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Quantum Metrology and Thermometry in Open Systems. Thermal ADVERTIMENT. Lʼaccés als continguts dʼaquesta tesi queda condicionat a lʼacceptació de les condicions dʼús establertes per la següent llicència Creative Commons: http://cat.creativecommons.org/?page_id=184 ADVERTENCIA. El acceso a los contenidos de esta tesis queda condicionado a la aceptación de las condiciones de uso establecidas por la siguiente licencia Creative Commons: http://es.creativecommons.org/blog/licencias/ WARNING. The access to the contents of this doctoral thesis it is limited to the acceptance of the use conditions set by the following Creative Commons license: https://creativecommons.org/licenses/?lang=en Universitat Autonoma` de Barcelona Quantum Metrology and Thermometry in Open Systems Thermal Equilibrium and Beyond by Mohammad Mehboudi under supervision of Dr. Anna Sanpera Trigueros A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy in Unitat de F´ısica Teorica:` Informacio´ i Fenomens` Quantics` Departament de F´ısica Facultat de Ciencies` Bellaterra, April 2017 iii Abstract In my thesis I explore quantum metrology in open systems, with a special focus on quantum temperature estimation, or thermometry. For this aim, I categorize my study in two different regimes of thermal equilibrium and beyond thermal equilibrium. In both cases, my collabora- tors and I, raise questions of fundamental and technological interest. Among such questions, I point out the followings: What are the ultimate precision bounds on thermometry with individual (single) probes? Is it possible to improve these bounds by using quantum resources such as quantum correlations and quantum criticality? We not only find the ultimate precision bound on thermometry, posed by physical laws of na- ture, but also show how to exploit quantum resources to surpass the classical bounds on preci- sion, even at finite temperature. Furthermore, we identify experimentally feasible measurements which can achieve these bounds. Specifically, our results show that in a many-body sample, the collective quantum correlations can become optimal observables to accurately estimate the tem- perature. In turn, the collective spin correlations can be read out with the non-demolishing quan- tum Faraday spectroscopy. Hence, our method offers inferring maximum information about the temperature, yet leaving the sample unperturbed. Out of thermal equilibrium, we address both static and dynamic systems. In the former case, we find the limitations/opportunities for estimation of low temperature, and small temperature gra- dient in a sample. Particularly, we identify that the thermometric precision at low temperature can be significantly enhanced by strengthening the probe-sample coupling. Our observations may find applications in practical nanoscale thermometry at low temperature—a regime which is particularly relevant to quantum technologies. With a more applied point of view, such non equilibrium protocols give rise to autonomous quantum heat pumps. Hence, we also give thought to probing the quality of such heat pumps with the tiniest probes, i.e., a single spin. Although at the first glance a spin seems to be a very small probe, we confirm its efficiency in probing quantum heat pumps. Our techniques may find applications in the emerging field of quantum thermal engineering, as they facilitate the diagno- sis and design optimization of complex thermodynamic cycles. When it comes to dynamic systems, we have formulate a (fluctuation-dissipation) theory with the help of which one can identify the smallest external perturbation which affects a quantum system. Our proposal might be found useful in quantum force detection, for instance, interfero- metric detection of gravitational waves. iv Resumen En mi tesis exploro la metrolog´ıa cuantica´ en sistemas abiertos, con especial enfoque en la es- timacion´ de temperatura cuantica´ o termometr´ıa. Para este objetivo, voy a clasificar mi estudio en dos reg´ımenes diferentes de equilibrio t´ermico y m´asall´adel equilibrio t´ermico. En ambos casos, mis colaboradores y yo planteamos cuestiones de interes´ fundamental y tecnologico.´ Entre estas preguntas, senalo˜ lo siguiente: ¿Cuales´ son los l´ımites de precision´ maximos´ en la termometr´ıa con sondas individuales? ¿Es posible mejorar estos l´ımites utilizando recursos cuanticos´ tales como las correlaciones cu´anticas y la criticalidad cu´antica? No solo´ encontramos la maxima´ precision´ ligada a la termometr´ıa, planteada por las leyes f´ısicas de la naturaleza, sino tambien´ muestramos como´ explotar los recursos cuanticos´ para superar los l´ımites clasicos´ en la precision,´ incluso a temperatura finita. Ademas,´ identificamos medidas experimentalmente viables que pueden alcanzar estos l´ımites. Espec´ıficamente, nuestros resul- tados muestran que en una muestra de muchos cuerpos, las correlaciones cuanticas´ colectivas pueden convertirse en observables optimos´ para estimar con precision´ la temperatura. A su vez, las correlaciones de spin colectivo se pueden observar con espectroscopia cu´anticade Faraday no destructiva. Por lo tanto, nuestro metodo´ ofrece la posibilidad de inferir la maxima´ infor- macion´ sobre la temperatura, dejando la muestra sin perturbar. Fuera del equilibrio termico,´ abordamos sistemas est´aticos y din´amicos. En el primer caso, encontramos limitaciones/oportunidades para la estimacion´ a baja temperatura, y bajo gra- diente de temperatura en una muestra. Particularmente, identificamos que la precision´ ter- mometrica´ a baja temperatura puede ser significativamente mejorada mediante el fortalec- imiento del acoplamiento sonda-muestra. Nuestras observaciones pueden encontrar aplicaciones practicas´ en termometr´ıa a nanoescala y baja temperatura—un regimen´ que es particularmente relevante para tecnolog´ıas cuanticas.´ Con un punto de vista mas´ aplicado, tales protocolos de no equilibrio dan lugar a bombas de calor cu´anticasaut´onomas. Por lo tanto, tambien´ pensamos en explorar la calidad de tales bom- bas de calor con las sondas mas´ pequenas,˜ es decir, un solo spin. Aunque a primera vista un spin parece ser una sonda muy pequena,˜ confirmamos su eficiencia explorando bombas de calor cuanticas.´ Nuestras tecnicas´ pueden encontrar aplicaciones en el campo emergente de la inge- nier´ıa termica´ cuantica,´ ya que facilitan el diagnostico´ y la optimizacion´ del diseno˜ de ciclos termodinamicos´ complejos. Cuando se trata de sistemas dinamicos,´ hemos formulado una teor´ıa (fluctuacion-disipaci´ on)´ con la ayuda de la cual se puede identificar la m´ınima perturbacion´ externa que afecta a un sistema cuantico.´ Nuestra propuesta puede ser util´ en la deteccion´ de la fuerzas cuanticas,´ como por ejemplo, la deteccion´ interferometrica´ de ondas gravitatorias. In memory of Mohammad Shafie Mehboudi I entwined mind and soul indeed Then planted the resulting seed In this marriage the outcome is joy Beauty and soulfulness employ Hafez Contents 1 Introduction1 1.1 The structure of thesis ............................... 6 2 An overview on quantum metrology 9 2.1 Error bounds on metrology ............................ 9 2.2 The QFI for a general density matrix: Classical and quantum contributions . 13 2.3 Some useful properties of the QFI ........................ 15 2.3.1 Additivity of the QFI ........................... 15 2.3.2 Extended convexity of the QFI ...................... 15 2.3.3 Relation to fidelity, fidelity susceptibility and the Bures distance . 16 3 Preliminaries from open quantum systems 20 3.1 General quantum maps .............................. 20 3.2 Lindbladian master equation ........................... 22 3.3 Bosonic heat baths ................................ 26 3.4 Two level system in a radiation field ....................... 28 3.5 Damped harmonic oscillator ........................... 29 3.6 Coupled harmonic oscillators in a common bath . 31 4 Quantum metrolgy at thermal equilibrium 34 4.1 The canonical distribution ............................ 35 4.1.1 Adiabatic driving and static susceptibilities . 36 4.2 Ultimate precision of temperature estimation ................... 37 4.2.1 Thermometry with a quantum harmonic oscillator . 40 4.2.2 Analysis: Temperature sensitivity of multi-level systems and harmonic oscillators ................................. 41 4.3 Metrology with bipartite systems: The role of interaction . 42 4.3.1 Thermometry precision .......................... 43 4.3.1.1 Local schemes for thermometry . 44 4.3.2 Estimation of the tunneling rate J .................... 44 4.3.2.1 Local estimation of J ..................... 46 4.4 Conclusions .................................... 47 5 Quantum metrolgy in many body systems 48 5.1 The model: XY spin chain in a transverse field . 50 5.2 Quantum Faraday Spectroscopy ......................... 52 5.3 Optimal strategy: lowest bound on the temperature error . 54 5.4 Thermometry in the XY model using a Faraday interface . 55 ix x 5.5 Estimation of the Hamiltonian couplings ..................... 59 5.6 Magnetometry in the XX model ......................... 60 5.7 Sub-shot-noise sensing in the XX model ..................... 62 5.7.1 Low-temperature approximation for F (h) . 62 5.7.2 Adaptive feedforward magnetometry . 63 5.7.3 Sub-shot-noise estimation of the coupling J . 65 5.8 Magnetometry beyond the XX model ....................... 66 5.9 Summary and Outlook .............................. 67 6 Quantum metrology beyond thermal equilibrium I: Static systems 69 6.1 Thermometry with strongly coupled probes ................... 71 6.2 Coupled harmonic oscillators with individual
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