A frequency-stabilized source of single photons from a solid-state qubit

Single quantum dots are solid-state emitters which mimic two-level atoms but with a highly enhanced spontaneous emission rate. A single quantum dot is the basis for a potentially excellent single photon source. One outstanding problem is that there is considerable noise in the emission frequency, making it very difficult to couple the quantum dot to another quantum system. We solve this problem here with a dynamic feedback technique that locks the quantum dot emission frequency to a reference. The incoherent scattering (resonance fluorescence) represents the single photon output whereas the coherent scattering (Rayleigh scattering) is used for the feedback control. The fluctuations in emission frequency are reduced to 20 MHz, just ~ 5% of the quantum dot optical linewidth, even over several hours. By eliminating the 1/f-like noise, the relative fluctuations in resonance fluorescence intensity are reduced to ~ 10E-5 at low frequency. Under these conditions, the antibunching dip in the resonance fluorescence is described extremely well by the two-level atom result. The technique represents a way of removing charge noise from a quantum device.

Single photons are ideal carriers of quantum information. A quantum state stored in one of the degrees of freedom of the photon's wave packet (polarization, phase or time-bin) can be maintained over long distances. Single photons are therefore important in quantum communication, for coupling remote stationary qubits, the basis of a quantum repeater, or for coupling different elements in a quantum device. Furthermore, single photons are the seed for a variety of quantum optics experiments.
A key challenge is to develop a single photon source [1]. Key parameters are fidelity of the antibunching, flux, wavelength and photon indistinguishability. Remarkably, solidstate emitters are presently better able to meet these demands than atomic systems (single atoms or parametric down conversion). In particular, spontaneous emission from individual quantum dots embedded in an inorganic semiconductor is a very promising source of highly antibunched, high flux, indistinguishable photons [2][3][4]. The antibunching, particularly with resonant excitation, is very high [5]. The radiative lifetime is very short, typically just less than 1 ns [6]. The flux is usually limited by the poor collection efficiency: most of the light is internally reflected at the GaAs-vacuum interface. However, this problem can be solved by nano-structuring the photonic modes to create a micro-cavity [7] or a photonic nanowire [8]. In the latter case, collection efficiencies of ∼ 70% have been achieved. The photon indistinguishability is very high for successive photons [3]. Based on the optical linewidth, typically a factor of two above the transform limit when measured with resonant excitation [9][10][11][12], the indistinguishability is also reasonably high for photons emitted widely separated in time. Furthermore, a single quantum dot has also been developed as a spin qubit [13], facilitating an interface between stationary qubits and photons [14][15][16].
Unlike a real atom, the exact transition wavelength of a quantum dot is not locked to any particular wavelength and varies considerably from quantum dot to quantum dot. However, the host semiconductor can be designed so that considerable possibilities for tuning the emission wavelength exist. Electric field tuning [17,18] and strain tuning [19,20] allow the emission wavelength to be tuned over several nanometres. A major problem remains. The emission wavelength is not constant: it varies randomly over time, even in very controlled environments at low temperature. The culprit at low frequency is electrical noise in the semiconductor which shifts the emission wavelength via the Stark effect [12]. This noise has a 1/f -like power spectrum resulting in, first, large and uncontrolled drifts at low frequencies and second, an undefined mean value. This noise, while poorly understood, is ubiquitous in semiconductors and makes it very difficult to couple an individual quantum dot to another quantum system, another quantum dot for instance, or an ensemble of cold atoms. We present here a new scheme which solves this problem: we create a stream of single photons with a wavelength which remains constant even over several hours.
The output of our quantum device is a stream of single photons generated by resonance fluorescence (RF) from a single quantum dot. RF has considerable advantages over non-resonant excitation of photoluminescence: the linewidth is much lower [11,12] and the antibunching is much better. We lock the wavelength of the quantum device to a stable reference. We generate an error signal, a signal with large slope at its zero-crossing, by measuring the differential transmission, ∆T /T , simultaneously [9,21,22]. The control variable is the voltage V g applied to a surface gate which influences the quantum dot frequency via the Stark effect. The performance of the feedback scheme is characterized by, first, measuring a series of snap-shots of the optical resonance to assess the residual frequency jitter; and second, by carrying out a full analysis of the noise in the RF. This scheme goes well beyond previous attempts at single emitter stabilization in the solid-state [23,24]. The absolute frequency of the quantum dot emission is locked with an uncertainty of just 20 MHz. We observe a reduction in the noise power up to a frequency of ∼ 100 Hz, high enough to eliminate the substantial drifts at low frequency. Arguably, these low frequency fluctuations have a classical nature, reflecting charge noise in our solid-state device on millisecond or second time-scales. On much shorter time scales, there are clear quantum effects: the intensity correlation coefficient exhibits a clear dip between 0 and ±2 ns; electron spins have decoherence times in the µs regime [13]. As such, this experiment represents a first step towards bridging these time scales, i.e. quantum control of a solid-state emitter. A sketch of the experimental concept is shown in Fig. 1(a). A linearly-polarized resonant laser is focused onto the sample surface and drives the optical transition. The resonance fluorescence of the quantum dot is collected with a polarization-based dark field technique [11,14,25], described in detail elsewhere [26]. Simultaneously, the optical resonance is detected in transmission by superimposing a sub-linewidth modulation to the gate. The transmission signal arises from an interference of quantum dot scattering with the driving laser [22]. The incoherent part, i.e. the resonance fluorescence, averages to zero in transmission; what is detected instead is the coherent scattering, i.e. the Rayleigh scattering. In this way, the experiment utilizes both incoherent and coherent parts of the scattered light, for the single photon output and con- trol, respectively. With a small modulation, the transmission signal has a large slope with zero crossing at zero detuning and is therefore ideal for the generation of an error signal.
∆T /T , the error signal, is recorded with a lock-in amplifier to reject noise and the lock-in output is fed into a classical feedback scheme. The feedback output is, like the modulation, applied to the gate electrode of the device. The set-point of the control loop is the zero crossing with the goal of locking the peak of the quantum dot RF spectrum to the laser.
The laser itself is locked to a HeNe laser reference.
The self-assembled InGaAs quantum dots, grown by molecular beam epitaxy, are integrated into a semiconductor charge-tunable heterostructure [27]. The quantum dots are A sub-linewidth square-wave modulation at 527 Hz is applied to the Schottky gate. This broadens both X 0 transitions slightly, here the "red" transition from Γ = 1.45 to Γ = 2.58 µeV. The transmitted light is detected with an in situ photodiode connected to a room temperature current-voltage preamplifier. Lock-in detection of the ∆T /T signal is shown in Fig. 1(c). With the sub-linewidth modulation, the ∆T /T resonance is proportional to the derivative of the RF spectrum [21]. There are two points which cross with high slope through zero, one for each X 0 transition. Both crossing points enable a feedback scheme: ∆T /T provides the error signal, V g the control parameter. For instance, if the transition energy increases due to electric fluctuations in the sample, ∆T /T moves away from zero.
Once this is detected, a modified V g is applied to the gate to bring the resonance back to versa. This is probably related to the so-called "dragging" [28] which is very pronounced on this quantum dot at high magnetic fields (above 0.1 T) [26]: the nuclear spins polarize in such a way as to maintain the resonance with the laser over large detunings. In other words, it is likely that the asymmetries in Fig. 2 The ultimate operation capability of the stabilization system is limited by the random noise in the output of the PID electronics. In Fig. 1(c) the noise in the ∆T /T signal is σ ∆T /T = 1.45 × 10 −4 . In the ideal case, this determines the energy jitter of the quantum dot resonance position [29], where δ is the detuning. This limit, ∼ 100 times smaller than the linewidth, shows the power of this technique. We have not yet reached this limit in practice. Nevertheless, stabilization with a residual jitter down to just σ f = 20 MHz is achieved.
The frequency locking feedback scheme was also tested regarding its long term behaviour and bandwidth. The RF signal was recorded over several hours, Fig. 3 RF signal S(t)/ S(t) provides a noise spectrum [12]: For the noise spectrum is dominated by spin noise [12].
The quantum dot noise N QD (f ) under feedback can be linked to the jitter in the energy detuning, σ E . The energy jitter is much less than the linewidth such that the change in the RF signal (∆RF) is related quadratically to the detuning for fluctuations around δ = 0. The variance of the RF noise, σ 2 RF , is related to an integral of the noise curve, [30]. Integrating up to frequency ∆f in the regime where N QD (f ) is approximately constant, With ∆f = 3.1 Hz, N QD (0) = 1.0 × 10 −5 , Γ = 2.58 µeV this predicts σ ON E = 0.073 µeV, in excellent agreement with the measurement from the stroboscopic experiment (0.089 µeV).
An intensity correlation measurement g (2) (t) was performed with a Hanbury Brown-Twiss interferometer. Low noise g (2) (t) can only be determined at these count rates (50 kHz per APD) by integrating over several hours and the feedback is therefore important to ensure that the detuning of the quantum dot with respect to the laser remains constant. g (2) (t) is shown in Fig. 4 from X 0 of the same quantum dot with zero detuning. g (2) (t) falls to 10% at t = 0. This does not reflect g (2) (0) of the quantum dot but rather the timing jitter of the detectors which is comparable to the radiative lifetime. We attempt to describe g (2) (t) with a convolution of g (2) (t) for an ideal two-level atom, g (2) atom (t), and the response of the detectors G(t): The detector response is a Gaussian function, g (2) atom (t) of a 2-level system with resonant excitation is, with λ = (Ω 2 − (1/4τ r ) 2 ) 1/2 . The temporal jitter of the detector τ D = 0.40 ns is measured independently. Ω and τ r are known from other experiments to within 10 − 20% and are allowed to vary in these windows by a fit routine. The convolution provides an excellent description of the measured g (2) (t) with Ω = (0.99 ± 0.1) µeV and τ r = (0.78 ± 0.05) ns. In particular, with low systematic error we can set an upper bound to the quantum dot g (2) (0) of 1-2%. The red curve shows a convolution of the two-level atom result with a Gaussian distribution which describes the timing jitter of the detectors. The blue curve shows the two-level atom response alone.
In conclusion, we have developed a dynamic method of locking the optical resonance of a single quantum dot to a stabilized laser in order to produce a stream of frequency-stabilized single photons via resonance fluorescence. Generally speaking, the scheme represents a way to reduce the local charge noise in a semiconductor. Now that the basic principle is established, there are options for improving the feedback scheme. First, the remaining jitter in the quantum dot resonance position can be reduced by reducing the noise in the transmission detection. Presently, we are far from the limit defined by the shot noise in the detector current. With lower noise, the feedback bandwidth can also be increased. The tantalizing prospect is to create transform-limited linewidths routinely with high bandwidth feedback. A bandwidth of about 50 kHz is required [12].
Secondly, the modulation required here to generate the error signal could be eliminated in a number of ways. For instance, a dispersive lineshape can arise naturally in reflectivity via weak coupling to a cavity [31]; or the Faraday effect in a small magnetic field [32] could be used.
We acknowledge support from the Swiss National Science Foundation (SNF) and NCCR QSIT. A.L., D.R. and A.D.W. acknowledge gratefully support from DFG SPP1285 and BMBF QuaHLRep 01BQ1035.