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Quantum evolution leading to classicality: a concrete example

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Abstract

Can certain degrees of freedom of a closed physical system, described by a time-independent Hamiltonian, become more and more classical as they evolve from 1 some state? This question is important because our universe seems to have done just that! We construct an explicit, simple, example of such a system with just two degrees of freedom, of which one achieves ‘spontaneous classicalization’. It starts from a quantum state and under the usual Hamiltonian evolution, becomes more and more classical (in a well-defined manner in terms of the Wigner function) as time progresses. This is achieved without the usual procedures of integrating out a large number of environmental degrees of freedom or conventional decoherence. We consider different ranges of parameter space and identify the conditions under which spontaneous classicalization occurs in our model. The mutual interaction between the sub-systems of a larger system can indeed drive some of the subsystems to a classical configuration, with a phase space trajectory of evolution. We also argue that the results of our toy model may well be general characteristics of certain class of interacting systems. Several implications are discussed.

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Notes

  1. The approaches of [26] and [38] also suggest the classicalization of the system even in this case, owing to the structure of the Wigner function which is still of the form Eq. (36).

  2. We could, for example, work with the first excited state of the normal mode instead of the ground state. This choice of initial state will include such configurations. First excited state of the coupled modes will just be a linear combination of such states.

  3. Sylvester’s Criterion: A real symmetric matrix with positive diagonal elements is positive definite if and only if each of its principal submatrix has positive determinant. Achieving criticality for just one pair denies the matrix the property of positive definiteness.

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Acknowledgments

The research of TP is partially supported by the J.C. Bose research grant of the Department of Science and Technology, Government of India. The research of K.P. is supported by the Shyama Prasad Mukherjee fellowship of the Council of Scientific and Industrial Research, Government of India.

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Correspondence to T. Padmanabhan.

Appendices

Appendix 1: General Wigner function for supercritical case

We consider a general Gaussian initial state, so as to be able to reach up to various cases we discussed previously

$$\begin{aligned} \Psi (X_-,X_+;0)=N \exp {\left\{ -\left( \alpha X_-^2+\tilde{\alpha } X_+^2 + 2 \beta X_+ X_-\right) \right\} }. \end{aligned}$$
(83)

Evolution of this general Gaussian state will be obtained from the Kernel of the system, obtained easily when written in normal modes

$$\begin{aligned} \psi (X_-,X_+;t)=\int dX_-'dX_+'G\left( X_-,X_-',X_+,X_+';t,0\right) \psi \left( X_-',X_+';0\right) , \end{aligned}$$
(84)

where,

$$\begin{aligned} G\left( X_-,X_-',X_+,X_+';t,0\right) \!=\! \left( \frac{\Omega _+}{2\pi i \hbar \sin {\Omega _+ t}}\right) ^{\frac{1}{2}} \left( \frac{\Omega _-}{2\pi i \hbar \sinh {\Omega _- t}}\right) ^{\frac{1}{2}}\!\times \!{\mathcal G}_+\!\times \!{\mathcal G}_-,\nonumber \\ \end{aligned}$$
(85)

with,

$$\begin{aligned} {\mathcal G}_+&= \exp {\left[ \frac{i \Omega _+}{2\hbar \sin {\Omega _+ t}}\left\{ \left( X_+^2+X_+'^2\right) \cos {\Omega _+ t}-2 X_+ X_+'\right\} \right] }, \nonumber \\ {\mathcal G}_-&= \exp {\left[ \frac{i \Omega _-}{2\hbar \sinh {\Omega _- t}}\left\{ \left( X_-^2+X_-'^2\right) \cosh {\Omega _- t}-2 X_- X_-'\right\} \right] }. \end{aligned}$$
(86)

For the initial state of the above mentioned kind, the time evolution gives

$$\begin{aligned} \Psi (X_-,X_+;t)\!&= \! N\frac{\sqrt{\alpha _1\alpha _2}}{\sqrt{\tilde{\alpha }k_1\!-\!\beta ^2\!-\!k_1\alpha _1\beta _1}}\exp {\left\{ \alpha _1\beta _1X_+^2+\alpha _2\beta _2X_-^2+ \frac{\alpha _2^2}{\alpha -\alpha _2\beta _2}X_-^2\right\} }\nonumber \\&\quad \times \exp {\left\{ \frac{\left( k_1\alpha _1X_++\beta \alpha _2X_-\right) ^2}{k_1\left( \tilde{\alpha }k_1-\beta ^2-k_1\alpha _1\beta _1\right) }\right\} }, \end{aligned}$$
(87)

with the condition of the state being normalizable imposed through

$$\begin{aligned} Re\left[ \frac{\beta ^2}{k_1}+\alpha _1\beta _1-\tilde{\alpha }\right] \le 0, \end{aligned}$$

where we have

$$\begin{aligned} \alpha _1&= \frac{i\Omega _+}{2\hbar \sin {\Omega _+t}}; \qquad \beta _1=\cos {\Omega _+t} \nonumber \\ \alpha _2&= \frac{i\Omega _-}{2\hbar \sinh {\Omega _-t}}; \qquad \beta _2=\cosh {\Omega _-t}\nonumber \\ k_1&= \alpha -\alpha _2\beta _2. \end{aligned}$$
(88)

Now, the time evolved state Eq. (87) can be rewritten in a more comprehensible form as

$$\begin{aligned} \Psi (X_-,X_+;t)\!=\!N\frac{\sqrt{\alpha _1\alpha _2}}{\sqrt{\tilde{\alpha }k_1-\beta ^2\!-\!k_1\alpha _1\beta _1}}\exp {\{-(B X_+^2\!+\!C X_-^2 \!+\! 2 D X_+X_-)\}}, \end{aligned}$$
(89)

where, we have

$$\begin{aligned} B&= -\left[ \frac{i\Omega _+}{2\hbar }\cot {\Omega _+t} -\frac{\left( \alpha -\frac{i\Omega _-}{2\hbar }\coth {\Omega _-t}\right) \frac{\Omega _+^2}{(2\hbar \sin {\Omega _+t})^2}}{\alpha \tilde{\alpha } -\beta ^2-\frac{\Omega _+\Omega _-}{4 \hbar ^2}\coth {\Omega _-t}\cot {\Omega _+t}-i \left( \frac{\Omega _-\tilde{\alpha }}{2\hbar }\coth {\Omega _-t}+ \frac{\Omega _+\alpha }{2\hbar }\cot {\Omega _+t}\right) } \right] ,\nonumber \\ \end{aligned}$$
(90)
$$\begin{aligned} D&= -\left[ \frac{\beta \frac{\Omega _+\Omega _-}{4 \hbar ^2} \mathrm{cosech }{\Omega _-t} \mathrm{cosec }{\Omega _+t} }{\alpha \tilde{\alpha } -\beta ^2-\frac{\Omega _+\Omega _-}{4 \hbar ^2}\coth {\Omega _-t} \cot {\Omega _+t}-i\left( \frac{\Omega _-\tilde{\alpha }}{2\hbar } \coth {\Omega _-t}+ \frac{\Omega _+\alpha }{2\hbar }\cot {\Omega _+t}\right) }\right] ,\end{aligned}$$
(91)
$$\begin{aligned} C&= -\left[ \frac{i\frac{\Omega _-\alpha }{2\hbar }\coth {\Omega _-t} + \frac{\Omega _-^2}{4 \hbar ^2}}{\alpha -i\frac{\Omega _-}{2\hbar } \coth {\Omega _-t}}+\frac{\beta ^2\alpha _2^2}{k_1\left( \tilde{\alpha }k_1-\beta ^2-k_1\alpha _1\beta _1\right) }\right] . \end{aligned}$$
(92)

This is the most generic form of the evolution of Eq. (83). From here we can easily obtain the late time limits of this state and obtain the corresponding Wigner function, which is given in Eq. (33). We see that coefficient \(D\) asymptotically gets suppressed and we essentially have a decoupled state in normal modes. The reduced Wigner function for modes \(x_1\) and \(x_2\) are given in the Morikawa form with parameters

$$\begin{aligned} \bar{\gamma }_1^2&= \frac{\left( B_R\cos ^2{\theta }+C_R\sin ^2{\theta }\right) }{2\left\{ (B_R-C_R)^2+(B_I-C_I)^2\right\} \sin ^2{\theta } \cos ^2{\theta }-2B_RC_R}\nonumber \\ \bar{\alpha }^2&= \frac{2B_RC_R\left[ \left\{ (B_R-C_R)^2+(B_I-C_I)^2\right\} \sin ^2{2\theta }+4B_RC_R\right] }{\left[ 2\left\{ (B_R-C_R)^2+(B_I-C_I)^2\right\} \sin ^2{\theta } \cos ^2{\theta }-2B_RC_R\right] \left( B_R\cos ^2{\theta }+C_R\sin ^2{\theta }\right) }\nonumber \\ \bar{\beta }&= \frac{2\left( B_RC_I\cos ^2{\theta }+C_R B_I\sin ^2{\theta }\right) }{\left( B_R\cos ^2{\theta }+C_R\sin ^2{\theta }\right) } \end{aligned}$$
(93)

and for mode \(x_2,\)

$$\begin{aligned} \bar{\gamma }_2^2&= \frac{\left( B_R\sin ^2{\theta }+C_R\cos ^2{\theta }\right) }{2\left\{ (B_R-C_R)^2+(B_I-C_I)^2\right\} \sin ^2{\theta }\cos ^2{\theta } -2B_RC_R}. \end{aligned}$$
(94)

Clearly for the super-critical limit \(C_R \rightarrow 0\) at late times, makes the parameter \(\bar{\alpha }\) vanish asymptotically for both the variables. Furthermore we can verify that coefficients \(B_R,B_I\) and \(C_I\) have strength of typical quantum systems. The interesting range \(f\gg \xi ^2\) either keeps \(\bar{\gamma }\) around the value of the corresponding quantum value (for \(x_1\)) or makes it much smaller than it (for \(x_2\)).

Appendix 2: General Wigner function for sub-critical case

The time evolution for the sub-critical case is governed by the standard normal mode propagators satisfying harmonic oscillator equations of motion classically. With a state of type Eq. (83) and the propagators

$$\begin{aligned} {\mathcal G}_+&= \exp {\left[ \frac{i \Omega _+}{2\hbar \sin {\Omega _+ t}}\left\{ \left( X_+^2+X_+'^2\right) \cos {\Omega _+ t}-2 X_+ X_+'\right\} \right] } \nonumber \\ {\mathcal G}_-&= \exp {\left[ \frac{i \Omega _-}{2\hbar \sin {\Omega _- t}}\left\{ \left( X_-^2+X_-'^2\right) \cos {\Omega _- t}-2 X_- X_-'\right\} \right] }, \end{aligned}$$
(95)

we obtain the a state of type Eq. (89) whose coefficients keep oscillating in this case without having any meaningful notion of an asymptotic value

$$\begin{aligned} B&= -\left[ \frac{i \Omega _+}{2\hbar }\cot {\Omega _+t}-\frac{\left( \alpha -\frac{i\Omega _-}{2\hbar }\cot {\Omega _-t}\right) \frac{\Omega _+^2}{h\hbar ^2} \mathrm{cosec }{\Omega _+t}}{\alpha \tilde{\alpha }-\beta ^2-\frac{\Omega _+\Omega _-}{4 \hbar ^2}\coth {\Omega _-t}\cot {\Omega _+t} -i\left( \frac{\Omega _-\tilde{\alpha }}{2\hbar }\coth {\Omega _-t}+ \frac{\Omega _+\alpha }{2\hbar }\cot {\Omega _+t}\right) } \right] ,\nonumber \\ D&= -\left[ \frac{\beta \frac{\Omega _+\Omega _-}{4 \hbar ^2} \mathrm{cosec }{\Omega _-t}\mathrm{cosec }{\Omega _+t} }{\alpha \tilde{\alpha }-\beta ^2 -\frac{\Omega _+\Omega _-}{4 \hbar ^2}\coth {\Omega _-t}\cot {\Omega _+t} -i\left( \frac{\Omega _-\tilde{\alpha }}{2\hbar }\cot {\Omega _-t}+ \frac{\Omega _+\alpha }{2\hbar }\cot {\Omega _+t}\right) }\right] ,\nonumber \\ C&= -\left[ \frac{i\frac{\Omega _-\alpha }{2\hbar }\cot {\Omega _-t} - \frac{\Omega _-^2}{4 \hbar ^2}}{\alpha -i\frac{\Omega _-}{2\hbar }\cot {\Omega _-t}}+ \frac{\beta ^2\alpha _2^2}{k_1\left( \tilde{\alpha }k_1-\beta ^2-k_1\alpha _1\beta _1\right) }\right] . \end{aligned}$$
(96)

These general expressions for the parameters can be extended analytically to the super-critical case by transformation \(\Omega _-\rightarrow i\Omega _-\). Thus the most general results can be obtained from this transformation for super-critical case as well. We can rewrite the Wigner function Eq. (33) in terms of \(x_1\) and \(x_2\) modes as

$$\begin{aligned} W(x_1,x_2,p_1,p_2)&= |A|^2\frac{2\pi }{\sqrt{B_RC_R-D_R^2}}\exp \left\{ -\left( \alpha _1 x_1^2+\alpha _2x_2^2\right. \right. \nonumber \\&\quad \left. \left. +\,\beta x_1 x_2+\sigma _1p_1^2+\sigma _2p_2^2+ \lambda p_1p_2\right) \right\} \nonumber \\&\quad \times \,\exp {\left\{ -\left( \gamma _1x_1p_1+\gamma _2x_2p_2+\delta _1 x_1p_2+\delta _2x_2p_1\right) \right\} }, \end{aligned}$$
(97)

where,

$$\begin{aligned}&\alpha _1=\frac{2}{B_RC_R-D_R^2}\left[ \left\{ B_R\left( C_R^2+C_I^2\right) -2D_ID_RC_I+C_R\left( D_I^2-D_R^2\right) \right\} \cos ^2{\theta }\right. \nonumber \\&\quad +\left\{ B_I^2C_R+B_R^2C_R-2D_ID_RB_I+B_R\left( D_I^2-D_R^2\right) \right\} \sin ^2{\theta }\nonumber \\&\quad \left. +\left\{ -\left( B_RC_I+B_IC_R\right) D_I+\left( B_IC_I-B_RC_R+D_I^2\right) D_R+D_R^3\right\} \sin {2\theta }\right] , \end{aligned}$$
(98)
$$\begin{aligned}&\alpha _2=\frac{2}{B_RC_R-D_R^2}\left[ \left\{ B_I^2C_R+B_R^2C_R-2D_ID_RB_I +B_R\left( D_I^2-D_R^2\right) \right\} \cos ^2{\theta },\right. \nonumber \\&\quad +\left\{ B_R\left( C_R^2+C_I^2\right) -2D_ID_RC_I+C_R\left( D_I^2-D_R^2\right) \right\} \sin ^2{\theta },\nonumber \\&\quad \left. +\left\{ \left( B_RC_I+B_IC_R\right) D_I-\left( B_IC_I-B_RC_R+D_I^2\right) D_R-D_R^3\right\} \sin {2\theta }\right] , \end{aligned}$$
(99)
$$\begin{aligned}&\beta =\frac{2}{B_RC_R-D_R^2}\left[ 2\left\{ (B_RC_I+B_IC_R)D_I\right. \right. \nonumber \\&\quad \left. -(B_IC_I-B_RC_R+D_I^2)D_R-D_R^3\right\} \cos {2\theta },\nonumber \\&\quad +\left\{ -B_I^2C_R-B_R^2C_R+2(B_I-C_I)D_ID_R+C_R\left( D_I^2-D_R^2\right) \right. \nonumber \\&\left. \left. +B_R\left( C_R^2+C_I^2-D_I^2+D_R^2\right) \right\} \sin {2\theta }\right] ,\end{aligned}$$
(100)
$$\begin{aligned}&\lambda =\frac{-D_R\cos {2\theta }+(B_R-C_R)\sin {\theta }\cos {\theta }}{B_RC_R-D_R^2}, \end{aligned}$$
(101)
$$\begin{aligned}&\gamma _1\!=\!\frac{B_RC_I\!+\!B_IC_R\!-\!2D_ID_R\!+\!(B_RC_I\!-\!B_IC_R)\cos {2\theta } \!+\!\left\{ \!-\!(B_R\!+\!C_R)D_I\!+\!(B_I\!+\!C_I)D_R\right\} \sin {2\theta }}{B_RC_R\!-\!D_R^2}, \nonumber \\\end{aligned}$$
(102)
$$\begin{aligned}&\gamma _2\!=\! \frac{B_RC_I\!+\!B_IC_R\!-\!2D_ID_R\!-\!(B_RC_I\!-\!B_IC_R)\cos {2\theta } \!-\!\left\{ -(B_R\!+\!C_R)D_I\!+\!(B_I\!+\!C_I)D_R \right\} \sin {2\theta }}{B_RC_R\!-\!D_R^2}, \nonumber \\\end{aligned}$$
(103)
$$\begin{aligned}&\delta _1\!=\!\frac{2\left[ (C_RD_I\!-\!C_ID_R)\cos ^2{\theta } \!+\!(B_RC_I\!-\!B_IC_R)\cos {\theta }\sin {\theta }\!+\!(B_ID_R\!-\!B_RD_I)\sin ^2{\theta }\right] }{B_RC_R-D_R^2}, \end{aligned}$$
(104)
$$\begin{aligned}&\delta _2\!=\!\frac{2\left[ (B_RD_I\!-\!B_ID_R)\cos ^2{\theta }\!-\!(C_RB_I\!-\!C_IB_R) \cos {\theta }\sin {\theta }\!+\!(C_ID_R\!-\!C_RD_I)\sin ^2{\theta }\right] }{B_RC_R-D_R^2},\end{aligned}$$
(105)
$$\begin{aligned}&\sigma _1= \frac{B_R\cos ^2{\theta }+C_R\sin ^2{\theta }+D_R\sin {2 \theta }}{2\left( B_RC_R-D_R^2\right) }, \end{aligned}$$
(106)
$$\begin{aligned}&\sigma _2= \frac{C_R\cos ^2{\theta }+B_R\sin ^2{\theta }-D_R\sin {2 \theta }}{2\left( B_RC_R-D_R^2\right) }. \end{aligned}$$
(107)

This expression is true in general for both the supercritical as well as the sub-critical case, only for the fact that in the supercritical case some of the parameters become much simplified. We obtain the expression for the reduced Wigner function for the general case by tracing over the \(x_1\) mode.

$$\begin{aligned} {\mathcal W}_R(x_2,p_2)&= \frac{2\pi }{\sqrt{4\alpha _1\sigma _1-\gamma _1^2}} \exp {\left[ -\left( \sigma _2-\frac{\delta _1^2}{4\alpha _1} +\frac{4\alpha _1(\gamma _1\delta _1-2\alpha _1\lambda )^2}{\gamma _1^2-4\alpha _1\sigma _1} \right) p_2^2 \right] }\nonumber \\&\quad \times \exp \left[ -\left( \delta _2-\frac{\sigma _1\beta }{2\alpha _1}+\frac{8\alpha _1(\gamma _1\delta _1-2\alpha _1\lambda )(\beta \gamma _1-2\alpha _1\gamma _2)}{\gamma _1^2-4\alpha _1\sigma _1} \right) p_2x_2\right. \nonumber \\&\quad \left. -\left( \alpha _2-\frac{\beta ^2}{4\alpha _1}+ \frac{4\alpha _1(\beta \gamma _1-2\alpha _1\gamma _2)^2}{\gamma _1^2-4\alpha _1\sigma _1} \right) x_2^2\right] . \nonumber \\ \end{aligned}$$
(108)

The variance in the correlation trajectory will now be determined by the inverse of the coefficient of \(p_2^2\) in the above expression. Thus, we have

$$\begin{aligned} \bar{\gamma }^2=\frac{1}{\left( \sigma _2-\frac{\delta _1^2}{4\alpha _1}+\frac{4\alpha _1(\gamma _1\delta _1-2\alpha _1\lambda )^2}{\gamma _1^2-4\alpha _1\sigma _1} \right) }. \end{aligned}$$

We can verify that for the supercritical limit the region \(f\gg \xi ^2\) sends the variance to the vanishingly small values asymptotically, whereas the sub-critical case does not contain any such regime. We can also obtain the results for the initially uncoupled state results, which is the case (\(D=0\)), from here.

Appendix 3: Wigner function for an arbitrary initial state for \(X_-\)

For the inverted mode \(X_-\) we have the expression for the Kernel in Eq. (86). We will try to obtain the Wigner function for a general initial state \(\Psi _0(X_-)\) which is not vanishing at \(X_-=0\). The time evolved state for this is given as

$$\begin{aligned} \Psi _t(X_-)=\int dX_-'{\mathcal G}_-(X_-,X_-')\Psi _0(X_-'). \end{aligned}$$
(109)

Since, \(\Psi _0(X_-)\) is a continuous differentiable function, we can Taylor expand it around \(X_-=0\) as

$$\begin{aligned} \Psi _0(X_-)=\Psi _0(0)+\sum _{n=1} c_n X_-^n, \end{aligned}$$
(110)

where \(c_n=\Psi ^{(n)}(X_-)|_0\). Therefore, the time evolution will be of the type

$$\begin{aligned} \Psi _t(X_-)\!=\! \exp {\left( \frac{i\Omega _-}{2\hbar }X_-^2\right) }\left[ c_0\!+\!\sum _n c_n \int dX_-'X_-^{'n}\exp {[i\tilde{a}X_-^{'^2}\!-\!i \tilde{b}X_-X_-']}\right] , \end{aligned}$$
(111)

where \(\tilde{a}=\frac{\Omega _-}{2\hbar }\coth {\Omega _- t}\) and \(\tilde{b}=\frac{\Omega _-}{\hbar \sinh {\Omega _- t}}\). We obtain the time evolved state, using

$$\begin{aligned}&\int dx x^{n}\exp {[i\tilde{a}x^2-i \tilde{b}x y]}=-\frac{1}{2}\tilde{a}(-i \tilde{a})^{-\frac{n}{2}-\frac{5}{2}}\nonumber \\&\quad \times \left[ \sqrt{-i \tilde{a}} \tilde{b} \left( 1-(-1)^n\right) y \Gamma \left( \frac{n}{2}+1\right) \, _1F_1\left( \frac{n}{2}+1;\frac{3}{2};-\frac{i \tilde{b}^2 y^2}{4 \tilde{a}}\right) \right] \nonumber \\&\quad -\frac{1}{2}\tilde{a}(-i \tilde{a})^{-\frac{n}{2}-\frac{5}{2}} \times \left[ \tilde{a} \left( (-1)^n+1\right) \Gamma \left( \frac{n+1}{2}\right) \, _1F_1\left( \frac{n+1}{2};\frac{1}{2};-\frac{i \tilde{b}^2 y^2}{4 \tilde{a}}\right) \right] ,\nonumber \\ \end{aligned}$$
(112)

where \({}_1F_1(a,b,z) \) is the Kummer confluent hypergeometric function and \(\Gamma (z) \) is the Gamma function. We realize that the parameter \(\tilde{b}\) becomes vanishingly small at late times and one can verify that in that limit the series expansion of Eq. (111) will have vanishing odd terms. The general structure of the time evolved state, however, will be

$$\begin{aligned} \Psi _t(X_-)= \exp {\left( \frac{i\Omega _-}{2\hbar }X_-^2\right) }\left[ \alpha _0+\sum _n \alpha _n\tilde{b}^{n} X_-^{2n} \right] , \end{aligned}$$
(113)

where \(\alpha _0\) receives contribution from all the even terms of the expansion of Eq. (110) and is non-zero generically. Therefore, the Wigner function will assume the structure

$$\begin{aligned} W[X_-,P_-]\sim |\alpha _0|^2\delta (P_--\Omega _-X_-) + \sum _n \tilde{b}^{n}f_n(X_-,P_-). \end{aligned}$$
(114)

Therefore, we see that unless the initial state is odd, the Wigner function at very late times is dominated by the Dirac delta distribution (upto a normalization).

Appendix 4: Classical equations of trajectories for coupled modes

The mode \(x_2\) is a linear combination of uncoupled modes \(X_+\) and \(X_-\). Now, \(X_{+}\) is a usual harmonic oscillator and hence \(X_{+}\) and \(P_{+}\) are bounded. But \(X_{-}\) and \(P_{-}\) will grow without bound. Therefore, \(x_{2}\) and \(p_{2}\), has the time dependent behavior as,

$$\begin{aligned} x_2&= \sin \theta (A_{-} e^{ \Omega _- t} + B_{-} e^{- \Omega _- t}) + \cos \theta (A_{+} e^{i \Omega _+ t} + A_{+} e^{-i \Omega _+ t} ) \end{aligned}$$
(115)
$$\begin{aligned} p_2&= \Omega _- \sin \theta \left( A_{-} e^{ \Omega _- t} - B_{-} e^{- \Omega _- t}\right) + i \Omega _+ \cos \theta \left( A_{+} e^{i \Omega _+ t} - A_{+} e^{-i \Omega _+ t}\right) . \end{aligned}$$
(116)

At late times, we may neglect the contribution of oscillatory terms (specially in the scenario when \(x_2\) is aligned dominantly along \(X_-\)) to approximate the trajectory as

$$\begin{aligned} p_{2}-\Omega _- x_2 \approx 0~. \end{aligned}$$
(117)

Note that this becomes precise in the limit where \(\cos \theta \) tends to zero, which is a limit that we shall consider later in the analysis. A similar analysis can be done for the \(x_1\) mode gives qualitatively similar results.

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Lochan, K., Parattu, K. & Padmanabhan, T. Quantum evolution leading to classicality: a concrete example. Gen Relativ Gravit 47, 1841 (2015). https://doi.org/10.1007/s10714-014-1841-9

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