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Physical Review A

Coherent densification of cold atoms in a Wannier-Stark ladder by spatially structured resonant coupling

Quentin Thommen

Published Aug 27, 2026• 0 citations

Abstract

We propose a mechanism for coherent densification in a tilted optical lattice. The starting point is a well-known property of Wannier-Stark ladders: a homogeneous resonant coupling between neighboring Wannier-Stark states restores coherent translation. When the coupling amplitude is position-dependent and has a node, the local transport velocity vanishes at the node. A single wave packet is thereby transported toward this point, while a cloud distributed around the node undergoes a reduction of its spatial extent. The aim is to provide a reversible density-redistribution tool in a lattice of fixed period, complementary to matter-wave lenses, accordion lattices, box traps, and coherent-transport protocols. We introduce a nearest-neighbor effective model in which the coupling varies linearly with the site index. This model admits an exact analytic solution via a generating function and the method of characteristics; in the continuum limit it predicts exponential contraction of distances to the node. We discuss a primary implementation based on a spatially shaped Raman or Bragg coupling, and an alternative based on temporal modulation of the lattice depth. Estimates for cesium indicate that densification times ranging from a few tens to a few hundreds of milliseconds are achievable with realistic parameters. The mechanism is not dissipative cooling, but a coherent compression step that can be halted, reversed, or followed by a controlled expansion stage.

DOI: 10.1103/tjgk-nv68

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We propose a mechanism for coherent densification in a tilted optical lattice. The starting point is a well-known property of Wannier-Stark ladders: a homogeneous resonant coupling between neighboring Wannier-Stark states restores coherent translation. When the coupling amplitude is position-dependent and has a node, the local transport velocity vanishes at the node. A single wave packet is thereby transported toward this point, while a cloud distributed around the node undergoes a reduction of its spatial extent. The aim is to provide a reversible density-redistribution tool in a lattice of fixed period, complementary to matter-wave lenses, accordion lattices, box traps, and coherent-transport protocols. We introduce a nearest-neighbor effective model in which the coupling varies linearly with the site index. This model admits an exact analytic solution via a generating function and the method of characteristics; in the continuum limit it predicts exponential contraction of distances to the node. We discuss a primary implementation based on a spatially shaped Raman or Bragg coupling, and an alternative based on temporal modulation of the lattice depth. Estimates for cesium indicate that densification times ranging from a few tens to a few hundreds of milliseconds are achievable with realistic parameters. The mechanism is not dissipative cooling, but a coherent compression step that can be halted, reversed, or followed by a controlled expansion stage.

Coherent transport of atoms trapped in an optical lattice can be controlled by microwave-induced spin flips that correlate with site-to-site hopping. We study the controllability of homogeneous one-dimensional systems of noninteracting atoms in the absence of site addressability. Given these restrictions, we construct a deterministic protocol to map an initially localized Wannier state to a wave packet that is coherently distributed over

n

sites. As an example, we consider a one dimensional quantum walk in the presence of both realistic photon scattering and inhomogeneous broadening of the microwave transition due to the optical lattice. Using composite pulses to suppress errors, fidelities of over

95%

can be achieved for a 25-step walk. We extend the protocol for state preparation to analytic solutions for arbitrary unitary maps given homogeneous systems and in the presence of time-dependent uniform forces. Such control is important for applications in quantum information processing, such as quantum computing and quantum simulations of condensed matter phenomena.

We analyze a method of compressing a cloud of cold atoms by dynamic control of a far off-resonance optical lattice. We show that by reducing the lattice spacing either continuously or in discrete steps while cooling the atoms with optical molasses large compression factors can be achieved. Particle motion in the time-dependent lattice is studied numerically using a three-dimensional semiclassical model. Two experimentally realistic models are analyzed. In the first we continuously vary the lattice beam angles to compress atoms initially in a Gaussian distributed cloud with standard deviation of 250 µm into a single site of a two-dimensional lattice of area A ∼ 35 × 35λ2, with λ the wavelength of the lattice beams. This results in an optical depth for an on-resonant probe beam >80 which is an increase by a factor of about 1800 compared to the uncompressed cloud. In the second approach we use a discrete set of lattice beam angles to decrease the spatial scale of the cloud by a factor of 500, and localize a few atoms to a single lattice site with an area A ≲ λ2.

Raman laser pulses are used to induce coherent tunneling between neighboring sites of a vertical 1D optical lattice. Such tunneling occurs when the detuning of a probe laser from the atomic transition frequency matches multiples of the Bloch frequency, allowing for a spectroscopic control of the coupling between Wannier-Stark (WS) states. In particular, we prepare coherent superpositions of WS states of adjacent sites, and investigate the coherence time of these superpositions by realizing a spatial interferometer. This scheme provides a powerful tool for coherent manipulation of external degrees of freedom of cold atoms, which is a key issue for quantum information processing.

We report on the realization of dynamical control of transport for ultra-cold 88 Sr atoms loaded in an accelerated amplitude-modulated one-dimensional (1D) optical lattice. We show that the behavior of the dynamical system can be viewed as if traveling wave packets were moving in a static lattice whose energy dispersion can be tailored at will in width, amplitude and phase. One basic control operation is a reversible switch between Wannier–Stark localization and driven transport based on coherent tunneling. Performing modulation sequences of this operation within a Loschmidt-echo scheme, we are able to reverse the atomic group velocities at once. We then apply the technique to demonstrate a novel mirror for matter waves working independently of the momentum state. We finally discuss advantages of amplitude over previously reported phase modulation techniques for applications in force measurements at micrometric scales.

We study, experimentally and with quantum Monte-Carlo wave-function simulations, the effects of dispersion and dissipation on two types of spatial wave-packets of atoms in 1D and 3D optical lattices. Breathing-mode oscillations are excited by suddenly changing the potential depth after equilibration of the atoms at the lattice sites. The subsequent breathing-mode oscillations are measured using Bragg scattering. The second type of oscillations is prepared by suddenly shifting the lattice, thus exciting coherent-state-like wave-packets. These sloshing-type oscillations manifest themselves in a periodic photon redistribution between the lattice beams. We measure the resultant intensity exchange, which is a sensitive probe for the wave-packet evolution. For both types of wave-packets we observe an initial collapse within about three to six oscillation periods, a fact which we attribute to dephasing of the components of the wave-packets caused by the anharmonicity of the lattice potential. The initial collapse is followed by revivals of the wave-packets. Dissipation not only reduces the amplitude of these revivals but also modifies their shape and time of occurrence. In shallow lattices we observe that tunneling of the atoms between neighboring potential wells influences the wave-packet oscillations.

Atomic wave packets loaded into a phase-modulated vertical optical-lattice potential exhibit a coherent delocalization dynamics arising from intraband transitions among Wannier-Stark levels. Wannier-Stark intraband transitions are here observed by monitoring the in situ wave-packet extent. By varying the modulation frequency, we find resonances at integer multiples of the Bloch frequency. The resonances show a Fourier-limited width for interrogation times up to 2 s. This can also be used to determine the gravity acceleration with ppm resolution.