Scanning tunnelling microscopy for ultracold atoms
Scanning tunnelling microscopy for ultracold atoms
Corinna Kollath,1 Michael Köhl,2, 3 and Thierry Giamarchi1
DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland
Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom
Institute of Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland
(Dated: October 19, 2021)
We propose a novel experimental probe for cold atomic gases analogous to the scanning tunnelling
microscope (STM) in condensed matter. This probe uses the coherent coupling of a single particle
to the system. Depending on the measurement sequence, our probe allows to either obtain the local
density, with a resolution on the nanometer scale, or the single particle correlation function in real
time. We discuss applications of this scheme to the various possible phases for a two dimensional
Hubbard system of fermions in an optical lattice.
PACS numbers: 73.43.Nq 03.75.Ss 71.10.Pm
Recent advances in the field of ultracold atoms have
led to a close connection between quantum gases and
condensed matter physics. The achievement of strongly
correlated systems and their remarkable tunability open
the possibility to realize ‘quantum simulators’ for quan-
tum many-body phenomena. To name one example, ul-
tracold fermionic systems clarified the crossover between
a BCS-state of paired fermions to a Bose-Einstein con-
densate of ultracold bosonic molecules [1, 2, 3, 4, 5]. Fur-
ther investigations of strongly correlated systems were
initialized by the successful loading of ultracold bosonic
[6] and fermionic [7] atoms into three-dimensional opti-
cal lattices. In these periodic lattice potentials created by
counter-propagating laser beams the physics of different
lattice models can be mimicked [8, 9]. In particular the
fermionic Hubbard model, which plays an important role
on the way of understanding high-temperature supercon-
ductivity, can be realized naturally given the short range
nature of the interactions between the neutral atoms.
Whereas achieving the exotic quantum phases exper-
imentally appears feasible with today’s technology their
clear identification remains an obstacle. Compared to
condensed matter the neutrality of the cold atoms is both
an advantage and a drawback since they cannot be per-
turbed as easily as electrons in a solid. Possible probes
are thus more sophisticated than their condensed mat-
ter counterparts. In addition, for ultracold atomic gases
an inhomogeneous confining potential causes the coex-
istence of different spatially separated quantum phases
[8, 10, 11]. This makes their realization and observation
in the presence of a trapping potential very involved and
creates a need for a method of probing the systems lo-
cally. However, the existing probes [10, 11] still involve
an averaging over regions of various densities and new
techniques which allow for a local detection need to be
developed.
In condensed matter physics, a remarkable local probe
was provided by the scanning tunneling microscope
(STM) [12]. It allowed to explore and image the sur-
face topography with atomic resolution, paving the way
to control and analyze quantum phenomena on solid sur-
faces [13]. In addition to the density analysis with un-
precedented resolution, the STM has also become a spec-
troscopic tool probing the local density of states. This
spectroscopic method had a major impact on the un-
derstanding of the physical properties of strongly corre-
lated systems for which the local density of states pro-
vides unique information on the physics of the system. In
particular the STM has made significant contributions to
the field of high temperature superconductors [14].
In this work we propose a novel experimental setup to
locally probe cold atomic systems in an approach similar
to and as versatile as the STM. The probe relies on the
coupling of a single particle to the system. Different ‘op-
erating modes’ yield either a measurement of the local
density or of the single particle Green’s function in time.
The realization of such a probe will open the possibili-
ties to investigate exotic quantum phases in great detail
as we show on the example of the Hubbard model. In
extension to the conventional STM in condensed matter
physics our scheme would allow for measurements in a
three-dimensional sample.
The key idea for the realization of an STM-like scheme
with cold atoms, the ‘cold atom tunneling microscope’ is
sketched in Fig. 1. A single trapped particle is used as a
probe of local quantities by inducing a controlled interac-
tion between the probe particle and the quantum many-
body state. To allow for a precise control over the motion
of the probe and to facilitate a convenient readout mech-
anism, we suggest to employ a single atomic ion trapped
in the vibrational ground state of a radio-frequency Paul
trap [15]. In this case the spatial resolution of the micro-
scope relies on the excellent control over the position and
motion of trapped ions on the sub-micron scale. However,
the working principle of the microscope does not depend
on the charge of the probe particle and it also applies to a
neutral atomic quantum dot [16] provided that the trap-
ping potential of the dot has only a negligible influence
http://arxiv.org/abs/0704.1283v1
FIG. 1: Sketch of the cold atom tunneling microscope. As an
example the application to an anti-ferromagnetic state with
alternating spin states labeled by different colors is shown.
on the quantum many-body system [17]. The controlled
interaction between the probe particle (the ion) and the
quantum many-body system could be provided by a two-
photon Raman coupling.
As we show below the cold atom tunnelling microscope
facilitates a local detection of the density on individual
lattice sites and, quite remarkably, it even allows to per-
form a spin-resolved detection of the density. The real-
ization of a spin-resolved STM is a long-sought goal in
condensed matter systems but has not yet been achieved.
The cold atom tunnelling microscope also allows to per-
form spectroscopy by observing the local single particle
Green’s function 〈c
σ,j(t0)cσ,j(0)〉F in time. Here cσ,j is
the annihilation operator for the neutral atom on a site j
with spin σ = {↑, ↓} and 〈·〉F stands for taking the expec-
tation value with respect to the atomic system only. The
temporal decay of this function directly reflects the na-
ture of the excitations and gives thus direct information
on the quantum phases present in the system.
We first show taking the example of fermionic atoms in
two different spin states ↑ and ↓ in an optical lattice how
a measurement of the local density, the ’scanning mode’,
can be achieved. It is facilitated by a two-photon Raman
coupling between the ion |i〉 and an atom |aj〉 in a lattice
well j by which a weakly bound molecular ion |i+aj〉 can
be created (see Fig. 2). This coupling can be described by
the expression
σ Ωσ(t)M
σIcσ,j + h.c.
. Here Mσ and
I are the annihilation operators for the molecular ion
and the atomic ion, respectively. The coupling strength
Ωσ(t) can be controlled experimentally. By choosing the
correct frequency and polarization of the laser fields, the
coupling is dependent on the atomic ‘spin’ state paving
the way for the spin-resolved microscopy.
The experimental sequence to detect the local density
is as follows: At time t = 0 the atomic many-body system
is prepared in its ground state |Ψ0〉. The ion is introduced
into the lattice well j in state |i〉 and the Raman cou-
pling is switched on for a duration δt, i.e. Ωσ(t) = Ωσ,0
if t ∈ [0, δt] and vanishes otherwise. The time δt has
to be short compared to the internal time-scales of the
internuclear
separation
Ii> x Iaj>
FIG. 2: Two-photon Raman coupling of the ion |i〉 and an
atom |aj〉 in a harmonic potential well to a molecular ion
bound state |i + aj〉. The single photon coupling is detuned
by ∆ from a resonant transition to suppress spontaneous emis-
sion from the intermediate excited state. The effective two-
photon Rabi frequency Ω0 is proportional to the coefficients
of the single photon transitions and inversely proportional to
the detuning ∆, i.e. Ω0 ∝ Ω1Ω2/∆.
probed system (i.e. time-scales set by the atom-atom in-
teraction U , the kinetic energy of the atoms J , and the
atom-ion contact interaction Uai) to avoid a change of
the many-body state during the probe sequence. The
Raman coupling generates a superposition of the initial
state |Ψ0〉 ⊗ |i〉 and the state cσ,j|Ψ0〉 ⊗ |i+ aj〉 in which
one atom is removed from the system and a molecular
ion is formed. The ratio between the amplitudes of the
two states depends on the density of atoms in the well j.
Hence detecting the probability (i.e. the average of the
outcome of several quantum measurements) for molecule
formation after the application of the Raman pulse mea-
sures the local density of atoms in the lattice well j
by the relation 〈
σMσ〉 =
σ sin
2 (Ωσ,0δt)〈nσ,j〉F
with 〈nσ,j〉 the local atomic density. The outcome of the
photo-association process can be detected by measuring
the changed oscillation frequency of the heavier molec-
ular ion in the Paul trap or by observing the absence
of resonant light scattering of the molecular ion and its
reappearance after photo-dissociation [18]. The proce-
dure can be repeated scanning different lattice sites as
sketched in Fig. 1 with a spatial resolution on the order
of 20 nm [19]. To facilitate the measurement of the den-
sity with a good signal to noise ratio, the lattice potential
could be increased such that the density profile on dif-
ferent lattice sites is frozen and sequential measurements
on single lattice site are feasible.
Using the cold atom tunnelling microscope with a dif-
ferent sequence, the ’tunneling mode’, allows to perform
spectroscopy and to measure time dependent correlations
locally. The experimental sequence is sketched in Fig. 3.
As in the scanning mode we start at t = 0 in the state
|Ψ0〉 ⊗ |i〉, i.e. the ground state |Ψ0〉 of the atomic sys-
tem and a single atomic ion. A two-photon Raman pro-
cess is applied over a short time interval δt1 to couple
the ion with an atom present in the lattice well. Sub-
sequently, the superposition state of the atomic and the
molecular ion |i〉+ α|i+ aj〉 is removed from the system
such that they are non-interacting with the remaining
quantum many-body system, for example their center-
of-mass position can be shifted by applying a small dc
voltage. After a variable time of free evolution t0 in this
isolated position they return into the addressed lattice
well and the application of the two-photon Raman pro-
cess is repeated for a time interval δt2. The outcome of
the molecule formation is detected afterwards [28]:
〈M †M〉 = A(δt1, δt2) + sin
2(δt2Ω0) [cos(δt1Ω0)− 1] {[cos(δt1Ω0)− 1]〈nj(0)nj(t0)nj(0)〉+ 〈nj(t0)nj(0)〉}
sin2(δt2Ω0) + sin
2(δt1Ω0)
〈nj(t0)〉+ sin
2(δt1Ω0) [cos(δt2Ω0)− 1] 〈c
j(0)cj(t0)c
j(t0)cj(0)〉 (1)
A(δt1, δt2) = 2 sin(δt1Ω0) sin(δt2Ω0) cos(δt2Ω0)ℜ
e−i(εM−εI )t0/h̄
j(t0)cj(0)〉
︸ ︷︷ ︸
+(cos(δt1Ω0)− 1) 〈nj(0)c
j(t0)cj(0)〉
︸ ︷︷ ︸
Ii > Ii >+α Ii+aj>
IΨ0>+β.cj IΨ0>IΨ0>
detection
Raman 
interaction
Raman 
interaction
FIG. 3: Schematics of the experimental sequence for the tun-
nelling mode. The atomic ion |i〉 is introduced at the lat-
tice site j into the many-body system in state |Ψ0〉. The
two-photon Raman process (red) couples an atom at this lat-
tice site |aj〉 to the ion with a certain amplitude. Subse-
quently, the ion and the many-body system are separated for
the probe time t0 during which they evolve individually. Af-
ter recombination, the Raman interaction is applied again and
the molecule formation is detected.
We supressed the spin index and on the right hand side
additionally the index F .
Using appropriate measurement sequences different
correlation functions can be extracted. To obtain the
temporal correlation function 〈c
σ,j(t0)cσ,j(0)〉F the de-
scribed measurement procedure is applied sequentially:
first, using δt1 = δt2 = δt for both pulses, second using
δt1 = δt for the first pulse and δt2 = 2π/Ωσ,0− δt for the
second pulse. Subtracting the outcome for the molecule
formation of the two measurements gives ∆〈M †σMσ〉 =
2A(δt, δt). For small values of (δtΩσ,0) the pre-factor of
the first summand A1 in A(δt, δt) is quadratic in (δtΩσ,0),
whereas the prefactor of the second term A2 is quar-
tic. Since additionally in many systems the decay of
the correlation function 〈nσ,j(0)c
σ,j(t0)cσ,j(0)〉 is faster
or comparable to the decay of the single particle correla-
tion function the second term can safely be neglected.
The expression (εI −εM )t0/h̄ represents the phase dif-
ference the atomic ion and the molecular ion collect dur-
ing the time t0. In principle this quantity could be zeroed
by choosing a suitable combination of the optical lattice
field and the ion trapping fields. However, this cancella-
tion is not necessary if εI − εM ≫ U, J because then the
temporal evolution of the correlation function is encoded
simply in the envelope of the detection signal.
One direct application of the cold atom tunnelling
microscope would be the identification of the quan-
tum many-body phases of the two dimensional Hubbard
model. In addition to the normal (Fermi liquid) quantum
fluid of fermions, this model can lead to broken symmetry
phases such as an anti-ferromagnet, and a strongly corre-
lated (Mott) insulator. An important and yet open ques-
tion is whether other more exotic phases can exist in this
model, such as inhomogeneous distribution of the den-
sity (stripes and checkerboards) or even superconducting
phases with d-wave symmetry for the pairing. Our local
probe directly detects symmetry broken phases such as
the anti-ferromagnet in which the spin density is modu-
lated (cf. Fig. 1) and even more inhomogeneous phases
with a modulation of the density (stripes and checker-
boards [20]).
Additionally, even for phases with homogeneous den-
sity and spin density, such as a quantum fluid or a su-
perconductor, the ’tunnelling mode’ reveals the nature
of the excitations by probing the single particle density
of states. This, for example, allows to characterize di-
rectly an s-wave or a d-wave superconductor. In Fig. 4
we plot the Fourier transform of the correlation function
σ,j(t0)cσ,j(0)〉F for both an s-wave superconducting and
a d-wave superconducting phase on a two-dimensional
lattice. Both are obtained using the phenomenological
BCS-approach using the energy dispersion on the lattice
−2J(cos(kxa) + cos(kya)). In the s-wave superconduct-
0 1 2 3
 ω /∆0
d-wave superconductor
s-wave superconductor
FIG. 4: The Fourier transform of the temporal correlation
function in an s-wave and a d-wave superconducting state is
shown for a gap value of ∆0 = 0.3J .
ing phase, the gap ∆s(k) ≡ ∆0, clearly leads to a strong
divergence of the correlation signal and a zero response
in the gap below ∆0. For the d-wave superconducting
phase with ∆d(k) =
(cos(kxa) − cos(kya)) [21], one
observes a quite different signal having a spectral weight
below the gap energy. Thereby the structure of the su-
perconducting order parameter and the size of the gap
can be extracted from the proposed measurement.
The independent control over the single particle and
the neutral atomic quantum gas lies at the heart of the
cold atom tunnelling microscope. To a good approxi-
mation the ion experiences only the ion trapping poten-
tial, the atom only the optical lattice potential and the
weakly bound molecular ion both potentials. This makes
the single ion a particularly attractive choice for shut-
tling the atomic and the molecular ion in and out of the
lattice without influencing the neutral atomic quantum
many-body state. For example a displacement of 1.2mm
within 50µs has been achieved without exciting vibra-
tional quanta [22].
The binding energies of the weakly bound states of the
atom-ion interaction potential (see Fig. 2) are determined
by its asymptotic behavior scaling as −C4/r
4. Here C4
is proportional to the electric dipole polarizability of the
neutral atom and r is the internuclear separation. The
binding energy of the most weakly bound molecular state
is two orders of magnitude less than for typical neutral
atom interactions [23, 24]. Several more deeply bound
states with binding energies in the 10-100MHz range are
available for Raman photo-association [25]. The gener-
ation of weakly bound molecules using two-photon Ra-
man coupling in optical lattices has already been demon-
strated for pairs of neutral atoms [26] and even the coher-
ent coupling of free atomic and bound molecular states
has been observed [27] which is the prerequisite for the
tunneling mode.
In order to probe the quantum many-body state with-
out perturbations the time scales set by the parameters
of the atomic system should be larger than the time inter-
vals of the Raman pulses. To realize a strongly correlated
phase in the lattice, the atom-atom scattering length aaa
needs to be enhanced by a Feshbach resonance. Assum-
ing aaa ≈ aai ≈ 10
3 a0 results in U ≃ Uai ≃ 20 kHz
for the fermionic isotope 40K, whereas J is typically one
order of magnitude smaller. Therefore the condition for
the proposed ‘scanning’ mode J , U , Uai ≪ 1/δt can for
example be fulfilled using an effective Raman coupling
Ωσ,0 = 2π× 10 kHz applied over a time interval δt = 5µs
resulting in a molecule formation probability of ≈ 0.1.
For the ‘tunnelling’ mode the Raman coupling needs to
be one order of magnitude stronger since the above condi-
tion has to be fulfilled for both Raman pulses of duration
δt and 2π/Ωσ,0 − δt, respectively. The shortness of the
photo-association pulse has other direct benefits: first,
the level shift due to the interaction Uai is not resolved
and thus the measurement is independent of the occupa-
tion of the lattice well by an atom in the second hyperfine
state which is not probed in the spin-resolved mode. Sec-
ondly, the short pulse and the subsequent removal of the
molecular ion from the quantum many-body system en-
sures also the stability of the microscope scheme against
three-body recombination in a lattice well.
In conclusion, we have proposed a novel experimental
setup, the cold atom tunnelling microscope, to observe
locally the (spin-resolved) density and the single parti-
cle Green’s function. In contrast to previous work this
measurement procedure does not average over spatially
different regions of the system with coexisting quantum
phases, but can resolve single lattice wells. A modifica-
tion of the proposed scheme would give also access to
nonlocal single particle correlation functions. The re-
quired modification consists of moving the probe particle
during the tunnelling mode scheme to a different lattice
well, say m, before the second Raman pulse is applied.
The outcome of the molecule formation then will be re-
lated to the correlation function 〈c†σ,m(t0)cσ,j(0)〉F of the
atomic system. Additionally the proposed setup opens
the possibility to create single particle excitations in a
controlled way.
We would like to thank C. Berthod and H. Häffner for
fruitful discussions. This work was partly supported by
the SNF under MaNEP and Division II.
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