Creating and measuring topological matter – with non-local order deeply embedded in the global structure of its quantum mechanical eigenstates – presents unique experimental challenges. Since this order has no signature in local correlation functions, it might seem experimentally inaccessible in any macroscopic system; however, as the precisely quantized Hall plateaux in integer and fractional quantum Hall systems show, topology can have macroscopic signatures at the system's edges. Ultracold atoms provide new experimental platforms where both the intrinsic topology and the edge behavior can be directly measured. This article reviews, using specific examples, how non-interacting topological matter may be created and measured in quantum gases.
Measurement. Science is rooted in measurement: it is from measurements, as unified by theory, that understanding is born. Our comprehension of the universe is therefore bounded by our ability to observe and shaped by human creativity. Scientific progress is driven by the identification of new physical systems and measurement techniques, leading to new conceptual understanding. This article focuses on systems of ultracold neutral atoms, quantum gases, that make quantum physics manifest in the laboratory. Many properties of these systems can be understood in the intellectual context of many-body physics which describes systems from the commonplace, such as crystals, fluids, and semiconductors, to the extreme, such as superconductors, quantum Hall systems, and neutron stars. Many-body physics asks how the properties of individual components – atoms, electrons, nucleons – give rise to the observed macroscopic phenomena. For example, classifying phases of matter and understanding the transitions between them is an essential part of condensed matter physics.
Matter, with all its myriad phases, surrounds us. Some phases are familiar, such as gases, liquids, or even ferromagnets, while others, such as antiferromagnets or nematics, are still ubiquitous but pass us by, quite unbeknown to our everyday existence. Each of these phases is characterized by a local order parameter that captures the type and degree of order associated with it. For example, the ferromagnetic order parameter is simply the local degree of magnetization and the superfluid order parameter is the (complex-valued) superfluid density. Microscopically, these phases of matter result from the complicated interaction between practically uncountable many elementary constituents, yet their basic physics can be encapsulated in reduced models that include only those degrees of freedom which are important to a specific phase or phase transition.
Unlike ferromagnets, antiferromagnets are difficult to identify macroscopically, though their local order parameters are almost the same: magnetization compared to staggered magnetization. While staggered magnetic order might be “hidden” from our daily existence, neutron scattering makes it easy to identify .
Quite in contrast, a new type of order first known in the B-phase [2, 3] of superfluid 3He and then found in the integer quantum Hall states [4, 5] in two-dimensional (2D) electron systems is hidden in a more fundamental way. This topological order cannot be described by any local order parameter; instead, topological order is encapsulated by global topological invariants such as Chern numbers or Z2 invariants . With the discovery of topological insulators  – zero-magnetic-field analogs to integer quantum Hall effects (IQHEs) that preserve time-reversal (TR) symmetry – interest in topological order has been rekindled, leading finally to a complete classification of topological insulators (TIs) and superconductors  (only some of which exist in current material systems , such as HgCdTe quantum wells or bulk Bi2Se3).
Ultracold atoms are very different from conventional materials, composed of a few  to a few hundred million atoms , with densities ranging from below up to approximately , and at temperatures from below 1 nK to a couple of microkelvins. These atomic systems are unique in the simplicity of their underlying Hamiltonians along with remarkable experimental flexibility, enabling experimentalists to control and engineer their quantum degrees of freedom. An almost unavoidable requirement for creating topological matter is the existence of static gauge fields in the microscopic Hamiltonian. Examples include the electromagnetic vector potential describing a large magnetic field in the case of quantum Hall systems, or spin–orbit couplings for TR invariant TIs. This article assumes the existence of these fields, and focuses on detecting the resulting phases of matter.
For a broad overview of artificial gauge fields, I refer the interested reader to one of several high-quality contemporary reviews or book chapters, either at a high level , or with a more technical focus [11-13]. In light of those reviews, this article will focus on a couple of specific examples of how ultracold atom systems can realize topological matter, and how the unique measurement opportunities thereby afforded open new windows into these systems. Topological matter can be manifested either by its transport properties – as is typically observed in condensed matter systems – or more directly by topological properties of the eigenstates. Measurement of both is possible with cold atoms.
2 Light–matter interaction
First, if artificial gauge fields are to be created using internal – spin – atomic degrees of freedom coupled by light, then we must understand the basic spin-dependent interaction between light and matter. The conventional spin-independent (scalar, Us) optical potentials acquire additional spin-dependent terms near atomic resonances [14-16]: the rank-1 (vector, Uv) and rank-2 tensor light shifts. For the alkali metal atoms illuminated by light near the (D1) and (D2) lines, the effective atom–light coupling Hamiltonian for the ground-state atoms (with ) is
where E is the complex-valued optical electric field. The rank-2 term is negligible for the parameters of interest and hence is not included in HL. Here determines the vector light shift; is the fine-structure splitting; and are the D1 and D2 transition energies; and is a suitable average. us sets the scale of the light shift and is proportional to the atoms’ AC polarizability.
The scalar and vector light shifts in HL can be independently specified with informed choices of laser frequency ω and intensity. Evidently, the vector light shift is a contribution to the total Hamiltonian acting like an effective magnetic field
that acts on and not the nuclear spin . At small magnetic fields when the Zeeman shifts are small in comparison with the hyperfine splitting – the linear, or anomalous, Zeeman regime – the effective Hamiltonian for a single manifold of total angular momentum states is
Here Beff acts as a true magnetic field and adds vectorially with B, and since in alkali metal atoms, we safely neglected a contribution to the Hamiltonian. We also introduced the hyperfine Landé g-factor . In the lowest energy manifold of 87Rb with , for which and , we get .
3 Basic concepts and direct measurement
Ultracold atomic systems are unique in their simplicity, allowing for the direct experimental measurement of specific physical phenomena uncomplicated by spurious effects. For example, particles moving in a 2D lattice potential are described in terms of a series of energy bands , where q is the crystal momentum. Each of these bands, labeled by index n, is associated with a Chern number , an integer characterizing its topological properties. To better understand this characterization of energy bands, we will consider a simple model of a spin-1/2 particle with band structure given by the Hamiltonian density
where and are the vector of Pauli matrices and the identity, respectively; summation over repeated indices is implied. Furthermore, defines the spin-dependent eigenstates comprising the lowest two Brillouin zones (BZs); is the reciprocal lattice wave vector; describes the creation of a particle with crystal momentum q in the mth spin state; and we define . This system is described by a pair of energy bands, at each q separated in energy by , with spin- and momentum-dependent eigenstates parallel to on the Bloch sphere. In terms of the normalized vector , the Chern number (the suitable topological invariant for this system)
is equal and opposite for the two bands, and counts the number of times the Bloch sphere is covered in the integration over the BZ. Direct measurement of this quantity is challenging.
Simple model. With this context, consider the spin-dependent band structure defined by
where we have taken the reciprocal lattice wave vector to be , implying the lowest BZ extends from to π along both and . Figure 1a shows that the map from the BZ to the Bloch sphere in Eq. (2) covers the sphere once, suggesting that the resulting band structure is topologically non-trivial with Chern numbers in the lowest two bands.
Because Eq. (2) includes only 0 and momentum components, the Hamiltonian's coordinate space representation (for )
requires simple nearest-neighbor – though spin-dependent – hopping, along with a local Zeeman shift. Here is a two-component field operator describing the creation of a particle at site . Extensions of the initial Jaksch and Zoller proposal for creating artificial abelian magnetic fields  using light- or radiofrequency-induced spin-dependent hopping can lead to Hamiltonians of this type ; indeed, Eq. (3) is a minimal case of the proposal previously [19, 20]. In an experimental realization, all hopping – between nearest neighbors of a spin-dependent bipartite lattice – would be generated by external control fields: and would be induced by light-induced hopping, while ΩRF would be generated by a uniform radiofrequency coupling.
Figure 1a shows schematically the geometry of this lattice, where induced hopping along or couples together sites of a bipartite lattice. This can generate tunable spin-dependent hopping matrix elements and , dependent on the phase, direction, and strength of the Raman lasers driving the hopping.
Chern number detection. With ultracold atoms in optical lattices, it is straightforward to measure the momentum distribution , where is the momentum, by abruptly removing the lattice and trapping potentials, allowing the atomic ensemble to expand in the absence of interactions for a sufficiently long time-of-flight (TOF) period , followed by absorption imaging. For deep lattices, when the tight-binding approximation is valid, we have
where is the Fourier transform of the localized Wannier function describing atoms localized in individual lattice sites and is the crystal momentum distribution.
The crystal momentum distribution can be experimentally measured using band mapping techniques [22, 23], where the optical lattice potential is ramped off as adiabatically as possibly just before TOF. Unfortunately, these techniques are plagued with unavoidable failures of adiabaticity near the BZ edges, or more generally, wherever the band gap closes. Equation (4) shows that the crystal momentum can be measured more directly, simply using , with k confined to the lowest BZ where is not small (experimentally, better results can be obtained by averaging over all k, suitably weighted to account for technical noise ). Relevant here, the spin-resolved momentum distribution can be obtained by introducing a magnetic field gradient during TOF to Stern–Gerlach separate the two spin components, and , with the normalized difference yielding the component of the magnetization
In this setting, the power of cold atom experiments is evident: simple rotations about either or just before TOF transform the or observable into . Sau et al.  noted that a band insulator – or any state where each momentum state in the lowest band is occupied with equal probability – allows the full magnetization vector to be obtained with just three measurements. Thus, the Chern number can be obtained directly by integrating experimental data .
Figure 2a depicts simulated spin-resolved momentum distributions including realistic technical noise, and Fig. 2b shows the resulting three components of . From the three magnetization components, the Chern number can be directly computed by applying a discretized version of Eq. (1), leading to the Chern numbers displayed in Fig. 2c at zero temperature and in Fig. 2d at . In both cases, the topological phase diagram displayed in Fig. 1d is clearly reproduced.
In many cases, the flatness of the band structure matters; for example, fractional quantum Hall effects (FQHEs) require that the band width be small compared to the typical interaction strength. As Yang et al.  show, it is possible to produce arbitrarily flat bands (also with arbitrary Chern number) by systematically including long-range (spin-dependent) hopping terms into the Hamiltonian. To understand this, again consider from Eq. (2) with , giving
Using the normalized vector , a new coupling gives rise to flat bands, gapped by Ω0 and with unchanged. This can be approximated order-by-order in powers α, giving spin-dependent hopping of arbitrary range. This type of engineering is in principle elegant, but even next-nearest-neighbor tunneling is difficult to engineer at will.
Flux lattices, as invented by Cooper and Dalibard , provide a direct pathway for engineering topological band structures (also see earlier work  invoking the same concept using magnetic coupling). While the systematic band-flattening procedure of Ref.  cannot be applied without additional lasers, the large number of parameters in flux-lattice configurations  in principle allow for the formation of nearly perfectly flat bands [31, 32] without additional experimental complexity. These conceptually simple lattice structures require as few as four laser fields, Raman-coupling together different internal atomic states.
4 Edge states
The Chern number of the nth band is a non-local measure of the eigenstate topology within that band. A remarkable result by Thouless, Kohmoto, Nightingale, and den Nijs  – the celebrated TKNN formula – showed that the Hall conductivity of a Chern insulator is given by
the sum of the Chern numbers of all bands below the Fermi energy. This simple observation connects topology inside a gapped insulator to the properties of its edge states, and explains the quantized Hall conductivity in both the IQHE  and the FQHE . In addition to the direct measurement of topological indices discussed above, cold atom systems allow for measurement of edge state transport . Here I illustrate the sharp contrast between edge and transport that can be observed in cold atom systems, by focusing on the type of artificial gauge fields already realized in the laboratory .
Many of the initial proposals for creating artificial gauge fields without optical lattices [37-39] – as realized at NIST/JQI  – create a synthetic magnetic field in a region of space bounded in one direction, here . With these experimental techniques, the unavoidable inhomogeneity makes a direct connection to topological invariants difficult. As I show below, these systems, with their natural boundaries, can be bulk insulators with chiral edge states, allowing a direct application of the TKNN formula.
Most studies of artificial gauge fields (see, for example, the outstanding review article by Dalibard et al. ) consider the properties of laser-dressed atoms moving adiabatically in just one “dressed” state. In this dressed state, Berry's phase effects can give rise to synthetic gauge fields. In what follows, I study (1) the adiabatic gauge field ; then (2) relax the adiabatic approximation somewhat ; (3) solve for the full eigenstate structure suitable for the NIST/JQI experiments without any adiabatic approximation; (4) show the system is a bulk insulator; and (5) identify the chiral edge states. This discussion follows the presentation first put forth in the methods of Beeler et al. .
Review of IQHE systems. Before discussing artificial gauge fields, consider the more commonplace problem of fermions with charge e moving in 2D, subject to a perpendicular magnetic field , confined in a box potential
along . This system, with single particle states described by the Landau gauge Hamiltonian
can be directly solved by taking the ansatz for the eigenfunctions. This gives the one-dimensional (1D) Schrödinger equation
for ; q, the plane-wave wave vector along , should be interpreted as being somewhat like a canonical momentum (and definitely not a mechanical momentum); is the magnetic length; and is the cyclotron frequency. When is absent, this gives harmonic oscillator solutions for labeled by the Landau level index n, spaced in energy by , centered at , and with oscillator width .
When , the eigenenergies remain close to the homogeneous system values provided y0 is more than from the boundaries. But as Fig. 3a shows, the energies rapidly increase as exceeds and the wave function is pressed against the system's walls. Here, we see that for any given Fermi energy (depicted for example in the bulk gap between the and Landau levels), states – one pair per Landau level – reside at the Fermi energy (two states in the pictured case) and the only low-energy excitations are near these edge states. In this geometry, these are the edge states predicted by the TKNN formula, each contributing a single quantum of conductance. We explicitly verify this interpretation in two ways. Firstly, Fig. 3b plots the electron density associated with these two edge states, showing that they reside at the system's edge. Secondly, Fig. 3c shows the time evolution of a numerical experiment where a weak attractive Gaussian potential centered at and extended along y is removed at . At (left-hand panel of Fig. 3c), we see localized regions of increased electron density at the system's edges, reflecting the incompressible bulk. As t increases, this density perturbation propagates in a chiral manner on the top and bottom edges, with group velocity simply given by . This same method of solution can be applied in the symmetric gauge case for the disk geometry (suitable for the first continuum gauge field proposals ).
Artificial gauge fields: adiabatic picture. Figure 4a shows the typical laser geometry and coupling scheme for the the three-level case realized in experiment. In the temporally rotating frame with the rotating wave approximation, and after a suitable gauge transformation, this system is described by the single-particle Hamiltonian (see Ref.  for a first-principles derivation)
for an atom with total angular momentum f in the – plane. are the total angular momentum operators; and are the single-photon recoil momentum and energy; ΩR is the Raman coupling strength; is a position-dependent detuning from Raman resonance; and is an external confining potential. In the various adiabatic approximations, Eq. (6) is solved, for example ignoring the spatial dependence of δ, giving a set of dressed states that contain a vector potential term that depends parametrically on δ and .
First, we can anticipate that this Hamiltonian might give rise to effective magnetic fields by assuming that is large compared to the kinetic energy, and so that motion is fully adiabatic within the resulting dressed states. This is solved by rotating about by an angle given by ; the dynamics in any given band labeled by mF is governed by the projected Hamiltonian
Using the formalisms of Dalibard et al.  (in terms of wave functions) or Madison et al. (in terms of unitary transformations), it is straightforward to show that each of these bands independently has a vector potential
along with adiabatic and geometric potentials
Typically experiments are performed in the lowest energy dressed state, where assumes its maximal magnitude. Equation (8) illustrates two important points: firstly, that the magnitude is defined by the recoil momentum of the lasers as well as mF; and secondly, that the vector potential is strictly bounded between . Suppose now that the vector potential depends on y, for example via a linear detuning gradient: this would result in a Landau gauge form vector potential with non-zero synthetic magnetic field [38, 39]. Thus the adiabatic approximation – if it should be trusted – leads us to expect a magnetic field (observed in Ref. ) in this geometry.
This kind of adiabatic approximation is far stronger than needed. A weaker adiabatic approximation takes the explicit matrix
for Eq. (6), in this case for an manifold, and exactly obtains its eigenvalues . Here we introduced , conforming to the notation of our earlier publications. Figure 4c shows these effective dispersion relations with minima shifted from zero (a vector potential, similar to the adiabatic result but quantitatively different) and with an effective mass larger than m (the effective mass is just m in the usual adiabatic approximation). Figure 4d then shows the experimentally measured synthetic vector potential, compared to theory.
Artificial gauge fields: numerically exact. Here we avoid adiabatic approximations altogether by solving the full Raman-coupled Hamiltonian for a system that is extended along ; has a detuning profile ; and is confined in a potential along . The compensation potential is described below and and describe the system's extent along and .
As with the solution of an electron moving in a uniform magnetic field expressed in the Landau gauge, this problem can be solved by taking a separable wave function of the form . With this ansatz, the full 1D spinful, coordinate representation Hamiltonian describing motion along is
where is a component spinor wave function and is the suitable identity. It is straightforward to numerically solve this 1D eigenvalue problem for any , , and .
The scalar potentials and – which depend on δ – complicate matters. For the linear detuning gradient we focus on, these can be well canceled by a Gaussian laser giving : to have a bulk cyclotron gap, the residual potential modulations must be small compared to the effective cyclotron energy. Figure 4a depicts the complete required geometry including a blue-detuned beam creating a box-shaped trap potential along and an elliptical Gaussian beam, both providing compensation along and weak overall confinement along ; also, not pictured, is a 1D lattice along confining the system into 2D systems.
The four smallest eigenenergies of this Hamiltonian (Eq. (10)) are plotted as a function of q in Fig. 5a, showing that each state is specified by two quantum numbers: q and a Landau-level-like index n. We numerically verified that the low-energy spectra of this exact Hamiltonian (Eq. (10)) and that resulting from projecting onto the lowest band of Eq. (6) are indistinguishable. This is because atoms always reside near the local minima in . Near these minima, the band spacing is large compared to the nK interval between eigenenergies depicted in Fig. 5a (using 87Rb parameters, the energy scale is twice as large as for 40K). Figure 5a illustrates a second issue regarding this type of scheme: the vector potential saturates at as becomes large. As a result the magnetic field decreases to zero, and the cyclotron gap closes. The beginning of this effect is visible in Fig. 5a near .
In systems of fermions with small Fermi energy EF (horizontal line in Fig. 5a), the full QHE analogy is revealed: the two points at the Fermi energy correspond to edge modes on the top and bottom of the system. Figure 5b plots the computed density distribution of those eigenstates, showing that they reside on the system's edge, and in addition the two edge states counterpropagate (opposite group velocities).
Lastly, following the example of electrons in a magnetic field, we computed the time evolution of the system following the release of a small Gaussian potential perturbation . Figure 5c shows the nearly non-dispersive motion of these states, counterpropagating on the two edges, with no visible perturbation in the bulk. Truly this system behaves as a quantum Hall system: application of the TKNN equation confirms the topological nature of these synthetic cyclotron bands.
Physical parameters. To estimate the number of fermionic 40K atoms required, we consider a uniform system with length , constraining q to be a multiple of . Here the edge states are well isolated for a Fermi momentum (solid red and blue circles in Fig. 5a). Since this energy is fully in the gap, it implies that states with are occupied. Thus, the number of states below the Fermi energy is . For example, when the number of atoms will be , and the proposed system with height has a reasonable aspect ratio. With these approximately 200 atoms, the Fermi energy will lie in the gap between the and bands, which are spaced by . As with existing proposals for creating topological matter with cold atoms [17-19, 28, 31, 43-46], the required energy scales and atom numbers are low, but within the scales that have already been realized in the laboratory [8, 47]. In this case, to identify the existence of edge states, the recent technique proposed by Goldman et al.  could be implemented to directly image the motion of edge states, more dramatically than what is shown in Fig. 5c.
The conceptually simple method for designing the vector potential discussed here relies on only two lasers and three atomic states, making the experimental implementation relatively easy. One of the major stumbling blocks in using alkali metal atoms for QHE physics is the large photon scattering rate from the Raman lasers for alkali metal fermions, which leads to significant heating of the sample . It is an ongoing research project to create schemes that are simple, have low spontaneous emission, and have unbounded vector potentials.
In this tutorial article, I have discussed two complementary methods by which a topological invariant – the Chern number – can be measured, either directly in terms of its definition or indirectly via edge-state transport using many-atom systems near thermodynamic equilibrium. These approaches – well defined in the context of the simple single-particle physics underlying topological insulators – give the basic context in which new methods, applicable for interacting systems such as p-wave superconductors or FQHEs, can be invented and implemented. For example, as coherent and isolated quantum systems, ultracold atoms are ideally placed to perform interference experiments – with bosons or fermions – both building on the ideas from quantum Hall systems (such as quasi-particle interferometers ) and also going in new directions (such as the recent experimental measurement of the Zak phase in a 1D ladder-type system ). An alternative promising dynamical technique is to look at the gauge field's effect on Bloch oscillations .
Ian B. Spielman received his Ph.D. degree in physics from the California Institute of Technology in 2004 for studying quantum Hall bilayers. As a postdoc he spent two years at NIST, Gaithersburg (USA) studying the physics of the superfluid-to-insulator transition in 2D atomic Bose gases. Since 2006 he is a physicist at NIST. His research focuses on realizing Hamiltonians familiar in condensed matter physics in atomic systems, including pioneering work creating artificial gauge fields for neutral atoms.