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A Quantum Point Contact for Neutral Atoms

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A Quantum Point Contact for Neutral Atoms View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server 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 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 [?] 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 mi- crofabricated 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 Land´e 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 ! = 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 and its normal derivative @ =@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 <zT). We found that, for | F | | cillation frequency of ! =2π 3:3 MHz and a root mean ` & 10bo, a two-dimensional version of the model could × squared (RMS) ground state width b = ~=2M! =8:1 reproduce the transmissions and spatial output distribu- nm in a waveguide with S =1µmandI=0:1A.The tions of a two dimensional split-operator FFT integration p 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 [?]. 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. ??a 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 2 2 1=4 ∼ ln(`=bo)bo=` , we can integrate over all solid angles z2 S(z)=S exp ; (1) and define a “conductance” Φ dependent only on param- o 2`2 eters of the constriction and the kinetic energy EI of the incident atoms: where So is the spacing at z =0,and`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) 1 2 4 curvature scale as S(z), S(z)− , S(z)− ,andS(z)− ,re- 1 spectively. For ` = 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 3`, allowing for a well-defined dent on-axis brightness (in cm− s− ). The transverse | | . waveguide potential over a factor of more than 103 in momentum distribution f(kx;ky) is defined as follows: level spacing (see Fig. ??b). 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,whereC =~Jo=2πEI, − o 2 2 2 1=2 as run to contact pads (necessary to connect the wires to ~kz = 2MEI ~ (kx +ky) ,andf(kx;ky) is nor- a power supply) far enough from the channel ( `)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 isotropic gas (f = 1) through a hole in a thin wall, Φ 2 R 10` 10`,orabout1mm ,forSo =1µmand`= 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. ??a. 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 )witha time-dependent Schr¨odinger equation in three spatial di- RMS spread ∆E, centered about EI . As an example, mensions. It is important to note that the nature of quan- the 23Na source described in Ref.
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