### Hamiltonian dynamics

A series RLC nanoelectronic circuit driven by an arbitrary power source is considered in this work. We assume that the capacitance of the system varies with time. If we denote the charge stored in the capacitor as *q*, we obtain a differential equation from Kirchhoff’s law as

$$\begin{aligned} \ddot{q} + \frac{R}{L} \dot{q} + \omega ^2(t) q = \frac{\mathcal {E} (t)}{L}, \end{aligned}$$

(1)

where \(\omega (t) = [LC(t)]^{-1/2}\) and \(\mathcal {E} (t)\) is the driving power source. We can say that the complete classical solution of this equation is represented as

$$\begin{aligned} Q(t)=Q_{c}(t) +Q_p(t), \end{aligned}$$

(2)

where \(Q_{c}(t)\) is a complementary function and \(Q_p(t)\) is a particular solution. We can do the same thing for the conjugate canonical current, such that

$$\begin{aligned} P(t)=P_{c}(t) +P_p(t). \end{aligned}$$

(3)

Once the solution of \(Q_{c}(t)\) and \(Q_p(t)\) are known, we can also have \(P_{c}(t)\) and \(P_p(t)\) from

$$\begin{aligned} P_{c}(t)=L e^{(R/L)t} d {Q}_c (t)/dt, \end{aligned}$$

(4)

$$\begin{aligned} P_{p}(t)= L e^{(R/L)t} d {Q}_p (t)/dt. \end{aligned}$$

(5)

The Hamiltonian that yields the classical equation of motion given in Eq. (1) can be written as

$$\begin{aligned} {\hat{H}} = e^{-(R/L)t} \frac{{\hat{p}}^2}{2L} + \frac{1}{2}e^{(R/L)t} \left[ \omega ^2(t)L {\hat{q}}^2 - 2 \mathcal {E} (t) {\hat{q}} \right] , \end{aligned}$$

(6)

where \({\hat{p}} = -i\hbar \partial / \partial q\), which stands for the operator of the canonical current.

We introduce an annihilation operator of the form

$$\begin{aligned} {\hat{A}} = \sqrt{\frac{1}{2\hbar L \omega _0}} \left\{ \left[ \frac{\rho _0 \omega _0}{\rho (t)} - i e^{(R/L)t} \frac{\dot{\rho }(t)}{\rho _0} \right] L [{\hat{q}}- Q_p(t)] + i \frac{\rho (t)}{\rho _0} [{\hat{p}}- P_p(t)] \right\} , \end{aligned}$$

(7)

where \(\omega _0 = \omega (0)\), \(\rho _0\) is an arbitrary real constant, and \(\rho (t)\) is a solution of the following differential equation

$$\begin{aligned} \ddot{\rho } (t) + \frac{R}{L} \dot{\rho } (t)+ \omega ^2(t) \rho (t) - e^{-2(R/L)t} \frac{(\omega _0\rho _0^2)^2}{\rho ^3(t)} =0 . \end{aligned}$$

(8)

Of course, the hermitian conjugate of Eq. (7), \({\hat{A}}\)
^{†}, is the creation operator. Notice that \({\hat{A}}\) and \({\hat{A}}\)
^{†} satisfy the boson commutation relation of the form [\({\hat{A}}, {\hat{A}}\)
^{†}] = 1. Now, we can establish an invariant operator in terms of \({\hat{A}}\) and \({\hat{A}}\)
^{†} such that [21]

$$\begin{aligned} {\hat{I}} = {\hbar {\omega }}_0 \left( {\hat{A}}^\dagger {\hat{A}} + \frac{1}{2} \right) . \end{aligned}$$

(9)

Due to the time dependence of parameters of the system, the Hamiltonian given in Eq. (6) is a somewhat complicated form. For this reason, it is favorable to simplify the problem. The time-dependent Hamiltonian can be transformed to a simple form by a suitable unitary operator \({\hat{U}}\). Let us consider the following unitary transformation of the Hamiltonian

$$\begin{aligned} {\hat{H}}' = {\hat{U}}^{-1} {\hat{H}} {\hat{U}} - i \hbar {\hat{U}}^{-1} \frac{\partial {\hat{U}}}{\partial t}, \end{aligned}$$

(10)

where a unitary operator \({\hat{U}}\) is chosen in the form [20]

$$\begin{aligned}{\hat{U}} &=\exp \left( \frac{iP_p(t) {\hat{q}} }{\hbar } \right) \exp \left( -\frac{iQ_p(t) {\hat{p}} }{\hbar }\right) \exp \left( \frac{i L \dot{\rho }(t) e^{(R/L) t} {\hat{q}}^2}{2\hbar \rho (t) }\right) \nonumber \\ & \quad \times \exp \left[ -\frac{i}{4\hbar } ({\hat{q}}{\hat{p}}+{\hat{p}}{\hat{q}}) \ln \left( \frac{\rho ^2(t)}{\rho _0^2} \right) \right] . \end{aligned}$$

(11)

Performing a straightforward algebra for Eq. (10) yields

$$\begin{aligned} {\hat{H}}{^\prime} = \frac{\rho _0^2 }{\rho ^2 (t)} e^{-(R/L)t} \left[ \hbar \omega _0 \left( {\hat{a}}^\dagger {\hat{a}} + \frac{1}{2} \right) \right] + \mathcal {L}_p (t), \end{aligned}$$

(12)

where

$$\begin{aligned}&{\hat{a}} = {\hat{U}}^{-1} {\hat{A}} {\hat{U}}, \end{aligned}$$

(13)

$$\begin{aligned}&{\hat{a}}^\dagger = {\hat{U}}^{-1} {\hat{A}}^\dagger {\hat{U}} , \end{aligned}$$

(14)

and \(\mathcal {L}_p (t)\) is a time function of the form

$$\begin{aligned} \mathcal {L}_p (t) = e^{-(R/L)t} \frac{P_p^2(t)}{2L} - \frac{1}{2} e^{(R/L)t} \omega ^2(t)L Q_p^2(t). \end{aligned}$$

(15)

From a direct evaluation with Eqs. (13) and (14), we easily see that \({\hat{a}} = X{\hat{q}} + {iY{\hat{p}}}\) and \({\hat{a}}^\dagger = X{\hat{q}} - {iY{\hat{p}}}\), where \(X=\sqrt{L\omega _0/(2\hbar )}\) and \(Y=1/{\sqrt{2\hbar L \omega _0}}\). These correspond to the ladder operators of the simple harmonic oscillator with frequency \(\omega _0\). We can also confirm that \([{\hat{a}},{\hat{a}}^\dagger ]=1\).

The equation for *q* in the transformed system is obtained by applying Hamiltonian dynamics with Eq. (12). Hence, from a minor calculation, the classical equation of motion in the transformed system is derived to be

$$\begin{aligned} \ddot{q} + \bigg ( \frac{R}{L} + 2\frac{\dot{\rho }(t)}{\rho (t)} \bigg ) \dot{q} + e^{-2(R/L)t}\frac{(\omega _0 \rho _0^2)^2}{\rho ^4(t)} q = 0 . \end{aligned}$$

(16)

Let us denote a classical solution (complementary functions) for this equation as \(Q_{\mathrm{t},c}(t)\). Then the corresponding classical solution for conjugate canonical current is obtained from

$$\begin{aligned} P_{\mathrm{t},c}(t) = \frac{\rho ^2(t)}{\rho _0^2}L e^{(R/L)t} d {Q}_{\mathrm{t},c} (t)/dt . \end{aligned}$$

(17)

The quantities \(Q_{\mathrm{t},c}(t)\) and \(P_{\mathrm{t},c}(t)\) will be used for developing a quantum theory of the system in the subsequent sections.

### Superposition of displaced squeezed number states

The DSNS is defined by first squeezing the number state and, then, displacing it. The superposition of DSNSs as well as that of DNSs also exhibits many nonclassical characteristics, such as interference and phase fluctuations that can be applied to implementing quantum information techniques, while its generation requires high technology and novel ideas that have yet to be developed. As a strategy for investigating this state, we first derive the DSNS in the transformed system, and then, we transform it inversely in order to obtain the DSNS in the original system.

The squeeze operator in the transformed system is given by

$$\begin{aligned} {\hat{S}} (z) = \exp \left [-\frac{1}{2}(z^* {\hat{a}}^2 - z {\hat{a}}^{\dagger 2})\right], \end{aligned}$$

(18)

where *z* is a squeezing parameter that can be represented in terms of its magnitude *r* and phase \(\phi\) such that

$$\begin{aligned} z = r e^{i \phi }. \end{aligned}$$

(19)

Using Eq. (65) in "Methods" (the last section), the squeeze operator can be rewritten as

$$\begin{aligned}{\hat{S}} (z) &= \frac{1}{\sqrt{s }} \exp \left[ \frac{i L \omega _0 }{2\hbar } \frac{\sin \phi \sinh r}{s} {\hat{q}}^2 \right] \exp \left[ -\frac{i}{\hbar } {\hat{q}} {\hat{p}}\ln s \right] \nonumber \\&\quad \times \exp \left [- \frac{i }{2 L \omega _0 \hbar } \frac{\sin \phi \sinh r }{s } {\hat{p}}^2 \right ], \end{aligned}$$

(20)

where

$$\begin{aligned} s= \cosh r + \cos \phi \sinh r . \end{aligned}$$

(21)

On the other hand, the displacement operator is defined to be

$$\begin{aligned} {\hat{D}} (\alpha ) = \exp (\alpha {\hat{a}}^\dagger - \alpha ^* {\hat{a}}) , \end{aligned}$$

(22)

where

$$\begin{aligned} \alpha = \sqrt{\frac{L \omega _0}{2\hbar }} Q_{\mathrm{t},c}(0) + \frac{i P_{\mathrm{t},c}(0)}{\sqrt{2\hbar L \omega _0}}. \end{aligned}$$

(23)

In actual evaluations, the expression of \({\hat{D}} (\alpha )\) given in "Methods" is useful.

Now let us consider the following transformation

$$\begin{aligned} \psi {^{\prime} _{\mathrm{s},n,\pm }} ({q},t) = {\hat{T}}{^\prime} ({\hat{q}},{\hat{p}},t) {\hat{D}} (\pm \alpha ) {\hat{S}} (z) \psi {^{\prime} _{n}} ({q},0) , \end{aligned}$$

(24)

where \(\psi {^{\prime} _{n}} ({q},0)\) are initial wave functions in number state in the transformed system and \({\hat{T}}{^\prime}\) is a time evolution operator of the form [20, 33]

$$\begin{aligned} {\hat{T}}{^\prime}({\hat{q}},{\hat{p}},t) = \exp \left( - \frac{i }{\hbar }\int _0^t {\hat{H}}{^\prime}({\hat{q}}, {\hat{p}}, \tau ) d \tau \right) . \end{aligned}$$

(25)

To derive \(\psi {^{\prime} _{\mathrm{s},n,\pm }} ({q},t)\) from Eq. (24), we first need the formulae \(\psi {^{\prime} _{n}} ({q},0)\). By solving the Schrödinger equation with Eq. (12) in the transformed system, we easily obtain the corresponding wave functions in the number state and confirm that their initial values are given by

$$\begin{aligned} \psi {^{\prime} _{n}} ({q},0) = \left( \frac{L \omega _0}{\hbar \pi }\right) ^{1/4} \frac{1}{\sqrt{2^nn!}} H_n \left[ \left( \frac{L \omega _0}{\hbar }\right) ^{1/2} {q}\right] \exp \left( -\frac{L \omega _0}{2\hbar } {q}^2\right) . \end{aligned}$$

(26)

These are the same as those of the simple harmonic oscillator with the angular frequency \(\omega _0\). The action of a squeezing operator in the initial number state gives

$$\begin{aligned}{\hat{S}} (z) \psi {^{\prime} _{n}} ({q},0) & = \left( \frac{L \omega _0}{\hbar \pi }\right) ^{1/4} \frac{1}{\sqrt{2^nn!}} \sqrt{\frac{G_{b}^n}{G_{a}}} H_n \left[ \left( \frac{L \omega _0}{\hbar G_{c}}\right) ^{1/2} {q}\right] \nonumber \\& \quad \times \exp \left( -\frac{L \omega _0}{2\hbar } G_{d} {q}^2\right) , \end{aligned}$$

(27)

where

$$\begin{aligned} G_{a}= & {} \cosh r + e^{i \phi } \sinh r , \end{aligned}$$

(28)

$$\begin{aligned} G_{b}= & {} \frac{\cosh r + e^{-i \phi } \sinh r}{ \cosh r + e^{i \phi } \sinh r} , \end{aligned}$$

(29)

$$\begin{aligned} G_{c}= & {} \cosh ^2 r + \hbox{sinh} ^2 r + 2\cos \phi \cosh r\sinh r , \end{aligned}$$

(30)

$$\begin{aligned} G_{d}= & {} \frac{1- i \sin \phi \sinh r(\cosh r + e^{i \phi } \sinh r)}{ (\cosh r + \cos \phi \sinh r)(\cosh r + e^{i \phi } \sinh r)}. \end{aligned}$$

(31)

Further action of \({\hat{D}} (\pm \alpha )\) and \({\hat{T}}{^\prime} ({\hat{q}},{\hat{p}},t)\), in turn, yields [7, 33]

$$\begin{aligned}\psi _{\mathrm{s},n,\pm }' ({q},t) &=\root 4 \of {\frac{L \omega _0}{\hbar \pi }} \frac{1}{\sqrt{2^nn!}} \sqrt{\frac{(h_{b} G_{b})^n}{h_{a} G_{a}}} \nonumber \\& \quad \times H_n \left [\sqrt{\frac{L \omega _0}{\hbar h_{a}^2 h_{b} G_{c}}} [{q} \mp Q_{\mathrm{t},c}(t) ]\right ] \nonumber \\&\quad \times \exp \bigg \{-\frac{L \omega _0}{2\hbar h_{a}} \bigg [ [G_{d}\cos \Omega (t) + i \sin \Omega (t) ]q^2 \nonumber \\& \qquad \qquad \mp 2q \bigg (G_{d} Q_{\mathrm{t},c}(0) + i \frac{P_{\mathrm{t},c}(0)}{\omega _0 L} \bigg ) \nonumber \\& \qquad \qquad + Q_{\mathrm{t},c}^2(0) G_{d} \cos \Omega (t) \bigg ]\bigg \} \nonumber \\&\quad \times \exp \left [ -\frac{i P_{\mathrm{t},c}^2(0) \sin \Omega (t)}{2 L \omega _0 h_{a}\hbar } - i \frac{Q_{\mathrm{t},c}(0)P_{\mathrm{t},c}(0)}{\hbar } \right. \\ & \qquad \qquad \left. \times \left ( \frac{1}{2} - i \frac{G_{d} \sin \Omega (t)}{h_{a}} \right ) \right ] \nonumber \\&\quad \times \exp \left [ -\frac{i}{\hbar } \int _0^t {\mathcal L}_{p} (\tau ) \text{ d } \tau \right ], \end{aligned}$$

(32)

where

$$\begin{aligned}\Omega (t) &= \rho _0^2 \omega _0 \int _0^t \frac{e^{-(R/L)\tau }}{\rho ^2 (\tau )} d\tau , \end{aligned}$$

(33)

$$\begin{aligned}h_{a} &= \cos \Omega (t) + i G_{d} \sin \Omega (t) , \end{aligned}$$

(34)

$$\begin{aligned}h_{b}& = 1- \frac{2 i \sin \Omega (t)}{h_{a} G_{c}} . \end{aligned}$$

(35)

Let us consider superposition states composed of the two DSNSs in the transformed system, which is

$$\begin{aligned} \psi {_{\mathrm{s},n}^{\prime \epsilon}} ({q},t) = \lambda _\mathrm{s}^\epsilon [\psi {^{\prime} _{\mathrm{s},n,+}} ({q},t) + \epsilon \psi {^{\prime} _{\mathrm{s},n,-}} ({q},t) ] , \end{aligned}$$

(36)

where \(\epsilon \) is given by \(\epsilon =|\epsilon |e^{i\varphi }\) and \(\lambda^{\epsilon}_s\) is a normalization constant of which the formula will be derived later. The superposition states in the original system are obtained by acting \({\hat{U}}\) in these states:

$$\begin{aligned} \psi _{\mathrm{s},n}^\epsilon ( {q},t) = {\hat{U}} \psi {_{\mathrm{s},n} ^ {\prime \epsilon}} ( {q},t) . \end{aligned}$$

(37)

A rigorous evaluation using Eq. (11) gives

$$\begin{aligned} \psi _{\mathrm{s},n}^\epsilon ( {q},t) = \lambda _\mathrm{s}^\epsilon [\psi _{\mathrm{s},n,+} ( {q},t) + \epsilon \psi _{\mathrm{s},n,-} ( {q},t)], \end{aligned}$$

(38)

where

$$\begin{aligned}\psi _{\mathrm{s},n,\pm } ({q},t) &= \root 4 \of {\frac{L \omega _0}{\hbar \pi }} \frac{\sqrt{{\rho _0}/{\rho (t)}}}{\sqrt{2^nn!}} \sqrt{\frac{(h_{b} G_{b})^n}{h_{a} G_{a}}} H_n \left [\frac{\xi _{\pm }(q,t)}{\sqrt{h_{a}^2 h_{b} G_{c}}}\right ] \nonumber \\&\quad \times \exp \left( \frac{i}{\hbar } P_{p}(t) {q} \right) \exp \left ( \frac{i L \dot{\rho }(t)e^{(R/L)t}}{2\hbar \rho (t)} [q - Q_{p}(t)]^2 \right) \nonumber \\&\quad \times \exp \bigg \{-\frac{L \omega _0}{2\hbar h_{a}} \bigg [ [G_{d}\cos \Omega (t) + i \sin \Omega (t) ] \frac{\rho _0^2}{\rho ^2(t)}[q-Q_p(t)]^2 \nonumber \\&\qquad \mp 2\frac{\rho _0}{\rho (t)}[q-Q_p(t)] \bigg (G_{d} Q_{\mathrm{t},c}(0) + i \frac{P_{\mathrm{t},c}(0)}{\omega _0 L} \bigg ) \nonumber \\&\qquad + Q_{\mathrm{t},c}^2(0) G_{d} \cos \Omega (t) \bigg ]\bigg \} \nonumber \\ & \quad \times \exp \left [ -\frac{i P_{\mathrm{t},c}^2(0) \sin \Omega (t)}{2 L \omega _0 h_{a}\hbar } - i \frac{Q_{\mathrm{t},c}(0)P_{\mathrm{t},c}(0)}{\hbar } \times \left( \frac{1}{2} - i \frac{G_{d} \sin \Omega (t)}{h_{a}} \right) \right] \nonumber \\ &\quad \times \exp \left [ -\frac{i}{\hbar } \int _0^t {\mathcal L}_{p} (\tau ) \text{ d } \tau \right] , \end{aligned}$$

(39)

with

$$\begin{aligned} \xi _{\pm }(q,t) = \sqrt{\frac{L\omega _0}{\hbar }} \bigg ( \frac{\rho _0}{\rho (t)} [{q}- Q_{p}(t) ] \mp Q_{\mathrm{t},c}(t) \bigg ). \end{aligned}$$

(40)

These are the full wave functions in the superposition states of the DSNSs. We can use them to derive an expectation value of various quantum observables in the superposition states.

From the absolute square of Eq. (38), we also have the probability densities as

$$\begin{aligned} \left| \psi _{\mathrm{s},n}^{\epsilon }( {q},t)\right| ^2 &= \sqrt{\frac{L\omega _0}{\hbar \pi }} \frac{1}{2^n n!}\frac{ \rho _0}{\rho (t)d } |\lambda _\mathrm{s}^\epsilon |^2 \nonumber \\ &\quad \times \exp \left \{ - \frac{L \omega _0 }{\hbar d^2 } \left [\frac{\rho _0^2}{\rho ^2(t)}[q-Q_p(t)]^2 +Q_{\mathrm{t},c}^2 (t) \right ] \right \} \nonumber \\&\quad \times \left \{ e^{Z(q,t)} \left[H_n \bigg (\frac{\xi _{+}(q,t)}{d}\bigg )\right]^2 + |\epsilon |^2 e^{-Z(q,t)}\left [H_n \bigg (\frac{\xi _{-}(q,t)}{d}\bigg )\right ]^2 \right. \nonumber \\&\qquad \left. +\, 2 |\epsilon |H_n \bigg (\frac{\xi _{+}(q,t)}{d}\bigg )H_n \bigg (\frac{\xi _{-}(q,t)}{d}\bigg ) \right. \nonumber \\ & \left. \qquad \times\, {\text{cos}} \left [\frac{2B(t)}{\hbar d^2 } \frac{\rho _0}{\rho (t)}[q-Q_p(t)] -\varphi \right ] \right \} , \end{aligned}$$

(41)

where

$$\begin{aligned}d^2 &= \cosh (2r) +\cos [2\Omega (t)-\phi ] \sinh (2r) \nonumber \\& = {s_0^2 \cos ^2 [\Omega ( t)- \phi /2] + s_0^{-2} \sin ^2 [\Omega ( t)}-\phi /2] , \end{aligned}$$

(42)

$$\begin{aligned}s_0 &= \cosh r + \sinh r = e^{r}, \end{aligned}$$

(43)

$$\begin{aligned}&Z(q,t) = \frac{2L\omega _0 Q_{\mathrm{t},c}(t) }{\hbar d^2}\frac{\rho _0}{\rho (t)}[q-Q_p(t)], \end{aligned}$$

(44)

$$\begin{aligned}B(t) &= P_{\mathrm{t},c}(t)\cosh (2r)+ \{ P_{\mathrm{t},c}(0) \cos [\Omega (t)-\phi ] \nonumber \\&\quad + L\omega _0 Q_{\mathrm{t},c}(0) \sin [\Omega (t) -\phi ] \} \sinh (2r). \end{aligned}$$

(45)

It is possible to represent *d* in terms of *s* given in Eq. (21) instead of \(s_0\). The transformation relation from \(s_0\) to *s* and vice versa are

$$\begin{aligned} s_0\rightarrow & {} \frac{s+\sqrt{s^2-\sin ^2 \phi }}{1+\cos \phi } , \end{aligned}$$

(46)

$$\begin{aligned} s\rightarrow & {} \frac{1-\cos \phi }{2s_0} + \frac{s_0(1+\cos \phi )}{2} . \end{aligned}$$

(47)

Now, the formula of \(\lambda _\mathrm{s}^\epsilon\) is derived from the normalization condition, \(\int _{-\infty }^\infty |\psi _{\mathrm{s},n}^{\epsilon }( {q},t)|^2 dq =1\). It results in

$$\begin{aligned} |\lambda _\mathrm{s}^\epsilon |^2 = \{1+ |\epsilon |^2 +2 |\epsilon | \exp (-\eta (t) ) L_n [2\eta (t)] \cos \varphi \}^{-1}, \end{aligned}$$

(48)

where

$$\begin{aligned} \eta (t) = \frac{1}{\hbar d^2} \left( L\omega _0 Q_{\mathrm{t},c}^2 (t) + \frac{B^2(t)}{L\omega _0} \right) . \end{aligned}$$

(49)

If we put \(r=0\), Eq. (48) reduces to Eq. (26) of Ref. [20], which is related to the superposition of DNSs. The probability densities, Eq. (41), consist of three terms. The first two terms correspond to the densities associated to \(\psi _{\mathrm{s},n,+} ({q},t)\) and \(\psi _{\mathrm{s},n,-} ({q},t)\), respectively. The last term that is represented in terms of a cosine function exhibits interference between the two components. This term signifies nonclassical properties of the quantum system, that do not appear in the counterpart classical system. The interference term is especially large when the two wave packets associated with \(\psi _{\mathrm{s},n,+} ({q},t)\) and \(\psi _{\mathrm{s},n,-} ({q},t)\) meet in space. From Eq. (41), we can confirm that the width of the packet is determined by the value of *d*. If \(d< 1\), the packet corresponds to that of the *q*-squeezing case, whereas if \(d> 1\), the packet belongs to that of the *p*-squeezing case. However, the degree of squeezing varies more or less with time according to the time variation of *d*.

For the case that *z* is real (\(\phi = 0\)), Eqs. (42) and (49) reduce to

$$\begin{aligned}&d_0^2 = {s_0^2 \cos ^2 \Omega ( t) + s_0^{-2} \sin ^2 \Omega ( t)}, \end{aligned}$$

(50)

$$\begin{aligned}&\eta _0 = \frac{1}{\hbar } \left( s_0^{-2}L\omega _0 Q_{\mathrm{t},c}^2 (0) + s_0^2 \frac{P_{\mathrm{t},c}^2(0)}{L\omega _0} \right) . \end{aligned}$$

(51)

We see that \(d_0\) varies with time for an arbitrary value of \(s_0\), while \(\eta _0\) is constant. However, for the case that \(s_0=1\), \(d_0\) does not vary with time and reduces to unity that corresponds to the situation of no squeezing at all.

Note that \(\eta _0\) can be rewritten in a simple form as

$$\begin{aligned} \eta _0 = 2 |f|^2 , \end{aligned}$$

(52)

where

$$\begin{aligned} f = \alpha \cosh r - \alpha ^* \sinh r . \end{aligned}$$

(53)

In fact, Eq. (53) is slightly different from that of the system proposed by El-Orany et al. (e.g. see Eq. (4) of Ref. [31]).

For the case that \(n=0\), \(\epsilon = \pm 1\), and \(\phi = 0\), the probability density becomes

$$\begin{aligned}\left| \psi _{\mathrm{s},n=0}^{\epsilon = \pm 1}( {q},t)\right| ^2 &= \sqrt{\frac{L\omega _0}{\hbar \pi }} \frac{2 \rho _0}{\rho (t)d_0 } \left| \lambda _{0,\mathrm s}^{\epsilon =\pm 1}\right| ^2 \exp \left \{ - \frac{L \omega _0 }{\hbar d_0^2 }\left [\frac{\rho _0^2}{\rho ^2(t)} [q-Q_p(t)]^2 +Q_{\mathrm{t},c}^2 (t) \right ] \right \} \nonumber \\&\quad \times \left \{ \cosh \left [ \frac{2L\omega _0 Q_{\mathrm{t},c}(t) }{\hbar d_0^2} \frac{\rho _0}{\rho (t)}[q-Q_p(t)] \right ] \pm \cos \left [ \frac{2}{\hbar d_0^2 s_0^2}\frac{\rho _0}{\rho (t)}[q-Q_p(t)] \right. \right. \nonumber \\&\quad \left. \Bigg. \times [P_{\mathrm{t},c}(0)s_0^4 \cos \Omega ( t) - L\omega _0 Q_{\mathrm{t},c} (0) \sin \Omega ( t)] \Bigg ] \right\} , \end{aligned}$$

(54)

where \(\lambda _{0, \mathrm s}^{\epsilon } = \lambda _\mathrm{s}^{\epsilon } |_{\phi =0}\). We can see from this expression that, for the non-squeezing case (\(s_0=1\)), the time dependence of phase of the interference term follows \([q-Q_p(t)]P_{\mathrm{t},c}(t)/\rho (t)\).

For a further simplified case which falls under \(n=0\), \(\epsilon = \pm 1\), \(\mathcal {E}(t) =0\), \(R=0\), and \(C(t) = C(0)\), we confirm that \(\Omega (t) \rightarrow \omega _0 t\) and the probability density reduces to

$$\begin{aligned}\left| \psi _{\mathrm{s},n=0}^{\epsilon = \pm 1}( {q},t)\right| ^2 &= \sqrt{\frac{L\omega _0}{\hbar \pi }} \frac{2}{d_0} \left| {\lambda _{0,\mathrm s}^{\epsilon =\pm 1}}\right| ^2 \exp \bigg ( - \frac{L \omega _0 [q^2 +Q_{\mathrm{t},c}^2 (t) ]}{\hbar d_0^2 } \bigg ) \nonumber \\&\quad \times \bigg \{ \cosh \bigg [ \frac{2L\omega _0 Q_{\mathrm{t},c}(t) q}{\hbar d_0^2} \bigg ] \pm \cos \bigg [ \frac{2q}{\hbar d_0^2 s_0^2}[P_{\mathrm{t},c}(0)s_0^4 \cos (\omega _0 t) \nonumber \\&\quad - L\omega _0 Q_{\mathrm{t},c}(0) \sin (\omega _0 t)] \bigg ] \bigg \}. \end{aligned}$$

(55)

In this case, the phase of the interference term is determined by \(q[P_{\mathrm{t},c}(0)s_0^2\cos (\omega _0 t) - L\omega _0 Q_{\mathrm{t},c}(0) \sin (\omega _0 t)/s_0^2]\). For the non-squeezing case, it is fixed simply by \(q P_{\mathrm{t},c}(t)\).

The probability density \(|\psi _{\mathrm{s},n}^\epsilon ( {q},t)|^2\) given in Eq. (41) is plotted in Figs. 1, 2, 3, and 4 as a function of *q* and *t* under the choice of parameters as Eqs. (81)–(87) in "Methods". The probability density given in Fig. 1 is for relatively small displacing parameters: \((Q_{\mathrm{t},c}(0), P_{\mathrm{t},c}(0)) =(1,1)\). By comparing Fig. 1a and b with each other, we can confirm that the width of the wave function becomes large as the value of *r* increases. Figure 1a is *q*-squeezing and Fig. 1b is *p*-squeezing. Figure 2a shows the time evolution of the wave packet that is dominated by \(\psi _{\mathrm{s},n,+} ( {q},t)\), while Fig. 2b is the case that \(\psi _{\mathrm{s},n,-} ( {q},t)\) is dominant. By adding these two packets, we may roughly obtain the normal wave packet, which is, for example, the one given in Fig. 1, with equal contribution of the two components. Figure 3 is the probability density for sufficiently high displacing parameters: \((Q_{\mathrm{t},c}(0), P_{\mathrm{t},c}(0)) =(5,5)\). We see by comparing Fig. 3a with Fig. 1a that the displacement of the wave packet is quite distinct when the displacing parameters are large, as expected. The effects of a driving electromotive force on the circuit can be identified from Fig. 4. The wave packets in Fig. 4 are distorted more or less significantly, due to the influence of a time-dependent electromotive force.

We can find nonclassical features of the system from quantum interference displayed in Figs. 1, 2, 3, and 4. Highly peaked ripples in density, formed near the center (\(q=0\)), signifies nonclassicality of the superposition state. The distribution of *q* exhibits peaks at the spots where the two or more non-zero line-shaped distributions intersect with each other. The distribution-lines are oscillating and mainly composed of two groups according to the two individual components of the superposition state. In the same situation, the distribution of the conjugate variable *p* oscillates as a reflection of quantum interference [34].

Notice that the concepts of the superposition principle and quantum interference are applied to developing fundamental mechanisms for quantum information theory. In connection with this, quantum physics and quantum field theory are undergoing a time of revolutionary change in these days.

### Time evolution of quantum observables

The time evolution of quantum observables in the quantum states developed in the previous sections can be found by evaluating the expectation value of them. Let us see for example the time evolution of charges and currents in the DSNS in the original system. Considering the fact that the notation of the wave functions in this state can be rewritten, without loss of generality, in the form

$$\begin{aligned} \psi _{\mathrm{s},n}^\epsilon (q,t) = \langle q | \psi _{\mathrm{s},n}^\epsilon (t) \rangle , \end{aligned}$$

(56)

and using the consecutive unitary transformation, the expectation value of an arbitrary quantum observable \(\hat{\mathcal O}\) can be evaluated from

$$\begin{aligned} \langle \psi _{\mathrm{s},n}^\epsilon (t) | \hat{\mathcal O} | \psi _{\mathrm{s},n}^\epsilon (t) \rangle = |\lambda _\mathrm{s}^\epsilon |^2 \langle \psi _{n}'(0) | \hat{\mathcal O}' | \psi {^{\prime} _{n}} (0) \rangle , \end{aligned}$$

(57)

where

$$\begin{aligned} \hat{\mathcal O}' = {\hat{S}}^\dagger [{\hat{D}}^\dagger (\alpha )+\epsilon ^* {\hat{D}}^\dagger (-\alpha )] {\hat{T}}{^{\prime \dagger}} {\hat{U}}^\dagger \hat{\mathcal O} {\hat{U}} {\hat{T}}{^\prime} [{\hat{D}}(\alpha )+\epsilon {\hat{D}}(-\alpha )] {\hat{S}} . \end{aligned}$$

(58)

Let us perform successive operations given in Eq. (58) after replacing \(\hat{\mathcal O}\) with \({\hat{q}}\) and \({\hat{p}}\). Then, with the use of Eqs. (67)–(80) in "Methods", we have

$$\begin{aligned}{\hat{q}}{^\prime} &= \sqrt{\frac{\hbar }{2L\omega _0}} \frac{\rho (t)}{\rho _0} \{({\hat{b}}+\alpha ) e^{-i\Omega (t)} +({\hat{b}}^\dagger +\alpha ^*) e^{i\Omega (t)} \nonumber \\&\quad +\epsilon [({\hat{b}}+\alpha )e^{2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{-i\Omega (t)} +({\hat{b}}^\dagger +\alpha ^*)e^{2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{i\Omega (t)} ] \nonumber \\&\quad +\epsilon ^*[({\hat{b}}-\alpha )e^{-2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{-i\Omega (t)} +({\hat{b}}^\dagger -\alpha ^*)e^{-2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{i\Omega (t)} ] \nonumber \\&\quad +|\epsilon |^2 [({\hat{b}}-\alpha ) e^{-i\Omega (t)} +({\hat{b}}^\dagger -\alpha ^*) e^{i\Omega (t)}] \} + Q_p(t), \end{aligned}$$

(59)

$$\begin{aligned}{\hat{p}}{^\prime} &= F(t) ({\hat{b}}+\alpha ) e^{-i\Omega (t)} +F^*(t)({\hat{b}}^\dagger +\alpha ^*) e^{i\Omega (t)} \nonumber \\&\quad +\epsilon [F(t)({\hat{b}}+\alpha )e^{2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{-i\Omega (t)} +F^*(t)({\hat{b}}^\dagger +\alpha ^*)e^{2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{i\Omega (t)} ] \nonumber \\&\quad+\epsilon ^*[F(t)({\hat{b}}-\alpha )e^{-2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{-i\Omega (t)} +F^*(t)({\hat{b}}^\dagger -\alpha ^*)e^{-2(\alpha ^* {\hat{b}}-\alpha {\hat{b}}^\dagger )}e^{i\Omega (t)} ] \nonumber \\&\quad +|\epsilon |^2 [F(t)({\hat{b}}-\alpha ) e^{-i\Omega (t)} +F^*(t)({\hat{b}}^\dagger -\alpha ^*) e^{i\Omega (t)}] + P_p(t), \end{aligned}$$

(60)

where \({\hat{b}}\) and \({\hat{b}}\)
^{†} are defined in "Methods" [see Eqs. (79) and (80)] and

$$\begin{aligned} F(t) = \sqrt{\frac{\hbar L}{2}} \left( \frac{\dot{\rho }(t)}{\rho _0 \sqrt{\omega _0}}e^{(R/L)t} - i \frac{\rho _0}{\rho (t)}\sqrt{\omega _0} \right) . \end{aligned}$$

(61)

Let us calculate the expectation value of canonical variables with an assumption that \(\alpha\) is sufficiently small than unity. If we consider up to \([\alpha ^{(*)}]^3\) terms, the expectation values are given by

$$\begin{aligned}\langle \psi _{\mathrm{s},n}^\epsilon (t) | {\hat{q}} | \psi _{\mathrm{s},n}^\epsilon (t) \rangle &=|\lambda _\mathrm{s}^\epsilon |^2\sqrt{\frac{\hbar }{2L\omega _0}} \frac{\rho (t)}{\rho _0} \{ (1-\epsilon ^*\epsilon ) [\alpha e^{-i\Omega (t)}+\alpha ^* e^{i\Omega (t)} ] \nonumber \\&\quad + (\epsilon -\epsilon ^*) [ { K} e^{-i\Omega (t)} - { K}^* e^{i\Omega (t)} ] \} +Q_p(t), \end{aligned}$$

(62)

$$\begin{aligned}\langle \psi _{\mathrm{s},n}^\epsilon (t) | {\hat{p}} | \psi _{\mathrm{s},n}^\epsilon (t) \rangle &=|\lambda _\mathrm{s}^\epsilon |^2 \{(1-\epsilon ^*\epsilon ) [F(t)\alpha e^{-i\Omega (t)}+F^*(t)\alpha ^* e^{i\Omega (t)} ] \nonumber \\&\quad + (\epsilon -\epsilon ^*) [F(t) { K} e^{-i\Omega (t)} -F^*(t) { K}^* e^{i\Omega (t)} ]\} +P_p(t), \end{aligned}$$

(63)

where

$$\begin{aligned}{ K} &= 2\{\alpha ^* e^{i\phi } (2n+1)\cosh r \sinh r -\alpha [(n+1)(\cosh r)^2+n(\sinh r)^2- 1/2]\} \nonumber \\&\quad +4\alpha ^{*3} e^{2i\phi } (2n^2 + 2n +1)(\cosh r)^2(\sinh r)^2 +2\alpha ^{*2}\alpha e^{i\phi } \{ (2n+1)\cosh r \sinh r \nonumber \\&\quad -2 [(3n^2+4n+2)(\cosh r)^3 \sinh r + (3n^2+2n+1)\cosh r(\sinh r)^3] \} \nonumber \\&\quad +2\alpha ^* \alpha ^2 \{ -(2n+1)[(\cosh r)^2+(\sinh r)^2] +2[(n^2+2n+1)(\cosh r)^4 \nonumber \\&\quad +n^2(\sinh r)^4 +(4n^2+4n+2)(\cosh r)^2(\sinh r)^2 ] \} +2\alpha ^3 e^{-i\phi } \{ (2n+1) \nonumber \\&\quad \times \cosh r \sinh r-2 [(n^2+2n+1)(\cosh r)^3 \sinh r + n^2\cosh r(\sinh r)^3 ] \} . \end{aligned}$$

(64)

To see the time behavior of canonical variables, let us see again for the case given in Eqs. (81)–(87). From Fig. 5, we can confirm that the expectation value of \({\hat{q}}\) is more or less distorted with time due to the influence of the time dependence of parameters.