB) approximation
V(r)
r
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
5 / 32
Structure of weakly bound molecules
What happens near the threshold? Near-threshold states have large quantum numbers We like to think that this justifies the semiclassical (WKB) approximation It does not. What really matters is the action: if S ~ the WKB is valid V(r)
r
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
5 / 32
Structure of weakly bound molecules
What happens near the threshold? Near-threshold states have large quantum numbers We like to think that this justifies the semiclassical (WKB) approximation It does not. What really matters is the action: if S ~ the WKB is valid V(r)
This is the case for a Coulomb potential
Semiclassical region
r
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
5 / 32
Structure of weakly bound molecules
What happens near the threshold? Near-threshold states have large quantum numbers We like to think that this justifies the semiclassical (WKB) approximation It does not. What really matters is the action: if S ~ the WKB is valid V(r)
This is the case for a Coulomb potential
Anticlassical region
r
But, for V (r) ∼ −Cn /rn with n > 2 the region near threshold is anticlassical! For molecules n = 6: WKB doesn’t work for weakly bound vibrational states
Harald Friedrich and Johannes Trost, “Working with WKB waves far from the semiclassical limit”, Physics Reports 397, 359 (2004) Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
5 / 32
Structure of weakly bound molecules
Vibrational structure of weakly bound molecules There were many attempts to describe vibrations of weakly bound molecules The first one: a classic paper by LeRoy and Bernstein:
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
6 / 32
Structure of weakly bound molecules
Vibrational structure of weakly bound molecules There were many attempts to describe vibrations of weakly bound molecules The first one: a classic paper by LeRoy and Bernstein:
They used WKB
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
6 / 32
Structure of weakly bound molecules
Vibrational structure of weakly bound molecules There were many attempts to describe vibrations of weakly bound molecules The first one: a classic paper by LeRoy and Bernstein:
They used WKB More or less good results for density of states dv/dEb , but not for absolute values of Eb Not applicable to really weakly bound molecules Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
6 / 32
Structure of weakly bound molecules
Improvements to the LeRoy-Bernstein quantization rule Stimulated by the cold-molecule research, many improvements have been proposed:
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
7 / 32
Structure of weakly bound molecules
Improvements to the LeRoy-Bernstein quantization rule
Patrick Raab and Harald Friedrich derived the so-called “quantization function”
Their theory describes the vibrational structure of weakly bound molecules pretty well... ...but what does the “quantization function” mean?
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
8 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich
Molecular potential
We consider a state with vibrational quantum number v and a binding energy Eb
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
Eb
r
v
DPG 2010
9 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich
We consider a state with vibrational quantum number v and a binding energy Eb
Molecular potential
We introduce a noninteger “threshold quantum number”, which corresponds to Eb = 0
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
vth Eb
r
v
DPG 2010
9 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich
We consider a state with vibrational quantum number v and a binding energy Eb
The difference, vth − v depends in some way on the binding energy Eb : vth − v = F (Eb ) F (Eb ) is called the quantization function
Mikhail Lemeshko (FHI)
Molecular potential
We introduce a noninteger “threshold quantum number”, which corresponds to Eb = 0
Probing weakly bound molecules
vth Eb
r
v
DPG 2010
9 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich
We consider a state with vibrational quantum number v and a binding energy Eb
The difference, vth − v depends in some way on the binding energy Eb : vth − v = F (Eb ) F (Eb ) is called the quantization function
Molecular potential
We introduce a noninteger “threshold quantum number”, which corresponds to Eb = 0
vth Eb
r
v
The quantization function gives positions of weakly bound vibrational levels
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
9 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector, and 1 1 κ1−2/n Γ( 2 + n ) — pure WKB term FWKB (κ) = √ 1 π(n − 2) Γ(1 + n )
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector, and 1 1 κ1−2/n Γ( 2 + n ) — pure WKB term FWKB (κ) = √ 1 π(n − 2) Γ(1 + n )
Fcr (κ) = −
1 u + — correction for long-range potential 2(n − 2) 2πκ1−2/n
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector, and 1 1 κ1−2/n Γ( 2 + n ) — pure WKB term FWKB (κ) = √ 1 π(n − 2) Γ(1 + n )
Fcr (κ) = −
Fip (κ) =
1 u + — correction for long-range potential 2(n − 2) 2πκ1−2/n
(Gκ)4 — interpolation between weakly- and deeply-bound states 1 + (Gκ)4
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector, and 1 1 κ1−2/n Γ( 2 + n ) — pure WKB term FWKB (κ) = √ 1 π(n − 2) Γ(1 + n )
Fcr (κ) = −
Fip (κ) =
1 u + — correction for long-range potential 2(n − 2) 2πκ1−2/n
(Gκ)4 — interpolation between weakly- and deeply-bound states 1 + (Gκ)4 Fth (κ) =
Mikhail Lemeshko (FHI)
2bκ − (pκ)2 — near-threshold dependence 2π [1 + (Gκ)4 ]
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
The quantization function of Raab and Friedrich The analytic expression for the quantization function is: h i F (Eb ) = Fth (κ) + Fip (κ) Fcr (κ) + FWKB (κ) , where κ ∼
√
Eb is the wavevector, and 1 1 κ1−2/n Γ( 2 + n ) — pure WKB term FWKB (κ) = √ 1 π(n − 2) Γ(1 + n )
Fcr (κ) = −
Fip (κ) =
1 u + — correction for long-range potential 2(n − 2) 2πκ1−2/n
(Gκ)4 — interpolation between weakly- and deeply-bound states 1 + (Gκ)4 Fth (κ) =
2bκ − (pκ)2 — near-threshold dependence 2π [1 + (Gκ)4 ]
The expressions are very accurate and can be used for any binding energy Eb Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
10 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules Now we know the vibrational structure. But what about rotation?
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
11 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules Now we know the vibrational structure. But what about rotation? Two years after introducing the quantization rule, LeRoy published an article:
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
11 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules Now we know the vibrational structure. But what about rotation? Two years after introducing the quantization rule, LeRoy published an article:
The WKB approximation was depressingly inaccurate, as LeRoy pointed out:
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
11 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution
We follow Raab and Friedrich to study rotation of weakly bound species We consider a molecule in ground rotational state, J = 0, with a binding energy Eb
V(r) Eb
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
v(J = 0)
r
J=0
DPG 2010
12 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution
We follow Raab and Friedrich to study rotation of weakly bound species We consider a molecule in ground rotational state, J = 0, with a binding energy Eb Rotation adds a centrifugal term to the potential, Vcent =
~2 J(J + 1) 2mr2
V(r) Eb
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
v(J = 0)
r
J=0
DPG 2010
12 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution
We follow Raab and Friedrich to study rotation of weakly bound species We consider a molecule in ground rotational state, J = 0, with a binding energy Eb Rotation adds a centrifugal term to the potential, Vcent =
~2 J(J + 1) 2mr2
If the angular momentum J is greater than some critical value J ∗ , the vibrational state is “pushed out” of the potential – the molecule dissociates V(r)
v(J > J *) J>J *
Integer part of J ∗ gives the number of
Eb
v(J = 0)
r
J=0
rotational states, supported by a given vibrational level
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
12 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution • We derived a simple analytic expression for a number of rotational states
supported by a weakly-bound molecule: J ∗ = F (Eb )(n − 2) F (Eb ) – the quantization function of Raab and Friedrich (you need C6 and Eb ), n – power of the potential V (r) = −Cn /rn
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
13 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution • We derived a simple analytic expression for a number of rotational states
supported by a weakly-bound molecule: J ∗ = F (Eb )(n − 2) F (Eb ) – the quantization function of Raab and Friedrich (you need C6 and Eb ), n – power of the potential V (r) = −Cn /rn • When the molecule is rotationless? We derived a simple criterion for that.
Molecule has only the ground rotational state if the binding energy satisfies: −1/2
Eb < d6 ~3 m−3/2 C6
(for n = 6)
m is the reduced mass, d6 ≈ 1.6 is a parameter
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
13 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution • We derived a simple analytic expression for a number of rotational states
supported by a weakly-bound molecule: J ∗ = F (Eb )(n − 2) F (Eb ) – the quantization function of Raab and Friedrich (you need C6 and Eb ), n – power of the potential V (r) = −Cn /rn • When the molecule is rotationless? We derived a simple criterion for that.
Molecule has only the ground rotational state if the binding energy satisfies: −1/2
Eb < d6 ~3 m−3/2 C6
(for n = 6)
m is the reduced mass, d6 ≈ 1.6 is a parameter • Rotational constants of weakly bound levels may be estimated as B =
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
Eb J ∗ (J ∗ + 1) DPG 2010
13 / 32
Structure of weakly bound molecules
Rotational structure of weakly bound molecules: our contribution
These expressions are surprisingly accurate, see PRA 79 050501(R) (2009) Last bound states of 85 Rb2 :
Mikhail Lemeshko (FHI)
v
J∗
∗ Jexact
123
0.22
0.22
122
4.25
4.25
121
8.28
8.48
Probing weakly bound molecules
DPG 2010
14 / 32
Molecules in nonresonant laser fields
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
15 / 32
Molecules in nonresonant laser fields
Is angular momentum always quantized? In the absence of a field, hJ2 i = J(J + 1) is an integer for states with J = 0, 1, 2 . . .
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
16 / 32
Molecules in nonresonant laser fields
Is angular momentum always quantized? In the absence of a field, hJ2 i = J(J + 1) is an integer for states with J = 0, 1, 2 . . . However, in the presence of a field, this is not true! An external field, such as a laser field, hybridizes rotational levels, forming a “pendular state”:
J=0 0.87
+ 0.48
~ J=0
J=4
J=2
+ 0.06
=
ε Field imparts a noninteger value of hJ2 i, in the example above hJ2 i = 1.47
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
16 / 32
Molecules in nonresonant laser fields
Is angular momentum always quantized? In the absence of a field, hJ2 i = J(J + 1) is an integer for states with J = 0, 1, 2 . . . However, in the presence of a field, this is not true! An external field, such as a laser field, hybridizes rotational levels, forming a “pendular state”:
J=0 0.87
~ J=0
J=4
J=2
+ 0.48
+ 0.06
=
ε Field imparts a noninteger value of hJ2 i, in the example above hJ2 i = 1.47 The molecule is shaken by the field Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
16 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
Mikhail Lemeshko (FHI)
hJ2 i~2 2mr2
r
Effective potential
potential is:
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
Mikhail Lemeshko (FHI)
r
Effective potential
potential is:
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
What happens when we apply the laser
r
Effective potential
potential is:
field? Let’s see...
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
r
Effective potential
potential is:
The laser is on...
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
r
Effective potential
potential is:
a bit more intensity...
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
r
Effective potential
potential is:
more...
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
r
Effective potential
potential is:
more!
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
r
Effective potential
potential is:
even more....
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
Dissociation!
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
The molecule is shaken enough by the
r
Effective potential
potential is:
field to dissociate
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
Molecules in nonresonant laser fields
Can one make use of it? Weakly bound molecules usually support no rotation (no states with J ≥ 1) For instance, the last vibrational state of 85 Rb2 dissociates for hJ2 i ≥ 0.27
Dissociation!
The laser field adds a centrifugal term to the potential V (r), so that the effective
Ueff (r) = V (r) +
hJ2 i~2 2mr2
We may tune it by changing the intensity
The molecule is shaken enough by the
r
Effective potential
potential is:
field to dissociate
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
17 / 32
What about experiments?
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
18 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 1
The intensity of some optical dipole traps
Mikhail Lemeshko (FHI)
r
Effective potential
reaches 106 –107 W/cm2
Probing weakly bound molecules
DPG 2010
19 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 1
The intensity of some optical dipole traps
This may have already dissociated some of the weakest-bound molecules
Mikhail Lemeshko (FHI)
r
Effective potential
reaches 106 –107 W/cm2
Probing weakly bound molecules
DPG 2010
19 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 2
Lasers in optical dipole trap change the
Mikhail Lemeshko (FHI)
r
Effective potential
effective potential...
Probing weakly bound molecules
DPG 2010
20 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 2
Lasers in optical dipole trap change the
...and therefore – the binding energy
Mikhail Lemeshko (FHI)
r
Effective potential
effective potential...
Probing weakly bound molecules
DPG 2010
20 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 2
Experimentalists measure this... Lasers in optical dipole trap change the
...and therefore – the binding energy
This may cause errors in measuring Eb
Mikhail Lemeshko (FHI)
r
Effective potential
effective potential...
Probing weakly bound molecules
DPG 2010
20 / 32
What about experiments?
Could the experimentalists have already seen the effect?
Yes, they could have: example 2
effective potential... ...and therefore – the binding energy
This may cause errors in measuring Eb
Mikhail Lemeshko (FHI)
Effective potential
Lasers in optical dipole trap change the
Probing weakly bound molecules
r ...here is unaffected E
DPG 2010
b
20 / 32
What about experiments?
Other possibilities
• Using nonresonant fields one can change the scattering length
• Thereby, one can tune the positions of Feshbach resonances
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
21 / 32
What about experiments?
Enhancing photoassociation of ultracold atoms
Collaboration with Ruzin Aganoglu and Christiane Koch, Freie Universität Berlin Ruzin’s talk: Thursday 11:00, E 001 Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
22 / 32
Probing weakly bound species by short laser pulses
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
23 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period”
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period” 40
100
2
60
For a cw-laser field, hJ2 i is constant
20 40 10
0
Intensity, Δω
80
30
20
1
2
3
4
5
6
Time, rot. periods Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period” 40
60
2
adiabatically
80
30
the rotational period, hJ2 i is transferred
20 40 10
0
Intensity, Δω
If the pulse duration is longer than
100
20
1
2
3
4
5
6
Time, rot. periods Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period” 40
80
30
60
2
the rotational period, hJ2 i is transferred adiabatically
20 40 10
Intensity, Δω
If the pulse duration is longer than
100
20
The molecule has no angular momentum after the pulse has passed 0
1
2
3
4
5
6
Time, rot. periods Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period” 40
60
2
nonadiabatic
80
30
the rotational period, the process is
20 40 10
Intensity, Δω
If the pulse duration is shorter than
100
20
A part of the angular momentum is transferred to the molecule forever 0
1
2
3
4
5
6
Time, rot. periods Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
What happens if a laser pulse is short?
“Short” means “shorter than the rotational period” In the case of very short pulses most of the
40
100
angular momentum is imparted forever
the system is perturbed
60
2
This angular momentum remains unless
20 40
If the transferred angular momentum
10
Intensity, Δω
80
30
20
exceeds some critical value, the molecule will be shaken enough by the pulse to
0
2
3
4
5
6
Time, rot. periods
dissociate
Mikhail Lemeshko (FHI)
1
Probing weakly bound molecules
DPG 2010
24 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
14
Consequently, the pulse intensity needed
10
for dissociation depends on the distance
10
the laser pulse struck
2
I, W/cm
which molecule had at the moment when
13 12
10
11
10
10
10
9
We can probe the vibrational dynamics!
10
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance For any intensity I there is some critical
14
10
distance r∗ .
13
I, W/cm
2
10
12
10
11
10
I
10
10
9
10
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance For any intensity I there is some critical
14
distance r∗ . If the internuclear distance is
10
smaller than r∗ at the moment when the
10 2
I, W/cm
pulse strikes, the molecule dissociates.
13 12
10
11
10
I
10
10
9
10
dissociation
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance For any intensity I there is some critical
14
distance r∗ . If the internuclear distance is
10
smaller than r∗ at the moment when the
10
No dissociation occurs for larger internuclear separations
2
I, W/cm
pulse strikes, the molecule dissociates.
13 12
10
11
10
I
10
10
9
10
dissociation
no dissociation
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance For any intensity I there is some critical
14
distance r∗ . If the internuclear distance is
10
smaller than r∗ at the moment when the
10
No dissociation occurs for larger internuclear separations
2
I, W/cm
pulse strikes, the molecule dissociates.
13 12
10
11
10
10
10
9
So, for any pulse intensity I the probability of dissociation is the probability to have
I
10
dissociation
r*
8
10
∗
internuclear distances smaller than r (I)
no dissociation
10
1
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
This is simply the integral of the squared
14
10
wavefunction:
13
10
r∗
Z
|φv (r)| dr.
F (r ) = 0
2
2
I, W/cm
∗
12
10
11
10
I
10
10
9
10
dissociation
no dissociation
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
This is simply the integral of the squared
14
10
wavefunction:
13
10
r∗
Z
2
2
|φv (r)| dr.
F (r ) = 0
Here comes the idea: In an experiment we can measure F (I)
I, W/cm
∗
12
10
11
10
I
10
10
9
10
dissociation
no dissociation
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
This is simply the integral of the squared
14
10
wavefunction:
13
10
r∗
Z
2
2
|φv (r)| dr.
F (r ) = 0
Here comes the idea: In an experiment we can measure F (I) ∗
We can calculate the dependence I(r )
I, W/cm
∗
12
10
11
10
I
10
10
9
10
dissociation
no dissociation
r*
8
10
1
10
2
3
10
10
r *, Å Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
25 / 32
Probing weakly bound species by short laser pulses
And... if the pulse is even shorter? If the pulse duration is shorter than the vibrational period, the transferred angular momentum depends on the internuclear distance
This is simply the integral of the squared
14
10
wavefunction:
13
10
r∗
Z
2
2
|φv (r)| dr.
F (r ) = 0
Here comes the idea: In an experiment we can measure F (I) ∗
We can calculate the dependence I(r )
I, W/cm
∗
12
10
11
10
I
10
10
9
10
dissociation
r*
8
10
Hence, we can obtain the square of the vibrational wavefunction! Mikhail Lemeshko (FHI)
no dissociation
10
1
2
3
10
10
r *, Å Probing weakly bound molecules
DPG 2010
25 / 32
Results for weakly bound 85 Rb2 molecules
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
26 / 32
Results for weakly bound 85 Rb2 molecules
Potential and wavefunction
We used a single Rb2 potential curve from ref. [Seto et al JCP, 113, 3067 (2000)], combining it with dispersion terms from ref. [van Kempen et al PRL 88, 093201 (2002)] The last vibrational state, v = 123, is bound by Eb = −237 kHz
E/h, MHz
2 1 0
-2
Mikhail Lemeshko (FHI)
v=123
-1 1
10
2
r, Å
10
Probing weakly bound molecules
3
10
DPG 2010
27 / 32
Results for weakly bound 85 Rb2 molecules
Dependence of dissociation probability from the intensity
The vibrational period of 85 Rb2 (v = 123) molecule is about 0.67 µs, so it can be probed by ns pulses. We performed the calculation for 50 ps Gaussian pulses. 0
Dissociation probability
10
-1
10
-2
10
-3
10
-4
10
8
10
10
9
10
10
11
10
12
10
13
10
14
10
2
Laser intensity, W/cm
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
28 / 32
Results for weakly bound 85 Rb2 molecules
Dependence of dissociation probability from the intensity
The vibrational period of 85 Rb2 (v = 123) molecule is about 0.67 µs, so it can be probed by ns pulses. We performed the calculation for 50 ps Gaussian pulses. 0
10
Dissociation probability
The maxima of F (I) reflect the nodes of the vibrational wavefunction
E/h, MHz
2 1 0 v=123
-1 -2
1
10
Mikhail Lemeshko (FHI)
2
r, Å
10
3
10
-1
10
-2
10
-3
10
-4
10
8
10
10
9
10
10
11
10
12
10
13
10
14
10
2
Laser intensity, W/cm
Probing weakly bound molecules
DPG 2010
28 / 32
Results for weakly bound 85 Rb2 molecules
Dependence of dissociation probability from the intensity
The vibrational period of 85 Rb2 (v = 123) molecule is about 0.67 µs, so it can be probed by ns pulses. We performed the calculation for 50 ps Gaussian pulses. 0
wavefunction’s main maximum
E/h, MHz
2 1 0 v=123
-1 -2
1
10
Mikhail Lemeshko (FHI)
2
r, Å
10
3
10
Dissociation probability
10
The “edge” of F (I) gives the position of the
-1
10
-2
10
-3
10
-4
10
8
10
10
9
10
10
11
10
12
10
13
10
14
10
2
Laser intensity, W/cm
Probing weakly bound molecules
DPG 2010
28 / 32
Conclusions and outlook
Outline
1
Structure of weakly bound molecules
2
Molecules in nonresonant laser fields
3
What about experiments?
4
Probing weakly bound species by short laser pulses
5
Results for weakly bound 85 Rb2 molecules
6
Conclusions and outlook
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
29 / 32
Conclusions and outlook
Conclusions
1
We showed that weakly bound molecules can be probed by “shaking” in nonresonant laser fields
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
30 / 32
Conclusions and outlook
Conclusions
1
We showed that weakly bound molecules can be probed by “shaking” in nonresonant laser fields
2
Using a cw-laser field one can control the atomic scattering length, positions of Feshbach resonances, and enhance the photoassociation yield
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
30 / 32
Conclusions and outlook
Conclusions
1
We showed that weakly bound molecules can be probed by “shaking” in nonresonant laser fields
2
Using a cw-laser field one can control the atomic scattering length, positions of Feshbach resonances, and enhance the photoassociation yield
3
Using short laser pulses, it is possible to map out the square of the vibrational wavefunction, and thus determine accurately the potential
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
30 / 32
Conclusions and outlook
Conclusions
1
We showed that weakly bound molecules can be probed by “shaking” in nonresonant laser fields
2
Using a cw-laser field one can control the atomic scattering length, positions of Feshbach resonances, and enhance the photoassociation yield
3
Using short laser pulses, it is possible to map out the square of the vibrational wavefunction, and thus determine accurately the potential
4
As an aside, we derived simple expressions for a number of rotational states, supported by a weakly bound molecule, and for the rotational constants.
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
30 / 32
Conclusions and outlook
Outlook
1
The experimentalists may have already observed some shaking due to the field of optical dipole traps
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
31 / 32
Conclusions and outlook
Outlook
1
The experimentalists may have already observed some shaking due to the field of optical dipole traps
2
Manipulating Feshbach resonances with a cw-laser might be a straightforward proof of the shaking mechanism
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
31 / 32
Conclusions and outlook
Outlook
1
The experimentalists may have already observed some shaking due to the field of optical dipole traps
2
Manipulating Feshbach resonances with a cw-laser might be a straightforward proof of the shaking mechanism
3
We look forward to the experiments with short laser pulses
Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
31 / 32
Conclusions and outlook
Thank you for your attention!
Further reading: – Phys. Rev. Lett. 103, 053003 (2009) (about probing weakly bound molecules) – Phys. Rev. A 79, 050501 (2009) (rotational states of weakly bound dimers) – J. At. Mol. Sci. 1, 39 (2010) (rotational states of weakly bound molecular ions) Mikhail Lemeshko (FHI)
Probing weakly bound molecules
DPG 2010
32 / 32
