An optically trapped exciton–polariton condensate forms states corresponding to excited energy levels of the confining potential. Recently, it was shown that non-uniformity of the ring-shaped trapping potential leads to the simultaneous occupation of two split energy levels. This results in the formation of an oscillating vortex cluster with periodically changing signs of topological charges. Here, we experimentally demonstrate the control of the topological charge oscillation frequency by tuning the ellipticity of the excitation profile. Our observations are corroborated using the linear Schrödinger equation for a two-dimensional quantum harmonic oscillator. Our findings open a promising avenue for the investigation of optical vorticity properties and their applications in controllable settings.

A quantum vortex is characterized by the hollow-core density distribution and quantized circulation of some physical quantity around the core. Initially predicted in liquid helium by Onsager,1 quantized vortices were observed in superconductors,2 superfluid helium,3 atomic Bose–Einstein condensates,4 and in laser systems.5,6 The optical vortices are characterized by the integer number of 2π phase winding (i.e., topological charge l) and associated with orbital angular momentum (OAM) or Laguerre–Gaussian higher-order modes.6 The growing interest in vortices has led to the manifestation of singular optics as a separate research field,7–9 spanning from studies of exotic fractional topological charges10 and half-quantized vorticity11,12 to practical applications. The latter include realization of vortex-based entangled states,13 optical tweezers,10 and 3D particle manipulation,14,15 as well as OAM entanglement-based cryptography16 and OAM-based optical communication line multiplexing.17 

Of particular interest in singular optics is the generation of vortex arrays: clusters8,18,19 and lattices.20–23 The former arise in the gain media of the optical cavity, which enables coexistence of two transverse modes of different parity with a fixed phase difference.19 Therefore, the generation of arrays with multiple arranged vortices would require precise control of the gain spatial distribution. In addition to pure photonic systems, hybrid light–matter systems, such as exciton–polaritons (polaritons hereafter), are suitable to study the vorticity. Polaritons are bosonic quasi-particles that appear in the strong coupling between confined photons and excitons inside the semiconductor microcavity. Due to their unique properties, namely strong inter-particle interaction, low effective mass ( 10 5 m e), and high mobility, polaritons are apt for the investigation of Bose–Einstein condensation,24,25 superfluidity,26,27 optical lattices,28–30 vortices31 including giant32 and stirring-induced33 ones, vortex–antivortex pairs,34 and finally, vortex clusters35,36 and lattices.37–39 Moreover, interactions of polaritons with the optically induced exciton reservoir40 allow for the realization of the arbitrary potential landscapes for condensates utilizing shaped excitation patterns. The malleability in the creation of optically induced potentials has resulted in reports on tunable “barriers” for coupling control,28 lattices of condensates,29 and optically confined condensates.41 

Recently, we experimentally demonstrated that optically trapped polariton condensate can spontaneously form two beating higher-order spatial modes with frequency splitting f originating from the small effective ellipticity, induced by pump non-uniformity.42 The mode splitting leads to the appearance of a cluster of four single-charged quantized vortices [two with l = + 1 and two with l = −1, see Fig. 1(a)] that periodically (twice per beating period) changes the signs of topological charges. This system, being an example of the multimode condensate43,44 demonstrates spontaneous formation and cyclic dynamical evolution, indicating a limit cycle. For simplicity, we will further refer to this object as an oscillating vortex cluster. Recently, this phenomenon was also observed experimentally, for a polariton condensate in micropillar cavities,45 and investigated theoretically.45–47 However, the control of the topological charge oscillating frequency has been lacking to date.

FIG. 1.

(a) Normalized measured spatial intensity distribution of the vortex cluster spontaneously arising in the optically trapped polariton condensate. Red, yellow, and violet dashed ellipses schematically denote the trapping potential of different ellipticities: ϵ1, ϵ2, and ϵ3. Intensity dips marked with red and black squares indicate the position of vortices with l = + 1 and l = −1, respectively, that oscillate at the frequency f. (b) Two frequency-detuned interfering Ince–Gaussian modes I G 33 ( e ) and I G 31 ( e ) form the spatial distribution of vortex cluster presented in panel (a). Color scale in (b) applies also to (a). (c) Energy splitting between Ince–Gaussian modes hf (where h is the Plank's constant), and frequency of vortex cluster oscillations is proportional to the trap ellipticity ϵ. (d) Schematic of the experimental setup based on Michelson interferometer for the measurement of | g ( 1 ) ( τ ) | of the condensate.

FIG. 1.

(a) Normalized measured spatial intensity distribution of the vortex cluster spontaneously arising in the optically trapped polariton condensate. Red, yellow, and violet dashed ellipses schematically denote the trapping potential of different ellipticities: ϵ1, ϵ2, and ϵ3. Intensity dips marked with red and black squares indicate the position of vortices with l = + 1 and l = −1, respectively, that oscillate at the frequency f. (b) Two frequency-detuned interfering Ince–Gaussian modes I G 33 ( e ) and I G 31 ( e ) form the spatial distribution of vortex cluster presented in panel (a). Color scale in (b) applies also to (a). (c) Energy splitting between Ince–Gaussian modes hf (where h is the Plank's constant), and frequency of vortex cluster oscillations is proportional to the trap ellipticity ϵ. (d) Schematic of the experimental setup based on Michelson interferometer for the measurement of | g ( 1 ) ( τ ) | of the condensate.

Close modal

In this Letter, we demonstrate that the excitation laser pattern ellipticity can be effectively utilized for the control of the vortex cluster oscillation frequency. By fine-tuning the ellipticity of the optical trap, we control the energy splitting between spatial modes and, therefore, the frequency of vortex cluster oscillation. This observation is qualitatively reproduced by the quantum harmonic oscillator model.

For our experiments, we use 2λ microcavity with InGaAs quantum wells.48 The sample is cooled down to 4 K in a closed-loop helium cryostat. For the non-resonant excitation, we use a continuous wave (CW) monomode Ti:sapphire laser tuned to one of the reflection minima of the distributed Bragg reflectors at 1.5578 eV. The exciton–photon detuning is set to Δ = −3.2 meV. For single-shot interferometric measurements (used for characterization of polariton condensates coherence properties), CW laser emission was transformed to 20 μs individual pulses, using an acousto-optic modulator. Time-integrated real-space condensate images and spectra were captured under excitation by 2 μs pulses at a repetition rate of 5 kHz. The excitation parameters were chosen in order to avoid the local heating of the sample and allow the condensate to reach a steady state on a timescale that is much longer than typical polariton scattering times.40 

In addition to the temporal modulation, we also implement a spatial modulation of the excitation laser beam in order to transform its initial Gaussian intensity profile into a ring-like profile with controllable ellipticity. This approach is utilized to inject excitons into the quantum well to form the trapping potential for polaritons.

Below, we discuss the impact of the spatial ellipticity parameter ϵ defined as a ratio of vertical Ly to horizontal Lx pump dimensions on the splitting of the condensate energy modes. First, we set L x = 20 μm, ϵ = 0.97, and the excitation power to P = 1.2 × P thr, where Pthr = 44 mW is the condensation threshold power. At these conditions, the condensate occupies the state depicted in Fig. 1(a). This photoluminescence (PL) intensity distribution corresponds to two frequency-detuned higher-order even Ince–Gaussian modes I G 31 ( e ) and I G 33 ( e ) [see Fig. 1(b)].42,49,50 Therefore, the temporal dynamics of this system obeys the following equation:
Ψ = I G 31 ( e ) e i π ft + I G 33 ( e ) e i π ft e i ϕ ,
(1)
where Ψ is the wave function of the condensate, and f is the beating frequency. Equation (1) allows us to reconstruct the spatial distribution of the condensate and the temporal dynamics of its intensity, phase, and topological charges. However, it does not reflect the functional dependence of the frequency f on the ellipticity of the optical trap.

In order to reveal the dependence of the vortex cluster oscillation frequency on the excitation pattern ellipticity, we measure the first-order coherence function modulus | g ( 1 ) ( τ , r ) | of the condensate. For this, we implement the Michelson interferometer with a tunable delay between the arms as shown in Fig. 1(d). Note that the real-space condensate PL is interfering with its retroreflected copy.

Scanning the time delay in the range from 0 to 650 ps for the fixed excitation conditions, we collect interference images using the single-shot imaging technique (the camera captures the interference patterns only for a single excitation pulse). For the analysis of interference images, we implement the off-axis digital holography technique51,52 and extract the modulus of the first-order coherence function averaged over the whole condensate area | g ( 1 ) ( τ ) |.

Furthermore, we vary the excitation pattern ellipticity in the range from 0.95 up to 1.02. The examples of | g ( 1 ) ( τ ) | as a function of the interferometer delay for ϵ = 0.97 and 1.00 are presented in Fig. 2(a) (red and gray dots, respectively). These two curves are characterized by an exponential amplitude decay with the rate dictated by the condensate coherence time τc. Moreover, both dependencies feature oscillations of the | g ( 1 ) ( τ ) | that arise because the condensate occupies two frequency-detuned modes of the confining potential. The frequency f of this beating equals the frequency of the periodical topological charges oscillation. To extract f, we fit the experimentally obtained | g ( 1 ) ( τ ) | with the first-order correlation function equation for the two-level Bose gas
| g ( 1 ) ( τ ) | = A f cos ( 2 π τ ) + B × e | τ | / τ c ,
(2)
where A and B are the fitting parameters.
FIG. 2.

(a) Experimentally measured | g ( 1 ) ( τ ) | for the trap with ellipticity of 0.97 and 1.00 (red and gray dots, respectively). Red and gray solid lines represent the fit of the experimental data with Eq. (2). The | g ( 1 ) ( τ ) | revivals appear due to the beating of Ince–Gaussian modes depicted in Fig. 1(b) and manifest the formation of the oscillating vortex cluster. (b) Measured frequency of the vortex cluster oscillation for different ellipticity values (violet dots). The error bars depict the error of the fit. Blue curve demonstrates extracted values of frequency splitting calculated with 2D quantum harmonic oscillator model. The right vertical axis demonstrates the frequency tuning range for both curves and absolute value of the splitting for theoretically produced blue curve. (c) Theoretically obtained frequencies of trap modes (red and black curves) and their splitting (blue curve) for ellipticity ϵ in the range from 0.5 to 1.5. Green square corresponds to the range of frequency splittings shown in panel (b). Black dashed line represents the low polariton branch frequency (350 THz) that corresponds to the zero frequency splitting between Ince–Gaussian modes.

FIG. 2.

(a) Experimentally measured | g ( 1 ) ( τ ) | for the trap with ellipticity of 0.97 and 1.00 (red and gray dots, respectively). Red and gray solid lines represent the fit of the experimental data with Eq. (2). The | g ( 1 ) ( τ ) | revivals appear due to the beating of Ince–Gaussian modes depicted in Fig. 1(b) and manifest the formation of the oscillating vortex cluster. (b) Measured frequency of the vortex cluster oscillation for different ellipticity values (violet dots). The error bars depict the error of the fit. Blue curve demonstrates extracted values of frequency splitting calculated with 2D quantum harmonic oscillator model. The right vertical axis demonstrates the frequency tuning range for both curves and absolute value of the splitting for theoretically produced blue curve. (c) Theoretically obtained frequencies of trap modes (red and black curves) and their splitting (blue curve) for ellipticity ϵ in the range from 0.5 to 1.5. Green square corresponds to the range of frequency splittings shown in panel (b). Black dashed line represents the low polariton branch frequency (350 THz) that corresponds to the zero frequency splitting between Ince–Gaussian modes.

Close modal

As a result, we obtain the dependence of frequency on the pump ellipticity, presented in Fig. 2(b) with violet dots. Big hollow core circles highlight the frequency values extracted from the data in Fig. 2(a): f = 4.962 ± 0.049 and f = 4.677 ± 0.068 GHz, respectively. The extracted frequency f decreases when the trap shape is tuned from being elliptical and vertically oriented ( ϵ < 1) to the circular one (ϵ = 1). Notably, for ϵ > 1, the oscillation frequency is growing. It is worth noting that the tuning range of oscillating vortex cluster frequency is  300 MHz.

The spectrum of the trapped polariton condensate can be qualitatively explained by analogy with a system localized in two-dimensional harmonic potential41 and described with the Schrödinger equation. Therefore, we adopt this approach to describe the dependence of oscillating vortex cluster frequency on the trap ellipticity. Since in our experiments, we use elliptical optical traps, the shape of the confining potential can be written as
V 2 D , harm = δ 2 [ ( x L x ) 2 + ( y L y ) 2 ] ,
(3)
where δ is the proportionality factor. Then, the Schrödinger equation is written as follows:
2 2 m * [ 2 Ψ ( x , y ) x 2 + 2 Ψ ( x , y ) y 2 ] + δ 2 [ ( x L x ) 2 + ( y L y ) 2 ] Ψ ( x , y ) = E n Ψ ( x , y ) .
(4)
Finally, the dependence of the energy levels on trap size, polariton mass, and mode indices reads as the following equation:
E ( L x , L y , n x , n y ) = 2 m * [ 1 L x 2 ( n x + 1 2 ) + 1 L y 2 ( n y + 1 2 ) ] + E LPB ,
(5)
where m * is the polariton effective mass, nx and ny are mode indices of the state. As far as Eq. (5) describes the energy levels relatively to the minimum of the parabolic potential, we add the energy offset corresponding to the minimum energy of the lower polariton branch ELPB. As a result, the frequency of the oscillating vortex cluster is defined by the energy difference between the spatial modes,
f = | E ( L x , L y , n x 1 , n y 1 ) E ( L x , L y , n x 2 , n y 2 ) | / h .
(6)

Note, that the order p and degree m of Ince polynomials in I G p m ( e ) are not the same as the indices nx and ny as it is for the Hermite–Gaussian and Laguerre–Gaussian modes. According to Refs. 49 and 50 for even (e) modes, the mode indices are defined as nx = m, n y = p m.

For the fitting of the experimental data, we use m * = 5.64 × 10 5 m e and ELPB = 1.4519 eV, retrieved from the polariton dispersion measurement. The horizontal size Lx of the excitation pattern is 20 μm, and the vertical size is calculated as L y = ϵ × L x (as in the experiment).

We calculate the vortex cluster oscillation frequencies for the range of ellipticities used in the experiment ( ϵ = 0.95…1.02). The simulation results are depicted in Fig. 2(b) as a blue line, which qualitatively matches the experimental observations. Both the experimental and theoretical curves have the same slopes for ϵ < 1 and ϵ > 1 with the local minimum at ϵ = 1. Moreover, we plot experimentally obtained values of frequency splitting as the difference between all f values and the minimum value from the dataset [for this, use right vertical axis in Fig. 2(b)]. Amazingly, we find that the experimentally obtained and calculated values of splittings are in a good agreement (violet and blue curves correspondingly). This could be explained by the presence of an additional frequency “offset” of  4.7 GHz between I G 31 ( e ) and I G 33 ( e ) induced by a not perfectly shaped potential inherited from azimuthal pump intensity non-uniformity.53 

Furthermore with our model, we find the frequencies of I G 31 ( e ) , I G 33 ( e ) spatial modes and their splitting in a wider range of ellipticity values from 0.5 to 1.5 depicted in Fig. 2(c). Note that, the splitting changes nonlinearly with the ellipticity. Therefore, the “linear-like” behavior demonstrated in Fig. 2(b) can be observed only in a very narrow range of ϵ, schematically denoted with a green square in Fig. 2(c). However, in the experiment, we do not observe such nonlinear dependence due to limited ellipticity range (from 0.95 to 1.02), ensuring the formation of the vortex cluster.

We find that when the ellipticity is decreased below 0.95, polaritons do not condense at such excitation conditions.54 On the other hand, at ϵ higher than 1.02, we observe the other polariton density distribution showcased in the inset of Fig. 3(a). This intensity distribution corresponds to the condensate occupying three consequent levels of the trapping potential separated by 26–30 GHz. This energy splitting can be resolved with direct measurement of the condensate spectrum depicted in Fig. 3(a). The corresponding first-order correlation function | g ( 1 ) ( τ ) | of the condensate is presented in Fig. 3(b). The condensate coherence time τc retrieved from the decay is 220 ps. The temporal resolution of our measurement apparatus does not allow us to clearly see the revivals of | g ( 1 ) ( τ ) | at bigger time delays. However, the fast-Fourier transformation (FFT) spectrum of the condensate | g ( 1 ) ( τ ) | in the inset of Fig. 3(b) features a resolvable peak at 28 GHz corresponding to energy splitting of condensate states. It is worth noting that a strong zero frequency component in the FFT spectrum is dictated by the finite condensate coherence time and unequal population of the trap modes. Note, that unlike the superposition of I G 31 ( e ) and I G 33 ( e ) modes, the condensates appearing at ellipticity larger than 1.02 do not carry any resolvable vorticity.

FIG. 3.

(a) Experimentally obtained spectrum of the polariton condensate created in an optical trap with ϵ = 1.03 (gray solid line), its cumulative fitting (red solid line) with three Gaussian peaks: blue, violet, and green dashed lines. The frequency splittings between the central and the side peaks is given with numbers. The inset shows the corresponding real-space condensate normalized intensity distribution (the scale bar length is 5 μm). (b) Experimentally measured | g ( 1 ) ( τ ) | (red line) for the trap with ϵ = 1.03. The inset depicts the corresponding spectrum retrieved with the fast-Fourier transformation (FFT) of the | g ( 1 ) ( τ ) | decay curve. The frequency peak at  28 GHz corresponds to the splitting between energy levels in panel (a).

FIG. 3.

(a) Experimentally obtained spectrum of the polariton condensate created in an optical trap with ϵ = 1.03 (gray solid line), its cumulative fitting (red solid line) with three Gaussian peaks: blue, violet, and green dashed lines. The frequency splittings between the central and the side peaks is given with numbers. The inset shows the corresponding real-space condensate normalized intensity distribution (the scale bar length is 5 μm). (b) Experimentally measured | g ( 1 ) ( τ ) | (red line) for the trap with ϵ = 1.03. The inset depicts the corresponding spectrum retrieved with the fast-Fourier transformation (FFT) of the | g ( 1 ) ( τ ) | decay curve. The frequency peak at  28 GHz corresponds to the splitting between energy levels in panel (a).

Close modal

In summary, we have demonstrated the control of oscillating vortex cluster frequency of the trapped polariton condensate occupying two energy levels. Varying the ellipticity of the optical trap, we achieved 300 MHz tuning range. As topological charge sign oscillation is nothing but the result of the beating between frequency detuned spatial modes, we have proposed an approach to precisely control the beating frequency through the trap ellipticity. The presented results are qualitatively supported by the quantum harmonic oscillator model, demonstrating a well-pronounced and predictable frequency tuning trend. Moreover, this simple model predicts the nonlinear behavior of the oscillating vortex cluster frequency dependence on the optical trap's ellipticity. Even though experimentally realized on the microcavity polaritons, the proposed technique can be extended to other physical systems, allowing the study of oscillating vortex clusters in laser diodes, fibers, VCSELs, etc. The OAM-carrying beams are promising for quantum key distribution (QKD) communication lines in free space.55 Their robustness in atmosphere is studied extensively in recent works.55–58 We believe that our findings are potentially significant for further QKD system development. Moreover, the condensate occupying two energy split spatial modes has been suggested as a potential platform for the realization of qubit gates.59 Therefore, our technique for controlling the energy splitting of the condensate modes could be a step toward the emerging applications of polaritons for unconventional computing.

This study was funded by the RFBR, Project No. 20-32-90128.

The authors have no conflicts to disclose.

Kirill A. Sitnik: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Ivan Gnusov: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Mikhail Misko: Investigation (equal); Software (equal); Validation (equal); Writing – review & editing (equal). Julian D. Töpfer: Investigation (equal); Software (equal). Sergey Alyatkin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Pavlos G. Lagoudakis: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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