Publications by Stephen Clark

Unifying neural-network quantum states and correlator product states via tensor networks

Journal of Physics A: Mathematical and Theoretical IOP Publishing 51 (2018) 135301-135301

SR Clark

The tensor network theory library

Journal of Statistical Mechanics: Theory and Experiment IOP Publishing 2017 (2018) 093102

S Al-Assam, D Jaksch

In this technical paper we introduce the tensor network theory (TNT) library - an open-source software project aimed at providing a platform for rapidly developing robust, easy to use and highly optimised code for TNT calculations. The objectives of this paper are (i) to give an overview of the structure of TNT library, and (ii) to help scientists decide whether to use the TNT library in their research. We show how to employ the TNT routines by giving examples of ground-state and dynamical calculations of one-dimensional bosonic lattice system. We also discuss different options for gaining access to the software available at

Non-linear quantum-classical scheme to simulate non-equilibrium strongly correlated fermionic many-body dynamics

Scientific Reports Nature Publishing Group 6 (2016) 32940-

JM Kreula, SR Clark, D Jaksch

We propose a non-linear, hybrid quantum-classical scheme for simulating non-equilibrium dynamics of strongly correlated fermions described by the Hubbard model in a Bethe lattice in the thermodynamic limit. Our scheme implements non-equilibrium dynamical mean field theory (DMFT) and uses a digital quantum simulator to solve a quantum impurity problem whose parameters are iterated to self-consistency via a classically computed feedback loop where quantum gate errors can be partly accounted for. We analyse the performance of the scheme in an example case.

Optical properties of a vibrationally modulated solid state Mott insulator.

Sci Rep 4 (2014) 3823-

S Kaiser, SR Clark, D Nicoletti, G Cotugno, RI Tobey, N Dean, S Lupi, H Okamoto, T Hasegawa, D Jaksch, A Cavalleri

Optical pulses at THz and mid-infrared frequencies tuned to specific vibrational resonances modulate the lattice along chosen normal mode coordinates. In this way, solids can be switched between competing electronic phases and new states are created. Here, we use vibrational modulation to make electronic interactions (Hubbard-U) in Mott-insulator time dependent. Mid-infrared optical pulses excite localized molecular vibrations in ET-F2TCNQ, a prototypical one-dimensional Mott-insulator. A broadband ultrafast probe interrogates the resulting optical spectrum between THz and visible frequencies. A red-shifted charge-transfer resonance is observed, consistent with a time-averaged reduction of the electronic correlation strength U. Secondly, a sideband manifold inside of the Mott-gap appears, resulting from a periodically modulated U. The response is compared to computations based on a quantum-modulated dynamic Hubbard model. Heuristic fitting suggests asymmetric holon-doublon coupling to the molecules and that electron double-occupancies strongly squeeze the vibrational mode.

Capturing Exponential Variance Using Polynomial Resources: Applying Tensor Networks to Nonequilibrium Stochastic Processes

Physical Review Letters American Physical Society 114 (2015) 090602

T Johnson, T Elliott, S Clark, D Jaksch

Estimating the expected value of an observable appearing in a nonequilibrium stochastic process usually involves sampling. If the observable’s variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high-variance observable exp(−βW), motivated by Jarzynski’s equality, with W the work done quenching from equilibrium at inverse temperature β, is exactly and efficiently captured by tensor networks.

Proposed parametric cooling of bilayer cuprate superconductors by terahertz excitation.

Physical review letters 114 (2015) 137001-

SJ Denny, SR Clark, Y Laplace, A Cavalleri, D Jaksch

We propose and analyze a scheme for parametrically cooling bilayer cuprates based on the selective driving of a c-axis vibrational mode. The scheme exploits the vibration as a transducer making the Josephson plasma frequencies time dependent. We show how modulation at the difference frequency between the intrabilayer and interbilayer plasmon substantially suppresses interbilayer phase fluctuations, responsible for switching c-axis transport from a superconducting to a resistive state. Our calculations indicate that this may provide a viable mechanism for stabilizing nonequilibrium superconductivity even above Tc, provided a finite pair density survives between the bilayers out of equilibrium.

Pressure-dependent relaxation in the photoexcited mott insulator ET-F2TCNQ: influence of hopping and correlations on quasiparticle recombination rates.

Physical review letters 112 (2014) 117801-

M Mitrano, G Cotugno, SR Clark, R Singla, S Kaiser, J Stähler, R Beyer, M Dressel, L Baldassarre, D Nicoletti, A Perucchi, T Hasegawa, H Okamoto, D Jaksch, A Cavalleri

We measure the ultrafast recombination of photoexcited quasiparticles (holon-doublon pairs) in the one dimensional Mott insulator ET-F(2)TCNQ as a function of external pressure, which is used to tune the electronic structure. At each pressure value, we first fit the static optical properties and extract the electronic bandwidth t and the intersite correlation energy V. We then measure the recombination times as a function of pressure, and we correlate them with the corresponding microscopic parameters. We find that the recombination times scale differently than for metals and semiconductors. A fit to our data based on the time-dependent extended Hubbard Hamiltonian suggests that the competition between local recombination and delocalization of the Mott-Hubbard exciton dictates the efficiency of the recombination.

Transport enhancement from incoherent coupling between one-dimensional quantum conductors


JJ Mendoza-Arenas, MT Mitchison, SR Clark, J Prior, D Jaksch, MB Plenio

Frozen photons in Jaynes-Cummings arrays

Journal of Physics B: Atomic, Molecular and Optical Physics 46 (2013)

N Schetakis, T Grujic, S Clark, D Jaksch, D Angelakis

We study the origin of 'frozen' photonic states in coupled Jaynes-Cummings-Hubbard arrays. For the case of half the array initially populated with photons while the other half is left empty, we show the emergence of a self-localized photon or 'frozen' states for specific values of the local atom-photon coupling. We analyse the dynamics in the quantum regime and discover important additional features that do not appear to be captured by a semi-classical treatment, which we analyse for different array sizes and filling fractions. We trace the origin of this interaction-induced photon 'freezing' to the suppression of the excitation of propagating modes in the system at large interaction strengths. We discuss in detail the possibility of experimentally probing the relevant transition by analysing the emitted photon correlations both in the idealized lossless case and more realistic scenarios when reasonable losses are included. We find a strong signature of the effect in the emitted photons statistics. © 2013 IOP Publishing Ltd.

Repulsively induced photon superbunching in driven resonator arrays

Physical Review A - Atomic, Molecular, and Optical Physics 87 (2013)

T Grujic, SR Clark, D Jaksch, DG Angelakis

We analyze the nonequilibrium behavior of driven nonlinear photonic resonator arrays under the selective excitation of specific photonic many-body modes. Targeting the unit-filled ground state, we find a counterintuitive "superbunching" in the emitted photon statistics in spite of relatively strong on-site repulsive interaction. We consider resonator arrays with Kerr nonlinearities described by the Bose-Hubbard model, but also show that an analogous effect is observable in near-future experiments coupling resonators to two-level systems as described by the Jaynes-Cummings-Hubbard Hamiltonian. For the experimentally accessible case of a pair of coupled resonators forming a photonic molecule, we provide an analytical explanation for the nature of the effect. © 2013 American Physical Society.

Extracting quantum work statistics and fluctuation theorems by single-qubit interferometry

Physical Review Letters 110 (2013)

R Dorner, SR Clark, L Heaney, R Fazio, J Goold, V Vedral

We propose an experimental scheme to verify the quantum nonequilibrium fluctuation relations using current technology. Specifically, we show that the characteristic function of the work distribution for a nonequilibrium quench of a general quantum system can be extracted by Ramsey interferometry of a single probe qubit. Our scheme paves the way for the full characterization of nonequilibrium processes in a variety of quantum systems, ranging from single particles to many-body atomic systems and spin chains. We demonstrate our idea using a time-dependent quench of the motional state of a trapped ion, where the internal pseudospin provides a convenient probe qubit. © 2013 American Physical Society.

Heat transport in the XXZ spin chain: From ballistic to diffusive regimes and dephasing enhancement

Journal of Statistical Mechanics: Theory and Experiment 2013 (2013)

JJ Mendoza-Arenas, S Al-Assam, SR Clark, D Jaksch

In this work we study the heat transport in an XXZ spin-1/2 Heisenberg chain with homogeneous magnetic field, incoherently driven out of equilibrium by reservoirs at the boundaries. We focus on the effect of bulk dephasing (energy-dissipative) processes in different parameter regimes of the system. The non-equilibrium steady state of the chain is obtained by simulating its evolution under the corresponding Lindblad master equation, using the time evolving block decimation method. In the absence of dephasing, the heat transport is ballistic for weak interactions, while being diffusive in the strongly interacting regime, as evidenced by the heat current scaling with the system size. When bulk dephasing takes place in the system, diffusive transport is induced in the weakly interacting regime, with the heat current monotonically decreasing with the dephasing rate. In contrast, in the strongly interacting regime, the heat current can be significantly enhanced by dephasing for systems of small size. © 2013 IOP Publishing Ltd and SISSA Medialab srl.

Non-equilibrium many-body effects in driven nonlinear resonator arrays

New Journal of Physics 14 (2012)

T Grujic, SR Clark, D Jaksch, DG Angelakis

We study the non-equilibrium behavior of optically driven dissipative coupled resonator arrays. Assuming each resonator is coupled with a two-level system via a Jaynes-Cummings interaction, we calculate the many-body steady state behavior of the system under coherent pumping and dissipation. We propose and analyze the many-body phases using experimentally accessible quantities such as the total excitation number, the emitted photon spectra and photon coherence functions for different parameter regimes. In parallel, we also compare and contrast the expected behavior of this system assuming the local nonlinearity in the cavities is generated by a generic Kerr effect as described by the Bose-Hubbard (BH) model rather than a Jaynes-Cummings interaction. We find that the behavior of the experimentally accessible observables produced by the two models differs for realistic regimes of interactions even when the corresponding nonlinearities are of similar strength. We analyze in detail the extra features available in the Jaynes-Cummings-Hubbard (JCH) model originating from the mixed nature of the excitations and investigate the regimes where the BH approximation would faithfully match the JCH physics. We find that the latter is true for values of the light-matter coupling and losses beyond the reach of current technology. Throughout the study we operate in the weak pumping, fully quantum mechanical regime where approaches such as mean field theory fail, and instead use a combination of quantum trajectories and the time evolving block decimation algorithm to compute the relevant steady state observables. In our study we have assumed small to medium size arrays (from 3 up to 16 sites) and values of the ratio of coupling to dissipation rate g/γ ∼ 20, which makes our results implementable with current designs in circuit QED and with near future photonic crystal set ups. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

Capturing long range correlations in two-dimensional quantum lattice systems using correlator product states

Physical Review B - Condensed Matter and Materials Physics 84 (2011)

S Al-Assam, SR Clark, CJ Foot, D Jaksch

We study the suitability of correlator product states for describing ground-state properties of two-dimensional spin models. Our ansatz for the many-body wave function takes the form of either plaquette or bond correlator product states, and the energy is optimized by varying the correlators using Monte Carlo minimization. For the Ising model, we find that plaquette correlators are best for estimating the energy, while bond correlators capture the expected long range correlations and critical behavior of the system more faithfully. For the antiferromagnetic Heisenberg model, however, plaquettes outperform bond correlators at describing both local and long range correlations because of the substantially larger number of local parameters they contain. These observations have quantitative implications for the application of correlator product states to other more complex systems, and they give important heuristic insights, in particular, the necessity of carefully tailoring the choice of correlators to the system considered, and its interactions and symmetries. © 2011 American Physical Society.

Quantum interference between charge excitation paths in a solid-state Mott insulator

Nature Physics 7 (2011) 114-118

S Wall, D Brida, SR Clark, HP Ehrke, D Jaksch, A Ardavan, S Bonora, H Uemura, Y Takahashi, T Hasegawa, H Okamoto, G Cerullo, A Cavalleri

Competition between electron localization and delocalization in Mott insulators underpins the physics of strongly correlated electron systems. Photoexcitation, which redistributes charge, can control this many-body process on the ultrafast 1,2 timescale. So far, time-resolved studies have been carried out in solids in which other degrees of freedom, such as lattice, spin or orbital excitations 3-5 , dominate. However, the underlying quantum dynamics of bareg electronic excitations has remained out of reach. Quantum many-body dynamics are observed only in the controlled environment of optical lattices 6,7 where the dynamics are slower and lattice excitations are absent. By using nearly single-cycle near-infrared pulses, we have measured coherent electronic excitations in the organic salt ET-F 2 TCNQ, a prototypical one-dimensional Mott insulator. After photoexcitation, a new resonance appears, which oscillates at 25THz. Time-dependent simulations of the Mottg Hubbard Hamiltonian reproduce the oscillations, showing that electronic delocalization occurs through quantum interference between bound and ionized holong doublon pairs. © 2011 Macmillan Publishers Limited. All rights reserved.

Categorical tensor network states

AIP Advances 1 (2011)

JD Biamonte, SR Clark, D Jaksch

We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state ψ, we present a new and general method to factor |ψ into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states. © 2011 Author(s).

Quantum interference between photo-excited states in a solid-state Mott insulator

Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference: 2010 Laser Science to Photonic Applications, CLEO/QELS 2010 (2010)

S Wall, D Brida, SR Clark, HP Ehrke, D Jaksch, A Ardavan, S Bonora, H Uemura, Y Takahashi, T Hasegawa, H Okamoto, G Cerullo, A Cavalleri

By exciting with sub-10-fs 1.6-μm pulses the quasi-one-dimensional Mott insulator ETF2TCNQ, we observe prompt collapse of the Mott gap modulated by 24-THz oscillations of the gap, which are assigned to quantum interference between holon-doublon excitations. © 2010 Optical Society of America.

Entanglement consumption of instantaneous nonlocal quantum measurements

New Journal of Physics 12 (2010)

SR Clark, AJ Connor, D Jaksch, S Popescu

Relativistic causality has dramatic consequences on the measurability of nonlocal variables and poses the fundamental question of whether it is physically meaningful to speak about the value of nonlocal variables at a particular time. Recent work has shown that by weakening the role of the measurement in preparing eigenstates of the variable, it is in fact possible to measure all nonlocal observables instantaneously by exploiting entanglement. However, for these measurement schemes to succeed with certainty, an infinite amount of entanglement must be distributed initially and all this entanglement is necessarily consumed. In this work, we sharpen the characterization of instantaneous nonlocal measurements by explicitly devising schemes in which only a finite amount of the initially distributed entanglement is ever utilized. This enables us to determine an upper bound to the average consumption for the most general cases of nonlocal measurements. This includes the tasks of state verification, where the measurement verifies if the system is in a given state, and verification measurements of a general set of eigenstates of an observable. Despite its finiteness, the growth of entanglement consumption is found to display an extremely unfavourable exponential of an exponential scaling with either the number of qubits needed to contain the Schmidt rank of the target state or the total number of qubits in the system for an operator measurement. This scaling is seen to be a consequence of the combination of the generic exponential scaling of unitary decompositions combined with the highly recursive structure of our scheme required to overcome the no-signalling constraint of relativistic causality. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.

Dynamical simulations of classical stochastic systems using matrix product states

Physical Review E 82 (2010) 036702

TH Johnson, SR Clark, D Jaksch

We adapt the time-evolving block decimation (TEBD) algorithm, originally devised to simulate the dynamics of one-dimensional quantum systems, to simulate the time evolution of nonequilibrium stochastic systems. We describe this method in detail; a system’s probability distribution is represented by a matrix product state (MPS) of finite dimension and then its time evolution is efficiently simulated by repeatedly updating and approximately refactorizing this representation. We examine the use of MPS as an approximation method, looking at parallels between the interpretations of applying it to quantum state vectors and probability distributions. In the context of stochastic systems we consider two types of factorization for use in the TEBD algorithm: non-negative matrix factorization (NMF), which ensures that the approximate probability distribution is manifestly non-negative, and the singular value decomposition (SVD). Comparing these factorizations, we find the accuracy of the SVD to be substantially greater than current NMF algorithms. We then apply TEBD to simulate the totally asymmetric simple exclusion process (TASEP) for systems of up to hundreds of lattice sites in size. Using exact analytic results for the TASEP steady state, we find that TEBD reproduces this state such that the error in calculating expectation values can be made negligible even when severely compressing the description of the system by restricting the dimension of the MPS to be very small. Out of the steady state we show for specific observables that expectation values converge as the dimension of the MPS is increased to a moderate size.

Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices

Physical Review A 82 (2010) 043617

M Bruderer, TH Johnson, SR Clark, D Jaksch, A Posazhennikova, W Belzig

We present an analysis of Bose-Fermi mixtures in optical lattices for the case where the lattice potential of the fermions is tilted and the bosons (in the superfluid phase) are described by Bogoliubov phonons. It is shown that the Bogoliubov phonons enable hopping transitions between fermionic Wannier-Stark states; these transitions are accompanied by energy dissipation into the superfluid and result in a net atomic current along the lattice. We derive a general expression for the drift velocity of the fermions and find that the dependence of the atomic current on the lattice tilt exhibits negative differential conductance and phonon resonances. Numerical simulations of the full dynamics of the system based on the time-evolving block decimation algorithm reveal that the phonon resonances should be observable under the conditions of a realistic measuring procedure.