Abstract
The geometric properties of energy bands underlie fascinating phenomena in many systems, including solidstate, ultracold gases and photonics. The local geometric characteristics such as the Berry curvature^{1} can be related to global topological invariants such as those classifying the quantum Hall states or topological insulators. Regardless of the band topology, however, any nonzero Berry curvature can have important consequences, such as in the semiclassical evolution of a coherent wavepacket. Here, we experimentally demonstrate that the wavepacket dynamics can be used to directly map out the Berry curvature. To this end, we use optical pulses in two coupled fibre loops to study the discrete time evolution of a wavepacket in a onedimensional geometric ‘charge’ pump, where the Berry curvature leads to an anomalous displacement of the wavepacket. This is both the first direct observation of Berry curvature effects in an optical system, and a proofofprinciple demonstration that wavepacket dynamics can serve as a highresolution tool for mapping out geometric properties.
Main
The Berry curvature is a geometrical property of an energy band, which plays a key role in many physical phenomena as it encodes how eigenstates evolve as a local function of parameters^{1}. In a twodimensional (2D) quantum Hall system, for example, the integral of the Berry curvature over the 2D Brillouin zone determines the Chern number: a global topological invariant that underlies the quantization of Hall transport for a 2D filled energy band^{2}. As first explained by Thouless^{3}, there can be an analogous topological quantization of particle transport in a 1D band insulator, when the lattice potential is ‘pumped’, that is, is slowly and periodically modulated in time. Also in this case, the geometrical and topological properties are defined for an effective 2D parameter space, but now one spanned by the 1D Bloch momentum and the external periodic pumping parameter.
A local nonzero Berry curvature can have striking physical effects in both 2D systems and 1D pumps, regardless of whether the global topological Chern number is nontrivial. In the simplest case, a semiclassical wavepacket moves as a coherent object governed by classical equations of motion with an additional ‘anomalous’ Hall velocity due to the geometrical Berry curvature at its centreofmass, as an external force is applied or as the control parameter is pumped^{1,4}. As highlighted further below, this anomalous transport can be understood physically as the Berry curvature acting like a magnetic field in parameter space^{5,6,7}.
In recent years, there have been many landmark experiments to engineer and study geometrical and topological energy bands in ultracold gases and photonics^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}. In photonics, for example, topological edge states have been studied in a wide variety of (effectively) 1D setups, such as quantum walks^{14} and pumping in optical quasicrystals^{13,29,30}, as well as in 2D quantum Halllike systems of photonic crystals^{9,10,11}, propagating waveguides^{12} and silicon ring resonators^{15,16}. However, while the existence of such edge states is linked to the nontrivial global topological properties of bulk bands, a direct measurement of the Berry curvature itself has so far not been realized. In ultracold atoms, instead, there has been much progress in this direction, for example, with experiments exploiting the effects of Berry curvature on particle transport to extract, in 2D systems, the Chern number^{19} and topological phase diagrams^{20}, and, in 1D systems, both the quantized ‘Thouless pumping’ of atoms in a filled band^{23,24} and the unquantized ‘geometrical pumping’ of a Bose–Einstein condensate^{25}.
In this Letter, we go beyond previous experiments to provide the first direct measurement of the Berry curvature in an optical system. We also demonstrate the first full tomography of the Berry curvature of a geometrical energy band from measurements of the anomalous displacement of a coherent classical wavepacket. This approach complements existing methods in ultracold gases to measure the Berry curvature via band projection^{26} or using Wilson lines^{27}, with the significant advantage that it does not require a full reconstruction of the band eigenstates as an intermediate step. Consequently, our method can be easily applied to complex systems with many bands and degrees of freedom, where a full determination of the eigenstates would be experimentally challenging.
In our ‘photonic mesh lattice’ setup^{31,32}, pulses are injected into two coupled optical fibre loops, so that the temporal evolution of the outgoing pulsetrain envelope simulates a wavepacket in a 1D lattice. More specifically, the position of the particle is encoded in the time at which a light pulse completes a round trip through one fibre loop. Tunnelling between time slots is achieved as the two coupled loops have slightly different lengths, allowing the pulse to ‘hop’ from one time slot to another. We then apply a periodic phase modulation in one loop to engineer Floquet quasienergy bands, in which the eigenstates depend on both the 1D ‘Bloch momentum’, Q, and on a control parameter, φ, given by the amplitude of the phase modulation. With respect to these Floquet bands, the Berry curvature takes the form^{1}
where ψ_{j}〉 are the eigenstates making up the jth band in the effective 2D parameter space. By preparing a wavepacket with a narrow spectral width and a welldefined centreofmass momentum Q, we realize a geometrical charge pump by slowly ramping the phase amplitude, φ, in time. Due to this pumping, the wavepacket moves along the 1D ‘photonic mesh lattice’, with an anomalous velocity contribution^{1,23,24,25} due to the Berry curvature Ω_{j}^{φ(t), Q}. Our highresolution observation of the wavepacket dynamics then allows us to map out the Berry curvature for an optical energy band for the first time. By bringing geometrical energy bands into the new setting of photonic mesh lattices, our work opens up new opportunities for engineering novel systems, combining topology, nonlinearity, gain or loss, which may then be studied via wavepacket dynamics.
In our experiment, we employ a timemultiplexing, linear optical scheme based on two coupled fibre loops of different lengths^{31,32} as shown in Fig. 1a. At the beginning of each measurement, an optical pulse with a wavelength λ = 1,555 nm is injected into the longer loop before being split by a 50/50 coupler into one pulse in each loop. After every round trip, each pulse is again split into two, although due to the different loop lengths, the shortloop pulse completes the round trip in a time, T_{2}, which is shorter than that taken in the long loop, T_{1}. Crucially, the average looplength is much larger than the difference in loop lengths; in particular, the former is about 1 km, corresponding to a ns average round trip time, while the latter is only about 7 m, corresponding to a time difference of ΔT = T_{1} − T_{2} ≈ 35.4 ns between round trips in the long and short loops. Furthermore, to avoid any overlap between distinct pulses, the pulse width T_{pulse} is set to 25 ns, such that it is shorter than the minimum pulse separation ΔT. Thanks to the separation of all of these timescales, we observe clearly separated sequences of pulses over time (see Fig. 1), so that each pulse can be straightforwardly labelled by two integers (m, n), with m being the total number of round trips and n counting how many more round trips were made in the long compared with the short loop. Of course, this picture is restricted to sequences shorter than the round trip time in order for the assignment of integers to be unambiguous; this limits the maximum number of pulses in a given sequence to .
To detect the average intensities of pulses in both loops, we use a combination of a photodiode and a digital sampling oscilloscope with a bandwidth of 200 MHz and a temporal distance of the samples of 0.4 ns. Thanks to the long pulsewidth, T_{pulse}, dispersive effects in the optical fibres are typically unimportant, even over 400 round trips, corresponding to 400 km of propagation. However, for removing noise and for modulating the amplitude and phase of the pulses, bandpass filters and modulators are needed, as discussed in detail in the Supplementary Information. These additional components introduce losses on the order of approximately 8 dB per round trip, which are compensated by erbiumdoped fibre amplifiers. In general, the use of these amplifiers destroys quantum correlations; however, all experiments here are performed with a classical, coherent bright light pulse and do not rely on quantum effects. Besides compensating any losses, the erbiumdoped fibre amplifiers also allow for the use of pulses with a peak power of up to 200 mW. In the future, when combined with dispersioncompensating fibres, this system can be used, therefore, for the measurement of nonlinear effects, for example, the formation of solitons based on nonlinear selfphase modulation, which will be of great interest to study in the context of geometrical or topological energy bands. Additionally, this experimental platform also allows for the investigation of the topology of nonHermitian systems, by including gain and loss in a controlled way. This can be achieved, by using fibrecoupled acoustooptical modulators, which are set to a transmission of 50%. Depending on an electrical radiofrequency signal, this value can be dynamically increased or decreased resulting in an effective gain or loss ^{32}.
Based on the labelling of pulses according to (m, n), the evolution of light can be mapped to a 1 + 1dimensional lattice as depicted in Fig. 1b, c, where m is the number of discrete time steps on the vertical axis, and n is the position along the horizontal axis, which can either increase or decrease by one at each time step. In our setup, the evolution between each time step is:
where u_{n}^{m} and v_{n}^{m} denote the pulse amplitudes incident on the beamsplitter from the short and long loops. As can be seen in these equations, pulse amplitudes at the m + 1 time step are an equal superposition of pulses at the previous time step m, with a relative π/2 phase shift for light that couples from one loop to the other, due to the 50/50 beamsplitter coupling the two loops. As mentioned above, we have neglected here dispersion effects and assumed that the pulse shape does not depend on the specific route taken. In the short loop, we have inserted an additional phase modulator that provides a timedependent phase shift Φ(m). A suitable choice of this phase shift will generate the nontrivial geometrical Berry curvature as discussed in the following and in the Supplementary Information.
Without the phase shift Φ(m), the lattice depicted in Fig. 1b and described by the above equations (2) and (3) is periodic under a double step in the time step m → m + 2 and the position n → n + 2. Guided by the tightbinding analogy above, we apply the socalled Floquet–Bloch ansatz for the eigenstates^{32,33}
where the parameter Q is the ‘Bloch momentum’ along the 1D lattice, introduced above, and θ is the propagation constant or ‘quasienergy’. Here, the periodic part of the eigenstates in the jth band is denoted by ψ_{j} = (U, V)^{T} and the quasienergy band structure satisfies^{33}
leading to two bands that are periodic in both Q and θ as expected in a Floquet system, with a bandtouching at the edge of the 1D Brillouin zone where Q = ±π (see Supplementary Note 2). In general, the number of bands can be increased by using a ‘spatially’structured phase modulation, that is, a modulation dependent on position n, which increases the size of the unit cell^{32}, or by adding additional fibre loops of different length, which can also increase the effective ‘spatial’ dimensionality.
We can engineer the timedependent phase shift Φ(m) to realize energy bands with nontrivial geometrical properties, by, for example, cyclically applying a positive phase +φ for even time steps and −φ for odd (see Fig. 2a), and hence breaking a symmetry of the 1D lattice (see Supplementary Methods for further discussion). Experimentally, this means switching the sign of the phase modulation every time that all pulses complete another round trip. Therefore, the necessary modulation frequency of the phase modulator is on the order of kHz. As the doublestep periodicity is maintained, we again use the above ansatz equation (4) and find two bands satisfying
as plotted in Fig. 2b, c (see also Supplementary Information). The manifold of eigenstates is now defined over the Bloch momentum Q and the periodic phase amplitude φ, allowing us to define the Berry curvature in equation (1). As shown in Fig. 2d, this Berry curvature is singular at points where the two bands touch but otherwise can take large nonzero positive and negative values across the effective 2D parameter space.
To probe the Berry curvature, we prepare an optical wavepacket by exciting a superposition of states in the jth band with a welldefined mean Bloch momentum Q and with a narrow momentum spread ΔQ ≈ 0.05π. This can be achieved in good approximation by initially generating a chain of pulses with a Gaussian envelope in the short and long loop, and by setting the correct phase modulation between the envelopes in both loops^{34}. Then, by slowly increasing the magnitude of the phase modulation after every second time step, we ‘pump’ the wavepacket along the band in the effective 2D parameter space. We distinguish therefore between two timescales; firstly, there is the fast timescale m, which counts the round trips and over which the sign of the phase flips as
while the phase magnitude φ, that is, the pumping parameter, is determined by the slower timescale M(m). Here, we let the phase magnitude increase linearly with M as
and set M(m) = ⌊m/2⌋. For simplicity, hereafter in our notation we drop the explicit dependence on m. Note that the fast phase variation in equation (7) is directly included in the bands thanks to the ansatz in equation (4), while the slow variation in equation (8) adiabatically ‘pumps’ the system, provided that the rate φ_{0} is sufficiently small compared with the bandgap.
The centreofmass of the Gaussian envelope then evolves along the 1D lattice with a total velocity
where the first term is the usual group velocity from the band dispersion
and the second is the anomalous velocity due to the Berry curvature. This is easiest to understand for a wavepacket in a 2D lattice, in which case , with the Berry curvature,
playing the role of a magnetic field in the 2D Brillouin zone, which is defined over the Bloch momenta Q_{x} and Q_{y} (refs 5,6,7). This anomalous velocity is then the momentumspace analogue of the Lorentz force , acting on a charged electron in a realspace magnetic field, , perpendicular to the 2D lattice. In a 1D pump, Q_{y} is replaced by the periodic pump parameter, in a procedure often known as dimensional reduction^{35}. As the pump parameter is externally controlled, the dynamics along one direction are frozen out, leaving only the anomalous term in equation (9).
Experimentally, what we measure is the shift in the centreofmass of the Gaussian envelope along the ‘position’ n as a function of the number of modulation periods M
We note that the limit of topological Thouless pumping can be recovered from this expression, if we consider the motion of a filled and gapped energy band after a complete adiabatic pump cycle. Then, the shift is quantized as an integer^{3}
where BZ stands for Brillouin zone, C_{j} is the topological Chern number, and where we have used that the group velocity integrates to zero over the full 2D parameter space. In our experiment, however, we consider a system in which the energy bands are not gapped and so the assumption of adiabatic evolution across the entire 2D parameter space breaks down and equation (12) is not applicable. Rather, we focus here on the local Berry curvature, and, in particular, on ways to exploit the local sensitivity of a spectrally narrow wavepacket to experimentally extract its value.
To cleanly separate Berry curvature from group velocity effects, we apply a variant of the ‘timereversal’ protocol^{36} and compare the propagation (see Fig. 3a) under a reversal of the sign of the phase magnitude (that is, +φ_{0} and −φ_{0}) to find
where we have used that, for this modulation, v_{j}^{G}(φ, Q) = v_{j}^{G}(−φ, Q) and Ω_{j}^{φ, Q} = Ω_{j}^{−φ, Q} (see Supplementary Information 4 and 5). By measuring the realspace propagation of the wavepackets for ±φ_{0}, the dynamical evolution and hence the influence of the nonvanishing Berry curvature is directly recorded. After exciting an initial eigenstate, no further information about the eigenstates is required as we directly measure the Berry curvature from its physical consequences on wavepacket dynamics; this is in contrast to other approaches for measuring the Berry curvature^{26,27} that require a full reconstruction of the band eigenstates as an intermediate step. The integrated Berry curvature over a complete phasepumping cycle is shown in Figs 2e and 3b. Note that this anomalous displacement can become very large as we approach the bandtouching points where the Berry curvature diverges and adiabaticity breaks down; experimentally we measure an anomalous displacement of up to approximately 1 position over 200 time steps for a wavepacket centred around Q = 0.1π subject to a phase gradient of φ_{0} = π/50. The general applicability of this method is further confirmed in Supplementary Information 8, where the same experimental protocol is applied to a second example of modulation and pumping sequence.
As the most important advance of our work, the wavepacket evolution can be measured after each discrete time step, allowing for unprecedented high time resolution of the dynamics. We exploit this to map out the local Berry curvature directly from anomalous transport for the first time, by performing a discrete differentiation of the lateral shift Δn_{j}(M) as (see Fig. 3c)
The result of this procedure is shown in Fig. 3d, e; as can be seen, the main features of the Berry curvature are well captured, such as the overall symmetry and sign, the quantitative magnitude over much of the band and the predicted complementarity of the two bands^{1,26} (that is, Ω_{1}^{φ, Q} = −Ω_{2}^{φ, Q}). Due to the ‘on the fly’ tracking of the wavepacket, we can measure the Berry curvature even along trajectories that eventually hit band singularities. Close to the centre of the Brillouin zone (Q ≈ 0), for example, we can extract the Berry curvature until the singularity is hit at φ = π; afterwards, part of the population is transferred to the other band and the measurement is no longer meaningful. For a gapped band, this technique would straightforwardly give the topological Chern number (equation (12)) when we integrate the measured Berry curvature over the effective 2D parameter space.
In conclusion, we have demonstrated that anomalous wavepacket dynamics can provide an important experimental tool for mapping out the local geometrical properties of energy bands. This versatility of this scheme should allow it to be applied in a variety of configurations, including systems with gain and loss^{32} and/or nontrivial topological invariants. In the presence of nonlinearities, this technique could provide a new approach for the creation and manipulation of different wavepacket structures, such as solitons. Given its simplicity, our experimental setup also holds the promise of transferring geometrical and topological concepts into the applied world of optoelectronics.
Methods
A detailed description of the experimental platform is included in Supplementary Note 1.
To reduce the noise limit of the data acquisition, the measured propagation was averaged over approximately 25 realizations. For each band, we sequentially prepared wavepackets with an initial mean Bloch momentum Q, which was scanned from −π to π in steps of 0.05π. For each value of Q, we performed measurements for −φ_{0} and +φ_{0} before then setting a new value of Q.
To start with a spatially broad and thus spectrally narrow distribution at a specific Bloch momentum Q, a single seed pulse is injected into the fibre loop system. By switching off one of the two loops in a cyclic fashion after every second time step m, the system is converted into a lossy mesh lattice as discussed in ref. 37. The resulting propagation is given by a diffusion equation leading to a coherent splitting of the initial pulse into a chain of pulses with a Gaussian envelope. Due to this diffusive motion, the width of the resulting wavepacket grows . Thus, for 200 time steps a wavepacket with a width of ≈8 positions with respect to the fullwidth at halfmaximum of the intensity is generated, which provides a sufficiently small spectral width ΔQ ≈ 0.05π to map out the local Berry curvature.
To excite a specific Bloch momentum Q, we apply a phase of φ = Q during the diffusive motion and a phase of φ = (Q ± π)/2 for the last time step before the actual propagation. The choice of the sign allows for a selection of either the upper or the lower band. As discussed in refs 34,37, in this way, the appropriate eigenstate corresponding to a Bloch momentum Q is excited. After the preparation of the Gaussian beam over the first 200 time steps, there is no further switchingoff of either loop and, instead, we measure the wavepacket dynamics during the next 200 time steps, using the phase modulator to adiabatically ‘pump’ the system as discussed in the main text.
The centreofmasses of the wavepackets were extracted by fitting Gaussian distributions to the crosssections for each time step. A detailed explanation of this fitting procedure, together with examples, is included in Supplementary Note 6.
The error bars depicted in Fig. 2e are estimated by the spectral width ΔQ ≈ 0.05π of the initial excitation in the horizontal direction and by the uncertainty of the fit procedure for a 95% confidence level in the vertical direction.
For the full reconstruction of the Berry curvature, it is necessary to differentiate the measured integrated Berry curvature with respect to the time step m as discussed in equation (14). Directly before evaluating the derivative, a moving average filter is applied to the data to decrease the noise level. A span of 5 values was chosen for the averaging. Raw data without averaging are provided in the Supplementary Methods.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.
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Acknowledgements
M.W. acknowledges financial support from the Erlangen Graduate School of Advanced Optical Technologies. Additionally, M.W. would like to thank M. Kremer and A. Bisianov for fruitful discussions. Furthermore, this project was supported by PE 523/141 and by the GRK2101 funded by the DFG. H.M.P. was supported by the EC through the H2020 Marie SklodowskaCurie Action, Individual Fellowship Grant No. 656093 SynOptic. I.C. was funded by the EUFET Proactive grant AQuS, Project No. 640800, and by Provincia Autonoma di Trento, partially through the project ‘On silicon chip quantum optics for quantum computing and secure communications (SiQuro)’.
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M.W. performed the experiments; all authors contributed to the theoretical background and the interpretation of the measurement.
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Wimmer, M., Price, H., Carusotto, I. et al. Experimental measurement of the Berry curvature from anomalous transport. Nature Phys 13, 545–550 (2017). https://doi.org/10.1038/nphys4050
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