A Quantum Point Contact for Neutral Atoms

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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 λ2 /π. The absence of frozen-in disorder, the low rate λ2 /π,whereλ is the de Broglie wavelength of the inci- dB dB dB of inter-atomic scattering (` 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 λ 50 nm [?,?]offerthepos- tum transport of neutral atoms in an aperiodic system. We dB ∼ 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 [?], 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 effects. 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 [?] 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 [?,?]. 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 [?] is absent for neutral particles. The quan- clear signature of dynamical Bloch band suppression [?], 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 [?]. interest in novel effects such as Fermionization [?]and Quantum transport also occurs in aperiodic systems. the formation of a Luttinger liquid [?]. For example, a quantum point contact (QPC) is a single Recently, several waveguides have been proposed constriction through which the conductance is always an [?,?,?] 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 [?] (see Fig. ??) which exploits recent advances in microfabricated atom optics [?,?]. A neutral atom with a comparable to the Fermi wavelength λF . Experimental realizations of a QPC include a sharp metallic tip con- magnetic quantum number m experiences a linear Zee- man potential U(r)=µ gm B(r) ,whereµ is the tacting a surface [?] and an electrostatic constriction B | | B in a two-dimensional electron gas [?,?]. Electron QPC’s Bohr magneton, g is the Landé g factor, and B(r)is 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 showninFig.??; however, they are free to move in than the mean free path between scattering events, `mfp. Geometric constraints are the limiting factor in the ac- the z direction, parallel to the wires. Non-adiabatic curacy of quantization in an electron QPC [?]. changes in m near the field minimum can be exponen- In this Letter, we present an experimentally realizable tially suppressed with a holding field Bh applied in the system that forms a QPC for neutral atoms — a con- axial direction z [?]. 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,whereλdB is the de Broglie wavelength of the

1 2 2 1/2 ω = µ gm(2µ I/πS ) /M B ,whereµ 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)andthe | | in the F =1,m =+1>state would have a classical os- modes of the waveguide ( z

2 it demonstrates the quantum mechanical nature of the one reservoir. We can redefine neutral atom conductance center-of-mass motion. For all of Fig. ?? we have as- as Γ = F/∆U,whereFis the transmitted atom flux, just 3 5 sumed ` =10So 10 bo; in the particular case of the as the electron conductance G is the ratio of electron flux Na source discussed≈ above, and assuming σ = 25 mrad, (current) to potential difference (voltage). One can show 2 the first step (Φ = λdB/π) corresponds to a transmit- that 1 ted flux of 500 atoms s− , which is a sufficient flux to ∼ N measure via photoionization. Γ= , (4) We can understand several features shown in Fig. ?? h by considering the adiabatic motion of atoms within assuming ∆U EI , all modes of the QPC are below cutoff and number of allowed propagating modes changes: the mth evanescent transmission is dominated by tunneling of step appears at ~ωo = EI /m. Note that this condition atoms occupying the (0, 0) mode. While the quantum can also be written bo = √mλdB/2π, demonstrating that dot between them is energetically isolated, atoms can still transverse confinement on the order of λdB/2π is essen- tunnel into and out of the dot. For cold Fermionic atoms, tial to seeing conductance steps in a QPC. Since low-lying the Pauli exclusion principle would enable a single atom modes occupy a circularly symmetric part of the poten- to block transmission through the trap, just as the charg- tial, the mth step involves m degenerate modes and is m ing energy of a single electron can block transmission in times as high as the first step. The large aspect ratio of electron quantum dots; such a blockade might be used the atom QPC allows for a sufficiently gentle constric- to make a single-atom transistor. In such a single-atom tion to suppress partial reflection at the entrance to the blockade regime, quantum dots can also show a suppres- guide, such that the sharpness of steps and flatness be- sion of shot noise below the Poissonian level [?]. Note tween them is limited only by the spread in incident atom that spectroscopic measurement of neutral atom traps energies. with resolvable energy levels has been suggested previ- It is interesting to compare the electron and atom QPC ously [?] in analogy to spectroscopic measurement of systems. If contact is made between two Fermi seas electron quantum dots. We emphasize that the loading whose chemical potentials differ by e∆V

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4 6 5 4 3 2 0.3 1 0.2 3.0 0.1 2.0 1.0

FIG. 3. Conductance Φ through a quantum point contact, as a function of average incident energy EI and energy spread ∆E. Φ is plotted in terms of the quantized unit of conduc- 2 tance, λdB /π,andEI and kBT are plotted in terms of ~ωo, the level spacing at the narrowest point of the constriction. The lowest ∆E shown, 0.02~ωo, corresponds to the example for 23Na discussed in the text.