A Quantum Point Contact for Neutral Atoms

arXiv:physics/9909020v1 [physics.atom-ph] 10 Sep 1999 λ ytmta om P o eta tm con- a — atoms width neutral state for ground QPC whose a striction forms that system [9]. QPC ac- electron the an in in factor quantization limiting of curacy the events, are scattering constraints between Geometric path free mean shorter phase- the be than must because channels 10 that requires than QPC’s transport coherent Electron less tip ratios [7,8]. length-to-width metallic gas have constriction electron sharp electrostatic two-dimensional an a a and include in [6] QPC surface a a contacting of realizations tal iaino lcrncnutnei utpe f2 of multiples quan- in The conductance electron conductance. of base some tization an of always multiple single is conductance a integer the is which (QPC) through contact constriction point quantum a example, For in observed yet not [5]. but transport for electron [4], predicted suppression originally band effect Bloch an dynamical a observe of to signature lengths (kHz coherence clear in scales longer time and atoms THz) slower neutral of their instead in with utilized has work as lattices Further op- well optical accelerating as [2,3]. through field, lattices atoms tical electric neutral of applied super- transport an through the electrons with of [1] were conduction lattices exam- ladders the For Wannier-Stark in and observed oscillations structures. demonstrated Bloch periodic ple, experiments in neutral- Pioneering transport and quantum electron mechanical both systems. quantum in by atom observed been dominated has is effects, particles of tion 73.40.Cg 32.80.Pj, 05.60.Gg, 03.75.-b, numbers: traps. PACS small quantum in the study gases to interacting used weakly be of can statistics constrictions such ho of discuss pair We and a reali waveguide, magnetic be system. microfabricated can aperiodic a that using geometry an constriction in practical atoms a quan- present neutral of of example of analogue an transport is atom and tum the contact point forms of quantum constriction electron units a an Such in atoms. quantized dent is λ constriction waveguide confining where osat sosre hog hneswoewdhis wavelength width Fermi whose the channels to through comparable observed is constant, dB 2 dB nti etr epeeta xeietlyrealizable experimentally an present we Letter, this In unu rnpr loocr naeidcsystems. aperiodic in occurs also transport Quantum mo- center-of-mass the which in transport, Quantum eso httecnutneo tm hog tightly a through atoms of conductance the that show We /π / 2 where , π e where , stecag fteeeto and electron the of charge the is λ dB λ dB sted rgi aeegho h inci- the of wavelength Broglie de the is eateto hsc,HradUiest,Cmrde Mas Cambridge, University, Harvard Physics, of Department sted rgi aeegho the of wavelength Broglie de the is unu on otc o eta atoms neutral for contact point quantum A .H Thywissen, H. J. b o λ scmaal to comparable is F Experimen- . h sPlanck’s is ∗ .M Westervelt, M. R. e Jl 9 1999) 19, (July ℓ 2 mfp zed /h w . , 1 h antcfil at field magnetic the rse ihahligfield holding a with pressed hw nFg ;hwvr hyaefe omv nthe in move to free are in magnitude they however, field 1; z in Fig. minimum in the shown near confined versely ormagneton, Bohr amncoclao ihcascloclainfrequency oscillation classical with simple oscillator two-dimensional harmonic symmetric cylindrically a forms rection a potential man arctdao pis[92] eta tmwt a with atom neutral A number quantum micro- [19,20]. magnetic in advances optics recent atom surface- exploits a which atoms fabricated 1) of for Fig. example (see waveguide the [18] on electromagnet focus four-wire will mounted we work, this In 1–8 hs ofieetmyb togeog omeet to constraint enough strong the be may confinement whose [16–18] [15]. liquid and Luttinger [14] a Fermionization of as formation such the theoretical effects novel aroused in quan- already interest The has to restricted spaces particles. energetically sub-dimensional neutral atoms neutral for quan- of absent statistics electron is tum an [13] of dot energetics en- tum the charging different Coulombic dominates fundamentally the that since is ergy electrons, atoms of such that for of from physics dot” The the “quantum in constrictions. a confined two QPC’s atoms the two of between energetics of trap in- the series For on a depend through gases. would transmission interacting weakly the of stance, physics ensembles the small exploring of for opportunities inherent excellent provide of is scales length and energy interest. new at quan- conductance tization of observation ef- the Furthermore, finite-temperature by fects. only conduc- limited transi- of accuracy quantization nonadiabatic for tance allowing inter-mode [12], minimized and are tions reflection effects deleterious as because such interesting is regime new This a osrcinwt eght-it ai of ratio length-to-width a cylindri- with a constriction through cal quantization conductance of sibility blt fnal oohoai atrwvswt de with waves matter wavelengths monochromatic Broglie nearly of ability fitraoi cteig( scattering inter-atomic of P o tm sqatzdi nee utpe of multiples integer in quantized is through atoms below, λ for defined as QPC “conductance”, a The atoms. dB 2 ieto,prle otewrs o-daai changes Non-adiabatic wires. the to parallel direction, eety eea aeudshv enproposed been have waveguides several Recently, would it realized, were atoms neutral for QPC a If m /π ertefil iiu a eepnnilysup- exponentially be can minimum field the near h bec ffoe-ndsre,telwrate low the disorder, frozen-in of absence The . † z n .Prentiss M. and 2] ertegiecne,tepotential the center, guide the Near [21]. b o U ahsts018 USA 02138, sachusetts . ( g r = ) λ steLn´ atr and factor, Landé g the is dB λ dB r / tm with Atoms . 2 µ ∼ π B ℓ gm o ogtdnlyfe atoms. free longitudinally for 0n 1,1 ffrtepos- the offer [10,11] nm 50 m mfp B h | xeine ierZee- linear a experiences B ple nteaildi- axial the in applied ∼ ( r ) n h avail- the and m), 1 ) | where , > m r trans- are 0 µ B B ∼ sthe is ( r 10 is ) 5 . 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 Schrödinger 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. 11 has a monochro- tum transport requires fully quantum-mechanical calcu- maticity E /∆E 50 for atoms traveling at 30 cm/s, or I ≈ lations, even for the longitudinal degree of freedom within λdB = 50 nm. To meet the constraint σ α [22], such a the waveguide. The Hamiltonian for an atom near the source can be reflected off of a diffuser [23],≫ such as the axis of the four-wire waveguide described by Eq. (1) is de-magnetized magnetic tape described in Ref. 24. As- suming the spatial density of atoms is preserved during 2 pˆ 1 2 2 Hˆ = + Mω2e−2ˆz /ℓ (ˆx2 +ˆy2), (2) propagation [25], such a source can have a flux density of QP C o 10 −2 −1 2M 2 Jo 2 10 cm s . The≈ quantized× conductance for atoms is shown in Fig. whereˆdenotes an operator, ωo is the transverse oscillation frequency at z = 0, and we have assumed the 3 and is the central result of this Letter. Conductance 2 ~ parallel-wire scaling of field curvature, S(z)−4. Since Φ/(λdB/π) is shown as a function of mean energy EI / ωo and energy spread ∆E/~ω . In the limits ~ω ∆E a direct numerical integration approach is computation- o o ≫ and ℓ bo, one can show analytically that the con- ally prohibitive, we developed a model that neglects ≫ 2 non-adiabatic propagation at the entrance and exit of ductance is Φ = (λdB /π)N, where N is the number of the channel. The waveguide potential is truncated at modes above cutoff at z = 0. The “staircase” of Φ versus EI /~ωo is a vivid example of quantum transport, as

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. 3 we have assumed asΓ= F/∆U, where F is the transmitted atom flux, just 3 5 ℓ = 10 So 10 bo; in the particular case of the Na as the electron conductance G is the ratio of electron flux source discussed≈ above, and assuming σ = 25 mrad, the (current) to potential difference (voltage). One can show 2 first step (Φ = λdB/π) corresponds to a transmitted flux that of 500 atoms s−1, which is a sufficient flux to measure ∼ N via photoionization. Γ= , (4) We can understand several features shown in Fig. 3 h by considering the adiabatic motion of atoms within assuming ∆U < kBT EF , where T is the temperature the waveguide. As atom waves propagate though the of the Fermi ensembles≪ and N is the number of modes constricting waveguide, modes with transverse oscillator above cutoff. 2 2 states (nx,ny) such that ~ωo(nx+ny+1) EI & 2M~ /ℓ Two QPC’s can form a trap between them, just as a − will contribute negligible evanescent transmission and pair of electron QPC’s form a quantum dot [13]. For adiabatically reflect before z = 0. Steps occur when the ~ωo > 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 [13]. 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 [17] 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 < kB T EF , and observation of such a small trap with two or more ≪ where ∆V is the applied voltage, T is the temperature QPC “leads” is a powerful configuration for atom optics, of the electron gas, and EF is the Fermi energy, then because loading a small, isolated trap is problematic, and the current that flows between them will be carried by because spectroscopy near the substrate is complicated electrons with an energy spread kB T and a mean energy by light scattering and inaccessibility. EF . For a cold atom beam, the particle flow is driven In conclusion, we show how an electromagnet wave- by kinetics instead of energetics. The incident kinetic guide could be used to create a quantum point contact energy EI corresponds to EF , and the energy spread for cold neutral atoms. This device is an example of a ∆E EI corresponds to kBT . The quantum of con- new physical regime, quantum transport within micro- ≪ ductance for both systems can be formulated in terms of fabricated atom optics. particle wavelength: the classical conductance of a point The authors thank A. Barnett, N. Dekker, M. Drndić, contact of area A connecting two three-dimensional gases E. W. Hagley, K. S. Johnson, M. Olshanii, W. D. Phillips, 2 2 2~ of electrons is G = (e kF A)/(4π ) [26], such that if M. G. Raizen, and G. Zabow for useful discussions. This 2 ~ 2 G = Ne /π , the effective area is A = NλF /π. work was supported in part by NSF Grant Nos. PHY- In order to determine the accuracy of conductance 9732449 and PHY-9876929, and by MRSEC Grant No. quantization, three measurements (F , Jof(0, 0), and DMR-9809363. J. T. acknowledges support from the λdB) are necessary for the atom QPC instead of two mea- Fannie and John Hertz Foundation. surements (current and ∆V ) for the electron QPC. The reduced number of degrees of freedom for electrons re- sults from their Fermi degeneracy: the net current is carried by electrons whose incident flux density Jo and wavelength λF are functions of EF and ∆V . As a thought experiment, the simplicity of an externally tuned J and o ∗ email:joseph [email protected] λ could also be extended to neutral atoms, if two de- † dB Also in the Division of Engineering and Applied Sciences, generate ensembles of Fermionic atoms were connected Harvard University. by a QPC and given a potential difference ∆U, such as could be induced by a uniform magnetic field applied to

3 [1] C. Waschke et al., Phys. Rev. Lett. 70, 3319 (1993); For a review, see E. E. Mendez and G. Bastard, Phys. Today 46, No. 6, 34 (1993) [2] P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37, 95 (1996), and references therein. 1 [3] M. B. Dahan et al., Phys. Rev. Lett. 76, 4508 (1996); S. R. Wilkinson et al., Phys. Rev. Lett. 76, 4512 (1996). [4] K. W. Madison et al., Phys. Rev. Lett. 81, 5093 (1998). 0 [5] D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 substrate (1986). -1 0 1 [6] J. K. Gimzewski and R. Möller, Phys. Rev. B 36, 1284 (1987); J. M. Krans et al, Nature 375, 767 (1995). [7] B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988); FIG. 1. Magnetic field contours above D. A. Wharam et al., J. Phys. C: Solid State Phys. 21, a micro-electromagnet waveguide. Four parallel wires, sep- L209 (1988). arated by a distance S and with anti-parallel current flow [8] C. W. J. Beenakker and H. van Houten, Solid State (marked “·” for +z and “×” for −z), are mounted on a sub- Physics 44, 1 (1991); and references therein. strate (crosshatched), which serves both to support the wires [9] See for instance D. P. E. Smith, Science 269, 371 (1995); mechanically and to dissipate the heat produced. A poten- H. van Houten and C. Beenakker, Physics Today 49, No. tial minimum is formed above the wires and can be used to 7, 22 (1996); S. Frank, et al., Science 280, 1744 (1999). guide atoms in the out-of-plane direction z. Twelve contours, [10] M.-O. Mewes et al., Phys. Rev. Lett. 78, 582 (1997); I. equally spaced by Bo/4, are shown, where Bo = µoI/2πS and Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. ±I (±2I) is the current in the inner (outer) wire pair. 82, 3008 (1999). [11] E. W. Hagley et al., Science 283, 1706 (1999). [12] A. Yacoby and Y. Imry, Phys. Rev. B 41, 5341 (1990). [13] L. L. Sohn et al. (eds.), Mesoscopic Electron Transport, NATO ASI Series E345 (Kluwer, Dordrecht, 1997), and references therein. [14] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [15] H. Monien, M. Linn, and N. Elstner, Phys. Rev. A 58, R3395 (1998). [16] E. A. Hinds, M. G. Boshier, and I. G. Hughes, Phys. Rev. Lett. 80, 645 (1998). [17] J. Schmiedmayer, Eur. Phys. J. D 4, 57 (1998). [18] J. H. Thywissen et al., Eur. Phys. J. D 7, No. 3 (1999), to appear. [19] J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52, 4004 (1995). [20] M. Drndić et al., Appl. Phys. Lett. 72, 2906 (1998). [21] For all calculations in this work, we apply a strong FIG. 2. (a) Top-down view of a waveguide wire geometry enough Bh that the spin-flip loss rate of the transverse −7 which creates a quantum point contact for atoms. The direc- ground state is . 10 ω. See C. V. Sukumar and D. M. tion of current flow is indicated on the wires (solid lines). A Brink, Phys. Rev. A 56, 2451 (1997). constriction with ℓ = 100So is shown. (b) Level spacing ~ω [22] If a source with σ . α were used, then Φ would still (in µK) of transverse oscillator states versus axial distance z. have steps at the same energies, but the plateaus in Fig. Points (⋄) are based on numerical calculations of the field cur- 3 would no longer be flat. vature at each z above the wire configuration shown in (a); [23] Similarly, a diffuser was necessary for the laser light the line is based on the parallel-wire scaling S(z)−2. Both source used to observe the quantization of photon cross- calculations assume Na atoms in the |F = 1,mF = +1i state, et al. section; see E. A. Montie , Nature 350, 594 (1991). So = 1 µm, I = 200 mA, and Bh = 35 G. [24] T. M. Roach et al., Phys. Rev. Lett. 75, 629 (1995). [25] K. Szymaniec, H. J. Davies, and C. S. Adams, Europhys. Lett. 45, 450 (1999). [26] Yu. V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov. Phys. JETP 21, 655 (1965).]

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, and EI 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.