We investigate the efficiency of a quantum Carnot engine based on open quantum dynamics theory. The model includes time-dependent external fields for the subsystems controlling the isothermal and isentropic processes and for the system–bath (SB) interactions controlling the transition between these processes. Numerical simulations are conducted in a nonperturbative and non-Markovian SB coupling regime by using the hierarchical equations of motion under these fields at different cycle frequencies. The work applied to the total system and the heat exchanged with the baths are rigorously evaluated. In addition, by regarding quasi-static work as free energy, we compute the quantum thermodynamic variables and analyze the simulation results by using thermodynamic work diagrams for the first time. Analysis of these diagrams indicates that, in the strong SB coupling region, the fields for the SB interactions are major sources of work, while in other regions, the field for the subsystem is a source of work. We find that the maximum efficiency is achieved in the quasi-static case and is determined solely by the bath temperatures, regardless of the SB coupling strength, which is a numerical manifestation of Carnot’s theorem.
REFERENCES
1.
S.
Carnot
, Reflexions sur la puissance motrice du feu
(ChexzBachelier Libraire, Quai des Augustins
, Paris
, 1824
).2.
B. P. E.
Clapeyron
, “Memoire sur la puissance motrice de la chaleur
,” J. Ec. R. Polytech.
14
, 153
–190
(1834
).3.
W.
Thomson
, “On an absolute thermometric scale founded on Carnot’s theory of the motive power of heat
,” in Cambridge Philosophical Society Proceedings, June 5
(Cambridge Philosophical Society
, 1848
), Vol. 100–106.4.
R.
Clausius
, “Ueber die bewegende kraft der wärme und die gesetze, welche sich daraus für die wärmelehre selbst ableiten lassen
,” Ann. Phys.
155
, 500
–524
(1850
).5.
R.
Clausius
, “Ueber die anwendung der mechanischen wärmetheorie auf die dampfmaschine
,” Ann. Phys.
173
, 513
–558
(1856
).6.
R.
Clausius
, “Ueber verschiedene für die anwendung bequeme formen der hauptgleichungen der mechanischen wärmetheorie
,” Ann. Phys.
201
, 353
–400
(1865
).7.
Thermodynamics in the Quantum Regime
, edited by F.
Binder
, L. A.
Correa
, G.
Gogolin
, J.
Anders
, and G.
Adesso
(Springer International Publishing
, 2018
).8.
R.
Kosloff
, “Quantum thermodynamics: A dynamical viewpoint
,” Entropy
15
, 2100
–2128
(2013
).9.
N.
Lambert
, S.
Ahmed
, M.
Cirio
, and F.
Nori
, “Modelling the ultra-strongly coupled spin-boson model with unphysical modes
,” Nat. Commun.
10
, 3721
(2019
).10.
R.
Härtle
, G.
Cohen
, D. R.
Reichman
, and A. J.
Millis
, “Decoherence and lead-induced interdot coupling in nonequilibrium electron transport through interacting quantum dots: A hierarchical quantum master equation approach
,” Phys. Rev. B
88
, 235426
(2013
).11.
M. F.
Ludovico
, J. S.
Lim
, M.
Moskalets
, L.
Arrachea
, and D.
Sánchez
, “Dynamical energy transfer in ac-driven quantum systems
,” Phys. Rev. B
89
, 161306
(2014
).12.
M.
Esposito
, M. A.
Ochoa
, and M.
Galperin
, “Quantum thermodynamics: A nonequilibrium Green’s function approach
,” Phys. Rev. Lett.
114
, 080602
(2015
).13.
G.
Cohen
and M.
Galperin
, “Green’s function methods for single molecule junctions
,” J. Chem. Phys.
152
, 090901
(2020
).14.
F.
Brandão
, M.
Horodecki
, N.
Ng
, J.
Oppenheim
, and S.
Wehner
, “The second laws of quantum thermodynamics
,” Proc. Natl. Acad. Sci. U. S. A.
112
, 3275
–3279
(2015
).15.
P.
Haughian
, M.
Esposito
, and T. L.
Schmidt
, “Quantum thermodynamics of the resonant-level model with driven system-bath coupling
,” Phys. Rev. B
97
, 085435
(2018
).16.
M.
Perarnau-Llobet
, H.
Wilming
, A.
Riera
, R.
Gallego
, and J.
Eisert
, “Strong coupling corrections in quantum thermodynamics
,” Phys. Rev. Lett.
120
, 120602
(2018
).17.
K.
Funo
and H. T.
Quan
, “Path integral approach to heat in quantum thermodynamics
,” Phys. Rev. E
98
, 012113
(2018
).18.
Á.
Rivas
, “Strong coupling thermodynamics of open quantum systems
,” Phys. Rev. Lett.
124
, 160601
(2020
).19.
L. M.
Cangemi
, V.
Cataudella
, G.
Benenti
, M.
Sassetti
, and G.
De Filippis
, “Violation of thermodynamics uncertainty relations in a periodically driven work-to-work converter from weak to strong dissipation
,” Phys. Rev. B
102
, 165418
(2020
).20.
P.
Talkner
and P.
Hänggi
, “Colloquium: Statistical mechanics and thermodynamics at strong coupling: Quantum and classical
,” Rev. Mod. Phys.
92
, 041002
(2020
).21.
A. M.
Zagoskin
, E.
Il’ichev
, and F.
Nori
, “Heat cost of parametric generation of microwave squeezed states
,” Phys. Rev. A
85
, 063811
(2012
).22.
A. M.
Zagoskin
, S.
Savel’ev
, F.
Nori
, and F. V.
Kusmartsev
, “Squeezing as the source of inefficiency in the quantum Otto cycle
,” Phys. Rev. B
86
, 014501
(2012
).23.
A.
Guthrie
, C. D.
Satrya
, Y.-C.
Chang
, P.
Menczel
, F.
Nori
, and J. P.
Pekola
, “Cooper-pair box coupled to two resonators: An architecture for a quantum refrigerator
,” Phys. Rev. Appl.
17
, 064022
(2022
).24.
K.
Funo
, N.
Lambert
, B.
Karimi
, J. P.
Pekola
, Y.
Masuyama
, and F.
Nori
, “Speeding up a quantum refrigerator via counterdiabatic driving
,” Phys. Rev. B
100
, 035407
(2019
).25.
Y.
Nomura
, N.
Yoshioka
, and F.
Nori
, “Purifying deep Boltzmann machines for thermal quantum states
,” Phys. Rev. Lett.
127
, 060601
(2021
).26.
K.
Funo
, N.
Lambert
, and F.
Nori
, “General bound on the performance of counter-diabatic driving acting on dissipative spin systems
,” Phys. Rev. Lett.
127
, 150401
(2021
).27.
Y.
Lu
, N.
Lambert
, A. F.
Kockum
, K.
Funo
, A.
Bengtsson
, S.
Gasparinetti
, F.
Nori
, and P.
Delsing
, “Steady-state heat transport and work with a single artificial atom coupled to a waveguide: Emission without external driving
,” PRX Q.
3
, 020305
(2022
).28.
S.
An
, J.-N.
Zhang
, M.
Um
, D.
Lv
, Y.
Lu
, J.
Zhang
, Z.-Q.
Yin
, H. T.
Quan
, and K.
Kim
, “Experimental test of the quantum Jarzynski equality with a trapped-ion system
,” Nat. Phys.
11
, 193
(2015
).29.
S.
An
, D.
Lv
, A.
del Campo
, and K.
Kim
, “Shortcuts to adiabaticity by counterdiabatic driving for trapped-ion displacement in phase space
,” Nat. Commun.
7
, 12999
(2016
).30.
I. A.
Martínez
, E.
Roldan
, L.
Dinis
, D.
Petrov
, J. M. R.
Parrondo
, and R. A.
Rica
, “Brownian Carnot engine
,” Nat. Phys.
12
, 67
(2016
).31.
N. V.
Vitanov
, A. A.
Rangelov
, B. W.
Shore
, and K.
Bergmann
, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond
,” Rev. Mod. Phys.
89
, 015006
(2017
).32.
E. A.
Hoffmann
, S.
Fahlvik
, C.
Thelander
, M.
Leijnse
, and H.
Linke
, “A quantum-dot heat engine operating close to the thermodynamic efficiency limits
,” Nat. Nanotechnol.
13
, 920
–924
(2018
).33.
J.
Bengtsson
, M. N.
Tengstrand
, A.
Wacker
, P.
Samuelsson
, M.
Ueda
, H.
Linke
, and S. M.
Reimann
, “Quantum Szilard engine with attractively interacting Bosons
,” Phys. Rev. Lett.
120
, 100601
(2018
).34.
M.
Josefsson
, A.
Svilans
, H.
Linke
, and M.
Leijnse
, “Optimal power and efficiency of single quantum dot heat engines: Theory and experiment
,” Phys. Rev. B
99
, 235432
(2019
).35.
D.
Prete
, P. A.
Erdman
, V.
Demontis
, V.
Zannier
, D.
Ercolani
, L.
Sorba
, F.
Beltram
, F.
Rossella
, F.
Taddei
, and S.
Roddaro
, “Thermoelectric conversion at 30 K in InAs/InP nanowire quantum dots
,” Nano Lett.
19
, 3033
–3039
(2019
).36.
Y.-H.
Ma
, R.-X.
Zhai
, J.
Chen
, C. P.
Sun
, and H.
Dong
, “Experimental test of the 1/τ-scaling entropy generation in finite-time thermodynamics
,” Phys. Rev. Lett.
125
, 210601
(2020
).37.
A. G.
de Oliveira
, R. M.
Gomes
, V. C. C.
Brasil
, N.
Rubiano da Silva
, L. C.
Céleri
, and P. H.
Souto Ribeiro
, “Full thermalization of a photonic qubit
,” Phys. Lett. A
384
, 126933
(2020
).38.
S.
Hernández-Gómez
, N.
Staudenmaier
, M.
Campisi
, and N.
Fabbri
, “Experimental test of fluctuation relations for driven open quantum systems with an NV center
,” New J. Phys.
23
, 065004
(2021
).39.
S.
Ryu
, R.
López
, L.
Serra
, and D.
Sánchez
, “Beating Carnot efficiency with periodically driven chiral conductorse
,” Nat. Commun.
13
, 2512
(2022
).40.
Y.
Tanimura
, “Stochastic Liouville, Langevin, Fokker–Planck, and master equation qpproaches to quantum dissipative systems
,” J. Phys. Soc. Jpn.
75
, 082001
(2006
).41.
Y.
Tanimura
, “Numerically ‘exact’ approach to open quantum dynamics: The hierarchical equations of motion (HEOM)
,” J. Chem. Phys.
153
, 020901
(2020
).42.
Y.
Tanimura
, “Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities
,” J. Chem. Phys.
141
, 044114
(2014
).43.
Y.
Tanimura
, “Real-time and imaginary-time quantum hierarchal Fokker-Planck equations
,” J. Chem. Phys.
142
, 144110
(2015
).44.
D.
Segal
, A.
Nitzan
, and P.
Hänggi
, “Thermal conductance through molecular wires
,” J. Chem. Phys.
119
, 6840
–6855
(2003
).45.
D.
Segal
and A.
Nitzan
, “Spin-boson thermal rectifier
,” Phys. Rev. Lett.
94
, 034301
(2005
).46.
D.
Segal
, “Heat flow in nonlinear molecular junctions: Master equation analysis
,” Phys. Rev. B
73
, 205415
(2006
).47.
N.
Boudjada
and D.
Segal
, “From dissipative dynamics to studies of heat transfer at the nanoscale: Analysis of the spin-Boson model
,” J. Phys. Chem. A
118
, 11323
–11336
(2014
).48.
A.
Dhar
, “Heat transport in low-dimensional systems
,” Adv. Phys.
57
, 457
–537
(2008
).49.
K. A.
Velizhanin
, H.
Wang
, and M.
Thoss
, “Heat transport through model molecular junctions: A multilayer multiconfiguration time-dependent Hartree approach
,” Chem. Phys. Lett.
460
, 325
–330
(2008
).50.
Y.
Dubi
and M.
Di Ventra
, “Colloquium: Heat flow and thermoelectricity in atomic and molecular junctions
,” Rev. Mod. Phys.
83
, 131
–155
(2011
).51.
C.
Wang
, J.
Ren
, and J.
Cao
, “Nonequilibrium energy transfer at nanoscale: A unified theory from weak to strong coupling
,” Sci. Rep.
5
, 11787
(2015
).52.
K.
Saito
and T.
Kato
, “Kondo signature in heat transfer via a local two-state system
,” Phys. Rev. Lett.
111
, 214301
(2013
).53.
L.
Song
and Q.
Shi
, “Hierarchical equations of motion method applied to nonequilibrium heat transport in model molecular junctions: Transient heat current and high-order moments of the current operator
,” Phys. Rev. B
95
, 064308
(2017
).54.
E.
Aurell
and F.
Montana
, “Thermal power of heat flow through a qubit
,” Phys. Rev. E
99
, 042130
(2019
).55.
M.
Kilgour
, B. K.
Agarwalla
, and D.
Segal
, “Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach
,” J. Chem. Phys.
150
, 084111
(2019
).56.
M.
Xu
, J. T.
Stockburger
, and J.
Ankerhold
, “Heat transport through a superconducting artificial atom
,” Phys. Rev. B
103
, 104304
(2021
).57.
A.
Kato
and Y.
Tanimura
, “Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence
,” J. Chem. Phys.
143
, 064107
(2015
).58.
M.
Esposito
, M. A.
Ochoa
, and M.
Galperin
, “Nature of heat in strongly coupled open quantum systems
,” Phys. Rev. B
92
, 235440
(2015
).59.
H. T.
Quan
, Y.-x.
Liu
, C. P.
Sun
, and F.
Nori
, “Quantum thermodynamic cycles and quantum heat engines
,” Phys. Rev. E
76
, 031105
(2007
).60.
N.
Brunner
, M.
Huber
, N.
Linden
, S.
Popescu
, R.
Silva
, and P.
Skrzypczyk
, “Entanglement enhances cooling in microscopic quantum refrigerators
,” Phys. Rev. E
89
, 032115
(2014
).61.
L.
Chotorlishvili
, Z.
Toklikishvili
, and J.
Berakdar
, “Thermal entanglement and efficiency of the quantum Otto cycle for the su(1,1) Tavis–Cummings system
,” J. Phys. A: Math. Theor.
44
, 165303
(2011
).62.
R.
Kosloff
and A.
Levy
, “Quantum heat engines and refrigerators: Continuous devices
,” Annu. Rev. Phys. Chem.
65
, 365
–393
(2014
).63.
R.
Uzdin
, A.
Levy
, and R.
Kosloff
, “Equivalence of quantum heat machines, and quantum-thermodynamic signatures
,” Phys. Rev. X
5
, 031044
(2015
).64.
A.
Kato
and Y.
Tanimura
, “Quantum heat current under non-perturbative and non-Markovian conditions: Applications to heat machines
,” J. Chem. Phys.
145
, 224105
(2016
).65.
Y. Y.
Xu
, B.
Chen
, and J.
Liu
, “Achieving the classical Carnot efficiency in a strongly coupled quantum heat engine
,” Phys. Rev. E
97
, 022130
(2018
).66.
S.
Restrepo
, J.
Cerrillo
, P.
Strasberg
, and G.
Schaller
, “From quantum heat engines to laser cooling: Floquet theory beyond the Born–Markov approximation
,” New J. Phys.
20
, 053063
(2018
).67.
M.
Kilgour
and D.
Segal
, “Coherence and decoherence in quantum absorption refrigerators
,” Phys. Rev. E
98
, 012117
(2018
).68.
A.
Das
and V.
Mukherjee
, “Quantum-enhanced finite-time Otto cycle
,” Phys. Rev. Res.
2
, 033083
(2020
).69.
P. A.
Camati
, J. F. G.
Santos
, and R. M.
Serra
, “Employing non-Markovian effects to improve the performance of a quantum Otto refrigerator
,” Phys. Rev. A
102
, 012217
(2020
).70.
R. S.
Johal
and V.
Mehta
, “Quantum heat engines with complex working media, complete Otto cycles and heuristics
,” Entropy
23
, 1149
(2021
).71.
K.
Ono
, S. N.
Shevchenko
, T.
Mori
, S.
Moriyama
, and F.
Nori
, “Analog of a quantum heat engine using a single-spin qubit
,” Phys. Rev. Lett.
125
, 166802
(2020
).72.
M.
Wiedmann
, J. T.
Stockburger
, and J.
Ankerhold
, “Non-Markovian dynamics of a quantum heat engine: Out-of-equilibrium operation and thermal coupling control
,” New J. Phys.
22
, 033007
(2020
).73.
M.
Wiedmann
, J. T.
Stockburger
, and J.
Ankerhold
, “Non-Markovian quantum Otto refrigerator
,” Eur. Phys. J. Spec. Tops
230
, 851
–857
(2021
).74.
M.
Xu
, J. T.
Stockburger
, G.
Kurizki
, and J.
Ankerhold
, “Minimal quantum thermal machine in a bandgap environment: Non-Markovian features and anti-Zeno advantage
,” New J. Phys.
24
, 035003
(2022
).75.
M.
Brenes
, J. J.
Mendoza-Arenas
, A.
Purkayastha
, M. T.
Mitchison
, S. R.
Clark
, and J.
Goold
, “Tensor-network method to simulate strongly interacting quantum thermal machines
,” Phys. Rev. X
10
, 031040
(2020
).76.
N.
Pancotti
, M.
Scandi
, M. T.
Mitchison
, and M.
Perarnau-Llobet
, “Speed-ups to isothermality: Enhanced quantum thermal machines through control of the system–bath coupling
,” Phys. Rev. X
10
, 031015
(2020
).77.
S.
Lee
, M.
Ha
, J. M.
Park
, and H.
Jeong
, “Finite-time quantum Otto engine: Surpassing the quasistatic efficiency due to friction
,” Phys. Rev. E
101
, 022127
(2020
).78.
S.
Hamedani Raja
, S.
Maniscalco
, G. S.
Paraoanu
, J. P.
Pekola
, and N.
Lo Gullo
, “Finite-time quantum Stirling heat engine
,” New J. Phys.
23
, 033034
(2021
).79.
F.
Ivander
, N.
Anto-Sztrikacs
, and D.
Segal
, “Strong system–bath coupling effects in quantum absorption refrigerators
,” Phys. Rev. E
105
, 034112
(2022
).80.
K.
Goyal
, X.
He
, and R.
Kawai
, “Entropy production of a small quantum system under strong coupling with an environment: A computational experiment
,” Physica A
552
, 122627
(2020
).81.
N.
Seshadri
and M.
Galperin
, “Entropy and information flow in quantum systems strongly coupled to baths
,” Phys. Rev. B
103
, 085415
(2021
).82.
S.
Sakamoto
and Y.
Tanimura
, “Numerically ‘exact’ simulations of entropy production in the fully quantum regime: Boltzmann entropy vs von Neumann entropy
,” J. Chem. Phys.
153
, 234107
(2020
).83.
R.
Schmidt
, M. F.
Carusela
, J. P.
Pekola
, S.
Suomela
, and J.
Ankerhold
, “Work and heat for two-level systems in dissipative environments: Strong driving and non-Markovian dynamics
,” Phys. Rev. B
91
, 224303
(2015
).84.
J.
Goold
, M.
Huber
, A.
Riera
, L. d.
Rio
, and P.
Skrzypczyk
, “The role of quantum information in thermodynamics—A topical review
,” J. Phys. A: Math. Theor.
49
, 143001
(2016
).85.
P.
Strasberg
, G.
Schaller
, T.
Brandes
, and M.
Esposito
, “Quantum and information thermodynamics: A unifying framework based on repeated interactions
,” Phys. Rev. X
7
, 021003
(2017
).86.
C.
Jarzynski
, “Nonequilibrium work theorem for a system strongly coupled to a thermal environment
,” J. Stat. Mech.: Theory Exp.
2004
, P09005
.87.
S.
Yukawa
, “A quantum analogue of the Jarzynski equality
,” J. Phys. Soc. Jpn.
69
, 2367
–2370
(2000
).88.
S.
Sakamoto
and Y.
Tanimura
, “Open quantum dynamics theory for non-equilibrium work: Hierarchical equations of motion approach
,” J. Phys. Soc. Jpn.
90
, 033001
(2021
).89.
M.
Campisi
, P.
Talkner
, and P.
Hänggi
, “Fluctuation theorem for arbitrary open quantum systems
,” Phys. Rev. Lett.
102
, 210401
(2009
).90.
M.
Esposito
, U.
Harbola
, and S.
Mukamel
, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems
,” Rev. Mod. Phys.
81
, 1665
–1702
(2009
).91.
U.
Seifert
, “Stochastic thermodynamics, fluctuation theorems and molecular machines
,” Rep. Prog. Phys.
75
, 126001
(2012
).92.
M.
Campisi
, P.
Hänggi
, and P.
Talkner
, “Colloquium: Quantum fluctuation relations: Foundations and applications
,” Rev. Mod. Phys.
83
, 771
–791
(2011
).93.
E.
Aurell
, B.
Donvil
, and K.
Mallick
, “Large deviations and fluctuation theorem for the quantum heat current in the spin-boson model
,” Phys. Rev. E
101
, 052116
(2020
).94.
R. S.
Whitney
, “Non-Markovian quantum thermodynamics: Laws and fluctuation theorems
,” Phys. Rev. B
98
, 085415
(2018
).95.
H. T.
Quan
, Y. D.
Wang
, Y.-x.
Liu
, C. P.
Sun
, and F.
Nori
, “Maxwell’s demon assisted thermodynamic cycle in superconducting quantum circuits
,” Phys. Rev. Lett.
97
, 180402
(2006
).96.
K.
Maruyama
, F.
Nori
, and V.
Vedral
, “Colloquium: The physics of Maxwell’s demon and information
,” Rev. Mod. Phys.
81
, 1
–23
(2009
).97.
H.
Leff
and A. F.
Rex
, Maxwell’s Demon 2 Entropy, Classical and Quantum Information, Computing
(CRC Press
, 2002
).98.
Y.
Tanimura
and R.
Kubo
, “Time evolution of a quantum system in contact with a nearly Gaussian-Markoffian noise bath
,” J. Phys. Soc. Jpn.
58
, 101
–114
(1989
).99.
Y.
Tanimura
, “Nonperturbative expansion method for a quantum system coupled to a harmonic-oscillator bath
,” Phys. Rev. A
41
, 6676
–6687
(1990
).100.
A.
Ishizaki
and Y.
Tanimura
, “Quantum dynamics of system strongly coupled to low-temperature colored noise bath: Reduced hierarchy equations approach
,” J. Phys. Soc. Jpn.
74
, 3131
–3134
(2005
).101.
S.
Koyanagi
and Y.
Tanimura
, “The laws of thermodynamics for quantum dissipative systems: A quasi-equilibrium Helmholtz energy approach
,” J. Chem. Phys.
157
, 014104
(2022
).102.
J.
Hu
, R.-X.
Xu
, and Y.
Yan
, “Communication: Padé spectrum decomposition of Fermi function and Bose function
,” J. Chem. Phys.
133
, 101106
(2010
).103.
A.
Lenard
, “Thermodynamical proof of the Gibbs formula for elementary quantum systems
,” J. Stat. Phys.
19
, 575
(1978
).104.
H.
Tasaki
, “Jarzynski relations for quantum systems and some applications
,” arXiv:cond-mat/0009244 [cond-mat.stat-mech] (2000
).105.
Y.
Oono
, Perspectives on Statistical Thermodynamics
(Cambridge University Press
, 2017
).106.
J.
Zhang
and Y.
Tanimura
, “Imaginary-time hierarchical equations of motion for thermodynamic variables
,” J. Chem. Phys.
156
, 174112
(2022
).107.
A.
Ishizaki
and G. R.
Fleming
, “On the adequacy of the Redfield equation and related approaches to the study of quantum dynamics in electronic energy transfer
,” J. Chem. Phys.
130
, 234110
(2009
).108.
Y.
Tanimura
and P. G.
Wolynes
, “The interplay of tunneling, resonance, and dissipation in quantum barrier crossing: A numerical study
,” J. Chem. Phys.
96
, 8485
–8496
(1992
).109.
T.
Ikeda
and Y.
Tanimura
, “Low-temperature quantum Fokker–Planck and Smoluchowski equations and their extension to multistate systems
,” J. Chem. Theory Comput.
15
, 2517
–2534
(2019
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
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