A quantum point contact for neutral atoms J. H. Thywissen,∗ R. M. Westervelt,† and M. Prentiss Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (July 19, 1999) atoms. The “conductance”, as defined below, through We show that the conductance of atoms through a tightly a QPC for atoms is quantized in integer multiples of confining waveguide constriction is quantized in units of 2 2 λdB/π. The absence of frozen-in disorder, the low rate λdB/π, where λdB is the de Broglie wavelength of the inci- of inter-atomic scattering (ℓ 1 m), and the avail- dent atoms. Such a constriction forms the atom analogue of mfp ∼ an electron quantum point contact and is an example of quan- ability of nearly monochromatic matter waves with de Broglie wavelengths λdB 50 nm [10,11] offer the pos- tum transport of neutral atoms in an aperiodic system. We ∼ present a practical constriction geometry that can be realized sibility of conductance quantization through a cylindri- using a microfabricated magnetic waveguide, and discuss how cal constriction with a length-to-width ratio of 105. a pair of such constrictions can be used to study the quantum This new regime is interesting because deleterious∼ effects statistics of weakly interacting gases in small traps. such as reflection and inter-mode nonadiabatic transi- tions are minimized [12], allowing for accuracy of conduc- PACS numbers: 03.75.-b, 05.60.Gg, 32.80.Pj, 73.40.Cg tance quantization limited only by finite-temperature ef- fects. Furthermore, the observation of conductance quan- Quantum transport, in which the center-of-mass mo- tization at new energy and length scales is of inherent tion of particles is dominated by quantum mechanical interest. effects, has been observed in both electron and neutral- If a QPC for neutral atoms were realized, it would atom systems. Pioneering experiments demonstrated provide excellent opportunities for exploring the physics quantum transport in periodic structures. For exam- of small ensembles of weakly interacting gases. For in- ple, Bloch oscillations and Wannier-Stark ladders were stance, the transmission through a series of two QPC’s observed in the conduction of electrons through super- would depend on the energetics of atoms confined in the lattices [1] with an applied electric field, as well as in trap between the two constrictions. The physics of such the transport of neutral atoms through accelerating op- a “quantum dot” for atoms is fundamentally different tical lattices [2,3]. Further work with neutral atoms in from that of electrons, since the Coulombic charging en- optical lattices has utilized their slower time scales (kHz ergy that dominates the energetics of an electron quan- instead of THz) and longer coherence lengths to observe a tum dot [13] is absent for neutral particles. The quan- clear signature of dynamical Bloch band suppression [4], tum statistics of neutral atoms energetically restricted to an effect originally predicted for but not yet observed in sub-dimensional spaces has already aroused theoretical electron transport [5]. interest in novel effects such as Fermionization [14] and Quantum transport also occurs in aperiodic systems. the formation of a Luttinger liquid [15]. For example, a quantum point contact (QPC) is a single Recently, several waveguides have been proposed constriction through which the conductance is always an [16–18] whose confinement may be strong enough to meet integer multiple of some base conductance. The quan- the constraint bo . λdB/2π for longitudinally free atoms. tization of electron conductance in multiples of 2e2/h, In this work, we will focus on the example of a surface- where e is the charge of the electron and h is Planck’s mounted four-wire electromagnet waveguide for atoms constant, is observed through channels whose width is [18] (see Fig. 1) which exploits recent advances in micro- fabricated atom optics [19,20]. A neutral atom with a comparable to the Fermi wavelength λF . Experimen- tal realizations of a QPC include a sharp metallic tip magnetic quantum number m experiences a linear Zee- man potential U(r) = µ gm B(r) , where µ is the contacting a surface [6] and an electrostatic constriction B | | B in a two-dimensional electron gas [7,8]. Electron QPC’s Bohr magneton, g is the Land´eg factor, and B(r) is arXiv:physics/9909020v1 [physics.atom-ph] 10 Sep 1999 have length-to-width ratios less than 10 because phase- the magnetic field at r. Atoms with m > 0 are trans- coherent transport requires that channels must be shorter versely confined near the minimum in field magnitude shown in Fig. 1; however, they are free to move in the than the mean free path between scattering events, ℓmfp. Geometric constraints are the limiting factor in the ac- z direction, parallel to the wires. Non-adiabatic changes curacy of quantization in an electron QPC [9]. in m near the field minimum can be exponentially sup- In this Letter, we present an experimentally realizable pressed with a holding field Bh applied in the axial di- system that forms a QPC for neutral atoms — a con- rection z [21]. Near the guide center, the potential forms a cylindrically symmetric two-dimensional simple striction whose ground state width bo is comparable to harmonic oscillator with classical oscillation frequency λdB/2π, where λdB is the de Broglie wavelength of the 1 2 2 1/2 ω = µ gm(2µ I/πS ) /MB , where µ is the per- z = zT , the planes between which atoms can propa- B 0 h 0 ± meability of free space, I is the inner wire current, 2I gate adiabatically in the waveguide, and the wavefunc- is the outer wire current, S is the center-to-center wire tion amplitude ψ and its normal derivative ∂ψ/∂z are 23 spacing, and M is the mass of the atoms. Sodium ( Na) matched between plane-wave states ( z > zT ) and the | | in the F =1,m = +1 > state would have a classical os- modes of the waveguide ( z < zT ). We found that, for | F | | cillation frequency of ω =2π 3.3 MHz and a root mean ℓ & 10bo, a two-dimensional version of the model could squared (RMS) ground state× width b = ~/2Mω =8.1 reproduce the transmissions and spatial output distribu- nm in a waveguide with S = 1 µm and pI = 0.1 A. The tions of a two dimensional split-operator FFT integration ˆ fabrication of electromagnet waveguides of this size scale of HQP C (ˆx, zˆ) with the full waveguide potential. This and current capacity has been demonstrated [20]. agreement gave us confidence in our three-dimensional A constriction in the waveguide potential can be cre- model of atom propagation through the constriction. ated by contracting the spacing between the wires of the The cross-section for an incident atomic plane wave waveguide. The constriction strength can be tuned dy- to be transmitted through a constriction is dependent namically by changing the current in the wires. Fig. 2a on the plane-wave energy EI and incident angle. How- shows a top-down view of a constriction whose wire spac- ever, if the RMS angular spread of incident plane waves ing S(z) is smoothly varied as σ is much greater than the RMS acceptance angle α 1/4 ∼ ln(ℓ/b )b2/ℓ2 , we can integrate over all solid angles 2 o o z and define a “conductance” Φ dependent only on param- S(z)= So exp 2 , (1) 2ℓ eters of the constriction and the kinetic energy EI of the incident atoms: where So is the spacing at z = 0, and ℓ is the character- istic channel length. Assuming the wires are nearly par- F Φ(EI )= , (3) allel, the guide width, depth, oscillation frequency, and Jof(0, 0) curvature scale as S(z), S(z)−1, S(z)−2, and S(z)−4, re- −1 spectively. For ℓ = 100So, field calculations above this where F is the total flux of atoms (in s ) transmit- curved-wire geometry show that the parallel-wire approx- ted though the constriction and Jof(0, 0) is the inci- 2 1 imation is valid for z . 3ℓ, allowing for a well-defined dent on-axis brightness (in cm− s− ). The transverse | | 3 waveguide potential over a factor of more than 10 in momentum distribution f(kx, ky) is defined as follows: level spacing (see Fig. 2b). Our particular choice of S(z) in the plane wave basis k , we consider a density {| i} is somewhat arbitrary but prescribes one way in which distribution of atoms on the energy shell a(k)dk = o o ~ wires can form a smooth, constricting waveguide as well (C/kz)δ [kz kz ] f(kx, ky)dk, where C = Jo/2πEI , − ~ o ~2 2 2 1/2 as run to contact pads (necessary to connect the wires to kz = 2MEI (kx + ky) , and f(kx, ky) is nor- a power supply) far enough from the channel ( ℓ) that malized such that− the incident flux density J = ≫ o their geometry is unimportant. The total “footprint” of dka(k)~kz/M. When applied to the diffusion of an this device (not including contact pads) is approximately Risotropic gas (f = 1) through a hole in a thin wall, Φ 2 10ℓ 10ℓ, or about 1 mm , for So =1 µm and ℓ = 100So. is equal to the area of the hole; for a channel with a × Atoms approach the constriction from the −z direc- small acceptance angle, α σ, Φ is the effective area tion, as shown in Fig. 2a. We calculate the propagation at the narrowest cross-section≪ of the channel. We con- of the atom waves through the constriction by solving the sider a distribution of incident energies g(EI ) with a time-dependent Schr¨odinger equation in three spatial di- RMS spread ∆E, centered about EI .