The complete understanding of thermodynamic processes in quantum scales is paramount to develop theoretical models encompassing a broad class of phenomena as well as to design new technological devices in which quantum aspects can be useful in areas such as quantum information and quantum computation. Among several quantum effects, the phase-space noncommutativity, which arises due to a deformed Heisenberg–Weyl algebra, is of fundamental relevance in quantum systems where quantum signatures and high energy physics play important roles. In low energy physics, however, it may be relevant to address how a quantum deformed algebra could influence some general thermodynamic protocols, employing the well-known noncommutative quantum mechanics in phase-space. In this work, we investigate the heat flow of two interacting quantum systems in the perspective of noncommutativity phase-space effects and show that by controlling the new constants introduced in the quantum theory, the heat flow from the hot to the cold system may be enhanced, thus decreasing the time required to reach thermal equilibrium. We also give a brief discussion on the robustness of the second law of thermodynamics in the context of noncommutative quantum mechanics.

1.
G.
Adesso
,
S.
Ragy
, and
A. R.
Lee
, “
Continuous variable quantum information: Gaussian states and beyond
,”
Open Syst. Inf. Dyn.
21
,
1440001
(
2014
).
2.
X.
Wang
,
T.
Hiroshima
,
A.
Tomita
, and
M.
Hayashi
, “
Quantum information with Gaussian states
,”
Phys. Rep.
448
,
1
(
2007
).
3.
R.
Nichols
,
P.
Liuzzo-Scorpo
,
P. A.
Knott
, and
G.
Adesso
, “
Multiparameter Gaussian quantum metrology
,”
Phys. Rev. A
98
,
012114
(
2018
).
4.
Y.
Yang
, “
Memory effects in quantum metrology
,”
Phys. Rev. Lett.
123
,
110501
(
2019
).
5.
J. P.
Dowling
and
K. P.
Seshadreesan
, “
Quantum optical technologies for metrology, sensing and imaging
,”
J. Light. Tech.
33
,
2359
(
2015
).
6.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information
(
Cambridge University Press
,
Cambridge
,
2010
).
7.
F.
Arute
,
K.
Arya
,
R.
Babbush
 et al, “
Quantum supremacy using a programmable superconducting processor
,”
Nature
574
,
505
510
(
2019
).
8.
K.
Wright
,
K. M.
Beck
,
S.
Debnath
 et al, “
Benchmarking an 11-qubit quantum computer
,”
Nat. Commun.
10
,
5464
(
2019
).
9.
P. A.
Camati
,
J. F. G.
Santos
, and
R. M.
Serra
, “
Coherence effects in the performance of the quantum Otto heat engine
,”
Phys. Rev. A
99
,
062103
(
2019
).
10.
R.
Dann
and
R.
Kosloff
, “
Quantum signatures in the quantum carnot cycle
,”
New J. Phys.
22
,
013055
(
2020
).
11.
O.
Abah
and
M.
Paternostro
, “
Implications of non-Markovian dynamics on information-driven engine
,”
J. Phys. Commun.
4
,
085016
(
2020
).
12.
J. P. S.
Peterson
,
T. B.
Batalhão
,
M.
Herrera
,
A. M.
Souza
,
R. S.
Sarthour
,
I. S.
Oliveira
, and
R. M.
Serra
, “
Experimental characterization of a spin quantum heat engine
,”
Phys. Rev. Lett.
123
,
240601
(
2019
).
13.
J.
Klatzow
,
J. N.
Becker
,
P. M.
Ledingham
 et al, “
Experimental demonstration of quantum effects in the operation of microscopic heat engines
,”
Phys. Rev. Lett.
122
,
110601
(
2019
).
14.
J. F. G.
Santos
, “
Gravitational quantum well as an effective quantum heat engine
,”
Eur. Phys. J. Plus
133
,
321
(
2018
).
15.
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
).
16.
M. J.
Kastoryano
,
F.
Reiter
, and
A. S.
Sørensen
, “
Dissipative preparation of entanglement in optical cavities
,”
Phys. Rev. Lett.
106
,
090502
(
2011
).
17.
J. M.
Raimond
,
M.
Brune
, and
S.
Haroche
, “
Colloquium: Manipulating quantum entanglement with atoms and photons in a cavity
,”
Rev. Mod. Phys.
73
,
565
(
2001
).
18.
D. F.
Walls
and
G. J.
Milburn
,
Quantum Optics
(
Springer
,
Berlin
,
2008
).
19.
K.
Zhang
,
F.
Bariani
, and
P.
Meystre
, “
Theory of an optomechanical quantum heat engine
,”
Phys. Rev. A
90
,
023819
(
2014
).
20.
T. P.
Purdy
,
P.-L.
Yu
,
R. W.
Peterson
,
N. S.
Kampel
, and
C. A.
Regal
, “
Strong optomechanical squeezing of ligh
,”
Phys. Rev. X
3
,
031012
(
2013
).
21.
Y.-D.
Wang
and
A. A.
Clerk
, “
Reservoir-engineered entanglement in optomechanical systems
,”
Phys. Rev. Lett.
110
,
253601
(
2013
).
22.
C. D.
Bruzewicz
,
J.
Chiaverini
,
R.
McConnell
, and
J. M.
Sage
, “
Trapped-ion quantum computing: Progress and challenges
,”
Appl. Phys. Rev.
6
,
021314
(
2019
).
23.
R.
Clausius
,
The Mechanical Theory of Heat
(
MacMillan
,
London
,
1879
).
24.
M. J.
de Oliveira
, “
Heat transport along a chain of coupled quantum harmonic oscillators
,”
Phys. Rev. E
95
,
042113
(
2017
).
25.
S.
Iubini
,
P.
Di Cintio
,
S.
Lepri
,
R.
Livi
, and
L.
Casetti
, “
Heat transport in oscillator chains with long-range interactions coupled to thermal reservoirs
,”
Phys. Rev. E
97
,
032107
(
2018
).
26.
A.
Xuereb
,
A.
Imparato
, and
A.
Dantan
, “
Heat transport in harmonic oscillator systems with thermal baths: Application to optomechanical arrays
,”
New J. Phys.
17
,
055013
(
2015
).
27.
W. T. B.
Malouf
,
J. P.
Santos
,
L. A.
Correa
,
M.
Paternostro
, and
G. T.
Landi
, “
Wigner entropy production and heat transport in linear quantum lattices
,”
Phys. Rev. A
99
,
052104
(
2019
).
28.
B. A. N.
Akasaki
,
M. J.
de Oliveira
, and
C. E.
Fiore
, “
Entropy production and heat transport in harmonic chains under time dependent periodic drivings
,”
Phys. Rev. E
101
,
012132
(
2020
).
29.
R.
Horodecki
,
P.
Horodecki
,
M.
Horodecki
, and
K.
Horodecki
, “
Quantum entanglement
,”
Rev. Mod. Phys.
81
,
865
(
2009
).
30.
A.
Streltsov
,
G.
Adesso
, and
M. B.
Plenio
, “
Colloquium: Quantum coherence as a resource
,”
Rev. Mod. Phys.
89
,
041003
(
2017
).
31.
H.-P.
Breuer
,
E.-M.
Laine
,
J.
Piilo
, and
B.
Vacchini
, “
Colloquium: Non-Markovian dynamics in open quantum systems
,”
Rev. Mod. Phys.
88
,
021002
(
2016
).
32.
K.
Modi
,
A.
Brodutch
,
H.
Cable
,
T.
Paterek
, and
V.
Vedral
, “
The classical-quantum boundary for correlations: Discord and related measures
,”
Rev. Mod. Phys.
84
,
1655
(
2012
).
33.
H. S.
Snyder
, “
Quantized space-time
,”
Phys. Rev.
71
,
38
(
1946
).
34.
S.
Doplicher
,
K.
Fredenhagen
, and
J. E.
Roberts
, “
The quantum structure of spacetime at the Planck scale and quantum fields
,”
Commun. Math. Phys.
172
,
187
(
1995
).
35.
N.
Seiberg
, “
Emergent spacetime
,” in
The Quantum Structure of Space and Time, The - Proceedings Of The 23rd Solvay Conference in Physics
(
World Scientific
,
2006
), pp.
163
178
.
36.
A. E.
Bernardini
and
O.
Bertolami
, “
Probing phase-space noncommutativity through quantum beating, missing information and the thermodynamic limit
,”
Phys. Rev. A
88
,
012101
(
2013
).
37.
M.
Rosenbaum
and
J. D.
Vergara
, “
The star-value equation and Wigner distributions in noncommutative Heisenberg algebras
,”
Gen. Relativ. Gravitation
38
,
607
(
2006
).
38.
O.
Bertolami
,
J. G.
Rosa
,
C. M. L.
de Aragão
,
P.
Castorina
, and
D.
Zappalà
, “
Noncommutative gravitational quantum well
,”
Phys. Rev. D
72
,
025010
(
2005
).
39.
R.
Banerjee
,
B. D.
Roy
, and
S.
Samanta
, “
Remarks on the noncommutative gravitational quantum well
,”
Phys. Rev. D
74
,
045015
(
2006
).
40.
K. P.
Gnatenko
and
V. M.
Tkachuk
, “
Upper bound on the momentum scale in noncommutative phase space of canonical type
,”
Eur. Phys. Lett.
127
,
20008
(
2019
).
41.
L.
Lawson
,
L.
Gouba
, and
G. Y.
Avossevou
, “
Two-dimensional noncommutative gravitational quantum well
,”
J. Phys. A: Math. Theor.
50
,
475202
(
2017
).
42.
P.
Leal
and
O.
Bertolami
, “
Relativistic dispersion relation and putative metric structure in noncommutative phase-space
,”
Phys. Lett. B
793
,
240
(
2019
).
43.
J. F. G.
Santos
and
A. E.
Bernardini
, “
Gaussian fidelity distorted by external fields
,”
Physica A
445
,
75
(
2016
).
44.
O.
Bertolami
,
A. E.
Bernardini
, and
P.
Leal
, “
Quantum information aspects of noncommutative quantum mechanics
,”
J. Phys.: Conf. Ser.
952
,
012016
(
2018
).
45.
P.
Leal
,
A. E.
Bernardini
, and
O.
Bertolami
, “
Quantum cloning and teleportation fidelity in the noncommutative phase-space
,”
J. Phys. A: Math. Theor.
52
,
375302
(
2019
).
46.
M. A. C.
Rossi
,
T.
Giani
, and
M. G. A.
Paris
, “
Probing deformed quantum commutators
,”
Phys. Rev. D
94
,
024014
(
2016
).
47.
P. R.
Giri
and
P.
Roy
, “
Non-Hermitian quantum mechanics in non-commutative space
,”
Eur. Phys. J. C
60
,
157
(
2009
).
48.
J. F. G.
dos Santos
,
F. S.
Luiz
,
O. S.
Duarte
, and
M. H. Y.
Moussa
, “
Non-Hermitian noncommutative quantum mechanics
,”
Eur. Phys. J. Plus
134
,
332
(
2019
).
49.
S.
Dey
,
A.
Fring
, and
L.
Gouba
, “
PT-symmetric noncommutative spaces with minimal volume uncertainty relations
,”
J. Phys. A: Math. Theor.
45
,
385302
(
2012
).
50.
S.
Dey
,
A.
Bhat
,
D.
Momeni
,
M.
Faizal
,
A. F.
Ali
,
T. K.
Dey
, and
A.
Rehman
, “
Probing noncommutative theories with quantum optical experiments
,”
Nucl. Phys. B
924
,
578
587
(
2017
).
51.
M.
Khodadi
,
K.
Nozari
,
S.
Dey
,
A.
Bhat
, and
M.
Faizal
, “
A new bound on polymer quantization via an opto-mechancial setup
,”
Nat. Sci. Rep.
8
,
1659
(
2018
).
52.
S.
Vinjanampathy
and
J.
Anders
, “
Quantum thermodynamics
,”
Contemp. Phys.
57
,
545
(
2016
).
53.
R.
Alicki
and
R.
Kosloff
, “
Introduction to quantum thermodynamics: History and prospects
,” in
Thermodynamics in the Quantum Regime
, edited by F. Binder et al. (
Springer, Cham
,
2019
), pp.
1
33
.
54.
S.
Deffner
and
S.
Campbell
,
Quantum Thermodynamics: An Introduction to the Thermodynamics of Quantum Information
(
Morgan & Claypool
,
2019
).
55.
J. F. G.
Santos
and
A. E.
Bernardini
, “
Quantum engines and the range of the second law of thermodynamics in the noncommutative phase-space
,”
Eur. Phys. J. Plus
132
,
260
(
2017
).
56.
T.
Pandit
,
P.
Chattopadhyay
, and
G.
Paul
, “
Non-commutative space engine: A boost to thermodynamic processes
,” arXiv:1911.13105.
57.
P.
Chattopadhyay
, “
Non-Commutative space: Boon or bane for quantum engines and refrigerators
,”
Eur. Phys. J. Plus
135
,
302
(
2020
).
58.
N. C.
Dias
and
J. N.
Prata
, “
Exact master equation for a noncommutative Brownian particle
,”
Ann. Phys.
324
,
73
(
2009
).
59.
W. O.
Santos
,
G. M. A.
Almeida
, and
A. M. C.
Souza
, “
Noncommutative Brownian motion
,”
Int. J. Mod. Phys. A
32
,
1750146
(
2017
).
60.
J. F. G.
Santos
, “
Noncommutative phase-space effects in thermal diffusion of Gaussian states
,”
J. Phys. A: Math. Theor.
52
,
405306
(
2019
).
61.
M. H.
Partovi
, “
Entanglement versus Stosszahlansatz: Disappearance of the thermodynamic arrow in a high-correlation environment
,”
Phys. Rev. E
77
,
021110
(
2008
).
62.
S.
Jevtic
,
D.
Jennings
, and
T.
Rudolph
, “
Maximally and minimally correlated states attainable within a closed evolving system
,”
Phys. Rev. Lett.
108
,
110403
(
2012
).
63.
S.
Das
and
E. C.
Vagenas
, “
Universality of quantum gravity corrections
,”
Phys. Rev. Lett.
101
,
221301
(
2008
).
64.
K.
Micadei
,
J. P. S.
Peterson
,
A. M.
Souza
 et al, “
Reversing the direction of heat flow using quantum correlations
,”
Nat. Commun.
10
,
2456
(
2019
).
65.
G.
Marcucci
and
C.
Conti
, “
Simulating general relativity and non-commutative geometry by nonparaxial quantumfluids
,”
New J. Phys.
21
,
123038
(
2019
).
66.
J.
Gamboa
,
M.
Loewe
, and
J. C.
Rojas
, “
Noncommutative quantum mechanics
,”
Phys. Rev. D
64
,
067901
(
2001
).
67.
C.
Bastos
,
O.
Bertolami
,
N. C.
Dias
, and
J. N.
Prata
, “
Weyl-Wigner formulation of noncommutative quantum mechanics
,”
J. Math. Phys.
49
,
072101
(
2008
).
68.
C. M.
Rohwer
,
K. G.
Zloshchastiev
,
L.
Gouba
, and
F. G.
Scholtz
, “
Noncommutative quantum mechanics—A perspective on structure and spatial extent
,”
J. Phys. A: Math. Theor.
43
,
345302
(
2010
).
69.
L.
Gouba
, “
A comparative review of four formulations of noncommutative quantum mechanics
,”
Int. J. Mod. Phys.
31
,
1630025
(
2016
).
70.
A.
Saha
,
S.
Gangopadhyay
, and
S.
Saha
, “
Noncommutative quantum mechanics of a harmonic oscillator under linearized gravitational waves
,”
Phys. Rev. D
83
,
025004
(
2011
).
71.
C.
Bastos
,
N. C.
Dias
, and
J. N.
Prata
, “
Wigner measures in noncommutative quantum mechanics
,”
Commun. Math. Phys.
299
,
3
(
2010
).
72.
C.
Bastos
,
O.
Bertolami
,
N.
Dias
, and
J.
Prata
, “
Noncommutative graphene
,”
Int. J. Mod. Phys. A
28
,
16
(
2013
).
73.
J. B.
Geloun
and
F. G.
Scholtz
, “
Coherent states in noncommutative quantum mechanics
,”
J. Math. Phys.
50
,
043505
(
2009
).
74.
T.
Harko
and
S.-D.
Liang
, “
Energy-dependent noncommutative quantum mechanics
,”
Eur. Phys. J. C
79
,
300
(
2019
).
75.
C.
Bastos
,
O.
Bertolami
,
N. C.
Dias
, and
J. N.
Prata
, “
Phase-space noncommutative quantum cosmology
,”
Phys. Rev. D
78
,
023516
(
2008
).
76.
A.
Serafini
,
Quantum Continuous Variables: A Primer of Theoretical Methods
(
CRC Press
,
Boca Raton
,
2017
).
77.
C.
Bastos
and
O.
Bertolami
, “
Berry phase in the gravitational quantum well and the Seiberg–Witten map
,”
Phys. Lett. A
372
,
34
(
2008
).
78.
J. F. G.
Santos
,
A. E.
Bernardini
, and
C.
Bastos
, “
Probing phase-space noncommutativity through quantum mechanics and thermodynamics of free particles and quantum rotors
,”
Physica A
438
,
340
(
2015
).
79.
J.
Gamboa
,
M.
Loewe
, and
J. C.
Rojas
, “
Noncommutative quantum mechanics
,”
Phys. Rev. D
64
,
067901
(
2001
).
80.
L.
Jacak
,
P.
Hawrylak
, and
A.
Wjs
,
Quantum Dots
(
Springer-Verlag
,
1998
).
81.
R.
Simon
,
E. C. G.
Sudarshan
, and
N.
Mukunda
, “
Gaussian-Wigner distributions in quantum mechanics and optics
,”
Phys. Rev. A
36
,
3868
(
1987
).
You do not currently have access to this content.