Quantum Beats of a Macroscopic Polariton Condensate in Real Space

Abstract

We experimentally observe harmonic oscillations in a bosonic condensate of exciton-polaritons confined within an elliptical trap. These oscillations arise from quantum beats between two size-quantized states of the condensate, split in energy due to the trap’s ellipticity. By precisely targeting specific spots inside the trap with nonresonant laser pulses, we control frequency, amplitude, and phase of these quantum beats. The condensate wave function dynamics is visualized on a streak camera and mapped to the Bloch sphere, demonstrating Hadamard and Pauli-Z operations. We conclude that a qubit based on a superposition of these two polariton states would exhibit a coherence time exceeding the lifetime of an individual exciton-polariton by at least two orders of magnitude.

1. Introduction

Quantum beats are a widely spread phenomenon that is characteristic of the coherent dynamics of a two-level quantum system excited in a superposition state [1,2,3]. The period of the beats is determined by the energy splitting of two participating quantum states [2], while their decay time characterizes the decoherence processes that are necessarily present in any quantum system coupled to an environment [3,4]. In condensed matter physics, quantum beats caused by size quantization of electron [5], hole [6] or exciton [7] wave functions, Zeeman splitting [8], spin-orbit [9], hyperfine [10] interactions etc have been documented. Quantum beats are typically detected by optical methods, including time-resolved photoluminescence [11], pump-probe [1], four-wave mixing [12], Faraday-rotation [13] spectroscopy, spin noise [14] and photon-echo [15] spectroscopies as well as several other techniques. These approaches enable detection of the oscillations of intensity, polarisation, coherence degree and/or other measurable characteristics of light caused by quantum beats in crystals, molecules and other matter objects coupled to light [16]. In this context, bosonic light-matter quasiparticles, namely, exciton-polaritons [17,18,19] represent a unique laboratory for studies of the optical manifestations of quantum beats. In the strong exciton-photon coupling regime in semiconductor microcavities, the quantum beats between exciton-polariton eigen-modes frequently referred to as Rabi-oscillations have been observed since mid-1990s [20,21]. The true quantum Rabi-oscillations were demonstrated by means of the solid-state cavity QED [22]. Next, a number of experiments demonstrated the complex coherent oscillatory dynamics of the polarization and intensity of light emitted by linear combinations of various polariton states [23,24] that e.g. led to generation of full Bloch light beams [23]. A fundamental factor that limits the decay time of polariton quantum beats is life-time of an individual exciton-polariton which is typically less than a hundred picoseconds even in high-quality semiconductor microcavities [25]. However, some oscillatory phenomena with much longer decay times in many-body bosonic condensates of exciton-polaritons have recently been theoretically discussed [26,27,28] and experimentally observed [29,30]. Specifically, the persistent Larmor precession driven by polariton–polariton interactions has been reported [29], and the continuous time-crystal behavior has been demonstrated in a polariton condensate coupled to phonons [31]. A complementary quantum-statistical analysis that explicitly treats fluctuations and coherence beyond the mean-field regime was presented in Ref. [32], highlighting the role of dark states in stabilising driven polariton condensates. It has been argued that a combination of strong optical non-linearity caused by repulsive polariton-polariton interactions, stimulated scattering of polaritons and dissipative coupling between different polariton condensates may lead to the limit-cycle solutions [26], dynamical attractors [27,33] and even to polariton time-crystals [28].

Related questions of how Bose–Einstein condensation in driven-dissipative media compares with Fröhlich-type condensation in other pumped bosonic systems have recently been addressed in Ref. [34]. In contrast to the quantum beats characteristic of a linear two-level quantum system whose frequency is simply given by the splitting between two participating energy levels, the limit cycle oscillations have their dynamics governed by a variety of factors including the polariton concentration, polariton-polariton interaction constants, spatial dependence of the polariton lifetime etc. [27]. The decay time of oscillations characteristic of limit cycles, theoretically, tends to infinity. In the present study, we experimentally observe long-lived oscillations of a macroscopic exciton-polariton condensate in real space that reveal features of quantum beats rather than limit-cycle oscillations.

2. Setting the Trap

We study bosonic condensates of exciton-polaritons created by a nonresonant $cw$ optical excitation in a planar GaAs-based microcavity. The description of the sample and its characteristic optical spectra can be found in [35]. We use a spatial-light modulator (SLM) (see Figure S1 of the Supplementary Material) to create an elliptical trap for exciton-polaritons shown in Figure 1a. We take advantage of the driven-dissipative nature of exciton-polariton condensates bosonic condensation at energy states other than the ground state. We carefully chose the design of the trap and the pumping intensity in order to make sure that the ensemble of exciton-polaritons populates a selected pair of energy levels of the trapping potential. The eigen-wave functions corresponding to these states represent two-dimensional p-shaped orbitals. The corresponding spatial distributions of the intensity and the phase of the polariton condensate are shown on Figure 1b,c, respectively.

To change the state of the condensate, or to put it into the oscillatory mode, we modify the profile of the potential trap with the use of nonresonant control pulses. These pulses excite electron and hole clouds that eventually form incoherent excitons that, in their turn, relax to the polariton modes. Together, this leads to the appearance of a localized repulsive potential acting on the polariton condensate. This potential eventually vanishes on a time scale of several hundred picoseconds (see Figure S7 of the Supplementary Material). The emission of excitons (exciton-polaritons) created by a control pulse can be directly observed with the use of the streak camera, as Figure 1a shows. One can see how its peak relaxes in energy, eventually reaching the condensate energy (Figure S7 of the Supplementary Material).

Elliptical traps offer the advantage of optical control over energy splitting between mentioned p-shaped orbitals. The experimentally obtained spatial distributions of the polariton densities corresponding to these orbitals are shown in Figure 1d,e. We map the true eigen functions of the trap by measuring the spatial maps of phase ${\varphi }_0\left(x,y\right)$ and amplitude $a\left(x,y\right)$ of the oscillations of intensity of the emitted light induced by quantum beats between the eigen states of the trap. These maps obtained both experimentally and theoretically are shown in panels of Figure 1f–i. The oscillations are the most pronounced in the areas of the strongest overlap of $p_x$ and $p_y$ orbitals. Varying the ellipticity of the trap we tune the splitting of $p_x$ and $p_y$ orbitals on a micro-electron-Volt scale as Figure 1l illustrates.

3. The Dynamics of Trapped Condensates

The oscillatory dynamics observed in our experiments can be analytically reproduced by a linear two-level model. We approximate the optically induced trap that confines the exciton-polaritons condensate by a modified potential of a two-dimensional harmonic oscillator $V=mf\left(\phi \right)\left({\omega }_x^2{x}^2+{\omega }_y^2{y}^2\right)/2$ (for details, see the Supplementary Materials). We solve the two-dimensional Schrödinger Equation (S7) and obtain the energy spectrum of the trapped condensate. We assume that the system occupied the $p_x$-eigen state prior to the arrival of the control pulse. The effect of this non-resonant pulse we model by a time-dependent perturbation potential. Figure 1k shows the theoretical calculation of the density dynamics of the polariton condensate at the specific spot (horizontal cross-section in panel (j)) of the elliptical trap (blue curve). These results appear to be in good agreement with the experimental data (orange curve).

Streak-camera measurements allow us to visualize time- and space-resolved images of polariton condensates. We find a very good agreement of the measurements with predictions of a two-level quasi-analytical model. After reaching the stationary oscillation regime, we observe pronounced intensity beats which may be fitted by a harmonic function: $a(x,y)sin[\omega t+{\varphi }_0(x,y)]$ at each spatial point. Figure 2 shows the spatial distribution of the polariton density obtained experimentally (panels (c–g)) and numerically (panels (h–l)) at different times within the period of quantum beats. In order to maximize the magnitude of intensity oscillations detected by streak-camera (panel (a)) we use a sequence of 8 control pulses incident to the same spot at equal time intervals, as dashed lines in Figure 2b show. The repetition frequency in this sequence of pulses was chosen equal to the condensate oscillation frequency to achieve a parametric resonance (see Supplementary Material, Figure S9).

4. Hadamard and Pauli Operations

It is instructive to map the observed quantum beats to a Bloch sphere. We do it by fitting the time-resolved tomography images recorded by the streak-camera with use of a two-level model introduced above. The oscillatory dynamics of the system manifests itself in the precession of its quantum state on the surface of the Bloch sphere as panel (e) (in a rest frame) and (f) (in a rotating frame) in Figure 3 show.

The poles of the Bloch sphere may be considered as the computational basis states $|0\rangle$ ($p_x$-orbital) and $|1\rangle$ ($p_y$-orbital). Once a polariton condensate is placed in a superposition of two eigen states of a trap, it may be considered as a qubit. Using nonresonant optical pulses focused at different locations of the trap, we were able to change the state of the trapped condensate. These changes manifest themselves in changing frequency, amplitude and phase of the quantum beats observed in the streak-camera images. Mapping the state of the condensate to the Bloch sphere we observed how the studied quantum system evolves from the pole of the sphere to its equatorial plane (Figure 3e,f). Panel (f) displays two curves: the experimental data is represented by the black curve, whereas the theoretical calculation is shown in blue-green. The two trajectories converge to the same final point, which is crucial for realizing a high fidelity quantum operation. This particular transformation corresponds to the Hadamard operation given by:

$$\begin{array}{c}\hfill H={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right].\end{array}$$

(1)

In order to implement the high-fidelity Hadamard operation, we have measured the dependence of the amplitude of the beats induced by the sequence of control pulses on the pump power and compared it with the theoretical prediction (Figure 3c). The black line in Figure 3c shows the predicted fidelity of the Hadamard operation for a pulse centered at the red spot in the panel (d). Carefully choosing the spot hit by the sequence of control pulses we have been able to achieve a fidelity of the Hadamard operation of over $0.95$. This number is obtained by averaging over 5 different initial states of the system. The theoretical estimate for the fidelity is much higher, about ∼1, as one can see from the calculated fidelity map shown in Figure 3d.

Once the Hadamard operation is implemented, we implement also Pauli-Z operation given by:

$$\begin{array}{c}\hfill Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].\end{array}$$

(2)

This is achieved by sending the 9th control pulse to a specific spatial location. It moves the system to an opposite end of the diameter in the equatorial plane of the Bloch sphere, as Figure 4 illustrates. The streak-camera images of the condensate density cross-section as a function of time are demonstrated in panels (a,b). 8 initializing pulses transfer the system from the pole to the equator ($|+\rangle$), then the 9th control pulse triggers the Pauli-Z gate operation. We have plotted the dependencies of the phase shifts induced by the control pulses as functions of the control pulse intensities (panel (c)). After optimization of the pump power the phase of the quantum beats is being shifted by this gate operation by $+\pi$ or $-\pi$ depending on the location of the control pulse. The final state of the system is the same in both cases, as panel (e) illustrates. This panel shows two calculated trajectories of the system, which coincide on the path from $|0\rangle$ to $|+\rangle$, and then move in opposite directions along the equator (solid and dotted lines). The experimental fidelity of the implemented Pauli-Z operation exceeds $0.97$ in both cases. We have obtained this estimate based on a series of experiments, where we initialized the system in >20 different points on the surface of the Bloch sphere. The studied set of initial locations is shown in of Figure 4g. In our implementation, the Pauli–Z gate flips the relative phase between $|0\rangle$ and $|1\rangle$ by $\pi$ for arbitrary superpositions prepared on the Bloch sphere, as illustrated in Figure 4e,g.

5. Discussion

In our experiments, streak-camera averages the emission intensity collected after millions of pulses. Still, we are able to observe the intensity beats whose amplitude and phase are characterized by well-defined patterns. The macroscopic many-body wavefunction of a polariton condensate changes on a length-scale of over $10 \mathrm{\mu }\mathrm{m}$ with a periodicity of about $100 \mathrm{p}\mathrm{s}$. Small discrepancies between the experimental traces and the predictions of the linear two-level model (e.g., in Figure 3c and Figure 4c) stem from weak nonlinear mixing of remote trap modes (1s, 3s, 3p, 3d) with the qubit basis (${\mathrm{2p}}_x$, ${\mathrm{2p}}_y$). This effect is minor, as corroborated by the high gate fidelities. This indicates that every initial pulse brings the condensate essentially to the same superposition of $p_x$- and $p_y$-orbitals. Regular oscillations of a trapped exciton-polariton condensate that persist for at least $1\mathrm{n}\mathrm{s}$. The phase locking of $p_x$- and $p_y$-orbitals imposed by the control pulse survives much longer than the phase of the condensate as a whole. This conclusion is consistent with recent experiments on persistent currents of exciton-polaritons in a ring geometry [36]. We note also that the real-space density dynamics of polariton condensates in traps have been studied in coupled systems [37,38]. These studies focused on nonlinear phenomena such as Lotka-Volterra population dynamics or collective Bogoliubov-like modes, where the interplay of interactions and dissipation drives pronounced density oscillations. In contrast, our present experiment demonstrates a linear phenomenon of quantum beats between the discrete $p_x$ and $p_y$ orbitals. Barrat et al. [30] have recently implemented a polariton-based qubit analog in an annular trap, exploiting two counter-circulating vortex states. Further exploring the potentiality of trapped condensates for quantum computation, here we harness the linear splitting between $p_x$ and $p_y$ orbitals confined by an elliptical trap and directly visualize the dynamics of the condensate wave function in real space. The frequency of the oscillations that we observe is defined by the splitting between size-quantization levels of the polariton condensate confined in a trap and it can be tuned either by changing the ellipticity of the trap or by modifying its potential with use of control pulses of light. In order to check that the observed dynamics is not linked with any dynamical attractor of a limit-cycle type, we excited the system in several initial states on the Bloch sphere. Shifting the initial condition we shift the whole trajectory on a sphere. Furthermore, we have perturbed the dynamics of the condensate by sending control laser pulses. In these studies, no trace of an attractor has been revealed. We conclude that the observed oscillations are quantum beats experienced by the polariton condensate as a whole entity. These beats have several unique features: (i) they persist over times orders of magnitude longer than the single-polariton lifetime, (ii) they involve several thousands of polaritons composing the condensate, (iii) a pronounced real space dynamics is observed. The observation of coherent real-space dynamics of a macroscopic many-body object paves the way for the realization of polariton qubits, as discussed in recent publications [30,39,40,41].

6. Conclusions

In conclusion, we have experimentally demonstrated long-living quantum beats of the trapped bosonic condensates of exciton-polaritons. By time-resolved-interferometry technique we have measured the coherence times T1 and T2 corresponding to the decay of the first order coherence of the eigen states of the polariton condensate in a trap and the decay of the coherence of a superposition of the two eigen states, respectively. Both T1 (1200 ps) and T2 (800 ps) exceed the single polariton life-time by two orders of magnitude. This proves that a superposition of two discrete states of a trapped exciton polariton-condensate may be used for quantum gate operations. We implemented the Hadamard and Pauli-Z operations by hitting specific spots on a trapped condensate with non-resonant light pulses that create time-dependent perturbation potentials for the exciton-polaritons. The experimentally measured fidelity of single-qubit operations on the trapped polariton condensates exceeds 0.95 for the Hadamard operation and 0.97 for the Pauli-Z operation. These results pave the way for the realization of quantum processors based on trapped exciton-polariton condensates.