A gated quantum dot far in the strong-coupling regime of cavity-QED at optical frequencies

A gated quantum dot far in the strong-coupling regime of cavity-QED at optical frequencies

Daniel Najer daniel.najer@unibas.ch Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Immo Söllner Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Pavel Sekatski Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Vincent Dolique Laboratoire des Matériaux Avancés (LMA), IN2P3/CNRS, Université de Lyon, F-69622 Villeurbanne, Lyon, France    Matthias C. Löbl Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Daniel Riedel Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Rüdiger Schott Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany    Sebastian Starosielec Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Sascha R. Valentin Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany    Andreas D. Wieck Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany    Nicolas Sangouard Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland    Arne Ludwig Lehrstuhl für Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany    Richard J. Warburton Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
December 29, 2018
Abstract

The strong-coupling regime of cavity-quantum-electrodynamics (cQED) represents light-matter interaction at the fully quantum level. Adding a single photon shifts the resonance frequencies, a profound nonlinearity. cQED is a test-bed of quantum optics Boca et al. (2004); Birnbaum et al. (2005); Hamsen et al. (2017) and the basis of photon-photon and atom-atom entangling gates Zheng and Guo (2000); Duan and Kimble (2004). At microwave frequencies, success in cQED has had a transformative effect Fink et al. (2008). At optical frequencies, the gates are potentially much faster and the photons can propagate over long distances and be easily detected, ideal features for quantum networks. Following pioneering work on single atoms Boca et al. (2004); Birnbaum et al. (2005); Hamsen et al. (2017); Kawasaki et al. (2018), solid-state implementations are important for developing practicable quantum technology Reithmaier et al. (2004); Yoshie et al. (2004); Faraon et al. (2008); Hennessy et al. (2007); Rakher et al. (2009); Reinhard et al. (2012); Volz et al. (2012); Ota et al. (2018). Here, we embed a semiconductor quantum dot in a microcavity. The microcavity has a -factor close to and contains a charge-tunable quantum dot with close-to-transform-limited optical linewidth. The exciton-photon coupling rate exceeds both the photon decay rate and exciton decay rate by a large margin (, ); the cooperativity is , the -factor 99.7%. We observe pronounced vacuum Rabi oscillations in the time-domain, photon blockade at a one-photon resonance, and highly bunched photon statistics at a two-photon resonance. We use the change in photon statistics as a sensitive spectral probe of transitions between the first and second rungs of the Jaynes-Cummings ladder. All experiments can be described quantitatively with the Jaynes-Cummings model despite the complexity of the solid-state environment. We propose this system as a platform to develop optical-cQED for quantum technology, for instance a photon-photon entangling gate.

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An excellent solid-state emitter of single photons is a self-assembled quantum dot in a semiconductor host Kuhlmann et al. (2013); Somaschi et al. (2016). An InGaAs semiconductor quantum dot in GaAs is a bright and fast emitter of highly indistinguishable photons, properties not shared by any other emitter. The challenge in pursuing the strong-coupling regime of cQED with such a quantum dot is to combine apparently contradictory elements.

First, the cavity must have an ultrahigh -factor yet a small mode volume, i.e. dimensions comparable to the optical wavelength. Nano-fabrication techniques are employed to create, for instance, micropillar Reithmaier et al. (2004); Somaschi et al. (2016) or photonic crystal cavities Yoshie et al. (2004); Faraon et al. (2008); Hennessy et al. (2007); Reinhard et al. (2012); Volz et al. (2012); Ota et al. (2018). The acute problem is that the -factor tends to deteriorate as the mode volume decreases. This is only partly a consequence of fabrication imperfections, sidewall roughness of a micropillar for example. An additional factor is the GaAs surface which pins the Fermi energy mid-gap resulting in surface-related absorption Guha et al. (2017). Achieving a low-volume, ultrahigh -factor cavity in GaAs has proved to be difficult. Secondly, a quantum dot benefits enormously from electrical control via the conducting gates of a diode structure. A gated quantum dot in high quality material gives close-to-transform-limited linewidths Kuhlmann et al. (2013) and control over both the optical frequency via the Stark effect and the quantum dot charge state Högele et al. (2004). A charge-neutral quantum dot operates as a two-level system and is ideal as source of highly indistinguishable photons; a single electron or hole allows the creation of entangled spin-photon pairs. However, the conducting layers of gated devices are not obviously compatible with an ultrahigh -factor cavity on account of significant free-carrier absorption in the doped layers and below-band-gap absorption via the Franz-Keldysh effect. Finally, the quantum dot in a microcavity must retain the close-to-transform-limited optical linewidths of the starting material. This is hard to achieve following aggressive nano-fabrication as the free surface can result in additional charge noise leading to blinking and spectral fluctuations of the quantum dot.

Figure 1: Gated quantum dot in a tunable microcavity: design and realisation. a, Simulation of the vacuum electric field in the microcavity (image to scale). The bottom mirror is a distributed Bragg reflector (DBR) consisting of 46 AlAs()/GaAs() pairs. ( refers to the wavelength in each material.) The top mirror is fabricated in a silica substrate Barbour et al. (2011); Hunger et al. (2012). It has radius of curvature m and consists of 22 silica()/tantala() pairs. The layer of quantum dots (QDs) is located at the vacuum field anti-node one wavelength beneath the surface. The vacuum-gap has the dimension of . () controls the lateral (vertical) position of the QD with respect to the fixed top mirror. b, The top part of the semiconductor heterostructure. A voltage is applied across the n-i-p diode. controls the QD-charge via Coulomb blockade and within a Coulomb blockade plateau the exact QD optical frequency via the dc Stark effect. Free-carrier absorption in the p-layer Casey, Sell, and Wecht (1975) is minimised by positioning it at a node of the vacuum field. A passivation layer suppresses surface-related absorption Guha et al. (2017). c, Laser detuning () versus cavity detuning () of a neutral QD exciton (X) and a positively-charged exciton (X) in one and the same QD (QD1). Cavity detuning is achieved by tuning the QD at fixed microcavity frequency (X); and by tuning the microcavity frequency at fixed QD frequency (X). For X, the weak signal close to the bare microcavity frequency arises from weak coupling to the other orthogonally-polarised X transition – it does not arise from blinking (see Supplementary III.E).

We present a resolution to these conundrums. We have found a way to create an ultrahigh -factor yet with small mode volume. The quantum dot is gated and exhibits close-to-transform-limited optical linewidths even in the cavity. On resonance with the microcavity, the quantum dot exciton is far in the strong-coupling regime. Strong coupling is achieved on both neutral and charged excitons in one and the same quantum dot by tuning the microcavity in situ. The output is close to a simple Gaussian beam allowing high efficiency collection. Notably, the solid-state feature which has complicated quantum dot cQED in the past – scattering from the bare cavity mode even at the quantum dot-cavity resonance Hennessy et al. (2007); Faraon et al. (2008); Reinhard et al. (2012); Volz et al. (2012); Ota et al. (2018); Kuruma et al. (2018); Greuter et al. (2015) – disappears. The system is an exemplary Jaynes-Cummings system despite the complexity of the solid-state environment.

Figure 2: Strong coupling of a QD exciton in the microcavity. The spectra were recorded by measuring the photons scattered by the microcavity–QD system at a temperature of 4.2 K, rejecting reflected laser light with a polarisation-based dark-field technique Kuhlmann et al. (2013). Data shown here were taken on the X transition (QD2). a, e signal with QD far-detuned from microcavity in order to determine the photon loss-rate , equivalently the quality factor ). b, f X at magnetic field  T showing strong coupling to one fine-structure-split (FSS) transition, weak coupling to the other (there is an almost perfect alignment of the X and microcavity axes). From the spectra, we determine the X–vacuum-field coupling rate () and the QD exciton decay rate into other photonic modes (). The cooperativity is defined as . c, d, g, X at  T: the magnetic field induces a large frequency separation between the fine-structure-split transitions. is smaller than at because the X transitions become circularly polarised and couple less strongly to the linear-polarised microcavity mode. The simple avoided-crossing in c enables a determination of and by using data at all values of . The dotted lines in c and solid lines in dg are fits to a solution of the Jaynes-Cummings Hamiltonian in the limit of very small average photon occupation Greuter et al. (2015). h, Summary of strong-coupling parameters recorded on X at  T on three separate QDs using the same microcavity mode. in all three cases.

We employ a miniaturised Fabry-Pérot cavity consisting of a semiconductor heterostructure and external top mirror (Fig. 1a and Supplementary section II). The -factor is as high as ; the mode volume just (where is the free-space wavelength). The heterostructure (see Supplementary section I) has an n-i-p design with the quantum dots in the intrinsic (i) region (Fig. 1b). Tunnel contact with the Fermi sea in the n-type layer establishes charge control via Coulomb blockade.

We excite the quantum dot–microcavity system with a resonant laser (continuous-wave) and detect the scattered photons. The average photon occupation is much less than one. When the microcavity and QD optical frequency come into resonance, we observe a clear avoided crossing in the spectral response (Fig. 1c) signifying strong coupling. We achieve strong coupling on different charge states in the same QD (Fig. 1c), also on many different QDs (Fig. 2h and Supplementary section III). The cavity-emitter detuning is controlled in situ either by tuning the QD (voltage ) or by tuning the microcavity (voltage ) (Fig. 1c). A full spectral analysis determines the exciton-photon coupling rate , the cavity-photon decay rate , and the exciton decay rate into non-microcavity modes, (Fig. 2). For QD2 at zero magnetic field, , corresponding to a cooperativity . The -factor Kuhn and Ljunggren (2010), the fraction of quantum dot emission funnelled into the cavity mode, is %.

Figure 3: Time-resolved vacuum Rabi oscillations. Intensity auto-correlation function as a function of delay on X in QD1 for (detuned via ) and . The inset shows the first few rungs of the Jaynes-Cummings ladder. The laser drives a two-photon transition . The solid red line is the result of calculating from the Jaynes-Cummings Hamiltonian using , and from the spectroscopy experiments (Fig. 2).
Figure 4: Strong coupling versus driving frequency and power. a, Signal versus for . At low power, LP1 and UP1 are clearly observed. As the power increases, the higher rungs of the Jaynes-Cummings ladder are populated. b, for and . c, for and . d, fast Fourier transform (FFT) of in b and c. e, f and g: , signal and FFT peak frequency of versus for . The solid red lines in bg (“model” in a) result from a calculation of (signal) from the Jaynes-Cummings Hamiltonian using , and from the spectroscopy experiments (Fig. 2). In a, the truncation of the Hilbert space to 15 rungs leads to a slight underestimation of the signal at high laser powers compared to the experiment. A signal-to-background ratio (SBR) of 85 was included. In e, the dashed red line shows the theoretical limit without the laser background.

To demonstrate a coherent atom-photon exchange, “vacuum Rabi oscillations” Kasprzak et al. (2010); Hamsen et al. (2017); Kuruma et al. (2018), we drive the system at a frequency slightly positively-detuned from the lower-frequency polariton (LP1) and record the two-photon auto-correlation (Fig. 3). Coherent oscillations are observed as a function of delay whose period, 220 ps, corresponds exactly to divided by the measured frequency splitting of the polaritons (Supplementary section III.F). These oscillations can be understood in terms of the Jaynes-Cummings ladder (Fig. 3 inset). The laser drives the two-photon transition weakly. decays by emitting two photons. Detection of the first photon leaves the system in a superposition of the eigenstates and such that a quantum beat takes place. Detection of the second photon projects the system into the ground state , stopping the quantum beat (Supplementary section V). The large (80 in this particular experiment) is confirmation that the states with are preferentially scattered Faraon et al. (2008); Reinhard et al. (2012).

The behaviour of depends strongly on the laser detuning and the cavity detuning (both defined with respect to the bare exciton). For , is highly bunched at the two-photon resonance, (Fig. 4b), yet highly anti-bunched at the single-photon resonance, (Fig. 4c). The anti-bunching is a demonstration of photon blockade in this system. The full dependence on is plotted in Fig. 4e. In principle, rises to extremely high values Birnbaum et al. (2005) as . In practice, the scattered signal becomes weaker and weaker as such that reaches a peak and is then pulled down by the poissonian statistics of the small leakage of laser light into the detector channel (Fig. 4e). is a rich function: its Fourier transform shows in general three peaks (Fig. 4d). The dependence on shows that these frequencies correspond to (see Supplementary section V.D.3), and (Fig. 4g). All this complexity is described by the Jaynes-Cummings model which, taking the parameters determined by the spectroscopy experiments and a numerical solution using the first fifteen rungs of the ladder (Supplementary section IV), gives excellent agreement with the experimental in all respects (Fig. 3, Fig. 4 and Supplementary section III). As the laser power increases, there is a spectral resonance at the LP2 and UP2 transitions, and, at the highest powers, a strong resonance at – this too is in agreement with the predictions of the model (Fig. 4a), and reflects the bosonic enhancement of the transitions between the higher lying rungs of the Jaynes-Cummings ladder.

In the experiments with a single laser, the second rung of the Jaynes-Cummings ladder is accessed by tuning the laser to a two-photon resonance (Fig. 4a). An alternative is to drive the system with two lasers in a pump-probe scheme. The strong transitions arise from the symmetric-to-symmetric and antisymmetric-to-antisymmetric couplings, e.g.  and , which lead to measurable changes in the populations of the states Fink et al. (2008). We present an alternative here, “-spectroscopy”. We present this experiment on the symmetric-to-asymmetric transition. The square of the matrix element is just 3% of that associated with the transition. A pump laser drives the transition on resonance, and a probe laser, highly red-detuned with respect to the pump, is scanned in frequency in an attempt to locate the transition (Fig. 5a). There is no resonance in the scattered intensity (Fig. 5c): any resonance lies in the noise (a few per cent). However, there is a clear resonance in at exactly the expected frequency (Fig. 5b): at the weak transition the number of scattered photons hardly changes but there are profound changes in their statistical correlations. Again, the Jaynes-Cummings model describes the experiment (Fig. 5b,c). Here, a short-time expansion in a truncated Hilbert space (first two rungs of the Jaynes-Cummings ladder) is used to calculate (Supplementary section VI).

Figure 5: -spectroscopy. a, Laser 1 is on resonance with the transition (black arrow, detuning ); laser 2 is scanned across the transition (blue arrow, detuning ). b, versus showing a pronounced resonance at . The red solid line is the result of an analytical calculation based on the Jaynes-Cummings Hamiltonian (Supplementary section VI). The offset in the experimental data with respect to the theory reflects additional coincidences arising from off-resonant, two-photon absorptions not included in the model. c, Signal versus . The signal increases with increasing due to off-resonant driving of the transition by laser 2. All data for X in QD2 at  T.

As an outlook, we offer some perspectives for future development. (a) The device is a potentially excellent single photon source. For a fixed and , the photon extraction efficiency via the cavity Cui and Raymer (2005) is maximised at the condition . For achieved here, this corresponds to . At this relatively low , the residual absorption losses in the semiconductor are negligible and the photon extraction efficiency should be as high as 90%. (b) An “atom drive” Law and Kimble (1997); Hamsen et al. (2017) can be engineered with a lateral waveguide. This is an excellent prospect for creating fast spin-photon entanglements, shaped-waveform single photons and, ultimately, a photon-photon gate. (c) An even higher is conceivable by decreasing via some lateral processing. (d) Two or more intra-cavity quantum dot spins can be entangled by common coupling to the cavity mode Imamoglu et al. (1999). (e) A monolithic design could exploit strain tuning of the quantum dot rather than position-based tuning of the cavity. Also, the splitting of the cavity mode (into two modes with linear, orthogonal polarisations) can be eliminated by applying a bias across the semiconductor DBR Frey et al. (2018).

Acknowledgements
We thank Ivan Favero for inspiration on surface passivation; Henri Thyrrestrup Nielsen for support in evaluating with very small binning times; Sascha Martin for engineering the microcavity hardware; and Melvyn Ho, Peter Lodahl and Philipp Treutlein for fruitful discussions. We acknowledge financial support from SNF projects 200020_156637 and PP00P2_179109, NCCR QSIT and EPPIC (747866). SRV, RS, AL and ADW acknowledge gratefully support from BMBF Q.com-H 16KIS0109.

Additional information
Further details of the experiment and the calculations based on the Jaynes-Cummings Hamiltonian are described in the Supplementary Information.

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