State preparation is a key tool in quantum science. To obtain quantum-enhanced precision in metrology, one needs to, for example, prepare a GHZ state or a squeezed state. To perform measurement-based quantum computation, one has to prepare a suitable resource state such as the cluster state. Quantum communication requires one to prepare entangled states, such as a Bell pair, and send one part to Alice and the other to Bob. The variational quantum eigensolver optimizes the energy given a variational class of states that can be prepared efficiently.
There are more examples, but the general underlying requirement for all states we consider is that they be efficiently preparable. For one-dimensional systems, one of the most important classes of states are matrix-product states (MPS). All MPS can be prepared with a sequential (staircase) quantum circuit (and thus a depth linear in system size) from a product state, and all sequential quantum circuits produce MPS. What about higher dimensions? Interestingly, there exist projected entangled-pair states (PEPS, the higher-dimensional generalization of MPS) that despite small bond dimension require exponential depth to prepare, so the answer is not as easy as in one dimension.
Together with Zhi-Yuan Wei and Ignacio Cirac, we introduced a natural generalization of sequentially prepared states to higher dimensions: the plaquette-PEPS . These are states that can be prepared with a quantum circuit comprising a number of mutually overlapping plaquette unitaries that scales linearly in system size. We derive some of the properties of these states, and find that they can represent a wide variety of known states. The big advantage of defining states via quantum circuits is that this automatically provides an algorithm to prepare them on a quantum processors.
In a second part, we put forward a way to produce these states as flying photons in a circuit QED platform . This will be useful to improve upon earlier experimental work of ours on preparing photonic MPS .
 Sequential generation of projected entangled-pair states; Zhi-Yuan Wei, DM, Ignacio Cirac, arXiv:2107.05873. Generation of photonic tensor network states with Circuit QED, Zhi-Yuan Wei, J. Ignacio Cirac, Daniel Malz, arXiv:2109.06781. Realizing a Deterministic Source of Multipartite-Entangled Photonic Qubits, J-C Besse, K Reuer et al, Nature Communications 11, 4877 (2020), arXiv:2005.07060.
p-PEPS are defined as plaquette unitaries (here 2x2) acting in a given order on a product state, such that later unitaries overlap with earlier ones and such that there is one unitary per lattice site.
Figure taken from Ref. .
As a photonic medium
Photonic metamaterials are materials structured on the nanometre scale and can coax light into adopting properties it wouldn't otherwise have. For example, light can be slowed done by many orders of magnitude, made massive, or completely trapped. We can use this to produce tailored interactions between embedded atoms (or atoms brought in close proximity), with many exciting avenues for quantum simulation.
Subwavelength atomic arrays levitated in free space are in some sense the most elementary metamaterial possible, in the simplest case consisting of only a simple cubic array of two-level systems with a dipole transition.
My master student Katharina Brechtelsbauer and I have shown that three-dimensional versions of such arrays can host complete band gaps of light, which would mean that they completely reflect any incident light. This implies that excited impurity atoms placed inside of these arrays no longer decay! This has never been observed in any other system, and we can show that it also enables long-range coherent interactions between the impurity atoms, and can therefore be used to realize the many proposals for quantum simulation with emitters coupled to nanophotonic structures.
 Quantum simulation with fully coherent dipole--dipole-interactions mediated by three-dimensional subwavelength atomic arrays; Katharina Brechtelsbauer, DM, Phys. Rev. A (2021), arXiv:2012.12771.
 Atomic waveguide QED with atomic dimers, David Castells-Graells, DM, Cosimo C. Rusconi, J. Ignacio Cirac, arXiv:2107.10813.
As a way to manipulate quantum states of light
Atoms (few-level systems / superconducting qubits) offer an exceptional level of control over their quantum states. While in free space they typically only interact weakly with each other, the light-matter interaction can be strongly enhanced with suitably confined light modes, such as cavities, waveguides, or subwavelength three-dimensional arrays.
With protocols how atomic states and photonic states can be exchanged or entangled, we can use the intrinsic nonlinearity of atoms to measure, control, and generate non-classical states of light.
Under this paradigm, we have been looking at photon detectors , generation of entangled photonic states , and quantum metrology .
 Generation of Photonic Matrix Product States with a Rydberg-Blockaded Atomic Array; Zhi-Yuan Wei, DM, Alejandro Gonzalez-Tudela, Ignacio Cirac, arXiv:2011.03919. Weakly invasive metrology: quantum advantage and physical implementations; M Perarnau-Llobet, DM, JI Cirac, arXiv:2006.12114. Nondestructive photon counting in waveguide QED, DM, JI Cirac, Phys. Rev. Research 2, 033091 (2020), arXiv:1906.12296.
The setups studied in Ref. . The array (gray) gives rise to omnidirectional photonic band gaps, which tailors the photonic environment for the impurity atoms (red).
A waveguide QED system. Atoms can decay into free-space (red), into the waveguide (blue) or into other tailored modes (green). There is a strong, superradiant coupling of the collective atomic mode to the waveguide, which enables higher-fidelity quantum protocols .
The first generation of quantum computers has arrived, in the form of superconducting qubits. While the current devices are still far away from fault tolerance, to get here, experimentalists around the world have already devoted intense research on fabricating devices with extremely low noise and an incredible level of control.
I'm interested in alternative ways one can use a quantum computer (beyond "digital quantum computation"), motivated by the large amount of freedom one has when interrogating these devices. Rather than understanding them as "computers", one can ask questions about the properties and dynamics of the underlying system -- arrays of qubits with tunable coupling -- and use them for analogue quantum simulation.
As a first step, Adam Smith and I did an experiment with continuous driving of a single superconducting qubit with two incommensurate frequencies. Confirming earlier predictions, this reveals a temporal topological band structure, which ultimately results in topological frequency conversion. The underlying physics is related to one of my other research interest: periodically modulated systems (below).
We are keen to extend this to multiple qubits and outlined some future steps in the manuscript .
 Topological two-dimensional Floquet lattice on a single superconducting qubit; DM & Adam Smith, arXiv:2012.01459.
Figure from Ref. , showing two distinct topological phases, alongside numerical simulations of a non-ideal setting (blue) and experimental data (orange dots).
There is an ongoing significant interest in nonreciprocal optomechanical devices for photons (and phonons). The motivation for this work comes from the shortcomings of current implementations of directional devices, such as isolators, circulators and directional amplifiers. Such devices are of crucial importance in measurement and communication systems. Conventionally, nonreciprocal devices rely on magneto-optic effects and strong magnetic fields. They are bulky, lossy, and cannot be integrated on a superconducting chip, which makes it difficult to combine them with modern quantum architecture. Moreover, stray magnetic fields due to these devices lead to additional decoherence, which exacerbates the problem.
In collaboration with Tobias Kippenberg's lab in Lausanne, we have demonstrated an magnetic-field-free optomechanical isolator for photons . The experiment is an optomechanical plaquette, which is a circuit where two cavity modes interact with two mechanical resonator modes (schematically shown on the right). There are two paths from one cavity to the other, and destructive interference can lead to isolation. Due to the dissipation in the intermediate mechanical modes, photons can still travel in the reverse direction, i.e., the device is non-reciprocal.
This work led on to a proposal for directional, quantum-limited, optomechanical amplifiers, both phase-preserving and phase-sensitive (shown on the right) , which have recently been realized in experiment [a].
[a] Realization of directional amplification in a microwave optomechanical device, Phys. Rev. Applied 11, 034027 (2019), arXiv:1811.06036.
 Nonreciprocal transport based on cavity Floquet modes in optomechanics, L Mercier de Lépinay, CF Ockeloen-Korppi, DM, MA Sillanpää, Phys. Rev. Lett. 125, 023603 (2020), arXiv:1912.10541. Quantum-limited directional amplifiers with optomechanics, Phys. Rev. Lett. 120, 023601 (2018). Nonreciprocal reconfigurable microwave optomechanical circuit, Nature Communications 8, 604 (2017).
A sketch of the optomechanical plaquette. The mechanical modes are represented by circles, the cavity modes by squares. Figure taken from Ref. .
Inspired by the results in optomechanics, which pertain to bosonic excitations, we have shown that in a similar way, directional transport of fermions can be achieved. Specifically, we proposed to use this mechanism to obtain current rectification in a double quantum dot .
The model we consider to achieve fermionic directional transport is shown on the right. The two green sites are two energy levels of a double quantum dot, the grey areas represent leads. An applied magnetic field Φ can be tuned to obtain directional transport. Figure taken from Ref. .
 Current rectification in double quantum dot through fermionic reservoir engineering; Phys. Rev. B 97, 165308 (2018).
Discrete time crystals
Time crystals are nonequilibrium phases of matter that spontaneously break time-translation symmetry. The classic example here is one of a spin chain with a pi-pulse applied every period. Two pi-pulses are equivalent to the identity, which means the systems recurs only after two periods [a].
In a generic closed system, one would expect the drive to heat up the system to a infinite temperature state, which is why it is understood that one needs to add MBL to stabilize the phase [a]. Yet, even in the absence of MBL gives a subharmonic response for time scales that are exponentially long in systems sizes, which would suggest time-crystalline order in the absence of MBL. In a numerical analysis, we have shown that a previously overlooked, subtle effect can explain why the apparent subharmonic response scaling is misleading, thus further clarifying the role of MBL in such systems .
In classical systems, it is less clear how time crystalline phases can arise, as also the "rules" are less clear. Previous work has suggested that in one-dimensional systems, short-range interactions are not sufficient [b], but long-range interactions may indeed stabilize DTC order [c]. In our work we take ask more questions about this classical setting, by studying the example of seasonal epidemic spreading on small-world graphs, and identify a setup where an arbitrarily small density of random (infinite-range) bonds is sufficient for a transition to a time-crystalline phase . In contrast to previous models, our model includes non-Markovianity in the form of "immune sites" that are non-dynamical for a set period of time. This opens an interesting parallel between research on DTCs and models that have been in use for decades in epidemiology.
[a] Else, Bauer, Nayak, "Floquet time crystals", PRL 117, 090402 (2016).
[b] Yao et al., "Classical discrete time crystals", Nat. Phys. 16, 438-447 (2020).
[c] Pizzi, Nunnenkamp, Knolle, "Bistability and time crystals in long-ranged directed percolation" arXiv:2004.13034.
 Time crystallinity and finite-size effects in clean Floquet systems; Phys. Rev. B 102, 214207 (2020), arXiv:2009.13527.
 Seasonal epidemic spreading on small-world networks: Biennial outbreaks and classical discrete time crystals; arxiv:2007.00979.
Illustration of our model for seasonal epidemic spreading .
Periodically driven optomechanical systems
Another common theme in my research are periodically modulated optomechanical systems. The quadratures of a mechanical resonator rotate in phase space with the mechanical frequency, which is why protocols to measure, amplify, or control single quadratures of the mechanical resonator, such as backaction-evading/quantum nondemolition measurements (experiments in Tobias Kippenberg's lab: [8,9]), phase-sensitive amplification, or entanglement schemes commonly employ periodically modulated drives. In the rotating-wave approximation there is often a frame in which the Hamiltonian becomes time-independent, but occasionally the off-resonant terms are important, too.
We have rigorously developed a framework to calculate noise spectra in such a situation , and used it to find a closed, exact solution to an optomechanical backaction-evading measurement outside the rotating-wave approximation . Together with Tania Monteiro's group at UCL we have checked our framework against related ones and developed it further .
Employing a backaction-evading measurement it is possible to produce a conditionally squeezed mechanical state -- essentially, acquiring a lot of information about a quadrature X reduces its uncertainty, thus squeezing it. Matteo Brunelli and Andreas Nunnenkamp and I have obtained analytical and numerical results for this case in a recent article .
In fact, our work on nonreciprocal optomechanical system also involves cavities with a modulated drive and some of the techniques developed in Refs [1,2] have been useful in analysing the effect of non-resonant terms in the optomechanical plaquette [3,4].
The theoretical framework to describe a periodically driven optomechanical system involves several copies of the system, which can conveniently be drawn as a lattice of modes. The figure has been taken from Ref. .
Topological Magnon Amplification
This project concerns the amplification of magnons in chiral topological edge modes . Such modes arise in certain (3D, but effectively 2D) magnetic insulators with Dzyaloshinskii-Moriya interaction, most notably with kagome [a] and honeycomb [b] lattices. We show that driving such systems with light can lead to edge mode instabilities and nonequilibrium steady states with large edge magnon population and further show that driving with a gradient leads to some sort of driven magnon Hall effect, two aspects that will aid their direct experimental detection. Beyond the characterisation these materials, our magnon amplification mechanism can be used to build a coherent magnon source (a magnon laser) and a travelling-wave magnon amplifier, two devices with great potential in the realm of magnon spintronics.
[a] H. Katsura, N. Nagaosa, and P. A. Lee, Physical Review Letters 104, 066403 (2010).[b] L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Physical Review X 8, 041028 (2018).
 Topological magnon amplification; DM, Johannes Knolle, Andreas Nunnenkamp, Nature Communications, 10, 3937 (2019), arXiv:1901.02282.
A kagome topological magnon insulator driven with a field gradient will exhibit a driven Hall effect, due to selective amplification of edge modes, as we show in our article .