控量子系統是量子科技發展中非常重要的一環。近幾年的研究發現，透過量子疊加路徑（superposition of trajectories）技術，能夠有效增加通訊任務中的通道容量（channel capacity）以及降低錯誤率（error rate）。然而，此疊加技術的大多數研究仍侷限於量子通訊（quantum communication）領域。因此，在本論文中，我們將此量子疊加路徑概念延伸應用於共振腔量子電動力學(cavity quantum electrodynamics)系統中，並探討其可能的影響。首先，我們發現量子疊加路徑能夠產生等效的超輻射現象（superradiance），並調節（增強或抑制）其等效輻射率。此外，我們發現量子疊加路徑能夠控制（增強或抑制）離散時間晶體（discrete time crystalline）效應。並且，在適當的條件下，能夠產生等效暗態（dark state）以及延長量子記憶體（quantum memory）之效應。

Controlling quantum system is a very important part for the development of quantum technology. Recently, it has been reported that superposition of trajectories can be used to increase channel capacity and suppress communication error rates. However, most of these studies are restricted in the domain of quantum communication. In this thesis, we aim to explore potential impacts of this engineering scheme on cavity QED systems. First, we show that superposition of trajectories can result in effective superradiant effect. Also, the effective decay rate can be modulated (enhanced or suppressed). Second, we show that superposition of trajectories can enhance or suppress the discrete time crystalline effect. Moreover, under suitable circumstances, an effective dark state can be created. This can further lead to an enhancement of the quantum memory effect.

[1] Rabi, I. I. On the process of space quantization. Physical Review 49, 324–328 (1936).
[2] Rabi, I. I. Space quantization in a gyrating magnetic field. Physical Review 51,652–654 (1937).
[3] Braak, D. Integrability of the Rabi model. Physical Review Letters 107, 100401(2011).
[4] Jaynes, E. T. & Cummings, F. W. Comparison of quantum and semiclassical radiationtheories with application to the beam maser. Proceedings of the IEEE 51, 89–109(1963).
[5] Nielsen, M. A. & Chuang, I. Quantum computation and quantum information (2002).
[6] Schmidt-Kaler, F. et al. Realization of the Cirac–Zoller controlled-not quantum gate.Nature 422, 408–411 (2003).
[7] Chow, J. M. et al. Universal quantum gate set approaching fault-tolerant thresholdswith superconducting qubits. Physical Review Letters 109, 060501 (2012).
[8] Barends, R. et al. Superconducting quantum circuits at the surface code thresholdfor fault tolerance. Nature 508, 500–503 (2014).
[9] Schuster, D. I. et al. ac Stark shift and dephasing of a superconducting qubit stronglycoupled to a cavity field. Physical Review Letters 94, 123602 (2005).
[10] Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449,443–447 (2007).
[11] Richerme, P. et al. Non-local propagation of correlations in quantum systems withlong-range interactions. Nature 511, 198–201 (2014).
[12] Romero, G., Ballester, D., Wang, Y. M., Scarani, V. & Solano, E. Ultrafast quantumgates in circuit QED. Physical Review Letters 108, 120501 (2012).
[13] Kyaw, T. H., Herrera-Mart´ı, D. A., Solano, E., Romero, G. & Kwek, L.-C. Creationof quantum error correcting codes in the ultrastrong coupling regime. Physical ReviewB 91, 064503 (2015).
[14] Ritter, S. et al. An elementary quantum network of single atoms in optical cavities.Nature 484, 195–200 (2012).
[15] Felicetti, S. et al. Dynamical casimir effect entangles artificial atoms. Physical ReviewLetters 113, 093602 (2014).
[16] Campagne-Ibarcq, P. et al. Deterministic remote entanglement of superconductingcircuits through microwave two-photon transitions. Physical Review Letters 120,200501 (2018).
[17] Oi, D. K. Interference of quantum channels. Physical Review Letters 91, 067902(2003).
[18] Gisin, N., Linden, N., Massar, S. & Popescu, S. Error filtration and entanglementpurification for quantum communication. Physical Review A 72, 012338 (2005).
[19] Chiribella, G. & Kristj´ansson, H. Quantum Shannon theory with superpositions oftrajectories. Proceedings of the Royal Society A 475, 20180903 (2019).
[20] Loizeau, N. & Grinbaum, A. Channel capacity enhancement with indefinite causalorder. Physical Review A 101, 012340 (2020).
[21] Abbott, A. A., Wechs, J., Horsman, D., Mhalla, M. & Branciard, C. Communicationthrough coherent control of quantum channels. Quantum 4, 333 (2020).
[22] Kristj´ansson, H., Chiribella, G., Salek, S., Ebler, D. & Wilson, M. Resource theoriesof communication. New Journal of Physics 22, 073014 (2020).
[23] Rubino, G. et al. Experimental quantum communication enhancement by superposingtrajectories. Physical Review Research 3, 013093 (2021).
[24] Foo, J., Onoe, S. & Zych, M. Unruh-DeWitt detectors in quantum superpositions oftrajectories. Physical Review D 102, 085013 (2020).
[25] Foo, J., Mann, R. B. & Zych, M. Entanglement amplification between superposeddetectors in flat and curved spacetimes. Physical Review D 103, 065013 (2021).
[26] Henderson, L. J. et al. Quantum temporal superposition: The case of quantum fieldtheory. Physical Review Letters 125 (2020).
[27] Ban, M. Two-qubit correlation in two independent environments with indefiniteness.Physics Letters A 385, 126936 (2021).
[28] Ban, M. Relaxation process of a two-level system in a coherent superposition of twoenvironments. Quantum Information Processing 19, 1–23 (2020).
[29] Siltanen, O., Kuusela, T. & Piilo, J. Interferometric approach to open quantumsystems and non-Markovian dynamics. Physical Review A 103, 032223 (2021).
[30] Dicke, R. H. Coherence in spontaneous radiation processes. Physical review 93, 99(1954).
[31] Gross, M. & Haroche, S. Superradiance: An essay on the theory of collective spontaneousemission. Physics Reports 93, 301–396 (1982).
[32] Chen, Y. N., Chuu, D. S. & Brandes, T. Current detection of superradiance andinduced entanglement of double quantum dot excitons. Physical Review Letters 90,166802 (2003).
[33] Brandes, T. Coherent and collective quantum optical effects in mesoscopic systems.Physics Reports 408, 315–474 (2005).
[34] Scheibner, M. et al. Superradiance of quantum dots. Nature Physics 3, 106–110(2007).
[35] Chan, H. W., Black, A. T. & Vuleti´c, V. Observation of collective-emission-inducedcooling of atoms in an optical cavity. Physical Review Letters 90, 063003 (2003).
[36] Wolke, M., Klinner, J., Keßler, H. & Hemmerich, A. Cavity cooling below the recoillimit. Science 337, 75–78 (2012).
[37] Norcia, M. A., Winchester, M. N., Cline, J. R. & Thompson, J. K. Superradianceon the millihertz linewidth strontium clock transition. Science advances 2, e1601231(2016).
[38] Norcia, M. A. et al. Frequency measurements of superradiance from the strontiumclock transition. Physical Review X 8, 021036 (2018).
[39] Chen, Y. N., Li, C. M., Chuu, D. S. & Brandes, T. Proposal for teleportation ofcharge qubits via super-radiance. New Journal of Physics 7, 172–172 (2005).
[40] Wagner, R. & Clemens, J. P. Performance of a quantum teleportation protocol basedon temporally resolved photodetection of collective spontaneous emission. PhysicalReview A 79, 042322 (2009).
[41] Chou, C. W., Polyakov, S. V., Kuzmich, A. & Kimble, H. J. Single-photon generationfrom stored excitation in an atomic ensemble. Physical Review Letters 92, 213601(2004).
[42] Reimann, R. et al. Cavity-modified collective Rayleigh scattering of two atoms. PhysicalReview Letters 114, 023601 (2015).
[43] Black, A. T., Thompson, J. K. & Vuleti´c, V. On-demand superradiant conversion ofatomic spin gratings into single photons with high efficiency. Physical Review Letters95, 133601 (2005).
[44] Higgs, P. W. Broken symmetries and the masses of gauge bosons. Physical ReviewLetters 13, 508–509 (1964).
[45] Wilczek, F. Quantum time crystals. Physical Review Letters 109, 160401 (2012).
[46] Shapere, A. & Wilczek, F. Classical time crystals. Physical Review Letters 109,160402 (2012).
[47] Li, T. et al. Space-time crystals of trapped ions. Physical Review Letters 109, 163001(2012).
[48] Bruno, P. Impossibility of spontaneously rotating time crystals: A no-go theorem.Physical Review Letters 111, 070402 (2013).
[49] Watanabe, H. & Oshikawa, M. Absence of quantum time crystals. Physical ReviewLetters 114, 251603 (2015).
[50] Sacha, K. & Zakrzewski, J. Time crystals: a review. Reports on Progress in Physics81, 016401 (2017).
[51] Sacha, K. Modeling spontaneous breaking of time-translation symmetry. PhysicalReview A 91, 033617 (2015).
[52] Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of drivenquantum systems. Physical Review Letters 116, 250401 (2016).
[53] Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Physical Review Letters117, 090402 (2016).
[54] von Keyserlingk, C. W. & Sondhi, S. L. Phase structure of one-dimensional interactingFloquet systems. ii. symmetry-broken phases. Physical Review B 93, 245146 (2016).
[55] Yao, N. Y., Potter, A. C., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals:Rigidity, criticality, and realizations. Physical Review Letters 118, 030401 (2017).
[56] Zhang, J. et al. Observation of a discrete time crystal. Nature 543, 217–220 (2017).
[57] Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolarmany-body system. Nature 543, 221–225 (2017).
[58] Abanin, D. A., Roeck, W. D. & Huveneers, F. Theory of many-body localization inperiodically driven systems. Annals of Physics 372, 1–11 (2016).
[59] Giergiel, K. et al. Creating big time crystals with ultracold atoms. New Journal ofPhysics 22, 085004 (2020).
[60] Gong, Z., Hamazaki, R. & Ueda, M. Discrete time-crystalline order in cavity andcircuit QED systems. Physical Review Letters 120, 040404 (2018).
[61] Duan, L.-M., Lukin, M. D., Cirac, J. I. & Zoller, P. Long-distance quantum communicationwith atomic ensembles and linear optics. Nature 414, 413–418 (2001).
[62] Horodecki, M., Shor, P. W. & Ruskai, M. B. Entanglement breaking channels. Reviewsin Mathematical Physics 15, 629–641 (2003).
[63] Hill, S. & Wootters, W. K. Entanglement of a pair of quantum bits. Physical ReviewLetters 78, 5022–5025 (1997).
[64] Fleischhauer, M. & Lukin, M. D. Quantum memory for photons: Dark-state polaritons.Physical Review A 65, 022314 (2002).
[65] Vinjanampathy, S. & Anders, J. Quantum thermodynamics. Contemporary Physics57, 545–579 (2016).
[66] Giovannetti, V., Lloyd, S. & Maccone, L. Quantum metrology. Physical review letters96, 010401 (2006).