## Abstract

Optical nonlinearities enable photon-photon interaction and lie at the heart of several proposals for quantum information processing, quantum nondemolition measurements of photons, and optical signal processing. To date, the largest nonlinearities have been realized with single atoms and atomic ensembles. We show that a single quantum dot coupled to a photonic crystal nanocavity can facilitate controlled phase and amplitude modulation between two modes of light at the single-photon level. At larger control powers, we observed phase shifts up to π/4 and amplitude modulation up to 50%. This was accomplished by varying the photon number in the control beam at a wavelength that was the same as that of the signal, or at a wavelength that was detuned by several quantum dot linewidths from the signal. Our results present a step toward quantum logic devices and quantum nondemolition measurements on a chip.

Photons are attractive candidates for quantum bits, because they do not interact strongly with their environment and can be transmitted over long distances. They are well suited for carrying information by means of polarization or photon number, and can be manipulated with great precision by optical elements (*1*). In addition, photonic qubits can be used to interconnect atom-like qubits realized in various systems (*2*–*6*). Quantum logic with photons requires a gate that facilitates an interaction between two coincident photons (*7*). A controlled-phase gate, which can be realized by an atom in a high-quality (Q) cavity (*2*), performs this function. In this gate, the accumulated phase of one beam is dependent on the total number of photons interacting with the atom, and the presence of other photons can be measured without destroying them (*8*–*10*).

Our nonlinear medium consisted of a three-hole–defect photonic crystal (PC) cavity (*11*) with a coupled InAs quantum dot (QD) (Fig. 1A). Because of the presence of a distributed Bragg reflector underneath the PC membrane, we treated the PC cavity as a one-sided system. The structure was thermally isolated, allowing us to control the cavity and QD resonances with a heating laser (*12*). The cavity field decay rate was κ/2π = 16GHz, corresponding to a quality factor *Q* = 10,000. The QD had an estimated spontaneous emission rate of γ/2π = 0.2GHz. In the described experiments, we employed two QDs: a strongly coupled QD with a vacuum Rabi frequency *g*/2π = 16 GHz and a weakly coupled QD with *g*/2π = 8 GHz.

We measured the phase of cavity-reflected photons by interfering them with a reference beam of known amplitude and phase (Fig. 1A). The reflectivity of the linearly polarized cavity was isolated from background laser scatter by means of a cross-polarized setup (*13*, *12*). The reference beam was introduced by inserting a quarter wave plate (QWP) between the beam-splitter and the cavity. The QWP converted the linearly polarized signal into an elliptically polarized beam with components parallel and orthogonal to the cavity polarization. After reflection from the sample, these two components acquired a relative phase. The detected signal *I*_{s} is an interference between the cavity-reflected component and the reference field (1) where *A*(θ) is a coefficient that depends on the QWP angle θ relative to the vertical polarization of the polarizing beam splitter (PBS), *r*(ω) is the frequency and power-dependent cavity reflectivity, and Ψ(θ) is the reference phase delay. *A*(θ), *r*(ω), and Ψ(θ) are given in the supporting online material (SOM).

We first performed phase measurements on a single (signal) beam reflected from the cavity with a QD. Interference between the QD-scattered field and the incident signal resulted in the rapidly varying feature in Fig. 1D. As the phase of the reference beam increased from 0° to 33°, this interference evolved from destructive to constructive, and the dip at θ = 0° changed to a peak at θ = 33°. We find that this interference is only explained by coherent light scattering from the QD. The experimental data are fit well by Eq. 1, as shown in Fig. 1D. Each fit gives the signal phase where and are the real and imaginary parts of the cavity reflectivity *r*(ω). The phase fits for 11 scans with different QWP angles θ are superposed in Fig. 1E. As the signal wavelength traverses the cavity resonance, ϕ changes from 0 to –π. An additional phase modulation occurs at the QD resonance, where the phase varies by almost π over the dot bandwidth (2*g*^{2}/κ = 2π × 32 GHz).

When measuring the controlled-phase shifts, we first considered the cases in which the control and signal have the same wavelength (they potentially could be distinguished by polarization or incident direction). When the control and signal are at the same wavelength, the nonlinear interaction between them (Fig. 2, A and B) arises from the saturation of the QD in the presence of cavity-coupled photons (*12*). Saturation occurs when the average photon number inside the cavity reaches approximately one photon per modified QD lifetime, given by κ/*g*^{2}. The cavity photon number is *n*_{c} = η*P*_{in}/[2 κħω_{c}], given the input power *P*_{in}, control frequency ω_{c}, and coupling efficiency η ≈ 2 to 5% in our experimental setup. The observed QD-induced dip does not fully reach zero at low powers, as expected from theory (*12*, *14*), because of QD-wavelength jitter and blinking (see SOM).

We observed a phase modulation of 0.24π (43°) when the control photon number was increased from *n*_{c} = 0.08 to 3 and the wavelength was set 0.014 nm (*g*/3.5) away from the anticrossing point (Fig. 2C). The reflectivity amplitude *R* normalized by the cavity reflectivity without a dot *R*_{0} is shown for the same detuning in Fig. 2E and changed from 50 to 100% at saturation. The excitation powers were 40 nW and 1.3 μW, measured before the objective lens (corresponding to *n*_{c} of 0.08 and 3, respectively), and indicate a coupling efficiency of up to 5%. However, the coupling efficiency fluctuated because of sample drift during the experiment. Therefore, we estimate control powers from fits to the data, and give power levels measured before the objective lens for reference.

In the context of quantum gates (*2*, *15*, *16*), we are interested in the signal photon's phase change caused by a single control photon. When the control and signal have the same wavelength (λ_{c} = λ_{s}) and the same duration, the change is given by the difference between the phase evaluated at *n*_{c} and 2*n*_{c} (Fig. 2C). We measured a maximum differential phase shift of 0.07π (12°) when *n*_{c} = 0.1. The differential amplitude is maximized at a higher *n*_{c} = 0.43, where it changes by 15% when *n*_{c} is increased to 2*n*_{c} (Fig. 2E). Theoretically, we estimate a maximum of 0.15π (27°) for phase and 20% amplitude modulation with our system parameters.

Conventionally, the intensity-dependent refractive index *n*_{2} or the Kerr coefficient χ^{(3)} describes the strength of a nonlinear medium in which the nonlinearity is proportional to the photon number (*17*). The cavity-embedded QD is highly nonlinear and is not well described as a pure Kerr medium. However, for weak excitations, we can still approximate the nonlinear index and susceptibility from the relationship between the acquired signal phase shift ϕ_{s} and *n*_{2} given by ϕ_{s} =(2π*n*_{2}/λ_{c})(*P*_{in}/*A*_{cav})(*c*/2κ*n*), where *A*_{cav} ≈ (λ/*n*)^{2} is the cavity area, and *c*/2κ*n* gives the propagation length in GaAs with refractive index *n* = 3.5. From our experimental data at very low values of control power, we infer *n*_{2} ≈ 2.7 × 10^{–5} cm^{2}/W and χ^{(3)} = 2.4 × 10^{–10} m^{2}/V^{2}. This value is many orders of magnitude larger than most fast optical nonlinearities in solid-state materials.

Spontaneous emission from the QD into modes other than the cavity reduces the performance of quantum gates because of photon losses. In Fig. 2D, we show a 1% photon loss due to incoherent fluorescent emission from the QD, which is driven 0.014 nm away from resonance by the signal laser. Fluorescence loss is expected to scale as *F*_{PC}/(*F* + *F*_{PC}) ≈ 0.15%, where *F* = 160 is the QD Purcell factor in the PC cavity and *F*_{PC} ≈ 0.25 is the suppression of the QD radiative rate due to the PC lattice (*18*). The observed 1% is higher than the expected value for losses, but within error, because *F*_{PC} strongly depends on the dot position and can at most be unity. Radiation from nearby emitters cannot be excluded from this signal and therefore fluorescence losses from the addressed QD may be lower (*18*).

For applications such as quantum nondemolition (QND) detection and optical control, it is advantageous to spectrally separate the control and signal beams. We detuned the control beam by Δλ = –0.027 nm (1.2*g*) with respect to the signal beam, which again was aligned to the QD-cavity intersection (Fig. 3A). The number of signal photons per QD lifetime (*n*_{s}) was fixed and the control photon number (*n*_{c}) was varied. In these measurements, a weakly coupled QD with *g*/2π ≈ κ/4π = 8 GHz was used. Saturation power scaled with the modified spontaneous emission rate *g*^{2}/κ, and so the smaller *g* value permitted lower control powers and reduced background noise. In Fig. 3, we show the principle of the measurement. First, the signal and control were turned on independently; the QD dip is visible in Fig. 3, B and C. The dip disappeared when the two beams were turned on simultaneously and interacted (Fig. 3D). For better visibility at high control powers, the signal power in Fig. 3 was set to 100 nW before the objective lens, corresponding to *n*_{s} = 0.2 signal photons in the cavity per cavity lifetime.

In Fig. 4, we show experimental results for phase shifts with control and signal beams at different wavelengths. Here, the signal phase was affected by the saturation of the QD and a frequency shift of the QD due to the ac Stark effect (*17*), which can be used to realize large phase shifts (*19*). The signal reflectivity and phase as functions of control-beam photon number are shown in Fig. 4, A and B. We fit both the signal and control data by a full quantum simulation and derived the underlying signal phase shift as a function of the control photon number (*20*) (see SOM). The reflectivity at the signal wavelength saturated completely when the control photon number reached *n*_{c} = 1.3, which corresponds to 1 μW of power measured before the objective lens. The associated phase modulation was 0.13π at the signal detuning of 0.009 nm (0.4*g*) from the dot resonance. The phase behavior in Fig. 4B is asymmetric with respect to the center of the QD-induced dip because the coupling of the control beam changed with the temperature scan.

We fixed the signal wavelength at 0.009 nm (*g*/3) away from the QD resonance and determined the phase and amplitude modulation for a range of values of *n*_{c}. The signal phase ϕ(*n*_{c}) relative to the signal phase with no control ϕ_{0} = ϕ(*n*_{c} = 0) is shown in Fig. 4D. The maximum observed phase shift when *n*_{c} = 1 was 0.16π (28.8°). The largest nonlinear phase change was observed for *n*_{c} = 0.05, for which ϕ (*n*_{c}) – ϕ(0) = 0.05π (9°). These values give a nonlinear index of *n*_{2} ≈ 1.8 × 10^{–5} cm^{2}/W, or χ^{(3)} ≈ 1.6 × 10^{–10} m^{2}/V^{2}, for a detuning of 0.027 nm (1.2*g*) between the signal and control. This value is similar to that of the QD with larger *g*. Numerical simulations indicate that the relative magnitude of nonlinearities due to these two QDs strongly depends on the laser frequency. The nonlinearities for the two cases are summarized in Table 1.

The current implementation of the QD/cavity system is already promising for low-power and QND photon detectors (*8*–*10*). We have shown that the phase and amplitude of the signal strongly depend on the control photon number when the signal and control photons are spectrally separated. Furthermore, the magnitude and bandwidth of the Kerr nonlinearity χ^{(3)} observed in this experiment are rivaled only by measurements in atomic ensembles (*21*, *22*).

To realize useful quantum logic gates, controlled π phase shifts are necessary (*23*). This will require repeated interactions. Such cascading requires coupling efficiencies that are higher than the observed 2 to 5%. This technical challenge can be overcome. We have already demonstrated architecture for a QD cavity-waveguide–coupled quantum network (*24*) with coupling efficiency above 50% between two nodes, and cavity-waveguide couplers (*25*) with coupling efficiency reaching 90%. The observed fluorescence losses are already sufficiently low to allow scalable computation (*26*), and can be further improved with increases in cavity Q. The ability to tailor photon-QD interactions by PC fabrication makes this a highly versatile platform for a variety of quantum optics experiments and has great potential for compact scalable quantum devices.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/320/5877/769/DC1

SOM Text

Figs. S1 and S2

References and Notes