Spatial Adiabatic Passage of Massive Quantum Particles

Spatial Adiabatic Passage of Massive Quantum Particles

Abstract

By adiabatically manipulating tunneling amplitudes of cold atoms in a periodic potential with a multiple sublattice structure, we are able to coherently transfer atoms from a sublattice to another without populating the intermediate sublattice, which can be regarded as a spatial analogue of stimulated Raman adiabatic passage. A key is the existence of dark eigenstates forming a flat band in a Lieb-type optical lattice. We also successfully observe a matter-wave analogue of Autler-Townes doublet using the same setup. This work shed light on a novel kind of coherent control of cold atoms in optical potentials.

pacs:
34.50.-s, 67.85.-d

Interference of probability amplitudes is one of the most significant properties of quantum mechanics. In the seminal work of an electron double-slit experiment Tonomura1989 (), building up of the interference pattern of electron wave function beautifully demonstrated nature of wave-particle duality. Quantum interference has been intensively utilized especially in the field of precision measurement such as superconducting magnetometer Jaklevic1964 () and atom interferometry Muller2010 (), and also lies at the heart of quantum information science.

A three-level system is a minimal example in which quantum interference takes place. Most commonly it is considered in a context of laser coupled atomic levels, and the Hamiltonian for a -type system (Fig. 1 (a)) in a rotating frame is written in the form

(1)

where () denotes a laser-induced Rabi frequency which couples basis states with ( with ), and () is the detuning of the corresponding laser 1 (2). A dark state () arises as one of the eigenstate of the Hamiltonian given by Eq. (1) if the Raman resonant condition is satisfied. By applying two laser pulses in so-called “counter-intuitive” order so that changes form to , the dark states smoothly evolve from into . This process is well known as Stimulated Raman Adiabatic Passage (STIRAP) Kuklinski1989 (); Gaubatz1990 (); Bergmann1998 (), and has been an important technique for robust population transfer between two atomic/molecular states.

Figure 1: (a) A -type three level system and a Lieb lattce. Gray shaded region indicates one of unit cells. (b) Energy band structure of the Lieb lattice with no distortion, calculated in the tight-binding limit. The corresponding potential landscape in our optical lattice setup is also shown below. (c) Energy band and potential landscape with distortion. If the phase of the diagonal lattice is shifted, band gaps become large and the Dirac cone disappears. Shown is the case of and , which is used in Fig. 2 (a).

Natural interests arise for a special case that the states of interest represent a matter wave of a quantum particle and the initial and the final states are spatially well isolated. Such processes, named spatial adiabatic passage (SAP), offer paradoxical “transport without transit” Rab2008 (); Benseny2012 () where matter waves are transported without populating the intermediate spatial region. Since the concept of SAP was introduced in the context of quantum dots Renzoni2001 (); Greentree2004 () and cold atoms Eckert2004 (), it has continuously attracted theoretical interests and various possibilities of its application have been discussed Menchon2016 (). Generalized to two-dimensions, SAP technique also enables preparation of states with angular momentum McEndoo2010 (); Menchon2014a (). It is also interesting to consider the case of interacting many-body systems such as Bose-Einstein condensates Graefe2006 (); Rab2008 (); Rab2012 (). Highly controllable and flexible systems of cold atoms are suitable to realize SAP and the above-mentioned applications.

A key for the realization of SAP is the existence of dark eigenstates. Interestingly, dark states also arise in eigenstates for a particle moving in a special kind of lattice structures. In this paper, we report on the successful realization of SAP by adiabatic control of matter-wave tunneling in an optical lattice. Our lattice consists of three sublattices (, , and shown in Fig. 1 (a)), forming a Lieb lattice. It is convenient to take plane waves on each sublattice , and with a momentum as a basis set. The existence of nearest-neighbor tunneling and induces coupling among these basis states given by and . The resulting Hamiltonian can be written as a matrix form

(2)

Now analogy to a -type system (1) is obvious: momentum-dependent couplings play a role of Rabi couplings in a three-level system and detunings can be mimicked by energy offsets , , and of each sublattice. All these parameters can be controlled by changing the lattice depth along each direction, which enables us to realize a coherent scheme to transport atoms among these sublattices. Note that classical waves traveling through coupled waveguides is known to obey coupled mode equations Hardy1985 () which are mathematically equivalent to Eq. (1) and SAP correspondence was demonstrated Menchon2012 (). However, the original concept, SAP with matter waves of a quantum particle has not been realized.

I Expermental Realization

Our experimental setup is similar to that described in Ref. Taie2015 (); Ozawa2017 (). In brief, an optical Lieb lattice is created by the combination of a 2D staggered superlattice with  nm spacing and a diagonal lattice with  nm periodicity. The resulting potential is given by , where is the recoil energy for the long lattice. Below we specify a lattice potential by -parameters and the relative phase of the diagonal lattice . Basically, tunneling amplitudes are determined by the depths of short lattices (, ) and the other lattice depths are responsible for the energy offsets , and . In our experiment, we use fermionic Yb with a small scattering length  nm to avoid interaction effects. The use of fermions introduces a complexity arising from the finite momentum spread due to the Pauli principle. In the absence of interactions and harmonic confinement, the dynamics conserves quasimomentum and states initially having well-defined quasimomentum evolve within a subspace spanned by three plane waves , and .

Adiabaticity of a process associated with a certain momentum is governed by the band gaps among the corresponding eigenstates. This implies that, for a Lieb lattice, adiabaticity cannot be maintained around the corner of the Brillouin zone where a Dirac cone exists (Fig. 1 (b)). To overcome this problem, we slightly deform the lattice structure by shifting the phase from an isotropic condition . The effectiveness of this scheme can be understood from the potential landscape shown in Fig. 1 (c). The deformation reduces the inter-unit-cell tunneling, therefore each cell becomes more like an isolated triple well. As a result, the momentum dependence of the dispersion curves is reduced and the Dirac cone is eliminated. Mathematically, this modifies the coupling term as (similar change applies for ), where denotes the imbalance between inter- and intra-unit-cell tunneling. For , along the Brillouin zone boundary () vanishes throughout the process. Introduction of can also suppress the breakdown of transport along this line.

Ii Spatial Adiabatic Passage

Figure 2: (a) Time evolution of sublattice occupancies , and during a SAP process. Time is rescaled by ms. Samples of absorption images used taken in the experiment are also shown in the right hand side. Labels (i)-(iii) represent the correspondence between images and data in the graph. (b) Band-mapping image at the half point (ii) of the SAP process (a). Boundaries for the first and second Brillouin zone are indicated by white lines. (c) Dependence of SAP efficiency on the ratio . (d) Adiabaticity of the SAP process. Sublattice occupancies after SAP are shown as a function of total sweep time .

A matter-wave analogue of a STIRAP in the Lieb lattice is to transport atoms between two sublattices ( and ) by a counter-intuitive temporal change of tunneling amplitudes. Throughout this process, the intermediate sublattice is not populated because the state adiabatically follows a dark state , with . First we load a sample of atoms at a temperature as low as of the Fermi temperature into the optical Lieb lattice. In the loading stage, the potential on a -sublattice is made much deeper than those of the others () to ensure that the initial state is only populated by . After that, we quickly change the lattice depths to in s. This is a starting point of STIRAP process, where the tunneling is much smaller than . To achieve a high tunneling rate, overall lattice depths are set relatively shallow, leading to an unwanted direct tunneling . We suppress by increasing the diagonal lattice depth beyond the equal-offset condition, i.e., . As long as is maintained, the dark state persists and STIRAP can be accomplished. We adiabatically sweep the lattice depths toward another limiting configuration , passing through the intermediate point corresponding to the potential shown in Fig. 1 (c). The time evolution during this process is monitored by mapping sublattice populations onto band populations followed by a standard band mapping technique Taie2015 (); Ozawa2017 (). The obtained time-of-flight images suffers from the blurring of the Brillouin zone boundaries due to unavoidable non-addiabaticity of the band mapping procedure and a harmonic confinement of the system. For a qualitative analysis of sublattice populations, we take a set of basis images in which all atoms resides on a specific sublattice and determine sublattice occupancies by projecting images onto each basis. Figure 2 (a) shows the time evolution of sublattice occupancies , and during the SAP process. The time scale is renormalized by where is the energy gap averaged over entire Brillouin zone at the half point of the process. Important features specific to SAP are well reproduced: initial population on the -sublattice, , is smoothly transfered into , but does not show increase throughout the process. From the final population we evaluate the efficiency of the process to be . During the SAP process the state is certainly following the dark state as one can see in the direct band-mapping image shown in Fig. 2 (b). Here, the concentration of the atomic distribution inside the second Brillouin zone indicates the state is kept in the second energy band which consists of the dark states. Usually, occupation of a certain energy band is accomplished by filling up all lower bands. The above SAP process provides an efficient way to prepare a non-equilibrium many-body state in which all fermions reside on the flat band of the Lieb lattice and the other bands are empty.

In our system, the SAP efficiency is limited by the existence of a trap-induced harmonic confinement. This causes inhomogeneity of energy offsets, especially around the edge of an atomic cloud. We change the ratio of to with denoting the mean trap frequency on plane, and monitor the change of SAP efficiency. As seen in Fig. 2 (c), the trap reduces the SAP efficiency.

We examine adiabaticity of the STIRAP process by changing a total sweep time (Fig. 2 (d)). As naively expected, for , adiabaticity is broken and significant populations in and are observed. Once the adiabatic condition is fulfilled for large , the final state is kept almost constant, with a large population in regardless of . This behavior is characteristic to a robust adiabatic process, in contrast to Rabi oscillations driven by a direct tunneling coupling.

Iii Bright State Transport

In atomic three-level systems on the one hand, an intermediate state generally suffers from significant loss due to spontaneous emission. One of the advantages for STIRAP is that the loss through state can be avoided by keeping the dark state. On the other hand, SAP with cold atoms does not suffer from any losses through intermediate states, which enables us to perform efficient transport through “bright” states where is significantly populated. We design an “intuitive” potential sweep for this bright state transport for the optical Lieb lattice. The sweep involves not only (tunneling) but also (detuning) to improve the transport efficiency. Figure 3 (a) shows the resulting time evolution during this sweep. Due to the requirement of minimizing unwanted excitations at the beginning of the sweep, initial localization on is not perfect in this experiment. However, the nature of the bright state transport is clearly visible as the increase of around the half point. As a reference, we also perform a “counter-intuitive” scheme under the same condition. In Fig. 3 (b) we start with a sample localized on , but apply the potential sweep in a time-reversal way compared with that of 3 (a) to transport atoms through the dark state. This scheme is essentially equivalent to the SAP process demonstrated in the previous section, except that the state does not always remain dark because changes during sweep. Similar performance of transport efficiency is obtained, but the behavior around the half point of the process is qualitatively different. At this point, the state becomes exactly dark and the certainly shows its minimum, in contrast to the bright state transport. The feasibility of accessing both bright and dark state transfer manifests the flexibility of our system in quantum state engineering.

Figure 3: (a) Transport from the - to -sublattice through the bright state. (b) Transport through the dark state. After loading to the -sulattice, we exactly reverse the potential sweep applied in (a).

Iv Autler-Townes doublet

Electromagnetically induced transparency (EIT) Boller1991 () is also an important process in a three-level system. In an EIT experiment, there is strong optical coupling between and which causes splitting of the transition by the Rabi frequency, known as Autler-Townes doublet Autler1955 (). As a result, the state becomes transparent for laser light driving the transition at a frequency region between the doublet.

To investigate a matter wave analogue of EIT physics, we carry out a measurement similar to a pump-probe experiment, as depicted in Fig. 4 (a). As before, we first prepare an initial state localized on . Then we allow weak tunneling coupling and after a fixed time, a fraction of atoms that tunneled into or is measured. Figure 4 (b) shows a set of tunneling spectra. In each spectrum we scan the “detuning” which determines the energy difference (). As we increase the coupling (decrease ), the spectrum drastically changes: for negligible , we can observe a single peak corresponding to , whereas a clear doublet structure appears for . The double peaks originate from tunnelings to and orbitals which are separated by the amplitude of tunneling coupling . The overall shift of the spectrum is caused by the change of the short lattice depth . While the short lattice creates the same potential curve for all sublattices, its effect on , and slightly differs depending on the configuration of other lattices. We estimate the shift of zero-point energy by a harmonic approximation. Whereas it fails to predict the position of resonance center, the tendency of the shift is well captured. The width of the observed resonance peaks is broadened by band dispersion and spatial inhomogeneity. In a typical EIT spectrum, a sharp dip can be observed even when the doublet splitting is smaller than the natural linewidth. This implies occurrence of coherent population trapping (CPT) Arimondo1996 () of a dark eigenstate. In the case of our system with no loss mechanism, CPT does not occur and hence a sharp EIT dip does not appear. However, the observed behavior exactly corresponds to a pump-probe detection of Autler-Townes doublet which is commonly observed in atom-field systems. As a future direction, it is interesting to introduce site-dependent loss (by e.g., high-resolution laser spectroscopy) and study CPT and related phenomena.

Figure 4: (a) Schematic of the experiment. (left) In the absence of , the weak tunneling coupling results in a fraction of atoms in . (right) In the presence of strong , tunneling only occurs when the energy resonates to bonding or anti-bonding orbitals, split by coupling . (b) Tunneling spectrum after a hold time of ms, at the lattice depths of . Fraction of atoms which tunnel from the -sublattice is shown. Vertical dashed lines show the rough estimation of the points where is satisfied, based on a harmonic approximation of the lattice potential.

V Conclusion

In conclusion, we have succeeded in demonstrating coherent tunneling processes of cold atoms in an optical Lieb lattice. The three-sublattice structure of the Lieb lattice has remarkable analogy to -type three level systems in quantum optics. By using this analogy and dynamical controllability of tunneling amplitudes, spatial addiabatic passage between two sublattice eigenstates was performed. We also observed an matter-wave analogue of Autler-Townes doublet in a tunnleing process from a sublattice into a strongly coupled pair of sublattices. The demonstrated techniques are useful to prepare exotic many body states in optical lattices. For example, at the half point of the SAP process in the Lieb lattice, all atoms are located on the flat band. This might be a general scheme applicable to other lattice with flat bands. Involving higher lattice orbitals is also interesting in connection with generation of angular momentum studied in Ref. Menchon2014a (). In addition, recent advances in fine potential engineering Liang2010 () combined with site-resolved imaging of lattice gases Bakr2009 () will greatly enlarge the application of SAP in cold atomic systems.

Acknowledgements.
We thank, T. Busch, J. Mompart, A. Greentree and V. Nesterenko for stimulating discussions, and H. Shiotsu for experimental help. This work was supported by the Grant-in-Aid for Scientific Research of JSPS (No. JP25220711, No. JP26247064 No. JP16H00990, No.JP16H01053), the Impulsing Paradigm Change through Disruptive Technologies (ImPACT) program, and JST CREST(No. JPMJCR1673). H.O. acknowledges support from JSPS Research Fellowships.

Vi Supplementary Information

vi.1 Sample Preparation

A quantum degenerate gas of Yb is produced by sympathetic evaporative cooling with fermionic Yb, in a crossed dipole trap operating at  nm. Taie2010 (). After evaporation, remaining Yb atoms are cleaned up by illuminating resonant laser light on the transition. All the data presented in the main text are taken with spin unpolarized samples.

vi.2 Optical Lieb Lattice

Our optical Lieb lattice consists of a short square lattice operating at a wavelength of  nm, a long square lattice at  nm and a diagonal lattice at  nm. The lattice beams for two square lattices are retroreflected by common mirrors. Therefore the relative phases between these lattices are determined by the relative frequencies between the short and long laser beams, which is actively stabilized to realize the staggered lattice configuration. The relative phase of the diagonal lattice is stabilized by a Michelson interferometer and the uncertainty is estimated to be during a typical measurement time Taie2015 ().

vi.3 Sublattice occupation measurent

To project sublattice occupations onto band occupations, the lattice potential is suddenly changed into . Without the diagonal lattice, the and directions are decoupled and atoms in () sublattice is mapped onto the 1st excited band of the () direction. Followed by the band mapping technique, atoms distribute over the () dimensional Brillouin zone with the one-to-one correspondence to sublattice occupancies (, and ).

To analyze data, we fit an empirical model function with taking , , and the background level as free parameters. Here, the basis functions , , is constructed by averaging more than 10 absorption images under the condition that almost all atoms localize on a sole sublattice.

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