## Tesi etd-09012015-204712 |

Thesis type

Tesi di laurea magistrale

Author

CUSUMANO, STEFANO

URN

etd-09012015-204712

Title

Thermodynamics and correlations in quantum cascade systems

Struttura

FISICA

Corso di studi

FISICA

Commissione

**relatore**Prof. Giovannetti, Vittorio

Parole chiave

- thermodynamics
- quantum physics
- cascade systems
- open systems
- correlations

Data inizio appello

23/09/2015;

Consultabilità

completa

Riassunto analitico

Interest in quantum cascaded systems first arose in the 80’, when new exotic<br>forms of light, like squeezed light, were discovered. This led to a strong demand<br>for a new quantum formalism able to describe the evolution of a total system in<br>which one subsystem is driven with the light from another quantum subsystem<br>(e.g. two atoms). Gardiner and Collet [15] and Charmichael [6] first developed<br>the so-called input-output formalism which allowed to describe the evolution<br>of the system’s operator through Langevin’s equations, and moreover led to<br>a standard method to derive from these equations a master equation for the<br>density matrix. This formalism has been subsequently developed for various<br>case of interests [14, 16], up to become a well-based theory presented in books<br>[17].<br>In recent years the interest towards cascaded system has undergone a revival<br>due to important application in quantum information theory and many-body<br>physics. In general quantum cascaded systems are studied in the wider context<br>of quantum open systems: while in the past the noise from an external environ-<br>ment was seen only as a detrimental feature causing decoherence [37], nowadays<br>it is considered as a tool to control system’s evolution [29, 34] in order to obtain<br>states of interest, like entangled states [24, 33] or particular many-body states<br>[32, 35].<br>In this thesis a quantum cascaded system composed of bosonic subsystems<br>will be studied, mainly focusing on his thermodynamics and on the correlations<br>arising during the evolution.<br>Thermodynamics has been since its dawn in the 19th a quite difficult sub-<br>ject: it started as a purely phenomenological science, until the atomic theory<br>became popular. It was then that Boltzmann made a first attempt to derive<br>thermodynamics entirely from classical mechanics. Even if his theory clarified<br>some points of thermodynamics, nonetheless it was still unsatisfactory from a<br>foundational point of view, because it relied on unproven assumptions like the<br>ergodicity postulate or the a priori probabilities hypothesis. Many other sci-<br>entists tried to solve this problem, but none of them gave a fully satisfactory<br>answer.<br>With the emergence of quantum theory the old image of a gas as a set of<br>balls in a box started to seem just a sketch of reality, so that many efforts were<br>given to establish the theory of quantum thermodynamics [18]. On one side it<br>is necessary to reconcile quantum mechanics and classical thermodynamics [36,<br>38], while on the other side the main thermodynamical quantities like work,<br>heat and entropy have to be redefined in the quantum framework [2–4, 22, 25].<br>Concerning cascaded systems the interest in thermodynamics is due to the<br>peculiar features that heat flux showed in the case of a quantum system com-<br>posed by two subsystem, as analyzed in [26], so that it is interesting to know<br>what happens if there are more subsystems, if the features remain unchanged<br>or they are sensible to the number of subsystems. Moreover one asks if it would<br>be possible to engineer these systems in order to create heat cells able to release<br>energy slower, faster or in another desired way. Moreover it is interesting to see<br>if it is possible to create a heat interferometer with this kind of systems [19, 27].<br>The interest in correlations is mainly due the importance that they have<br>in several information theory protocols: it is well known that entanglement is<br>considered the main source for many quantum computation tasks [9, 10, 30],<br>and one of the most striking feature of quantum mechanics [37]. Many efforts<br>have been given all over the years to find methods for generating [24, 33, 35]<br>and distributing [5, 23] entangled states.<br>Moreover in recent years correlations have been analyzed in the new frame-<br>work of quantum discord [21, 31], a quantity that allows to measure all quantum<br>correlations of a state beyond entanglement. One of the most striking feature<br>of quantum discord is that while every entangled state has a non-null discord,<br>there exist non-entangled states also with non-zero discord [11, 13]. In recent<br>years several studies have been done to understand quantum discord [8, 12], to<br>define it operationally [7], to quantify it [28] and at least in some cases to find<br>closed formulas for its evaluation [1, 11, 20].<br>The first chapters of this thesis are focused on the basic arguments necessary<br>to understand the following chapters, such as open systems dynamics, cascaded<br>systems definition and Gaussian states. The last chapters are instead focused<br>on deriving and explaining all the results obtained about thermodynamics and<br>correlations. Here it is a detailed outline of the thesis.<br>In the second chapter a brief introduction about quantum mechanics and<br>open system dynamics is given. Dynamical semigroups and the concept of mas-<br>ter equation are introduced, together with the assumptions usually made in the<br>analysis of such systems.<br>The third chapter is focused on cascade systems, how they are defined and<br>which are the physical assumption made. The collisonal model used to find the<br>master equation is explained, and then the master equation is derived under<br>different physical assumptions.<br>Chapter four deals with Gaussian states and the formalism used to describe<br>them. The covariance matrix formalism is introduced and the dynamics of the<br>covariance matrix is derived through the master equation.<br>In the fifth chapter the thermodynamics of the system is analyzed, focusing<br>on the behaviour of the heat flux. First all the quantities used, such as heat<br>flux and transferred heat, are defined. Then, using the master equation, it is<br>studied how heat flux is affected by the system’s characteristics, like the number<br>of subsystems or their temperature.<br>Fynally in the sixth chapter correlations in the system are studied, observing<br>how they can arise during the transient dynamics. All the necessary formulas<br>to compute the amount of correlations in the system are explained and then<br>correlations dynamics is examined.<br><br>References<br>1. G. Adesso, A. Datta, Phys. Rev. Lett. 105, 030501 (2010).<br>2. R. Alicki, Journal of Physics A: Mathematical and General 12, L103<br>(1979).<br>3. J. Anders, V. Giovannetti, New Journal of Physics 15, 033022 (2013).<br>4. F. Binder, S. Vinjanampathy, K. Modi, J. Goold, Phys. Rev. E 91, 032119<br>(2015).<br>5. H.-J. Briegel, W. Dür, J. I. Cirac, P. Zoller, Phys. Rev. Lett. 81, 5932–5935<br>(1998).<br>6. H. J. Carmichael, Phys. Rev. Lett. 70, 2273–2276 (1993).<br>7. D. Cavalcanti et al., Phys. Rev. A 83, 032324 (2011).<br>8. F. Ciccarello, V. Giovannetti, Phys. Rev. A 85, 022108 (2012).<br>9. R. Cleve, H. Buhrman, Phys. Rev. A 56, 1201–1204 (1997).<br>10. R. Cleve, D. Gottesman, H.-K. Lo, Phys. Rev. Lett. 83, 648–651 (1999).<br>11. B. Dakić, V. Vedral, Č. Brukner, Phys. Rev. Lett. 105, 190502 (2010).<br>12. B. Dakič et al., Nat. Phys. 8, 666–670 (2012).<br>13. A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti, A. Acín, Phys. Rev.<br>A 81, 052318 (2010).<br>14. C. W. Gardiner, Phys. Rev. Lett. 70, 2269–2272 (1993).<br>15. C. W. Gardiner, M. J. Collett, Phys. Rev. A 31, 3761–3774 (1985).<br>16. C. W. Gardiner, A. S. Parkins, Phys. Rev. A 50, 1792–1806 (1994).<br>17. C. W. Gardiner, P. Zoller, Quantum Noise, ed. by Springer.<br>18. J. Gemmer, M. Michel, G. Mahler, Quantum Thermodynamics, ed. by<br>Springer, vol. 784.<br>19. F. Giazotto, M. J. Martìnez-Pérez, Nature 492, 401–405 (2012).<br>20. P. Giorda, M. G. A. Paris, Phys. Rev. Lett. 105, 020503 (2010).<br>21. L. Henderson, V. Vedral, Journal of Physics A: Mathematical and General<br>34, 6899 (2001).<br>22. M. Horodecki, J. Oppenheim, Nat. Comm. 4, 2059 (2013).<br>23. B. Kraus, J. I. Cirac, Phys. Rev. Lett. 92, 013602 (2004).<br>24. H. Krauter et al., Phys. Rev. Lett. 107, 080503 (2011).<br>25. S. Lloyd, Nat. 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