4.4 Holographic QET at finite temperature
4.4.2 Energy extraction at finite temperature
Given the Banados geometry due to the LPO acting on the thermal state, the procedure for evaluating the extraction energy density for the holographic QET protocol is universal in our framework, so is the formal form the result.
To recapitulate, we first evaluate the total energy density after LPO and LU for the thermal state, and the result denoted by TttPUβ is
TttPUβ = TttU+ (1− F[ ϕ′2]
)TttPβ. (4.36)
We then evaluate the extraction energy density ∆ρβB := TttPβ − TttPUβ and obtain
∆ρβB =−TttU + F[ ϕ′2]
TttPβ. (4.37)
Finally we subtract the divergent counter term−TttUto obtain the regularized extraction energy density for QET, and the result is
∆ρ(reg)βB = F[ ϕ′2]
TttPβ. (4.38)
Note that it takes the same form as (4.26) except by replacing TttP with TttPβ as expected by the universality of our framework. Obviously, the energy extraction is positive definite as guaranteed by the positive-definiteness of both F[
ϕ′2]
and TttPβ.
In next chapter, we give the conclusion for this thesis.
Appendix
4.A Energy density profile for infinitesimal holo-graphic local operations
Due to the divergent expression (4.25) for the energy density profile caused by the inhomogeneous infinitesimal LU, it is hard to find out the typical pattern of this energy density profile. However, we can consider the infinitesimal LU which has the amplitude of UV cutoff to see the pattern. For this purpose, we can expand the expression of (4.25) up toO(ϕ2), and the result is
TttU = lim
Furthermore, we require the overall size of ϕ isO(ϵ), i.e., ϕ(x) = ϵh(x), and then
This can be understood as the energy density profile induced by LUs on a region with few lattice sites if we discretize the space where CFT lives with lattice spacing ofO(ϵ).
If we consider the bump to be in the Gaussian form h(x) = exp{−x2
σ2 } , (4.41)
then the typical energy density is shown in Fig. 4.7, where we see some region of negative energy density, but overall it is positive. This means that the Gaussian LU injects energy. This is in contrast to the energy density of LPO, e.g., (4.21), which is extensive and positive definite, but with vanishing regularized energy.
-4 -2 2 4 x
-0.5 0.5 1.0 1.5 2.0
Ttt U
Figure 4.7: The energy density profile TttU(x) for ϕ(x) = ϵ e−x2.
Chapter 5
Conclusion
In this final chapter, we give the conclusion for the topics in chapter 3 and in chapter 4.
In chapter 3, we take the advantage of exact solvable reduced dynamics of the topological qubit and study the decoherence patterns when the topo-logical qubit is subjected to the linear or circular motions. Our results show interesting interplay of relativistic quantum information and the topological ordered states manifested in the Majorana zero modes.
Though our results are pertinent to topological qubit, we believe the following aspects are generic and should hold even for the usual qubit. They are (1) thermalization due to acceleration; (2) “anti-Unruh" phenomenon and decoherence impedance shown in chapter 3 and their intimate relation as they are both related to the short-time scale non-equilibrium effect; (3) information backflow invoked by the time modulation of coupling constant and accelerations. We think it deserves more study to clarify the underlying physics for some of the phenomenon and check their generality.
Besides, by exploiting the nonlocal feature of the topological qubit, we find that some incoherent accelerations of its constituent Majorana zero modes will preserve the coherence. This novel feature may help to develop the robust qubit in the future. Moreover, in chapter 3 we only consider the reduced dynamics of single topological qubit. It is interesting to study the multiple topological qubits, especially the evolution of their entanglement.
Besides entanglement, for multiple topological qubits, another quantum
cor-relation called quantum discord also can be considered. As a possible future work, we describe it briefly in the following.
We have discussed a lot about quantum decoherence so far. A quan-tum system loses its information into the environment, then finally becomes classical. From quantum to classical, it is intuitive to think that in the inter-mediate process the system is losing “quantumness”. Then a very interesting question is: How to quantify the quantumness of the system? As the sugges-tion provided by [128] , a quantity called “quantum discord” can characterize the quantumness of the system. Actually quantum discord is the quantum correlation part of the quantum mutual information of the system. The quantum mutual information of bipartite system is
I(ρ) = S(ρA) + S(ρB)− S(ρ). (5.1) It can be expressed as the classical correlation part C(ρ) plus the quantum correlation part Q(ρ) (quantum discord) as
I(ρ) = C(ρ) + Q(ρ) (5.2)
The classical correlation is defined as
C(ρ) = max{Bk}I(ρ|{Bk}) = S(ρA)− min{Bk}S(ρ|{Bk}), (5.3) where {Bk} is a complete set of one-dimensional projection operators on subsystem B. Different projectors give different values ofI(ρ|{Bk}). To get the classical correlation, we need to get the maximum value of I(ρ|{Bk}) over all possible projectors. It is highly nontrivial to find such maximum value, hence so far there is no general method to find it. But in some specific cases, e.g. two-qubit X state[129]1 , the quantum discord is available by a standard recipe.
1The form of the density matrix of two-qubit X state is as
ρX =
Although the entanglement entropy is also a kind of quantum correla-tion; therefore, it is intuitive to ask if it is closely related to the quantum discord. However it has been shown that for separable mixed states (hence no entanglement) the quantum discord does not vanish. It is also found that in some cases, the entanglement entropy is larger than the quantum discord.
Hence it can be said if any connection between the entanglement entropy and the quantum discord exists, it must be very nontrivial.
Such nontrivial relation between quantum discord and entanglement en-tropy bring a question to us: If these two quantities are applied to char-acterize the evolution of decoherence patterns, will they vanish (therefore completely decoherent) at the same time or not? It is interesting to extend the framework mentioned in chapter 3 to two qubits case to obtain the quan-tum discord and entanglement entropy. To obtain the quanquan-tum discord, we can follow the recipe provided in [129] for two-qubit X state case.
Next, let’s discuss the future work relevant to the topic in chapter 3.
The highlight of this study mentioned in chapter 3 is that exact solvable reduced dynamics of the topological qubit. To achieve it, two properties of the topological qubit are crucial: One is that the Majorana modes are the zero energy mode; therefore their kinetic terms don’t exist. The other is that the non-locality of the topological qubit requires the correlations of different channels vanish.
On the other hand, because of the elegance of the exact solvable reduced dynamics of the topological qubit., it is impossible to stop thinking about how to extend it to more general cases. In our further study, as any extension to more general cases , e.g. for usual qubit case, violates one of the two particular properties mentioned above, the dressed Green’s functions in our formalism won’t all obey bosonic time-ordering rule, but some of them obey fermionic time-ordering rule, e.g., whereTf denotes the fermionic time-ordering operator.
Such existence of the mixed time-ordering rules in the dressed Green’s
unavailable. Therefore to find the revised version of the Wick’s formalism is the key point to this problem. As long as this key point is achieved, the exact solvable reduced dynamics formalism can apply to deal with problems in various topics.
In chapter 4, we propose a tentative holographic scheme for the QET protocol. In the spirit of holography we try to realize all the quantum op-erations involved in the protocol as geometrical as possible. As we have no concrete way at this moment to realize POVM in CFTs or the holography, we choose to modify the conventional QET protocol by considering extraction of energy out of a particular excited state obtained by local projection. In this way we do not need the correlated the measurement outcome with Bob’s local operation when extracting energy. Despite that, our protocol still suc-ceeds in gaining the energy extraction. The success of our simplifying QET protocol can be attributed to the following facts: (a) the injected energy density due to local projection is positive semi-definite; and (b) the propor-tionality between Bob’s extraction energy density and the energy density of the excited state after Alice’s LPO is also positive definite. Moreover, this proportionality is universal in the sense that it does not depend on Alice’s local projection. It is not clear if such positive semi-definiteness is universal or not. If so, it may be related to some sort of the positive energy condition.
In chapter 4, we have shown that the positive semi-definiteness holds even with the presence of bulk black hole. It is very interesting to see the possi-ble connection between the quantum information inequality and the positive energy condition.
Another interesting perspective of our study is the demonstration how the surface/state duality (or AdS/MERA duality) can help to realize some quantum information task. In our case, we argue that the deformation of the UV surface can be thought as the local (unitary) operations in accordance with the surface/state duality. It would be interesting to understand more precisely what is the dual operation in the CFT side for these surface defor-mation. This reinforces the emerging point of view of bulk gravity from the quantum circuit based on the underlying CFT. Besides, the consideration of deforming UV surface also rises the new technical issue of finding a more covariant way of regularizing the stress tensor associated with such kind of
surface. For convenience we adopt the old regularization method as Brown and York [124]. We believe that the tentative scheme mentioned in chapter 4, in connection of quantum information tasks to holography, is just part of the first step along this direction and more studies are needed.
As the epilogue of this thesis, let’s talk a little bit more about entangle-ment. The topics in chapter 3 and in chapter 4 are both related to entan-glement as mentioned as follows: For the topic in chapter 3, the system is entangled with the environment and losing its cohrerence. One of the moti-vation of the topic in chapter 3 is to find a robust qubit which maintains its coherence against the environment. In other words, such robust qubit should not be easy to entangle with the environment. For the topic in chapter 4, the two separated subsystems with real space entanglement are subjected to the local projective operation and unitary transformation respectively. The entanglement between the two separated subsystems is a prerequisite for the protocol of QET. It is intuitive to respect that as long as such two subsys-tems entangle more with the environment, they lose more entanglement with each other, such that the efficiency of the extraction of energy is effected.
Therefore even though it is not emphasized in the framework of QET, the requirement of robust entangled subsystems against environment can’t be ignored from the practical perspective. From the above descriptions, the importance of robust qubits against the environment is apparent. As the results shown in this thesis, the topological qubit is a promising candidate as such robust qubit. It is also expected that as two entangled topological qubits are regarded as the two subsystems in QET, due to their robustness against environment, the efficiency of the extraction of energy is effected less than the one of usual qubits under the disturbance of the environment. Due to such robustness, the topological qubit always has great importance in the studies of quantum information. We expect that in the near future, there will be more significant progresses in these relevant topics.
Bibliography
[1] Pei-Hua Liu and Feng-Li Lin, “Decoherence of topological qubit in linear and circular motions: decoherence impedance, anti-Unruh and informa-tion backflow,” JHEP 1607 (2016) 084 [arXiv:1603.05136 [quant-ph]].
[2] Dimitrios Giataganas, Feng-Li Lin, and Pei-Hua Liu , "Towards holo-graphic quantum energy teleportation," Phys. Rev. D 94, 126013 (2016) [arXiv:1608.06523 [hep-th]].
[3] Shih-Hao Ho, Sung-Po Chao ,Chung-Hsien Chou and Feng-Li Lin, “De-coherence patterns of topological qubits from Majorana modes," New J. Phys. 16, no. 11, 113062 (2014) [arXiv:1406.6249 [cond-mat.str-el]].
[4] H. D. Zeh,“On the Interpretation of Measurement in Quantum Theory,”
Found. Phys.1: 69 (1970).
[5] W. H. Zurek, Pointer Basis of Quantum Apparatus: Into What Mixture Does the Wave Packet Collapse?, Phys. Rev. D 24, 1516 (1981).
W. H. Zurek, Environment induced superselection rules, Phys. Rev. D 26, 1862 (1982).
[6] W. H. Zurek, “Decoherence and the transition from quantum to classi-cal,” Physics Today 44 (10) 36-44 (1991).
W. H. Zurek,“Decoherence and the transition from quantum to classical - Revisited,” Los Alamos Science 27, 86-109 (2002).
W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Reviews of Modern Physics, 75, 715-765 (2003).
[7] W. H. Zurek,“Preferred states, predictabilty, classicality, and the envi-ronmentinduced decoherence,” Prog. Theor. Phys. 89, 281312 (1993).
[8] W. H. Zurek, S. Habib, J. P. Paz,“Coherent states via decoherence,”
Phys. Rev. Lett. 70, 11871190 (1993).
[9] S. -H. Ho, W. Li, F. -L. Lin and B. Ning,“QuantumDecoherence with Holography," JHEP 1401, 170 (2014). [arXiv:1309.5855 [hep-th]].
[10] R. P. Feynman and F. L. Vernon, Jr., “The theory of a general quantum system interacting with a linear dissipative system, Annals Phys. 24, 118 (1963).
[11] J. S. Schwinger, “Brownian motion of a quantum oscillator,” J. Math.
Phys. 2, 407 (1961).
L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Zh.
Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP 20, 1018 (1965)].
[12] Z. -b. Su, L. -y. Chen, X. -t. Yu and K. -c. Chou,“Influence functional and closed-time-path Green’s function,” Phys. Rev. B 37, 9810 (1988).
[13] R. Kubo,“Statistical mechanical theory of irreversible processes. 1. Gen-eral theory and simple applications in magnetic and conduction prob-lems,” J. Phys. Soc. Jap. 12, 570 (1957). P. C. Martin and J. S.
Schwinger,“Theory of many particle systems. 1.,” Phys. Rev. 115, 1342 (1959).
[14] A. Einstein,“Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen,” Annalen der Physik 17, 549 (1905).
[15] M. Smoluchowski,“Zur kinetischen Theorie der Brownschen Molekular-bewegung und der Suspensionen,” Annalen der Physik 21, 756 (1906).
[16] P. Langevin,“ Sur la theorie du mouvement brownien,” Comptes rendus de l’Academie des Sciences (Paris), 146, 530 (1908).
[17] A. O. Caldeira and A. J. Leggett,“Influence of dissipation on quantum tunneling in macroscopic systems,” Phys. Rev. Lett. 46, 211(1981).
[18] A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum
[19] B. L. Hu, J. P. Paz and Y. -h. Zhang, “Quantum Brownian motion in a general environment: 1. Exact master equation with nonlocal dissipa-tion and colored noise,” Phys. Rev. D 45, 2843 (1992).
[20] W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14, 870 (1976).
[21] B. S. DeWitt, “Quantum gravity: the new synthesis," in General rela-tivity: An Einstein centenary survey, edited by S. W. Hawking and W.
Israel, Cambridge University Press, Cambridge, 680-745 (1979).
[22] E. G. Brown, E. Martin-Martinez, N. C. Menicucci and R. B. Mann,
“Detectors for probing relativistic quantum physics beyond perturbation theory,” Phys. Rev. D 87, 084062 (2013) [arXiv:1212.1973 [quant-ph]].
[23] A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires," Phys.
Usp. 44, 131 (2001) [arXiv:cond-mat/0010440].
[24] N. Read and D. Green,“Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect,” Phys. Rev. B 61, 10267 (2000).
[25] G. ’t Hooft, “Dimensional Reduction in Quantum Gravity,” (1993) [arXiv:gr-qc/9310026].
[26] L. Susskind, “The World as a Hologram,” Journal of Mathematical Physics. 36(11): 6377-6396(1995)[arXiv:hep-th/9409089].
[27] J. M. Maldacena, “The large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor.
Phys. 38, 1113 (1999)][arXiv:hep-th/9711200].
[28] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory cor-relators from non-critical string theory,” Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109].
E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math.
Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
[29] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement en-tropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep-th/0603001].
[30] S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement En-tropy,” JHEP 0608 (2006) 045 [arXiv:hep-th/0605073].
[31] A. Lewkowycz and J. M. Maldacena, “Generalized gravitational en-tropy,”JHEP 1308 (2013) 090 [arXiv:1304.4926 [hep-th]].
[32] A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys.
Rev. Lett. 96, 110404 (2006).
[33] G. Vidal, “Class of Quantum Many-Body States That Can Be Efficiently Simulated,” Phys. Rev. Lett. 101, 110501 (2008).
[34] G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett. 99, 220405 (2007).
[35] B. Swingle, “Entanglement Renormalization and Holography,” Phys.
Rev. D 86, 065007 (2012) [arXiv:0905.1317 [cond-mat.str-el]].
[36] S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863 (1992).
[37] S. Rommer, S. Ostlund, “A class of ansatz wave functions for 1D spin systems and their relation to DMRG,” Phys. Rev. Lett. 75, 3537 (1995).
[38] F. Verstraete and J.I. Cirac, “ Renormalization Algorithms for Quantum-Many Body Systems in Two and Higher Dimensions,”
arXiv:0407066[cond-mat](2004).
[39] G. Evenbly and G. Vidal, “Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz,”
Phys.Rev. B 89 (2014) 235113, [arXiv:1310.8372].
[40] J. Haegeman, T. J. Osborne, H. Verschelde and F. Verstraete, “En-tanglement Renormalization for Quantum Fields in Real Space,” Phys.
[41] M. Miyaji and T. Takayanagi, “Surface/State Correspondence as a Generalized Holography,” PTEP 2015 (2015) no.7, 073B03 [arXiv:1503.03542 [hep-th]].
[42] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe,
“Continuous Multiscale Entanglement Renormalization Ansatz as Holo-graphic Surface-State Correspondence,” Phys. Rev. Lett. 115, no. 17, 171602 (2015) [arXiv:1506.01353 [hep-th]].
[43] M. Nozaki, S. Ryu and T. Takayanagi, “Holographic Geometry of En-tanglement Renormalization in Quantum Field Theories,” JHEP 1210 (2012) 193 [arXiv:1208.3469[hep-th]].
[44] L. Fu and C. L. Kane, “Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator," Phys. Rev. Lett.
100, 096407 (2008).
[45] M. Z. Hasan, C. L. Kane, “Topological Insulators," Rev. Mod. Phys. 82, 3045 (2010).
[46] X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors", Rev. Mod. Phys. 83, 1057 (2011).
[47] M. W.-Y. Tu and W.-M. Zhang, “Non-Markovian decoherence the-ory for a double-dot charge qubit", Phys. Rev. B 78, 235311 (2008) [arXiv:0809.3490 [cond-mat.meshall]].
J. S. Jin, M. W.-Y. Tu, W.-M. Zhang and Y. J. Yan, “Non-equilibrium quantum theory for nanodevices based on the Feynman Vernon influence functional", New J. Phys. 12 083013 (2010) [arXiv:0910.1675 [cond-mat.meshall]].
[48] H.-B. Liu, J.-H. An, C. Chen, Q.-J. Tong, H.-G. Luo and C. H. Oh,
“Anomalous decoherence in a dissipative two-level system", Phys. Rev.
A 87, 052139 (2013).
[49] S. T. Wu, “Quenched decoherence in qubit dynamics due to strong amplitude-damping noise”, Phys. Rev. A 89,034301 (2014) [arXiv:1310.6843[quant-ph]].
[50] J. Polchinski, “String theory. Vol. 1: An introduction to the bosonic string,” Cambridge University Press, 2007.
[51] S.-P. Chao, S. A. Silotri, and C.-H. Chung, “Nonequilibrium transport of helical Luttinger liquids through a quantum dot," Phys. Rev. B. 88, 085109 (2013).
[52] S. -H. Ho, W. Li, F. -L. Lin and B. Ning, “Quantum Decoherence with Holography," JHEP 1401, 170 (2014) [arXiv:1309.5855 [hep-th]].
[53] D. T. Son and A. O. Starinets, “Minkowski space correlators in AdS/CFT correspondence: Recipe and applications," JHEP 0209, 042 (2002) [arXiv:hep-th/0205051].
[54] S. Hill, W. K. Wootters, “Entanglement of a Pair of Quantum Bits,"
Phys. Rev. Lett. 78, 5022 (1997).
W. K. Wootters, “Entanglement of a Pair of Quantum Bits", Phys. Rev.
Lett. 80, 2245 (1998).
[55] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, “The Unruh effect and its applications,” Rev. Mod. Phys. 80, 787 (2008) [arXiv:0710.5373 [gr-qc]].
[56] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math.
Phys. 43, 199 (1975) [Commun. Math. Phys. 46, 206 (1976)].
[57] A. Peres and D. R. Terno, “Quantum information and relativity theory,”
Rev. Mod. Phys. 76, 93 (2004) [quant-ph/0212023].
[58] P. M. Alsing and G. J. Milburn, “Teleportation with a Uniformly Ac-celerated Partner," Phys. Rev. Lett. 91, 180404 (2003).
P. M. Alsing, D. McMahon, and G. J. Milburn, “Teleportation in a non-inertial frame," J. Opt. B: Quantum Semiclass. Opt. 6 (2004), S834.
[59] I. Fuentes-Schuller and R. B. Mann, “Alice falls into a black hole: En-tanglement in non-inertial frames,” Phys. Rev. Lett. 95, 120404 (2005) [quant-ph/0410172].
[60] P. M. Alsing, I. Fuentes-Schuller, R. B. Mann and T. E. Tessier, “Entan-glement of Dirac fields in non-inertial frames,” Phys. Rev. A 74, 032326 (2006) [quant-ph/0603269].
[61] S. Y. Lin, C. H. Chou and B. L. Hu, “Disentanglement of two har-monic oscillators in relativistic motion,” Phys. Rev. D 78, 125025 (2008) [arXiv:0803.3995 [gr-qc]].
[62] E. Martin-Martinez, “Relativistic Quantum Information: develop-ments in Quantum Information in general relativistic scenarios,”
arXiv:1106.0280 [quant-ph].
[63] D. C. M. Ostapchuk, S. Y. Lin, R. B. Mann and B. L. Hu, “Entanglement Dynamics between Inertial and Non-uniformly Accelerated Detectors,”
JHEP 1207, 072 (2012) [arXiv:1108.3377 [gr-qc]].
[64] B. Richter and Y. Omar, “Degradation of entanglement between two accelerated parties: Bell states under the Unruh effect,” Phys. Rev. A 92, 2, 022334 (2015) [arXiv:1503.07526 [quant-ph]].
[65] S. Y. Lin and B. L. Hu, “Backreaction and the Unruh effect: New insights from exact solutions of uniformly accelerated detectors," Phys.
Rev. D, 76, 064008 (2007) [arXiv:gr-qc/0611062].
[66] J. S. Bell and J. M. Leinaas, “Electrons As Accelerated Thermometers,”
Nucl. Phys. B 212, 131 (1983).
[67] J. S. Bell and J. M. Leinaas, “The Unruh Effect and Quantum Fluctu-ations of Electrons in Storage Rings,” Nucl. Phys. B 284, 488 (1987).
[68] J. R. Letaw and J. D. Pfautsch, “The Quantized Scalar Field in Rotating Coordinates,” Phys. Rev. D 22, 1345 (1980). “The Quantized Scalar Field in the Stationary Coordinate Systems of Flat Space-time,” Phys.
Rev. D 24, 1491 (1981).
J. R. Letaw, “Vacuum Excitation of Noninertial Detectors on Stationary World Lines,” Phys. Rev. D 23, 1709 (1981).
P. C. W. Davies, T. Dray and C. A. Manogue, “The Rotating quantum vacuum,” Phys. Rev. D 53, 4382 (1996) [gr-qc/9601034].
O. Levin, Y. Peleg and A. Peres, “Unruh effect for circular motion in a cavity,” J. Phys. A 26, 3001 (1993).
[69] J. Doukas, S. Y. Lin, B. L. Hu and R. B. Mann, “Unruh Effect under Non-equilibrium conditions: Oscillatory motion of an Unruh-DeWitt detector,” JHEP 1311, 119 (2013) [arXiv:1307.4360].
[70] D. Kothawala and T. Padmanabhan, “Response of Unruh-DeWitt de-tector with time-dependent acceleration,” Phys. Lett. B 690, 201 (2010) [arXiv:0911.1017 [gr-qc]].
[71] N. Obadia and M. Milgrom, “On the Unruh effect for general
[71] N. Obadia and M. Milgrom, “On the Unruh effect for general