In order to construct time-dependent pseudo-bosonic coherent states, first, we extend the non-Hermitian integrals of motion method to cases where the quantum systems are described by time-dependent non-Hermitian Hamiltonians; second, we introduce a pseudo-bosonic annihilation operator defined as a time-dependent non-Hermitian linear invariant. The pseudo-bosons operators are a pseudo-Hermitian extension of usual boson operators. In fact, they are obtained from the modification of usual boson commutation relations where the annihilation and creation operators are related to their adjoint operators via the bounded Hermitian invertible operator or metric operator. Thus, the pseudo-bosonic coherent states are just obtained as eigenstates of the pseudo-bosonic annihilation operator. As an illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian and we compare the obtained results with those in the literature (Swanson model but time-independent).

1.
E.
Schrödinger
, “
Der stetige Übergang von der Mikro-zur Makromechanik
,”
Naturwissenschaften
14
,
664
(
1926
).
2.
R. J.
Glauber
, “
The quantum theory of optical coherence
,”
130
,
2529
(
1963
);
R. J.
Glauber
, “
Photon correlations
,”
10
,
84
(
1963
);
R. J.
Glauber
, “
Coherent and incoherent states of the radiation field
,”
Phys. Rev.
131
,
2766
(
1963
).
3.
J. R.
Klauder
, “
Continuous-representation theory. I. Postulates of continuous-representation theory
,”
4
,
1055
(
1963
);
J. R.
Klauder
, “
Continuous-representation theory. II. Generalized relation between quantum and classical dynamics
,”
J. Math. Phys.
4
,
1058
(
1963
).
4.
E. C. G.
Sudarshan
, “
Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams
,”
Phys. Rev. Lett.
10
,
277
(
1963
).
5.
I. A.
Malkin
,
V. I.
Man’ko
, and
D. A.
Trifonov
, “
Coherent states and transition probabilities in a time-dependent electromagnetic field
,”
Phys. Rev. D
2
,
1371
(
1970
).
6.
I. A.
Malkin
,
V. I.
Man’ko
, and
D. A.
Trifonov
, “
Linear adiabatic invariants and coherent states
,”
J. Math. Phys.
14
,
576
(
1973
).
7.
V. V.
Dodonov
,
I. A.
Malkin
, and
V. I.
Man’ko
, “
Integrals of the motion, Green functions and coherent states of dynamical systems
,”
Int. J. Theor. Phys.
14
,
37
(
1975
).
8.
V. V.
Dodonov
and
V. I.
Man’ko
, “
Coherent states and the resonance of a quantum damped oscillator
,”
Phys. Rev. A
20
,
550
(
1979
).
9.
V. G.
Bagrov
,
D. M.
Gitman
,
E. S.
Macedo
, and
A. S.
Pereira
, “
Coherent states of inverse oscillators and related problems
,”
J. Phys. A: Math. Theor.
46
,
325305
(
2013
).
10.
F. G.
Scholz
,
H. B.
Geyer
, and
F. J.
Hahne
, “
Quasi-Hermitian operators in quantum mechanics and the variational principle
,”
Ann. Phys.
213
,
74
(
1992
).
11.
C. M.
Bender
, “
Making sense of non-hermitian Hamiltonians
,”
Rep. Prog. Phys.
70
,
947
(
2007
).
12.
A.
Mostafazadeh
, “
Pseudo-Hermitian representation of quantum mechanics
,”
Int. J. Geom. Methods Mod. Phys.
07
,
1191
(
2010
).
13.
D. A.
Trifonov
, “
Pseudo-boson coherent and Fock states
,” in
Differential Geometry, Complex Analysis and Mathematical Physics
, edited by
K.
Sekigawa
, et al.
(
World Scientific
,
Singapore
,
2009
), pp.
241
250
; arXiv:0902.3744.
14.
F.
Bagarello
, “
Pseudobosons, Riesz bases, and coherent states
,”
J. Math. Phys.
51
,
023531
(
2010
).
15.
F.
Bagarello
, “
(Regular) pseudo-bosons versus boson
,”
J. Phys. A: Math. Gen.
44
,
015205
(
2011
).
16.
F.
Bagarello
, “
Linear pseudo-fermions
,”
J. Phys. A: Math. Theor.
45
,
444002
(
2012
).
17.
F.
Bagarello
, “
Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states
,”
J. Math. Phys.
57
,
103501
(
2016
).
18.
F.
Bagarello
and
A.
Fring
, “
From pseudo-bosons to pseudo-Hermiticity via multiple generalized Bogoliubov transformations
,”
Int. J. Mod. Phys. B
31
,
1750085
(
2017
).
19.
C.
Figueira de Morisson Faria
and
A.
Fring
, “
Time evolution of non-Hermitian Hamiltonian systems
,”
J. Phys. A: Math. Gen.
39
,
9269
(
2006
).
20.
C.
Figueira de Morisson Faria
and
A.
Fring
, “
Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: From the time-independent to the time-dependent quantum mechanical formulation
,”
Laser Phys.
17
,
424
(
2007
).
21.
A.
Mostafazadeh
, “
Time-dependent pseudo-Hermitian Hamiltonians defining a unitary quantum system and uniqueness of the metric operator
,”
Phys. Lett. B
650
,
208
(
2007
).
22.
M.
Znojil
, “
Time-dependent version of crypto-Hermitian quantum theory
,”
Phys. Rev. D
78
,
085003
(
2008
).
23.
M.
Znojil
, “
Three-Hilbert-space formulation of quantum mechanics
,”
SIGMA
5
,
001
(
2009
).
24.
H.
Bila
, “
Adiabatic time-dependent metrics in PT-symmetric quantum theories
,” arXiv:0902.0474.
25.
J.
Gong
and
Q.-h.
Wang
, “
Geometric phase in PT-symmetric quantum mechanics
,”
Phys. Rev. A
82
,
012103
(
2010
).
26.
J.
Gong
and
Q.-h.
Wang
, “
Time-dependent PT-symmetric quantum mechanics
,”
J. Phys. A: Math. Theor.
46
,
485302
(
2013
).
27.
M.
Maamache
, “
Periodic pseudo-Hermitian Hamiltonian: Nonadiabatic geometric phase
,”
Phys. Rev. A
92
,
032106
(
2015
).
28.
A.
Fring
and
M. H. Y.
Moussa
, “
Unitary quantum evolution for time-dependent quasi-Hermitian systems with non-observable Hamiltonians
,”
Phys. Rev. A
93
,
042114
(
2016
).
29.
A.
Fring
and
M. H. Y.
Moussa
, “
Non-Hermitian Swanson model with a time-dependent metric
,”
Phys. Rev. A
94
,
042128
(
2016
).
30.
B.
Khantoul
,
A.
Bounames
, and
M.
Maamache
, “
On the invariant method for the time-dependent non-Hermitian Hamiltonians
,”
Eur. Phys. J. Plus
132
,
258
(
2017
).
31.
A.
Fring
and
T.
Frith
, “
Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
,”
Phys. Rev. A
95
,
010102
(
2017
).
32.
F. S.
Luiz
,
M. A.
Pontes
, and
M. H. Y.
Moussa
, “
Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps
,”
Phys. Scr.
95
,
065211
(
2020
).
33.
F. S.
Luiz
,
M. A.
Pontes
, and
M. H. Y.
Moussa
, “
Gauge linked time-dependent non-Hermitian Hamiltonians
,” arXiv:1703.01451.
34.
M.
Maamache
,
O.
Kaltoum Djeghiour
,
N.
Mana
, and
W.
Koussa
, “
Pseudo-invariants theory and real phases for systems with non-Hermitian time-dependent Hamiltonians
,”
Eur. Phys. J. Plus.
132
,
383
(
2017
).
35.
W.
Koussa
,
N.
Mana
,
O. K.
Djeghiour
, and
M.
Maamache
, “
The pseudo Hermitian invariant operator and time-dependent non-Hermitian Hamiltonian exhibiting a SU(1, 1) and SU(2) dynamical symmetry
,”
J. Math. Phys.
59
,
072103
(
2018
).
36.
M.
Maamache
, “
Non-unitary transformation of quantum time-dependent non-Hermitian systems
,”
Acta Polytech.
57
,
424
(
2017
).
37.
H. R.
Lewis
and
W. B.
Riesenfeld
, “
An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field
,”
J. Math. Phys.
10
,
1458
(
1969
).
38.
W.
Koussa
,
M.
Attia
, and
M.
Maamache
, “
Pseudo-fermionic coherent states with time-dependent metric
,”
J. Math. Phys.
61
,
042101
(
2020
).
39.
M. S.
Swanson
, “
Transition elements for a non-Hermitian quadratic Hamiltonian
,”
J. Math. Phys.
45
,
585
(
2004
).
40.
Z.
Ahmed
, “
Pseudo-Hermiticity of Hamiltonians under gauge-like transformation: Real spectrum of non-Hermitian Hamiltonians
,”
Phys. Lett. A
294
,
287
(
2002
).
41.
H. F.
Jones
, “
On pseudo-Hermitian Hamiltonians and their Hermitian counterparts
,”
J. Phys. A: Math. Theor.
38
,
1741
(
2005
).
42.
B.
Bagchi
,
C.
Quesne
, and
R.
Roychoudhury
, “
Pseudo-Hermiticity and some consequences of a generalized quantum condition
,”
J. Phys. A: Math. Gen.
38
,
L647
(
2005
).
43.
D. P.
Musumbu
,
H. B.
Geyer
, and
W. D.
Heiss
, “
Choice of a metric for the non-Hermitian oscillator
,”
J. Phys. A: Math. Theor.
40
,
F75
(
2007
).
44.
C.
Quesne
, “
Non-Hermitian oscillator Hamiltonian and SU(1, 1): A way towards generalizations
,”
J. Phys. A: Math. Theor.
40
,
F745
(
2007
).
45.
A.
Sinha
and
P.
Roy
, “
Generalized Swanson models and their solutions
,”
J. Phys. A: Math. Theor.
40
,
10599
(
2007
).
46.
E.-M.
Graefe
,
H. J.
Korsch
,
A.
Rush
, and
R.
Schubert
, “
Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator
,”
J. Phys. A: Math. Theor.
48
,
055301
(
2015
).
47.
M.
Maamache
,
Y.
Bouguerra
, and
J. R.
Choi
, “
Time behavior of a Gaussian wave packet accompanying the generalized coherent state for the inverted oscillator
,”
Prog. Theor. Exp. Phys.
2016
,
063A01
.
48.
F.
Bagarello
,
F.
Gargano
, and
S.
Spagnolo
, “
Bi-squeezed states arising from pseudo-bosons
,”
J. Phys. A: Math. Theor.
51
,
455204
(
2018
).
49.
F.
Bagarello
,
F.
Gargano
, and
S.
Spagnolo
, “
Two-dimensional non commutative Swanson model and its bi-coherent states
,” in
Geometric Methods in Physics, XXXVI
, Trends in Mathematics, edited by
P.
Kielanowski
,
A.
Odzijiwicz
, and
E.
Previato
(
Birkhäuser
,
2019
), pp.
9
19
.
50.
M.
Maamache
,
J.-P.
Provost
, and
G.
Vallée
, “
Unitary equivalence and phase properties of the quantum parametric and generalized harmonic oscillators
,”
Phys. Rev. A
59
,
1777
(
1999
).
51.
P.
Caldirola
, “
Forze no-conservative nella meccanica quantistica
,”
Nuovo Cimento
18
,
393
(
1941
).
52.
E.
Kanai
, “
On the quantization of the dissipative systems
,”
Prog. Theor. Phys.
3
,
440
(
1948
).
You do not currently have access to this content.