molecules represent ideal candidates for storage
qubits and time keepers in future precision measurements on dipolar quantum matter.
The experiment starts with the creation of 23Na40K
molecules in the singlet rovibrational ground state
X1Sþjv ¼ 0; J ¼ 0i (17, 27). Here, v denotes the
vibrational quantum state, and J is the total angular
momentum quantum number, neglecting nuclear
spins. To reach the ground state, weakly bound
Feshbach molecules of 23Na40K are created and
subsequently transferred to the singlet rovibrational ground state by means of stimulated Raman
adiabatic passage (STIRAP) (17, 28). By choosing
the polarization of the Raman beams appropriately,
we create a pure ensemble of 2 × 103 ground-state
molecules, all in the lowest energy hyperfine level. The molecular ensemble typically has an average density of n = 2 × 1010 cm–3 and a temperature
of 300 nK and is trapped in a crossed optical dipole trap operating at a wavelength of l = 1064 nm.
Finally, the ground-state molecules are detected by
a reverse STIRAP transfer back to Feshbach mole-
cules and subsequent imaging of the 40K com-
ponent by using light resonant on the atomic
cycling transition. This procedure selectively de-
tects molecules in the lowest hyperfine level (17).
After the creation of ground-state molecules, a
two-photon microwave pulse is applied to prepare each of the molecules in a superposition of
two hyperfine levels within jv ¼ 0; J ¼ 0i (Fig. 1).
At a magnetic field of 85.6 G, where the experiment operates, the 36 hyperfine levels in J ¼ 0
are split by the nuclear Zeeman effect (17, 20).
Hence, the nuclear spin projections mINa and mIK
are good quantum numbers. The two levels that
form the superposition state are the lowest hyperfine level jmINa ; mIK i ¼ j3=2; –4i and the first
excited hyperfine level j3=2; –3i, denoted by j↓i
and j↑i, respectively. For the coherent transfer
between j↓i and j↑i, a hyperfine level with mixed
mIK = −4 and −3 character in the rotationally
excited jv ¼ 0; J ¼ 1i state (20) is used as an intermediate state to drive two-photon Rabi oscillations (27).
Ramsey precession is initiated by applying a
p/2-pulse near the microwave two-photon resonance. Subsequently, the superposition state undergoes field-free time evolution with respect to the
unperturbed resonance frequency wres. After a
hold time T, we apply a second p/2-pulse and
record the number of molecules in j↓i (Fig. 2). By
fitting the observed Ramsey precession with a
model that incorporates the decay of molecule
number and coherence (27), we extract the molecule lifetime T1 and the coherence time T ; 2. At a
molecular lifetime of T1 = 1.9(5) s, we observe a
coherence time on the scale of a second, T ; 2 =
0.7(3) s, which is a thousand times longer than
rotational coherence times measured similarly
without spin-echo techniques. The oscillation frequency of the Ramsey precession is given by the
difference of the two-photon microwave drive,
w1 ; w2, and the unperturbed resonance frequency, wres. From this, we extract the energy difference
between j↓i andj↑i to be wres = 2p × 20.514(10) kHz
at a magnetic field of 85.6 G, which allows us to
Fig. 2. Coherent Ramsey precession of nuclear spin states in
23Na40K. An initial p/2-pulse, resonant on the dressed two-photon
transition, creates a superposition of the j↓i and j↑i states in the
equatorial plane of the Bloch sphere (see inset). The Bloch vector
precesses at a frequency jðw1 ; w2 Þ ; wres j for a variable precession time
T until a second resonant p/2-pulse completes the Ramsey sequence.
The solid blue line is a fit of the complete data set with a single oscillation
frequency and phase (27), indicating the phase coherence of the
Ramsey precession. The bottom row shows magnified sections of the
full data set above. Data points correspond to the average of typically
three experimental runs; the error bars denote the standard deviation
of the mean.
Fig. 3. High-resolution Ramsey spectroscopy. Ramsey fringes are recorded
as a function of two-photon drive frequency w1 ; w2, keeping the precession
time T = 112 ms fixed. The distance between adjacent Ramsey fringes is 1/T ≈
8.9 Hz, resulting in hertz-level precision. The solid blue line shows a fit with a
Ramsey line shape function that includes, as a free parameter, the two-photon
Rabi coupling W (27), which determines the overall envelope of the spectrum.