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Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians

Abstract

We extend the Dodonov–Malkin–Man’ko–Trifonov (DMMT) invariant method (Malkin et al. Phys. Rev. D 2, 1371 1, J. Math. Phys. 14, 576 2; Dodonov et al. Int. J. Theor. Phys. 14, 37 3; Dodonov and Man’ko Phys. Rev. A 20, 550 4) to time-dependent pseudo-fermionic systems by introducing ladder invariant operators (time-dependent integrals of motion) which play the role of time-dependent pseudo-fermionic operators and constructing time-dependent pseudo-fermionic coherent states (PFCS) for such systems. As illustrative example, we study in details the time-dependent parity-time-symmetric two-level system under synchronous combined modulations. We explicitly determine time-dependent pseudo-fermionic operators and construct time-dependent PFCS for this physical system. We show that our approach can be extended to time-dependent pseudo-bosonic systems.

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Appendix: The phase αζ(t) Vanishes

Appendix: The phase αζ(t) Vanishes

Our purpose is to show that the phase αζ(t) given in (23) is equal to zero for any pseudo-fermionic system. From (23), the dynamic and geometric phases are, respectively, given as

$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{d} & =&-\left \langle \psi_{\zeta}(t)\right \vert \eta H(t)\text{\ }\left \vert \psi_{\zeta}(t)\right \rangle , \end{array} $$
(65)
$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{g} & =&i\left \langle \psi_{\zeta}(t)\right \vert \eta \frac{\partial}{\partial t}\text{\ }\left \vert \psi_{\zeta} (t)\right \rangle , \end{array} $$
(66)

where the overdot denotes differentation with respect to time.

From (30) the PHCS \(\left \vert \psi _{_{\zeta }}(t)\right \rangle \) and \(\left \langle \psi _{\zeta }(t)\right \vert \) are given by,

$$ \left \vert \psi_{\zeta}(t)\right \rangle =e^{-\tfrac{1}{2}\zeta^{\ast}\zeta }\left( |\psi_{0}(t)\rangle-\zeta|\psi_{1}(t)\rangle \right) , $$
(67)
$$ \left \langle \psi_{\zeta}(t)\right \vert =e^{-\tfrac{1}{2}\zeta^{\ast}\zeta }\left( \left \langle \psi_{0}(t)\right \vert +\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \right) . $$
(68)

By using the following properties of Grassmann variable,

$$ \zeta|\psi_{0}(t)\rangle=|\psi_{0}(t)\rangle \zeta,\text{ }\zeta|\psi_{1}(t)\rangle=-|\psi_{1}(t)\rangle \zeta, $$

one finds that the phases (65) and (66) are equal to

$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{d} & =&-(1-\zeta^{\ast}\zeta)\left \langle \psi_{0}(t)\right \vert \eta H(t)|\psi_{0}(t)\rangle-\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \eta H(t)|\psi_{0}(t)\rangle \\ & &-\zeta \left \langle \psi_{0}(t)\right \vert \eta H(t)|\psi_{1}(t)\rangle -\zeta^{\ast}\zeta \left \langle \psi_{1}(t)\right \vert \eta H(t)|\psi_{1}(t)\rangle. \end{array} $$
(69)

and

$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{g} & =&ie^{-\zeta^{\ast}\zeta}(\left \langle \psi_{0}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle-\left \langle \psi_{0}(t)\right \vert \eta \zeta \dot{\psi}_{1}(t)\rangle \\ &&+\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle-\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \eta \zeta \dot{\psi}_{1}(t)\rangle) \end{array} $$

since

$$ \frac{\partial}{\partial t}(\zeta|\psi_{1}(t)\rangle)=\frac{\partial}{\partial t}(-|\psi_{1}(t)\rangle \zeta), $$
(70)

then

$$ \zeta|\dot{\psi}_{1}(t)\rangle=-|\dot{\psi}_{1}(t)\rangle \zeta, $$
(71)

thus

$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{g} & =i(1-\zeta^{\ast}\zeta)\left \langle \psi_{0}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle+i\zeta \left \langle \psi_{0}(t)\right \vert \eta \dot{\psi}_{1}(t)\rangle \\ & +i\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle+i\zeta^{\ast}\zeta \left \langle \psi_{1}(t)\right \vert \eta \dot{\psi}_{1}(t)\rangle). \end{array} $$

We evaluate the four contributions of the geometrical phase, namely, \(\left \langle \psi _{0}(t)\right \vert \eta \dot {\psi }_{0}(t)\rangle , \left \langle \psi _{0}(t)\right \vert \eta \dot {\psi }_{1}(t)\rangle , \left \langle \psi _{1}(t)\right \vert \eta \dot {\psi }_{0}(t)\rangle \) and \(\left \langle \psi _{1}(t)\right \vert \eta \dot {\psi }_{1}(t)\rangle \).

We have

$$ B\left \vert \psi_{0}\right \rangle =0,\Rightarrow \dot{B}\left \vert \psi_{0}\right \rangle +B\left \vert \dot{\psi}_{0}\right \rangle =0,\Rightarrow \left \vert \dot{\psi}_{0}\right \rangle =-B^{-1}\dot{B}\left \vert \psi_{0}\right \rangle , $$
(72)

since B is an invariant associated to H(t), i.e

$$ \dot{B}=iBH-iHB, $$
(73)

then

$$ \left \vert \dot{\psi}_{0}\right \rangle =-B^{-1}(iBH-iHB)\left \vert \psi_{0}\right \rangle =-iH\left \vert \psi_{0}\right \rangle +iB^{-1}HB\left \vert \psi_{0}\right \rangle , $$
(74)

because \(B\left \vert \psi _{0}\right \rangle =0,\) then

$$ \left \vert \dot{\psi}_{0}\right \rangle =-iH\left \vert \psi_{0}\right \rangle , $$
(75)

which means that \(\left \vert \psi _{0}\right \rangle \) obeys to the Schrödinger equation, then

$$ i\left \langle \psi_{0}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle=\left \langle \psi_{0}(t)\right \vert \eta H|\psi_{0}(t)\rangle, $$
(76)

and

$$ i\left \langle \psi_{1}(t)\right \vert \eta \dot{\psi}_{0}(t)\rangle=\left \langle \psi_{1}(t)\right \vert \eta H|\psi_{0}(t)\rangle. $$
(77)

Now we calculate \(\left \vert \dot {\psi }_{1}\right \rangle ,\) we have

$$ \bar{B}\left \vert \psi_{0}\right \rangle =\left \vert \psi_{1}\right \rangle,\Rightarrow\left( \frac{d}{dt}\bar{B}\right)\left \vert \psi_{0}\right \rangle +\bar {B}\left \vert \dot{\psi}_{0}\right \rangle =\left \vert \dot{\psi}_{1}\right \rangle , $$
(78)

Equation (75) leads to

$$ \left \vert \dot{\psi}_{1}\right \rangle =\left( \frac{d}{dt}\bar{B}\right)\left \vert \psi_{0}\right \rangle -i\bar{B}H\left \vert \psi_{0}\right \rangle , $$
(79)

\(\bar {B}\) is also an invariant associated to H(t), i.e

$$ \frac{d}{dt}\bar{B}=i\bar{B}H-iH\bar{B}, $$
(80)

then

$$ \left \vert \dot{\psi}_{1}\right \rangle =-iH\bar{B}\left \vert \psi_{0}\right \rangle , $$
(81)

and since \(\bar {B}\left \vert \psi _{0}\right \rangle =\left \vert \psi _{1}\right \rangle ,\) thus

$$ \left \vert \dot{\psi}_{1}\right \rangle =-iH\left \vert \psi_{1}\right \rangle , $$
(82)

which means that \(\left \vert \psi _{1}\right \rangle \) obeys also to the Schrödinger equation, then

$$ i\left \langle \psi_{1}(t)\right \vert \eta \dot{\psi}_{1}(t)\rangle=\left \langle \psi_{1}(t)\right \vert \eta H|\psi_{0}(t)\rangle, $$
(83)

and

$$ i\left \langle \psi_{0}(t)\right \vert \eta \dot{\psi}_{1}(t)\rangle=\left \langle \psi_{0}(t)\right \vert \eta H|\psi_{0}(t)\rangle, $$
(84)

one can deduce that the geometrical phase is given by

$$ \begin{array}{@{}rcl@{}} \dot{\alpha}_{\zeta}^{g} &=&(1-\zeta^{\ast}\zeta)\left \langle \psi_{0}(t)\right \vert \eta H(t)|\psi_{0}(t)\rangle+\zeta^{\ast}\left \langle \psi_{1}(t)\right \vert \eta H(t)|\psi_{0}(t)\rangle \\ & &+\zeta \left \langle \psi_{0}(t)\right \vert \eta H(t)|\psi_{1}(t)\rangle +\zeta^{\ast}\zeta \left \langle \psi_{1}(t)\right \vert \eta H(t)|\psi_{1}(t)\rangle, \end{array} $$
(85)
$$ \begin{array}{@{}rcl@{}} & =&-\dot{\alpha}_{\zeta}^{d}. \end{array} $$
(86)

Thereby the total phase \(\dot {\alpha }_{\zeta }^{g}+\dot {\alpha }_{\zeta }^{d}=0\).

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Zenad, M., Ighezou, F.Z., Cherbal, O. et al. Ladder Invariants and Coherent States for Time-Dependent Non-Hermitian Hamiltonians. Int J Theor Phys (2020). https://doi.org/10.1007/s10773-020-04401-8

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Keywords

  • Coherent states
  • Invariant theory
  • Non-Hermitian Hamiltonians
  • Grassmann variables