Photonic quantum simulations of SSH-type topological insulators
with perfect state transfer
Topological insulators could profoundly impact the fields of spintronics, quantum computing and low-power electronicsKhang et al. (2018); Nayak et al. (2008). To enable investigations of these non-trivial phases of matter beyond the reach of present-day experiments, quantum simulations provide tools to exactly engineer the model system and measure the dynamics with single site resolutionGeorgescu et al. (2014); Goldman et al. (2016); King et al. (2018); Lu et al. (2014); Meier et al. (2018); Stützer et al. (2018). Nonetheless, novel methods for investigating topological materials are needed, as typical approaches that assume translational invariance are irrelevant to quasi-crystalsKraus et al. (2012); Bandres et al. (2016) and more general non-crystalline structuresAgarwala and Shenoy (2017). Here we show the quantum simulation of a non-crystalline topological insulator using multi-photon interference. The system belongs to the same chiral orthogonal symmetry class as the SSH model, and is characterised by algebraically decaying edge states. In addition, our simulations reveal that the Hamiltonian describing the system facilitates perfect quantum state transfer of any arbitrary edge state. We provide a proof-of-concept experiment based on a generalised Hong–Ou–Mandel effect, where photon-number states impinge on a variable coupler.
Contribution of NIST, an agency of the U.S. government, not subject to copyright.
Topological insulators are materials with an insulating bulk, and topologically protected states on their boundaries which are robust to impuritiesHasan and Kane (2010). Such exotic electronic properties originate in the symmetries of physical systems and are captured by topological invariants. The simplest model presenting topological behaviour is the Su–Schrieffer–Heeger (SSH) modelSu et al. (1979) that describes a one-dimensional dimerised chain of atoms. The particular flavour of symmetries (chiral, time-reversal and charge-conjugation symmetry) endow the topology of the system with a non-trivial character, which has the physical consequence of topologically protected zero-energy edge states. This seminal model which explained the formation of solitons in long polyacetylene chainsHeeger et al. (1988) has been investigated in several physical systemsAtala et al. (2013); Downing and Weick (2017) and generalised to various models with different interactions and symmetriesLi et al. (2014); Maffei et al. (2018). Despite promising results Hasan and Kane (2010), topological insulators have only been identified and studied in a fraction of a percent of the known crystal structures Bradlyn et al. (2017). Moreover, there is a strong interest in investigating topological systems that lack translational invariance Kraus et al. (2012); Bandres et al. (2016); Agarwala and Shenoy (2017). Thus, simulation techniques that treat topological insulators with rare symmetries are a welcome addition to the field. Whilst 2D and 3D topological insulators are renowned for their robust edge conduction channels Hasan and Kane (2010), there is a growing interest in the transport of quantum states in 1D topological systems. The prediction that the state of a single qubit can be perfectly transported across a 1D chain Christandl et al. (2004) has recently been demonstrated for photons Chapman et al. (2016) and superconducting charge qubits Li et al. (2018). It has been proposed to extend this process to topological systems, as they provide protection against perturbation and disorder Mei et al. (2018).
Quantum simulations exploit well-controlled platforms to deliver new insights into complex phenomena which have so far evaded experimental realisation. They have allowed researchers to observe the dynamics of topological currents and defects in topological insulators, and directly measure their topological invariantsGoldman et al. (2016); King et al. (2018); Lu et al. (2014). A recent striking example is the simulation of the topological Anderson insulator phaseMeier et al. (2018); Stützer et al. (2018). One frequent approach to quantum simulations is quantum walks (QWs). QWs facilitate building complex evolutions from simple steps, each performed by a local unitary. Quantum integrated optics is very well suited to this task because it permits the creation, manipulation and readout of photonic quantum states in a highly controlled manner, with high speeds and low losses Tanzilli et al. (2012). Indeed, topologically protected bound states have been demonstrated with photonic QWsKitagawa et al. (2010, 2012).
QWs are the quantum analogy of random walks in which an initially localised particle (a walker) evolves into a superposition of many positions in spaceAharonov et al. (1993); Farhi and Gutmann (1998); Kempe (2003). One important kind of QWs are continuous time QWs (CTQWs) where space is discrete and represented by a graph, and time is continuousFarhi and Gutmann (1998). CTQWs describe an evolution of the walker generated by a Hamiltonian . The Hamiltonian sets the amplitudes of hopping from a graph vertex to per unit time, . The amplitudes are zero for disjoint vertices. The walker, initially in , is in a superposition state at time . The probability distribution of its final position is peaked at the graph edges and has a variance which scales quadratically with time (ballistic spread). This is in contrast to a random walk where the distribution is binomial (linear spread). Most implementations of QWs that allow a single particle to take steps, require at least detectors to measure the walker’s final position Sansoni et al. (2012). Thus, they are quite resource-ineffective and suffer from error accumulations that prevent the simulations of longer times.
Here we simulate a generalised SSH topological insulator which is capable of perfect quantum state transfer, by means of multi-particle bosonic quantum interference. The model belongs to the chiral orthogonal class of topological insulators Kitaev (2009) and reveals weakly localised edge states. This new system features similarities to the SSH topological insulator brought close to its topological phase-transition. The modification w.r.t. the original SSH model lies in fully site-dependent couplings between the chain sites which deprives the system of translational invariance. Our simulation is an analogue of a CTQW that requires only a single step to calculate the outcome of an arbitrarily long QW. It avoids error-accumulation losses which occur during numerous steps in QWs and brings the number of optical components to a minimum at the expense of increasing the number of interfering particles. It is experimentally implemented by interference of photon-number states (light pulses with a definite particle number) on a beam splitter with an adjustable splitting ratio. It relies on a multi-particle Hong–Ou–Mandel effect, which we experimentally observed for states with up to four photons. The total number of interfering particles allows us to simulate a generalised SSH topological insulator with up to sites.
A pair of photon-number (Fock) states, and , interfering on a beam splitter show a multi-photon Hong–Ou–Mandel effect, Fig. 1a. Here and denote photonic creation operators which act on the beam splitter input modes. For a given total number of photons , the beam splitter input state is fully determined by the imbalance between the occupation of the input modes, which we denote as . A beam splitter interaction between these inputs, , is governed by a Hamiltonian
where is the beam splitter reflectivity (defined as the probability of reflection of a single photon) and is the phase difference between the reflected and transmitted fieldsKim et al. (2002). In the case of a balanced beam splitter (), two photons arriving at the input ports will leave through the same exit port. This is known as photon bunchingHong et al. (1987). Similar effects hold for multi-photon-number states Campos et al. (1989) with two equally populated modes, where the most probable event is for all photons to exit together from one port, with decreasing probabilities for less bunched states. This is reflected in the probabilities of detecting the states and behind the beam splitter, , where denotes a Kravchuk functionStobińska et al. (2018); Atakishiyev and Wolf (1997). Since is independent of , for convenience we set to keep the matrix representation of real-valued in the Fock-state basis. For more information see Supplementary Sections I and II.
Multiphoton Hong–Ou–Mandel interference allows one to complete a CTQW generated by in a single step. After the beam splitter, the infinitesimal evolution turns into a superposition
Equation (2) provides the physical interpretation of as the starting position of the walker in the graph, from where it moves to the right or left with jump probability amplitudes
The total number of particles confines the region of evolution to points, from to . Distribution of the walker’s final position is given by , where the beam splitter reflectivity controls the evolution time of the quantum walk which is simulated. The spectrum of is harmonic and the evolution is periodic, and thus we need only consider values , for which . The variance reveals a ballistic spread, which with this approximation reads
(see Supplementary Section III for derivation). Sample distributions are depicted in Fig. 2.
Decoherence deforms the QW statistics towards the binomial distribution Kendon (2007). This effect is also present in our simulation where the degree of coherence can be controlled by the distinguishability of particles (see Supplementary Section IV). If one of the beams comes in a superposition of two orthogonal polarisations , whereas the other beam is in a single polarisation , then ‘tunes’ the distinguishability between the modes. For they are indistinguishable, while makes and orthogonally polarised. Since a beam splitter sees orthogonal modes independently, the model can be extended to higher dimensions, for example by including spectral degrees of freedom.
We now show how Fock state interference on a beam splitter facilitates the quantum simulation of a generalised SSH model. First, consider a general chiral XY spin-chain represented by the Hamiltonian
where and are the Pauli operators acting on the th spin. When restricted to the single excitation subspace spanned by the states where is the raising operator, this model has matrix elements . Couplings of the form would correspond to the original SSH model, where governs the dimerisation. For it reveals topologically-protected edge states, e.g. near the left end of the chain of the form , where is the localisation lengthNevado et al. (2017).
Now consider the matrix elements of the photonic Hamiltonian in the Fock state basis, . We find that the photonic and XY matrix representations are identical when we set the spin-couplings to Christandl et al. (2004). In this case the system completely lacks translational invariance, but also belongs to the chiral orthogonal (BDI) class of Altland–Zirnbauer symmetry classes which are characterised by a topological invariant (see Supplementary Section V). Thus, rather remarkably, we see that two-mode Fock state interference is capable of simulating a non-crystalline topological material. Its zero-energy eigenmode is of the form (see Supplementary Section VI for derivation). This weakly localised edge state features an algebraically decaying envelope given by , Fig. 2(d). We note that the original SSH model can also present weakly localised edge states with a numerically comparable envelope, when brought close to its topological transition (see Supplementary Section V).
Moreover, we discover that the Hamiltonian in Eq. (5) facilitates perfect quantum state transfer Christandl et al. (2004) of any arbitrary edge state over a finite chain. To demonstrate this behaviour we performed an experimental simulation of the state transfer of an SSH-like strongly localised edge state. Such a quantum simulation has been visualised in Fig. 3, where the initial Fock state gradually transforms into for (which plays the role of time) changing from to . Mathematically, the beam splitter interaction performs an -fractional Quantum Kravchuk-Fourier transform (-QKT) of the input state with fractionality Stobińska et al. (2018). The -QKT is the spatial inversion operator Atakishiyev and Wolf (1997) (mirror reflection), and thus impinging any edge state onto a beam splitter with reflectivity corresponds to a perfect state transfer of that state to the other end of the chain.
Fig. 1b shows the experimental schema used for quantum simulations by means of the multi-photon Hong–Ou–Mandel effect. Two pulsed spontaneous parametric down-conversion (SPDC) sources each generate two-mode photon-number correlated states (see Supplementary Section VII). The modes are separated with a polarizing beam splitter into the spatial modes –. The idler beams and are used for heralding the creation of the signal Fock states in and in which interfere in a variable ratio fibre coupler (the beam splitter). Photon-number statistics are measured with photon-number-resolved detection implemented by transition edge sensors with efficiency exceeding Humphreys et al. (2015). The optimal temporal overlap at the beam splitter was achieved by adjusting an optical path delay .
For a small parametric gain, the probability of creating a pair of -photon states is approximately . The average photon number was , which provided sufficient multi-photon events. For example, as the repetition rate of the pump was , approximately four-photon Fock states were generated per minute in each arm of the SPDC, of which about 6 reached the detectors due to losses in the setup. There was no post-selection, rather we recorded those events which were multi-photon, as heralded by the idler beam.
We interfered a multi-photon Fock state on a coupler with splitting ratios (green), (red), (blue) and (grey), and measured photon-number statistics, Fig. 4a–d. They correspond to CTQWs generated by for , , and , respectively. Variances of the distributions are , , and , which is in agreement with Eq. (4) and proves the ballistic spread. The result obtained for (blue) shows a zero-energy eigenmode of a generalised SSH model described by , which reveals two weakly localised edge states with an algebraically decaying envelope. Errors were estimated as a square root inverse of the number of measurements.
Next, the vacuum was impinged with multi-photon Fock states (giving an input state of ) at the beam splitter with splitting ratios (green), (red), (blue) and (grey), Fig. 4e–h. This process simulates perfect state transfer of the first spin in the chain of particles for subsequent time instances , , , and , respectively.
In conclusion, we have demonstrated that the quantum interference of multi-photon Fock states on a beam splitter is capable of simulating a non-crystalline topological material. The Hamiltonian of our simulation corresponds to a Kravchuk transform Stobińska et al. (2018) that is computationally expensive to perform classically. Whereas in the quantum domain this simulation has a computational complexity of just , which gives a clear quantum advantage. The system is characterised by a lack of translational invariance, and also belongs to the same chiral orthogonal symmetry class of 1D systems as the SSH model Dumitrescu et al. (2015). Examples of such materials include superconducting nanowiresDiez et al. (2012); Dumitrescu et al. (2015), disordered graphene quasi-1D nanoribbonsHan et al. (2007), disordered cold atomsPinheiro and Larson (2015) and quasi-1D organic superconductorsDumitrescu and Tewari (2013). These may find applications in next generation electronicsMcCaughan and Berggren (2014) and spintronicsNautiyal et al. (2004) operating with almost no energy dissipation and offering speeds exceeding 100GHz. In addition, our system is capable of the perfect quantum state transfer of an arbitrary quantum state, which is a requirement for information interchange in quantum computers operating on localised qubitsBanchi et al. (2017); Mei et al. (2018). We showed this experimentally for a state initially localized on one site. From the point of view of fundamental research a beam splitter implements an exchange interaction, one of the most ubiquitous interactions in condensed matter systems. Thus, our simulations could be further used to model, for example, exciton-polariton superfluids Kavokin et al. (2017) with sub-Poissonian statistics, which is also beyond the capabilities of current technology.
Acknowledgements.Acknowledgements MS, TS and AB were supported by the Foundation for Polish Science “First Team” project No. POIR.04.04.00-00-220E/16-00 (originally: FIRST TEAM/2016-2/17). AE and IW were supported by the Engineering and Physical Sciences Research Council project No. EP/K034480/1. MS thanks Paweł Kurzynski for discussions on quantum walks. Contributions MS, TS, AB and PPR developed the theory while AE, WRC, WSK, and IAW were responsible for realization of the experiment. JJR, SWN, TG, and AL delivered and maintained the transition edge sensor detection system. AB and AE developed the software and performed numerical computations. AB prepared the plots. All the co-authors wrote up the manuscript. Competing Interests The authors declare that they have no competing financial interests. Correspondence All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data available from authors upon request.
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Supplementary information: Photonic quantum simulations
of SSH-type topological insulators with perfect state transfer
Appendix A Multiphoton quantum interference
We examine photon-number statistics behind the beam splitter with reflectivity , see Fig.1. The probability amplitude of detecting and photons provided that and were injected is as follows
Here are orthonormal Kravchuk functions, see section B for details. Thus, photon-number statistics reads as follows
Please note that while is governed by the reflectivity , it is independent of the phase .
The output state which exits the beam splitter takes the form
Appendix B Kravchuk functions
Kravchuk functions are a set of orthonormal discrete polynomials Atakishiyev and Wolf (1997)
For large they limit to Hermite–Gauss polynomials. They may be expressed by means of the Gauss hypergeometric function as
Thus, we obtain an alternative expression for the photon-number statistics
Appendix C Ballistic spread of quantum walk variance
We now show that the spread of the variance of a walker’s final position in the Hong-Ou-Mandel CTQW is ballistic. To do so, we will perform a mapping from the beam-splitter Hamiltonian to the Schwinger representation of the algebra. It allows one to associate two quantum-harmonic oscillator modes with spin operators as follows
where is the Casimir operator and the standard commutation relations hold
By using Eq. (S7) one can show that a two-mode Fock state corresponds to an Dicke state, and also that
It is therefore straightforward to show that in the Heisenberg picture evolves as
Note that is a linear combination of raising and lowering operators, , therefore for the states the average value of is zero and we obtain
Due to the same reason as above, for the states the average value of equals
In addition, since are eigenstates of , we have . The value can be evaluated in the following way. At first, we note that
For the states we have
By approximating we get
which confirms ballistic spreading. In the main text we denote by . Fig. S1 shows the variance of the probability distributions of final positions of a walker as a function of the time of the walk, for various values of and .
Appendix D Distinguishability as a form of decoherence
In the presence of decoherence, QW statistics turn into the binomial distribution Kendon (2007), characteristic of a random walk. We expect this effect also for the Hong–Ou–Mandel CTQW, where the degree of coherence is controlled by the distinguishability of interfering particles. Consider one beam in a particular polarisation , and the other beam in a superposition of the same polarisation and an orthogonal one
The parameter introduces weights between the modes, and ‘tunes’ the distinguishability. The particles are fully indistinguishable if , whereas if they are maximally distinguishable: and are orthogonally polarized. Transformation (S19) leads to the interference of with a two-mode Fock-state superposition , instead of the single-mode Fock state , as before.
In Fig. S2 we show how the doubly-peaked distribution from Fig. 2 of the main text changes if the particles become distinguishable. With increasing , full cancellation of certain events is impossible and the two peaks gradually shift to the centre of the line and merge to create the binomial distribution for . Distinguishability mimics decoherence because it leads to interference of the multi-photon states with the vacuum state thus, it implements the usual model describing particle loss. It causes quantum particles to behave as classical ones, whose statistics cannot mimic quantum coherence and therefore we observe a transition from the quantum to the classical domain.
Appendix E SSH and generalised SSH spin chain
As we saw in the main text, a general chiral XY spin chain is represented by the Hamiltonian
In the single excitation subspace this Hamiltonian has matrix elements
where , , and , and are Pauli operators acting on the th spin.
e.1 SSH model
Couplings of correspond to the SSH model. Figure S3 shows the zero-energy state close to the topological transition. It is a superposition of two edge states which are of the form (e.g. the left edge) , where is the localisation length Nevado et al. (2017). In much the same way as the topological edge states of the generalised SSH model discussed in the main text, here we see that the conventional SSH model can also present weakly localised edge states that are nonetheless topological, see Fig. S3.
e.2 Generalised SSH model
The photonic system simulates a spin chain with couplings . This model belongs to a non-trivial class in the periodic table of topological insulators, as defined by the Altland–Zirnbauer symmetry classes Kitaev (2009). A system is categorised according to the properties of its time-reversal operator , charge-conjugation operator , and chiral-symmetry operator , where is a unitary operator and is complex conjugation. If there exists a ( or ) that commutes (anti-commutes) with than the system is said to possess the respective symmetry and is classified according to the square of that operator. is real-valued and thus its time-reversal symmetry operator is simply . Due to the absence of couplings beyond nearest-neighbour the chiral symmetry operator is given by , and finally . Our system possesses all three symmetries with . Thus it belongs to the BDI (chiral orthogonal) class of topological insulators which in one-dimension is characterised by a topological invariant. The zero-energy eigenmode for a chain of length (discussed more in section F) is shown in Figure 2 of the main text.
Appendix F Edge states
Let us find the eigenstates of , Eq. 1 and see if there are any edge states among them. To this end we employ the following transformation
where , and diagonalize this Hamiltonian in the basis of Fock states
Thus, the eigenstates of are two-mode Fock states
Thus, are eigenstates of with eigenvalues .
Please note that for , the eigenstate is a zero-energy mode
Appendix G Experimental methods
In the experiment, Fig. 1(b), a light pulse from a titanium-sapphire laser at (FWHM of ; repetition rate ) pumped two collinear type-II phase-matched -long SPDC waveguides written in a periodically poled KTP (PP-KTP) crystal. They generated independent two-mode squeezed vacuum states , where and denote two output modes, named the signal and idler, is a probability of creation of a pair of photons and is the parametric gain. The average photon number in each mode equals . We achieved an average photon number of , which was sufficient to ensure the emission of multi-photon pairs. A polarizing beam splitter split the squeezed vacua into four spatial modes –. They were filtered by bandpass filters with FWHM angle-tuned to the central wavelength ( – signal modes and , – idler modes and ) in order to reduce the background Eckstein et al. (2011). The modes and were used for heralding and conditional creation of Fock states in modes and . They interfered in a variable ratio phase-matched fiber coupler, which allows one to set the ratio between and with an error of . We employed transition-edge sensors running at which allow for photon-number resolved measurements in all modes Gerrits et al. (2011).
Measurements for each setting of the splitting ratio were taken over seconds, giving samples for . The measurement errors for each mode were estimated to , where is the number of events for a given number of photons in modes and .