# Day 2 – VI Paraty Quantum Information Workshop

**Posted:**August 22, 2017

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The morning session started with a talk by Juliette Monsel (Inst. Nèel, Grenoble), on the thermodynamical study of a hybrid optomechanical system. More precisely, she studies the same experimental system described in Pierre-Louis’ talk: a quantum dot at the basis of an oscillating nanowire. They investigate a quarter of an oscillation. While the wire oscillates, the quantum dot does random jumps between ground and excited states. So the trajectory will be a patchwork of the different trajectories associated with different frequencies. In this way, jump history can give information about trajectory. This allows for the estimate of the entropy production.

In simulations, they verified Jarzynski’s equality, the second law of thermodynamics and Crooks relation, even after including in the simulations the effect of experimental errors. While the results were primarily from simulations, I gather this is an investigation that supports future experimental explorations of thermodynamical quantities in these physical systems.

Cecilia Cormick (Córdoba) is a Paraty veteran, having attended 4 instalments of the Paraty event. She did a postdoc at Ulm, and continues the collaboration with Martin Plenio’s group, studying theoretical proposals to simplify quantum simulations in ion traps. Her talk focused on the simulation of the spin-boson model, which is used to model chemical reactions, motion of defects in solids, among other phenomena. The model features a “spin” (actually a two-level subsystem of a possibly larger system), coupled to a bunch of harmonic oscillators. Position-dependent forces push the system. This tends to localize system in left/right position (or up/down states of the “spin”). The two competing dynamics are tunnelling, and “freezing” in one of the configurations. This model has no closed analytic solutions.

The characterization of the model crucially involves the spectral density, which determines how the system is affected by the bath. Cecilia simplified an earlier proposal by Porras (PRA 2008). In that proposal, the spin is modelled by the ion’s internal state, and vibrational modes simulate the boson bath. Spin-dependent forces couple spin and modes. Lasers couple to the ion in an internal state-dependent way, realizing the model’s Hamiltonian. Changing the laser changes the spectral density. Each model’s oscillator is modelled by one physical vibrational mode, so the scaling is not very favourable, experimentally.

The basic idea of Cecilia’s proposal involves damping oscillators to widen the peaks of their spectral density, so composing fewer of these widened Lorentzians can then approximate the desired spectral density. To actually make this work, Cecilia uses influence functionals to show that there are parameter regimes in which ion cooling is actually described by a Lorentzian espectral density. The simplest system for experimental implementation involves two ions and one mode, and Cecilia showed detailed simulations which suggest this would recover the interesting dynamics featured by the model.

Fernando Nicácio (UFRJ) delivered a talk with results on how to identify various characteristics of a quantum map directly from the dynamics. He reviewed Lyapunov’s early result on necessary and sufficient conditions for asymptotic stable dynamics, and reviewed how the covariance matrix can be used to characterize the state. Then Fernando reviewed in a single slide a number of different results using the so-called bona-fide relation (new to me!), which introduces another matrix from which various theorems give necessary and/or sufficient conditions for properties of the stable solutions: separability, steering, classicality, etc. I wish Fernando had spent a bit more time on this, perhaps we can check his paper and the references therein for more information. He then presented new results. One of them was a set of necessary and sufficient conditions for classicality of stable solutions, same for separability of such solutions. Fernando also worked out a couple of examples: two oscillators attached to different heat baths; and cascade OPO with baths.

After a lively coffe break, we returned to hear a talk by Daniel Felinto (UFPE). Daniel described the research efforts of 3 different labs housed in the physics department at UFPE.

In the quantum networks lab, headed by Daniel, they have been using an ensemble of cold atoms to store excitations caused by a write laser pulse, and return the photons a little while later, prompted by a read pulse. This type of system is one of the most promising ways to enable quantum repeaters for long-distance quantum communication. Besides storing a single excitation, Daniel and collaborators have managed to store two excitations, which is a new regime and could be useful as a starting point for a source of indistinguishable photons.

In the cold atoms lab, headed by prof. Tabosa, they explore chi^5 and chi^7 processes that store excitations provided by light states with orbital angular momentum. Interestingly, depending on the nonlinearity of the medium, the topological charge is changed during storage, so this is a way to both store and process the information stored.

In the ultrafast quantum optics lab (together with prof. Acioli), the basic system is a PDC source with a femtosecond pump. They used the system to do some fundamental studies on the collapse of the wavefunction (collaboration with Fernando Parisio). The focus on the longer term, however, involves a collaboration with Marco Bellini, and consists in exploring the strong interaction of PDC photons with a narrowband atomic medium (a hot Rb cell). This cell strongly deforms the single-photon pulse.

Next we had a talk by Pablo Saldanha (UFMG), who reported a theoretical treatment of the single- and two-photon superradiance experiments reported in the previous talk by Daniel Felinto. For superradiance to happen it’s necessary to have a positive interference between the many atoms that can be excited in the sample, even if only a single photon is absorbed by the sample. Pablo described the read/write process in detail, and showed us one of the main theoretical predictions of his model, which is the explicit expression for the probability of photon emission as a function of time. This is a sinusoidal oscillation in time, damped by the decay, whose rate evidences the possibility of superradiance. The models fit well with the experimental data from Felinto’s group. They manage to fit Rabi oscillations, and also the way the photo-detection probabilities vary as the pump power changes.

The last talk of the morning was given by Gabriel Landi (USP), who studies phase-space thermodynamical measures of irreversibility. The first important point Gabriel recalled is that the time derivative of the entropy equals the entropy production rate minus the entropy flux, i.e. the rate in which entropy is flowing to the environment.

To set up the stage, he described the example of a damped harmonic oscillator, which starts and is always in a coherent state. The standard formulation would predict infinite entropy production and entropy flux. He then proposed the Wigner entropy as a good quantifier for entropy production. That’s defined as the Shannon entropy of the Wigner function. He went on to obtain the formula for the entropy production in terms of the Wigner function W, later generalizing the result for different types of bath (dephasing and squeezing).

In the second half of the talk, he discussed the role of coherence in thermodynamics, using the spin Husimi function. He obtains formulas for the entropy production for a dephasing bath and the amplitude damping bath, and in both the dephasing current plays an important role.

After the (3 1/2h long) lunch break extravaganza, we reconvened at Casa de Cultura for the afternoon session. The three talks of the afternoon dealt with related topics.

The first talk of the afternoon was by Adam Sawicki (Polish Academy of Sciences), and described an algorithm that is guaranteed to terminate, and which decides whether a set of qudit gates is universal, i.e. able to approximate arbitrarily well any unitary. The work relies on group theory, and first identifies a necessary and sufficient condition (involving the adjoint commutant of the set of gates) for a gate set to be universal, provided it generates an infinite group. Then he described a way to find out whether the generated group is infinite, which works for almost all matrices (except some with special spectral properties). He delineated an algorithm capable of dealing even with the exceptions, and gave a few examples of how long it was expected to run in some simple cases.

I was curious about the computational complexity of the algorithm, which was also the question asked by a workshop participant.

The next talk was by Michal Oszmaniec (Gdansk), and concerned universality of “gate sets“ involving bosonic and fermionic linear optics. More precisely, he considered adding extra gates to these sets, and characterizing what set of operations becomes available via this gate extension. The results proved curious. For N bosons in d>2 modes, any extra unitary V creates a universal set (aided by all passive linear-optical operations). The weird part is what happens when d=2, in which case the situation when N=6 demands extra care – that such apparently arbitrary numbers crop up in such a clearly stated problem is indeed curious.

For fermions the situation is similar (it is easy to upgrade fermionic linear optics to full universality), except when the interferometer is exacly half-filled – another curious result. Michal then described an application of these results, by promoting a simple set of 3 passive linear optical elements with a cross-Kerr element to achieve universality, useful in metrology with photons.

The last talk of the day was by Daniel Brod (UFF), who described how to create a passive CPHASE gate using cross-Kerr nonlinear media. Daniel started by reviewing the dual rail encoding for a qubit using two photon propagation modes, and the difficulty of achieving entangling two-qubit gates. The solution: cross-Kerr interactions, also known as cross-phase modulation, chi^3 media, as a kind of four-wave mixing.

The initial proposals for using cross-Kerr media for gates had many perceived shortcomings: non-linearities were very weak, photons were multimode, etc. In particular Shapiro in 2006 studied some regimes for the propagation of photons in these media, and pointed out difficulties. These regimes comprised very short and very long pulses.

Daniel managed to revisit this problem using the S matrix formalism. During the propagation in these media, photons change pulse shape, but this can be taken into account in quantum computation schemes, in particular due to measurement-based schemes which don’t require too many concatenated two-qubit gates. His scheme uses a string of interaction sites, where the photons can interact with two-level systems, which mediate the interaction between photons. A fidelity of 0.99 can be achieved with just 12 such interaction sites, and they proved that F->1 as the number of sites increases. Daniel also explained how his results are compatible with previous no-go results, by exploring regimes in which those lost their validity (such as medium-width photon pulses).

In the evening we had a very pleasant poster session, with finger food, drinks and very lively physics discussions going into the night. Paraty is living up to its reputation as a great place to gather, have fun and collaborate!

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