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
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.
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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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
Evolution of this general Gaussian state will be obtained from the Kernel of the system, obtained easily when written in normal modes
where,
with,
For the initial state of the above mentioned kind, the time evolution gives
with the condition of the state being normalizable imposed through
where we have
Now, the time evolved state Eq. (87) can be rewritten in a more comprehensible form as
where, we have
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
and for mode \(x_2,\)
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
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
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
where,
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.
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
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
Since, \(\Psi _0(X_-)\) is a continuous differentiable function, we can Taylor expand it around \(X_-=0\) as
where \(c_n=\Psi ^{(n)}(X_-)|_0\). Therefore, the time evolution will be of the type
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
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
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
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,
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
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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DOI: https://doi.org/10.1007/s10714-014-1841-9