Colloquium: Quantum matter built from nanoscopic lattices of atoms and photons

This Colloquium describes a new paradigm for creating strong quantum interactions of light and matter by way of single atoms and photons in nanoscopic lattices. Beyond the possibilities for quantitative improvements for familiar phenomena in atomic physics and quantum optics, there is a growing research community that is exploring novel quantum phases and phenomena that arise from atom-photon interactions in one- and two-dimensional nanophotonic lattices. Nanophotonic structures offer the intriguing possibility to control atom-photon interactions by engineering the medium properties through which they interact. An important aspect of these new research lines is that they have become possible only by pushing the state-of-the-art capabilities in nanophotonic device fabrication and by the integration of these capabilities into the realm of ultracold atoms. This Colloquium attempts to inform a broad physics community of the emerging opportunities in this new field on both theoretical and experimental fronts. The research is inherently multidisciplinary, spanning the fields of nanophotonics, atomic physics, quantum optics, and condensed matter physics.

Figure 1

Overview. Recent developments in experimental and theoretical techniques bring forth new atom-light interfaces (Sec. 3) that can simultaneously achieve stable atom trapping and strong atom-photon interactions beyond conventional settings (Sec. 2), and offer surprising new paradigms in atomic physics, cavity QED, and waveguide QED (Sec. 4). In this Colloquium we discuss these new possibilities, including a hybrid atom trap and vacuum lattices (Sec. 6), collective dissipation engineering (Sec. 7), chiral quantum optics (Sec. 8), and many-body physics with atom-atom (Sec. 9), spin-motion (Sec. 10), and photon-photon (Sec. 11) interactions.

Figure 2

Conventional approaches in quantum optics to achieve strong atom-photon interactions. (a) Diffraction-limited focusing of an optical beam onto a tightly trapped atom, (b) cavity QED, where the interaction is enhanced by a large number of photon round-trips, (c) atomic ensemble, where a large atom number results in high probability of interaction with a single photon, and (d) atomic ensemble of Rydberg atoms.

Figure 3

(a) Schematic of a ${\mathrm{SiO}}_2$ nanofiber of radius $r_{\text{fiber}}=250\mathrm{nm}$, with atoms trapped approximately 230 nm from the surface. Adapted from [240]. (b) Dispersion relation of the guided mode frequency $f$ vs axial wave vector $k$ (in units of $2\pi /r_{\text{fiber}}$) for the nanofiber. For sufficiently small fibers or low frequencies, all modes except the HE11 mode are cut off. The transition frequencies associated with the Cs D1 and D2 lines are also shown for reference. (c) Intensity distribution $|E{|}^2$ (in arbitrary units) for the fundamental HE11 mode of the nanofiber ([135]). (d) (Left) Ratio of emission rate ${\mathrm{\Gamma }}_{1\mathrm{D}}$ into the guided mode to total emission ${\mathrm{\Gamma }}_{\mathrm{tot}}$ of an atom trapped near a nanofiber of radius $r_{\text{fiber}}=200\mathrm{nm}$, as a function of dimensionless distance $r/r_{\text{fiber}}$. The curves correspond to different magnetic sublevels of Cs. (Right) Same ratio, calculated as a function of fiber radius (normalized by the resonant wave vector $k_0={\omega }_{eg}/c$ in vacuum), for an atom on top of the fiber surface. From [136].

Figure 4

The quasi-1D “alligator” photonic crystal waveguide (APCW). (a) Schematic illustration of a section of the APCW (${\mathrm{Si}}_3{\mathrm{N}}_4$ dielectric) and its dimensions, with one atom trapped per unit cell (green spheres). (b) Dispersion relation of frequency $\nu$ vs wave vector $k_x$ (normalized by $\pi /a$, where $a$ is the lattice constant) for the lowest order TE and TM modes (black and gray curves, respectively). A band gap (red shaded region) appears for the TE mode near the D1 and D2 transition frequencies of atomic Cs (dotted lines), in which the TE field cannot propagate but evanescently decays. The decay constant ${\kappa }_x$ (normalized by $a^{-1}$) is shown in red. From [107]. (c) Intensity profile $|E{|}^2$ (in arbitrary units) for the TE mode at the lower band edge. (d) Total spontaneous emission rate ${\mathrm{\Gamma }}_{\mathrm{tot}}$ (into both free space and guided modes), normalized by the vacuum rate ${\mathrm{\Gamma }}_0$, for a single atom located at the center of a unit cell in the middle of a APCW composed of $N_{\text{cells}}=150$ unit cells. The dashed vertical line indicates the frequency of the band edge.

Figure 5

Photonic crystal cavities. (a) A scanning electron microscope (SEM) image (upper panel) and simulation of mode intensity (lower) of a defect cavity defined in a one-dimensional photonic crystal beam. On the left of the upper panel is a tapered nanofiber used for coupling light into and out of the cavity. In [232], atoms were optically trapped nearby in the interference fringe formed by reflecting a tightly focused beam from the structure (inset). (b) An “optomechanical crystal” SEM image of a defect cavity in a one-dimensional photonic crystal beam. The panel below shows a zoom-in of the defect region, with variations in the sizes of the holes. This defect creates localized resonances both for photons and phonons as seen in the two bottom panels illustrating the cavity mode intensity and acoustic “breathing” modes. From [201]. (c) Cavity mode defined in a 2D photonic crystal membrane. In [250], the defect is formed by the omission of three holes from the otherwise periodic structure.

Figure 6

Comparison of cooperativity factors $C=g_{\mathrm{JC}}^2/\kappa {\mathrm{\Gamma }}^{\prime }$ for different cavity QED and atomic interfaces. With initial designs and reasonable $Q\approx 20000$, a nanoscopic photonic crystal cavity (red diamond) ([235]) and an APCW band-gap atom cavity (red star) ([107]) have already achieved cooperativity parameters $C>1$ that are comparable with state-of-the-art macroscopic and microscopic resonators, which include mirror Fabry-Perot cavities [triangles, 1 ([159]); 2 ([203])], fiber-based cavities [circles, 1 ([47]); 2 ([121])], microtoroid, microsphere, and bottle whispering-gallery-mode resonators [squares, 1 ([8]); 2 ([217]); 3 ([171]); open squares: projections using intrinsic $Q$]. Here the dashed lines mark constant quality factors $Q={10}^4$, ${10}^5$, ${10}^6$, ${10}^7$, and ${10}^8$ in increasing order. Blue stars mark projected improvements by (1) improved trap loading to minimize the effective mode area ([109]; [82]); (2) increased quality factor to $Q=2\times {10}^5$ as well as 10-fold reduction in the band curvature to reduce effective mode volume ([57]; [252]); (3) further suppression of ${\mathrm{\Gamma }}^{\prime }$ ([109]; [86]).

Figure 7

(a) Atoms coupled to a PCW with the atomic transition frequency ${\omega }_{eg}$ lying within a band of guided modes. Strong quantum coherence effects can emerge from collective dissipation through the waveguide, particularly when atoms are trapped at a spacing equal to integer multiples of half the resonant guided wavelength $\pi /k({\omega }_{eg})$, such that the phase factor $e^{ik({\omega }_{eg})|z_j-z_l|}=\pm 1$ between any pair of atoms. (b) Experimentally measured reflectance from an atomic ensemble coupled to an optical nanofiber when atoms are trapped near the ideal spacing. The reflectance (red curve) and optical depth (gray) are shown as a function of time. Both quantities decrease in time due to a decrease in the number of atoms (number of atoms shown in the inset). From [49].

Figure 8

(a) Interplay between the direction of propagation and polarization of the guided modes of an optical nanofiber, as illustrated in [179]. A field propagating in one direction can have near circular polarization of different orientations on opposite sides of the fiber. The orientation of circular polarization is reversed in the lab frame for fields propagating in the opposite direction. (b) Measured (circles) and predicted (bars) difference in right- and left-going emitted intensities from different Zeeman levels $m_{F^{\prime }}$ ($|m_{F^{\prime }}|\le 5$ for atomic Cs), normalized by the total guided intensity. From [161].

Figure 9

Quantum many-body physics for atomic transition frequencies in a band gap. (a) Schematic illustration of excited atom and associated photon bound state, with spatial extent $L$. A second atom nearby in ground state $g$ can be excited by an “exchange” interaction with the first atom. Inset: Band structure with atomic frequency ${\omega }_{eg}$ within a band gap, detuned by ${\mathrm{\Delta }}_{\mathrm{BE}}$ from the band edge. (b) Measurements showing the collectively enhanced emission rates ${\overline{N}}_a{\mathrm{\Gamma }}_{1\mathrm{D}}$ (green curve) and coherent interaction strengths ${\overline{N}}_a{J}_{1\mathrm{D}}$ (blue curve) near a band edge, both normalized by the vacuum emission rate ${\mathrm{\Gamma }}_0$. For detunings ${\mathrm{\Delta }}_{\mathrm{BE}}>0$, the atomic frequency lies within a band gap, which strongly suppresses emission into the waveguide while coherent interactions persist. Here ${\overline{N}}_a\simeq 3$ atoms. From [107].

Figure 10

A hybrid optical and Casimir-Polder trap in a double-beam PCW. (a) Schematic illustration of the waveguide structure. (b) Intensity profile of the blue-detuned guided mode. Green spheres indicate the location of the trap minima. (c) Casimir-Polder potential (top) and the total trap potential (bottom) in the $y\text{−}z$ plane. (d)–(f) Line cuts of the Casimir-Polder potential (solid red lines), FORT (dashed blue lines), and total potential (solid blue lines) through the trap center. Adapted from [109].

Figure 11

(a) Schematic of a Rb atom sitting at a fixed distance $z_A$ above a one-dimensional dielectric grating made of silicon. (b) Calculation of Casimir-Polder potential in this geometry along the lateral $x$ direction. The potential $U(x_A,z_A)$ is normalized by the value for a flat interface $U^{(0)}(z_A-a/2)$. The dimensions taken for the calculation are $a=100\mathrm{nm}$ and $d=2s=2z_A=6a$. (a), (b) From [48]. (c) Schematic of a GaP dielectric slab of thickness $W$ with dielectric posts of height $h$. (d) Cut along $y$ of the total potential (solid black) and side-illumination potential (dotted blue) at $z_A=32.5\mathrm{nm}$ for the case of a ${}^{87}\mathrm{Rb}$ atom. The chosen dimensions of the structure are $d=50\mathrm{nm}$, $R=0.2d$, and $W=h=119\mathrm{nm}$. (c), (d) From [86].

Figure 12

(a) Schematic of two two-level atoms coupled to a waveguide and individually addressed by lasers ${\mathrm{\Omega }}_{1,2}$. The single-atom level structure consists of atomic ground and excited states $|g,e⟩$, where $|e⟩$ decays into the waveguide and free space with rates ${\mathrm{\Gamma }}_{1\mathrm{D}}$ and ${\mathrm{\Gamma }}^{\prime }$, respectively. (b) Scheme for quantum state engineering: one single atom is separated several wavelengths apart such that it can be individually addressed with respect to an atomic ensemble of $N$ atoms that is collectively driven.

Figure 13

(a) Dynamical control of atom-atom interactions using a $\mathrm{\Lambda }$ atomic level scheme. The transition $|g⟩\text{−}|e⟩$ is coupled to the modes of the PCW, while the transition $|s⟩\text{−}|e⟩$ is off-resonantly driven (detuning ${\delta }_L$) by an external laser $\mathrm{\Omega }$. Atoms $j$ and $l$ may then exchange excitations in level $s$ via the illustrated two-photon processes. When this exchange occurs within the band gap, the detuning ${\mathrm{\Delta }}_L$ from the band edge (blue curve) tunes the range of the effective spin-exchange interaction. Furthermore, for full control of atom-atom interactions, arbitrary long-range spin-exchange interaction between atoms $j$ and $l$ may be engineered. In particular, a relative shift of the $s$ level of atom $l$ by ${\mathrm{\Delta }}_{jl}$ matched by a sideband of the Raman laser $\tilde{\mathrm{\Omega }}$ (shown in gray), selectively creates an interaction between these two atoms. (b) By engineering the interactions (red arrows mark nearest-neighbor interactions) between the atoms at each end of a finite linear chain using this full control technique, the system can be mapped to a chain with periodic boundary conditions.

Figure 14

(a) Trapping scheme to explore spin-motion coupling. The atoms (green) are trapped by an external potential (blue) at every second node of the Bloch function (red) of the PCW. (b) Considering a pair of atoms, interactions mediated by the photonic band gap lead the atoms to have lower energy if they displace from the nodes of the Bloch function (to the left $|L⟩$ or right $|R⟩$) and form either a spin triplet ($|+⟩$) or singlet ($|-⟩$). Without the exponential decay of the band-gap interactions the four possible configurations shown here would be degenerate. The exponential dependence of the band-gap interactions causes the state $|RL,+⟩$ to be the lowest-energy configuration. (c) Many-body phase diagram. For weak band-gap interactions, the atomic wave packets are centered around the minimum of the external trapping potential, and the spins exhibit an antiferomagnetic Néel ordering. For larger interaction strengths, a spontaneously broken symmetry emerges where consecutive pairs of atoms form spatial dimers, while the internal spin state of the pair exhibits a large spin triplet fraction.

Figure 15

(a) Imaginary part of the linear susceptibility (in arbitrary units) of an atom in the EIT configuration as a function of input probe detuning ${\mathrm{\Delta }}_p={\omega }_p-{\omega }_{eg}$. The imaginary part of the susceptibility is zero at zero detuning, indicating zero absorption of a continuous probe. Inset: The level structure of EIT, showing a probe field (red) on the $|g⟩\text{−}|e⟩$ transition and a control field on the $|s⟩\text{−}|e⟩$ transition. (b) A shift of the level $s$ (inset) leads to a shifted atomic response where zero absorption is now achieved at a detuning ${\mathrm{\Delta }}_p\ne 0$. (c) In a PCW, the shift of level $s$ can occur via interactions with other atoms in level $s$. These interaction-induced shifts create new transparencies for different components of the input probe. Here we see the single-photon transmission $I_1$ is unaffected by the shift, while the two-photon transparency is shifted to the right as a result of uniform interactions $U=0.5\mathrm{\Gamma }$. Transmission is shown for an input coherent state and normalized by the input single-photon and two-photon intensities for control Rabi frequency $\mathrm{\Omega }=1.9\mathrm{\Gamma }$ and an ensemble optical depth of 400.