### Numerical methods for Quantum Optics 2017

This workshop will be an informal meeting of scientists working in Quantum Optics, Quantum Information and Many-body Physics, with the aim of discussing numerical methods based techniques such as matrix product states, DMRG, tensor-network states, stochastic unravellings, mean-field or cluster decompositions, etc, in particular with applications to open quantum systems.

The workshop is a continuation of a previous meeting that took place at the Atominstitut in Vienna, on January 2016. This edition happens at the premises of CSIC (Spanish Research Council) in Madrid

### Program

Thursday 26 | Friday 26 | |

9:00 | Mari Carmen Bañuls Tensor networks, dynamics and the many body localization problem | Darrick Chang Atom-light interactions as a quantum spin model |

9:35 | Simone Montangero Open many-body quantum systems dynamics via LPTN | Ines de Vega Exact dynamics of impurities in photonic crystals and waveguides |

10:10 | Coffee break & discussion | |

10:50 | Jonathan Keeling Suppressing and restoring the Dicke superradiance transition by dephasing and decay | Eduardo Sánchez-Burillo Waveguide QED with Matrix Product States |

11:25 | Andrew Daley Non-equilibrium spin dynamics in AMO systems | Eduardo Mascarenhas Matrix-Product-State Simulations of Open Quantum Systems: Boundary Driven Disordered Chains and 2D Dissipative criticality |

12:00 | Jorge Yago Dissipative preparation of spin-entangled states in fermionic ultracold gases | Diego Porras Quantum lattice laser: quantum sensing and non-equilibrium phase transition |

13:00 | Lunch | |

15:00 | Riccardo Rota Dissipative phase transitions in 2D lattices studied via the corner-space renormalization method | Discussions |

15:35 | Davide Rossini Linked cluster expansions for dissipative quantum systems | |

16:10 | Román Orús A simple tensor network algorithm for 2d steady states | |

16:45 | Coffee break & discussion | |

17:25 | Edmund Owen Dissipation-Induced Mobility in Frustrated Lattices | |

18:00 | Dieter Jaksch Tensor Network Theory for Strongly Driven Many-body Quantum Systems | |

18:35 | Poster session | |

21:00 | Dinner |

### Logistics

The workshop will take place at the Residencia de Estudiantes, located in Calle del Pinar, 21-23, Madrid. Follow these instructions to locate our premises and find connections to and from the airport.

There are a large number of hotels around the CSIC campus. We have worked with NH Zurbano, NH Balboa, and many others. If needed, we can also request a preferred accommodation rate at the Students' Residence of CSIC, a historical building right within our premises. If you need this, please contact the residence by email stating that you are going to participate in a CSIC activity and therefore need the reduced rate.

### Participants

- Alberto Biella, EPFL LPTN
- Oliver Brown, Heriot-Watt Univ.
- Alexandra Nagy, EPFL LPTN
- Mari Carmen Bañuls, MPQ (Germany)
- Darrick Chang, ICFO
- Cristiano Ciuti, MPQ (France)
- Andrew Daley, Univ. Strathclyde
- James Douglas, ICFO
- Ines De Vega, LMU
- Samuel Fernández-Lorenzo, Univ. Sussex
- Juanjo García-Ripoll, IFF-CSIC
- Giacomo Giudice, EPFL
- Dieter Jaksch, Univ. Oxford
- Michael Hartmann, Univ. Heriot-Watt.
- Julian Huber, Atominstitut
- Jonathan Keeling, Univ. St. Andrews
- Eduardo Mascarenhas, EPFL
- Marco Manzoni, ICFO
- Pietro Silvi, Univ. Ulm
- Edmund Owen, Heriot-Watt Univ.
- Diego Porras, Univ. Sussex
- Peter Rabl, Atominstitut
- Davide Rossini, SNS
- Riccardo Rota, MPQ (France)
- Vincenzo Savona, EPF Laussane
- Eduardo Sanchez-Burillo, ICMA (CSIC-Unizar)
- Emanuele Tirrito, ICFO
- Jorge Yago, Univ. Strathclyde
- David Zueco, ICMA (CSIC-Unizar)

### Contributions

#### Talks

**Andrew Daley**(University of Strathclyde)

*Non-equilibrium spin dynamics in AMO systems*

Over the past few years, the possibility to control and measure atomic and molecular systems time-dependently has generated a lot of progress in exploring out-of-equilibrium dynamics for strongly interacting many-particle systems. This connects directly to fundamental questions relating to the relaxation of such systems to equilibrium, as well as the spreading of correlations and build-up of entanglement. While ultracold atoms allow for exceptional microscopic control over quantum gases with short-range interactions, experiments with polar molecules and chains of trapped ions now also offer the possibility to investigate spin models with long-range interactions. I will discuss recent developments in this area, illustrated with our recent theoretical work in two directions: (i) the new opportunities to compare dynamics with short and long-range interactions, especially using systems of trapped ions, where it is possible to control the effective range of interactions, and (ii) spin dynamics with bosonic atoms in tilted optical lattices.

**Riccardo Rota (Univ. Paris Diderot)**

*Dissipative phase transitions in 2D lattices studied via the corner-space renormalization method*

The study of dissipative phase transitions is an emerging topic of research for non-equilibrium quantum many-body systems, which can be realized in artificial platforms using Rydberg atoms, semiconductor microstructures or superconducting circuits. Recently, unconventional magnetic phase transitions have been predicted in spin lattices described by a dissipative Heisenberg model with anisotropic spin-spin coupling and incoherent spin relaxation: in particular, the predictions have been based on single-site [1] and cluster mean-field [2] theory. A crucial problem is to explore the physical properties beyond mean-field. By applying the corner-space renormalization method [3], we have explored the critical behavior of such class of spin systems [4]. We have been able to investigate the finite-size scaling and to calculate the critical exponent of the magnetic linear susceptibility. We show that the Von Neumann entropy increases across the critical point, revealing a strongly mixed character of the ferromagnetic phase. At the same time, the quantum Fisher information, an entanglement witness, exhibits a critical behavior at the transition point, showing that quantum correlations play a crucial role. Our results suggest that dissipative phase transition can share properties of both thermal and quantum phase transitions.

[1] T. E. Lee, S. Gopalakrishnan, and M. D. Lukin, Phys. Rev. Lett. 110, 257204 (2013).

[2] J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, Phys. Rev. X 6, 031011 (2016) [3] S. Finazzi, A. Le Boité, F. Storme, A. Baksic and C. Ciuti, Phys. Rev. Lett. 115, 080604 (2015).

[4] R. Rota, F. Storme, N. Bartolo, R. Fazio and C. Ciuti, arXiv:1609.02848 [quant-ph]

**Darrick Chang**(ICFO)

*Atom-light interactions as a quantum spin model*

Atomic systems interacting with propagating optical fields constitute a promising platform to generate many-body states of light. The numerical simulation of such systems is difficult, however, due to the large number of degrees of freedom associated with the atoms and the field continuum. Here, we show that many problems of interest can be mapped onto an effective one-dimensional open interacting spin system, where the "spins" represent atomic internal degrees of freedom interacting via photon exchange. Correlations between the spins in turn can be used to re-construct any field correlations. We show that matrix product states can be used to solve for the dynamics of the spin model, and apply this technique to a specific problem involving vacuum induced transparency, in which atoms coupled to a cavity give rise to pulse propagation with photon number dependent group velocity.

**Davide Rossini**(Scuola Normale Superiore, Pisa, Italy)

*Linked cluster expansions for dissipative quantum systems*

We discuss how to employ linked-cluster algorithms in the context of dissipative quantum many-body systems on a lattice. In combination with perturbation theory applied to open quantum systems, these methods allow to obtain a property in the thermodynamic limit up to a given order, by summing up the contributions coming from finite small-size clusters. Close to dissipative phase transitions it is possible to get information on the critical behavior, by means of a Padé analysis of the obtained data. We apply this approach to a quantum spin-1/2 anisotropic Heisenberg model on a two-dimensional lattice, in the presence of incoherent spin flips.

**Diego Porras**(University of Sussex)

*Quantum lattice laser: quantum sensing and non-equilibrium phase transition*

I will present recent theoretical results on the theoretical description of a set of single-quibt lasers in coupled optical cavities. I will show that this is not only a challenging theoretical problems, but it may also have important implications for quantum sensing and metrology.

**Dieter Jaksch**(University of Oxford)

*Tensor Network Theory for Strongly Driven Many-body Quantum Systems*

Recent experiments in quantum materials and ultracold gases indicate that selective driving may generate or enhance ordered phases of matter. In this talk I will show how tensor network based numerical methods can help explain and engineer such phenomena. Specifically, I will consider a driven fermionic Hubbard model in the strongly correlated limit where the onsite interaction dominates over the kinetic energy [1]. I will show how this modulation can be handled numerically and identify changes of the nature of the system under driving into an attractive Luttinger liquid. I will furthermore discuss driving enhanced fermion pairing in one spatial dimension and present results at zero and finite temperatures.

**Edmund Owen**(Heriot-Watt University)

*Dissipation-Induced Mobility in Frustrated Lattices*

Engineered quantum systems allow us to investigate rare condensed matter phenomena which are not easily examined in crystal lattices. One such behaviour is geometric frustration, whereby a particle cannot move due to destructive interference between tunnelling processes. I will show that purely local, Markovian dissipation can break this frustration and induce mobility by introducing incoherent tunnelling terms into the master equation for the frustrated states. We simulate the effect of interactions using a variational Matrix Product Operator (MPO) method and show how this changes the mobility of the excitations.

**Eduardo Mascarenhas**(EPFL)

*Matrix-Product-State Simulations of Open Quantum Systems: Boundary Driven Disordered Chains and 2D Dissipative criticality*

I will review our latest results on two dissipative spin-1/2 systems obtained through matrix product state (MPS) simulations. The first system under study is the boundary driven disordered XXZ chain. Using time-dependent MPS simulations of the nonequilibrium steady state (NESS), we gain evidence for the vanishing of spin diffusion along the chain, hinting at the many-body localization transition. The second system that we consider is the anisotropic 2D XYZ spin lattice, which has been pointed out recently as a paradigmatic example for the study of dissipative phase transitions in 2-D. This system is in particular known to display a dissipative phase transition at a finite value of the anisotropy parameter, characterized by the divergence of a magnetic susceptibility typical of a ferromagnetic transition. Here we show how the MPS approach allows to accurately model this critical behaviour.

**Eduardo Sánchez-Burillo**(ICMA, CSIC-Unizar)

*Waveguide QED with Matrix Product States*

The impressive development of quantum technologies in the last years has allowed to manipulate few photons interacting with qubits (or, more generally, few-level systems) in the laboratory. One of the most promising platforms for studying these interactions is waveguide QED (wQED), where photons propagate through a one-dimensional system. The huge confinement of the photons in these systems permits to access to the strong coupling regime between the photons and the qubits, generate effective photon-photon or qubit-qubit interactions, etc. In our work, we have applied Matrix Product States, more common in condensed matter physics, to this kind of problems. In my talk, I will review some of our results, as scattering of few photons under the ultrastrong coupling regime, where the rotating-wave approximation breaks down, deterministic two-photon generation, or the emergence of linear scattering in nonlinear systems.

**Ines de Vega**(Ludwig Maximilian University of Munich)

*Exact dynamics of impurities in photonic crystals and waveguides*

In this talk we will analyze the light-matter interaction of a one dimensional array of impurities in an anisotropic photonic crystal or equivalently, a waveguide. We will analyze the accuracy of a weak coupling assumption between the impurities and the electromagnetic field within those materials, by comparing the solution of a weak coupling master equation with numerically exact results of matrix product states. Considering such numerically exact method, we will also analyze the collective dynamics of the impurities when located at different distances, as well as their entanglement dynamics and preservation.

**Jonathan Keeling**(University of St Andrews)

Suppressing and restoring the Dicke superradiance transition by dephasing and decay,"We show that dephasing of individual atoms destroys the superradiance transition of the Dicke model, but that adding individual decay toward the spin down state can restore this transition. To demonstrate this, we present a method to give an exact solution for the N atom problem with individual dephasing which scales polynomially with N. By comparing finite size scaling of our exact solution to a cumulant expansion, we confirm the destruction and restoration of the superradiance transition holds in the thermodynamic limit.

**Jorge Yago**(University of Strathclyde)

*Dissipative preparation of spin-entangled states in fermionic ultracold gases*

The reliable performance of quantum metrology and quantum simulation in the context of ultracold atomic gases requires the efficient generation of highly entangled states in many cases. We present a novel scheme, based on dissipative state preparation, to systematically prepare spin-symmetric states by making use of the statistics of fermionic atoms in the optical lattice coupled to a bosonic reservoir gas [1]. We exploit the correspondence between spin and spatial symmetries in fermions to target and eliminate the spin-antisymmetric sector, by combining a Raman coupling between lattice bands and the dissipative coupling with a BEC reservoir. The scheme dynamically drives the system towards an entangled state which is symmetric in spin. This procedure represents a significant improvement with respect to previous proposals based on the symmetric nature of s-wave collisions, where the state was prepared at the cost of a significant decrease in the particle number[2].

[1] A. J. Daley, P. O. Fedichev, and P. Zoller, Phys. Rev. A 69, 022306 (2004).

[2] M. Foss-Feig, A. J. Daley, J. K. Thompson, and A. M. Rey, Phys. Rev. Lett. 109, 230501 (2012).

**Mari Carmen Bañuls**(Max Planck Institut für Quantenoptik)

*Tensor networks, dynamics and the many body localization problem*

Matrix Product States (MPS) are a very powerful tool to study ground states of one dimensional quantum systems, but a full description of the most general out of equilibrium setup is often out of reach. Their extension to operators (MPO) nevertheless offer several ways of numerically exploring out-of-equilibrium problems. One example application includes approximating the steady state of a dissipative quantum system. Another one is simulating the evolution of mixed states, and identifying the operators that show the slowest evolution and thus will give rise to large time scales in the system. A well suited scenario for such studies is that of many body localization. Combining tensor network techniques and quantum information concepts, we can explore the characteristics of this kind of systems from a new perspective.

**Pietro Silvi**(Ulm university)

*Open many-body quantum systems dynamics via LPTN*

We review a recently introduced method to simulate open many-body quantum systems dynamics via locally purified tensor networks and present some improvements on the original algorithm. Finally, we present preliminary results on the Kibble-Zurek mechanism at finite temperature and in presence of dissipation.

#### Posters

**James Douglas**(ICFO)

*Simulating quantum light propagation using matrix product states*

Recent experiments with Rydberg atoms have demonstrated that non-linear effects in light propagation are observable at the level of individual photons. These and related non-linearities can be approximately treated using effective theories and numerics in the limit where on average much less than one photon propagates in the system. However, more general numerical methods that can capture the quantum behaviour of the photons at higher flux are lacking. Here, we describe an approach to this problem by using a "spin model" that reduces the light propagation problem to a system of interacting spins. The dynamics of the light can then be found by finding the time evolution of the spins using matrix product states. As an example we use this model to simulate the number dependent pulse velocity in vacuum induced transparency.

**Albert H. Werner**(QMath University Copenhagen)

*Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems*

Open quantum many-body systems play an important role in quantum optics and condensed matter physics, and capture phenomena like transport, the interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behavior, for various open quantum systems ranging from few to many bodies.

**Alexandra Nagy**(EPFL LTPN)

*A Full Configuration Interaction Monte Carlo approach to the nonequilibrium steady state of open quantum systems*

Many-body open quantum systems have attracted increasing attention in recent years. From a theoretical viewpoint, these systems call for new effective methods for the simulation of the dynamics and of the nonequilibrium steady state (NESS). In this contribution, we will discuss our recent progress in the development of a projector Monte Carlo approach to stochastically sample the time evolution of the density matrix – as dictated by the Liouville-von-Neumann equation – towards the NESS. For closed, Hamiltonian systems, various quantum Monte Carlo approaches have been the election tool to stochas- tically sample system properties, both at zero and finite temperature. Modeling the ground state properties at zero temperature in particular, is made possible by stochastically sampling the time evolution of the imaginary-time Schrdinger equation, with a class of methods generally known as projector Monte Carlo [1]. The Liouvillian dynamics towards the steady state shares with the imaginary-time schrdinger equation the fact that, in the long-time limit, the eigenstate with the smallest-real-part-eigenvalue will dominate. In the Liouvillian case, this corresponds to the NESS. It is therefore natural to attempt an extension of projector Monte Carlo techniques to the simulation of the NESS properties. However, the complex-valued density matrix follows an oscillatory dynamics which may easily result in the well known sign problem affecting most Monte Carlo algorithms. Recently, a new projector Monte Carlo approach – called Full Configuration Interaction Quantum Monte Carlo (FCIQMC) – has been developed for quantum chemistry simulations, and was found to alleviate significantly the sign problem [2]. We present a proof of principle of the possibility to apply FCIQMC to the real-time evolution of the Liouville-von-Neumann equation towards the NESS. We study in particular the properties of the NESS of simple nonlinear arrays, where the FCIQMC results can be compared with exact numerical results obtained using quantum trajectories, and assess the accuracy and extent of the method. FCIQMC holds promise as a computationally effective tool to address open quantum system independently of their dimensionality.

**Emanuele Tirrito**(ICFO)

*Criticality and Excitation Gap in Quantum Systems*

We demonstrate an efficient method that allows for simultaneous determination of the ground state, low energy excitation properties and excitation gap in quantum many body systems. To this aim we first use tensor networks (TN) language to show that the infinite density matrix renormalization group (iDMRG) in the real space is associated in a natural manner to the infinite time-evolving block decimation (iTEBD) implemented on a continuous matrix product state (MPS), and defined in imaginary time. We illustrate this association showing that the (imaginary) time MPS in iTEBD reproduces accurately the properties of the two-dimensional (2D) classical Ising model, verifying in this way that the time MPS corresponds to a well-defined physical many- body state. We apply then our scheme to the one-dimensional (1D) quantum Ising chain, where the time MPS is defined in continuous imaginary time." Julian Huber,Atominstitut,Pt-symmetry breaking in open quantum systems,"The phenomenon of PT-symmetry breaking in classical systems with balanced loss and gain is associated with a sharp transition from a purely real to complex eigenvalue spectrum of the underlying dynamical matrix. Over the past years this phenomenon has been extensively studied, for example, using coupled optical modes, where however, the system is always in a large amplitude classical state. In this work we study for the first time the effect of PT-symmetry breaking in the quantum regime where the effects of non-linearities and intrinsic quantum noise become important. We analyze the stationary states of two coupled harmonic oscillators with engineered loss and gain. By applying different numerical techniques to solve the corresponding master equation for this system we observe an unconventional transition from a high-noise symmetric state to a parity-broken lasing state with strongly reduced fluctuations. Moreover, we show that the transition point strongly depends on the quantumness of the system. Additionally, we apply numerical techniques for the simulation of extended PT-symmetric spin chains, which we use to demonstrate a crossover from a symmetric to a symmetry broken phase also for finite dimensional quantum systems.

**Marco Manzoni**(ICFO)

*Simulating quantum light propagation using matrix product states*

Recent experiments with Rydberg atoms have demonstrated that non-linear effects in light propagation are observable at the level of individual photons. These and related non-linearities can be approximately treated using effective theories and numerics in the limit where on average much less than one photon propagates in the system. However, more general numerical methods that can capture the quantum behaviour of the photons at higher flux are lacking. Here, we describe an approach to this problem by using a "spin model" that reduces the light propagation problem to a system of interacting spins. The dynamics of the light can then be found by finding the time evolution of the spins using matrix product states. As an example we use this model to simulate the number dependent pulse velocity in vacuum induced transparency.

**Oliver Brown**(Heriot-Watt University)

*Driven-dissipative system of strongly interacting photons*

In this poster we present ongoing progress in the investigation of the stationary state of a one-dimensional lattice of driven-dissipative systems linked by some hopping rate. In particular, we are seeking to balance the drive strength and dissipation rate such that there is an effective photonic chemical potential which leaves the system with one excitation per site. We then look for a build up in correlations brought about by an increase in the hopping rate between sites, in analogy to the Mott insulator to superfluid phase transition.

**Samuel Fernández-Lorenzo**(University of Sussex)

*Quantum sensing close to a dissipative phase transition: symmetry breaking and criticality as metrological resources*

We study the performance of a single qubit-laser as a quantum sensor to measure the amplitude and phase of a driving field. By using parameter estimation theory we show that certain suitable field quadratures are optimal observables in the lasing phase. The quantum Fisher information scales linearly with the number of bosons and thus the precision can be enhanced by increasing the incoherent pumping acting on the qubit. If we restrict ourselves to measurements of the boson number observable, then the optimal operating point is the critical point of the lasing phase transition. Our results point out an intimate connection between symmetry breaking, dissipative phase transitions and efficient parameter estimation.

**Vincenzo Savona**(Institute of Physics, Ecole Polytechnique Fédérale de Lausanne)

*A Full Configuration Interaction Monte Carlo approach to the nonequilibrium steady state of open quantum systems*

Many-body open quantum systems have attracted increasing attention in recent years. From a theoretical viewpoint, these systems call for new effective methods for the simulation of the dynamics and of the nonequilibrium steady state (NESS). In this contribution, we will discuss our recent progress in the development of a projector Monte Carlo approach to stochastically sample the time evolution of the density matrix -- as dictated by the Liouville-von-Neumann equation -- towards the NESS. For closed, Hamiltonian systems, various quantum Monte Carlo approaches have been the election tool to stochastically sample system properties, both at zero and finite temperature. Modeling the ground state properties at zero temperature in particular, is made possible by stochastically sampling the time evolution of the imaginary-time Schroedinger equation, with a class of methods generally known as projector Monte Carlo. The Liouvillian dynamics towards the steady state shares with the imaginary-time schroedinger equation the fact that, in the long-time limit, the eigenstate with the smallest-real-part-eigenvalue will dominate. In the Liouvillian case, this corresponds to the NESS. It is therefore natural to attempt an extension of projector Monte Carlo techniques to the simulation of the NESS properties. However, the complex-valued density matrix follows an oscillatory dynamics which may easily result in the well known sign problem affecting most Monte Carlo algorithms. Recently, a new projector Monte Carlo approach -- called Full Configuration Interaction Quantum Monte Carlo (FCIQMC) -- has been developed for quantum chemistry simulations, and was found to alleviate significantly the sign problem. We present a proof of principle of the possibility to apply FCIQMC to the real-time evolution of the Liouville-von-Neumann equation towards the NESS. We study in particular the properties of the NESS of simple nonlinear arrays, where the FCIQMC results can be compared with exact numerical results obtained using quantum trajectories, and assess the accuracy and extent of the method. FCIQMC holds promise as a computationally effective tool to address open quantum system independently of their dimensionality.