content_image/384598.jpg content_image/384599.jpg content_image/384600.jpg content_image/384601.jpg content_image/384602.jpg content_image/384603.jpg content_image/384604.jpg content_image/384607.jpg content_image/384605.jpg content_image/384610.jpg content_image/384611.jpg content_image/384606.jpg
and p$_{c}$ ( t ) = m ˙ x$_{c}$ ( t ) the corresponding momentum, and S$_{c}$ ( t ) = ∫ t 0 dt ( p 2 $_{c}$/ 2 m − mω $^{2}$x2 $_{c}$/ 2 + fx$_{c}$ ) the classical ac- tion. The two variances are given by 

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The Leipnik’s joint entropy 

<img src='content_image/384599.jpg'>

of the Gaussian wave packet (2) is determined by the uncertainty relation as 

Note that the centroid ( x$_{c}$, p$_{c}$ ) does not give any contri- bution to the joint entropy. So, the question is how to find the variances for the position and momentum. There is arbitrariness in choosing a complex solution u to Eq. (3) since any linear superposition of u and u ∗ is also a solution. For instance, the Gaussian wave packet for a free particle with the minimum uncertainty at t = 0 is obtained by the solution 

and for a harmonic oscillator by 

Then, the most general solution satisfying Eq. (4) may involve two parameters r and ϑ as 

where r ≥ 0 and 2 π > ϑ ≥ 0. Here, the r and ϑ have the interpretation as the squeeze parameter and angle [10, 11], which are equivalent to determining two integration constants, Δ x (0) and Δ p (0). In fact, the general solution (11) leads to the Bogoliubov transformation between the Fock bases constructed from u ( t ) and u$_{0}$ ( t ): 

Though the free particle allows a shift of time, which corresponds to choosing a set of r and ϑ , the Hamiltonian (1) does not preserve the time-translational symmetry, so we select the wave packets at t = 0 by adjusting r and ϑ . 

First, we consider the joint entropy for the free parti- cle with ω = 0. As the external force only governs the centroid of the Gaussian wave packet, the joint entropy 

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<img src='content_image/384602.jpg'>

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does not depend on the details of the external force. We thus obtain the Leipnik’s joint entropy 

<img src='content_image/384607.jpg'>

The initial entropy at t = 0 

<img src='content_image/384605.jpg'>

depends not only on the squeeze parameter r but also the squeeze angle ϑ , which is drawn in Fig. 1. One interest- ing point is that the minimum entropy may be provided by the zero-squeeze angle ( ϑ = 0) even for the non-zero squeeze parameter r . The entropy (13) monotonously increases against time in the range of squeeze angle ( π ≥ ϑ ≥ 0) and has the minimum entropy S (0). This is numerically shown in Fig. 2. 

However, in the other range (2 π > ϑ > π ) the entropy has the lower bound 

<img src='content_image/384610.jpg'>

<img src='content_image/384611.jpg'>

<img src='content_image/384606.jpg'>

