Purification of a single-photon nonlinearity

Abstract

Single photon nonlinearities based on a semiconductor quantum dot in an optical microcavity are a promising candidate for integrated optical quantum information processing nodes. In practice, however, the finite quantum dot lifetime and cavity-quantum dot coupling lead to reduced fidelity. Here we show that, with a nearly polarization degenerate microcavity in the weak coupling regime, polarization pre- and postselection can be used to restore high fidelity. The two orthogonally polarized transmission amplitudes interfere at the output polarizer; for special polarization angles, which depend only on the device cooperativity, this enables cancellation of light that did not interact with the quantum dot. With this, we can transform incident coherent light into a stream of strongly correlated photons with a second-order correlation value up to 40, larger than previous experimental results, even in the strong-coupling regime. This purification technique might also be useful to improve the fidelity of quantum dot based logic gates.

ARTICLE

Received 1 Apr 2016 | Accepted 15 Jul 2016 | Published 30 Aug 2016

H. Snijders1, J.A. Frey2, J. Norman3, M.P. Bakker1, E.C. Langman2, A. Gossard3, J.E. Bowers3, M.P. van Exter1,D. Bouwmeester1,2 & W. Lfer1

Single photon nonlinearities based on a semiconductor quantum dot in an optical microcavity are a promising candidate for integrated optical quantum information processing nodes. In practice, however, the nite quantum dot lifetime and cavity-quantum dot coupling lead to reduced delity. Here we show that, with a nearly polarization degenerate microcavity in the weak coupling regime, polarization pre- and postselection can be used to restore high delity. The two orthogonally polarized transmission amplitudes interfere at the output polarizer; for special polarization angles, which depend only on the device cooperativity, this enables cancellation of light that did not interact with the quantum dot. With this, we can transform incident coherent light into a stream of strongly correlated photons with a second-order correlation value up to 40, larger than previous experimental results, even in the strong-coupling regime. This purication technique might also be useful to improve the delity of quantum dot based logic gates.

DOI: 10.1038/ncomms12578 OPEN

Purication of a single-photon nonlinearity

1 Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands. 2 Department of Physics, University of California, Santa Barbara, California 93106, USA. 3 Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106, USA. Correspondence and requests for materials should be addressed to W.L. (email: mailto:[email protected]

Web End [email protected] ).

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Single-photon nonlinearities enabled by quantum two-level systems are essential for future quantum information technologies, as they are the building block of quantum

photonics logic gates1, deterministic entanglers of independent photons2 and for coupling distant nodes to form a quantum network3. Near-unity delity interaction of photons with a two-level system such as an atom or quantum dot (QD) is enabled by embedding it into an optical cavity4. Next, the electronic and photonic states become bound and form the dressed states5 of cavity quantum electrodynamics (CQED). A hallmark of single-photon nonlinearities is the modication of the photon statistics of a quasi-resonant weak coherent input beam6: the transmitted photon statistics can become antibunched due to the photon blockade effect1,7,8. The anharmonicity of the JaynesCummings (JC) ladder911 can also be used to reach the regime of photon tunnelling6,12 where the single-photon component is reduced, leading to enhanced photon correlations, or the appearance of N41 multiphoton bundles13,14.

In terms of the second-order photon correlation function g2(0), values up to B2 (refs 1518) have been obtained experimentally with QDs, which hardly exceeds even the classical case of thermal light following Bose statistics of g2(0) 2. In atomic

systems with much longer coherence times, values up to B50 have been obtained6 and it is known19 that strict two-photon light sources exhibit diverging g2(0) if the two-photon ux is reduced. Most related QD experiments to date have been operating in the strong-coupling regime of CQED, which is considered to be essential due to its photon-number-dependent energy structure6,17,18. In the weak-coupling regime, the energy

structure is not resolved and it is an open question whether photon-number-dependent JC effects can still be observed20. The strong coupling regime, however, requires a small optical mode volume, which in turn makes it extremely hard to achieve polarization degeneracy of the fundamental cavity mode. This is due to unavoidable deviations from the ideal shape and intrinsic birefringence21,22 on the GaAs platform, precluding implementation of deterministic polarization-based quantum gates2,23,24.

Here we show, using a nearly polarization-degenerate cavity in the weak coupling CQED regime, that we can transform incident coherent light into a stream of strongly correlated photons with g2(0) 25.70.9, corresponding to \40 in the absence of detector

jitter. The polarization-degenerate cavity enables us to choose the incident polarization yin 45 such that both ne-structure split

QD transitions along yXQD 0 and yYQD 90 are excited, and we

can use a postselection polarizer behind the cavity (yout) to induce quantum interference of the two transmitted orthogonal polarization components (Fig. 1a). This leads to the appearance of two special postselection polarizer angles y out (depending on sample parameters), which can be used to restore perfect QD contrast (red curves in Fig. 1b). This compensates fully for reduced QD cavity coupling due to nite QD lifetime and QD cavity coupling strength, leading to complete suppression of transmission of the single-photon component in the low excitation limit. The transmission of higher-photon number states remains largely intact, allowing us to observe in Fig. 1c the strongest photon correlations to date in a solid-state system, reaching the range of strongly coupled atomic systems6. In the following, a detailed

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Figure 1 | The purication technique. (a) Cartoon of the experiment: polarization pre- and postselection in a resonant transmission CQED experiment enables tuning of the photon statistics from antibunched to bunched. (b) Theoretical resonant transmission spectra for coherent light with mean photon number oo1, with and without the QD, comparing the conventional case (parallel polarizers) with the case of special polarization postselection along y out: close to one of the QD resonances, single-photon transmission is perfectly suppressed, despite the nite lifetime and cavity coupling of the QD transition.

(c) Second-order correlation function for the special polarization angle case, comparing theory and experiment using two different sets of single photon counters (SPCs) with different timing jitter, 50 ps and 500 ps.

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experimental and theoretical investigation of this effect, which can be seen as a purication of a single-photon nonlinearity, will be presented.

ResultsDevice structure. Our device consists of self-assembled InAs/GaAs QDs embedded in a micropillar FabryPerot cavity grown by molecular beam epitaxy25 (see Supplementary Fig. 1 and Supplementary Note 1). The QD layer is embedded in a PIN junction, separated by a 35 nm-thick tunnel barrier from the electron reservoir, to enable tuning of the QD resonance frequency by the quantum-conned Stark effect. For transverse mode connement and to achieve polarization degenerate cavity modes, we rst ion-etch micropillars of large diameter (35 mm) and slightly elliptical shape, then we use wet-chemical oxidation of an AlAs layer26 to prepare an intra-cavity lens for transverse-mode connement27, avoiding loss by surface scattering at the side walls. Finally, we ne-tune the cavity modes by laser induced surface defects28,29 to obtain a polarization mode splitting much smaller than the cavity linewidth.

Device parameters and theoretical model. The system we study here is tuned to contain a single neutral QD within the cavity linewidth. The excitonic ne-structure splitting leads to 4.8 GHz splitting between the orthogonally polarized QD transitions at 0

oYQD and 90 oXQD. The fundamental cavity modes show a

residual polarization splitting of 4 GHz (f Xc 0 GHz, f Yc 4

GHz) and the cavity axes are rotated by 5 with respect to the QD axes. To determine further system parameters, we model our QD cavity system by a two-polarization JC Hamiltonian coupled to the incident coherent eld and take care of cavity and QD dissipation by the quantum master equation formalism30,31. We compare experiment and theory for six different inputoutput polarizer settings to faithfully determine the model parameters, these measurements were performed for an input power of 100 pW to avoid saturation effects32. We obtain (see Supplementary Fig. 2 and Supplementary Note 2) a cavity decay rate k 1053 ns 1, QD relaxation rate g|| 1.00.4 ns 1, QD

pure dephasing g* 0.60.0 ns 1 and QD cavity coupling

rate g 140.1 ns 1; from which we can calculate the

device cooperativity C g

in Fig. 2, we nd excellent agreement between experiment and theory.

Now we perform photon correlation measurements; instead of tuning the laser, we now tune the QD, the reference are the cavity modes. As the cavity linewidth is large compared with the QD tuning range in Fig. 3, there is nearly no difference compared with tuning the laser. Experimentally, using an external electric eld to tune the QD via the quantum conned Stark effect is much more robust than laser frequency tuning. Figure 3 shows the false-colour map of g2(0) as function of output polarization yout and QD detuning. We see clearly that the enhanced bunching occurs under the special polarization condition in the low-transmittivity regions indicated in Fig. 2. This is expected as in weak coherent light beams, the P1 single-photon component is dominating and removal thereof should lead to enhanced bunching. The theoretical simulation (Fig. 3b) shows a maximal photon bunching of g2(0)E40. Compared with this, the experimentally observed photon correlations are less (g2(0)E6), which is due to the detector response: Fig. 3a was recorded with a 500 ps timing-jitter detector, if we repeat the measurement at the special polarization angle with a 50 ps timing-jitter detector (the corresponding g2(t) measurements are compared in Fig. 1c), we obtain g2(0) 25.70.9. Both results agree very well to

the convolution of the theoretically expected g2(t) with the detector responses (Fig. 1c; see also Supplementary Fig. 3 and Supplementary Note 3).

DiscussionWe have shown by experiment and theory that the reduced delity of a QD nonlinearity, caused by imperfect QD-cavity coupling, can be strongly enhanced by pre- and post-selection of specic polarization states. This enables transformation of a weak coherent input beam into highly bunched light with g2(0)\40, a value that has not been reached before, not even in the strong coupling regime. How is it possible to reach such high photon correlations, how does the polarization-based purication technique work?

We consider incident light with a frequency in the vicinity of one of the QD resonances, say oXQD, and let us decompose the

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Figure 2 | Coherent probing of the QD cavity system. Experimental(a) and theoretical (b) false colour plot of the columnwise normalized optical transmission as a function of the laser detuning DfLaser and the polarization

yout (yin 45). The ne-split QD transition frequencies are at

fXQD 2:4 GHz and fYQD 2:4 GHz. The red circles indicate the special

polarization conditions; the white square indicates the area explored in Fig. 3.

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Figure 3 | Photon bunching and purication. Experimental (a) and theoretical (b) data of the second-order correlation function as a function of the QD frequency and output polarization (yout), taken in the area marked with a white rectangle in Fig. 2. The vertical dashed lines indicate the special polarization angle and the QD axis, and the horizontal line indicate the QD resonance frequency.

Figure 4 | Special polarization and cooperativity. Black: numerically determined special polarization angle, where photon bunching in transmission is maximized, as a function of the cooperativity C, which in turn is modied by varying only the QD lifetime g||. The green curve is given by the phenomenological expression 45exp ( C): in the limit of high C,

the QD alone can efciently lter out single-photon states, leading to photon bunching. However, for low cooperativity C, it is advantageous to mix the quantum-dot scattered light with a local oscillator provided by orthogonal polarization. The error bars (all s.d.) are due to numerical errors in optimizating the laser frequency. The red data point corresponds to the sample presented here; the blue data corresponds to another device with lower cooperativity.

electromagnetic eld transmitted through the cavity in two orthogonally polarized components: the signal Field ES polarized along the QD resonance polarization yXQD 0 and the local

oscillator ELO, which has interacted with an empty cavity, because it is polarized orthogonally to yXQD. Now, we consider three cases: (i) efcient interaction of the QD with incident light (cooperativity C41), (ii) intermediate interaction (CE1) and (iii)

weak interaction (C-0). The special polarization angles for various cooperativities are shown in Fig. 4.

In case (i), the QD leads to a nearly complete removal of the single-photon component from the incident coherent light polarized along the QD polarization: these photons are in principle perfectly reected from the cavity and we simply have to detect along the same axis (y out yXQD 0 , see Fig. 4) to

observe strong photon correlations. A signicant proportion of higher photon number states are transmitted. As the second-order correlation function can be expressed in terms of the photon number distribution as g20 / 2P2=P21

(ignoring N42 photon number states), which for P2ooP1 and PN42ooP2, this leads to diverging photon correlations such as g20 / 1=a2 if the single-photon component is

attenuated as P1-aP1.

Now in case (ii), for realistic systems, the nite lifetime of the QD transition and/or limited QD-cavity coupling g leads to a reduced cooperativity: even in the low-excitation limit, not every single-photon state is ltered out. Therefore, the signal eld ES contains a fraction of coherent light reducing the photon bunching along the QD polarization yXQD, compare Fig. 3. This effect has been called self-homodyning in literature33,34. With the purication technique, we now rotate the postselection polarizer to interfere a portion of the local oscillator eld ELO

with the signal eld, leading to the superimposed eld ESL eij

S

ES eij

LO

ELO35. The polarizer angle controls the relative intensity of the two components and we can control the transmission phases fs and fLO by adjusting the laser frequency, because the phases vary strongly in the vicinity of the QD and cavity resonances. We simply have to choose the local oscillator intensity that it matches the intensity of the portion of ES and adjust the phases for destructive interference. The result is that we detect in transmission mainly the single-photon ltered

portion of ES, which leads to very high photon correlations in the transmitted light despite limited cooperativity.

Finally, in case (iii) for C-0, only a vanishing fraction of the photons have interacted with the QD. We have to tune the postselection polarizer to 45 to destructively interfere nearly

equal amounts of ES and ELO to observe enhanced photon correlations. This case is similar to that recently investigated in ref. 36, where (weak) photon bunching is observed for a relative phase of p (fs fLO p). We have a high-nesse (FE800)

cavity and signicant cooperativity, which enables us to observe much stronger photon correlations (Supplementary Fig. 4 and Supplementary Note 4).

The special postselection angle y out and laser frequency have to be optimized numerically in principle, because pure dephasing cannot be taken care of in a semiclassical model. Despite this, we found that the special polarization angle shows approximately a very simple dependency on the cooperativity: y out 45 exp C, see Fig. 4, which agrees well to our

intuitive explanation here.

As a last point, we analyse the strong photon bunching in terms of the photon number distribution Pn. We use our theoretical model to determine Pn, as direct experimental determination thereof is strongly complicated by its sensitivity to loss. However, also the simulation of narrow-band photon number Fock input states is challenging in the quantum master model37. Therefore, we use coherent input light and analyse the intra-cavity light in terms of its polarized photon number distribution, taking care of quantum interference at the postselection polarizer acting on the intra-cavity eld. This is an approximation, because imperfect transmission through the cavity reshapes Pn. We found that the photon statistics Pn can be calculated best by projection on the required Fock states using polarization-rotated Fock space ladder operators bwx=y awx=ycosyout awy=xsinyout and tracing out

the undesired polarization component afterwards. With the numerically31 calculated steady-state density matrix operator r of

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100

102

104

106

108

1010

Special output polarization

90 60 30 0 30

P0

P n

120

out ()

Figure 5 | Photon number distributions. Calculated photon number distribution after the polarizer, with (through curves) and without (dashed curves) coupling to the QD in the cavity, the laser frequency is set to one of the QD resonances. With QD, we clearly see the photon-number-dependent shift of the transmission dip. Only the photon number distribution of the detected polarization component is shown; therefore, the total number of photons in case with QD can exceed the case without QD due to polarization conversion by the dot. For clarity, pure dephasing has been neglected here, making the special polarization angle different from the other simulations and experimental results.

our system (Supplementary Note 2), we obtain the photon number distribution after the polarizer:

Pn X

N

m0

x mj0x0yi 1

Figure 5 shows the four lowest photon number probabilities as a function of the polarizer angle yout, for the case with and

without QD. In the empty-cavity case we see, as expected, lowest transmission under the cross-polarization condition (yout 45). For the case with the QD, we observe a photon-

number-dependent shift of the transmission dip. At the special polarization angle y out, we see that the one-photon component reaches a minimum, while the higher-photon number states do not, which explains the enhanced photon bunching enabled by the purication technique.

It is important to note that also the two-photon transmission dip (P2) is not exactly at cross-polarization, which suggests the following intuitive explanation: apparently, in the photon number basis, the different Fock states pick up a different phase during transmission through the QD cavity system. In the weak-coupling regime, but often also in the strong coupling regime, the individual JC dressed states cannot be resolved spectrally, because gtk. However, the CQED system is still photon-number sensitive, which implies lifetime-dependent JC effects in the weak coupling regime: the decay rate of the CQED system increases with the number of photons in the cavity20,38. As consequence, higher photon-number states have a modied interaction cross-section and experience a reduced phase shift. The dip in P2 in Fig. 5 is already very close to the cross-polarization angle yout 45, whereas the dips for

higher photon number states Pn42 are indistinguishable from yout

45.

In conclusion, we found that the nonlinear response of a lossy cavity QD system can be strongly enhanced by postselection of a particular polarization state. This leads to interference between Fock states that experienced different modications by the QD nonlinearity and results in strong photon correlations of the transmitted light. As the underlying effect, interference of the two polarizations modes leads to high-delity cancellation of the

single-photon transmission for the special polarization postselection condition. By correlating the results with a theoretical model, we found indications of photon-number sensitive JC physics in the weak coupling regime of CQED.

Data availability. All relevant data are available on request.

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Acknowledgements

We thank G. Nienhuis for fruitful discussions. We acknowledge funding from FOM-NWO (08QIP6-2), from NWO/OCW as part of the Frontiers of Nanoscience program, and from the National Science Foundation (NSF) (0901886, 0960331).

Author contributions

J.A.F., J.N., E.C.L., A.G., J.E.B., D.B. and W.L. designed and fabricated the devices. M.P.B., H.S., M.P.v.E., D.B. and W.L. conceived and conducted the optical experiments. All authors contributed to the manuscript.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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How to cite this article: Snijders, H. et al. Purication of a single-photon nonlinearity. Nat. Commun. 7:12578 doi: 10.1038/ncomms12578 (2016).

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