Multi-dimensional photonic states from a quantum dot

TL;DR

This paper demonstrates a method to generate high-dimensional quantum states, specifically W-states, across multiple time bins using a charged quantum dot, scalable without loss of coherence, advancing quantum information processing.

Contribution

It introduces a scalable technique to produce high-dimensional time-bin superposed quantum states from a quantum dot using repeated Raman scattering.

Findings

Successfully created W-states superposed across multiple time bins.

Method maintains coherence and single-photon purity at larger dimensions.

Framework for deterministic creation of arbitrary time-bin encoded qudits.

Abstract

Quantum states superposed across multiple particles or degrees of freedom are of crucial importance for the development of quantum technologies. Creating these states deterministically and with high effciency is an ongoing challenge. A promising approach is the repeated excitation of multi-level quantum emitters, which have been shown to naturally generate light with quantum statistics. Here we describe how to create one class of higher dimensional quantum state, a so called W-state, which is superposed across multiple time bins. We do this by repeated Raman scattering of photons from a charged quantum dot in a pillar microcavity. We show this method can be scaled to larger dimensions with no reduction in coherence or single photon character. We explain how to extend this work to enable the deterministic creation of arbitrary time-bin encoded qudits.

Tables

Table 1. Table 1: Coherent single-photon states

# Time-bins	$g^{\left(2\right)}\left(0\right)$	Visibility
1	$0.0233\pm 0.0009$	-
2	$0.0463\pm 0.0012$	$65.6\pm 4.3\%$
3	$0.0309\pm 0.0004$	$69.6\pm 5.4\%$
4	$0.0577\pm 0.0004$	$67.7\pm 5.7\%$

Equations

$$|W⟩=\frac{|001⟩+|010⟩+|100⟩}{\sqrt{3}}.$$

Full text

I Introduction

Photonic states with multiple components are an essential resource for secure quantum relays Collins et al. (2005), measurement based quantum computers Raussendorf et al. (2003) and quantum enhanced sensors Giovannetti et al. (2004). Many early demonstrations of these technologies have made use of the polarisation degree of freedom, which is simple to produce and manipulate but is limited to two dimensions Nagali et al. (2009). The challenge of extending these quantum states to greater dimensions in a scalable way, through adding extra quantum bits or degrees of freedom, promises new functionality and greater resistance to errors. In some cases replacing the ‘quantum bit’ with a three dimensional qutrit or a d-dimensional qudit has clear advantages, for instance in quantum communication where the larger alphabet of characters allows transmission of more than one bit of classical information per photon Bechmann-Pasquinucci and Tittel (2000). Such higher dimensional states can be encoded in the photon path, orbital angular momentum Mair et al. (2001), the radial degree of freedom Karimi et al. (2014), temporal modes Polycarpou et al. (2012), or perhaps most naturally in separate time bins Brendel et al. (1999); Jayakumar et al. (2014).

One interesting class of higher dimensional photonic state is the W-state, which is a state of the form:

[TABLE]

In the case of a single photon W-state, this can take the form of a single photon superposed across multiple modes. W-states have uses ranging from fundamental investigations of quantum mechanics to imaging and random number generationChaves and Brask (2011); Heaney et al. (2011); Gottesman et al. (2012); Gräfe et al. (2014).

There are several approaches to realising higher dimensional photonic states. For example, it has been demonstrated that the widely studied parametric generation of entangled photon pairs can be extended to photon triplets Hübel et al. (2010). It is also possible to fuse together Browne and Rudolph (2005) or entangle O’Brien et al. (2003) smaller photonic states in a gate operation, but even for state-of-the-art technologies these approaches are probabilistic or have limited fidelities Hacker et al. (2016), which reduces the efficiency. An attractive option is to directly generate a complex photonic state from a single quantum emitter Schön et al. (2005, 2007), which in principle allows the deterministic generation of multi-qubit photonic states. In fact, it has been argued that creating entangled photonic states on demand would be the final enabling technology in the development of a photonic quantum computer Gimeno-Segovia et al. (2015); Rudolph (2016). Trapped spins in quantum dots have been shown to be suitable multi-level emitters for this purpose Lindner and Rudolph (2009); Schwartz et al. (2016). In addition, they have been used to demonstrate coherent spin manipulations Press et al. (2008), spin-photon entanglement De Greve et al. (2012); Gao et al. (2012) and distant entanglement between two spins Delteil et al. (2015).

In this work, we use a cavity enhanced Raman transition in a quantum dot (QD) to show enhanced spin preparation and then to sequentially generate time-bin-encoded single photon W-states. Firstly, we demonstrate that a high Q-factor micropillar cavity allows us to observe cavity stimulated Raman emission. We use this effect to perform spin state preparation Atatüre et al. (2006) on a trapped hole spin over an order of magnitude faster than in the non cavity-enhanced case. We then demonstrate our scheme for W-state generation and show its scalability by producing photons superposed across up to four time bins. Finally, we explain how the techniques demonstrated in this work could allow the deterministic generation of arbitrary single photon time bin encoded states. We anticipate that this capability will prove useful for single-mode quantum computation Humphreys et al. (2013) and for maximising the key rates of QKD protocols over long distances Zhong et al. (2015); Sasaki et al. (2014); Takesue et al. (2015).

II Cavity-enhanced spin preparation

We perform our experiments using a single-hole charged quantum dot in a micropillar cavity (FIG 1b), a system which has been shown to be a good source of indistinguishable photons Bennett et al. (2016). Applying a Voigt geometry magnetic field results in a double-$\Lambda$ system. The magnetic field and temperature are tuned so that a single vertical transition is cavity-enhanced (FIG 1c). The states of the system are aligned parallel/anti-parallel to the magnetic field. We label the hole spin states, $\ket{h}/\ket{\bar{h}}=\frac{\ket{\Uparrow}\pm\ket{\Downarrow}}{\sqrt{2}}$ and the trion states as $\ket{T}/\ket{\bar{T}}=\frac{\ket{\Uparrow\Downarrow\uparrow}\pm\ket{\Uparrow\Downarrow\uparrow}}{\sqrt{2}}$.

In order to maximise the number of operations that can be performed on a spin per unit time, it is desirable to perform spin state preparation as rapidly as possible. In the Voigt configuration the spin preparation typically takes several nanoseconds Emary et al. (2007).

In our experiments we use a micropillar cavity to decrease the spin preparation time. Driving the $\ket{\bar{h}}\rightarrow\ket{\bar{T}}$ transition we observe cavity-stimulated Raman emission Sweeney et al. (2014). This increases the speed of spin preparation because system is more likely to decay via the enhanced vertical transition than the non-enhanced diagonal transition. As a result we can perform spin preparation in $270\pm 6$ ps - over an order of magnitude faster than is typically seen for similar non-cavity enhanced systems (FIG 2). We also note that the selective enhancement of one transition of the lambda system means that it can function as a cycling transition to be used for spin readout Carter et al. (2013). The ability to perform rapid spin preparation and reliable spin readout in the Voigt geometry will help make trapped spins in quantum dots appealing candidates for stationary qubits and light matter interfaces.

The coherence time of the Raman scattered photons is determined by the coherence of the long lived ground state hole spin and the laser rather than the shorter coherence time of the exciton Fernandez et al. (2009); Sun et al. (2016). A further useful property of Raman emission is that the wavelength of the emitted photons can be tuned by tuning the wavelength of the excitation laser He et al. (2013); Sweeney et al. (2014). In our system we observe a tuning range of over 65 $\mu$eV. This would allow for the generation of indistinguishable photons from different sources - a key requirement for several quantum photonic technologies Browne and Rudolph (2005); Knill et al. (2001).

III The sequential generation of time-bin encoded W-States

Here we implement a scheme for the generation of a single photon W-state. W-states are of particular interest as they represent one of two types of maximally entangled tripartite states and maintain their entanglement in the presence of dissipation Dür et al. (2000). The concept can be generalized to include W-states with more than three qubits.

We use a series of weak resonant pulses (shown in FIG 3a) to drive the $\ket{\bar{h}}\rightarrow\ket{\bar{T}}$ transition (shown as the red arrow in FIG 1c). The power of these pulses is tuned so that they each have the same probability of driving the transition and generating a photon. When the $\ket{\bar{h}}\rightarrow\ket{\bar{T}}$ transition is driven, the system preferentially decays vertically (shown as the blue transition in FIG 1c). The emitted photon has an equal probability of being measured in each time bin (FIG 3b). The light produced by this mechanism is emitted as single quanta because the detection of a photon at the wavelength of the enhanced transition heralds the preparation of the hole spin in the $\ket{h}$ state. Once in the $\ket{h}$ state the system cannot be re-excited by the laser driving the $\ket{\bar{h}}\rightarrow\ket{\bar{T}}$ transition until a spin-flip occurs. Spin-flips typically take three orders of magnitude longer than the W-state generation process and so are not of concern here Heiss et al. (2007). In contrast, the single photon emission that is usually observed from quantum dots relies on the spontaneous decay time of a transition being greater than the applied pulse length Hours et al. (2003).

The setup for our scheme is shown in FIG 1a. A non-resonant pulse ensures that there is a non-zero population in the $\ket{\bar{h}}$ state. Then resonant pulses produced using a continuous wave laser and an electro-optic modulator (EOM) are used to drive the Raman transition.

First, we use a single resonant pulse and second order correlation function measurement to demonstrate that the system can act as a single photon source. FIG 4a shows a time resolved measurement of the output photons and FIG 4e shows the second order correlation function measurement. The low $g^{(2)}(0)\approx 0.02$ indicates that the output light is primarily composed of single photons.

We then use a two pulse sequence to create a photon superposed across two time bins (FIG 4b). Extending this to a three pulse sequence we generate a single photon three-part W-state (FIG 4c). Finally, we use a four pulse sequence to create a four part W-state and demonstrate the scalability of our scheme (FIG 4d). The output light for all of the pulse sequences have a greatly reduced peak in the second order correlation function at $\tau=0$, indicating that the output is dominated by single photon emission (FIG 4e-h).

We also observe anti-bunching over tens of nanoseconds; the peaks at $\tau=\pm 12.5$ ns are lower than the other peaks away from $\tau=0$. This indicates that the non-resonant pulse does not completely randomise the hole spin state meaning that a photon is less likely to be generated if a photon has been produced during the preceding excitation sequence. This effect could be removed by deterministic preparation of the spin in the $\ket{\bar{h}}$ state prior to the incidence of the resonant laser pulses.

Finally, we probe the coherence between neighbouring time-bins of the three part W-state by time resolving the output of an unbalanced Michelson interferometer. This measurement shows that the photons are in a coherent superposition between different time-bins and not simply emitted into one time-bin or another probabilistically. The long arm of the interferometer delays the light by one time-bin relative to the short arm (FIG 4i&j). This means that we expect to see interference between time-bin 1 and time-bin 2, and between time-bin 2 and time-bin 3 of a single 3-time-bin photon. By varying the phase difference between the two arms we observe interference between the overlapping time-bins (FIG 4k). The interference measurements were also performed for the 2 and 4 part W-states; the visibilities obtained by fitting a sinusoidal function to the measured intensities of the overlapping time bins as a function of phase are shown in Table 1. We attribute the deviation from perfect interference visibility to the decoherence of the hole spin. Using the 3-part W-state interference visibility measurement we estimate the hole $T_{2}^{*}$ time to be $\sim 3.7$ ns - within the range of previously measured hole inhomogeneous dephasing times Carter et al. (2014).

IV Proposal for the deterministic generation of single photon W-States

As quantum dots in high Q cavities can act as deterministic photon sources Nowak et al. (2014), we propose an extension to our scheme to generate generate W-States deterministically. Our proposal is as follows:

Ensure that the system in the state $\ket{\bar{h}}$ by first preparing the $\ket{h}$ state as demonstrated earlier, then using an off-resonant pulse to perform a $\pi$-rotation of the hole spinPress et al. (2008). 2. 2.

Apply a $\frac{\pi}{3}$ pulse to the diagonal transition indicated in FIG 2a. The resulting Raman emission leaves us with the state: $\sqrt{\frac{2}{3}}\ket{\bar{h}}\ket{0_{\tau=1}}+\frac{1}{\sqrt{3}}\ket{h}\ket{1_{\tau=1}}$, where $\ket{0_{\tau=1}}$ $(\ket{1_{\tau=1}})$ indicates the absence (presence) of a photon in time-bin 1. 3. 3.

A $\frac{\pi}{2}$ pulse leaves us with the state: $\frac{1}{\sqrt{3}}\ket{\bar{h}}\ket{0_{\tau=2}0_{\tau=1}}+\frac{1}{\sqrt{3}}\ket{h}\ket{1_{\tau=2}0_{\tau=1}}+\frac{1}{\sqrt{3}}\ket{h}\ket{0_{\tau=2}1_{\tau=1}}$. 4. 4.

A final $\pi$ pulse leaves us with the state $\frac{1}{\sqrt{3}}\ket{h}\ket{1_{\tau=3}0_{\tau=2}0_{\tau=1}}+\frac{1}{\sqrt{3}}\ket{h}\ket{0_{\tau=3}1_{\tau=2}0_{\tau=1}}+\frac{1}{\sqrt{3}}\ket{h}\ket{0_{\tau=3}0_{\tau=2}1_{\tau=1}}$.

Ignoring the hole state and considering the state of the photon in isolation it is clear that this gives us with a photon in the state $\frac{1}{\sqrt{3}}(\ket{001}+\ket{010}+\ket{100})$ - a single photon W-state.

Conclusions and outlook

We have generated time-bin encoded single photon W-states from the cavity enhanced Raman emission of a quantum dot. The use of an EOM in this scheme allows us to make use of flexible electronic triggering Dada et al. (2016) to determine the probability amplitudes of the photon for each time bin.

A phase modulator could be used to control the relative phase of the laser between two resonant pulses to create a qubit suitable for time-bin-encoded quantum key distribution Nisbet-Jones et al. (2013). This phase modulation could also be achieved by using a detuned pulse to rotate the hole spin about the z-axis of the Bloch sphere Delteil et al. (2015). In combination with the demonstrated ability to control the amplitude of each time bin, this would allow the creation of arbitrary single photon time-bin-encoded states, which have uses in quantum communication and computation.

Using continuously varied excitation rather than pulsed excitation would allow the generation of arbitrarily shaped photons Nisbet-Jones et al. (2011). This, in addition to phase modulation would enable to encode quantum information in the temporal mode of a photon Brecht et al. (2015). In combination with the wavelength tuning made possible by the cavity enhanced Raman emission process this will allow the generation of photons from QD sources that are indistinguishable from photons from other light sources such as lasers and trapped atoms. We expect our results to pave the way for solid state sources of on-demand photonic qubits and efficient interfaces between quantum dots and other quantum optical systems.

Methods

The experiments were performed using an InAs quantum dot in a GaAs/AlGaAs micropillar cavity with a Q-factor of $\sim 7500$. The dot-cavity system was cooled to 5K. A pulsed non-resonant 850 nm laser is used to non-resonantly excite the quantum dot and ensure that there is a finite population in the $\ket{\bar{h}}$ state before the any resonant pulses are applied. The resonant pulses are generated using a continuous wave diode laser at $\sim 940$ nm and electro-optic modulator. The resonant pulses are separated from the emitted light by both polarisation and spectral filtering. The emitted light is the detected by fibre coupled avalanche photodiodes. To record the HBT measurements we time-tag each photon, this allows us to remove the effect of photons generated from the non-resonant pulses on the correlation measurements through temporal filtering. The unbalanced Michelson interferometer used was kept at a constant temperature in order to minimise drift.

Acknowledgments

The authors acknowledge funding from the EPSRC for MBE system used in the growth of the micropillar cavity. J. L. gratefully acknowledges financial support from the EPSRC CDT in Photonic Systems Development and Toshiba Research Europe Ltd.

Data Access

The experimental data used to produce the figures in this paper is publicly available at [Information will be made available on Cambridge Data Repository before publication].