Contents - Max-Planck-Institut für Physik komplexer Systeme

2.2 Molecular bond by internal quantum reflection
WEIBIN LI, ELIAS DIESSEN, THOMAS POHL, JAN M ROST
Recently, the existence of ultralong range molecules
has been demonstrated experimentally [1]. Reproducing
theoretically the measured spectral lines of
the Rb ∗ 2 Rydberg molecule quantitatively [2], turned
out to be much more difficult than anticipated with
the benefit of a few surprises on the way: (i) The
spectrum of excited Rb ∗ 2 is due to internal quantum
reflection (IQR) stabilizing the molecule against fast
decay into Rb + 2 + e− . (ii) The experiment did not only
generate Rydberg dimers but also trimers. (iii) The
homonuclear Rb ∗ 2 molecule has a permanent dipole
moment [3].
Proposed with a simple consideration [4] and supported
by extensions to trimers [5] and borromean
dimers [6], the measurement of the entire excitation
spectrum of a long range Rydberg dimer motivated us
to investigate the spectrum theoretically. The difficulties
encountered as well as the properties (i) and (iii) are
related to the fact that the low energy collision Rb+e −
has a p-wave shape resonance which means that one
has to include not only s-wave scattering but also the
p-wave scattering to describe the collision of the electron
at position r with the ground state atom at R.
Theoretical approach The hamiltonian reads
H = p 2 /2 − 1/r + V ≡ H0 + V (r,R) (1)
with the additional pseudo potential due to the ground
state atom
V (r,R) = 2πδ(r − R) As(k) + 3Ap(k) 3←− ∇ · −→
∇ , (2)
where As(k) and Ap(k) are the energy dependent scattering
lengths for s- and p-wave scattering and k is
semiclassically defined through the energy of the isolated
Rydberg atom En = k 2 /2 − 1/R. Since the potential
is energy dependent, a formally correct solution
must proceed via the construction of a Green function
[2] which produces for fixed R molecular potential
curves U(R), see Fig. 2.
To get the spectrum (Fig. 1) of the Rydberg molecule,
one needs to solve the vibrational Schrödinger equation
with U(R) which is far from the standard routine
for two reasons: First, we are at a highly excited
manifold and when the two atoms come together, the
molecule decays into Rb + 2 + e− . Hence one has at
R = 0 an open boundary condition whose specific nature
cannot be determined within the present description.
However, what comes as a rescue is the form of
the potential: Due to the p-wave resonance it exhibits a
steep cliff like shape falling about three orders of magnitude
(see inset of Fig. 2) compared to the modulation
depth of the potential shown in Fig. 2. This allows
one to treat the vibrational motion as a resonance
which can decay inward as compared to the usual situation
of decay when the particles move away from
each other. The simplest way to determine the resonance
positions is the stabilization method of Hazy and
Taylor [7] where one artificially encloses the system in a
box with variable size. Here we only put an inner limit
R0 of the box (the outer one is provided by the potential
well). The corresponding energies of the bound states
are shown in Fig. 3. One clearly sees the resonances as
the lines connecting the avoided crossings. The widths
can be determined either by binning (Fig. 2b) or from
the Wigner time delay [8] (Fig. 1c) in good agreement
with each other.
I on
signal
( arb.
units)
15
10
5
0
5
10
15
theory: dimer
theory: trimer
35s
Rb +
+
Rb2 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10
Relative frequency (MHz)
Figure 1: Spectrum for Rb with principal quantum numbers n =
35. Frequencies are measured with respect to the atomic transition
5s1/2 → ns1/2. The upper parts show the atomic ion spectrum
Rb + (black) and the lower ones the spectra of the molecular ions Rb +
2
(blue). Calculated binding energies (table 1) are indicated by vertical
lines, for the dimer in black, for the trimer in blue.
Dimers and trimers The positions of the resonances
agree well with the experimental ones, see table 1. The
situation is similar to other manifolds n = 36,37, also
measured in the experiment. The experimental lines
missing in the theoretical dimer spectrum are in fact
spectral lines of a trimer, formed with two ground state
and the Rydberg Rb atoms. The ground state energy
E00 is to a very good approximation twice the dimer,
E00 = 2E0, and the missing lines are different excitations
of the trimer in its two dimer-like legs. Hence, we
can infer from our theoretical analysis that the experiment
did not only produce dimers but also trimers in
single photon association [2]. The energy spectrum of
the latter is very well approximated by the sum of two
dimer energies Eν ′ ν = Eν ′ + Eν.
44 Selection of Research Results