Probing halo molecules with nonresonant light Mikhail Lemeshko and Bretislav Friedrich Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany We propose a new technique to probe weakly-bound species, by “shaking”: using short laser pulses, one can map out the square of the vibrational wavefunction Is angular momentum always quantized? J=0

2

• In the absence of a field it is: J = J(J+1) is an integer for states with J=0,1,2... • However, in the presence of a the field, this is not true! An external field, such as a laser field, hybridizes rotational levels, forming a “pendular state”:

+ 0.48

0.87

~ J=0

J=4

J=2

=

+ 0.06

ε

• The field provides a molecule with a noninteger value of J2 . The molecule is aligned and its axis librates about the field vector: the laser field “shakes” the molecule

Can one make use of it? 2

J

ħ2

2mr • If the imparted angular momentum J exceeds a critical value J*, the molecule is shaken enough by the field to dissociate

Dissociation by shaking

, which we may tune by changing the laser intensity

r J > J*

Veff (r )

Yes, one can. • The laser field adds a centrifugal term to the moleculat potential V(r), so that the effective potential is Veff (r) = V(r) +

2

• Usually, the values of J* are quite small, since most of halo molecules are rotationless. For example the last vibrational state of Rb 2 dissociates for J ≥ 0.27 ħ 3 , where m is the reduced mass and C pertains to the asymptotic behavoir • A molecule is rotationless if its binding energy satisfies E b < d 6 3/2 1/2 6 m C6 of the molecular potential, V(r > ∞) = −C6 /r 6 ; d6 ≈ 1.6 is a universal dimenstionless parameter, which can be evaluated analytically

J=0

What happens if a laser pulse is short? “Short” means “shorter than the rotational period”. The figure shows the time-dependence of transferred angular momentum J2 (solid lines) for pulses of the same intensity, Δω =100, but of different duration (dotted lines)

• For a cw-laser field, J

2

• If the pulse duration is longer than the rotational period, J is imparted adiabatically, following the pulse shape. The molecule has no angular momentum after the pulse has passed

60 2

• If the pulse duration is shorter than the rotational period, the process is nonadiabatic. The molecule still has some angular momentum left after the pulse

J

20 40

• In the case of a very short pulse, most of the angular momentum is imparted forever. This angular momentum remains in place unless the system is perturbed If the nonadiabatically imparted J

80

30

10

exceeds some critical value, the molecule will be shaken enough to dissociate

0

20 0 0

1

2

3

4

5

6

Time, rotational periods

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. Consequently, for different internuclear distances, we need different intensities to dissociate the molecule. Thus, we can probe the vibrational dynamics.

14

10

For any intensity I there is some critical distance r*. If the internuclear distance is smaller than r* at the moment when the pulse strikes, the molecule dissociates.

13

10

r*

F(r*) = |ψv (r)|2 dr 0

12

10

I, W / cm

Here comes the idea: • In the experiment we can measure F(I) • We can calculate the dependence I(r*) • Hence, we can obtain the square of the vibrational wavefunction!

This probability is simply the integral of the squared wavefunction:

∫

2

No dissociation occurs for larger internuclear separations. So, for any intensity I the probability of dissociation is the probability of hitting at an internuclear distance smaller than r*(I).

~

I

11

10

Dissociation ~ rp ! r *

10

10

9

10

No dissociation ~ rp > r * ~

8

r*

10

* rmin

10

100

1000

r *, Å

Results for 85Rb2 halo molecules 85 • The vibrational period of Rb2 (v =123) is about

Analytic

• The last vibrational state, v =123, is bound by E b = −237 kHz

[van Kempen et al, PRL 88, 093201 (2002)]

E / h, MHz

• We used a single Rb2 potential curve [Seto et al, JCP 113, 3067 (2000)], combining it with dispersion terms

0.67μ s, so it can be probed by ns pulses.

1 Exact

• We performed the calculation for 50 ps pulses.

0 v=123

• The oscillations of F(I) reflect the nodes of the

-1

vibrational wavefunction

V (r) -2

10

r, Å

100

0

10

Dissociation probability

2

-1

10

-2

10

-3

10

-4

10

1000

8

10

9

10

10

11

10

10

12

13

10

Laser intensity, W/cm

10

14

10

2

An analytic model of shaking The process of dissociation by shaking is amenable to an analytic treatment, one based on a near-threshold expansion of the wavefunction

• When the energy is small, we may take into account only the asymptotic part of the potential, V(r > ∞) = −C6 /r .6 In general, for arbitrary energy, there is no analytic solution of the Schrödinger equation with this potential

( (

1/4

( (

m C6 −1/4 C ξ J ( ) ξ ξ where are the Bessel functions and , = • However, there are solutions for a zero energy: ψ0 = 21/4 Γ(5/4) ξ −1/4J1/4(ξ ) and ψ1 = 2−3/4Γ(3/4) 2 m J (ξ) 6 2 2 4 −1/4 ±1/4 ħ

( (

Eb • Then the wavefunction of a bound state near threshold can be written as: ψv (ξ ) = ψ0 (ξ ) − k ψ1 (ξ ) , where k = 2 m 2

1/2

1/2

2ħ r

ħ

( (

• The model allows to understand the behavior of F(I). The oscillations correspond to the nodes of |ψv(r)|, which are solutions of the equation J1/4( ξ ) = λ J−1/4(ξ ) , with λ = k Γ(3/4) 2 m C6 2 2

2 Γ(5/4)

1/4

ħ

• The maximum of the wavefunction corresponds to “the edge” of F(I), which is given by the first root of the equation J5/4( ξ ) = −λ J−5/4(ξ ), near ξ = 0

What about experiments? 0.1

• One can control the molecular size: the nonresonant field pushes a bound state upwards, thereby increasing the mean radius of the wavefunction 85

8

Rb 2 in the least-bound state is already a long-range molecule. However, it may be transferred to a “pure halo regime” using a cw-laser field. The intensity needed is 6.5 ×10 W/cm2

• Polar molecules, such as KRb, can be probed by half-cycle pulses of a much smaller intensity, due to a permanent dipole interaction • We look forward to experiments with short laser pulses

E / h, MHz

0

r =1320 Å cw-laser

• Nonresonant fields of optical dipole traps are sufficient to dissociate some of the weakest-bound molecules. Experimentalists might have already observed some trap losses due to shaking -0.1 -0.2 -0.3

r =167 Å

1

10

2

3

10

10

r, Å

References M. Lemeshko, B. Friedrich. Probing halo molecules with nonresonant light, submitted (2009); arXiv:0903.0811 M. Lemeshko, B. Friedrich. Rotational and rotationless states of halo molecules, submitted (2009); arXiv:0904.0567

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4

10

Intensity, !"

2

2

100

40

is constant