Supersymmetry identifies molecular Stark states whose eigenproperties can be obtained analytically Mikhail Lemeshko,a Mustafa Mustafa,b Sabre Kais,b Bretislav Friedricha a Fritz

Haber Institute of the Max Planck Society, Berlin, Germany b Purdue

University, West Lafayette, Indiana, USA

Friday seminar February 18, 2011

Supersymmetry (SUSY) in particle physics Assumption: for every boson of the Standard Model there is a SUSY partner – a fermion of the same mass

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

2/9

Supersymmetry (SUSY) in particle physics Assumption: for every boson of the Standard Model there is a SUSY partner – a fermion of the same mass Supersymmetric partners were never

If SUSY exists, it should be broken

observed in nature

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

2/9

Supersymmetry (SUSY) in quantum mechanics Witten proposed to consider the ‘simplest example’ of SUSY in zero-dimensional field theory, i.e. quantum mechanics [Nucl. Phys. B 185, 513 (1981)]

Ed Witten

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

3/9

Supersymmetry (SUSY) in quantum mechanics Witten proposed to consider the ‘simplest example’ of SUSY in zero-dimensional field theory, i.e. quantum mechanics [Nucl. Phys. B 185, 513 (1981)]

Ed Witten

Matt Witten Dr. House Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

3/9

Supersymmetry (SUSY) in quantum mechanics

• Supersymmetry is closely related to exact solvability. All exactly solvable

potentials (oscillator, Morse, Coulomb, . . . ) exhibit SUSY and shape-invariance

• Do molecules in fields exhibit supersymmetry?

• If SUSY yielded analytic solutions for molecules in fields, this would allow to

reverse-engineer the problem, i.e. to obtain the field parameters needed to create molecular states with desired properties

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

4/9

Supersymmetry (SUSY) and shape-invariance All we need to construct SUSY is the ground-state wavefunction ϕ0 (x) E4–

Superpotential: W (x) = −ϕ00 (x)/ϕ0 (x) d + W (x), Intertwining operators: A± ≡ ∓ dx

E3+ A

E

–

– 3

E2+ A+

E2–

E1+

E1– E

– 0

E0+ H–

ϕ0 (x)

H+

F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995) Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

5/9

Supersymmetry (SUSY) and shape-invariance All we need to construct SUSY is the ground-state wavefunction ϕ0 (x) E4–

Superpotential: W (x) = −ϕ00 (x)/ϕ0 (x)

A

d + W (x), Intertwining operators: A± ≡ ∓ dx

Superpartner Hamiltonians: H∓ = Superpartner potentials: V± (x) ≡

A± A∓

W 2 (x)

− + Intertwining relations: En+1 = En ; + − ψn−1 ∼ A− ψn ;

=

±

E3+

d2 − dx 2

E

E2+

+ V∓ (x)

W 0 (x)

E0− = 0; − + ψn ∼ A+ ψn−1

–

– 3

A+

E2–

E1+

E1– E

– 0

E0+ H–

ϕ0 (x)

H+

F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995) Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

5/9

Supersymmetry (SUSY) and shape-invariance All we need to construct SUSY is the ground-state wavefunction ϕ0 (x) E4–

Superpotential: W (x) = −ϕ00 (x)/ϕ0 (x)

A

d + W (x), Intertwining operators: A± ≡ ∓ dx

Superpartner Hamiltonians: H∓ = Superpartner potentials: V± (x) ≡

A± A∓

W 2 (x)

− + Intertwining relations: En+1 = En ; + − ψn−1 ∼ A− ψn ;

=

±

E3+

d2 − dx 2

E

E2+

+ V∓ (x)

W 0 (x)

E0− = 0;

–

– 3

A+

E2–

E1+

E1– E

– 0

− + ψn ∼ A+ ψn−1

E0+ H–

ϕ0 (x)

H+

If the superpartner potentials are shape-invariant, V+ (x, a0 ) + g(a0 ) = V− (x, a1 ) + g(a1 ), (a is a parameter, a1 = f (a0 ), and g(a) is independent of x), the problem is analytically solvable: − + E1− = g (a1 ) − g(a0 ); ϕ− 1 (x, a0 ) = A (x, a0 )ϕ0 (x, a1 ), and so on

F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251, 267 (1995) Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

5/9

SUSY of the molecular Stark effect Effective potential for a linear molecule in collinear electrostatic and laser fields: Vµ,α (θ) =

m2 − ω cos θ − ∆ω cos2 θ, sin2 θ

with ω ≡ µε/B; ∆ω ≡ 2π∆αI/(Bc), µ – dipole moment, ∆α – polarizability anisotropy ~ J

|m|

4

3

2

3

1

[

2 1 0

n

0

[

3

[

0 1 2

β =0

β =1

2

1 0 4 0 3 1 2 1 0

β =5

8

0

β

m – good quantum number J˜ – adiabatic label Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

6/9

SUSY of the molecular Stark effect Effective potential for a linear molecule in collinear electrostatic and laser fields: Vµ,α (θ) =

m2 − ω cos θ − ∆ω cos2 θ, sin2 θ

with ω ≡ µε/B; ∆ω ≡ 2π∆αI/(Bc), µ – dipole moment, ∆α – polarizability anisotropy Solutions of the SE are the “pendular states”: ψ ω,∆ω (θ, φ) = ˜

∞ X

J,m

˜

cJ,m Jm (ω, ∆ω)YJm (θ, φ),

J=m

~ J

|m|

4

3

2

3

1

[

2 1 0

n

0

[

3

[

0 1 2

β =0

β =1

2

1 0 4 0 3 1 2 1 0

β =5

8

0

β

m – good quantum number J˜ – adiabatic label Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

6/9

SUSY of the molecular Stark effect Effective potential for a linear molecule in collinear electrostatic and laser fields: Vµ,α (θ) =

m2 − ω cos θ − ∆ω cos2 θ, sin2 θ

with ω ≡ µε/B; ∆ω ≡ 2π∆αI/(Bc), µ – dipole moment, ∆α – polarizability anisotropy ~ J

Solutions of the SE are the “pendular states”: ∞ X

J,m

˜

cJ,m Jm (ω, ∆ω)YJm (θ, φ),

SUSY allows to solve the problem exactly for the “stretched” states, J˜ = m:

1

[

2 1 0

3

2

3

J=m

n

0

[

3

[

0 1 2

β =0

β =1

2

1 0 4 0 3

Ansatz: W (θ) = − (m + 1/2) cot θ + β sin θ

1 2

→ ψ0 (θ) = N (−1)m (sin θ)m eβ cos θ

1 0

In which case the field strengths satisfy: ∆ω =

ω2 4(m+1)2

≡

β2

β =5

0

β

Weak-field and strong-field limits are also solvable exactly Mikhail Lemeshko (FHI)

Supersymmetry

8

ψ ω,∆ω (θ, φ) = ˜

|m|

4

m – good quantum number J˜ – adiabatic label Friday Seminar

6/9

An inverse problem for molecules in fields Molecular properties can be derived in closed form, which allows to reverse-engineer the problem and create quantum states with preordained characteristics 0

For instance, for m = 0: E0

-20

• Space-fixed dipole moment:

|m| = 0 1 2 3

-60 -80

1 2β

The states are strongly oriented, e.g. for

40 K87 Rb

µZ = 0.9µ is achieved at ε = 38 kV/cm I = 1.75 · 109 W/cm2

µΖ /µ

0.8 0.6

0 1

2

3

0

1

2

0.4 0.2 0.8



µZ /µ ≡ hcos θi = coth(2β) −

-40

0.6 0.4

3

0.2

2



8 6 4 2 0 0

01

23

1

2

3

4

5

6

7

8

9

10

β M. Lemeshko, M. Mustafa, S. Kais, B. Friedrich, Phys. Rev. A, accepted (2011); New. J. Phys., submitted (2011) Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

7/9

An inverse problem for molecules in fields Molecular properties can be derived in closed form, which allows to reverse-engineer the problem and create quantum states with preordained characteristics 0

For instance, for m = 0: E0

-20

• Space-fixed dipole moment:

-80

1 2β

µZ = 0.9µ is achieved at ε = 38 kV/cm I = 1.75 · 109 W/cm2 1 coth(2β) − 2β 2 β

• Expectation value of the angular momentum:

2

3

0

1

2

0.6 0.4

3

0.2 8 6 4 2

1 2

0 1

0.4

0.8

2

• Alignment cosine: hcos2 θi = 1 +

0.6 0.2



40 K87 Rb

µΖ /µ

0.8

The states are strongly oriented, e.g. for

hJ2 i = β coth(2β) −

|m| = 0 1 2 3

-60



µZ /µ ≡ hcos θi = coth(2β) −

-40

0 0

01

23

1

2

3

4

5

6

7

8

9

10

β

M. Lemeshko, M. Mustafa, S. Kais, B. Friedrich, Phys. Rev. A, accepted (2011); New. J. Phys., submitted (2011) Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

7/9

Conclusions and outlook

• Supersymmetry allows to identify exactly solvable cases for molecules in fields

• With analytic solutions in hand we can reverse-engineer the problem: design the

molecular states with desired characteristics

• The analytic solutions can be used to solve complex problems, like many-body

models of ultracold molecular gases

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

8/9

Thank you for your attention!

Mikhail Lemeshko (FHI)

Supersymmetry

Friday Seminar

9/9