Abstract
We compute how an accelerating qubit coupled to a scalar field — i.e. an Unruh-DeWitt detector — evolves in flat space, with an emphasis on its late-time behaviour. When calculable, the qubit evolves towards a thermal state for a field prepared in the Minkowski vacuum, with the approach to this limit controlled by two different time-scales. For a free field we compute both of these as functions of the difference between qubit energy levels, the dimensionless qubit/field coupling constant, the scalar field mass and the qubit’s proper acceleration. Both time-scales differ from the Candelas-Deutsch-Sciama transition rate traditionally computed for Unruh-DeWitt detectors, which we show describes the qubit’s early-time evolution away from the vacuum rather than its late-time approach to equilibrium. For small enough couplings and sufficiently late times the evolution is Markovian and described by a Lindblad equation, which we derive in detail from first principles as a special instance of Open EFT methods designed to handle a breakdown of late-time perturbative predictions due to the presence of secular growth. We show how this growth is resummed in this example to give reliable information about late-time evolution including both qubit/field interactions and field self-interactions. By allowing very explicit treatment, the qubit/field system allows a systematic assessment of the approximations needed when exploring late-time evolution, in a way that lends itself to gravitational applications. It also allows a comparison of these approximations with those — e.g. the ‘rotating-wave’ approximation — widely made in the open-system literature (which is aimed more at atomic transitions and lasers).
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References
S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics and Particle Creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
C. Kiefer, Is there an information loss problem for black holes?, Lect. Notes Phys. 633 (2003) 84 [gr-qc/0304102] [INSPIRE].
D. Giulini, C. Kiefer, E. Joos, J. Kupsch, I.O. Stamatescu and H.D. Zeh, Decoherence and the appearance of a classical world in quantum theory, Springer, Germany (2003).
A.A. Starobinsky, Stochastic de Sitter (inflationary) stage in the early universe, Lect. Notes Phys. 246 (1986) 107 [INSPIRE].
A.A. Starobinsky and J. Yokoyama, Equilibrium state of a selfinteracting scalar field in the de Sitter background, Phys. Rev. D 50 (1994) 6357 [astro-ph/9407016] [INSPIRE].
N.C. Tsamis and R.P. Woodard, Stochastic quantum gravitational inflation, Nucl. Phys. B 724 (2005) 295 [gr-qc/0505115] [INSPIRE].
E.B. Davies, Quantum Theory of Open Systems, Academic Press, London (1976).
R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Springer (1987).
R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer (1995).
C.W. Gardiner and P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer (2000).
U. Weiss, Quantum Dissipative Systems, World Scientific (2000).
H.P. Breuer and F. Petruccione, The theory of open quantum systems, Oxford University Press (2002).
A. Rivas and S.F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).
G. Schaller, Open Quantum Systems Far from Equilibrium, Springer (2014).
A.H. Nayfeh, Perturbation Methods, John Wiley & Sons, Inc. (1973).
F. Tanaka, Coherent Representation of Dynamical Renormalization Group in Bose Systems, Prog. Theor. Phys. 54 (1975) 1679 [INSPIRE].
L.Y. Chen, N. Goldenfeld and Y. Oono, Renormalization Group Theory for Global Asymptotic Analysis, Phys. Rev. Lett. 73 (1994) 1311 [cond-mat/9407024] [INSPIRE].
L.-Y. Chen, N. Goldenfeld and Y. Oono, The Renormalization group and singular perturbations: Multiple scales, boundary layers and reductive perturbation theory, Phys. Rev. E 54 (1996) 376 [hep-th/9506161] [INSPIRE].
C.M. Bender and L.M.A. Bettencourt, Multiple scale analysis of the quantum anharmonic oscillator, Phys. Rev. Lett. 77 (1996) 4114 [hep-th/9605181] [INSPIRE].
J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739 (2004) 3 [hep-ph/0409233] [INSPIRE].
Y. Urakawa and T. Tanaka, Influence on Observation from IR Divergence during Inflation. I., Prog. Theor. Phys. 122 (2009) 779 [arXiv:0902.3209] [INSPIRE].
C.P. Burgess, R. Holman and G. Tasinato, Open EFTs, IR effects & late-time resummations: systematic corrections in stochastic inflation, JHEP 01 (2016) 153 [arXiv:1512.00169] [INSPIRE].
C.P. Burgess, R. Holman, G. Tasinato and M. Williams, EFT Beyond the Horizon: Stochastic Inflation and How Primordial Quantum Fluctuations Go Classical, JHEP 03 (2015) 090 [arXiv:1408.5002] [INSPIRE].
M.-a. Sakagami, Evolution From Pure States Into Mixed States in de Sitter Space, Prog. Theor. Phys. 79 (1988) 442 [INSPIRE].
L.P. Grishchuk and Y.V. Sidorov, On the Quantum State of Relic Gravitons, Class. Quant. Grav. 6 (1989) L161 [INSPIRE].
R.H. Brandenberger, R. Laflamme and M. Mijic, Classical Perturbations From Decoherence of Quantum Fluctuations in the Inflationary Universe, Mod. Phys. Lett. A 5 (1990) 2311 [INSPIRE].
E. Calzetta and B.L. Hu, Quantum fluctuations, decoherence of the mean field and structure formation in the early universe, Phys. Rev. D 52 (1995) 6770 [gr-qc/9505046] [INSPIRE].
C. Kiefer, D. Polarski and A.A. Starobinsky, Quantum to classical transition for fluctuations in the early universe, Int. J. Mod. Phys. D 7 (1998) 455 [gr-qc/9802003] [INSPIRE].
C. Kiefer and D. Polarski, Emergence of classicality for primordial fluctuations: Concepts and analogies, Annalen Phys. 7 (1998) 137 [gr-qc/9805014] [INSPIRE].
C. Agon, V. Balasubramanian, S. Kasko and A. Lawrence, Coarse Grained Quantum Dynamics, Phys. Rev. D 98 (2018) 025019 [arXiv:1412.3148] [INSPIRE].
D. Boyanovsky, Effective field theory during inflation: Reduced density matrix and its quantum master equation, Phys. Rev. D 92 (2015) 023527 [arXiv:1506.07395] [INSPIRE].
D. Boyanovsky, Effective field theory during inflation. II. Stochastic dynamics and power spectrum suppression, Phys. Rev. D 93 (2016) 043501 [arXiv:1511.06649] [INSPIRE].
E. Nelson, Quantum Decoherence During Inflation from Gravitational Nonlinearities, JCAP 03 (2016) 022 [arXiv:1601.03734] [INSPIRE].
T.J. Hollowood and J.I. McDonald, Decoherence, discord and the quantum master equation for cosmological perturbations, Phys. Rev. D 95 (2017) 103521 [arXiv:1701.02235] [INSPIRE].
S. Shandera, N. Agarwal and A. Kamal, Open quantum cosmological system, Phys. Rev. D 98 (2018) 083535 [arXiv:1708.00493] [INSPIRE].
C. Agón and A. Lawrence, Divergences in open quantum systems, JHEP 04 (2018) 008 [arXiv:1709.10095] [INSPIRE].
J. Martin and V. Vennin, Observational constraints on quantum decoherence during inflation, JCAP 05 (2018) 063 [arXiv:1801.09949] [INSPIRE].
J. Martin and V. Vennin, Non Gaussianities from Quantum Decoherence during Inflation, JCAP 06 (2018) 037 [arXiv:1805.05609] [INSPIRE].
G. Lindblad, On the Generators of Quantum Dynamical Semigroups, Commun. Math. Phys. 48 (1976) 119 [INSPIRE].
V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E.C.G. Sudarshan, Properties of Quantum Markovian Master Equations, Rept. Math. Phys. 13 (1978) 149 [INSPIRE].
R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570 [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1., Phys. Rev. 115 (1959) 1342 [INSPIRE].
S. Nakajima, On Quantum Theory of Transport Phenomena, Prog. Theor. Phys. 20 (1958) 948.
R. Zwanzig, Ensemble Method in the Theory of Irreversibility, J. Chem. Phys. 33 (1960) 1338.
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
B.S. DeWitt, Quantum Gravity: The New Synthesis, in General Relativity, An Einstein Centenary Survey, S.W. Hawking and W. Israel eds., Cambrdige University Press (1979).
D.W. Sciama, P. Candelas and D. Deutsch, Quantum Field Theory, Horizons and Thermodynamics, Adv. Phys. 30 (1981) 327 [INSPIRE].
C.P. Burgess, J. Hainge, G. Kaplanek and M. Rummel, Failure of Perturbation Theory Near Horizons: the Rindler Example, JHEP 10 (2018) 122 [arXiv:1806.11415] [INSPIRE].
G. Kaplanek and C.P. Burgess, Hot Cosmic Qubits: Late-Time de Sitter Evolution and Critical Slowing Down, JHEP 02 (2020) 053 [arXiv:1912.12955] [INSPIRE].
S. Takagi, Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-Unruh Effect in Rindler Manifold of Arbitrary Dimension, Prog. Theor. Phys. Suppl. 88 (1986) 1 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press (1982).
K.J. Hinton, Particle detector equivalence, Class. Quant. Grav. 1 (1984) 27 [INSPIRE].
J. Hadamard, Lectures on Cauchy’s problem in linear partial differential equations, Yale University Press, New Haven (1923).
B.S. DeWitt and R.W. Brehme, Radiation damping in a gravitational field, Annals Phys. 9 (1960) 220 [INSPIRE].
B.S. Kay and R.M. Wald, Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate Killing Horizon, Phys. Rept. 207 (1991) 49 [INSPIRE].
M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996) 529 [INSPIRE].
S.A. Fulling, M. Sweeny and R.M. Wald, Singularity Structure of the Two Point Function in Quantum Field Theory in Curved Space-Time, Commun. Math. Phys. 63 (1978) 257 [INSPIRE].
B.F. Svaiter and N.F. Svaiter, Inertial and noninertial particle detectors and vacuum fluctuations, Phys. Rev. D 46 (1992) 5267 [Erratum ibid. D 47 (1993) 4802] [INSPIRE].
A. Higuchi, G.E.A. Matsas and C.B. Peres, Uniformly accelerated finite time detectors, Phys. Rev. D 48 (1993) 3731 [INSPIRE].
L. Sriramkumar and T. Padmanabhan, Response of finite time particle detectors in noninertial frames and curved space-time, Class. Quant. Grav. 13 (1996) 2061 [gr-qc/9408037] [INSPIRE].
J. Louko and A. Satz, How often does the Unruh-DeWitt detector click? Regularisation by a spatial profile, Class. Quant. Grav. 23 (2006) 6321 [gr-qc/0606067] [INSPIRE].
E. Montroll, Nonequilibrium Statistical Mechanics, in Lectures in Theoretical Physics: Vol. III, W.E. Britten, B.W. Downs and J. Downs eds., Interscience (1961).
S.M. Barnett and S. Stenholm, Hazards of Reservoir Memory, Phys. Rev. A 64 (2001) 033808.
P. Langlois, Causal particle detectors and topology, Annals Phys. 321 (2006) 2027 [gr-qc/0510049] [INSPIRE].
K. Fredenhagen and R. Haag, Generally Covariant Quantum Field Theory and Scaling Limits, Commun. Math. Phys. 108 (1987) 91 [INSPIRE].
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Corporation (1965).
F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Function, Cambridge University Press (2010).
W. Troost and H. Van Dam, Thermal Effects for an Accelerating Observer, Phys. Lett. 71B (1977) 149 [INSPIRE].
W.G. Unruh and R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D 29 (1984) 1047 [INSPIRE].
F. Benatti, R. Floreanini, Entanglement generation in uniformly accelerating atoms: Reexamination of the Unruh effect, Phys. Rev. A 70 (2004) 012112 [quant-ph/0403157].
H.W. Yu, J. Zhang, H.-w. Yu and J.-l. Zhang, Understanding Hawking radiation in the framework of open quantum systems, Phys. Rev. D 77 (2008) 024031 [Erratum ibid. D 77 (2008) 029904] [arXiv:0806.3602] [INSPIRE].
H. Yu, Open quantum system approach to Gibbons-Hawking effect of de Sitter space-time, Phys. Rev. Lett. 106 (2011) 061101 [arXiv:1101.5235] [INSPIRE].
J. Hu and H. Yu, Entanglement generation outside a Schwarzschild black hole and the Hawking effect, JHEP 08 (2011) 137 [arXiv:1109.0335] [INSPIRE].
J. Hu and H. Yu, Geometric phase for an accelerated two-level atom and the Unruh effect, Phys. Rev. A 85 (2012) 032105 [arXiv:1203.5869] [INSPIRE].
M. Fukuma, Y. Sakatani and S. Sugishita, Master equation for the Unruh-DeWitt detector and the universal relaxation time in de Sitter space, Phys. Rev. D 89 (2014) 064024 [arXiv:1305.0256] [INSPIRE].
G. Menezes, N.F. Svaiter and C.A.D. Zarro, Entanglement dynamics in random media, Phys. Rev. A 96 (2017) 062119 [arXiv:1709.08702] [INSPIRE].
Z. Tian, J. Wang, J. Jing and A. Dragan, Entanglement Enhanced Thermometry in the Detection of the Unruh Effect, Annals Phys. 377 (2017) 1 [arXiv:1603.01122] [INSPIRE].
G. Menezes, Entanglement dynamics in a Kerr spacetime, Phys. Rev. D 97 (2018) 085021 [arXiv:1712.07151] [INSPIRE].
A. Chatterjee, S. Saha and C. Singha, How the mass of a scalar field influences Resonance Casimir-Polder interaction in Schwarzschild spacetime, arXiv:1912.07502 [INSPIRE].
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 (2007) 064008 [gr-qc/0611062] [INSPIRE].
D. Moustos and C. Anastopoulos, Non-Markovian time evolution of an accelerated qubit, Phys. Rev. D 95 (2017) 025020 [arXiv:1611.02477] [INSPIRE].
R. S. Whitney, Staying positive: going beyond Lindblad with perturbative master equations, J. Phys. A 41 (2008) 175304 [arXiv:0711.0074].
V. Gorini, A. Kossakowski and E.C.G. Sudarshan, Completely Positive Dynamical Semigroups of N Level Systems, J. Math. Phys. 17 (1976) 821 [INSPIRE].
A.G. Redfield, The Theory of Relaxation Processes, Advances in Magnetic and Optical Resonance 1 (1965) 1.
I. Gradshteyn and M. Ryzhik, Table of Integrals Series and Products, Academic Press (1965).
G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press (1995).
A. Erdelyi, Higher Transcendental Functions: Volume 1, McGraw-Hill Book Company (1953).
R.G. Bartle, The Elements of Integration and Lebesgue Measure, Wiley (1995).
R. Dumcke, H. Spohn, The Proper Form of the Generator in the Weak Coupling Limit, Z. Phys. B 34 (1979) 419.
E. Davies, Markovian Master Equations, Commun. Math. Phys. 39 (1974) 91.
E. Davies, Markovian Master Equations. II, Math. Ann. 219 (1976) 147.
S. De Bìevre and M. Merkli, The Unruh effect revisited, Class. Quant. Grav. 23 (2006) 6525 [math-ph/0604023] [INSPIRE].
D. Moustos, Asymptotic states of accelerated detectors and universality of the Unruh effect, Phys. Rev. D 98 (2018) 065006 [arXiv:1806.10005] [INSPIRE].
B.A. Juárez-Aubry and D. Moustos, Asymptotic states for stationary Unruh-DeWitt detectors, Phys. Rev. D 100 (2019) 025018 [arXiv:1905.13542] [INSPIRE].
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Kaplanek, G., Burgess, C.P. Hot accelerated qubits: decoherence, thermalization, secular growth and reliable late-time predictions. J. High Energ. Phys. 2020, 8 (2020). https://doi.org/10.1007/JHEP03(2020)008
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DOI: https://doi.org/10.1007/JHEP03(2020)008