# Dark state experiments with ultracold, deeply-bound triplet molecules

###### Abstract

We examine dark quantum superposition states of weakly bound Rb Feshbach molecules and tightly bound triplet Rb molecules in the rovibrational ground state, created by subjecting a pure sample of Feshbach molecules in an optical lattice to a bichromatic Raman laser field. We analyze both experimentally and theoretically the creation and dynamics of these dark states. Coherent wavepacket oscillations of deeply bound molecules in lattice sites, as previously observed in Ref. [1], are suppressed due to laser-induced phase locking of molecular levels. This can be understood as the appearance of a novel multilevel dark state. In addition, the experimental methods developed help to determine important properties of our coupled atom / laser system.

## 1 Introduction

Very recently, several groups have produced dense, ultracold ensembles of molecules that are deeply bound [1, 2, 3, 4, 5] and in a ro-vibrational ground state [1, 2, 4, 5]. This was achieved by binary association of alkali atoms in ultracold ensembles via two different pathways: (1) photoassociation [6, 7] and (2) magneto-association at Feshbach resonances [7, 8] combined with stimulated Raman adiabatic passage (STIRAP) [9], a special coherent optical transfer method. In contrast to photoassociation, magneto-association only produces weakly-bound Feshbach molecules [7, 8]. STIRAP can then be used to transfer these weakly-bound molecules to the rovibrational ground state. This method is coherent, efficient, fast, reversible, and highly selective. STIRAP is based on a counter-intuitive light pulse sequence giving rise to a dynamically changing dark superposition state (Fig. 1a)

(1) |

In this paper, we deliberately replace the efficient but complex STIRAP transfer of Ref. [1] with a simple square laser pulse scheme. This reveals interesting fundamental processes and dynamics in the coupled atom / laser system, that would otherwise be hidden. In addition, this procedure allows us to determine important properties and parameters of our system and to check for consistency with our theoretical model. We study the creation and lifetime of dark superposition states that contain a sizeable fraction of deeply bound molecules. These molecules are held in a 3D optical lattice. Because the lattice potential is much shallower for the deeply bound molecules than for the Feshbach molecules, and because the transfer is fast, the deeply bound molecules coherently populate several Bloch bands. In contrast to Ref. [1], where similar circumstances lead to coherent oscillations in the lattice, oscillations are suppressed in the experiment described here due to phase locking of all quantum levels involved. A novel dark state appears which is a superposition of up to 8 quantum levels. We investigate the limiting conditions under which oscillations set in.

## 2 Experimental setup and initial preparation of molecules

We carry out our dark state experiments with a 50 m-size pure ensemble of weakly bound Rb Feshbach molecules.
The molecules are trapped in the lowest Bloch band of a cubic 3D
optical lattice with no more than a single molecule per lattice
site [10] and an effective lattice filling factor of about
0.3. The lattice depth for the Feshbach molecules is 60 ,
where is the recoil energy, with the
mass of the molecules and nm the lattice period. Such
deep lattices suppress tunneling between different sites. A pure
ensemble of Feshbach molecules has been produced as follows. We
prepare a cold cloud of Rb atoms that are
either Bose condensed or *nearly*^{1}^{1}1It turns out
that this increases the number of Feshbach molecules. Bose condensed in a
Ioffe-type magnetic trap with trap frequencies Hz. Within 100 ms we adiabatically load the atoms
into the 3D optical lattice. After turning off the magnetic trap, we
flip the spins of our atoms from their initial state to by suddenly reversing the
bias magnetic field of a few G. This spin state features a 210 mG-wide Feshbach resonance at 1007.40 G [11]. By
adiabatically ramping over this resonance, we efficiently convert
atoms at multiply occupied lattice sites into Rb
Feshbach molecules. After conversion, inelastic collisions occur at
lattice sites that contain more particles than a single Feshbach
molecule, leading to vibrational relaxation of these molecules,
release of binding energy into kinetic energy and removal of all
particles from these sites. A subsequent combined microwave
and optical purification pulse removes all remaining chemically
unbound atoms, creating a pure sample of
Feshbach molecules. Afterwards, the magnetic field is set to
1005.8 G, where the Feshbach molecules are in a quantum state
which correlates with at 0 G. Here, and are the total
angular momentum quantum numbers for the molecule and its atomic
constituents, respectively, and is the total magnetic
quantum number; is the vibrational quantum number for the
triplet ground state potential () and is the
quantum number for rotation.

The bichromatic Raman laser field for the creation of the molecular dark states is based on two lasers (1 and 2) which connect the Feshbach molecule level , via an excited level , to the absolute lowest level in the triplet potential (Fig. 1a). Laser 1 is a Ti:Sapphire laser and laser 2 is a grating-stabilized diode laser. Both lasers are Pound-Drever-Hall locked to a single cavity which itself is locked to an atomic Rb-line. From the lock error signals, we estimate frequency stabilities on a ms-timescale of 40 kHz and 80 kHz for lasers 1 and 2, respectively. Both laser beams have a waist of 130 m at the location of the molecular sample, propagate collinearly, and are polarized parallel to the direction of the magnetic bias field. Thus, the lasers can only induce transitions.

The ground state has a binding energy of 7.0383(2) THz and can be described by the quantum numbers where and are the total electronic and nuclear spins of the molecule, respectively. At 1005.8 G is separated by hundreds of MHz from any other bound level, so that there is no ambiguity as to which level is addressed. The level is located in the vibrational manifold of the electronically excited (5s + 5p) potential and has character. It has an excitation energy of 294.6264(2) THz with respect to , and a width = 8 MHz. The Rabi frequencies of the two lasers depend on their respective intensities , i. e., 0.4 MHz and 30 MHz , and are typically chosen to be in the MHz regime.

## 3 Dark state evolution within a square pulse

Our square pulse projection experiments are carried out as follows. We expose the Feshbach molecules in the lattice to square pulses of Raman lasers 1 and 2 of variable pulse duration. Laser 2 is switched on about 1 s before laser 1 to avoid excitation from to due to jitter in the laser pulse timing. The Raman lasers are resonant () and the Rabi frequency MHz while is varied (Fig. 2). After the pulse, we measure the fraction of molecules remaining in state by dissociating them into pairs of atoms at the Feshbach resonance, releasing them from the lattice and applying standard absorption imaging. It is important to note that we actually only count atoms in the lowest Bloch band of the lattice. The release from the optical lattice is done as described in [12], where after 13 ms of ballistic expansion we map out the Bloch bands in momentum space (see Appendix for details).

Figure 2 shows the remaining fraction of molecules in state versus pulse duration. Within 1 s we observe a rapid loss of molecules that depends on the ratio . The remaining molecules are stable on a much longer timescale. This can be understood in terms of formation of a dark state . We can write

(2) |

where

(3) |

is a bright state which quickly decays via
resonant excitation to level . The dark state remains after the lasers are switched on and can be detected as a fraction
of molecules projected back to after switching off the lasers^{2}^{2}2This fact can be used to conveniently calibrate the Rabi frequency ratio . We found good agreement with other calibration methods for the Rabi frequencies..

Also, after the pulse a fraction of the initial molecules are in state with a maximum of 25% for . Thus, a sizeable fraction of the molecules can be coherently transferred to the ground state. Remarkably, this transfer takes place in less than 1 s! Such short transfer times cause Fourier broadening, resulting in considerably reduced laser stability requirements. In addition, due to the formation of a dark state, there is still a well-defined phase relation between the and molecules.

As can be seen from Fig. 2, the dark state slowly decays. Its lifetime is shortest for , where we measure it to be 50 s. The decay of the dark state is likely due to phase fluctuations of the Raman lasers. Phase fluctuations lead to an admixture of a bright state component to the otherwise dark state, which causes losses. In Sec. 4 we will show that these fluctuations can be expressed in terms of the short-term relative linewidth of the lasers, , which we find to be about kHz. In principle, the decay of the dark state could be due to other effects, such as coupling to levels other than , , and . However, we have verified that this is not the case, because losses due to optical excitation are completely negligible on the 100 s-timescale when we expose a pure ensemble of () molecules to only laser 2 (1).

We also searched for laser power dependent shifts of the two-photon resonance. Using the Raman square pulse measurements, we scanned the relative detuning of the lasers for a fixed pulse duration and various laser powers. Within the accuracy of our measurements of kHz, we could not detect any shifts of the resonance.

The behavior in Fig. 2 is described well by a closed three-level model (a system) and its dynamics can be simulated with a master equation which we describe in the following.

## 4 Three-level model and master equation

Neglecting lattice effects, we can describe the internal dynamics of the molecules as they are subjected to the Raman laser fields with a three-level model. We use a master equation [13, 14] which takes into account decoherence due to phase fluctuations of the Raman lasers. We consider the case where laser 1 is kept on resonance and laser 2 has a detuning (Fig. 3). Identifying the levels , , with numbers 1, 2, 3, respectively, we can write the master equation as,

(4) |

where is the density matrix, are the Rabi frequencies, is the spontaneous decay rate of the excited level , and is the relative linewidth of the two Raman lasers. The matrices and are ladder operators and each is the transpose of the other. For example

(5) |

Setting the linewidth of the excited level = 8 MHz, the detuning and Rabi frequencies MHz and to give the ratios in Fig. 2, we fit all the data with a single fit parameter . As a best fit, we obtain a relative linewidth of the two Raman lasers kHz, which is a reasonable value for our laser system.

## 5 Coherent oscillations and their suppression

In reference [1] coherent oscillations of molecular wavepackets of molecules in the optical lattice were observed. We now investigate how these observations fit together with the experimental results of the square pulse projection experiments presented here. For clarity, the oscillation data from Ref. [1] are presented again in Fig. 4 and briefly discussed.

Using an STIRAP pulse sequence, Feshbach molecules are
efficiently transferred to level . The Raman lasers are extinguished and the molecules are held for a time , after which they are transferred back to
with a reverse STIRAP pulse. The number of recovered Feshbach
molecules is counted. However, we only detect atoms that end up in
the lowest Bloch band after dissociation of the Feshbach molecules (see Appendix). The oscillation can be
understood as follows. We consider the localized spatial center-of-mass (c.o.m.) wavepacket of a Feshbach molecule at a particular lattice site in the lowest
Bloch band. The first STIRAP transfer projects this wavepacket onto
the much shallower^{3}^{3}3Due to a smaller dynamic polarizability, the lattice depth for the tightly bound molecules is shallower than for the Feshbach molecules by a factor of 10. lattice potential felt by the molecules
(Fig. 5) without changing its shape. As
a consequence, molecules are coherently spread over
various Bloch bands, and the wavepacket undergoes “breathing”
oscillations with the lattice site trap frequency . These
coherent oscillations (period 80 s ) are damped by
tunneling of molecules in higher Bloch bands to
neighboring lattice sites. The reverse STIRAP transfer maps this
periodic oscillation back to the Feshbach molecule signal in Fig. 4. Higher Bloch
bands are populated here as well, but are at most partially counted
in our scheme (see Appendix), which leads to an apparent decrease in
transfer efficiency.

The question arises why similar oscillations are not observed in our square pulse projection measurements shown in Fig. 2, especially for the case where 50% of the population is in state . One might assume that the spatial wavepackets of the molecules undergo similar breathing oscillations. These oscillations would then periodically break up the dark superposition state and lead to corresponding losses. They would also periodically produce population in higher Bloch bands of the Feshbach molecule lattice. As we will see, the oscillations are suppressed because the Raman lasers phase lock the involved quantum levels which stops, in a sense, the free evolution of the wavepackets. We can understand this behavior in detail with the help of a multi-band model, which we describe in the following.

## 6 Multi-band model

In an optical lattice the molecular levels , and from the previous model have a substructure given by the lattice Bloch bands. Because the lattice depths for the levels , and are in general different, the respective band structures will also vary. This combination of external (c.o.m. motion in the lattice) and internal degrees of freedom gives rise to a number of new quantum levels which are coupled by the laser fields (Fig. 6). We assume each Feshbach molecule to be initially localized in a singly-occupied lattice site. The corresponding localized molecular wavepacket can be described by Wannier functions [15] which form a complete set of orthonormal functions. In the following we will denote the Wannier function for level and band as . We note that for deep lattices, these Wannier functions closely resemble harmonic oscillator wavefunctions.

The Raman lasers couple different according to
the respective wavefunction overlaps (Fig. 6). Since
the initial wavepackets of the Feshbach molecules are symmetric,
only even bands will be populated. We restrict our calculations to
the four lowest Bloch bands with even symmetry,^{4}^{4}4The effect
of including higher bands into the model was found to be
negligible. corresponding to the band indices . The
dynamics in each of the three lattice directions is then described by a
12-level model, which can in principle be solved in terms of a
master equation (Sec. 4). However, we have
used a Schrödinger equation-based model since the numerical code
is less involved. In this approach, laser phase fluctuations are
neglected, and we introduce a lattice site tunnel rate for each
band. These tunnel rates are chosen to match the expected tunnel rates for the different bands and are slightly adjusted for a better
fit of the data in Fig. 4. We note that the
results of the model calculations are essentially independent of the
excited state lattice depth, which is not well known.

The Hamiltonian of our time dependent Schrödinger equation

(6) |

has the form of a matrix,

(7) |

Here and are the energy and tunnel matrix element respectively for the Wannier function in band of level . is the overlap integral of the respective Wannier functions.

Diagonalizing this Hamiltonian, we find twelve “eigenstates” of
the coupled system which in general have complex eigenvalues. In the
following, we study the case of strong coupling (^{5}^{5}5For our experiments where MHz and kHz this condition is satisfied. which is the regime for phase locking.
In this regime, four of these eigenstates have negligible
contribution from the exited level and thus a long
lifetime. These 4 quasi-dark states essentially correspond to the 4
lattice bands in our model and will be denoted as
with n = 0, 2, 4, 6. We now study the spatial waveforms of these
dark states (Fig. 7) and compare the components
with and character. Neglecting a small component the dark superposition state, can be written as

(8) |

As an example () Fig. 7 shows that the wavepackets and have the same shape. This is not surprising since this ensures that the ratio of the and amplitudes equals everywhere, as in Eq. 1.

Let us now discuss the formation and evolution of the dark state that we have observed in the square pulse experiments of Sec. 3. A dark state is formed in less than 1 by subjecting Feshbach molecules to a square Raman laser pulse. As in the STIRAP transfer (discussed in Sec. 5) the initial projection onto will not change the shape of the Feshbach molecule wavepacket, given by the Wannier function . The dark state can be expressed as a coherent superposition of the four dark eigenstates of the 12-level Hamiltonian

(9) |

The subsequent coherent evolution of these dark states will again in principle lead to breathing oscillations. The amplitude of these oscillations depends on the extent to which higher bands (i. e., , ) are excited. The excitation increases with increasing deviation of from the initial state , i. e., with rising .

This can also be understood from another point of view. The effective
lattice potential felt by the molecules in such a superposition
state is the weighted average of the potentials for the two contributing
states and .
For the case this effective potential is about half as deep as that for the Feshbach molecules. Compared to the case of pure ground state molecules (Fig. 4) where the lattice potential is reduced by a factor of 10, the oscillations of the wavepacket are strongly suppressed and cannot be observed with our current experimental precision. For , the dark state has a dominant contribution from state , and the effective
lattice potential essentially corresponds to the one for ground
state molecules. In this case oscillations appear despite
the strong coupling, a fact which we also have experimentally
verified^{6}^{6}6For these experiments we have ramped into the dark
state and back in a fashion similar to STIRAP transfer pulses to
avoid strong losses caused by direct projection into ..

### Conclusion

We have analyzed coherent wavepacket dynamics and their suppression in a 3D optical lattice. We observed optically induced phase locking of a number of quantum levels, which can also be viewed as the appearance of a novel multi-level dark state. The experiments were carried out with tightly bound molecules as a component of a dark quantum superposition state. Thus, the experiments demonstrate control of molecular motion in an optical lattice for the first time. In addition, different models have been introduced and discussed in detail, with which the lattice dynamics can be understood and quantitatively described.

### Acknowledgements

The authors thank Helmut Ritsch for very helpful discussions and support in model calculations. We also thank Gregor Thalhammer for early assistance in the lab, and Florian Schreck for loaning us a Verdi V18 pump laser. This work was supported by the Austrian Science Fund (FWF) within SFB 15 (project part 17).

## Appendix: Theoretical band population analysis

As stated before, our signals only include molecules for which the constituent atoms end up in the lowest Bloch band of the lattice. A controlled lattice rampdown in a few milliseconds maps the bands and quasi-momentum distribution of the atoms into momentum space [16, 12]. We image these distributions after 13 ms of time-of-flight via absorption imaging. Fig. 8 shows a typical distribution.

The dotted square region corresponds to the lowest Bloch band and is dominantly populated.

An important question is how the Bloch bands for the Feshbach molecules map onto the Bloch bands for the atoms. In other words, if we measure the atomic population of the Bloch bands – do we know what the band population for the molecules was? As the lattice is very deep for the Feshbach molecules and atoms, we can approximate the potential at an individual lattice site as harmonic with trap frequency . In one dimension, the eigenfunctions of the harmonic oscillator are

(10) |

where is the oscillator length and H is the n Hermite polynomial. We assume that we have two atoms in a lattice site with coordinates . The relative and c.o.m. coordinates of the atom pair are

(11) | |||||

(12) |

The c.o.m.. In one dimension, the eigenfunctions of the harmonic oscillator are We assume that we have two atoms in a lattice site with coordinates . The relative and c.o.m. coordinates of the atom pair are

(13) | |||||

(14) |

The c.o.m. potential for the pair will be harmonic with trap frequency and the potential for the relative coordinate will be a sum of the harmonic potential and the interaction potential (Fig. 9).

When we form or dissociate a molecule by adiabatically ramping across a Feshbach resonance, only the quantum level in the potential will change – from a molecular bound state to an unbound atomic pair state in the lowest Bloch band. The wavefunction in the c.o.m. coordinate remains unchanged. We can now calculate how band populations of Feshbach molecules converted to atomic band populations by using the coordinate transformations. As an example: A Feshbach molecule in the lowest Bloch band (i. e., center-of-mass coordinate) will produce an atom pair with the following wavefunction: 1 gives the conversion amplitudes from molecular to atomic bands for the four lowest symmetric molecular bands. Correlations between the two constituent atoms of a molecule are not discussed here. . This means that both atoms will also end up in the lowest Bloch band of the lattice. This analysis can be extended to any band. Table

We finally note that when we apply absorption imaging, the optical density of the atomic sample is integrated in the direction of observation. Thus in this direction no band population analysis is possible. We accounted for this in our multi-band model described in Sec. 6.

atomic Bloch band | ||||||||||

0 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | … | ||

0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | |

2 | 0 | 0 | 0 | 0 | 0 | … | ||||

4 | 0 | 0 | 0 | … | ||||||

6 | 0 | … | ||||||||

{rotate}90 molecular band | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

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