Quantum Point Contacts for Neutral Atoms

VOLUME 83, NUMBER 19 PHYSICAL REVIEW LETTERS 8NOVEMBER 1999

J. H. Thywissen,* R. M. Westervelt,† and M. Prentiss Department of Physics, Harvard University, Cambridge, Massachusetts 02138 (Received 19 July 1999) We show that the conductance of atoms through a tightly conﬁning waveguide constriction is 2 quantized in units of ldB͞p, where ldB is the de Broglie wavelength of the incident atoms. Such a constriction forms the atom analog of an electron quantum point contact and is an example of quantum transport of neutral atoms in an aperiodic system. We present a practical constriction geometry that can be realized using a microfabricated magnetic waveguide, and discuss how a pair of such constrictions can be used to study the quantum statistics of weakly interacting gases in small traps.

PACS numbers: 03.75.–b, 05.60.Gg, 32.80.Pj, 73.40.Cg

Quantum transport, in which the center-of-mass motion to-width ratio of ϳ105. This new regime is interesting of particles is dominated by quantum mechanical effects, because deleterious effects such as reflection and inter- has been observed in both electron and neutral-atom mode nonadiabatic transitions are minimized [12], allow- systems. Pioneering experiments demonstrated quantum ing for accuracy of conductance quantization limited only transport in periodic structures. For example, Bloch by finite-temperature effects. Furthermore, the observa- oscillations and Wannier-Stark ladders were observed in tion of conductance quantization at new energy and length the conduction of electrons through superlattices [1] with scales is of inherent interest. an applied electric field, as well as in the transport of If a QPC for neutral atoms were realized, it would neutral atoms through accelerating optical lattices [2,3]. provide excellent opportunities for exploring the physics Further work with neutral atoms in optical lattices has of small ensembles of weakly interacting gases. For utilized their slower time scales (kHz instead of THz) and instance, the transmission through a series of two QPC’s longer coherence lengths to observe a clear signature of would depend on the energetics of atoms confined in the dynamical Bloch band suppression [4], an effect originally trap between the two constrictions. The physics of such a predicted for but not yet observed in electron transport [5]. “quantum dot” for atoms is fundamentally different from Quantum transport also occurs in aperiodic systems. that of electrons, since the Coulombic charging energy For example, a quantum point contact (QPC) is a single that dominates the energetics of an electron quantum constriction through which the conductance is always an dot [13] is absent for neutral particles. The quantum integer multiple of some base conductance. The quan- statistics of neutral atoms energetically restricted to sub- tization of electron conductance in multiples of 2e2͞h, three-dimensional spaces has already aroused theoretical where e is the charge of the electron and h is Planck’s interest in novel effects such as fermionization [14] and constant, is observed through channels whose width is the formation of a Luttinger liquid [15]. comparable to the Fermi wavelength lF. Experimental Recently, several waveguides have been proposed realizations of a QPC include a sharp metallic tip con- [16–18] whose confinement may be strong enough to tacting a surface [6] and an electrostatic constriction in a meet the constraint b0 &ldB͞2p for longitudinally free two-dimensional electron gas [7,8]. Electron QPC’s have atoms. In this Letter, we will focus on the example of a length-to-width ratios less than 10 because phase-coherent surface-mounted four-wire electromagnet waveguide for transport requires that channels must be shorter than the atoms [18] (see Fig. 1) which exploits recent advances mean-free path between scattering events, ᐉmfp. Geomet- in microfabricated atom optics [19,20]. A neutral atom ric constraints are the limiting factor in the accuracy of with a magnetic quantum number m experiences a linear quantization in an electron QPC [9]. Zeeman potential U͑r͒ ෇ mBgmjB͑r͒j, where mB is In this Letter, we present an experimentally realizable the Bohr magneton, g is the Landé g factor, and B͑r͒ system that forms a QPC for neutral atoms—a constriction is the magnetic field at r. Atoms with gm . 0 are whose ground state width b0 is comparable to ldB͞2p, transversely confined near the minimum in field mag- where ldB is the de Broglie wavelength of the atoms. The nitude shown in Fig. 1; however, they are free to move “conductance,” as defined below, through a QPC for atoms in the z direction, parallel to the wires. Nonadiabatic 2 is quantized in integer multiples of ldB͞p. The absence changes in m near the field minimum can be exponentially of frozen-in disorder, the low rate of interatomic scattering suppressed with a holding field Bh applied in the axial (ᐉmfp ϳ 1 m), and the availability of nearly monochro- direction z [21]. Near the guide center, the potential matic matter waves with de Broglie wavelengths ldB ϳ forms a cylindrically symmetric two-dimensional simple 50 nm [10,11] offer the possibility of conductance quan- harmonic oscillator with classical oscillation frequency 2 2 1͞2 tization through a cylindrical constriction with a length- v ෇ ͓mBgm͑2m0I͞pS ͒ ͞MBh͔ , where m0 is the

FIG. 1. Magnetic field contours above a microelectromagnet waveguide. Four parallel wires, separated by a distance S and with antiparallel current flow (marked “?” for 1z and “3” for 2z) are mounted on a substrate (crosshatched), which serves FIG. 2. (a) Top-down view of a waveguide wire geometry both to support the wires mechanically and to dissipate the heat which creates a quantum point contact for atoms. The direction produced. A potential minimum is formed above the wires of current flow is indicated on the wires (solid lines). A and can be used to guide atoms in the out-of-plane direction constriction with ᐉ ෇ 100S0 is shown. (b) Level spacing h¯v z. Twelve contours, equally spaced by B0͞4, are shown, where (in mK) of transverse oscillator states versus axial distance B0 ෇ m0I͞2pS and 6I (62I) is the current in the inner (outer) z. Points (¶) are based on numerical calculations of the field wire pair. curvature at each z above the wire configuration shown in (a); the line is based on the parallel-wire scaling S͑z͒22. Both calculations assume Na atoms in the jF ෇ 1, mF ෇ 11͘ state, ෇ ෇ ෇ permeability of free space, I is the inner wire current, S0 1 mm, I 200 mA, and Bh 35 G. 2I is the outer wire current, S is the center-to-center wire spacing, and M is the mass of the atoms. Sodium 10ᐉ 3 10ᐉ, or about 1 mm2, for S ෇ 1 mm and ᐉ ෇ (23Na) in the jF ෇ 1, m ෇ 11͘ state would have a 0 F 100S . classical oscillation frequency of v ෇ 2p33.3 MHz 0 Atoms approach the constriction from the 2z direc- andp a root mean squared (rms) ground state width tion, as shown in Fig. 2a. We calculate the propagation b ෇ h¯͞2Mv ෇ 8.1 nm in a waveguide with S ෇ 1 mm of the atom waves through the constriction by solving and I ෇ 0.1 A. The fabrication of electromagnet wave- the time-dependent Schrödinger equation in three spatial guides of this size scale and current capacity has been dimensions. It is important to note that the nature of quan- demonstrated [20]. tum transport requires fully quantum-mechanical calcula- A constriction in the waveguide potential can be tions, even for the longitudinal degree of freedom within created by contracting the spacing between the wires the waveguide. The Hamiltonian for an atom near the axis of the waveguide. The constriction strength can be of the four-wire waveguide described by Eq. (1) is tuned dynamically by changing the current in the wires. 2 Figure 2a shows a top-down view of a constriction whose pˆ 1 2 22ˆz2͞ᐉ2 2 2 Hˆ QP C ෇ 1 Mv e ͑xˆ 1 yˆ ͒ , (2) wire spacing S͑z͒ is smoothly varied as 2M 2 0 ∑ ∏ z2 where ^ denotes an operator, v0 is the transverse os- S͑z͒ ෇ S exp , (1) cillation frequency at z ෇ 0, and we have assumed the 0 ᐉ2 2 parallel-wire scaling of field curvature, S͑z͒24. Since a where S0 is the spacing at z ෇ 0 and ᐉ is the characteristic direct numerical integration approach is computationally channel length. Assuming the wires are nearly paral- prohibitive, we developed a model that neglects nona- lel, the guide width, depth, oscillation frequency, and diabatic propagation at the entrance and the exit of the curvature scale as S͑z͒, S͑z͒21, S͑z͒22, and S͑z͒24, channel. The waveguide potential is truncated at z ෇ respectively. For ᐉ ෇ 100S0, field calculations above this 6zT , the planes between which atoms can propagate adia- curved-wire geometry show that the parallel-wire approxi- batically in the waveguide, and the wave function am- mation is valid for jzj & 3ᐉ, allowing for a well-defined plitude c and its normal derivative ≠c͞≠z are matched 3 waveguide potential over a factor of more than 10 in between plane-wave states (jzj . zT ) and the modes of level spacing (see Fig. 2b). Our particular choice of S͑z͒ the waveguide (jzj , zT ). We found that, for ᐉ * 10b0, is somewhat arbitrary but prescribes one way in which a two-dimensional version of the model could reproduce wires can form a smooth, constricting waveguide as well the transmissions and spatial output distributions of a two- as run to contact pads (necessary to connect the wires to dimensional split-operator fast Fourier transform integra- a power supply) far enough from the channel (¿ᐉ) that tion of Hˆ QPC͑xˆ,ˆz͒ with the full waveguide potential. This their geometry is unimportant. The total “footprint” of agreement gave us confidence in our three-dimensional this device (not including contact pads) is approximately model of atom propagation through the constriction.

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The cross section for an incident atomic plane wave to be transmitted through a constriction is dependent on the plane-wave energy EI and incident angle. How- ever, if the rms angular spread of incident plane waves 6 s is much greater than the rms acceptance angle a ϳ 2 2 1͞4 5 ͓ln͑ᐉ͞b0͒b0 ͞ᐉ ͔ , we can integrate over all solid angles and define a conductance F dependent only on parameters 4 3 of the constriction and the kinetic energy EI of the incident atoms: 2 0.3 1 0.2 ͑ ͒ ෇ F F EI , (3) 0.1 J0f͑0, 0͒ 3.0 2.0 where F is the total flux of atoms (in s21) transmitted 1.0 ͑ ͒ through the constriction and J0f 0, 0 is the incident on- FIG. 3. Conductance F through a quantum point contact, as 2 21 axis brightness (in cm s ). The transverse momen- a function of average incident energy EI and energy spread tum distribution f͑kx, ky͒ is defined as follows: in the DE. F is plotted in terms of the quantized unit of conductance, 2 plane-wave basis ͕jk͖͘, we consider a density distribution ldB͞p, and EI and kBT are plotted in terms of h¯v0, the ͑ ͒ ෇ ͑ ͞ 0͒ ͓ level spacing at the narrowest point of the constriction. The of atoms on the energy shell a k dk C kz d kz 2 k0͔f͑k , k ͒dk, where C ෇ hJ¯ ͞2pE , hk¯ 0 ෇ ͓2ME 2 lowest DE shown, 0.02h ¯v0, corresponds to the example for z x y 0 I z I 23Na discussed in the text. 2͑ 2 2͔͒1͞2 ͑ ͒ h¯ kx 1 ky , and f kx, ky Ris normalized such that the incident flux density J0 ෇ dk a͑k͒hk¯ z͞M. When applied to the diffusion of an isotropic gas (f ෇ 1) through tribute negligible evanescent transmission and adiabati- a hole in a thin wall, F is equal to the area of the hole; cally reflect before z ෇ 0. Steps occur when the number for a channel with a small acceptance angle, a ø s, F of allowed propagating modes changes: the mth step ap- ෇ ͞ is the effective area at the narrowest cross section of the pears at h¯v0 pEI m. Note that this condition can also be channel. We consider a distribution of incident energies written b0 ෇ m ldB͞2p, demonstrating that transverse ͑ ͒ g EI with a rms spread DE, centered about EI .As confinement on the order of ldB͞2p is essential to seeing an example, the 23Na source described in Ref. [11] has conductance steps in a QPC. Since low-lying modes oc- a monochromaticity EI ͞DE ഠ 50 for atoms traveling at cupy a circularly symmetric part of the potential, the mth 30 cm͞s, or ldB ෇ 50 nm. To meet the constraint s ¿ step involves m degenerate modes and is m times as high a [22], such a source can be reflected off of a diffuser as the first step. The large aspect ratio of the atom QPC [23], such as the demagnetized magnetic tape described in allows for a sufficiently gentle constriction to suppress par- Ref. [24], or translated laterally to reduce time-averaged tial reflection at the entrance to the guide, such that the spatial coherence. Assuming the spatial density of atoms sharpness of steps and flatness between them is limited is preserved during propagation [25], such a source can only by the spread in incident atom energies. 10 22 21 have a flux density of J0 ഠ 2 3 10 cm s . It is interesting to compare the electron and atom QPC The quantized conductance for atoms is shown in Fig. 3 systems. If contact is made between two electron reser- and is the central result of this Letter. Conductance voirs whose chemical potentials differ by eDV , kBT ø 2 F͑͞ldB͞p͒ is shown as a function of mean energy EF, where DV is the applied voltage, T is the temperature EI ͞h¯v0 and energy spread DE͞h¯v0. In the limits of the electron gas, and EF is the Fermi energy, then the h¯v0 ¿ DE and ᐉ ¿ b0, one can show analytically that current that flows between them will be carried by elec- 2 the conductance is F ෇ ͑ldB͞p͒N, where N is the num- trons with an energy spread kBT and a mean energy EF. ber of modes above cutoff at z ෇ 0. The “staircase” of F For a cold atom beam, the particle flow is driven by kinet- versus EI ͞h¯v0 is a vivid example of quantum transport, ics instead of energetics. The incident kinetic energy EI as it demonstrates the quantum mechanical nature of corresponds to EF, and the energy spread DE ø EI cor- the center-of-mass motion. For all of Fig. 3, we have responds to kBT. The quantum of conductance for both 3 5 assumed ᐉ ෇ 10 S0 ഠ 10 b0; in the particular case of the systems can be formulated in terms of particle wavelength: Na source discussed above, and assuming s ෇ 25 mrad, the classical conductance of a point contact of area A con- 2 the first step (F ෇ ldB͞p) corresponds to a transmitted necting two three-dimensional gases of electrons is G ෇ 21 2 2 2 2 flux of ϳ500 atoms s , which is a sufficient flux to ͑e kFA͒͑͞4p h¯͒ [26], such that, if G ෇ Ne ͞ph¯, the ef- 2 measure via photoionization. fective area is A ෇ NlF͞p. We can understand several features shown in Fig. 3 by In order to determine the accuracy of conductance quan- considering the adiabatic motion of atoms within the wave- tization, three measurements [F, J0f͑0, 0͒, and ldB] are guide. As atom waves propagate through the constricting necessary for the atom QPC instead of two measurements waveguide, modes with transverse oscillator states ͑nx, ny͒ (current and DV) for the electron QPC. The reduced num- 2 2 such that h¯v0͑nx 1 ny 1 1͒ 2 EI * 2Mh¯ ͞ᐉ will con- ber of degrees of freedom for electrons results from their

3764 VOLUME 83, NUMBER 19 PHYSICAL REVIEW LETTERS 8NOVEMBER 1999

Fermi degeneracy: the net current is carried by electrons Sciences, Harvard University, Cambridge, Massachusetts whose incident flux density J0 and wavelength lF are 02138. functions of EF and DV. As a thought experiment, the [1] C. Waschke et al., Phys. Rev. Lett. 70, 3319 (1993). For a review, see E. E. Mendez and G. Bastard, Phys. Today simplicity of an externally tuned J0 and ldB could also be extended to neutral atoms, if two degenerate ensembles 46, No. 6, 34 (1993). of fermionic atoms were connected by a QPC and given [2] P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37, 95 (1996), and references therein. a potential difference DU. We can redefine neutral atom ෇ ͞ [3] M. B. Dahan et al., Phys. Rev. Lett. 76, 4508 (1996); S. R. conductance as G F DU, where F is the transmitted Wilkinson et al., Phys. Rev. Lett. 76, 4512 (1996). atom flux, just as the electron conductance G is the ratio [4] K. W. Madison et al., Phys. Rev. Lett. 81, 5093 of electron flux (current) to potential difference (voltage). (1998). One can show that [5] D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 N (1986). G ෇ , (4) h [6] J. K. Gimzewski and R. Möller, Phys. Rev. B 36, 1284 assuming DU , k T ø E , where T is the temperature (1987); J. M. Krans et al., Nature (London) 375, 767 B F (1995). of the Fermi ensembles and N is the number of modes [7] B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988); above cutoff. D. A. Wharam et al., J. Phys. C, Solid State Phys. 21, Two QPC’s can form a trap between them, just as a pair L209 (1988). of electron QPC’s form a quantum dot [13]. For h¯v0 . [8] C. W. J. Beenakker and H. van Houten, Solid State Phys. EI , all modes of the QPC are below cutoff and evanescent 44, 1 (1991), and references therein. transmission is dominated by tunneling of atoms occupy- [9] See, for instance, D. P. E. Smith, Science 269, 371 (1995); ing the ͑0, 0͒ mode. While the quantum dot between them H. van Houten and C. Beenakker, Phys. Today 49, No. 7, is energetically isolated, atoms can still tunnel into and out 22 (1996); S. Frank et al., Science 280, 1744 (1999). of the dot. For cold fermionic atoms, the Pauli exclusion [10] M.-O. Mewes et al., Phys. Rev. Lett. 78, 582 (1997); I. principle would enable a single atom to block transmission Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. through the trap, just as the charging energy of a single 82, 3008 (1999). [11] E. W. Hagley et al., Science 283, 1706 (1999). electron can block transmission in electron quantum dots; [12] A. Yacoby and Y. Imry, Phys. Rev. B 41, 5341 (1990). such a blockade might be used to make a single-atom tran- [13] Mesoscopic Electron Transport, edited by L. L. Sohn sistor. In such a single-atom blockade regime, quantum et al., NATO ASI, Ser. E, Vol. 345 (Kluwer, Dordrecht, dots can also show a suppression of shot noise below the 1997), and references therein. Poissonian level [13]. Note that spectroscopic measure- [14] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). ment of neutral atom traps with resolvable energy levels [15] H. Monien, M. Linn, and N. Elstner, Phys. Rev. A 58, has been suggested previously [17] in analogy to spectro- R3395 (1998). scopic measurement of electron quantum dots. We empha- [16] E. A. Hinds, M. G. Boshier, and I. G. Hughes, Phys. Rev. size that the loading and observation of such a small trap Lett. 80, 645 (1998). with two or more QPC “leads” is a powerful configuration [17] J. Schmiedmayer, Eur. Phys. J. D 4, 57 (1998). for atom optics, because loading a small, isolated trap is [18] J. H. Thywissen et al., Eur. Phys. J. D 7, 361 (1999). [19] J. D. Weinstein and K. G. Libbrecht, Phys. Rev. A 52, problematic, and because spectroscopy near the substrate 4004 (1995). is complicated by light scattering and inaccessibility. [20] M. Drndic´ et al., Appl. Phys. Lett. 72, 2906 (1998). In conclusion, we show how an electromagnet wave- [21] For all calculations in this work, we apply a strong enough guide could be used to create a quantum point contact for Bh that the spin-flip loss rate of the transverse ground state cold neutral atoms. This device is an example of a new is &1027v. See C. V. Sukumar and D. M. Brink, Phys. physical regime, quantum transport within microfabricated Rev. A 56, 2451 (1997). atom optics. [22] If a source with s&awere used, then F would still The authors thank A. Barnett, N. Dekker, M. Drndic´, have steps at the same energies, but the plateaus in Fig. 3 E. W. Hagley, K. S. Johnson, M. Olshanii, W. D. Phillips, would no longer be flat. M. G. Raizen, C. I. Westbrook, and G. Zabow for useful [23] Similarly, a diffuser was necessary for the laser light discussions. This work was supported in part by NSF source used to observe the quantization of photon cross section. See E. A. Montie et al., Nature (London) 350, Grants No. PHY-9732449 and No. PHY-9876929, and by 594 (1991). MRSEC Grant No. DMR-9809363. J. T. acknowledges [24] T. M. Roach et al., Phys. Rev. Lett. 75, 629 (1995). support from the Fannie and John Hertz Foundation. [25] K. Szymaniec, H. J. Davies, and C. S. Adams, Europhys. Lett. 45, 450 (1999). *Email address: [email protected] [26] Yu. V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov. †Also in the Division of Engineering and Applied Phys. JETP 21, 655 (1965)].

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