Abstract
Quantum dots in photonic crystals are interesting because of their potential in quantum information processing^{1,2} and as a testbed for cavity quantum electrodynamics. Recent advances in controlling^{3,4} and coherent probing^{5,6} of such systems open the possibility of realizing quantum networks originally proposed for atomic systems^{7,8,9}. Here, we demonstrate that nonclassical states of light can be coherently generated using a quantum dot strongly coupled to a photonic crystal resonator^{10,11}. We show that the capture of a single photon into the cavity affects the probability that a second photon is admitted. This probability drops when the probe is positioned at one of the two energy eigenstates corresponding to the vacuum Rabi splitting, a phenomenon known as photon blockade, the signature of which is photon antibunching^{12,13}. In addition, we show that when the probe is positioned between the two eigenstates, the probability of admitting subsequent photons increases, resulting in photon bunching. We call this process photoninduced tunnelling. This system represents an ultimate limit for solidstate nonlinear optics at the singlephoton level. Along with demonstrating the generation of nonclassical photon states, we propose an implementation of a singlephoton transistor^{14} in this system.
Main
The optical system consists of a selfassembled InAs quantum dot with decay rate γ/2π≈0.1 GHz coupled to a threehole defect cavity^{15} in a twodimensional GaAs photonic crystal, as described in ref. 5. The quantumdot/cavity coupling rate g/2π=16 GHz equals the cavity field decay rate κ/2π=16 GHz (corresponding to a cavity quality factor Q=10,000), which puts the system in the strong coupling regime^{10,11}. We first characterize the system in photoluminescence by pumping the structure above the GaAs bandgap. The photoluminescence scans in Fig. 1b show the anticrossing characteristic of strong coupling between the quantum dot and the cavity. Here, the quantum dot is tuned into resonance using local temperature tuning^{16} around an average temperature of 20 K maintained in a continuous He flow cryostat. To generate nonclassical light, we coherently probe the system with linearly polarized laser beams (Fig. 1a) and observe the crosspolarized output, as described in our previous work^{5}. The crosspolarized setup enables us to separate the cavitycoupled signal from the direct probe reflection, which is essential for achieving large signaltonoise ratios needed in autocorrelation measurements. Our setup is such that the measurement on the reflected port from this singlesided cavity is analogous to a transmission measurement in a Fabry–Perot arrangement.
The energy eigenstates of a twolevel system strongly coupled on resonance to an optical resonator are grouped into twolevel manifolds denoted n,±〉, with energies , where n is the number of energy quanta in the system and ω_{0} is the barecavity frequency (Fig. 2a). The anharmonic energylevel spacing causes phenomena such as photon blockade^{12} or photoninduced tunnelling. To observe photon blockade, a coherent probe beam (frequency ω_{p}) tuned to ω_{1,±}=ω_{0}±g is coupled to the cavity. This probe is resonant with the firstorder manifold, but detuned from transitions to the second manifold, , as shown in Fig. 2a. Consequently, once a photon is coupled into the system, it suppresses the probability of coupling a second photon with the same frequency. As a result, the output field acquires subPoissonian statistics. In addition to photon blockade, photoninduced tunnelling is expected near the barecavity resonance (ω_{p}−ω_{0}=Δω_{p}→0): the absorption of a first photon enhances the absorption of subsequent photons owing to resonance with higherorder manifolds, so the output consists of ‘photon bunches’. These phenomena are purely quantum effects that cannot be explained using semiclassical theories. These effects can be probed by measuring the secondorder correlation function, g^{(2)}(τ). The signature of the photon blockade effect is the antibunching in g^{(2)}(τ) (that is, g^{(2)}(0) is a local minimum, g^{(2)}(0)<g^{(2)}(τ)), as recently demonstrated by Birnbaum et al. ^{12} in an experiment with neutral atoms. In the case of photoninduced tunnelling, g^{(2)}(0) is a local maximum.
In Fig. 2b, we simulate the theoretical output spectrum as a singlefrequency probe beam is tuned through the cavity and indicate the resonance of the transitions 0〉→1,+〉 and 1,+〉→2,+〉. The simulated driving field injects an average cavity photon number 〈n〉∼0.4 when resonant with the polaritons in the first manifold, and slightly saturates the quantum dot. The expected secondorder correlation function for our system is shown in Fig. 2c, where we plot the dependence of g^{(2)}(0) on the detuning Δω_{p} of the probe from the anticrossing frequency ω_{0}. As expected from the intuitive argument above, the simulation predicts photon bunching as Δω_{p}→0. Photon blockade is evident in the antibunched region near Δω_{p}∼±1.5g (Fig. 2c, inset). The blockade does not occur at Δω_{p}=±g as previously explained because the linewidth of the eigenstates (∼κ) is comparable to the splitting of the manifolds (∼2g), which results in a significant overlap of the allowed transitions between consecutive manifolds. As Δω_{p}→0, the probability of absorbing the first photon decreases. However, if a photon is absorbed, it enhances the probability of capturing subsequent photons, and produces a photonbunched output.
We measure the timedependent autocorrelation g^{(2)}(τ) using the Hanbury–Brown–Twiss (HBT) setup shown in Fig. 1a and described in refs 17 and 18. The relevant features occur at timescales that correspond to the quantumdot/cavity coupling rate g, enveloped by the coherence time^{19}, as shown in Fig. 2d. The coherence time for our system is given by the cavity photon lifetime 1/2κ∼5 ps. Hence, the timedependent features in g^{(2)}(τ) occur much faster than the 300 ps timeresolution of the singlephoton counting modules in the HBT setup. To resolve the relevant features, we sample the autocorrelation function by short pulses (Δt_{FWHM}∼40 ps, Δω_{FWHM}/2π∼12 GHz, where FWHM stands for fullwidth at halfmaximum) with a repetition rate of 12.5 ns. This probe pulse duration represents a compromise between fast sampling and a linewidth that is narrow enough to resolve the relevant spectral features. In the remainder of this letter, we present the measurements of g^{(2)}(τ) for different detunings of the probe beam, denoted as g^{(2)}(τ,Δω_{p}/g).
To observe photon blockade and photoninduced tunnelling, we measured the unnormalized secondorder correlation function at detunings Δω_{p}/g=1.5 and Δω_{p}/g=0 as shown in Fig. 3. The expected photon antibunching and bunching behaviour is clearly visible at zero time delay (Fig. 3b,d). The histograms also show bunching over timescales of hundreds of nanoseconds. This bunching is a purely classical effect that results from the Poissonian blinking of the quantum dot. As reported by Santori et al. ^{20}, such blinking is caused by quantumdot transitions between a bright and a dark state, and results in bunching near τ=0 that falls off with the mean switching rate. Our observations indicate that the blinking rates vary for different quantum dots. The quantum dot measured in this experiment spends ∼80% of the time in the bright state.
Photon blockade and photoninduced tunnelling are quantified by the normalized secondorder correlation function g^{(2)}(τ,Δω_{p}/g). Each peak in the histogram of Fig. 3 represents the unnormalized value of the secondorder correlation averaged over the pulse duration of 40 ps. We express this time averaging by using the notation . The data are normalized such that . We stress that captures both the quantum and classical nature (blinking) of the output field. To find the normalization constant , we fit the histogram with the function for m≥1. The quantity represents the number of counts at time m T_{0}, where m indexes the peak number with m=0 corresponding to τ=0 and T_{0}=12.5 ns is the pulse repetition period. The normalized secondorder correlation at τ=0 is (see Fig. 3 for details). In the case of photon blockade, , showing the antibunched quantum nature of the system. For photoninduced tunnelling, (Fig. 3d), which indicates bunching.
There are several factors that account for the difference between the theoretically predicted (Fig. 2c) and measured values for : background due to imperfect extinction of the crosspolarized experimental setup (signaltonoise ratio ∼6:1), quantumdot blinking and finite bandwidth of the probe that affects the spectral resolution. Both the background and the output signal when the quantum dot is in the dark state result in a flat secondorder correlation with (coherent light). With the quantum dot in the dark state, the cavity reflectivity becomes that of an empty cavity, as shown by the dashed line in Fig. 2b. Near Δω_{p}/g=0, the empty cavity (quantum dot in dark state) has high transmission, so the observed signal has a large coherentstate component. As a result, the region near Δω_{p}/g=0 is expected to show the largest deviation in g^{(2)}(0) compared with the ideal (nonblinking) dot in Fig. 2c. This deviation will bring the observed g^{(2)}(0) closer to g^{(2)}(0)=1 of a coherent beam. At the blockade frequency (Δω_{p}/g∼1.5), the transmitted intensity in the dark state decreases relative to the brightstate intensity (Fig. 2b), and coherent light represents a smaller fraction of the collected signal.
We repeated the autocorrelation measurements for a large set of detunings to map the full spectrum of . The measurement of the full autocorrelation spectrum entails several challenges such as sample drift resulting in fluctuating coupling intensity into the cavity, and fluctuating temperature. To map the dependence of on probe detuning, we maintained constant coupling into the cavity mode for the full duration of the experiment. In Fig. 4 we plot for different detunings of the probe frequency. To emphasize the nonclassicality of the signal, we plot in the same figure , the bunched secondorder correlation resulting from quantumdot blinking. For every autocorrelation measurement, the nonclassical and classical contributions were easily distinguished by their greatly differing timescales as in Fig. 3c,d. The plots in Fig. 4 show the transition from the blockade regime () to the photoninduced tunnelling regime . The values for the classical bunching were obtained by taking the ratio (see Fig. 3). As expected, is higher as Δω_{p}/g→0 because the intensity fluctuations due to blinking are largest at this detuning. While taking the data, we kept a constant probe power of ∼1.0 nW before the objective lens (〈n〉∼0.4 at the polariton frequency), and the coupling was reoptimized for every data point. The lowest value for obtained in this data set is not as antibunched as the value reported in Fig. 3b, because we could not reproduce exactly the same coupling conditions. We found that the experimental data is well fitted by a numerical model that takes into account pulses of finite bandwidth, quantumdot blinking and background from the imperfect extinction of the crosspolarized setup (see Fig. 4 and the Methods section for details).
The experimental data in Fig. 4 show that, starting from a coherent state, the strongly coupled system enables control of the statistics of the output field from subPoissonian to superPoissonian. Thus, by engineering the parameters of the system and by choosing the appropriate probe frequency, various nonclassical states of light could be generated on demand. One of the most useful states is the singlephoton state that has applications in quantum cryptography and distributed quantum networking. To achieve efficient singlephoton sources based on photon blockade in strongly coupled solidstate systems, the quality factor (Q) and the coupling strength (g) need to be higher than those in our current work. In Fig. 5a, we show the expected secondorder coherence g^{(2)}(0) when operating in the blockade regime for the range of parameters 1.6<κ/2π[GHz]<16 (that is, 10^{4}<Q<10^{5}) and 16<g/2π[GHz]<64. These estimations show that with Q=25,000 and g/2π=48 GHz, values achievable in photonic crystals with InAs quantum dots^{21}, the singlephoton source should exhibit antibunching with g^{(2)}(0)∼0.15 (Fig. 5b). For even higher quality factors (Q=10^{5}), almost complete antibunching is expected (g^{(2)}(0)∼0.01). These simulations were carried out assuming continuouswave weak excitation (average cavity photon number 〈n〉∼0.01) of the system.
Using the anharmonicity of the eigenenergy spacing in this system, a singlephoton transistor^{14} could be implemented. In our transistor scheme, the frequency of the gate field is resonant with one of the polaritons in the firstorder manifold, say ω_{0}+g. A photon injected at ω_{0}+g increases the probability of absorbing photons that are resonant with the 1,+〉→2,+〉 transition at . If the signal is tuned to this frequency, the presence of the gate field enhances the transmission of the signal field^{22}. The photonic crystal architecture enables easy integration of such a singlephoton transistor with photonic crystal waveguides^{23,24} so the singlephoton switching is done directly on the chip. The most straightforward configuration would be a photonic crystal cavity buttcoupled in between two photonic crystal waveguides^{25}. For a practical implementation, it is desirable that both the singlephoton source and the singlephoton transistor operate in pulsed mode, with one photon emitted (or switched) per pulse. The performance of the device depends on the coupling efficiencies in and out of the cavity, the bandwidth and the intensity of the pulse (a detailed analysis of the device performance will be the subject of further publications).
Methods
Autocorrelation measurement
We scan several cavities until we find one that contains a strongly coupled quantum dot, as determined by the anticrossing behaviour in photoluminescence between the quantum dot and the cavity during temperature tuning. Then we direct the pulsed laser beam at the cavity and observe the reflected beam in crosspolarization. While tuning the local temperature with an extra heating beam, we adjust the probe beam coupling to optimize the quantumdotinduced reflectivity drop, as described for the continuouswave beam in ref. 5. Then we stop scanning and temperaturetune the quantum dot and cavity into resonance. With the pulsed probe beam at different detunings with respect to the anticrossing point, we measure the autocorrelation signal by passing the reflected probe through a grating filter (to remove stray light) followed by the HBT setup. To limit sample drift, the alignment procedure is repeated for every data point in Fig. 4.
Data analysis
The numerical model for the secondorder coherence in Fig. 4 is based on numerical integration of the quantum master equation. A timedependent driving term in the Hamiltonian represents the 40 ps excitation pulses. The intensity of the drive field matches the intensity used in the experiment, representing onethird of the quantumdot saturation intensity. In our experiment, this intensity was ∼1 nW for the incident beam, measured before the objective lens. The state of the quantum dot/cavity is timeevolved using a quantum Monte Carlo approach, which we based on the qotoolbox of ref. 26. The Hamiltonian is given by
where the field E(t) represents the timedependent driving field (frequency ω) of the cavity and is given by a sequence of Gaussian pulses. a,a^{†} denote the annihilation and creation operators of the cavity mode, and σ_{+,−} are the raising and lowering operators of the quantum dot. The quantum dot can emit into free space or into the cavity mode, which in turn leaks photons into the output channel at the loss rate ω/Q. We then compute the autocorrelation on the output channel, as described in greater detail in ref. 27. The simulation also accounts for quantumdot blinking and laser background. The full secondorder coherence is calculated as a weighted sum of the different contributions,
where the autocorrelation function G_{B}^{(2)}(τ) accounts for the quantumdot bright state, G_{D}^{(2)}(τ) for the quantumdot dark state (calculated using g→0), G_{BG}^{(2)}(τ) for background laser signal (a coherent state) and p_{B}, p_{D}, p_{BG} are the corresponding probabilities. The secondorder correlation function for zero time delay is computed as . Here, G^{(2)}(1) is the autocorrelation of the nearestneighbour peak to τ=0 in the simulation. Owing to computational constraints, this is not evaluated at the actual pulse repetition time τ=12.5ns but at τ=300 ps, a sufficient separation that amounts to nearly 60 coherence lengths. The factor is estimated from each autocorrelation measurement as .
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Acknowledgements
Financial support was provided by the MURI Center for Photonic Quantum Information Systems (ARO/IARPA Program), ONR Young Investigator Award, I.F. was supported by the NDSEG fellowship and D.E. was supported by the NSF and NDSEG fellowships. Part of the work was carried out at the Stanford Nanofabrication Facility of NNIN supported by the National Science Foundation.
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Faraon, A., Fushman, I., Englund, D. et al. Coherent generation of nonclassical light on a chip via photoninduced tunnelling and blockade. Nature Phys 4, 859–863 (2008). https://doi.org/10.1038/nphys1078
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