Abstract
Transport phenomena on a quantum scale appear in a variety of systems, ranging from photosynthetic complexes to engineered quantum devices. It has been predicted that the efficiency of coherent transport can be enhanced through dynamic interaction between the system and a noisy environment. We report an experimental simulation of environmentassisted coherent transport, using an engineered network of laserwritten waveguides, with relative energies and interwaveguide couplings tailored to yield the desired Hamiltonian. Controllablestrength decoherence is simulated by broadening the bandwidth of the input illumination, yielding a significant increase in transport efficiency relative to the narrowband case. We show integrated optics to be suitable for simulating specific target Hamiltonians as well as open quantum systems with controllable loss and decoherence.
Introduction
Recent research into photosynthetic antenna complexes has shown evidence of coherence in excitonic energy transport^{1,2,3,4}, despite the noisy cellular environment in which such complexes are found. Indeed, environmental decoherence has been credited with increasing the efficiency of transport through these systems, an effect known as environmentassisted quantum transport (ENAQT)^{5} or dephasingassisted transport^{6,7}. While ENAQT has been the subject of many theoretical studies—whether in the photosynthetic context^{8,9,10,11,12,13,14,15,16} or in other nanoscale transport systems^{17,18,19,20}—and despite its potential for improving transport in artificial quantum systems, it has so far never been directly observed.
ENAQT can be understood by considering a single excitation on a network of N coupled sites, governed by a tightbinding Hamiltonian^{5}
where denotes the excitation being localized at site m, ɛ_{m} the energy of that site and V_{mn} the coupling between sites m and n. Although ENAQT can occur on an ordered lattice where all the energies ɛ_{m} are equal^{21}, transport enhancement was first explained in disordered systems, which we consider here.
In most studies, ENAQT is about the efficiency of transport from a particular initial site to a particular target site, where the excitation is irreversibly trapped. In the case of a photosynthetic complex (Fig. 1a), trapping describes the transfer of excitons to a reaction center, where they drive charge separation. It can be modelled as irreversible coupling of the target site to a sink at rate κ. The efficiency is usually defined as the probability of finding the exciton in the sink after a certain time or, more commonly, in the longtime limit.
ENAQT occurs when adding decoherence increases the trapping probability above the fully coherent case. In general, decoherence results from coupling of a quantum system to inaccessible degrees of freedom. For example, in photosynthetic antenna complexes, the energies of chromophores are coupled to molecular vibrations; tracing out this environment results in decoherence in the excitonic subspace.
ENAQT can be understood in several different ways: as suppression of destructive interferences, as bringing of neighbouring sites into resonance, or as transitions between otherwisestationary eigenstates. The first of these interpretations starts from the fact that energetic disorder tends to localize the wavepacket through processes such as destructive interference or Anderson localization^{22,23,24}, thus preventing it from reaching the target. Adding decoherence diminishes these coherent processes, increasing the transport efficiency to the target. The second view is that decoherence—if it is in the site basis, as is often assumed—can be modelled as fluctuations of site energies. On that view, increasing fluctuations enhance sitetosite transfer by bringing neighbouring sites into transient resonance more frequently. The third approach is to note that the eigenstates of H are stationary in the absence of decoherence, making it difficult to reach the target if the initial and target sites differ in energy. Incoherent processes, however, permit transitions between the eigenstates of H, yielding greater mobility. Finally, in some specialized cases ENAQT can also be understood as an instance of a phonon antenna^{25} or of momentum rejuvenation^{26}.
The earliest theoretical studies of ENAQT focused on the case where the decoherence takes the form of siteindependent, Markovian, pure dephasing^{5,6}. However, any form of decoherence, including nonMarkovian ones^{27,28}, can increase transport efficiency as long as it allows population transfer between otherwisestationary eigenstates.
Here we use an integrated photonic simulator to demonstrate the first implementation of ENAQT. Our simulator was fabricated using femtosecondlaser direct writing, which allows waveguides to be drawn directly into glass using a focused, pulsed laser. This permits the creation of threedimensional waveguide arrays, as well as precision and repeatability in engineering interactions^{29,30,31,32,33,34,35,36}. We use control over the wavelength and bandwidth of the guided light to simulate effective decoherence, thereby enhancing transport efficiency.
Results
Theoretical model
We simulate ENAQT in an array of coupled singlemode optical waveguides obeying the equation^{37,38}
where the light is propagating in the z direction, is a creation operator for a photon in waveguide m at position z, and β_{m} and C_{mn} are the propagation constants of the waveguides and the couplings between them, respectively. The former are determined by the waveguides’ refractive index profiles, while the latter also depend on the separations between them. Light propagation governed by this Schrödingerlike equation directly simulates evolution under H, with C_{mn} replacing V_{mn} and the β_{m} replacing ɛ_{m}. We can thus simulate different Hamiltonians by controlling the number, position and refractive indices of the waveguides.
The intrinsic stability of laserwritten waveguide arrays renders decoherence challenging to simulate. One approach is to stochastically modulate the index of refraction along each waveguide^{39}. Although every realization with a particular longitudinal index profile will be fully coherent, decoherence can be simulated by averaging over the recorded optical outputs from many realizations in postprocessing. This approach was recently used to simulate decoherenceenhanced navigation of a maze^{40}.
By contrast, we simulate decoherence by averaging over the results from a single array illuminated with many optical wavelengths. Although each individual wavelength propagates through the waveguide array coherently, effective decoherence can be achieved using broadband illumination and a single output intensity measurement that does not resolve wavelength. In other words, the wavelength degree of freedom is traced out yielding a partially mixed state. Similar approaches are well established in optics, where, for instance, thick birefringent quartz plates followed by wavelengthinsensitive measurements have been used to decohere the polarization states of single photons^{41,42}.
As an example of our approach to simulating decoherence, we consider two uncoupled waveguides a and b that have a difference Δβ in their propagation constants at a wavelength λ_{0}. As light propagates along the waveguides, it will accumulate a phase difference zΔβ between them. Broadband illumination can then be seen to cause effective decoherence on a length scale comparable to the illumination coherence length. Any initial coherence ρ_{ab} between the waveguides will decay as
where g^{(1)}(t) is the normalized firstorder temporal correlation function of the light, which—being proportional to the Fourier transform of the spectrum—decays to zero faster for spectrally broader illumination. Because Markovian dephasing causes exponential decays of coherences, the dephasing in equation (3) is nonMarkovian unless the light has a Lorentzian spectrum.
The overall strength γ of the decoherence can be quantified as the inverse of the optical coherence length, , and is usually proportional to the fullwidth at halfmaximum bandwidth Δλ. For a uniform distribution centred at λ_{0},
In the case of coupled waveguides, wavelength dependence affects couplings in addition to the propagation constants. The resulting decoherence will therefore not only have characteristics of pure dephasing (as in equation (3)), but will also include offdiagonal terms. However, this kind of decoherence can also result in ENAQT. Indeed, the offdiagonal terms imply that our decoherence could, in principle, be used to simulate ENAQT in ordered systems^{21}, but the effect would be weaker because of the absence of the puredephasing contribution when Δβ=0.
Experimental results
For our simulation, we chose the network of four sites shown in Fig. 1b because it is one of the smallest systems in which ENAQT is possible and because it can give significant enhancements even with relatively weak decoherence. We simulated this network using waveguides arranged as in Fig. 1c, where waveguide 1 was the input, waveguide 3 the target and the sink consisted of a long linear array of tightly coupled waveguides. The coupling between waveguides in this sink was significantly higher than between the four main waveguides, so that any light entering the sink from waveguide 3 was largely transported away^{43,44,45}. The sink need only be long enough to prevent light reflected from the far end from returning into the main simulation waveguides.
We chose propagation constants and separations among the four main waveguides in order to best approximate the Hamiltonian corresponding to Fig. 1b,
where all parameters depend on the wavelength λ of the input light. Due to their wide separations, couplings between nonneighbouring sites are negligible (<5% of neighbouringsite couplings). Our simulator is designed so that, at our central simulation wavelength λ_{0},
In this case, one of the eigenstates of H has no support on site 3, , while the remaining three eigenstates all have substantial support on site 3. Because cannot couple to the sink—at least at λ_{0}—the maximum trapping efficiency at infinite time is .
Considering the wavelength dependence of H provides a different way to see the decohering effects of broadband illumination. Due to the dependence of β and C on the wavelength, equation (6) only holds at a particular wavelength λ_{0}. At other wavelengths, will have some support on waveguide 3 and thus be able to couple to the sink, increasing the efficiency above 2/3, as shown in Fig. 1d. Unlike in other examples of ENAQT^{5}, the efficiency increases monotonically with the strength of decoherence, meaning there is no optimal level of decoherence in this model.
On the basis of measurements of couplings and propagation constants in isolated pairs of waveguides (see Methods xsection), we selected the following design parameters: Δβ=C=1.0 cm^{−1}, C_{trap}=1.5 cm^{−1}, and C_{sink}=1.75 cm^{−1} (Fig. 1c). Figure 2a shows numerical modelling of light propagation given these parameters. Although these were designed for a center wavelength of 800 nm, variations in the implementation of the waveguide parameters resulted in equation 6 being satisfied at λ_{0}=792.5 nm.
Our experimental setup is shown in Fig. 2b. We measured the efficiency using narrowband light (less than 1 nm bandwidth and always horizontally polarized for consistency) from a tunable Ti:sapphire laser (SpectraPhysics Tsunami) in quasicw mode (Fig. 2b). The output was imaged using a custombuilt 14 × magnifying telescope and the optical power was measured using a largearea powermeter after isolating either the system or sink waveguides using a variable slit. Examples of the output distribution are given in Fig. 2c, showing the significant difference between illumination at λ_{0} and an offcentre wavelength. Figure 3a shows the measured efficiency (fraction of light output in the sink modes) for wavelengths ranging from 745 to 835 nm.
ENAQT—shown in Fig. 3b—is the average enhancement in efficiency over the spectral band of interest, relative to the efficiency at λ_{0},
where the average 〈⋯〉 over λ is taken over the tophat distribution on . As predicted, it is an increasing function of the bandwidth, that is, of the decoherence strength γ. The highest ENAQT observed was (7.6±1.2)% at a bandwidth of 95 nm. We thus demonstrated that coherent transport in a coupled, statically disordered system can be enhanced through decoherence.
Discussion
The theoretical prediction in Fig. 3b contains no free parameters. It is a simulation of the dynamics under the simulated Hamiltonian H, together with trapping from site 3 at the rate κ, discussed in Methods section, and pure dephasing between waveguide 4 and the other three waveguides at the rate γ. The disagreement between theory and experiment is small considering the number of possible contributing factors. These include the offdiagonal decoherence when the waveguides are coupled, the fact that the trapping is not perfectly exponential, errors in the measurements of the coupling constants, optical losses and error in satisfying equation 6. As an example, the shaded band in Fig. 3b represents the error that would arise if Δβ(λ_{0}) deviated from C(λ_{0}) by up to 10%.
In our experiment, the magnitude of ENAQT was limited by the maximum achievable decoherence, which at γ=0.02 cm^{−1} was small compared with the inverse of the propagation length. We were limited by two components of equation (4): the tunability of our laser limited Δλ, while Δβ was limited (via equation (6)) by the need to keep C small enough to stay in the tightbinding approximation. These are not fundamental limitations, and future improvements that increase γ would result in significantly larger transport enhancement; as shown in Fig. 1d, the transport efficiency of this model can get arbitrarily close to 1 for sufficiently long propagation distances and decoherence strengths. Stronger decoherence would also allow ENAQT to be observed in networks that are less sensitive to decoherence than our model.
Our results demonstrate that integrated photonics is wellsuited for simulating open quantum systems and capable of implementing a disordered target Hamiltonian with controllable loss. Using broadband excitation to introduce tunable levels of decoherence is the first technique to simulate an openquantum system in a photonic architecture; previously, low intrinsic noise in photonic devices rendered decoherence difficult to realize in integrated optics, particularly without averaging over results from many different device realizations. Future work will determine the full range of decoherence processes that can be simulated in this way (for example, by modifying the illumination spectrum) as well as ways to extend it to enable the simulation of a greater range of openquantum systems. Controllable decoherence will not only enhance photonic simulation, but will also permit photonic implementations of quantumcomputational methods that take advantage of decoherence^{46,47,48}. In addition, analogies with and extensions of our approach will aid in the experimental optimization of transport in other engineered quantum systems.
Methods
Waveguide fabrication
The waveguides were fabricated in highpurity fused silica (Corning 7980) using a laser directwrite technique whereby Ti:sapphire laser pulses are tightly focused into the sample, which is then translated in three dimensions to yield continuous regions of positive refractive index change, which act as waveguides^{30}. To obtain singlemode waveguides with the desired propagation and coupling characteristics, laser pulses of duration 150 fs, energy 400 μJ, central wavelength 800 nm and repetition rate 100 kHz were focused 400 μm below the surface using a × 40 microscope objective and translated at 75 mm min^{−1} (for waveguides other than 4).
Permanent index changes result in the focus, yielding elliptical, vertically oriented, singlemode waveguides with modes ∼17 × 19 μm in size at 800 nm. Waveguide 3 and the sink waveguides are in a plane parallel to the surface, while angles of 120° between this plane and the other system waveguides minimize nonneighbour coupling. Because couplings between elliptical waveguides are dependent on angular orientation^{29}, we determined the couplings by writing pairs of waveguides oriented at the specified angles. Couplings as a function of separation were determined by writing the pairs at different separations and measuring the output intensities after a known propagation length when only one is optically excited.
The final waveguide separations were chosen to ensure that the tightbinding approximation is maintained, that C can be matched by Δβ, that C_{sink}>C_{trap}>C and that the number of sink modes remains manageable. The systemsink population transfer can be modelled as a constant effective rate^{43} if . In our case, although x=0.86 meant that the decay was not perfectly exponential, it was nevertheless effectively irreversible, considering that sufficiently many sink modes were present to prevent reflection from the far end and back into the main waveguides.
For waveguide 4, the writing translation speed was decreased in order to increase the propagation constant. In a pair of waveguides with mismatched propagation constants, the maximum power transfer depends only on the ratio Δβ/C. We used this fact to determine the translation speed decrease necessary to set Δβ=C at 800 nm. Due to the measurement uncertainty (about 8%) and daytoday variability in the waveguide writing process, several waveguide arrays were written with slightly different writing speeds for waveguide 4 to ensure that one set would yield Δβ=C in our desired wavelength range. In the best sample, the writing speed was decreased by 12%.
Additional information
How to cite this article: Biggerstaff, D. N. et al. Enhancing coherent transport in a photonic network using controllable decoherence. Nat. Commun. 7:11282 doi: 10.1038/ncomms11282 (2016).
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Acknowledgements
D.N.B., A.A.Z., M.A.B., A.F., A.G.W. and I.K. were supported by Australian Research Council (ARC) Centres of Excellence for Engineered Quantum Systems (CE110001013) and Quantum Computation and Communication Technology (CE110001027). R.H., M.G., S.N. and A.S. thank the German Ministry of Education and Research (Center for Innovation Competence program, 03Z1HN31), the Deutsche Forschungsgemeinschaft (NO462/61, SZ 276/91, SZ 276/71) and the GermanIsraeli Foundation for Scientific Research and Development (1157127.14/2011). A.F. and I.K. acknowledge ARC Discovery Early Career Researcher Awards (DE130100240 and DE140100433).
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I.K. developed the theory. D.N.B., I.K., A.F. and A.G.W. conceived the experiment. R.H., M.G. and A.S. designed and fabricated the integrated waveguides. D.N.B., A.A.Z., M.A.B. and R.H. performed the measurements and analysed the data. All authors contributed to writing the manuscript.
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Biggerstaff, D., Heilmann, R., Zecevik, A. et al. Enhancing coherent transport in a photonic network using controllable decoherence. Nat Commun 7, 11282 (2016). https://doi.org/10.1038/ncomms11282
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DOI: https://doi.org/10.1038/ncomms11282
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