Trapping Atoms Using Nanoscale Quantum Vacuum Forces

ARTICLE

Received 1 Nov 2013 | Accepted 9 Jun 2014 | Published 10 Jul 2014 DOI: 10.1038/ncomms5343 OPEN Trapping atoms using nanoscale quantum vacuum forces

D.E. Chang1, K. Sinha2, J.M. Taylor2,3 & H.J. Kimble4,5

Quantum vacuum forces dictate the interaction between individual atoms and dielectric surfaces at nanoscale distances. For example, their large strengths typically overwhelm externally applied forces, which makes it challenging to controllably interface cold atoms with nearby nanophotonic systems. Here we theoretically show that it is possible to tailor the vacuum forces themselves to provide strong trapping potentials. Our proposed trapping scheme takes advantage of the attractive ground-state potential and adiabatic dressing with an excited state whose potential is engineered to be resonantly enhanced and repulsive. This procedure yields a strong metastable trap, with the fraction of excited-state population scaling inversely with the quality factor of the resonance of the dielectric structure. We analyse realistic limitations to the trap lifetime and discuss possible applications that might emerge from the large trap depths and nanoscale conﬁnement.

1 ICFO—Institut de Ciencies Fotoniques, Mediterranean Technology Park, Castelldefels, 08860 Barcelona, Spain. 2 Joint Quantum Institute, College Park, Maryland 20742, USA. 3 National Institute of Standards and Technology, 100 Bureau Dr MS 8410, Gaithersburg, Maryland 20899, USA. 4 IQIM, California Institute of Technology, Pasadena, California 91125, USA. 5 Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125, USA. Correspondence and requests for materials should be addressed to D.E.C. (email: [email protected]).

NATURE COMMUNICATIONS | 5:4343 | DOI: 10.1038/ncomms5343 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5343

ne of the spectacular predictions of quantum electro- an atom only weakly dressed by its excited state can be trapped by dynamics is the emergence of forces that arise purely properly tailoring the dispersion of the underlying structure. Ofrom quantum fluctuations of the electromagnetic We begin by briefly reviewing how quantum vacuum vacuum1. Also known as London-van der Waals2 or Casimir fluctuations give rise to forces on the ground and excited states. forces3 in different regimes, these forces are often dominant at Within an effective two-level approximation of an isotropic short distances and can give rise to undesirable effects in atom with ground and excited states |gS,|eS, respectively, the nanoscale systems, such as nanomechanical stiction4. These interaction between the electromagnetic field and an atom at forces have also attracted increasing attention in the fields of position r is given by the dipole Hamiltonian H ¼d E(r). atomic physics and quantum optics. In particular, significant Conceptually, following a complete decomposition of the efforts have been made in recent years to interface cold atoms electric field into its normal modes k, one can write P y with the evanescent fields of dielectric micro- and nano-photonic H ¼ g ðrÞðs^ þ s^ Þðâ þ â Þ, where s^ ¼jiihjj. The systems5–14. These systems are expected to facilitate strong, k k eg ge k k ij y tunable interactions between individual atoms and photons for energy non-conserving terms (sêgâk and s^geâk) enable an atom applications such as quantum information processing15 and the in its ground state |g,0S to couple virtually to the excited state 16–18 investigation of quantum many-body physics . In practice, and create a photon, |e,1kS, which is subsequently re-absorbed. efficient atomic coupling to the evanescent fields of the The corresponding frequency shift for the ground stateP within 2 nanophotonic systems requires that atoms be trapped within second-order perturbation theory is dog(r) ¼ k gk(r) / sub-wavelength distances of these structures. (o0 þ ok), where o0 is the unperturbed atomic transition At these scales, quantum vacuum forces can overwhelm the frequency. This shift results in a vacuum-induced mechanical forces associated with conventional optical dipole traps, typically potential when translational symmetry is broken owing to resulting in a loss of trap stability at distances dt100 nm from dielectric surfaces. Using a quantization technique for electro- dielectric surfaces. Given the ability to engineer the properties of dynamics in the presence of dispersive dielectric media, the shift nanophotonic structures, an interesting question arises as to can be expressed in terms of the scattered component of the whether such systems could be used to significantly modify dyadic electromagnetic Green’s function evaluated at imaginary vacuum forces, perhaps changing their sign from being attractive frequencies o ¼ iu (ref. 27) (also see Methods), to repulsive, or even creating local potential minima. The strength Z1 of nanoscale vacuum forces should lead to unprecedented energy 3cG u2 do ðrÞ¼ 0 du Tr G ðr; r; iuÞ; ð1Þ and length scales for atomic traps, which would find use beyond g 2 2 2 sc o0 o0 þ u nanophotonic interfaces, such as in quantum simulation proto- 0 cols based on ultracold atoms19 and control of inter-atomic 20 where G0 is the free-space spontaneous emission rate of the atom. interactions . For simplicity, we have given the result in the zero-temperature Here we propose a novel mechanism in which engineered limit. Finite-temperature corrections28,29 associated with atomic vacuum forces can enable the formation of a nanoscale atomic 21 interactions with thermal photons are only relevant for distances trap. While a recent no-go theorem forbids a vacuum trap for from surfaces comparable with or larger than the thermal atoms in their electronic ground states, tailored nanophotonic ‘ c blackbody wavelength, d\lT, where lT ¼ 7:6 mmat systems can yield strong repulsive potentials for atomic excited kBT states. We show that a weak external optical field can give rise to T ¼ 300 K. an overall trapping potential for a dressed state. Remarkably, A similar process can occur for an atom in its excited state |e,0S, in which the atom virtually emits and re-absorbs an off- absent fundamental limits on the losses of the surrounding S dielectric structure, the fraction of excited-state population in the resonant photon (thus coupling to the state |g,1k ). However, the excited state is unique in that it can also emit a resonant dressed state can become infinitesimal, which greatly enhances 27 the trapping lifetime and stability. We identify and carefully photon. The total excited-state shift is given by analyse the actual limiting mechanisms. While we present dogðrÞ G0pc calculations on a simple model where analytical results can be doeðrÞ¼ Tr Re Gscðr; r; o0Þ: ð2Þ 3 o0 obtained, we also discuss the generality of our protocol to realistic systems such as photonic crystal structures. The first term arises from off-resonant processes, while the second term can be interpreted as the interaction between the atom and its own resonantly emitted photon. Naturally, the dielectric Results environment can modify the spontaneous emission rate as well, Quantum interactions between atom and surface. The no-go 2G pc GðrÞ¼G þ 0 Tr Im G ðr; r; o Þ: ð3Þ theorem for non-magnetic media states that a dielectric object in 0 o sc 0 vacuum cannot be stably trapped with vacuum forces for any 0 surrounding configuration of dielectric objects, provided that the Note that the resonant shifts and modified emission rates are system is in thermal equilibrium21. In analogy with Earnshaw’s complementary, in that they emerge from the real and imaginary theorem, which prohibits trapping of charged objects with static parts of the Green’s function. Micro- and nano-photonic systems electric potentials, at best one can create a saddle-point potential are already widely exploited to modify emission rates of atoms 30–33 (see, for example, ref. 22 involving metal particles and refs 13,23 and molecules in a variety of contexts . We propose that these for a hybrid vacuum/optical trap for atoms). For dielectric same systems can be used to tailor excited-state potentials to objects, potential loopholes involve embedding the system in a facilitate vacuum trapping. high-index fluid24 or using blackbody radiation pressure associated with strong temperature gradients25. Applied to Interactions with a Drude material. In what follows, we present atoms, this theorem essentially forbids the stable vacuum a simple model system that enables one-dimensional trapping in trapping near a dielectric structure when the atom is in its a plane parallel to a semi-infinite dielectric slab, as illustrated in electronic ground state. No such constraint exists for atoms in Fig. 1. The use of a homogeneous dielectric (as opposed to, for their excited states26, although the robustness of any possible trap example, a photonic crystal) greatly simplifies the calculation would be limited by the excited-state lifetime. Here, we show that and enables one to understand the relevant trap properties

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L L () e (z) (z) e L F 0 F zt zb g (z) g (z) z Zb Ud (z)

Figure 1 | Schematic of vacuum trapping mechanism. (a) A semi-infinite dielectric with specially tailored frequency-dependent permittivity E(o) enables large repulsion in the excited state oe(z), in contrast to the attractive vacuum potential experienced by an atom in its ground state. An overall trapping potential normal to the surface (along z) can be generated by driving the atom with an external laser of frequency oL, which is greater than the natural resonance frequency o0 of the atom. The laser comes into local resonance with the atom at zb. An atom far from zb is unaffected owing to the large laser detuning oL o0 and feels an attractive force F toward the dielectric interface. (b) The atom comes closer to local resonance with the laser as z-zb. This creates an atomic-dressed state, whose excited-state component yields a net repulsive force on the atom. (c) A slowly moving atom experiences the adiabatic dressed-state potential Ud(z) shown in red, which characterizes the position-dependent mixing between ground and excited states. The minimum of the dressed potential is located at zt4zb. analytically, although we later argue why the qualitative features photon scattering. The ground-state shift for atomic frequencies B 3G0 should still hold in more complex, realistic settings. In the o0 opl and in the near-field is given by dog 3.An 32ðk0zÞ dielectric slab model, dispersion engineering will be realized via additional condition for the validity of the short-distance results the frequency-dependent electric permittivity E(o). While this pffiffiffiffiffiffiffiffiffiffiffiffi D = particular system achieves a one-dimensional trap along z, our obtained for the excited state is k0zt p Q, as shown in the electrodynamic calculations are performed in three dimensions. Methods. We continue to present approximate results in this A short-distance expansion of the Green’s function, valid for regime owing to its simplicity, although all of our numerical sub-wavelength scales, shows that the excited-state emission rate results (such as in Figs 2c,d and 3) are calculated using the full ðrÞ Green’s functions. G(z) and resonant contribution doe ðzÞ to the shift are given by We also briefly comment on two effects in realistic systems that GðzÞ 1 Ea 1 can be safely excluded from our analysis. First, in principle, the 3 Im ; ð4Þ G0 4ðk0zÞ Ea þ 1 excited-state shift can be affected by coupling to even higher electronic levels. However, a careful calculation of this contribu- ðrÞ 34 doe ðzÞ 1 Ea 1 tion shows that it is negligible compared with the resonantly 3 Re ; ð5Þ enhanced term of equation (5), provided that we work in the G0 8ðk0zÞ Ea þ 1 regime Q44Dp. Second, we have ignored corrections to the shifts where the permittivity Ea ¼ E(o0) is evaluated at the atomic owing to surface roughness. It can be shown that the length scale resonance frequency, z is the distance from the surface and for modifications to the shifts is roughly set by the size of the 35 k0 ¼ o0/c is the resonant free-space wavevector. A large repulsive surface variation itself . The realistic regimes of operation of our potential is generated when Ea approaches 1 from above, trap will be at d\10 nm from surfaces, which allows us to safely - þ Ea 1 , which is analogous to the interaction between a ignore the angstrom-scale surface roughness achievable in state- classical dipole at position z and its large induced image dipole in of-the-art dielectric structures36. the dielectric. The modified spontaneous emission rate originates from the absorption or quenching of the atomic emission owing Formation of a vacuum trap. We now describe how a trap can to material losses. o2 We choose a Drude model, EðoÞ¼1 p , as a simple form for an atom subject to these potentials, and in the presence o2 þ iog of a weak driving laser of frequency o and Rabi frequency O (see dielectric function. Note that this function satisfies Kramers– L Kronig relations (causality), as a choice of a non-causal function Fig. 1). For conceptual simplicity, we choose the Rabi frequency to be spatially uniform, such that all atomic forces are attributable could result in an apparent violation of the no-go theorem21 as well. In the limit of vanishing material loss parameter g, the to the vacuum potentials alone. Qualitatively, the strong position dependence of the ground- and excited-state frequencies causes system passes through E ¼1 at the plasmon resonance pffiffiffi the laser to come into resonance with the atom at a single, tunable frequency opl ¼ op= 2. The system is conveniently point z ¼ zb. Under certain conditions (described in detail later), parameterized by the quality factor Qopl/g and a an atom starting at position z44zb will be far detuned, such that dimensionless detuning Dp ¼ (o opl)/g. For Q44Dp, the it is essentially in the ground state and is attracted by the pure Ea 1 factor Q=ði þ 2DpÞ resembles the complex Ea þ 1 ground-state potential. However, moving closer to zb brings the susceptibility of a simple resonator, which in the far-detuned atom closer to resonance. The dressing of the atom with a small ðrÞ G 1 Q doe 1 Q fraction of excited-state population causes a repulsive barrier to limit (Dp441) yields G 3 D2 and G 3 D . The 0 16ðk0zÞ p 0 16ðk0zÞ p form near zBz , yielding a metastable trap. Significantly, our 1 2 b dispersion and dissipation scale like Dp and Dp , respectively. approach does not attempt to directly counteract the attractive Thus, for high Q, it is possible to choose detunings where the ground-state potential with large external optical potentials, in atom still sees significant repulsive forces, but where spontaneous sharp contrast with other trapping schemes near dielectric sur- emission into the material is not significantly enhanced (in faces8,9,11,12,37–39. addition to material-induced emission of equation (4), there still The order of our calculation of the trap properties is as follows. B is emission into free-space at a rate G0). This scaling behaviour We first find the adiabatic potential experienced by a slowly is in close analogy with conventional optical trapping of atoms, moving atom. We derive relevant properties around the location where simultaneously using large trapping intensity and detuning of the trap minimum zt (Fig. 1). In particular, we show that maintains reasonable trap depths but suppresses unwanted absent material constraints (arbitrarily high Q), the dressed state

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5 1 60 4 10 20 Δ 0 0 A: z =0.4 nm 0.8 3 –20 B –40 40 0.6 U ~ –60 0 2 Ω 104

(z) 10 12 14 (nm) (mk) z

U 0.4 20 U 20 Δ A Δz 1 0

0.2 ) (mK) –20 Umax z ( –40 0 0 3 B: Δz =0.6 nm 10 0 U 10 20 30 40 50 –60 z z 103 104 105 106 10 12 14 b z (nm) Δ b p z (nm)

Figure 2 | Trap properties. (a) Ideal limit of trapping potential, corresponding to the pure ground-state potential :dog(z) and an inﬁnite barrier at z ¼ zb. 2 The maximum depth of the potential is denoted by Umax:|dog(zb)|. The quantum wavefunction amplitudes |c(z)| (in arbitrary units) for the two lowest eigenstates are illustrated in red, while the binding energy U0 and position uncertainty Dz of the ground state are depicted as well. (b) Properties of the ideal trapping potential versus barrier position zb, for the case of a Caesium atom. The blue solid and dashed curves give the potential depth at the barrier position, Umax, and the ground-state-binding energy U0, respectively. The green curve depicts the ground-state uncertainty Dz.(c) For the non-ideal trapping potential, the depth of the classical potential Udepth is numerically calculated. Here we plot it normalized by the maximum possible ~ value Umax:|dog(zb)|, versus detuning parameter Dp and normalized Rabi freuqency O O=Omin.(d) The trapping potential, ground-state-binding 2 energy U0, and wavefunction amplitude |c(z)| for the parameter sets A, B, as illustrated in c. The colour coding is the same as in a.

105 20 / / / NT NT total r total ad total trans 105 15 ~ Ω Ω~ 10 4 104 10 5 103 104 105 106 103 104 105 106 103 104 105 106 Δ Δ Δ <1 ms p p p 3 4 5 6 10 10 10 10 01 Δ p

~ Figure 3 | Trap lifetime. (a) Trap lifetime ttotal versus dimensionless Rabi frequency O ¼ O=Omin and detuning Dp. Here we have assumed a plasmon resonance quality factor of Q ¼ 107 and atomic properties corresponding to Caesium. The regions of parameter space where no trap exists are labelled ‘NT’ in the plot. (b) For the same parameters, the individual contributions of recoil heating, anti-damping and transient heating are plotted as a fraction of the total heating rate. Tunnelling contributes less than 10 2 to the total heating over this range of parameters and is not plotted here.

can in principle have infinitesimal excited-state population and calculation reveals that the Green’s function Gsc(r, r, o) for the scattering. We then quantize the motion to find the motional excited-state resonant shift and emission rate in equations (2) and eigenstates, binding energy and position uncertainty Dz. Finally, (3) must in fact be evaluated at the laser frequency in this regime, we analyse the mechanisms that limit the trap lifetime. which reflects that the atom primarily acts as a Rayleigh scatterer 40 Following standard procedures , the average force (see Methods). Consequently, Ea in equations (4) and (5) is experienced by the atom is given by the expectation value of replaced with E(oL). TheR adiabatic potential seen by a slowly ^ do z ^ ~ dp ‘ doe g ~ dhpðzÞi the Heisenberg equation dt ¼ ð dz sêeðtÞþ dz s^ggðtÞÞ, while moving atom is UdðzÞ¼ 1 dz dt . In principle, the Heisen- the internal atomic dynamics satisfies the usual Bloch equations, d^p berg equation for dt should also include the effects of spontaneous for example, emission. It can be shown, however, that this term contributes a ds O zero average force (see Methods), and only results in a heating ge ¼ðid ðzÞGðzÞ=2Þs þ i ðs s Þ: ð6Þ term to be analysed later. dt a ge 2 ee gg Three independent frequencies characterize the system (oL, op Here, for notational simplicity, sij ¼hsîji denotes the expecta- and o0), and two relative frequencies must be specified. One is tion value of the atomic variables, and we have defined da(z) ¼ determined by selecting the trap barrier position, determined by oL (oe(z) og(z)) as the detuning between the laser and the the relation da(zb)0. A second is the dimensionless detuning local atomic resonance frequency. When the atom moves between the plasmon resonance and laser, Dp ¼ (oL opl)/g, slowly on the timescales of the internal dynamics, the atomic which we treat as a free parameter that determines the relative coherence can be adiabatically eliminated, dsge 0, such that strengths of the excited-state shift and dissipation. In the near- dt field, the excited-state repulsion exceeds the ground-state the atomic populations are functions of position alone, O2 attraction by a factor 2Q/3Dp. The population at zb must then see ¼ 2 2 2. We are primarily interested in the regime GðzÞ þ 4daðzÞ þ 2O exceed the inverse of this quantity, see43Dp/2Q, to provide a where the atom is weakly driven, seeoo1. A more careful classical barrier in the adiabatic potential, which translates to a

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qffiffiffiffiffiffi Omin 1 3Q best possible scaling of the trap properties. In particular, the trap minimum Rabi frequency 3 3 . Since DptQ, this G0 16ðk0zbÞ 2Dp depth Udepth (the energy required for a classical particle to scheme potentially enables atom trapping near surfaces with become unbound) cannot exceed the value of the ground-state greatly reduced intensities as compared with conventional : E potential at zb, Umax |dog(zb)| (for example, Umax 60 mK trapping. In the latter case, an optical dipole potential requires when zb ¼ 10 nm for a Caesium atom). This trapping potential Omin 1 a minimum driving field of 3 to overcome the can be easily quantized numerically. In Fig. 2b, we plot the G0 ðk0zbÞ attractive ground-state potential. quantum-binding energy U0 (where 0oU0oUdepthrUmax) and The excited-state population at the trap minimum is fixed by position uncertainty Dz for the motional ground-state wave- 3Dp function. For the numerical results in Figs 2 and 3, we use atomic the ratio of ground to excited-state forces, seeðztÞ¼ 2Q , and is surprisingly independent of O (provided that O4O ). An properties corresponding to Caesium (l0 ¼ 852 nm, G0/2p ¼ 5.2 min 4 increasing Rabi frequency has the effect of increasing the local MHz, recoil frequency or/G0 ¼ 4 10 ). The strength of the detuning da(zt) to preserve the same population. This manifests vacuum potentials is reflected in the strong confinement of the itself as a self-selection of the trapping position and an increasing wavefunctions (for example, Dzo4 nm for distances zbo50 nm), distance from the barrier with increasing intensity, which are an order of magnitude smaller than in conventional ÀÁ optical traps. To contrast our results for Caesium with those of a z z 1 1=2 t b O~ 2 1 ; ð7Þ less massive atom, we have considered Lithium as well. Results for zb 6Dp Li similar to those for Cs in Fig. 2a,b show that at zb ¼ 10 nm, the potential depth is smaller by roughly a factor of two, while the where we have defined the dimensionless Rabi frequency confinement Dz sees roughly a fivefold increase. Generally, within O~ O=O ¼ min. The ability to tune the trap position using both the Wentzel–Kramers–Brillouin approximation and to lowest the laser amplitude and its frequency (which determines zb) B order in Dz, the ground-state localization scales like k0Dz enables the adiabatic transformation between deep but small traps (o (k z )4/G )1/3, while the binding energy U EU close to the surface to larger, shallow traps B100 nm from the r 0 b 0 0 max (1 3Dz/zb) can be nearly the entire classical potential depth. surface. This is expected to greatly facilitate trap loading, as This ideal model describes qualitatively well the actual techniques to trap atoms at these larger distances are already well 8,11,12,41 potentials over a large parameter regime. In Fig. 2c, we plot the established . ratio of the numerically obtained trap depth to the theoretical The photon-scattering rate at the trap position, Rsc ¼ maximum, U /U , as functions of D and dimensionless G s depth max p (zt) ee(zt), leads to lifetime limitations of our trapping scheme. driving amplitude O~. Here, we have chosen a quality factor of D Fixing other parameters, the detuning p from the plasmon Q ¼ 107 and a barrier position of z ¼ 10 nm. A trap depth resonance can be optimized to yield the minimum scattering rate, b comparable with Umax is achievable over a large range of values. R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3G0 . Thus, absent fundamental limits on the sc;min 3=2 For low values of O~, the trap depth is weakened owing to escape 4 2Qðk0ztÞ magnitude of Q, the photon-scattering rate can in principle be over the potential barrier, while for large values, the trap becomes suppressed to an arbitrary degree. This implies that an shallow as its position zt is pulled away from zb (as discussed infinitesimal violation of the assumptions underlying the no-go previously). Two representative trap potentials, along with the theorem can in fact allow for trapping. ground-state wavefunctions and confinement Dz are illustrated in In a conventional optical dipole trap, the potential U ¼ Fig. 2d. 2 a(oL)|E(r, oL)| /2 seen by an atom depends on the local field intensity and is proportional to a spatially constant atomic Analysis of trap heating. We have identified four sources of polarizability evaluated at the laser frequency. In contrast, here we motional heating—recoil heating, non-adiabatic motion of the exploit a novel mechanism arising from strong spatial variations atom within the trap, fluctuations of the atomic coherence and in the polarizability itself, induced through the vacuum level tunnelling over the finite-height potential barrier—that limit the shifts. This creates a back-action effect analogous to that trap lifetime in absence of external cooling mechanisms. Tun- producing a dynamical ‘optical spring’ in opto-mechanical 42 nelling is exponentially suppressed with barrier height and thus is systems , and enables extremely steep trap barriers around zb easily suppressed with increasing Rabi frequency. This calculation (as in Fig. 1c). In particular, the repulsive force is given by can be found in the Methods, while we discuss the more ‘ doe doe F ¼ dz seeðzÞ, where dz is engineered to be large. However, important mechanisms here. O2 the excited-state population s ðzÞ 2 itself varies strongly As in conventional optical trapping, the random momentum ee 4d ðzÞ a recoil imparted on the atom by scattered photons yields an with position through the spatially dependent detuning 2 ð‘ keff Þ (for simplicity, here we omit the varying linewidth G(z)). increase in motional energy at the rate dE=dt ¼ 2m Rsc. Two Taking into account the spatial variation of see around the qualitative differences emerge relative to free space, however. G0see Q equilibrium position, one finds an overall repulsive force that First, the photon-scattering rate Rsc ¼ GðrÞsee 3 2 is 16ðk0zÞ D scales like ðdoeÞ2, p dz modified in the presence of dielectric surfaces. Second, the : 2 effective momentum keff imparted by a photon is enhanced ‘ doeðztÞ 2ðz ztÞseeðztÞ F : ð8Þ compared with the free-space momentum.pffiffiffi This momentum dz daðztÞ scales at close distances like keff 3=z (see Methods), owing to emission of high-wavevector photons into the dielectric. Starting from the motional ground state, this heating causes the atom to Results for trapping of caesium. Given the possible steepness of E 2 this barrier, an ideal limit for the overall potential is shown in become unbound over a timescale tr (k0zt) Dp/(3or). It should Fig. 2a. The attractive part consists of the pure ground-state be noted that the heating rate dE/dt does not depend on the atom potential and arises from quantum fluctuations, while an infinite confinement (that is, whether we operate in the Lamb-Dicke hard wall is created at z ¼ z owing to the combination of strong limit), as the suppression of motional jumps in tight traps is b compensated by the increase in energy gained per jump43. resonant excited-state shifts (originating from the interaction of 2 The increase in lifetime with trap position, tr / zt , is attributable the atom with a classical image dipole) and opto-mechanical 2 ð‘ keff Þ back-action. This idealized model enables one to understand the entirely to the reduction in recoil energy 2m associated with a

NATURE COMMUNICATIONS | 5:4343 | DOI: 10.1038/ncomms5343 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5343 photon emitted into the surface. In particular, the decreasing trap An additional heating mechanism, which does not appear in 3 depth Udepth / zt versus distance is exactly compensated for by our simple model of an isotropic two-level atom, arises from the 3 possibility of an excited state in a realistic multi-level atom to the reduced surface-enhanced emission rate G / zt . Our derivation of the atomic potential assumes that the atomic emit photons of different polarizations and transition into E coherence adjusts rapidly to the local trap properties, dsge/dt 0, different states in a ground-state manifold. The different which results in a purely conservative potential. As we now Clebsch–Gordan coefficients of these transitions, along with the show, non-adiabatic corrections result in a momentum- symmetry breaking introduced by the nearby dielectric structure, dependent force, dp/dtpp (anti-damping). We can solve imply that different excited states can experience different equation (6) for the atomic motion perturbatively by writing vacuum shifts. This would appear as different trapping potentials s ðtÞ¼sð0ÞðzðtÞÞ þ sð1ÞðtÞ, where sð0Þ is the adiabatic solution. for the various dressed ground-state sub-levels absent of some ge ge ge ge special engineering. As a result, spontaneous changes in the sub- It depends only on the instantaneous position and is given by levels (such as arising from spontaneous photon scattering) ð0Þ O sge in the weak saturation limit. Here, d (z) ¼ d (z) þ 2dcðzÞ c a would induce fluctuations of the trapping potential and lead to iG(z)/2 is the complex detuning. Substituting this solution motional heating. One possibility to prevent these issues in an into equation (6) yields a velocity-dependent correction to the experiment is to utilize the effective two-level structure provided ð0Þ ð1Þ 1 dsge O ddc dz by a cycling transition. coherence, sge ¼ðidcÞ ¼ 3 . This in turn yields a dt 2idc dz dt new contribution to the force, F(1) ¼ bp, where the anti-damping doe see 2 2 dG dda rate is given by b o 2 4 ½ðG 4d Þ þ 8d G Discussion r dz 2k j dc j a dz a dz 0 We have described a protocol for an atomic trap based on evaluated at the trap position zt. Intuitively, the atom is anti- damped (b40) because the laser frequency is blue-detuned engineered vacuum forces, and have analysed in detail a model case of one-dimensional trapping near a Drude material, where relative to the atomic frequency at zt, giving rise to preferential Stokes scattering over anti-Stokes. Aside from the term analytical results are possible to obtain. The Drude response proportional to dG/dz, the heating rate is exactly analogous to approximates well a number of metals, such as silver and gold, in 44 that occurring in an opto-mechanical system in the blue-detuned the optical domain . In practice, however, these metals are 42 limited by their low-quality factors (Q 102), and the thermal regime . We define a characteristic time tad needed for the o energy increase produced by anti-damping to equal the ground- fluctuations of such conducting materials can give rise to strong noise-induced trap heating and decoherence45. On the other hand, state-binding energy U0. Thus far, we have focused on the motion of the atomic-dressed photonic crystal structures enable engineering of resonances B 7 state, in which a trap emerges owing to weak mixing with the through geometry, and quality factors approaching Q 10 have 46 excited-state potential. Following a spontaneous emission event, been observed . We anticipate that our trapping protocol could however, the atom returns to the ground state and sees the pure be quite generally applied, and the resulting trap behaviour would B 1 qualitatively remain the same. In particular, the spatial ground-state potential for a transient time ttrans |dc(zt)| until dependence of the excited-state properties would no longer scale the atomic coherence is restored, during which the atomic wave 3 packet accelerates toward the surface and gains energy. While a like (k0z) as in the case of a simple dielectric surface, but be characterized by a more complex spatial function f(r) that reflects simulation of the full dynamics of the atom (including the spatial 23 and internal degrees of freedom) is quite challenging, here we the mode shape of the underlying structure . However, the p estimate the heating rate from these transient processes. In scaling of the excited-state shift (doe Q/Dp) and emission G =D2 particular, we first find the spatial propagator under the pure ( / Q p) is generic to coupling between an atom and resonator ground-state potential, U(z , z , t , t ) (see Methods). Then, of Lorentzian lineshape, and thus the capability of achieving f i f i highly stable traps with high-Q structures using our proposed assuming that a spontaneous emission event occurs at ti ¼ 0 with the atom initially in the motional ground state c(z ), the new techniques is maintained. Photonic crystals thus offer great i flexibility in tailoring the properties of vacuum traps (such as wavefunction thatR emerges once the coherence equilibrates is given by c(z )B dz U(z , z , t ,0)c(z ). The probability of dimensionality), which we plan to investigate in detail in future f i f i trans i work. For example, it has been shown that dielectric gratings can atom loss P per spontaneous emission event is estimated from trans produce quantum vacuum forces that trap atoms along the lateral the overlap between the initial and final states, P ¼ 1 trans directions of a dielectric interface35, which when combined with |/z |z S|2, with a corresponding trap lifetime of t 1 ¼ P R . f i trans trans sc our previous techniques could yield full three-dimensional traps. The overall trap lifetime is obtained by adding the individual We note that theoretical23 and experimental efforts13 have already lifetimes in parallel, t 1 ¼ðt 1 þ t 1 þ t 1 þ t 1 Þ, and is total r ad trans tunnel been made to achieve realistic hybrid atom traps in photonic plotted in Fig. 2a. The overall trap lifetime exhibits a complicated crystal structures, where a combination of vacuum forces and O D dependence on Rabi frequency and detuning p owing to the optical forces is required to stabilize the trap in all dimensions. In different scalings of the constituent heating mechanisms (see addition, while we have focused on trapping of atoms weakly Fig. 3b). As an example, however, for external parameters 5 4 dressed by an external laser, it would be interesting to investigate Dp ¼ 2.5 10 and O/Omin ¼ 1.5 10 for atomic Cs, one E the feasibility of laser-free traps, where atoms pumped into achieves a lifetime of ttotal 15 ms, ground-state confinement of E E metastable electronic levels are stably confined using engineered Dz 1 nm, trap depth of Udepth 10 mK and trap distance of E resonant shifts alone. zt 15 nm. This Rabi frequency corresponds to an incident The ability to create atomic traps with parameter sets (such as intensity of just 0.05 mW mm 2. To compare, nanofibre-based 8,11 depth, confinement and proximity to surfaces) that are not optical traps employ guided intensities three orders of possible in conventional traps (whether optical, electrical or magnitude larger to achieve confinement and surface distances magnetic) should lead to a number of intriguing applications. For D B of z 20 nm (corresponding to trap frequencies of example, it has been proposed that the trapping of atoms near om ‘ ¼ 2 100 kHz) and z B200 nm, respectively. 2p 4pmðDzÞ t dielectric surfaces with nanoscale features (such as a sub- E Trapping frequencies of om/2p 40 MHz would be needed to wavelength lattice) would facilitate large interaction strengths reach single-nanometer confinement, far beyond what is realistic and long-range interactions for ultracold atoms39.However,the in optical traps. major limitation of optical trapping is the divergent intensity

6 NATURE COMMUNICATIONS | 5:4343 | DOI: 10.1038/ncomms5343 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5343 ARTICLE needed to overcome the ground-state vacuum forces of these this interaction induces a shift of the bare ground-state energy by an amount Z Z structures. The use of resonantly enhanced excited-state repulsion X 2 1 he; 1r;o;j jHaf jg; 0ij dogðraÞ¼ 2 dr do ; ð16Þ to create a trap could significantly reduce the power requirements ‘ o þ o0 and enable smaller lattice constants. Furthermore, traps with j localization on the nanometer scale can induce inter-atomic forces where o0 is the bare resonance frequency of the atom. Substituting equations (10) and (13) into the expression above, and using analyticity of the integrand to rotate that are of comparable strength with molecular van der Waals the frequency integral onto the positive imaginary axis o ¼ iu, one finds27 forces, which enables novel opportunities to control ultracold Z1 20 2 atomic collisions and realize exotic interactions such as p-wave m0 u o0 do ðr Þ¼ du G ðr ; r ; iuÞ : ð17Þ 47 g a ‘ 2 2 Yi sc;ij a a Yj scattering of fermions . Atoms trapped near surfaces could also p o0 þ u 0 act as exquisite nanoscale probes of surface physics, including the Here, we have only included the scattered component of the Green’s function, precision measurement of vacuum forces themselves. as the free component gives a contribution that is independent of position. While we have considered a two-level atom for simplicity, a more realistic model of an Methods isotropic atom consists of multiple excited states and equal polarizabilities in each direction (that is, the ground state can emit virtual photons of any polarization Electromagnetic field quantization In this section, we briefly describe the . with equal strength). Since the contribution to the ground-state shift from each quantization scheme of the electromagnetic field in the presence of arbitrary linear excited-state transition is additive, equation (17) can be modified to account for dielectric media, which follows that of ref. 27. The free field is characterized this by setting Y ¼ Y for all i and associating o3Y2=ð3pE ‘ c3Þ¼G with the by a set of bosonic modes with frequencies o and annihilation operators a^ ðr; oÞ i 0 0 0 0 0 j free-space spontaneous emission rate of any excited state. This yields equation (1). (j ¼ x, y, z), with a corresponding Hamiltonian X Z Z ‘ y Excited-state interactions We want to analyse the effective excited-state prop- Hf ¼ dr dooa^j ðr; oÞâjðr; oÞ: ð9Þ . j erties in the specific case that an atom is weakly driven by a classical field of o The bosonic operators satisfy canonical commutation relations ½â ðr; oÞ; frequency L. We thus transform to a rotating frame where the free atomic evo- y j ds^ d o o â ðr0; o0Þ ¼ d dðr r0Þdðo o0Þ and are physically associated with noise lution is given by Ha ¼ ee and ¼ L 0 is the detuning between the laser k jk o polarization sources in the material. The electric field operator at frequency o is frequency and atomic resonance frequency 0. driven by the sources at the same frequency and is given in the Coulomb gauge by27 The standard technique to derive the excited-state shift and decay rate follows rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that of the ground state and uses time-independent second-order perturbation Z 27 ‘ E theory . In this approach, however, one cannot find the jump operator associated 2 0 0 0 0 0 Ejðr; oÞ¼im o dr Gjkðr; r ; oÞ Im Eðr ; oÞa^kðr ; oÞ: ð10Þ with photon emission, which is needed to calculate the effect of emission on atomic 0 p motion (for example, recoil heating). To rectify this, we employ an alternative Here and in the following, it is assumed thatP all repeated vector or tensor approach based on deriving an atomic master equation within the Born–Markov indices are summed over, for example, Gija^j ¼ Gijâj: Eðr;oÞ is the frequency- approximation48. In particular, the equation of motion for the reduced atomic j density matrix ra is given by dependent permittivity at position r, and Gij is the dyadic Green’s function which is the (gauge-invariant) solution to Z1 1 ÂÃÂÃ r_ ¼ Tr dt H~ ðtÞ; H~ ðt tÞ; r j0ih0j ; ð18Þ 2 a ‘ 2 f af af a o 0 0 ðÞrr Eðr; oÞ Gðr; r ; oÞ¼dðr r ÞI: ð11Þ 0 c2 R where H~ is the interaction Hamiltonian in the interaction picture (with respect to The total electric field is E(r) ¼ do E(r, o) þ h.c. Similarly, the magnetic field af R the free Hamiltonian H ¼ H þ H ) and |0S is the electromagnetic vacuum state. is BðrÞ¼ do 1 rEðr; oÞþh:c. It can be verified that these expressions 0 a f io Any exponentialR of time appearing in the integral can be evaluated using the preserve the same field commutation relations as in free-space field quantization. 1 relation dt eiot ¼ pdðoÞþiP 1. Using the field expansion of equation (10) in We now consider a two-level atom at position r and with ground and excited 0 o a H~ and following some manipulation, equation (18) can be written in the form states |gS,|eS, respectively, interacting with the electromagnetic field within the af i electric dipole approximation. The interaction Hamiltonian is given by _ ra ¼‘ ½Heff ; raþL½ra: ð19Þ Haf ¼d EðraÞð12Þ The effective Hamiltonian describes coherent interactions between the atom and vacuum field and takes the form ¼YiðsêgEiðraÞþh:c:Þ; ð13Þ Heff ¼ ‘ doeðraÞsêe; ð20Þ where is the atomic dipole matrix element. Yi which can be interpreted as a position-dependent energy shift of the excited state. Taking a simple isotropic two-level model for an atom, the excited-state shift is Plane-wave expansion. We consider an interface with vacuum in the region z40 found to be and a dielectric with permittivity E(o) in the region zo0. The solution to Z1 2 4 cG0 u G0pc equation (11) in the region z 0 can be written as the sum of a free component do ; ; ; ; o ; eðraÞ¼ 2 du 2 2 Tr Gscðra ra iuÞ Tr Re Gscðra ra LÞ (that is, the solution in isotropic vacuum) and a scattered component. In the region o0 o0 þ u oL 0rzrz0, they can be expanded in plane waves, which yields 0 Z ð21Þ 2 c 0 i 0 ik jj ðq q Þ d 0 ^^ ik? ðz zÞ ; Gfree ¼ 2 2 dkjj e ðz z Þzz þ e k0k0 as stated in the main text. 4p o 2k? The Liouvillian term L[ra] can be written in a Lindblad form ð14Þ X Z y 1 y y Z L½ra¼ dk Okjra Okj ðOkj Okjra þ ra Okj OkjÞ: ð22Þ 2 2 ic 0 0 j G ¼ dk eik jj ðq q Þeik?ðz þ z Þ sc 8p2o2 jj ð15Þ The jump operators take the form 1 o2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ð ^ ^ Þð ^ þ ^ Þþ ð^^ Þð^^ Þ : 2 4 ikr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rp k jj z k?k jj k jj z k?k jj rs 2 z k jj z k jj 2E0m0oL e k? c OkjðraÞ¼ dr ImEðr; oLÞYiG ðra; r; oLÞs^ge: ð23Þ ‘ ð2pÞ3=2 ij Here k|| ¼ (kx, ky) and q ¼ (x, y) are the wavevector and position along the The excited-state spontaneous emission rate is in turn given through the direction parallel to the interface, while the full wavevector is given by k0 ¼ P R 2 y ‘ k k ^z with the relation k2 þ k2 ¼ðo=cÞ . Further defining q ¼ relation j dk Okj Okj ¼ GðraÞsêe,or qjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi? jj ? ? 2 2 2G pc EðoÞðo=cÞ k as the perpendicular component of the wavevector in the Gðr Þ¼G þ 0 Tr Im G ðr ; r ; o Þ: ð24Þ jj a 0 o sc a a L k? q? k?E q? L medium, rs ¼ and rp ¼ are the Fresnel reflection coefficients for s k? þ q? k?E þ q? This again averages over dipole orientations and reproduces equation (3), with - and p polarized waves, respectively. the replacement of o0 oL. To derive this, we have used the identity Z o2 dr00 Im Eðr00; oÞG ðr; r00; oÞG ðr00; r0; oÞ¼Im G ðr; r0; oÞ: ð25Þ Ground-state interactions. The interaction Hamiltonian Haf couples the ground c2 ij jk ik state of the bare system |g,0S (consisting of the atomic ground state and vacuum) The expressions for the excited-state emission rate G(ra) and level shift doe(ra) y 27 to states je; 1r;o;jiâj ðr; oÞje; 0i. Within second-order perturbation theory, agree with previous derivations , if the laser frequency oL is replaced by the

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atomic resonance frequency o0. The appearance of oL physically describes the Here, zmax is the position corresponding to the maximum of the metastable effect of the weakly driven atom (that is, a linear optical element) interacting with barrier. The ground-state energy of Us(z) is solved numerically. its own Rayleigh-scattered field, which can be altered owing to the presence of Once the approximate ground-state-binding energy is obtained, the tunnelling nearby dielectric surfaces. This is in accordance with the well-known result for the probability Pt for a particle hitting the metastable barrier is calculated using the fluorescence spectrum of a driven atom, which is dominated by the Rayleigh Wentzel–Kramers–Brillouin approximation, 0 1 contribution for atomic populations seeoo1 (ref. 48). More importantly, we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zz2 obtained for the first time a decomposition of the jump operator associated with 2mðU ðzÞ‘ o Þ @ d b A: photon emission for arbitrary dielectric surroundings, which enables one to Pt ¼ 4 exp 2 dz ‘ 2 ð31Þ calculate recoil heating. z1 : Here, z1,2 are the solutions to Ud(z) ¼ ob and are thus the classical turning : Short-distance limit. We now derive the short-distance expansion of the excited- points of a particle with energy ob. The rate that the trapped atom collides with state properties. The resonant term in equations (2) and (3) depend on the quantity the barrier is approximated by Dpz/mDz, where the position and momentum uncertainties are obtained from the numerical ground-state solution. We then Tr Gsc(ra, ra, oL), which for planar interfaces is given by estimate the inverse of the tunnelling-limited lifetime to be t 1 ¼ Pt Dpz . Z1 tunnel mDz 2 i k jj 2ik?z c 2 2 Tr Gscðra; ra; oLÞ¼ dk jj e rs þ 2 ðk jj k?Þrp : ð26Þ 4p k? oL Recoil heating. Recoil heating can be calculated by treating the atomic position ra 0 in equation (23) as an operator. For notational simplicity, we shall consider recoil E For short distances, the expression is dominated by large k|| ik>, which gives heating only along the trapping direction z, although our results can be easily 2ik z 2k z rise to a long exponential tail e ? e jj . Expanding to second order for large generalized. The increase in momentum uncertainty owing to photon scattering is k||, this tail contributes as given by Z1 "# 2 2 1 c k jj E 1 EðE 1Þ d 2 2 2k jj z hp i¼Trðp r_ Þ: ð32Þ Tr Gscðra; ra; oLÞ dk jj e 2 þ 2 : ð27Þ z z a 2p oL E þ 1 ðE þ 1Þ dt 0 Here we will focus solely on the Liouvillian term in the density matrix evolution We now apply these results to the Drude model. First, we note the exact relation (that is, momentum incurred from spontaneous emission). It can be shown that the 2Q2 d EðoÞ¼1 2 . For frequencies within a range DptQ of the plasmon Liouvillian term produces no net force, that is, hpzi¼0. Using the fact that ðQ þ Dp Þ þ iðQ þ Dp Þ @f dt E 1 Q ½pz; f ðzaÞ ¼ i‘ for any function f, one can derive the following general resonance, one readily finds that the response is well approximated @za E þ 1 i þ 2Dp expression for the recoil heating rate in terms of Green’s functions, by a complex Lorentzian. For larger detunings Dp\Q, the exact expression of E(o) d ÀÁ must be used. Furthermore, within the range DptQ, one must compare the size of 2 ‘ 2 2 2 hpz i¼ m0oLYiYj @1 þ 2@1@2 @2 hIm Gijðza; za; oLÞsêei: ð33Þ the first and second terms on the right-handp sideffiffiffiffiffiffiffiffiffiffiffiffi of equation (27). The first term dt dominates the integral provided that k0zt Dp=Q, in which case Here, the derivatives @1 and @2 act on the first and second spatial arguments of the Green’s functions, respectively, and we have suppressed the atomic spatial kL Q variables in the directions that are not of interest. We further assume that the Tr Gscðra; ra; oLÞ 3 ; ð28Þ 8pðkLzÞ i þ 2Dp internal and spatial degrees of freedom can be de-correlated, hGijsêeihGijisee. which reproduces the asymptotic expressions for the excited-state shift and This is valid in our regime of interest where properties such as the excited-state linewidth in the main text. decay rate have negligible variation over the spatial extent of the atomic wave packets. As in the main text, we use the notation sij ¼hsîji to denote the expectation value of atomic operators. Formally, we can write the above equation in Tunnelling. The atomic escape rate from the metastable potential formed by the the convenient form vacuum trapping scheme owing to quantum tunnelling is calculated in the d Wentzel–Kramers–Brillouin approximation in two steps. First, we find an hp2ið‘ k Þ2Gðr Þs : ð34Þ : z eff a ee approximation to the real part of the bound-state energy ob. This is implemented dt by replacing the actual metastable-dressed-state potential Ud(z) with a stable one, This form is intuitive as it describes the momentum increase arising from a Us(z), according to the formula (also see Fig. 4) random walk process, where the atom experiences random momentum kicks of : size ± keff at a rate G(ra)see (that is, the photon-scattering rate). keff is a derived UsðzÞ¼UdðzÞðz4zmaxÞ; ð29Þ quantity that thus characterizes the effective momentum associated with scattered photons in the vicinity of a dielectric surface. : ¼ UdðzmaxÞðz zmaxÞ ð30Þ As a simple example, we consider an atom in free space polarized along x. Evaluating equation (33) withpffiffiffiffiffiffiffi the free-space Green’s function Gfree,xx from equation (14) yields keff ¼ 2=5ðoL=cÞ (and G(ra) ¼ G0), recovering the known 43 result . This recoil momentum is smaller than the full momentum oL/c of the scattered photon because the atom emits in a dipole pattern, but only the Z max projection of the photon momentum along z contributes to motional heating in that direction. Applying the same formalism to an atom near a dielectric described by the Drude model, we find that the scattered component of the Green’s functionpffiffiffi dominates the recoil heating, leading to the asymptotic expansion of keff 3=za given in the main text. In all of our numerical calculations, the full Green’s function is used rather than asymptotic results. (z)

U Transient heating. Following a spontaneous emission event, the spatial wavefunction of the atom is temporarily untrapped and evolves under the pure ground- state potential for a characteristic time ttrans before the internal dynamics equilibrates. Here we derive an approximate propagator U(z , z , t , t ) for the wave- Z f i f i Z1 2 function evolving under the ground-state potential. : The problem significantly simplifies if the ground-state potential Ug(z) ¼ og(z) is linearized around the atom trapping position z ¼ zt, p2 H z þ U ðz ÞþU0 ðz Þðz z Þ: ð35Þ 2m g t g t t This linearization is justified in our regime of interest where the atom is tightly Z trapped and the wavefunction undergoes little evolution in the time ttrans. The constant energy and position offsets in the Hamiltonian play no role, and thus we 2 Figure 4 | Tunnelling. Black curve: metastable-dressed-state potential pz 0 can consider the simplified Hamiltonian H ¼ 2m þ UgðztÞz. The propagator for Ud(z) (in arbitrary units) seen by the atom. The maximum of the this Hamiltonian is given by metastable barrier is located at zmax. Red curve: stable potential Us(z) used ! 0 0 2 3 : iUgtðzf þ ziÞ iðUgÞ t to approximately calculate the binding energy ob (blue line) of the Uðz ; z ; t; 0Þ¼U ðz ; z ; tÞ exp ; ð36Þ f i free f i 2‘ 24‘ m atom in the trap. The classical turning points z1, 2 of a particle with this pffiffiffiffiffiffiffiffi 2 energy, which are used to evaluate the tunnelling rate in the Wentzel– m imðzf zi Þ where Ufreeðzf ; zi; tÞ¼ 2pi‘ t exp 2‘ t is the well-known propagator for a Kramers–Brillouin approximation, are labelled as well. free particle49.

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Phys. 2, 856–861 (2006). Acknowledgements 17. Hartmann, M. J., Brandao, F. G. S. L. & Plenio, M. B. Strongly interacting We thank N. Stern and O. Painter for helpful discussions. D.E.C. acknowledges support polaritons in coupled arrays of cavities. Nat. Phys. 2, 849–855 (2006). from Fundacio´ Privada Cellex Barcelona. K.S. was funded by the NSF Physics Frontier 18. Chang, D. E. et al. Crystallization of strongly interacting photons in a nonlinear Center at the JQI. J.M.T. acknowledges funding from the NSF Physics Frontier Center at optical fibre. Nat. Phys. 4, 884–889 (2008). the JQI and the US Army Research Office MURI award W911NF0910406. H.J.K. 19. Lewenstein, M., Sanpera, A. & Ahufinger, A. Ultracold Atoms in Optical acknowledges funding from the IQIM, an NSF Physics Frontier Center with support of Lattices: Simulating Quantum Many-Body Physics (Oxford University Press, the Moore Foundation, by the AFOSR QuMPASS MURI, by the DoD NSSEFF program, 2012). and by NSF PHY-1205729. 20. Bolda, E. L., Tiesinga, E. & Julienne, P. S. Effective-scattering-length model of ultracold atomic collisions and Feshbach resonances in tight harmonic traps. Author contributions Phys. Rev. A 66, 013403 (2002). D.E.C. and H.J.K. conceived of the idea. D.E.C. and K.S. performed the calculations. All 21. Rahi, S. J., Kardar, M. & Emig, T. Constraints on stable equilibria with authors contributed to the elaboration of the project and helped in editing the manuscript. fluctuation-induced (Casimir) forces. Phys. Rev. Lett. 105, 070404 (2010). 22. Levin, M., McCauley, A. P., Rodriguez, A. W., Reid, M. T. H. & Johnson, S. G. Additional information Casimir repulsion between metallic objects in vacuum. Phys. Rev. Lett. 105, Competing financial interests: The authors declare no competing financial interests. 090403 (2010). 23. Hung, C.-L., Meenehan, S. M., Chang, D. E., Painter, O. & Kimble, H. J. Reprints and permission information is available online at http://npg.nature.com/ Trapped atoms in one-dimensional photonic crystals. New J. Phys. 15, 083026 reprintsandpermissions/ (2013). 24. Munday, J. N., Capasso, F. & Parsegian, V. A. Measured long-range repulsive How to cite this article: Chang, D. E. et al. Trapping atoms using nanoscale quantum Casimir-Lifshitz forces. Nature 457, 170–173 (2009). vacuum forces. Nat. Commun. 5:4343 doi: 10.1038/ncomms5343 (2014). 25. Antezza, M., Pitaevskii, L. P. & Stringari, S. New asymptotic behavior of the surface-atom force out of thermal equilibrium. Phys. Rev. Lett. 95, 113202 This work is licensed under a Creative Commons Attribution- (2005). NonCommercial-NoDerivs 4.0 International License. The images or 26. Failache, H., Saltiel, S., Fichet, M., Bloch, D. & Ducloy, M. Resonant van der other third party material in this article are included in the article’s Creative Commons Waals repulsion between excited Cs Atoms and sapphire surface. Phys. Rev. license, unless indicated otherwise in the credit line; if the material is not included Lett. 83, 5467–5470 (1999). under the Creative Commons license, users will need to obtain permission from the 27. Buhmann, S. Y. & Welsch, D.-G. Dispersion forces in macroscopic quantum license holder to reproduce the material. To view a copy of this license, visit http:// electrodynamics. Prog. Quant. Electron. 31, 51–130 (2007). creativecommons.org/licenses/by-nc-nd/4.0/