## Final Report Summary - QTHERM (Quantum Thermodynamics of Mesoscopic Systems)

In the following, each scientific outcome is presented more in detail and its importance for the objectives of the project explained.

1) A new interferometric technique for the detection of the characteristic function of the probability distribution of work.

The ingredients of a typical quantum thermodynamics setting are a quantum system, a bath at a certain temperature, and an external driving. Initially the system is in contact with the bath, once the system is prepared in a thermal state at the bath temperature, the system is detached from the bath, and the “protocol” can take place. The protocol consists of the controlled variation of the system Hamiltonian induced by the external driving and represented in the Hamiltonian by a protocol parameter that is function of time. The first thermodynamic question one can ask is: “what is the work done (made) on (by) the system?”. The possibility of answering in a satisfactory way to this question is paramount for the characterization of almost any other thermodynamic quantities, as the work dissipated and entropy production rate, and the possibility of testing important fluctuation relations like Jarzynski’s and Tasaki-Crooks’. A complete knowledge of work is obtained by detecting the probability distribution of work or analogously its characteristic function. Up to now the detection of such probability distribution relied on measurements of energy of the total system to be performed at the beginning and at the end of the protocol. However, this procedure becomes highly inconvenient for increasing “dimension” of the quantum system. As the objects of our investigation are quantum systems at the border between micro and macroscopic physics, a simpler scheme for accessing the thermodynamic properties of interest is paramount. The ability to access easily the work properties of a quantum thermodynamics process is crucial for systems that are mesoscopic in nature and/or endowed with many degrees of freedom.

To solve this problem we have designed a new interferometric technique for the detection of the characteristic function of work. The scheme relies on the suitably engineered coupling between the quantum system undergoing the protocol and a two-level ancillary system, whose state is the only quantity that needs to be accessed. Through the introduction of the ancillary system, the detection procedure is dramatically simplified, consisting now only of single-qubit tomographies performed at different times. The original paper introducing the technique and chosen as an Editor suggestion in Physical Review Letters is

L. Mazzola, G. De Chiara, and M. Paternostro,

“Measuring the Characteristic Function of the Work Distribution”

Phys. Rev. Lett. 110, 230602 (2013).

There we demonstrated the feasibility of this protocol in two different and timely experimental setups, namely optomechanical cavities and nanomechanical resonators coupled to Couper-pair box.

We continued the development and theoretical testing of our technique in the following contribution published in International Journal of Quantum Information,

L. Mazzola, G. De Chiara, and M. Paternostro,

“Detecting the Work Statistics through Ramsey-like interferometry”

Int. J. Quantum Inform. 12, 1461007 (2014)

where we expanded our study in two different directions:

a) We demonstrated the versatility of our interferometric approach to the study of the statistics of work done by an external driving potential that changes the frequency of a harmonic oscillator. This problem is key in the current theoretical design of Otto cycles based on trapped-ion technology, and this physical situation is indeed encountered in a number of experimental scenarios, from cavity-quantum electrodynamics to its superconducting-circuit counterpart.

b) We provided first results showing the approach that should be used in order to reconstruct the characteristic function of work distribution for a system weakly coupled to an environment. While we acknowledge that the formulation of an unambiguous concept of work (and heat) in the open quantum scenario is a diﬃcult problem that has not yet found solution, we show how our interferometric scheme can be applied to infer the generalized open-case probability of work distribution, introduced in T. Albash et al. Phys. Rev. E 88, 032146 (2013).

Engaging in collaborations with experimental groups has been a fundamental objective of the training envisaged in the proposed project. Being specifically designed to address the experimental need to measure the probability distribution of work, our interferometric technique has found the interest of various experimental groups. Among them, the Centro Brasileiro Pesquisa Fisica in Rio de Janeiro and Universidade do ABC in Sao Paulo has been able to actually implement the scheme experimentally. The corresponding research outputs have been reported in the following manuscript that is now under review in Physical Review Letters.

Tiago Batalhão, Alexandre M. Souza, Laura Mazzola, et al.

“Experimental reconstruction of work distribution and veriﬁcation of ﬂuctuation relations at the full quantum level”

arXiv:1308.3241

This work provided the ﬁrst experimental veriﬁcation of the quantum Jarzynski identity and the Tasaki-Crooks relation following a quantum process implemented in a Nuclear Magnetic Resonance (NMR) system. The study of fluctuation relations, initially in the classical and more recently in the quantum scenario, has found lots of interest in the last decade. The power of such fluctuation relations is that, defeating classical thermodynamics textbooks knowledge, they provide exact links between equilibrium and out-of-equilibrium thermodynamical quantities. For example using the Jarzynski equality, one can obtain free-energy differences from measurements of the work associated to an irreversible process. Crooks equality instead gives a relation between the work distribution measured along a given (forward) protocol and its time-reversed counterpart, and demonstrate that this relation depends on free-energy differences. Again, measuring the work statistics of forward and backward process will be in most cases simpler than accessing the free-energy difference. We demonstrated that our interferometric scheme come in handy for this task.

So far only the classical version of these relations had been experimentally tested. We have gone a step forward and demonstrated such relations in a fully quantum mechanical setup. Our experiment was carried out using liquid-state NMR spectroscopy of the 1H and 13C nuclear spins of a chloroform-molecule sample. The nuclear spin 13C played the role of a driven system, while the 1H one embodied the ancillary system necessary for the reconstruction of the probability distribution of work of 13C. The quantum process implemented consisted of a rapid change in a time-modulated radio frequency ﬁeld at the frequency of the frequency of the 13C nuclear spin. Using our technique, we could detect the work statistics associated to the forward and backward process, consider their ratio, and compare it with the theoretically predicted value for the free energy, finding that they were in agreement. This procedure allowed us to check the Tasaki-Crooks and Jarzynski fluctuation relations.

2) Thermodynamics of paradigmatic quantum many-body systems.

The investigation of the thermodynamics of systems with many-degrees of freedom has been a central objective of the project. The following contribution, which is under consideration in Physical Review E, is concerned with the thermodynamics of a paradigmatic continuous variable quantum many-body system.

A. Carlisle, L. Mazzola, M. Campisi, et al.

“Out of equilibrium thermodynamics of quantum harmonic chains”

arXiv:1403.0629

We consider an array of nearest-neighbor coupled quantum harmonic oscillators interacting with a thermal bath, and studied its thermodynamics when subjected to a global quench of the inter-oscillator coupling strength. Various noticeable results emerged from this investigation: we derived the characteristic function of the probability distribution of work for the chain of harmonic oscillators, calculated the reversible and dissipated work together with the variation of free energy. This allowed us to study quantum fluctuations identities in relation to the degree of squeezing induced by the dynamics. For the specific case of two harmonic oscillators we presented a direct quantitative link between the non-equilibrium lag (generally referred to as non-equilibrium entropy production) produced by the global quench and the quantum correlations shared by the two oscillators. This interesting link, which has been unveiled in the bipartite scenario, paves the way to the investigation of multipartite figure of merits for quantum correlations and thermodynamic quantities.

While on one hand we have been interested in thermodynamics of paradigmatic many-body systems, on the other we have striven at working out general system-independent thermodynamical relations. The following paper, under consideration in Physical Review X, reports on these two different types of contributions.

L. Fusco et al.

“Assessing the non-equilibrium thermodynamics in a quenched quantum many-body system

via single projective measurements”

arXiv:1404.3150

a) We analysed in detail the full statistics of the work distribution in a quantum many-body systems. In particular we focused on the case of a sudden quench of a Hamiltonian parameter and derived an explicit expression for all the moments and cumulants of the work distribution. In the case where the system is subjected to a sudden switch of an external magnetic field described by an operator that commutes with the unperturbed part of the Hamiltonian, we demonstrate that the momenta of work probability distribution are directly proportional to the momenta of the magnetization, therefore have a fairly intuitive physical meaning. We also obtained a relation that links the cumulant of the distribution generated by the system magnetization to the derivatives of the average value of the magnetization itself. The importance of these results stem from the fact that various non-equilibrium thermodynamics quantities can be written as function of these cumulants.

b) The connections described above became even more meaningful when studying a system characterized by criticality. We studied the thermodynamics of a spin chain described by the Ising model in a transverse field, and observed signature of the phase transition characterizing this system in high order moments of the work distribution. In particular we observed the signature of quantum phase transition in the variance and skewness of the probability distribution of work. Our results suggest the possibility to experimentally detect quantum phase transitions by looking at the full statistics of work.

3) System-environment correlations and non-Markovianity.

In the thermodynamics setting described above we assumed that the bath is used for the preparation of a thermal state of the system and that is later detached, so that during the protocol the system is externally driven (the Hamiltonian depends on time) but effectively closed (there are no losses). However, this situation is not necessarily realistic in any physical system, often the non-trivial effect of the environments cannot be neglected. The generalization of the concept of work to the open quantum system case is a key issue that we are currently investigating. Probably a clear answer to this issue can be offered only elucidating the role of system-environment correlations in thermodynamic quantities. Understanding the role of system-environment correlations is fundamental for the generalization of thermodynamical concepts to the open quantum system case. These are difficult issues that have been tackled by many experts of the field without founding yet a universally accepted solution. Therefore it is important to have a specific training on the topic and see how system-environment correlations are relevant in the emergence of other closely related phenomena, such as non-Markovianity of a system. The main scientific outcome of this study is the demonstration that creation of dynamical system-environment correlations is necessary for the appearance of non-Markovian effects. Namely, we have discovered an upper bound to the rate of change of a non-Markovianity indicator clearly dependent on the dynamics of system-environment correlations. A further connection with a witness of initial system-environment correlations was unveiled. This work that has been essential part of the training on the thematic explained above has been reported in the following two publications:

L. Mazzola, C. A. Rodriguez-Rosario, K. Modi, M. Paternostro

“Dynamical-role of system-environment correlations in non-Markovian dynamics”

Physical Review A 86, 010102(R) 2012.

C. A. Rodriguez-Rosario, K. Modi, L. Mazzola, A. Aspuru-Guzik

“Unification of witnessing initial-system environment correlations and witnessing non-Markovianity”

Europhysics Letters 99, 20010 (2012).

Further analysis lead us to find a lower bound to the rate of change of non-Markovianity, reported in the following paper:

Interaction-induced correlations and non-Markovianity of quantum dynamics

A. Smirne, L. Mazzola, M. Paternostro, and B. Vacchini

Phys. Rev. A 87, 052129 (2013).

4) Quantum information theory in optomechanical systems.

Optomechanics is one of the timely test-beds we chose for our investigation on quantum thermodynamics in mesoscopic systems. Not only we are going to exploit the single optomechanical cavities as a thermal machine, but also we will consider systems endowed with many optomechanical cavities (such as an optomechanical crystal) to study the transmission of heat in mesoscopic quantum systems. The approach used in this kind of study will be quantum-information oriented. The simplest system where to start from is a small network of three optomechanical cavities. The idea is to understand how quantities representing correlations shared by the different parts of the system behave. We focus on the problem of entanglement distribution in such optomechanical network driven by multipartite entangled optical resources. We relate the performance of the distribution of quantum correlations and entanglement to the symmetry of the optical resource state, identifying also the optimal resources to use at a given optomecahnical working point. These results have been published in a special issue, “Loss of coherence and memory effects in quantum dynamics”, in Journal of Physics B:

M. Paternostro, L. Mazzola, J. Lie

“Driven optomechanical systems for mechanical entanglement distribution”

Journal of Physics B: Atomic, Molecular and Optical Physics 45, 154010 (2012).

5) Experimental collaboration on non-Markovianity and quantum correlations

We have given a lot of importance to the development of research ideas and active collaborations with experimental groups. Part of the training had consisted in working at strict contact with the experimental group of Professor P. Mataloni at University “La Sapienza” in Rome, on specific topics connected to the themes of the project, namely characterization of correlations and non-Markovian quantum dynamics in linear optics setups.

Starting from a quadripartite photonic entangled state, the tomographic characterization of three-, two- and one-photon state was performed. The states experimentally measured exhibited high fidelity to the theoretical states obtained from the original four-qubit resource. The high quality of three-photon states allowed the experimental verification of the monogamy of correlations embodied by the Koashi-Winter relation, obtained using only a small number of experimental correlators for the reconstruction of such expression. These results have been published on a Special Issue of New Journal of Physics called “Focus on Quantum Tomography”:

A. Chiuri, L. Mazzola, M. Paternostro and P. Mataloni,

“Tomographic characterization of correlations in a photonic tripartite state”

New Journal of Physics 14, 085006 (2012).

The second project carried in collaboration with this experimental group was devoted to the design and realization of an open quantum system photonic simulator. The non-Markovian process realized coupled system and environment through a transverse Ising model. The setup could be adapted to engineer also Markovian dynamics and study the cross-over between the two non-Markovian/Markovian regimes. This investigation allowed us to explore experimentally the link between the emergence of non-Markovianity and system-environment correlations in strict connection with the outcome ii) of this report. This work was published in a new journal of the prestigious Nature Publishing Group.

A. Chiuri, C. Greganti, L. Mazzola, M. Paternostro, P. Mataloni

“Linear Optics Simulation of Quantum non-Markovian Dynamics”

Scientific Reports 2, 968 (2012).

A crucial part of the training had consisted in acquiring competences and expertise on quantum thermodynamics. This has been achieved through constant study of review and research papers and through the interaction with other experts in the field.

Finally, resources have been used for the purchase of a laptop, for participation to various international conference, such as the “kTlog2 Quantum Fluctuations and Information” Cuenca, Spain, the “Frontiers of quantum and mesoscopic thermodynamics”, Prague, Czech Republic, the “VI Quantum Information Workshop”, Paraty, Brazil and for research visits to Universidade do ABC, Sao Paulo and Centro Pesquisa Fisica Brasileiro, Rio de Janeiro, Brazil, Oxford University and University “La Sapienza”, Rome.