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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 17 — Aug. 17, 2009
  • pp: 14744–14760
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Van der Waals enhancement of optical atom potentials via resonant coupling to surface polaritons

Joseph Kerckhoff and Hideo Mabuchi  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 14744-14760 (2009)
http://dx.doi.org/10.1364/OE.17.014744


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Abstract

Contemporary experiments in cavity quantum electrodynamics (cavity QED) with gas-phase neutral atoms rely increasingly on laser cooling and optical, magneto-optical or magnetostatic trapping methods to provide atomic localization with sub-micron uncertainty. Difficult to achieve in free space, this goal is further frustrated by atom-surface interactions if the desired atomic placement approaches within several hundred nanometers of a solid surface, as can be the case in setups incorporating monolithic dielectric optical resonators such as microspheres, microtoroids, microdisks or photonic crystal defect cavities. Typically in such scenarios, the smallest atom-surface separation at which the van der Waals interaction can be neglected is taken to be the optimal localization point for associated trapping schemes, but this sort of conservative strategy generally compromises the achievable cavity QED coupling strength. Here we suggest a new approach to the design of optical dipole traps for atom confinement near surfaces that exploits strong surface interactions, rather than avoiding them, and present the results of a numerical study based on 39K atoms and indium tin oxide (ITO). Our theoretical framework points to the possibility of utilizing nanopatterning methods to engineer novel modifications of atom-surface interactions.

© 2009 Optical Society of America

1. Introduction

Over the past decade there have been numerous experimental and theoretical studies in atomic, molecular and optical physics [1

1. R. Folman, P. Krüger, D. Cassettari, B. Hessmo, T. Maier, and J. Schmiedmayer, “Controlling Cold Atoms using Nanofabricated Surfaces: Atom Chips,” Phys. Rev. Lett. 84, 4749–4752 (2000). [CrossRef] [PubMed]

, 2

2. B. Lev, Y. Lassailly, C. Lee, A. Scherer, and H. Mabuchi, “Atom mirror etched from a hard drive,” Appl. Phys. Lett. 83, 395 (2003). [CrossRef]

, 3

3. Y. Wang, D. Anderson, V. Bright, E. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. Saravanan, S. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005). [CrossRef] [PubMed]

, 4

4. S. Ghanbari, T. D. Kieu, A. Sidorov, and P. Hannaford, “Permanent magnetic lattices for ultracold atoms and quantum degenerate gases,” J. Phys. B: At. Mol. Opt. Phys. 39, 847–860 (2006). [CrossRef]

, 5

5. P. Treutlein, D. Hunger, S. Camerer, T. Hänsch, and J. Reichel, “Bose-Einstein condensate coupled to a nanomechanical resonator on an atom chip.” Phys. Rev. Lett. 99140403, (2007). [CrossRef] [PubMed]

, 6

6. T. P. Purdy and D. M. Stamper-Kurn, “Integrating cavity quantum electrodynamics and ultracold-atom chips with on-chip dielectric mirrors and temperature stabilization,” Appl. Phys. B 90, 401–405 (2008). [CrossRef]

] and in quantum information science [7

7. H. Mabuchi, M. Armen, B. Lev, M. Loncar, J. Vuckovic, H. J. Kimble, J. Preskill, M. Roukes, and A. Scherer, “Quantum networks based on cavity QED” Quantum Information and Computation , 1, 7–12 (2001).

, 8

8. M. Trupke, J. Metz, A. Beige, and E. A. Hinds, “Towards quantum computing with single atoms and optical cavities on atom chips,” J. Mod. Opt. 54, 1639–1655 (2007). [CrossRef]

, 9

9. B. Dayan, A. S. Parkins, T. Aoki, E. P. Ostby, K. J. Vahala, and H. J. Kimble, “A Photon Turnstile Dynamically Regulated by One Atom,” Science 319, 1062–1065 (2008). [CrossRef] [PubMed]

] focusing on the interaction of gas-phase neutral atoms with static or electromagnetic fields projected by planar micro- or nanofabricated structures. The tight dimensional control that can be achieved via modern lithographic techniques generally enables new regimes of performance and scalability in integrated atomic devices, which hopefully can be used to further the success of recent technology breakthroughs such as chip-scale atomic clocks [11

11. P. Schwindt, S. Knappe, V. Shah, L. Hollberg, and J. Kitching, “Chip-scale atomic magnetometer.” Appl. Phys. Lett. 856409 (2004). [CrossRef]

] and magnetometers [10

10. S. Knappe, P. Schwindt, V. Shah, L. Hollberg, J. Kitching, L. Liew, and J. Moreland, “A chip-scale atomic clock based on 87Rb with improved frequency stability ,” Opt. Express , 131249–1253 (2005). [CrossRef] [PubMed]

].

In many such scenarios—in particular those exploiting strong coupling of trapped atoms with optical fields confined in photonic waveguides or resonators—robust and accurate localization of atoms near solid surfaces emerges as a critical requirement. Indeed, the prospect of tight control over atomic ensembles alone has driven much of the development of integrated atomic platforms. Often atomic localization is achieved by “scaling down” trapping techniques first developed in free-space, as in the manipulation of neutral atoms by magnetic field gradients created by current-carrying microwires [1

1. R. Folman, P. Krüger, D. Cassettari, B. Hessmo, T. Maier, and J. Schmiedmayer, “Controlling Cold Atoms using Nanofabricated Surfaces: Atom Chips,” Phys. Rev. Lett. 84, 4749–4752 (2000). [CrossRef] [PubMed]

] or by optical dipole forces induced by the evanescent tails of confined laser fields [12

12. H. Mabuchi and H. J. Kimble, “Atom galleries for whispering atoms: binding atoms in stable orbits around an optical resonator.” Opt. Lett. 19, 749–751 (1994). [CrossRef] [PubMed]

]. Robust and conceptually simple, these techniques typically improve in strength and efficiency as the atoms are brought closer to the defining structure(s). However, bulk surface effects become dominant at short distances. For small values of the atom-surface separation r, energy potentials for optical dipole traps and magnetostatic microwire traps are roughly proportional to 1-r/r 0 and r -1, respectively, whereas van derWaals potentials have a much stronger variation ~r -3. Hence many integrated atomic systems are expected to exist in an awkward regime where atoms are kept close to trap-defining structures to achieve tight potentials, but not so close that van der Waals effects become problematic. In the context of cavity quantum electrodynamics (cavity QED) with monolithic dielectric resonators this can be a particularly significant compromise, as it generally precludes realization of the strongest possible atom-photon coupling parameters, which would require stable placement of atoms nearly in contact with the resonator surface. For this reason and others [13

13. C. Henkel, B. Power, and F. Sols, “New light on cavity QED with ultracold atoms,” J. Phys.: Conf. Series 19, 34–39 (2005). [CrossRef]

], van der Waals interactions can thus become a limiting factor in the achievable performance of a wide range of integrated atomic devices, and we are motivated to investigate new approaches to atom manipulation that can compete with the strong onset of common r -3 surface effects. It may be anticipated that such approaches will rely on the physics particular to atoms in the near field of bulk structures, rather than the reapplication of techniques developed in vacuo.

A reasonable first step in this direction could be to explore the design of trapping schemes that attempt to exploit serendipitous aspects of the atom-surface interaction. For example, the demonstration of a positive van der Waals energy shift of a particular excited state of Cs over an oriented sapphire crystal that is orders of magnitude larger than typical, attractive van der Waals potentials [14

14. H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant van der Waals repulsion between excited Cs atoms and sapphire surface,” Phys. Rev. Lett. 83, 5467–5470 (1999). [CrossRef]

, 15

15. H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D 23, 237–255 (2003). [CrossRef]

, 16

16. H. Failache, S. Saltiel, A. Fischer, D. Bloch, and M. Ducloy, “Resonant quenching of gas-phase Cs atoms induced by surface polaritons,” Phys. Rev. Lett. 88, 243603 (2002). [CrossRef] [PubMed]

] suggests a possible means of surface-specific manipulation of atoms. If the van der Waals, repulsive character of an excited atomic state may be accessed appropriately in the near-field of a surface, it would provide a valuable tool for the efficient localization of atoms very close to structures of interest. The anomalous van der Waals perturbation demonstrated in [15

15. H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D 23, 237–255 (2003). [CrossRef]

] arises from the off-resonant coupling of an atomic transition to a surface polariton resonance in the sapphire crystal, suggesting that analogous and potentially improved phenomena could be accessed in alternative systems chosen or even created using modern methods and insights from materials science and nanofabrication.

2. Atom-surface interactions

QED interactions between atoms and/or bulk materials have been analyzed in many contexts over almost 80 years (e.g. [17

17. R. Eisenschitz and F. London, “Über das Verhältnis der van der Waalsschen Kräfte zu den homöopolaren Bindungskräften,” Z. Physik , 60491–527 (1930). [CrossRef]

, 18

18. I. E. Dzyaloshinskii, E. Lifshitz, and L. Pitaevshkii, “General theory of van der Waals’ forces,” Sov. Phys. Uspekhi 4, 153–176 (1961). [CrossRef]

]), but this section will largely summarize relevant results in work by Wylie and Sipe [19

19. J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near and interface,” Phys. Rev. A 30, 1185–1193 (1984). [CrossRef]

, 20

20. J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near and interface II,” Phys. Rev. A 32, 2030–2043 (1985). [CrossRef] [PubMed]

, 21

21. J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981). [CrossRef]

] (whose perspective we adopt throughout the article) as they naturally highlight the importance of surface excitations in producing anomalous van der Waals effects.

An atom near a macroscopic structure experiences effects often dominated by the perturbing interaction of the atomic dipole and radiation field

Hi=μ·D,
(1)

where µ⃗ is the atomic dipole operator and D⃗ is the transverse displacement field operator at the atom’s position. Thus, just as atomic level shifts and the spontaneous decay of excited states arise from interactions with radiation modes in a vacuum, van derWaals effects are produced by similar physical processes, with contributions from a near field, planar surface. For example, in a purely semi-classical description, an excited atomic dipole near an interface will have a reaction to the reflected portion of its own induced radiation field. As the amplitude of the reflected field at the atom’s position increases, so do these reactive, surface-induced effects.

Without having to specify any geometry or optical properties of the system yet, the energy shift of atomic level a may be approximated by second order perturbation theory

δEa=h̅1PΣB,N,np(B)DαBNDβNBμαanμβna(ωNωB)+(ωnωa)
(2)

where P indicates the principal part, the summation is over field states B and N and atomic states n, h̄ωi is the energy eigenvalue for un-perturbed field or atomic state i, p(B) denotes an assumed thermal distribution of the field, and the summation over vector components α and β is implied. Also, the transition rate from atomic state a to n is given by Fermi’s golden rule

Rna=2πh̅ΣB,Np(B)n,Nμ.Da,B2δ((ωNωB)+(ωmωa)).
(3)

We introduce the correlation functions

G˜αβ(t)=ih̅1[Dα(t),Dβ(0)]Θ(t)
α˜αβa(t)=ih̅1a[μα(t),μβ(0)]aΘ(t),
(4)

where Dα (t) and µα (t) are interaction picture operators with respect to Hi and Θ(t) is the Heaviside step function, and also specify their Fourier transforms: Gαβ (ω) and αaαβ (ω). Equations (4) describe the linear response of operator expectation 〈X(t)〉, given a perturbation proportional to X(0). And, from linear response theory, Gαβ (ω) and αaαβ (ω) are the generalized susceptibilities for the field and the atom. In particular, Gαβ (ω) may be identified with the expected displacement field induced by an oscillating classical dipole, and αaαβ (ω) the polarizability of the a th atomic state by the field. The fluctuation-disspation theorem, a zero temperature approximation (appropriate for optical frequencies at room temperature), and the analytic properties of the susceptibilities allow Eq. (2) and (3) to be written

δEa=δEavf+δEar
=h̅2π0dζGαβ(iζ)ααβa(iζ)
ΣnμαanμβnaReGαβ(ωan)Θ(ωan)
(5)
Rna=2h̅μαanμβnaImGαβ(ωan)Θ(ωan)
(6)

where ωan=ωa-ωn.

One of the advantages of this approach is that the contribution to Gαβ (ω) from an interface may be simply added that of free space by the superposition principle, and therefore the surface effects can be considered separately from the (formally divergent) free space effects. From here on, we make this separation and only consider the effects from field modes emanating from the surface. The first term on the RHS of Eq. (5), δEavf, may be associated with the polarization energy of the atom due to quantum fluctuations in these surface modes, in analogy to the Lamb shift in free space. As the term integrates the susceptibilities over imaginary frequencies, this shift is always real, usually negative 1, and generally insensitive to resonances in the environment, giving rise to the familiar, attractive van der Waals potential for ground state atoms. This is in contrast to the last term in Eq. (5), δEra, which arises from the residue of the principal part integration in Eq. (2). This term has none of the analytic constraints of the first term: it may be either positive or negative, and is highly sensitive to environmental resonances near ωan. Due to the zero temperature field assumption, this residue exists only for excited atomic states, which may be degenerate with other states of the atom plus field. As it originates from global energy degeneracies, this term admits a semi-classical physical interpretation of the energy associated with an atomic dipole interacting with the in phase portion of its own, expected reflected field. Similarly, Eq. (6) may be viewed in analogy to the dephasing rate of an excited dipole from the interaction with the out of phase portion of the expected reflected field.

The atomic polarizability can be shown to be

ααβa(ω)=2h̅Σnωnaμαanμβnaωna2ω2,
(7)

while the field susceptibility may be constructed from a plane wave decomposition of the reflected electromagnetic field produced by an oscillating dipole above a particular surface. To specify now some general aspects of the system, the interface is assumed to be translationally invariant in the (xy) plane (see Fig. 1) which gives

Gαβ(ω)=iω˜22πdκW0(ŝαŝβRs+p̂α0+p̂β0Rp)exp(2iW0z)
(8)

where ω̃=ωc -1, κ⃗ and W 0 are the components of the vacuum wave vector at the transfer region-vacuum interface perpendicular and parallel to z⃗, respectively, ŝ and p̂ are the s- and p-polarized electric field vectors in the vacuum for upward (+) and downward (-) propagating fields, Rs,p are the Fresnel coefficients of the transfer region for the respective polarizations, and z is the height of the atom above the surface of the transfer region.

Fig. 1. The assumed geometry of an oriented atomic dipole in vacuum at height z above a planar, transfer region interface that separates an underlying bulk substrate with dielectric constant ε 1 from the vacuum. Adapted from [21].

Although surface interactions are typically considered in the near field limit, which leads to a dramatic simplification of Gαβ (ω), Eq. (8) is especially useful for engineering considerations, as it directly connects the effects on the atom by the surface with an intuitive picture of the dipole interacting with all possible, reflected surface modes at frequency ω. In particular, if the Fresnel coefficients are large over a large range of κ⃗ at an atomic transition frequency ωan, the corresponding effects described by Eq. (6) and the last term of Eq. (5) will be likewise large and carry the sign of the integrated Fresnel coefficients.

The common case of an atom above a bulk substrate (no transfer region) with separations zω̃na may be considered by taking the near-field limit (c) in Eq. (8). In this limit, the Fresnel coefficients become κ⃗-independent, and the integral may be simply evaluated:

Gzz(ω)=2Gxx(ω)=2Gyy(ω)
=14z3ε1(ω)1ε1(ω)+1
(9)

where the final “image factor” is the familiar, “quasi-static,” p-polarized Fresnel coefficient of the interface (and all other Gαβ (ω) components are 0). In this limit, all surface modes coupling to the atom are evanescent (ReW0=0), and therefore bound to the surface. Resonant surface excitations in all κ⃗ modes are excited by the atom as ε 1(ωan)→-1, leading to a divergent response in δ E r a and Rna. Significantly, an anomalously large, positive, van derWaals-like energy shift of excited atomic states may be realized by off-resonant coupling to these surface excitations in the near field as ε (ωan)+1→0+. An intuitive explanation for this effect is that the atomic dipole may interact with a near field image dipole (without retardation) that because of material resonances is both larger than and ~180°-shifted from the image dipole in an ordinary reflector.

For example, consider a Drude metal with the dielectric function

εD(ω)=1ωp2ω(ω+iγ)
(10)

where ωp is the metal’s plasma frequency and γ the electron collision rate. Using εD, the real part of the image factor in Eq. (9) reaches a minimum of 12(2γωp+(γωp)2)1Imin when ω=(ωp22+γωp2)12ωmin.. Compared to 1, image factor for a perfect conductor, this minimum coefficient has the opposite sign and is far larger in magnitude when ωpγ. The corresponding imaginary part of the minimum image factor is Imin(1+2γωp)12,, which notably also diverges for a “dissipation free” metal, γ→0. An excited atom with a downward transition frequency near ωmin in the near field of this Drude metal will experience a positive energy shift proportional to z -3 that will dominate δEvfa when ωpγ. Also, the spontaneous emission rate of that particular transition will likewise be enhanced by a factor of order h̄-1 δEra, proportional to z -3.

3. Van der Waals atom mirrors

The flexibility of δEra suggests that a resonant material interface may be used to engineer the energy potential of an atom in its near field. However, to do so we are required to find a suitable means of accessing these excited state effects, while keeping the atomic heating from spontaneous emission manageable.

Previously used to measure the attractive van der Waals potential for alkali atoms near transparent dielectrics [22

22. A. Landragin, J.-Y. Courtois, G. Labeyrie, N. Vansteenkiste, C. Westbrook, and A. Aspect, “Measurement of the van der Waals force in an atomic mirror,” Phys. Rev. Lett. 77, 1464–1467 (1996). [CrossRef] [PubMed]

], optical atom mirrors could provide a simple platform to study and begin to exploit anomalous interactions of atoms and surface polaritons. In a traditional optical atom mirror, an evanescent, coherent field is produced by total internal reflection of a laser beam from a planar, vacuum-dielectric interface (see Fig. 2). If the frequency of the laser is far blue-detuned from a ground state transition, an atom approaching the evanescent field experiences a repulsive optical dipole force proportional to the intensity of the field:

Udh̅Ω024Δe2κzz
(11)

where Ω0 is the Rabi oscillation frequency the atom would undergo at the dielectric surface if the laser was on resonance, Δ is the detuning of the field from the transition frequency, and κz is the field amplitude decay rate (typically of order the angular wave number of the laser). For sufficiently large Ud, this optical potential may repel incident, cold thermal atoms (e.g. falling in vacuum under the influence of gravity) with minimal heating from spontaneous emission. The inclusion of van der Waals effects adds an attractive, ~z -3, potential associated with the red-shifting of the atomic ground state (as measured in [22

22. A. Landragin, J.-Y. Courtois, G. Labeyrie, N. Vansteenkiste, C. Westbrook, and A. Aspect, “Measurement of the van der Waals force in an atomic mirror,” Phys. Rev. Lett. 77, 1464–1467 (1996). [CrossRef] [PubMed]

]), but also modifies the laser detuning as both ground and excited states are perturbed. These surface-induced modifications to the optical potential may be used to produce van der Waals-enhanced optical potentials, where the repulsive energy is derived from the laser fields, but the spatial variation of the potential carries the sharp character of the surface interactions: as the van derWaals, ~z -3, modifications occur predominantly in the near field where the un-perturbed Ud is roughly linear, the enhancement will be sharply peaked. As a simple example, if the excited state experiences a very large, positive van derWaals shift from coupling to resonant surface excitations, the field detuning will decrease faster than the red-shifting of the ground state. In the perturbative regime, Δδ≫Ei, the van der Waals contributions to the potential are

UvdW~Ω024Δ2δEer+δEgvf,
(12)

and so an appropriately large and positive δEre will provide a steep (~z -3) and repulsive UvdW. Also note that the probe need not be evanescent in this case to achieve near field repulsion. However, it is important to recognize that this system will also experience an enhanced rate of spontaneous emission in the near field Ω024Δ2Re, which will be comparable to the optical potential enhancement if δEe~h̄Re, as suggested in the end of section 2. Although this necessarily increases the heating rate, surface-induced spontaneous emission enhancement can sharply redistribute atomic populations in the near field [16

16. H. Failache, S. Saltiel, A. Fischer, D. Bloch, and M. Ducloy, “Resonant quenching of gas-phase Cs atoms induced by surface polaritons,” Phys. Rev. Lett. 88, 243603 (2002). [CrossRef] [PubMed]

] to advantageous effect. For example, enhanced emission may “turn off” target optical dipole transitions and thus quench dipole forces. If a ground state atom experiences balanced attractive and repulsive optical dipole forces at moderate atom-surface separations [12

12. H. Mabuchi and H. J. Kimble, “Atom galleries for whispering atoms: binding atoms in stable orbits around an optical resonator.” Opt. Lett. 19, 749–751 (1994). [CrossRef] [PubMed]

], greatly enhanced spontaneous emission on the “attractive” dipole transition can produce a net repulsive force in the surface near field, producing a sharp barrier and preventing collisions with the surface. As this general approach is particularly valid when spontaneous emission enhancement dominants any energy level shifts, its application may be particularly useful when the stability of atomic transitions is required. Moreover, in comparison, to other “repulsive” van derWaals interactions that rely on the reflection of atomic wavepackacts from steeply attractive van der Waals potentials [23

23. T. A. Pasquini, T. Pasquini, Y. Shin, C. Sanner, M. Saba, A. Schirotzek, D. Pritchard, and W. Ketterle, “Quantum reflection from a solid surface at normal incidence,” Phys. Rev. Lett. 93223201, (2004). [CrossRef] [PubMed]

], both approaches outlined above are designed to manipulate the classical center of mass, and thus may be performed with atoms with relatively small de Broglie wavelengths.

More accurate and general optical forces for thermal atoms may be predicted from the steady state solution to the atomic master equation (h̄=1) [24

24. C. Cohen-Tannoudji, J. Doupont-Roc, and G. Grynberg. Atom-Photon Interactions. Wiley-VCH, Weinheim (2004).

]:

ρ˙=(ρ)
(ρ)=i[ρ,H]+Σa,nDna(ρ)
H=12(Σa,nΩanσnaΣaΔaσaa)+h.c.
Dan(ρ)=Rnatot(σnaρσna12{σnaσna,ρ})
(13)

where ρ is the atomic state density matrix, σna is the atomic lowering operator |n〈〉a|, Ωan is the Rabi frequency induced between states a and n, Δa is the energy of the laser photons near-resonant with the transition of a ground state atom to level a minus the energy of state a, Rtotna is the total decay rate of state a to n, and ({·, ·}) [·, ·] is the (anti-)commutator. In the cases considered below, the nullspace of the superoperator contains a unique atomic steady state ρss, to which an arbitrary initial atomic state will decay in time of order max{R -1 na}. Typically, the internal state of the atom relaxes much faster than the external parameters change (i.e. the velocity of the atom times the spatial derivative of the coefficients in Eq. (13) is much less than Rna). In such regimes, use of ρss to estimate the force on the atom as a function position may be justified. To associate an internal atomic state to a force on the center of mass (CM) motion of the atom, we adopt a semi-classical approximation that assumes quantized internal dynamics (Eq. (13)), but classical CM dynamics. In particular, as the expected force on the center of mass of an atom by Ehrenfest’s theorem is - Tr([RCM,H]ρ) (where R CM is the center of mass position operator), declaring the center of mass a “classical” variable, rCM, allows the construction of an effective potential for the CM motional dynamics

Ueff(rCM)=rCMdrTr(δH(r)δrρss(r)).
(14)

This description should be most accurate at capturing the essential contributions to the mean force on slowly moving atoms with small de Broglie wavelengths from reactive (“dipole”) interactions with the laser fields [24

24. C. Cohen-Tannoudji, J. Doupont-Roc, and G. Grynberg. Atom-Photon Interactions. Wiley-VCH, Weinheim (2004).

]. Similarly, it is straightforward to show that the steady state heating rate of the atom due to isotropic spontaneous emission is

h̅23c2kBmΣa,nωna2RnatotTr(σaaρss(rCM))
(15)

where m is the mass of the atom.

Fig. 2. Left: diagram of a typical optical atom mirror, as described in section 3. Right: the 39K fine structure energy levels relevant to the example in section 4.

4. Example

4.1. 39K and ITO

As described in section 2 and appendix A, resonant van der Waals effects are a product of matched, high quality surface polariton and strong optical dipole transition resonances. As is suggested by Eq. 20, dipole transition moments tend to increase rapidly with wavelength (~|ωna|-3). Similarly, compared to plasmonic metals like Au and Ag, relatively stiff materials like sapphire, Y2O3, and SiC tend to feature high quality surface phonon-polariton excitations near crystal lattice vibrational modes in the far infra-red (<1000cm-1) [25

25. A. D. Rakić, A. Djurišic, J. Elazar, and M. Majewski, “Optical properties of metallic films for vertical-cavity opto-electronic devices,” Appl. Opt. 37, 5271–5283 (1998). [CrossRef]

, 26

26. S. Saltiel, D. Bloch, and M. Ducloy, “A tabulation and critical analysis of the wavelength- dependent dielectric image coefficient for the interaction excerted by a surface onto a neighbouring excited atom,” Opt. Comm. 265, 220–233 (2006). [CrossRef]

, 27

27. E. Palik (Ed.), Optical Handbook of the Optical Constants of Solids, Academic Press, New York, 3 volumes (1998).

]. Thus, strong resonant interactions with common bulk materials are likely to occur in multiply-excited states of alkali atoms, which have strong, ~10µm wavelength transitions [28

28. O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. 51, 1058–1061 (1961). [CrossRef]

, 29

29. A. Lindgård and S. E. Nielsen, “Transition probabilities for the alkali isoelectronic sequences Li I, Na I, K I, Rb I, Cs I, Fr I,” Atomic Data and Nuclear Data Tables 19, 533–633 (1977).

]. However, working with these highly excited atomic states is often very difficult. For example, although the resonant van derWaals interaction of the Cs 6D 3/2→7P 1/2 (12-µm) transition and oriented crystal sapphire is very large [15

15. H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D 23, 237–255 (2003). [CrossRef]

], an even partially excited 6D 3/2 state will populate so many lower-lying levels through spontaneous emission [28

28. O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. 51, 1058–1061 (1961). [CrossRef]

] that construction of an enhanced optical potential is difficult both to model and achieve.

However, there are combinations of common materials and convenient alkali states that admit a significant interaction without large complications from the multi-level atomic structure. The relatively strong, 3D 5/2→4P 3/2 1.178µm transition in neutral 39K [30

30. C. Corliss and J. Sugar, “Energy levels of potassium, KI through KXIX,” J. Phys. Chem. Ref. Data , 8, 1109–1145 (1979). [CrossRef]

] is slightly to the blue of a surface polariton resonance recently identified in indium tin oxide (ITO) [31

31. C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. Aspnes, J.-P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103, 093108 (2008). [CrossRef]

, 32

32. S. Franzen, “Surface plasmon polaritons and plasma absorption in indium tin oxide compared to silver and gold,” J. Phys. Chem. C 112, 6027–6032 (2008). [CrossRef]

], which suggests that the 3D 5/2 state may experience a net positive van derWaals energy shift in the vicinity of ITO. Due to dipole transition selection rules, 3D 5/2 states spontaneously decay only to 4P 3/2, and 4P 3/2 decays only to the 4S 1/2 ground states [28

28. O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. 51, 1058–1061 (1961). [CrossRef]

]. Thus optically probing the 4P 3/2↔4S 1/2 and 3D 5/2↔4P 3/2 transitions forms a closed system between these fine-structure states. Moreover, cycling transitions between the |4S 1/2,F=2,mF=±2〉↔|4P 3/2,F=3,mF=±3〉↔|3D 5/2,F=4,mF=±4〉 hyperfine magnetic sublevels are possible (39K has a 3/2 nuclear spin [33

33. E. Arimondo, M. Inguscio, and P. Volino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys. 49, 31–76 (1977). [CrossRef]

]). Calculation of the van der Waals effects on these fine structure states by a near field planar ITO film is done in appendix A and presented in table 2. The results anticipate a net positive energy shift of the 3D 5/2 state, as well as a significantly enhanced spontaneous emission rate on the 1.178µm transition.

Fig. 3. Left: effective, steady state optical potentials as a function of 39K-surface separation for both ITO and (dispersionless) ITO* materials with {Ω11,κ 12,Δ2}/2π={100MHz,50MHz, (767nm)-1,100MHz,0MHz} and atomic parameters from tables 1 and 2. Right: steady state atomic level populations for the same.

In Fig. 3 we plot the steady state energy expectation (Eq. (14)) of a van der Waals-enhanced optical atom mirror for 39K above an ITO interface as a function of atom-surface separation. Employing a probe configuration that exploits both the van der Waals energy shifts and enhanced emission to produce a sharply peaked potential in the surface near field, the 4P 3/2↔4S 1/2 766.7nm transition is driven with maximum Rabi frequency, detuning, and field amplitude decay {Ω11,κ 1}/2π={100MHz,50MHz, (767nm)-1} and the 3D 5/2↔4P 3/2 1.178µm transition is driven with {Ω22}/2π={100MHz,0MHz}. As ITO is quite lossy at 1.178µm (see Eq. 21), the second transition is assumed to be propagating in vacuum (κ 2=0) rather than evanescent. The free-space, spontaneous decay rates of these levels are listed in table 1 and the additive enhancement of spontaneous emission rates and energy level shifts from van der Waals interactions are listed in table 2. To highlight the significance of the ITO polariton resonance, Fig. 3 depicts the effective, steady state optical potential for a 39K atom both in the near field of an ITO surface and in the near field of a similar, but dispersionless dielectric “ITO*” (see appendix A). The ITO potential exhibits a greatly enhanced potential barrier over the dispersionless surface, capable of deflecting cold atoms falling from mesoscopic heights.

We begin to understand the origins of the enhanced potential by diagonalizing the “force operator” F=δHδr for the two potentials as a function of position and considering its eigen-values. As plotted in Fig. 4, in its eigenbasis, the diagonal elements of F from the far field up to about .2µm suggest that the two systems feature identical repulsive and attractive optical dipole force eigenstates, as well as a state that experiences no force. In the near field, each force eigenstate attains a van der Waals-like character, with the most significant difference between the ITO and ITO* systems emerging from the initially force-less state, due to the difference in ΔE3D52 calculated for the two surfaces (table 2): the polariton resonance in ITO produces a positive ΔE3D52 and thus a repulsive F 22, while F 22 for the dispersionless ITO* has a familiar, attractive, van derWaals character. The total expected, steady state forces, and thus the effective potentials depicted in Fig. 3, may be constructed from these position-dependent force eigenvalues and the diagonal elements of the steady state atomic density matrix in the force eigenbases, ηii. The right half of Fig. 4 depicts the steady state η diagonal elements. As to be expected, both the ITO effective potential in Fig. 3 and the population in the van derWaals-repulsed state η22 in Fig. 4 begin to decrease with decreasing separation at about the same point (~80nm). It may appear from Fig. 4 that the enhancement of the optical forces in the ITO system is due to the near-field repulsive F 22 eigenvalue. While this is largely true, the close correspondence of η 11 in the two material systems is a coincidence caused by the large enhanced decay of the 3D 5/2 state in the ITO system. If this decay enhancement is artificially removed in the ITO system, ITO’s η 11 is significantly smaller over the 60nm-250nm separation range, where F 11 is still repulsive. In this respect, the optical forces in the ITO near field can be considered van der Waals-enhanced by both ΔE3D52 and R4P323D52.

Fig. 4. Left: eigenvalues of the force operator as a function of separation for the ITO and ITO* systems depicted in firgure 3. Right: steady state atomic populations in the (position-dependent) force eigenbasis for the same.

We conclude this subsection with a mention of the optical powers required to achieve the drives in the above example. It is informative to calculate the saturation intensity for an an electric dipole transition [34

34. D. Steck, “Alkali D line data,” http://steck.us/alkalidata/.

]

Isat=h̅Rnafsωna312πc22Ja+12Jn+1.
(16)

Fig. 5. Quantum trajectory simulations of 39K incident on the optical potentials represented in Fig. 3 after falling from a height of 1mm. The left (right) graph depicts 100 trajectories of the atoms falling towards ITO (ITO*) surfaces. The highlighted green and red trajectories have the median reflected and absorbed escape times, respectively. The ITO potential reflected 88% of the atoms, while the ITO* potential reflected only 7%.

4.2. Quantum trajectory simulations

After falling 1mm, an atom initially at rest accelerates to .14m/s (for 39K, this corresponds to a kinetic energy of .96MHz). Because the internal state of the 39K atom should relax at a rate of order 24.5µs-1 (see section 3 and table 1) and the near field enhancement feature depicted in Fig. 3 is much broader than the ~10nm the atom would be expected to travel before reaching steady state, we can partially justify the use of the steady state approximation. Similarly, the thermal de Broglie wavelength for a 1µK (polarization gradient cooled [35

35. J. Dalibard and C Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am B 62023–2045 (1989). [CrossRef]

]) 39K atomic ensemble is only h(2πmkBT)-1/2=7nm, which supports the classical CM approximation for a typical cold atom ensemble. On the other hand, as the laser fields in this example excite the atoms significantly (see Fig. 3), incident atoms will heat from spontaneous emission scattering, complicating use of the steady state approximation. To demonstrate the effects of spontaneous emission, illustrate some dynamics connecting the internal and external atomic variables, and numerically evaluate the steady state approximation introduced above, we consider quantum trajectory simulations [36

36. H. J. Charmichael, An Open Systems Approach to Quantum Optics, (Springer-Verlag, Berlin, 1993).

] of 39K atoms falling from a height of 1mm onto the one-dimensional ITO and ITO* optical potentials represented by Fig. 3.

Rather than describing the state evolution of a single atom, it is more accurate to interpret the atomic master equation, Eq. (13), as evolving the mean internal state of an ensemble of independent atoms, without any measurements being made on the system [37

37. L. Bouten, R. van Handel, and M. James, “An introdution to quantum filtering,” SIAM J. Control Optim. , 46, 2199–2241, (2007). [CrossRef]

]. This “ensemble average” evolution is ill-suited to demonstrate the dynamics between an individual atomic state and the forces on the atomic mass. In contrast to its associated master equation, an ensemble of quantum trajectories presume that at least partial measurements are being made on the system, which add a stochastic element to the dynamics, but helps preserve the state purity of each trajectory [36

36. H. J. Charmichael, An Open Systems Approach to Quantum Optics, (Springer-Verlag, Berlin, 1993).

]. In this subsection, we construct such trajectories by simulating experiments with photon counting on all radiation fields with perfect efficiency, spectroscopically filtered with a bandwidth small enough to perfectly distinguish spontaneous emission from the two atomic excited states, but much larger than any of the other dynamical rates in Eq. (13). Although not experimentally realistic, these trajectory simulations in particular help elucidate the effects of spontaneous emission on the motional dynamics of the atoms. As purity is preserved in these simulations, the (unnormalized) evolution of a pure, internal atomic state is guided by the stochastic Schrödinger equation (h̄=1) [38

38. R. van Handel and H. Mabuchi, “Quantum projection filter for a highly non-linear model in cavity QED,” J. Opt. B: Quantum Semiclass. Opt. , vol. 7, pp. S226–S236, 2005. [CrossRef]

]

dψ={iHdt+Σa,n((σna1)dQna12Rnatotσnaσnadt)}ψ
(17)

where |ψ〉 is the internal state vector of the atom and dQna=1 in each time step with probability Rtotnaψ|σ na σ na|ψdt and is zero otherwise. And as above, we use a semiclassical approximation that the evolution of the (classical) atomic center of mass is driven by the force 〈ψ|F(rCM)|ψ〉, that spontaneous emission events (dQna=1) impart a random momentum kick with a probability distribution associated with the photon scattering randomly into 4π steradians, and that rCM determines the dynamical rates in Eq. (17) according to the atom-surface separation.

Fig. 6. A .5µs slice of a representative ITO trajectory. Left: the instantaneous force eigen-state populations ηii as a function of time. Right: the expected force on the atom as a function of time. Purple and cyan lines indicate different types of photon counting events in the trajectory.

The K-ITO and -ITO* separation as a function of time in 100 quantum trajectories each is plotted in Fig. 5. Despite an average 114 spontaneous emission events per drop, the ITO potential reflects 88% of incident atoms with a turning point that suggests the steady state effective potential of Fig. 3. Similarly, the ITO* potential reflects a minority (7%) of incident atoms, as to be expected from the initial kinetic energy (.96MHz) and Fig. 3. The imperfect reflection or transmission of atoms in the two cases is due to “heating” from non-deterministic scattering events.

Fig. 7. Thick lines: quasi-mean ηii from a single quantum trajectory as a function of separation. Solid (dashed) lines indicate the incident (reflected) portions of the trajectory. Thin lines: steady-state ηii from Fig. 4.

Considering whole trajectories again, we re-examine the use of steady state approximations. Thick lines in Fig. 7 depict time-filtered ηii as a function of separation in a representative trajectory. Low-pass, post-filtered with a cutoff frequency roughly 1/10th the free space spontaneous emission rates, these ηii represent the quasi-mean atomic populations relevant to the transient, atom mirror interaction. While both incident and reflecting, the quasi-mean ηii follow the underlayed, steady state ηii from Fig. 4 (thin lines) with ~.1 precision. While Fig. 5 suggested the validity of Ueff at a single position (the classical turning point) over many trajectories, Fig. 7 suggests the steady state potential for a single trajectory over a wide range of separations.

Finally, we note that other useful, van der Waals-enhanced optical potentials may also be constructed. For example, for the same optical parameters as in Fig. 3, but with an additional, -3MHz detuning of the 3D 5/2↔4P 3/2 transition, a signifiant Ueff minimum forms in the near field, as depicted in Fig. 8. Quantum trajectory simulations of atoms initially at rest at the potential minimum suggest a mean trapping time of 20µs before heating effects eject the atom (Fig. (8)), whereas ITO* potentials produce no such trapping and lose atoms within µs’. Although these trapping times are modest in this (non-optimized) potential, they are long compared to many fast atomic processes that characterize near-field atomic systems (e.g. [39

39. B. Lev, K. Srinivasan, P. Barclay, O. Painter, and H. Mabuchi, “Feasibility of detecting single atoms using photonic bandgap cavities” Nanotechnology 15, S556S561, (2004). [CrossRef]

]). Traps and barriers with significantly lower heating rates would be possible for systems with anomalous van der Waals interactions that are an order of magnitude or more larger than the ground state perturbations (e.g. effects on the order of demonstrated in Cs and sapphire interactions [15

15. H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D 23, 237–255 (2003). [CrossRef]

]).

Fig. 8. Left: effective, steady state optical potentials with {Ω11,κ 122}/2π={100MHz,50MHz, (767nm)-1,100MHz,-3MHz} that exhibit a near field potential minimum. Right: 100 quantum trajectory simulations for 39K atoms initially at rest in the 39K-ITO potential minimum. The highlighted green and red trajectories have the median reflected and absorbed escape times, respectively.

5. Conclusion

Harnessing surface effects for strong, near field manipulation of gas-phase atoms would enable integrated atomic systems and (at least in this implementation) employ an interesting intersection of fundamental QED, atomic physics, material science and surface polaritonics. Although exceptional, anomalous van der Waals effects admit an intuitive physical picture (section 2) that should facilitate their generalization. And as suggested by section 4, the effects need not be large for significant enhancement of near field potentials, and efficient steady state analysis may be reasonably accurate. However, we emphasize again that the identification of a broad class of surfaces with resonant excitations tunable to atomic transitions would open the door to general application of van der Waals-enhanced optical dipole potentials for atoms.

Appendix

A. Calculation of van der Waals effects

Estimating the influence of a surface on atomic states is complicated, even in the near field where the field susceptibilities may be approximated by Eq. (9). In this limit, with a simple planar interface, the energy shifts and spontaneous emission enhancement are

δEa=Σnμan2+μzan216z3(2π0dζε1(iζ)1ε1(iζ)+1ωnaωna2+ζ2+2Reε1(ωan)1ε1(ωan)+1Θ(ωan))
ΣnMan(Δnavf+Δnar)z3
(18)
Rna=μan2+μzan216h̅z34Imε1(ωan)1ε1(ωan)+1Θ(ωan)h̅1Manrnaz3.
(19)

We can therefore break up our calculation into separate parts: the dipole transition strengths, Man, and the image factors, Δna and rna.

We will neglect any atomic hyperfine structure and assume an isotropically prepared atomic dipole for calculation and conceptual clarity. The widths of ε (ω) features will be on the order of .1eV, while typical alkali hyperfine splittings are only of order 10GHz or less [33

33. E. Arimondo, M. Inguscio, and P. Volino, “Experimental determinations of the hyperfine structure in the alkali atoms,” Rev. Mod. Phys. 49, 31–76 (1977). [CrossRef]

]. Thus image factors should not vary significantly between hyperfine states. However, these splittings are of order the energy scales of typical laser Rabi frequencies and detunings. Moreover, atomic dipole moments vary greatly between different hyperfine spin projections. A more thorough analysis of Man should be considered in any experimental implementation.

Under these approximations, we may estimate the atomic dipole strength with [46

46. M. Chevrollier, M. Fichet, M. Oria, G. Rahmat, D. Bloch, and M. Ducloy, “High resolution selective reflection spectroscopy as a probe of long-range surface interaction: measurement of the surface van der Waals attraction exerted on excited Cs atoms,” J. Phys. II France 2, 631–657 (1992). [CrossRef]

]

μan2+μzan2=h̅R{n,a}fs(cωan)3(1+2JnJa2Ja+1Θ(ωna))
(20)

where Rfs {n,a} is the free space decay rate for either an or an, and Ji is the spin-orbit quantum number of state i. Using atomic data from [30

30. C. Corliss and J. Sugar, “Energy levels of potassium, KI through KXIX,” J. Phys. Chem. Ref. Data , 8, 1109–1145 (1979). [CrossRef]

] and [28

28. O. S. Heavens, “Radiative transition probabilities of the lower excited states of the alkali metals,” J. Opt. Soc. Am. 51, 1058–1061 (1961). [CrossRef]

], we list the calculated Man for the 4S 1/2, 4P 3/2, and 3D 5/2 levels of neutral 39 K in table 1.

The image factors may likewise be calculated from the bulk material’s dielectric function. In modeling the response of indium tin oxide (ITO), we follow [31

31. C. Rhodes, M. Cerruti, A. Efremenko, M. Losego, D. Aspnes, J.-P. Maria, and S. Franzen, “Dependence of plasmon polaritons on the thickness of indium tin oxide thin films,” J. Appl. Phys. 103, 093108 (2008). [CrossRef]

] and use an empirical Drude free-electron model

ε1(ω)=εωp2ω(ω+iγ)
(21)

with measured material constants ε =3.8, ωp=2.19eV, and γ=.111eV. Using Eq. (21), we list the calculated Δna and rna in table 1.

The aggregate van der Waals effect for each atomic level is determined by Eq. (18) and Eq. (19), and the entries in table 1. The results are presented in table 2 in comparison to a dispersionless dielectric called “ITO*” with ε 1(ω)=ε . The largest effect is the enhanced decay R4P323D52. The nearby surface polariton resonance in ITO (ε 1(1.0524eV)=-0.4826+0.4517i) and the relatively strong 3D 5/2 →4P 3/2 dipole moment (see table 1) combine for a net positive energy shift and a significantly enhanced spontaneous emission rate.

Table 1. Calculation of the transition dipole strengths and image factors for the 4S1/2, 4P3/2, and 3D5/2 levels of neutral 39K and ITO. Data taken from [30, 28, 31].

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Table 2. Total van der Waals energy shifts and enhanced decay rates for 39K, ITO, and ITO* as determined by Eq. (18) and Eq. (19), and table 1.

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Acknowledgments

This work has been supported by an HP Labs Innovation Research Award and by the NSF under CCF-0622246. JK thanks Young Chul Jun, Jon Schuller and Mark Brongersma for many useful discussions.

Footnotes

1Each n-level’s contribution to δEvfa carries the sign of ωna, as is suggested from Eq. (7). Thus for downward transitions, δEvfa will be positive, but the net shift will often be negative because δEra~-2|δEvfa| for any interface that is moderately reflective at |ωna| (see Eq. 18).

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H. Failache, S. Saltiel, M. Fichet, D. Bloch, and M. Ducloy, “Resonant coupling in the van der Waals interaction between an excited alkali atom and a dielectric surface: an experimental study via stepwise selective reflection spectroscopy,” Eur. Phys. J. D 23, 237–255 (2003). [CrossRef]

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OCIS Codes
(240.5420) Optics at surfaces : Polaritons
(270.5580) Quantum optics : Quantum electrodynamics
(020.1335) Atomic and molecular physics : Atom optics
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Quantum Optics

History
Original Manuscript: May 26, 2009
Manuscript Accepted: June 25, 2009
Published: August 5, 2009

Citation
Joseph Kerckhoff and Hideo Mabuchi, "Van derWaals enhancement of optical atom potentials via resonant coupling to surface polaritons," Opt. Express 17, 14744-14760 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14744


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References

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