Introduction
Media with long 
ph values10, 11, 15, 16, 17, 18, 19 are attractive for optical storage, and for optical functional devices based on the interaction between light and matter. For both applications, the system size should be as small as possible because a long ph per shorter length means a large storage capacity density, and a long ph per smaller volume means strong interaction enhancement. Of the promising high-Q cavity systems2, 4, 19, 20, 21, photonic-crystal cavities have the smallest size and a high value of Q over the mode volume V. This is particularly important because large values of Q/V enable various forms of enhancement in light–matter interactions, such as a reduction in switching power5, 8. Indeed, cavities with a Q of 9
105 and V of 
1.7(
/n)3 where n is the refractive index of silicon have been fabricated in two-dimensional silicon photonic-crystal slabs11, and all-optical switching5 or bistable memories22 have been demonstrated at an extremely low input. The ultrahigh Q of these photonic-crystal cavities has only been determined by measuring the transmission spectrum width, which is only a few picometres wide. In contrast, Q can be directly derived by measuring ph, which is a more reliable way to determine an ultrahigh Q. Although such measurements have been performed for other systems19, 23, for semiconductor photonic-crystal cavities they have only been performed for relatively low Q (ref. 24). This is because ultrasmall high-Q semiconductor photonic-crystal cavities exhibit a strong nonlinearity25, 26, which disturbs measurement even at a relatively low input. Time-domain measurements are also crucial for slow-light applications. Although photonic crystals are considered to be promising candidates for all-dielectric slow-light media, and exhibit significantly slow light (10-2c) in the frequency domain17, 18, it is difficult to demonstrate ultraslow light in the time domain because of the dispersion problem. High-Q and ultrasmall cavities are building blocks for coupled-resonator optical waveguides (CROW), which can potentially overcome this dispersion problem27. Therefore, time-domain measurements of pulse propagation by means of a cavity will demonstrate the possibility of the slow-light application of nanocavities. In addition, the time-domain measurement provides a further benefit. The adiabatic dynamic control of light12, 13, 14, which is a novel light manipulation technique, is now being discussed in relation to such ultrasmall high-Q systems. By abruptly changing the optical properties of the system within ph, we can convert the frequency of light or change its speed. The physical mechanism of this new control technique is simple, but it becomes meaningful only in small systems with a long ph. This demonstration essentially requires time-domain observation.
On the basis of the above considerations, we performed a series of time-domain measurements of high-Q nanocavities. For this study, we used waveguide-width modulated photonic-crystal nanocavities that can exhibit extremely high Q. We have already reported the cavity design in detail11. Figure 1a and b shows the design and a scanning electron microscope (SEM) image of the device we used for our study. Figure 2a shows the continuous wave (CW) transmission spectrum measured through input and output waveguides. The loaded Q derived from the spectrum width is extremely high at 1.2106, which is the largest Q reported for any wavelength-sized optical cavity. The estimated unloaded Q is 1.3106. The near-field top-view image of the device at the resonance injection clearly shows that the light is confined in the cavity (Fig. 1c), where the calculated V is only 1.7(/n)3. For the time-domain measurement, we used a time-correlated single-photon counting (TCSPC) system28, which uses a near-infrared photomultiplier tube. This approach enables us to measure light with an extremely weak signal. Rectangular pulses with a sudden ON to OFF transition are launched into the nanocavity through the input waveguide. Figure 2b shows the measured time-domain signal of the cavity shown in Fig. 1. The photon lifetime is an extremely long 1.01 ns, which is equivalent to Q=1.2106. This is the first direct time-domain demonstration of an ultrahigh-Q measurement in wavelength-scale optical cavities, and the obtained value agrees reasonably well with that derived using a spectral-domain measurement. This demonstration suggests the possibility of storing photons for a significantly long time. In comparison with optical bistable memories, in principle, this kind of photon-trapping memory can store both the coherence and quantum state of the light. Note that the input and output are all in plane for our device, which will be crucial for on-chip applications.
Figure 1: Photonic-crystal nanocavity with a locally modulated waveguide width.
a, The design parameters, where a is the lattice constant, r is the hole radius and t is the slab thickness. b, An SEM image of the device, which was fabricated in a silicon photonic-crystal slab and consists of a line-defect nanocavity with local width modulation, which is coupled to input and output waveguides. Although it is difficult to discern, the holes in the circled region are shifted slightly towards the outside, which enables mode-gap confinement to be obtained. The cavity can be charged and discharged through the input and output waveguides. L is the distance between the waveguides. c, A near-field image acquired using a near-infrared camera from the top of the sample when a resonant wavelength light was injected through the input waveguide. The waveguide regions are indicated by the red squares.
Full size image (54 KB) (54 KB)Figure 2: Photon lifetime and pulse delaying observed at an ultrahigh-Q photonic-crystal nanocavity.
a, Transmission spectrum of the nanocavity measured using a wavelength-tuneable CW light source. The corresponding Q value is 1.2106. Full width at half maximum (FWHM) of the transmission spectrum is 1.3 pm. b, The time-domain signal from the photonic-crystal nanocavity (red curve) with a rectangular pulse input that has a sharp ON to OFF transition. The fitted exponential curve (blue line) has a decay time of 1.01 ns. The signal is acquired using TCSPC. The black line is the waveform of the output from a straight photonic-crystal waveguide without a cavity as the reference data. The decay time was 70 ps, which gives the measurement resolution. c, A 1.07-ns-wide gaussian-like optical pulse is generated by using an electrical pulse shaper and an electro–optic modulator, and launched into the photonic crystals. The transmitted pulse from a photonic-crystal nanocavity that has a Q of 7.4105 is indicated by the red line. The black line indicates a reference pulse transmitted from a photonic-crystal waveguide without a cavity. d, The same arrangement as c, but the input pulse width is 1.90 ns.
We next measured the delay time in a cavity by using a pulse input. The black curves in Fig. 2c and d are the output from samples without a cavity (a straight photonic-crystal waveguide) for two different input pulse widths, and the red curves are from samples with a cavity (as shown in Fig. 1) for the same pulse input. The dispersion in the waveguides was negligible because of their short length. The time difference between the red and black curves, therefore, corresponds to the group delay time delay due to the cavity. This delay is mainly determined by the phase shift caused by the cavity resonance. Note that the delay time depends on the input pulse width. A coupled-mode theory simulation predicts that one can achieve delay 
2ph for a relatively long input pulse without noticeable distortion, and that the delay becomes smaller for shorter input pulses. When the input pulse is shorter than ph, the pulse shape becomes distorted because the frequency width of the input pulse is wider than the cavity resonance. The result obtained in the simulation is almost identical to the measured one. The delay time of 1.45 ns is the longest yet reported for slow-light systems in photonic crystals. In addition, the transit speed of the pulse from the input waveguide end to the output waveguide end, of length L=8.4 
m (Fig. 1b), is estimated to be 5,800 m s-1, which indicates that the light velocity was reduced to 210-5c. This is the slowest value reported for any dielectric slow-light media. As we pointed out earlier, our device (a cavity coupled to waveguides) is a basic building block for a CROW. The group velocity obtained with a CROW does not depend on its length and should be similar to that of a single-cavity system. Thus, our result provides the ultimate limit for the lowest group velocity in a CROW using existing cavities. For a practical slow-light application, we may have to cascade these cavities to form a CROW mode to extend the time delay or expand the bandwidth depending on the use, or we can introduce the dynamic tuning to break the delay-bandwidth product limitation29.
We next investigate the optical nonlinear effect on Q and ph. The red line in Fig. 3b–d shows the transmitted signal waveform, at different input powers, from the waveguide-width-modulated cavity having a ph of 1 ns. We observed bistable nonlinear behaviour that resulted from the thermo–optic effect induced by two-photon absorption in silicon25 at a power of more than 8 W in CW measurements (Fig. 3a). Notice that this value is the power at the input photonic-crystal waveguide. Because the transmittance of the cavity through the input to output photonic-crystal waveguides is 1%, the estimated power that couples to the cavity is only 800 nW. When we increase the input power in the time-domain measurements, we expect ph to be shorter because of the increased two-photon and free-carrier absorption. Indeed, we observed shortening of ph when the input amplitude was larger than 20 W, and ph decreases from 1 ns to 470 ps as the input power increases (Fig. 3b–d). To investigate this more closely, we fitted Fig. 3c with a double exponential curve at 580 ps and 1.19 ns. Because this fast decay curve recovers quickly to a slow decay curve, we believe it to be somehow related to the dynamics of the carriers5. By measuring the dynamics in the time domain, we managed to separate the information on carrier-based nonlinear effects from the linear response of the cavity. This kind of detailed information cannot be obtained from spectral-domain measurements. These results demonstrate that time-domain measurements are a particularly powerful tool for investigating the dynamic properties of nanocavities.
Figure 3: Nonlinear response of a waveguide-width-modulated photonic-crystal nanocavity.
a, The transmission spectrum of the photonic-crystal nanocavity at different CW input powers. The centre wavelength shift is caused by the thermo-optic effect and the sharp edge observed at an input power of 8 W is caused by the bistability resulting from the thermo–optic effect generated by two-photon absorption carriers. b–d, The reference waveform of the output from a photonic-crystal waveguide without a cavity (black line). The decay time was 190 ps, which gives the measurement resolution. The red line is the measured ON/OFF edge of the rectangular pulse of the output from a photonic-crystal nanocavity at different input amplitudes (20 W, 80 W and 170 W, as shown in panels b, c and d, respectively). The wavelength of the input light is set to the cavity resonance. The fitted exponential curves are shown by the blue lines. At a low input power the decay is a smooth exponential curve, but a faster decay is observed for a higher input power. The slight intensity modulation at the flat top of the rectangular pulse for the 170-W input results from the cavity resonance shift caused by the plasma dispersion effect of the generated carriers.
Finally, we demonstrated dynamic Q tuning using timing-controlled pump pulses. The purpose of this study is to change Q dynamically within ph, which is normally difficult for small cavities. We pumped the photonic-crystal cavity from the top of the slab. As the absorbed energy creates free carriers, Q can be dynamically tuned when the timing is adequately adjusted. We set the timing of the pump pulse immediately after the ON to OFF transition of the rectangular probe and recorded the decay. The result is shown in Fig. 4, where the decay times of the probe waveform with and without pumping are obviously different. This is a clear demonstration of the dynamic control of high-Q cavities within ph. Control of the optical cavity within ph will enable us to study various interesting aspects of the dynamic control of light12, 13, 14, which has recently been subject to extensive discussion. When we consider buffering a light signal in a cavity, the discharge time is usually limited by ph, which will limit the speed of the buffering operation. This experiment opens the possibility of eliminating photons faster than ph. Although it is not a direct demonstration of the read–write memory operation, in the future, coupled cavities will allow the sophisticated trapping/release of photons dynamically from/into the input/output waveguide14. This will constitute a true all-optical on-chip memory.
Figure 4: Dynamic tuning of the photon lifetime of a photonic-crystal nanocavity.
Output waveform of a Q=2.7105 waveguide-width-modulated cavity measured using a digital sampling oscilloscope with a fast InGaAs detector. The red curve shows the ring-down waveform of this cavity where the photon lifetime is 250 ps. The black curve is the ring-down waveform when the cavity is irradiated with a 2-ps pump pulse at 800 nm and t=350 ps to generate carriers. The decay immediately after the pump pulse is 30 ps, which is close to the temporal resolution of the sampling oscilloscope. We roughly estimate that the optical pumping energy absorbed in the cavity volume is 12 fJ.
The time-domain measurement performed in this work directly showed that photons are trapped for 1 ns in an extremely small space. In addition, we showed, through this measurement, that the photonic-crystal nanocavity can delay a pulse for 1.45 ns, which is equivalent to 5,800 m s-1 (210-5c). This technique also constitutes a powerful tool for investigating the dynamic behaviour of photonic-crystal nanocavities.
Methods
Our typical setup has a time resolution of 70 ps, as shown by the black line in Fig. 2b. (Note that the time resolution for the Fig. 3b–d experiment is slightly lower, at 190 ps, as shown by the black line in Fig. 3b, because we used a slower pulse pattern generator.) This value incorporates various timing jitters of the pulse generator (<15 ps), fibre vibration, detector and TCSPC circuits (<25 ps). By using an electrical pulse pattern generator and a 40-GHz lithium-niobate intensity modulator, we generate a 10-ns-wide rectangular optical pulse having a falling time of less than 25 ps. The optical power of the pulse is adjusted using a variable attenuator before injecting it into the photonic-crystal nanocavity. We typically set the photon counting rate of the TCSPC to below 0.05 photons per incidence, and performed accumulated measurement until the number of photons at the peak was of the order of a few hundreds.
Author contributions
T.T. performed the experiment, M.N. planned the project, E.K. fabricated the sample, and both A.S. and H.T. supported the numerical calculation and discussion.
ek, J. & Polzik E. S. 
i
, M. & Joannopoulos, J. D. 
