Electrically tunable artificial gauge potential for polaritons

Neutral particles subject to artificial gauge potentials can behave as charged particles in magnetic fields. This fascinating premise has led to demonstrations of one-way 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 Harper-Hofstadter and Haldane models. Here we show that application of perpendicular electric and magnetic fields effects a tunable artificial gauge potential for two-dimensional 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 non-equilibrium dynamics of strongly correlated photons.

To confirm that the metal pad configuration described in the Methods section of the main text provides a uniform and controllable electric field distribution we performed a 2D electrostatic simulation of the dielectric environment in COMSOL. We modelled the AlAs/GaAs/InGaAs environment as a single dielectric with = 12.9. The gates are modelled as 200 nm thick metals with a 30 µm gap, and the electric potential is constant and fixed on their boundary. The electric potential, and the electric field distribution when one of the gates is applied 1 V and the other is kept at 0 V is illustrated in Supplementary Figure 1. In the simulations we find that the electric field within a few microns from the centre of the gap of the gates at a depth that corresponds to the quantum well (QW) position should be relatively uniform and along the x direction. In this uniform region the electric field is 0.69 V G /(30 µm) where V G is the applied voltage between the gates.

Supplementary Note 2. EXPERIMENTAL DETAILS
We find that in measurements where an electric field is applied, the results are affected by the duration, intensity, and energy of excitation lasers. We believe these changes are due to excess charge carriers that are optically created that screen the externally applied electric field. In order to avoid such effects, in experiments where an applied electric field is needed, we perform experiments with a weak laser (New Focus TLB-6716, < 100 pW for reflection experiments and ≤ 40 nW for interference experiments) and, to reduce light exposure further, we apply a sequence of low duty cycle laser pulses. These laser pulses are timed with a voltage pulse sequence that alternates the sign of the applied voltage between the gates to ensure that the average electric field applied to the sample remains 0.
For reflection and interference experiments we use the pulse sequence illustrated in Supplementary Figure 2. To acquire reflection or interference data corresponding to a target voltage V target we ensure that the counter or the camera (PointGray Grasshopper 3 41C6NIR) is gated so that data from the time interval when the applied voltage is V target is recorded. Voltage pulse lengths are chosen to be 22.4 µs. To limit the exposure time of the laser on the sample, the excitation laser is modulated using an acousto-optic modulator (AOM, double pass, extinction ≥ 50 dB).
The excitation laser is switched on by the AOM only around the time interval where V target is applied. Following these voltage pulses we apply 20 pairs of the high voltage pulse ±V high = ±10 V to remove the remaining charge carriers created during the measurement so that these excessive charge carriers do not affect the following measurement. Our data is not sensitive to the exact values used in the sequence, for example voltage value used for V high pulses, the number of pairs of V high pulses used, voltage pulse lengths, and laser illumination duration. Hence these values can be varied without affecting the results, but some form of this pulse sequence is necessary to obtain consistent electric field-dependent data. For all the electric field-dependent data reported in this paper, we use the sequence described in this paragraph.
Each data point measured presented in the main text is an average of data recorded in multiple runs of the pulse sequence. For example the reflection data shown in Figure 1 (main text) consists of 4,500 repetitions of the pulse sequence for each target voltage value, whereas the interference image shown in Figure 3 (main text) is formed by acquiring and summing 107,000 frames from the camera; each frame corresponds to a repetition of the pulse sequence. Data acquisition from the camera is carried out in blocks of 1,000 frames where pixel values from all 1,000 frames are added to form a single image. Following a block of data acquisition we do an equal acquisition with 1,000 frames with the illumination laser turned off and record the difference between the two images to remove any noise due to a persistent background in the images.
Polarization of the excitation beams is fixed to be linear (close to p-polarization) for all experiments. In the reflection measurement we measure the reflected light intensity without any additional polarization filtering. In polariton interference experiment the polarizer in front of the camera transmits the light nearly orthogonally polarized to the incident beams to observe high visibility interference images. This nearly orthogonal polarization configuration significantly at-AOM Counter Camera Acquisition We then fit the exciton-polariton energy dispersion with respect to the in-plane momentum k y to: where Ω is the exciton cavity coupling strength that we refer to as Rabi energy, and we assume that exc (k y ) exc (0) and cav (k y ) cav (0) + 1 2mcav ( k y ) 2 . Based on this relation, we can extract Due to the finite size of the beams used, a k y uncertainty, which we quantify below, is present for both excitation and detection beams. With this uncertainty, we expect emission spectra measured around a mean k y value to be asymmetric, and the linewidth of the spectra to depend on this mean value. In particular our collection fibre acts as a filter and selects a distribution of k y values. We model the field distribution as a probability density function given by We assume a polariton resonance at LP (k y ) leads to an emission intensity lineshape given by A is a constant characterizing intensity of polariton emission. The measured spectrum is then modelled by With σ k = 0.5 × 10 6 m −1 , and the linewidth Γ = 120 µeV we obtain a reasonable agreement between the model and the observed lineshapes. Supplementary Figure 5 illustrates the resulting asymmetric lineshapes as well as the change in the expected linewidths with k y . Note that at high k y values there is a high energy tail in the emission lineshapes that is not captured by this model.
This tail occurs at energies close to the bare exciton resonance. To measure the k y distribution of the excitation beams used for the interference experiment (main text Figure 3) we slightly modify the interference experiment performed. We detune the excitation laser to the red of the polariton resonances so that the laser does not excite any polaritons, and we change the polarizer angle in front of the camera such that two beams have equal detected intensities. We also change the distance between the high NA lens and the sample so that the two beams overlap at the sample surface. With the sample illuminated by the k − y beam and the k = 0 beam, an image of the light intensity at the surface shows an interference pattern illustrated in Supplementary Figure 6. We emphasize again that, unlike the experiments in the main text, here polaritons have not been excited, and the image purely shows the interference of two laser beams.
This interference pattern is due to the spatially varying phase between two beams, for simplicity (considering only y direction) we model this interference as the interference of two beams that are defined by their field distributions f k − y (k y ) and f 0 (k y ). To extract k − y and σ − k we take a 2D Fourier Transform of the intensity image, and analyze the line cut at k x = 0. From the position of the two Gaussian peaks we find k − y = −2.9 × 10 6 m −1 . From their widths we find σ − k = 0.4 × 10 6 m −1 . Acquiring and analyzing a dataset using k + y excitation beam we find k + y = 2.7 × 10 6 m −1 and σ + k = 0.5 × 10 6 m −1 .
x (μm) Extracting electric field dependent energy shift of polaritons Due to the fibre coupling we employ, the excitation laser intensity exhibits significant changes (up to 25 %) as its energy is tuned. Since the excitation intensity is less than 100 pW, we do not expect any changes in the polariton behaviour due to these intensity variations. To be able to fit, and visualize the underlying changes in intensity due to the excitation of polariton resonances, we calculate and plot the ratio of the recorded intensity to the average of the intensity of the data obtained for the six highest voltage values applied for each laser energy. For example in Figure 1d in the main text each horizontal line is obtained by dividing the reflected intensity for each voltage by the average intensity of the reflection at 11.04, 11.52, 12, 12.48, 12.96 and 13.44 V.
This normalization procedure however affects the lineshape of the reflection data. The procedure above (due to the Stark shift) effectively calculates the ratio by dividing the average of data points of reflected intensity at high energy tails of the polariton resonance. If these points overlap with the tail due to the asymmetric lineshape (observed in the PL spectra, Supplementary Figure 5) that extends to high energies, or overlaps with the polariton resonance, the reflection ratio at low energies can have values higher than 1. To account for this lineshape that is asymmetric in energy relative to the polariton resonance, we include a smooth step function in our fits. The fitting function we use at low magnetic field is given by : We use for our fits step = 0.2 meV. At high magnetic fields the data exhibits two closely spaced dips that we attribute to the interplay of the Zeeman splitting of the exciton transitions and the TE-TM splitting of the photonic modes, and use: We expect, due to the finite TE-TM splitting at the k ± y values that we perform the measurements that the transitions will be elliptically polarized and the two dips might be of different amplitude.

Supplementary Note 4. MODEL OF POLARITONS UNDER MAGNETIC AND ELECTRIC FIELDS
We model the behaviour of polaritons in our system by numerically determining the eigenvalues and the associated wavefunctions of a Schrödinger equation for the excitons in the system. We compare the measured observables obtained in various experiments with the results of calculations based on the numerical study both to extract parameters that describe the exciton system, as well as to verify the physical origin of the change in the observables.
The Hamiltonian for the 2D exciton relative motion wave function ψ (r) is given by [2,3] where the Hamiltonian is described in SI unit and r = r e − r h is the relative coordinate, R = (m e r e + m h r h ) / (m e + m h ) is the centre of mass (CM) coordinate,L z = i ẑ × ∇ r is the angular momentum operator in the z (growth) direction, P is the exciton magnetic CM momentum, which is an eigenvalue of the magnetic momentum operatorP = −i ∇ R − eB × r/2. Note that P is the conserved momentum of the exciton in a magnetic field and associated with translational invariance in the plane of the quantum well and is identical to the CM momentum at B = 0 [2].
Hence the exciton excited by a photon has P = k, which is the same as the in-plane momentum of the photon. E extx is the applied electric field, B z the magnetic field in the growth direction, We ignore the eB 2ηL z term in Supplementary Equation 5 since we are dealing with the ground state of the 2D exciton wave function, which is spatially symmetric (1s state for B z = 0). We also ignore the centre of mass kinetic energy term ( k) 2 /(2M ) since we fix the magnetic CM momentum k in the experiment. Thus, this term gives a constant energy shift.
The effective Coulomb potential for a QW with finite width d is [4][5][6] where ε is the relative permittivity of the material, ε 0 is the vacuum permittivity, and we assume that the QW has infinite barriers. The wavefuntions of the electron and hole in the growth direction is given by U i (z i ) = 2 d sin z i d π where i is either e or h. If we introduce the variable u = (z e − z h ) /d and v = (z e + z h ) /d, then Supplementary Equation 6 can be written as: We use Mathematica's finite element method for numerically solving the Schrödinger equation with the Hamiltonian given by Supplementary Equation 5 to obtain the energy and the wave function of the lowest energy state. The computation region is limited to r max = 10 a 0 (0 ≤ |r| ≤ r max , and a 0 = 4πεε 0 2 /µe 2 , 3D Bohr radius of the exciton), which is large enough that the lowest eigenvalue of the equation does not change (up to six digits) as r max increases.
The physical parameter values used in this calculation are summarized in Supplementary Table I. m 0 is the electron mass and we choose the remaining parameters such as k y , µ, and M , as we discuss below, to best match with our experimental data.

Parameter Symbol Value
Relative permittivity (GaAs) ε 12.9 Quantum well width Then, the binding energy of the exciton with no external magnetic and electric field is 7.94 meV.

Effect of the applied electric field on polaritons
An electric field alters the exciton energy thereby changing the polariton resonance energy. In addition polarization of the exciton due to E ext reduces the electron hole overlap, leading to a reduced Ω , which also changes the polariton resonance energy. To model the polariton behaviour we use Supplementary Equation 1 where Ω and exc depend on both B z and E ext . Shifts in  increase in Ω leads to red-shift of the LP energy.
Dependence of the polarizability α on B z is depicted in Supplementary Figure 10b. We compare the theoretical result of α for polaritons with the experimental results by varying the reduction factor γ of the applied electric field E ext = γ V G 30 µm where V G is the potential applied to the gate. We find a good agreement with the experimental results when γ = 0.84. Note that the COMSOL simulation gives γ ∼ 0.69 as discussed in Section I. As shown in Supplementary Figure 10b, both polariton and exciton polarizabilities decrease with B z but the polarizability of the polariton is smaller than that of the exciton.
Here, as in the main text, we calculate the difference of the effective electric fields ∆E eff between the case of k ± y = ±2.8×10 6 m −1 , this is shown as Supplementary Figure 10c. Unlike Supplementary  Figures 10a and b, ∆E eff for polariton and the exciton cases are very similar to each other and the small discrepancy is due to the fact that the Rabi energy is reduced by E ext . The electric field difference ∆E eff shows linear dependence on B z . Note that we find that the total exciton mass M ∼ 0.07 m 0 gives a good agreement between the theoretical and the experimental results. See Supplementary Figure 11 for the simulation results with three different M values.
The difference of the induced dipole moments ∆d between k ± y with B z is shown in Supplementary  Figure 10d. d corresponds to the induced dipole moment of the exciton or polariton due to the in-plane momentum and B z . Supplementary Figure 10d shows that the dipole moment of the polariton increases with B z up to around 3 -4 T and then decreases with B z for higher magnetic fields. A remarkable feature is that the magnetic field at which the dipole moment reaches the maximum value is different for the exciton and the polariton cases, in our sample, the polariton has the maximum value around 3 -4 T while the exciton has its maximum value around 4 -5 T. This is due to the fact that the exciton content of the lower polariton decreases with B z .

Effective vector potential for polaritons
For polaritons at arbitrary detuning, it is not generally possible to write the lower (or upper) polariton dispersion as a free particle with single effective mass. However for a small range of  wavevector values (δk y ) around a particular wavevector k y (k y = k y + δk) we will show that it is possible to describe polaritons as particles moving in a vector potential. Using the description in the main text, the energy difference between the cavity mode and the exciton mode is: ∆(k y ) = cav (k y ) − exc (k y ) = cav (0) − exc (0) + 1 2m cav ( k y ) 2 − 1 2M ( k y − qA eff ) 2 (8) The energy eigenvalues in Supplementary Equation 1 can be re-written as: LP,UP (k y ) = 1 2 ∆(k y ) ± ∆ 2 (k y ) + Ω 2 + exc (0) + 1 2M ( k y − qA eff ) 2 (9) Assuming k y δk y we do a linear expansion in δk y of LP (k y ) and find: where A = m M 2αB z E x |X k | 2 and (k y ) = exc (k y ) + 1 2 ∆(k y ) − ∆ 2 (k y ) + Ω 2 − 1 2m k y − qA 2 , and |X k | 2 = 1 2 1 + ∆(k y ) √ ∆ 2 (k y )+Ω 2 the cavity and exciton Hopfield coefficients.