Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements
<p>Quantum-heat characteristic function <math display="inline"><semantics> <mrow> <mo>〈</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>β</mi> <mi>Q</mi> </mrow> </msup> <mo>〉</mo> </mrow> </semantics></math> for a two-level quantum system as a function of <math display="inline"><semantics> <msub> <mi>c</mi> <mn>1</mn> </msub> </semantics></math> in Equation (<a href="#FD8-condensedmatter-05-00017" class="html-disp-formula">8</a>) for three values of <math display="inline"><semantics> <msup> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mn>2</mn> </msup> </semantics></math>, which characterizes the initial state. The function is obtained from numerical simulations performed for a system with a Hamiltonian <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mi>J</mi> <mo>(</mo> <mo>|</mo> <mn>0</mn> <mo>⟩</mo> <mo>⟨</mo> <mn>1</mn> <mo>|</mo> <mo>+</mo> <mo>|</mo> <mn>1</mn> <mo>⟩</mo> <mo>⟨</mo> <mn>0</mn> <mo>|</mo> <mo>)</mo> </mrow> </semantics></math> subject to a sequence of <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> projective measurements. The latter are separated by a fixed waiting time <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, averaged over 2000 realizations, with <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. Units are used with <math display="inline"><semantics> <mrow> <mi>ℏ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 2
<p>Histogram of the initial (dashed) and final (solid) energy outcomes for the TPM scheme described in the text performed over <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math> realizations. While the initial state is non-uniform, the final state is practically uniform over the three energy levels.</p> "> Figure 3
<p>Comparison of the analytic expression (<a href="#FD22-condensedmatter-05-00017" class="html-disp-formula">22</a>) of the asymptotic (large-<span class="html-italic">M</span>) quantum-heat characteristic function <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>ϵ</mi> <mo>)</mo> </mrow> </semantics></math> (blue solid lines) with the numerical results averaged over <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math> realizations (red dots). The initial state is the same as in <a href="#condensedmatter-05-00017-f002" class="html-fig">Figure 2</a> and, again, <math display="inline"><semantics> <mrow> <mi>O</mi> <mo>=</mo> <msub> <mi>S</mi> <mi>z</mi> </msub> </mrow> </semantics></math>. In panel (<b>a</b>), the Hamiltonian is the same as in <a href="#condensedmatter-05-00017-f002" class="html-fig">Figure 2</a>, while in panel (<b>b</b>) the Hamiltonian is <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <msubsup> <mi>S</mi> <mi>z</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msub> <mi>ω</mi> <mn>2</mn> </msub> <msub> <mi>S</mi> <mi>x</mi> </msub> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>ω</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 4
<p>Behavior of <math display="inline"><semantics> <msub> <mi>β</mi> <mi>eff</mi> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mi>α</mi> </semantics></math> for different values of <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2.5</mn> <mo>]</mo> </mrow> </semantics></math>. We have chosen: (<b>a</b>) <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>3</mn> <mo>}</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics></math>, respectively.</p> "> Figure 5
<p>(<b>a</b>) Behavior of the asymptotic value <math display="inline"><semantics> <msub> <mover accent="true"> <mi>β</mi> <mo>¯</mo> </mover> <mi>eff</mi> </msub> </semantics></math> for a large positive <math display="inline"><semantics> <mi>α</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>) as a function of <math display="inline"><semantics> <msub> <mi>E</mi> <mn>3</mn> </msub> </semantics></math> (solid blue line) with <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. We compare the curve with its limiting (lower and upper) values, defined in Equation (<a href="#FD26-condensedmatter-05-00017" class="html-disp-formula">26</a>) (dash-dotted red lines). (<b>b</b>) Behavior of the asymptotic slope <span class="html-italic">r</span>, rescaled for <span class="html-italic">v</span>, for a large negative <math display="inline"><semantics> <mi>α</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mn>20</mn> </mrow> </semantics></math>) as a function of <math display="inline"><semantics> <msub> <mi>E</mi> <mn>3</mn> </msub> </semantics></math>. In both cases, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 6
<p>Behavior of <math display="inline"><semantics> <msub> <mi>β</mi> <mi>eff</mi> </msub> </semantics></math> as a function of <span class="html-italic">q</span>, which parametrizes the initial state <math display="inline"><semantics> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </semantics></math> as in Equation (<a href="#FD30-condensedmatter-05-00017" class="html-disp-formula">30</a>), in each of the three cases (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>></mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo><</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. The Protocol
Intermezzo on Two-Level Quantum Systems
3. Parametrization of the Initial State
4. Large Limit
5. Estimates of
5.1. Asymptotic Behavior for a Large Positive
5.2. Asymptotic Behavior for a Large Negative
5.3. Limits of the Adopted Parametrization
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Esposito, M.; Harbola, U.; Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 2009, 81, 1665. [Google Scholar] [CrossRef] [Green Version]
- Campisi, M.; Hänggi, P.; Talkner, P. Colloquium: Quantum fluctuations relations: Foundations and applications. Rev. Mod. Phys. 2011, 83, 1653. [Google Scholar] [CrossRef]
- Seifert, U. Stochastic thermodynamics, fluctuation theorems, and molecular machines. Rep. Prog. Phys. 2012, 75, 126001. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Deffner, S.; Campbell, S. Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information; Morgan & Claypool Publishers: Williston, VT, USA, 2019. [Google Scholar]
- Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 1997, 78, 2690. [Google Scholar] [CrossRef] [Green Version]
- Crooks, G. Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences. Phys. Rev. E 1999, 60, 2721. [Google Scholar] [CrossRef] [Green Version]
- Collin, D.; Ritort, F.; Jarzynski, C.; Smith, S.B.; Tinoco, I.; Bustamante, C. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 2005, 437, 231–234. [Google Scholar] [CrossRef]
- Toyabe, S.; Sagawa, T.; Ueda, M.; Muneyuki, E.; Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality. Nat. Phys. 2010, 6, 988–992. [Google Scholar] [CrossRef] [Green Version]
- Kafri, D.; Deffner, S. Holevo’s bound from a general quantum fluctuation theorem. Phys. Rev. A 2012, 86, 044302. [Google Scholar] [CrossRef] [Green Version]
- Albash, T.; Lidar, D.A.; Marvian, M.; Zanardi, P. Fluctuation theorems for quantum process. Phys. Rev. A 2013, 88, 023146. [Google Scholar]
- Rastegin, A.E. Non-equilibrium equalities with unital quantum channels. J. Stat. Mech. 2013, 6, P06016. [Google Scholar] [CrossRef]
- Sagawa, T. Lectures on Quantum Computing, Thermodynamics and Statistical Physics; World Scientific: Singapore, 2013. [Google Scholar]
- An, S.; Zhang, J.N.; Um, M.; Lv, D.; Lu, Y.; Zhang, J.; Yin, Z.; Quan, H.T.; Kim, K. Experimental test of the quantum Jarzynski equality with a trapped-ion system. Nat. Phys. 2015, 11, 193–199. [Google Scholar] [CrossRef] [Green Version]
- Batalhão, T.B.; Souza, A.M.; Mazzola, L.; Auccaise, R.; Sarthour, R.S.; Oliveira, I.S.; Goold, J.; Chiara, G.D.; Paternostro, M.; Serra, R.M. Experimental Reconstruction of Work Distribution and Study of Fluctuation Relations in a Closed Quantum System. Phys. Rev. Lett. 2014, 113, 140601. [Google Scholar] [CrossRef] [Green Version]
- Cerisola, F.; Margalit, Y.; Machluf, S.; Roncaglia, A.J.; Paz, J.P.; Folman, R. Using a quantum work meter to test non-equilibrium fluctuation theorems. Nat. Comm. 2017, 8, 1241. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Bartolotta, A.; Deffner, S. Jarzynski Equality for Driven Quantum Field Theories. Phys. Rev. X 2018, 8, 011033. [Google Scholar] [CrossRef] [Green Version]
- Hernández-Gómez, S.; Gherardini, S.; Poggiali, F.; Cataliotti, F.S.; Trombettoni, A.; Cappellaro, P.; Fabbri, N. Experimental test of exchange fluctuation relations in an open quantum system. arXiv 2019, arXiv:1907.08240. [Google Scholar]
- Talkner, P.; Lutz, E.; Hänggi, P. Fluctuation theorems: Work is not an observable. Phys. Rev. E 2007, 75, 050102(R). [Google Scholar] [CrossRef] [Green Version]
- Campisi, M.; Talkner, M.; Hänggi, P. Fluctuation Theorem for Arbitrary Open Quantum Systems. Phys. Rev. Lett. 2009, 102, 210401. [Google Scholar] [CrossRef]
- Mazzola, L.; De Chiara, G.; Paternostro, M. Measuring the characteristic function of the work distribution. Phys. Rev. Lett. 2013, 110, 230602. [Google Scholar] [CrossRef] [Green Version]
- Allahverdyan, A.E. Nonequilibrium quantum fluctuations of work. Phys. Rev. E 2014, 90, 032137. [Google Scholar] [CrossRef] [Green Version]
- Talkner, P.; Hänggi, P. Aspects of quantum work. Phys. Rev. E 2016, 93, 022131. [Google Scholar] [CrossRef] [Green Version]
- Jaramillo, J.D.; Deng, J.; Gong, J. Quantum work fluctuations in connection with the Jarzynski equality. Phys. Rev. E 2017, 96, 042119. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Deng, J.; Jaramillo, J.D.; Hänggi, P.; Gong, J. Deformed Jarzynski Equality. Entropy 2017, 19, 419. [Google Scholar] [CrossRef] [Green Version]
- Jarzynski, C.; Wojcik, D.K. Classical and Quantum Fluctuation Theorems for Heat Exchange. Phys. Rev. Lett. 2004, 92, 230602. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Campisi, M.; Pekola, J.; Fazio, R. Nonequilibrium fluctuations in quantum heat engines: Theory, example, and possible solid state experiments. New J. Phys. 2015, 17, 035012. [Google Scholar] [CrossRef]
- Campisi, M.; Pekola, J.; Fazio, R. Feedback-controlled heat transport in quantum devices: Theory and solid-state experimental proposal. New J. Phys. 2017, 19, 053027. [Google Scholar] [CrossRef] [Green Version]
- Batalhao, T.B.; Souza, A.M.; Sarthour, R.S.; Oliveira, I.S.; Paternostro, M.; Lutz, E.; Serra, R.M. Irreversibility and the arrow of time in a quenched quantum system. Phys. Rev. Lett. 2015, 115, 190601. [Google Scholar] [CrossRef] [Green Version]
- Gherardini, S.; Müller, M.M.; Trombettoni, A.; Ruffo, S.; Caruso, F. Reconstructing quantum entropy production to probe irreversibility and correlations. Quantum Sci. Technol. 2018, 3, 035013. [Google Scholar] [CrossRef] [Green Version]
- Manzano, G.; Horowitz, J.M.; Parrondo, J.M. Quantum Fluctuation Theorems for Arbitrary Environments: Adiabatic and Nonadiabatic Entropy Production. Phys. Rev. X 2018, 8, 031037. [Google Scholar] [CrossRef] [Green Version]
- Batalhão, T.B.; Gherardini, S.; Santos, J.P.; Landi, G.T.; Paternostro, M. Characterizing irreversibility in open quantum systems. In Thermodynamics in the Quantum Regime; Springer: Berlin/Heidelberger, Germany, 2018; pp. 395–410. [Google Scholar]
- Santos, J.P.; Céleri, L.C.; Landi, G.T.; Paternostro, M. The role of quantum coherence in non-equilibrium entropy production. npj Quant. Inf. 2019, 5, 23. [Google Scholar] [CrossRef] [Green Version]
- Kwon, H.; Kim, M.S. Fluctuation Theorems for a Quantum Channel. Phys. Rev. X 2019, 9, 031029. [Google Scholar] [CrossRef] [Green Version]
- Rodrigues, F.L.; De Chiara, G.; Paternostro, M.; Landi, G.T. Thermodynamics of Weakly Coherent Collisional Models. Phys. Rev. Lett. 2019, 123, 140601. [Google Scholar] [CrossRef] [Green Version]
- Campisi, M.; Talkner, P.; Hänggi, P. Fluctuation Theorems for Continuously Monitored Quantum Fluxes. Phys. Rev. Lett. 2010, 105, 140601. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Campisi, M.; Talkner, P.; Hänggi, P. Influence of measurements on the statistics of work performed on a quantum system. Phys. Rev. E 2011, 83, 041114. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Yi, J.; Kim, Y.W. Nonequilibirum work and entropy production by quantum projective measurements. Phys. Rev. E 2013, 88, 032105. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Watanabe, G.; Venkatesh, B.P.; Talkner, P.; Campisi, M.; Hänggi, P. Quantum fluctuation theorems and generalized measurements during the force protocol. Phys. Rev. E 2014, 89, 032114. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Hekking, F.W.J.; Pekola, J.P. Quantum jump approach for work and dissipation in a two-level system. Phys. Rev. Lett. 2013, 111, 093602. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Alonso, J.J.; Lutz, E.; Romito, A. Thermodynamics of weakly measured quantum systems. Phys. Rev. Lett. 2016, 116, 080403. [Google Scholar] [CrossRef] [Green Version]
- Gherardini, S.; Buffoni, L.; Müller, M.M.; Caruso, F.; Campisi, M.; Trombettoni, A.; Ruffo, S. Nonequilibrium quantum-heat statistics under stochastic projective measurements. Phys. Rev. E 2018, 98, 032108. [Google Scholar] [CrossRef] [Green Version]
- Elouard, C.; Herrera-Martí, D.A.; Clusel, M.; Auffèves, A. The role of quantum measurement in stochastic thermodynamics. npj Quantum Info. 2017, 3, 9. [Google Scholar] [CrossRef]
- Gherardini, S.; Gupta, S.; Cataliotti, F.S.; Smerzi, A.; Caruso, F.; Ruffo, S. Stochastic quantum Zeno by large deviation theory. New J. Phys. 2016, 18, 013048. [Google Scholar] [CrossRef]
- Gherardini, S.; Lovecchio, C.; Müller, M.M.; Lombardi, P.; Caruso, F.; Cataliotti, F.S. Ergodicity in randomly perturbed quantum systems. Quantum Sci. Technol. 2017, 2, 015007. [Google Scholar] [CrossRef] [Green Version]
- Piacentini, F.; Avella, A.; Rebufello, E.; Lussana, R.; Villa, F.; Tosi, A.; Marco, G.; Giorgio, B.; Eliahu, C.; Lev, V.; et al. Determining the quantum expectation value by measuring a single photon. Nat. Phys. 2017, 13, 1191–1194. [Google Scholar] [CrossRef] [Green Version]
- Wolters, J.; Strauß, M.; Schoenfeld, R.S.; Benson, O. Quantum Zeno phenomenon on a single solid-state spin. Phys. Rev. A 2013, 88, 020101(R). [Google Scholar] [CrossRef] [Green Version]
- Doherty, M.W.; Manson, N.B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollenberg, L.C. The nitrogen-vacancy colour centre in diamond. Phys. Rep. 2013, 528, 1–45. [Google Scholar] [CrossRef] [Green Version]
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Giachetti, G.; Gherardini, S.; Trombettoni, A.; Ruffo, S. Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements. Condens. Matter 2020, 5, 17. https://doi.org/10.3390/condmat5010017
Giachetti G, Gherardini S, Trombettoni A, Ruffo S. Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements. Condensed Matter. 2020; 5(1):17. https://doi.org/10.3390/condmat5010017
Chicago/Turabian StyleGiachetti, Guido, Stefano Gherardini, Andrea Trombettoni, and Stefano Ruffo. 2020. "Quantum-Heat Fluctuation Relations in Three-Level Systems Under Projective Measurements" Condensed Matter 5, no. 1: 17. https://doi.org/10.3390/condmat5010017