Abstract
Neutral particles subject to artificial gauge potentials can behave as charged particles in magnetic fields. This fascinating premise has led to demonstrations of oneway waveguides, topologically protected edge states and Landau levels for photons. In ultracold neutral atoms, effective gauge fields have allowed the emulation of matter under strong magnetic fields leading to realization of HarperHofstadter and Haldane models. Here we show that application of perpendicular electric and magnetic fields effects a tunable artificial gauge potential for twodimensional microcavity exciton polaritons. For verification, we perform interferometric measurements of the associated phase accumulated during coherent polariton transport. Since the gauge potential originates from the magnetoelectric Stark effect, it can be realized for photons strongly coupled to excitations in any polarizable medium. Together with strong polariton–polariton interactions and engineered polariton lattices, artificial gauge fields could play a key role in investigation of nonequilibrium dynamics of strongly correlated photons.
Introduction
Synthesis of artificial gauge fields for photons have been demonstrated in a number of different optical systems. In most cases, the implementation is achieved through the design of the optical system^{1,2,3,4}, leaving little or no room for fast control of the magnitude of the effected gauge field after sample fabrication is completed. For many applications on the other hand, it is essential to be able to tune or adjust the strength of the gauge field during the experiment^{5,6,7}; this is particularly the case for nanophotonic structures^{8} where fast local control of the gauge field strength may open up new possibilities for investigation of manybody physics of light^{9}.
Cavitypolaritons are hybrid lightmatter quasiparticles arising from nonperturbative coupling between quantum well (QW) excitons and cavity photons. A magnetic field applied along the growth direction influences polaritons through their excitonic nature and leads to a diamagnetic shift, Zeeman splitting of circularly polarized modes and enhancement of exciton–photon coupling strength^{10}. For excitons with a nonzero momentum, the applied B_{z} also induces an electric dipole moment^{11,12,13,14}. In a classical picture, this dipole moment is due to the Lorentz force that creates an effective electric field E_{eff} (ref. 15) that is proportional to k × B. For excitons with small momentum and polarizability α, this E_{eff} causes an induced dipole moment (d∝αk × B). An additional external electric field E_{ext} in the QW plane then alters the dispersion of excitons as it leads to energy changes proportional to k:−d·E_{ext}∝−α(k × B)·E_{ext}, so that the energy minimum is no longer at k=0 as would be the case for a free particle with an effective mass but at finite k. This simple modification of the dispersion relation is equivalent to an effective gauge potential A_{eff} for excitons; due to their partly excitonic character, the Hamiltonian describing the dynamics of polaritons also contains an effective gauge potential term proportional to A_{eff}.
In the realization we describe here, based on the effective gauge potential A_{eff}, the strength and direction of the effected gauge potential is controlled electrically. Moreover, since our scheme relies on the magnetoelectric Stark effect, time reversal symmetry^{16,17,18} of the optical excitations is broken. This realization, in combination with strong polariton–polariton interactions^{19,20,21}, has the potential to open up new possibilities of investigating manybody physics of light.
Results
Demonstration of the effective electric field E _{eff}
The cavity polariton sample we use is illustrated in Fig. 1a. Our demonstration relies on the fact that it is possible to excite polaritons with a welldefined inplane wavevector k by appropriately choosing the angle and energy of the excitation beam. As illustrated in Fig. 1a, we choose . Two metal gates deposited 30 μm apart allow us to apply an electric field in the x direction such that for polaritons propagating with a wavevector along the y direction, will add or cancel due to the Lorentz force. By recording changes in the transition energy of polaritons as a function of E_{ext}, we can determine the strength of E_{eff}. In our experiments, we only address the lower polariton branch.
Changes in the reflected intensity of a laser beam probing polaritons at B_{z}=0 T with =2.7 μm^{−1} as a function of E_{ext} is shown in Fig. 1d. For each E_{ext}, the reflected intensity shows a dip at an energy corresponding to the polariton resonance (Fig. 1e). The spectral centre of the dip shifts to lower energies with E_{ext} in a way that is well described by a secondorder polynomial. The expected behaviour for a neutral polarizable quasiparticle (that is, exciton or polariton) that is subject to E_{ext}, would be to have a dcStark shift equal to −αE_{ext}^{2} (ref. 22); here the coefficient α is the quasiparticle polarizability. Therefore, the electric field that yields the maximum lowerpolariton energy identifies that exactly cancels any internal or effective electric fields E_{eff} . To quantify E_{eff} due to the Lorentz force described above, we extract the difference of at a fixed B_{z} for polaritons excited with =2.7 μm^{−1} and =−2.9 μm^{−1}. This approach also allows us to exclude influences of builtin electric fields. At B_{z}=0 T, we find that the difference of V_{G} values that correspond to the maximum energy with excitation is 0.02 V, indicating that E_{eff} is negligible at 0 T (Fig. 1f).
In stark contrast, for B_{z}=5 T, the energy shift of polaritons with E_{ext} displays a significant difference between for and , as illustrated in Fig. 2a. The magnetic field dependence of the difference of for , Fig. 2b, shows a behaviour that is well described by a linear increase with B_{z}, demonstrating E_{eff} due to the Lorentz force.
B _{ z }induced changes to polaritons
With increasing B_{z}, the polarizability decreases and the polariton energy at E_{ext}=0 increases, as illustrated in Fig. 2c. The polariton energy at E_{ext}=0 and the polarizability can be extracted from a fit to a secondorder polynomial: −dE_{ext}−α. The firstorder coefficient of the polynomial in turn, yields the induced dipole moment of the polaritons (excitons). The difference of the firstorder coefficient for ±k_{y} excitation beams (twice the induced dipole moment) is shown in Fig. 2d: we observe that the induced dipole moment first increases with B_{z} and then starts to slowly decrease for B_{z}≥4 T. We expect that for the high magnetic field regime, the induced dipole moment decreases as (ref. 12). We find very good agreement with a theoretical model of the polaritons (shown as solid lines in Fig. 2) that allows us to identify the physical mechanism underlying the overall B_{z} and E_{ext} dependence (see Supplementary Note 4 for detailed information about the model). The change in the lower polariton energy is due to an interplay between a diamagnetic blueshift of the exciton transition energy and a redshift due to an increase in the cavityexciton coupling strength^{10}. Concurrently, the polarizability decreases with B_{z}, as would be expected in a classical picture as the size of the polarizable particle decreases. We emphasize that due to the change in the exciton energy and cavityexciton coupling strength the exciton content of the polaritons changes as B_{z} is varied. We also find that the energy shift of polaritons with E_{ext} is influenced by the decrease in the electron–hole overlap due to E_{ext}induced polarization, which leads to a reduction of the cavityexciton coupling strength. In the E_{ext} range that we probe the decrease in the cavityexciton coupling leads to an effective smaller polarizability for polaritons as compared to excitons, due to a reduction on the excitonic character of the polaritons. Moreover, the model also captures the presence of E_{eff} due to the centre of mass motion of polaritons and confirms that even with changes in exciton content with B_{z}, its expected dependence on B_{z} is still linear. Finally, we emphasize that while E_{eff} arises from exciton physics, the strong coupling between excitons and photons has to be taken into account to obtain a quantitative agreement between the model and our experiments.
Demonstration of a gauge potential for polaritons
Remarkably, our findings demonstrate that polariton energy under perpendicular B and E_{ext} depends not only on the magnitude of k but also on its direction: this nonreciprocal flow of light is an indication of the presence of a gauge field for polaritons. The dispersion relation for free excitons propagating in the QW plane with wavevector in the presence of an electric field is:
where M is the total exciton mass, and is the exciton energy at k_{y}=0 and E_{ext}=0. is an effective vector potential for excitons, with and . The strong coupling to the cavity further changes the dispersion relation, but polaritons in a narrow energymomentum range can still be treated as free particles under the influence of an effective vector potential for polaritons, qA, whose value depends on the detuning between the excitons and polaritons as well as their effective masses (see Supplementary Note 4).
For our experiments, both B_{z} and E_{ext} are uniform over the area of our experiment yielding a constant gauge potential qA. Since physical observables are gauge invariant, we might expect that a constant gauge potential would not have any observable experimental signatures, and our measurements to be independent of qA. In practice, our experiments involve propagation of photons between two regions with different constant gauge potentials; when calculating the magnitude of qA above, we fixed a gauge by choosing the constant gauge potential for photons outside the cavity to be qA_{out}=0. The difference qA−qA_{out}≠0 is gauge invariant and stems from an effective (sheet) magnetic field at the interface between the cavity and vacuum. The experimental signatures of qA, such as shift of the energy minimum of polariton dispersion away from k=0, are observable in our experiments due to this effective magnetic field that alters the kinetic momentum as photons/polaritons pass through the interface^{23,24,25}.
To demonstrate the validity of the description of photon/polariton dynamics as determined by a gauge potential, we perform an interference experiment where we measure the phase accumulated by polaritons propagating for a length l inside the sample relative to a fixed phase reference (see Fig. 3a and ‘Methods’ section for detailed information about the interference experiment). An interference image obtained at E_{ext}=0 is shown in Fig. 3b. Changes in the interference pattern with y at two different values of E_{ext} are illustrated in Fig. 3c.
A plot of the phase change of polaritons with E_{ext} for different directions of excitation at B_{z}=6 T is shown in Fig. 3d. The data show that at a fixed E_{ext} and at the same excitation frequency, the phase accumulated for polaritons propagating along directions differ, and that this phase difference depends on the sign of E_{ext} and B_{z}. The kinetic momentum of polaritons with a fixed energy is modified by the absence or presence of a constant vector potential so that the additional accumulated phase for travelling a distance l is given by . At a fixed E_{ext}, the difference between the phase accumulated when exciting with allows us to avoid phase changes due to the Stark effect and to directly obtain . The fact that depends both on the sign of the electric field as well as the magnetic field is another manifestation of the validity of the description with qA. Since the phase shift as a function of E_{ext} can be modelled with a secondorder polynomial in E_{ext}, the dcStark effect is the dominant underlying mechanism for the phase shift. Figure 3d shows that polaritons acquire a phase shift of ∼0.25 radians due to qA as they propagate an average distance l=9 μm.
Discussion
The parameters of our experiment already ensure significant phase accumulation over the lattice constant (≥2.4 μm) of stateoftheart polariton lattices^{8} or average separation (∼4 μm) of coupled polariton microcavities^{26}. Choosing detunings that yield predominantly excitonlike lower polaritons will further increase qA. In addition, employing structures with longer polariton lifetimes^{27} will allow the realization of polariton lattices with larger lattice constants. With these advances, it should be possible to design structures where the accumulated polariton phase due to qA in a unit cell could be on the order of π.
Together with strong photon–photon interactions, realization of tunable artificial gauge fields is key for ongoing research aimed at the observation of topological order in drivendissipative photonic systems^{28,29}. Cavitypolaritons constitute particularly promising candidates in this endeavour since their partly excitonic nature enhances interactions^{19,20,21}, and in an external magnetic field interplay of exciton Zeeman splitting and photonic spinorbit coupling, can lead to opening of nontrivial energy gaps^{30,31}. Our work demonstrates that the same polariton system may be complemented by the realization of tunable gauge fields, if an inplane electricfield gradient is introduced in the QW. While such gradients are manifest in most gated structures, creation of a constant artificial magnetic field over a region of several microns should be possible in specially designed structures. We envision that lattices of dipolaritons^{32} would provide a very promising avenue towards this endeavour: the use of an inplane magnetic field in this case would effect a gauge potential that is substantially enhanced by the large dipolariton polarizability and strong external electric fields along the growth direction^{13,14}. The gradients of the latter in turn could be used to generate fluxes approaching magnetic flux quantum in a plaquette of area ∼5 × 5 μm^{2}. (Di)Polaritons that are subject to complex electric field distributions enabling large, tunable and inhomogeneous effective gauge field profiles^{33} would constitute a novel feature in the exploration of topologically nontrivial nonequilibrium quantum systems.
Methods
Sample
Our sample consists of three layers of 9.6 nmthick In_{0.04}Ga_{0.96}As QWs sandwiched between two distributed Bragg reflectors (DBRs) formed by 20 (top) and 22 (bottom) pairs of λ/4 thick AlAs/GaAs layers. The GaAs spacer layer containing QWs between the top and bottom DBRs is 1λ thick. Spectral linewidths, measured at a position where the cavity mode is far reddetuned from the exciton mode, indicate that the cavity has a Q factor exceeding 10^{4} (fullwidth at halfmaximum linewidth 0.11 meV). The exciton emission spectrum at the same position has a fullwidth at halfmaximum linewidth less than 1.2 meV.
Using electron beam lithography and liftoff techniques, we have defined rectangular metal (10 nm Ti, 210 nm Au) pads that are 700 × 120 μm. All of the experiments reported in this paper were carried out between two such pads that are separated by 30 μm. These pads are wire bonded onto a chip carrier, and were driven by a highvoltage amplifier (Falco Systems WMA300) to apply electric fields to the QW. The position on the sample is chosen such that for and B_{z}=0 the detuning of the cavity mode energy from the exciton resonance is 0.09 meV, which is small compared to the exciton cavity coupling strength 5.2 meV.
Inplane electric field
As described in Fig. 1a, we apply an electric potential V_{G} between two gates to create E_{ext}. We find a good agreement between our data and a numerical calculation for the polariton behaviour (see Supplementary Notes 1 and 4) with the electric field E_{ext}=0.84 V_{G}/(30 μm). Since the two quantities are linearly related, we use E_{ext} and V_{G} interchangeably. For all experiments that involve a nonzero inplane electric field, we acquire data in a pulse sequence (see Supplementary Note 2). This sequence is applied to avoid charge accumulationinduced variation in the actual electric field.
kresolved photoluminescence spectra
To measure the kresolved photoluminescence (PL) spectra, polaritons are created nonresonantly by a 780 nm continuouswave diode laser. The incoming collimated laser beam passes through a high numerical aperture (NA) lens (NA=0.68) so that the beam is focused on the sample plane. The PL signals from the sample also pass through the same lens. The back aperture of the high NA lens is the Fourier plane of emission. Emission around a particular inplane momentum (k_{x}, k_{y}) is collected by coupling the light from a small spot in the Fourier plane into a singlemode fibre which is connected to a spectrometer. To obtain the polariton energy dispersion with respect to the inplane wavevector k_{y}, we varied the position of the singlemode fibre, which changes the collection spot on the Fourier plane. For each position that corresponds to k_{x}=0 and a k_{y} value, we record the PL spectrum on the spectrometer. The incident 780 nm laser is blocked using a 800 nm longpass filter at the entrance to the spectrometer.
Polariton interference measurement
We use the setup depicted in Fig. 3a to create two beams that are incident with different inplane momenta ( and k=0) and are separated by l. The laser energy is chosen to be nearly resonant with the polariton modes with . Owing to the steep polariton dispersion, at the same energy, the k=0 beam is off resonance from the polariton modes; the photons in this beam are therefore reflected by the top mirror without being subject to qA. Photons in the beam are converted into a propagating polariton cloud. As polaritons propagate towards the position of the k=0 beam, they accumulate an additional phase due to qA. Propagating polaritons continue to emit photons out of the sample, and when these photons overlap with those from the k=0 beam they interfere, allowing us to measure changes in the accumulated phase.
We model the light intensity in the region where the interference occurs as the sum of a Gaussian (k=0 beam) and a plane wave (polaritons propagating with k_{y}):
We independently determine b_{0}, σ_{y} and y_{0} using images obtained at very high applied electric field, for which the polariton term is negligible , and fix k_{y}. For each E_{ext}, we fit the detected polariton intensity to equation (2) to find values of a_{0}, b_{1} and φ.
Since our excitation beams have a finite size, they have a finite width in k_{y} (0.45 μm^{−1}), therefore, the exact k_{y} value of the polaritons that are excited is determined by the energy of the excitation beam. For a fixed excitation laser energy, changes in E_{ext} will shift the dispersion relation so that Δk_{y} is resonant with the excitation laser. These changes (due to qA or Stark effect) in the dispersion relation show up as phase changes in our experiments, Δφ=lΔk_{y} where l is the mean distance from the excitation spot to the interference region. Since we are interested in how the phase changes with E_{ext} and the absolute phase between the two beams is not determined, we set the phase at E_{ext}=0 to 0 radian.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Additional information
How to cite this article: Lim, H.T. et al. Electrically tunable artificial gauge potential for polaritons. Nat. Commun. 8, 14540 doi: 10.1038/ncomms14540 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Rechtsman, M. C. et al. Straininduced pseudomagnetic field and photonic Landau levels in dielectric structures. Nat. Photon 7, 153–158 (2012).
 2.
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
 3.
Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon 7, 1001–1005 (2013).
 4.
Schine, N., Ryou, A., Gromov, A., Sommer, A. & Simon, J. Synthetic Landau levels for photons. Nature 534, 671–675 (2016).
 5.
Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).
 6.
Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).
 7.
Dalibard, J., Gerbier, F., Juzelinas, G. & Öhberg, P. Colloquium: artificial gauge potentials for neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).
 8.
Jacqmin, T. et al. Direct observation of dirac cones and a flatband in a honeycomb lattice for polaritons. Phys. Rev. Lett. 112, 116402 (2014).
 9.
Carusotto, I. & Ciuti, C. Quantum fluids of light. Rev. Mod. Phys. 85, 299–366 (2013).
 10.
Pietka, B. et al. Magnetic field tuning of excitonpolaritons in a semiconductor microcavity. Phys. Rev. B 91, 075309 (2015).
 11.
Kallin, C. & Halperin, B. I. Excitations from a filled Landau level in the twodimensional electron gas. Phys. Rev. B 30, 5655–5668 (1984).
 12.
Paquet, D., Rice, T. M. & Ueda, K. Twodimensional electronhole fluid in a strong perpendicular magnetic field: exciton Bose condensate or maximum density twodimensional droplet. Phys. Rev. B 32, 5208–5221 (1985).
 13.
Lozovik, Y. E., Ovchinnikov, I. V., Volkov, S. Y., Butov, L. V. & Chemla, D. S. Quasitwodimensional excitons in finite magnetic fields. Phys. Rev. B 65, 235304 (2002).
 14.
Butov, L. V. et al. Observation of magnetically induced effectivemass enhancement of quasi2D excitons. Phys. Rev. Lett. 87, 216804 (2001).
 15.
Thomas, D. G. & Hopfield, J. J. A magnetoStark effect and exciton motion in CdS. Phys. Rev. 124, 657–665 (1961).
 16.
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 17.
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon 6, 782–787 (2012).
 18.
Tzuang, L. D., Fang, K., Nussenzveig, P., Fan, S. & Lipson, M. Nonreciprocal phase shift induced by an effective magnetic flux for light. Nat. Photon 8, 701–705 (2014).
 19.
Amo, A. et al. Superfluidity of polaritons in semiconductor microcavities. Nat. Phys. 5, 805–810 (2009).
 20.
Ferrier, L. et al. Polariton parametric oscillation in a single micropillar cavity. Appl. Phys. Lett. 97, 031105 (2010).
 21.
Ferrier, L. et al. Interactions in confined polariton condensates. Phys. Rev. Lett. 106, 126401 (2011).
 22.
Miller, D. A. B. et al. Electric field dependence of optical absorption near the band gap of quantumwell structures. Phys. Rev. B 32, 1043–1060 (1985).
 23.
Fang, K. & Fan, S. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013).
 24.
LeBlanc, L. J. et al. Gauge matters: observing the vortexnucleation transition in a Bose condensate. New J. Phys. 17, 065016 (2015).
 25.
Kennedy, C. J., Burton, W. C., Chung, W. C. & Ketterle, W. Observation of BoseEinstein condensation in a strong synthetic magnetic field. Nat. Phys. 11, 859–864 (2015).
 26.
Rodriguez, S. R. K. et al. Interactioninduced hopping phase in drivendissipative coupled photonic microcavities. Nat. Commun. 7, 11887 (2016).
 27.
Steger, M., Gautham, C., Snoke, D. W., Pfeiffer, L. & West, K. Slow reflection and twophoton generation of microcavity excitonpolaritons. Optica 2, 1–5 (2015).
 28.
Hafezi, M., Lukin, M. D. & Taylor, J. M. Nonequilibrium fractional quantum Hall state of light. New J. Phys. 15, 063001 (2013).
 29.
Umucalilar, R. O. & Carusotto, I. Fractional quantum Hall states of photons in an array of dissipative coupled cavities. Phys. Rev. Lett. 108, 206809 (2012).
 30.
Nalitov, A. V., Solnyshkov, D. D. & Malpuech, G. Polariton Z topological insulator. Phys. Rev. Lett. 114, 116401 (2015).
 31.
Karzig, T., Bardyn, C.E., Lindner, N. H. & Refael, G. Topological Polaritons. Phys. Rev. X 5, 031001 (2015).
 32.
Cristofolini, P. et al. Coupling quantum tunneling with cavity photons. Science 336, 704–707 (2012).
 33.
Imamoglu, A. Inhibition of spontaneous emission from quantumwell magnetoexcitons. Phys. Rev. B 54, R14285–R14288 (1996).
Acknowledgements
The authors acknowledge many insightful discussions with Iacopo Carusotto. This work is supported by SIQURO, NCCR QSIT and an ERC Advanced investigator grant (POLTDES).
Author information
Author notes
 HyangTag Lim
 & Emre Togan
These authors contributed equally to this work
Affiliations
Institute of Quantum Electronics, ETH Zurich, CH8093 Zurich, Switzerland
 HyangTag Lim
 , Emre Togan
 , Martin Kroner
 , Javier MiguelSanchez
 & Atac Imamoğlu
Authors
Search for HyangTag Lim in:
Search for Emre Togan in:
Search for Martin Kroner in:
Search for Javier MiguelSanchez in:
Search for Atac Imamoğlu in:
Contributions
H.T.L. and E.T. designed the experiments, carried out the measurements and analysed the results. J.M.S. and E.T. fabricated the sample. M.K. helped with the experiments. A.I. supervised the project. H.T.L., E.T. and A.I. wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Emre Togan or Atac Imamoğlu.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary Notes, Supplementary Figures and Supplementary References
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Further reading

Observation of localized modes at effective gauge field interface in synthetic mesh lattice
Scientific Reports (2019)

Interacting Floquet polaritons
Nature (2019)

Onchip polariton generation using an embedded nanograting microring circuit
Results in Physics (2018)

Enhanced Interactions between Dipolar Polaritons
Physical Review Letters (2018)

Floquet topological polaritons in semiconductor microcavities
Physical Review B (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.