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\centerline{\medium On an example of genuine quantum chaos }\par
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{
\itmed
\centerline{M. Kuna}\par
\centerline{Department of Physics}\par
\centerline{Pedagogical College of S\l upsk}\par
\centerline{ul. Arciszewskiego 25, 76-200 S\l upsk, Poland}\par
}
\vskip 2.0 cm
{
\itmed
\centerline{W. A. Majewski}\par
\centerline{Institute of Theoretical Physics and Astrophysics}\par
\centerline{Gda\'nsk University,}\par
\centerline{ul.Wita Stwosza 57, 80-952 Gda\'nsk, Poland} \par
\centerline{e-mail: fizwam@halina.univ.gda.pl} \par
}
\vfil
\noindent
{\medium Abstract}: The first example of a quantum system with the
genuine quantum chaos is presented.
\vfil
\noindent
PACS numbers: 03.65.Db,05.45.+b
\vfil
\eject
\normal
Much progress has been achieved in understanding and calculating various
properties of quantal and semiquantal models with chaotic behaviour during the last two
decades. A~prominent part of these investigations deals with the study
of semiquantal models. Here by a
semiquantal model we understand a classical dynamical system
which can be derived from a purely quantum--mechanical system by
limiting procedure. It still remains ,
however, to determine whether or not the chaotic behaviour of the semiquantal model
reflects ``true" chaos in the quantum system.
The first question we meet concerns
the definition of ``quantum chaos". Several definitions exist and their
interconnections have not been fully elucidated yet.
In the present paper, by genuine quantum chaos we understand the
positivity of the quantum characteristic exponents which were
recently introduced and discussed by us (cf~[1],~[2],~[3]).
Although it is easy to give a (mathematical) model with
positive quantum characteristic exponent (see [2]), there still
is no model of a purely quantum--mechanical system with genuine
quantum chaos. In order to improve this situation we reexamine the Milburn
model (see [4], [5], [6]).
We will show that this model can be considered as an example of a
quantum system with genuine quantum chaos. For this
reason, one can say that our approach to the description of
chaotic behaviour of quantum dynamical systems seems to be fully justified.
Let us turn to the description of the Milburn model. It is
the quantum optical counterpart of a parametrically kicked nonlinear
oscillator. Its hamiltonian $H$ is of the form (cf.~[4], [6]):
$$H = H_{NL} + H_{PA} \eqno(1)$$
where
$$H_{NL} = {\chi \over 2}(a^{\ast})^{2}a^2 \eqno(2)$$
and
$$H_{PA} = i{\hbar}{\kappa \over2}[(a^{\ast})^{2} -a^{2}]\cdot\sum_{n=-\infty}^{+\infty} \delta (t - n\tau ) \eqno(3)$$
In above formulae, $\chi$ is a constant proportional to the
third order nonlinear susceptibility of the medium, $a$ stands for
the boson annihilation operator, $\kappa$ is the coupling
constant (the product of the pump field amplitude and the
second--order nonlinearity in the parametric gain medium).
Finally, $\tau$ is the period of free evolution (i.e. the
evolution described by $H_{NL}$) between each
pump pulse. Let us remark that these pump pulses mimick
kicks of the harmonic oscillator. The Heisenberg equation of motion
for the nonlinear hamiltonian $H_{NL}$ gives the following
evolution for $a$:
$$a(t) = e^{-i{\chi}ta^{\ast}a}a(0). \eqno(4)$$
Then, the time evolution of the system can be described (see [4], [6])
by the equation:
$$a(t_{n}^{+}) = a(t_{n}^{-})\cosh r + a^{\ast}(t_{n}^{-})\sinh r \eqno(5)$$
where $t_{n}^{+}\,(t_{n}^{-})$ is the time just after (before) the
passage of the n-th pulse, r is the effective constant for the
kick and the time dependence of $a$ on $t$ is given by (4).
Now, we want to calculate quantum characteristic exponents
${\lambda}^q$ for the evolution ${\cal U}$ given by (4) and (5), i.e.
${\cal U}a(t_{n}^{+}) = a(t_{n+1}^{+})$ and a quantum
characteristic exponent is defined by
$${\lambda}^{q}({\cal U}; x, y) {\buildrel \rm def \over =} \lim\limits_{n\to\infty}\,{1\over n}\,\log\Vert(D_{x}{\cal U}^{n})(y)\Vert \quad $$%(\equiv \lambda^q) \eqno (4)$$
where $(D_{x}{\cal U}^{n})(y)$ denotes derivatives of ${\cal U}$
composed with itself $n$ times at $x$ in direction $y$. To do
so, let us rewrite (4) and (5) in terms of selfadjoint operators
$\Phi$ and $\Pi$, where $\Phi$ and $\Pi$ are defined by:
$$a = \Phi + i\Pi \eqno(6)$$
Then, (4), (5) and their conjugate equations can be written as:
$$\left( \matrix{{\Phi}(t_{n}^{+})\cr
{\Pi}(t_{n}^{+})\cr}
\right)
= e^{-i{\mu}\over 2}
\left( \matrix{ e^r \cos {\mu}B_0 & e^r \sin {\mu}B_0 \cr
-e^{-r} \sin {\mu}B_0 & e^{-r} \cos {\mu}B_0\cr}
\right)
\left( \matrix{{\Phi}(t_{n-1}^{+})\cr
{\Pi}(t_{n-1}^{+})\cr}
\right) \eqno(7)$$
where ${\mu} = {\chi}{\tau}$ and $B_0 = a^{*}a = ({\Phi}^2 + {\Pi}^2 -{1\over 2})$.
Now, let us change the time evolution (7) slightly. Namely, we
will impose a control over the possible number of photons in the
propagator of free evolution. We ought to do this modification
in order to have a real influence of kicks on the mode. In order
to carry out the considered change let us denote by $p_n$ the
projection on the $n$--photon state. Clearly
$$[ H_{NL}, P_N ] = 0 = [ a^{*}a, P_N ]$$
where $P_N = \sum_{n=1}^N p_n$. Thus we replace (7) by
$$\left( \matrix{{\Phi}(t_{n}^{+})\cr
{\Pi}(t_{n}^{+})\cr}
\right)
= e^{-i{\mu}\over 2}
\left( \matrix{ e^r \cos {\mu}B & e^r \sin {\mu}B \cr
-e^{-r} \sin {\mu}B & e^{-r} \cos {\mu}B\cr}
\right)
\left( \matrix{{\Phi}(t_{n-1}^{+})\cr
{\Pi}(t_{n-1}^{+})\cr}
\right) \eqno(8)$$
with $B = \left( B_0\cdot P_N \right){\oplus}P^{\perp}_N$.
Next, let us consider the case:
$$A^2 \equiv {\cos}^{2}{\mu}B{\cosh}^{2} r - 1 > {\epsilon}1 \eqno(9)$$
with arbitrary small ${\epsilon} > 0$. The opposite case $A^2 <
-{\epsilon}1$ can be treated in a similar way. Let us observe that the
following condition for the parameter ${\mu}$:
$${\mu} \in X = \{ x \in R; x > 0, x\cdot n \ne (2k -1)\cdot
{{\pi}\over 2}\,\, \rm for \,\, any \,\,k \in {\cal N} \,\, and
\,\,n \in N_0 \} \eqno(10)$$
where $N_0 = \{1, \dots N\}$ implies the nontriviality of the
operator $A$ $(A \ne -1)$. Then, for large enough $r$, the
equality (9) is satisfied. Moreover, let us assume that
$${\mu} \in X \cap Y \equiv Z = \{ x \in R; x > 0, x\cdot n \ne
{k\over 2}\cdot {\pi}\,\, \rm for \,\, any \,\,k \in {\cal N} \,\, and
\,\,n \in N_0 \} \eqno(11)$$
where
$$Y = \{ x \in R; x > 0, x\cdot n \ne k\cdot {\pi}\,\, \rm for \,\, any \,\,k \in {\cal N} \,\, and
\,\,n \in N_0 \} \eqno(12)$$
The condition ${\mu} \in Y$ implies that the function
${\sin}^{-1}{\mu}B$ is well defined (note that the spectrum of
$B$ is equal to $\{1, \dots N\}$). Under the above conditions
we can rewrite (8) as:
$$\left( \matrix{{\Phi}(t_{n}^{+})\cr
{\Pi}(t_{n}^{+})\cr}
\right)
= e^{-i{\mu}\over 2}
{\bf PDP}^{-1}
\left( \matrix{{\Phi}(t_{n-1}^{+})\cr
{\Pi}(t_{n-1}^{+})\cr}
\right) \eqno(13)$$
where
$${\bf P} = \left( \matrix{ 1 & 1 \cr
-e^{-r}({\sin}{\mu}B)^{-1}({\cos}{\mu}B{\sinh}r + A) & -e^{-r}({\sin}{\mu}B)^{-1}({\cos}{\mu}B{\sinh}r - A) \cr}
\right) \eqno(14)$$
$${\bf P}^{-1} = \left( \matrix{ - {\cos}{\mu}B{\sinh}r(2A)^{-1} + {1\over 2} & -e^{r}{\sin}{\mu}B(2A)^{-1} \cr
{\cos}{\mu}B{\sinh}r(2A)^{-1} + {1\over 2} & e^{r}{\sin}{\mu}B(2A)^{-1} \cr}
\right) \eqno(15)$$
and
$${\bf D} = \left( \matrix{ {\cos}{\mu}B{\cosh}r - A & 0 \cr
0 & {\cos}{\mu}B{\cosh}r + A \cr}
\right)
\equiv \left( \matrix{ {\Lambda}_1 & 0 \cr
0 & {\Lambda}_2 \cr}
\right) \eqno(16)$$
Therefore
$$\left( \matrix{{\Phi}(t_{n}^{+})\cr
{\Pi}(t_{n}^{+})\cr}
\right)
= e^{-in{\mu}\over 2}
{\bf P{D}^{n}P}^{-1}
\left( \matrix{{\Phi}(0)\cr
{\Pi}(0)\cr}
\right) \eqno(17)$$
where we have used the fact that ${\bf D}$ does not depend on
$t_n$.
As the next step in exhibiting a positive quantum characteristic
exponent for our model let us consider a ``rotation'' in $({\Phi}, {\Pi})$
variables. To do so let us define:
$${\Phi}^{\epsilon} = {1\over 2}[e^{i{\epsilon}}a + e^{-i{\epsilon}}a^{\ast}] \eqno(18)$$
and
$${\Pi}^{\epsilon} = {1\over 2i}[e^{i{\epsilon}}a - e^{-i{\epsilon}}a^{\ast}] \eqno(19)$$
Clearly
$$({\Phi}^{\epsilon})^2 + ({\Pi}^{\epsilon})^2 = a^{\ast}a + {1\over 2} = ({\Phi})^2 + ({\Pi})^2 .\eqno(20)$$
Moreover
$$\left( \matrix{{\Phi}^{\epsilon}(t_{n}^{+}) - {\Phi}(t_{n}^{+})\cr
{\Pi}^{\epsilon}(t_{n}^{+}) - {\Pi}(t_{n}^{+})\cr}
\right)
= e^{-in{\mu}\over 2}
{\bf P{D}^{n}P}^{-1}
\left( \matrix{{\Phi}(0)^{\epsilon} - {\Phi}(0)\cr
{\Pi}(0)^{\epsilon} - {\Pi}(0)\cr}
\right). \eqno(21)$$
Therefore:
$$\left( \matrix{D_{\epsilon}({\Phi}^{\epsilon})^{(n)}\cr
D_{\epsilon}({\Pi}^{\epsilon})^{(n)}\cr}
\right)
= e^{-in{\mu}\over 2}
{\bf P{D}^{n}P}^{-1}
\left( \matrix{-{\Pi}(0)\cr
{\Phi}(0)\cr}
\right) \eqno(22)$$
where $D_{\epsilon}{\Phi}^{\epsilon} = \lim_{{\epsilon}\to 0}
{{{\Phi}^{\epsilon} - {\Phi}}\over{\epsilon}}$\quad ( $D_{\epsilon}{\Pi}^{\epsilon} = \lim_{{\epsilon}\to 0}
{{{\Pi}^{\epsilon} - {\Pi}}\over{\epsilon}}$).
Remarks:
\item{i)} Let us note that $D_{\epsilon}{\Phi}^{\epsilon}$
is well defined as the
derivative with respect to the parameter ${\epsilon}$ of the
one--parameter family ${\Phi}^{\epsilon}$
of closed operators. Similary, $D_{\epsilon}{\Pi}^{\epsilon}$ is
related to ${\Pi}^{\epsilon}$.
\item{ii)}The operators ${\Phi}^{\epsilon}$,
${\Pi}^{\epsilon}$ and the phase angle ${\epsilon}$ can be
used to characterize squeezed states (cf. [7]).
\item{iii)} Let us recall that the basic motivation for the study
of characteristic exponents is the problem of stability. Thus,
here we study the stability properties of the evolution of
Milburn model with respect to the phase angle ${\epsilon}$.
As the final step of calculating a positive quantum exponent for
considered model of the kicked oscilator let us introduce the
following cut--off in the $({\Phi}, {\Pi})$ variables. We replace
${\Phi} ({\Pi})$ by ${\Phi}_{\delta} = \int_{-{\delta}}^{+{\delta}}\,\,{\lambda}dE_{\Phi}({\lambda})$
\quad (${\Pi}_{\delta} = \int_{-{\delta}}^{+{\delta}}\,\,{\lambda}dE_{\Pi}({\lambda}) \,\,)$
where ${\delta} \in R^{+}$ and $\{E_{\Phi}({\lambda})\}$ $(\{E_{\Pi}({\lambda})\})$
stands for the spectral resolution of ${\Phi} ({\Pi})$ respectively.
Consequently, we will consider
$$\left( \matrix{D_{\epsilon}({\Phi}_{\delta}^{\epsilon})^{(n)}\cr
D_{\epsilon}({\Pi}_{\delta}^{\epsilon})^{(n)}\cr}
\right)
= e^{-in{\mu}\over 2}
{\bf P{D}^{n}P}^{-1}
\left( \matrix{-{\Pi}_{\delta}(0)\cr
{\Phi}_{\delta}(0)\cr}
\right) \eqno(23)$$
Let us remarks that (23) strongly converges to (22) as ${\delta}
\to\infty$. Consequently, (23) is well defined approximation of
the genuine dynamics.
Then, introducing new variables: $\tilde{\Phi}, \tilde{\Pi}$
$$\left( \matrix{\tilde{\Phi}_{\delta}\cr
\tilde{\Pi}_{\delta}\cr}
\right)
{\buildrel \rm def \over =}\,\,
{\bf P}^{-1}
\left( \matrix{{\Phi}_{\delta}\cr
{\Pi}_{\delta}\cr}
\right) \eqno(24)$$
one has (cf. the second equality in (16))
$$\Vert D_{\epsilon}\tilde{\Phi}_{\delta}^{\epsilon} \Vert = \Vert {\Lambda}_{1}^{n}\tilde{\Pi}_{\delta}(0) \Vert \eqno(25)$$
$$\Vert D_{\epsilon}\tilde{\Pi}_{\delta}^{\epsilon} \Vert = \Vert {\Lambda}_{2}^{n}\tilde{\Phi}_{\delta}(0) \Vert \eqno(26)$$
Further, let us remark that 0 does not belongs to the spectrum
of ${\Lambda}_2$. Hence, 0 is in the resolvent set of
${\Lambda}_2$ and ${\Lambda}_2$ is a bijection. Therefore,
$$\Vert {\Lambda}_{2}^{n}\tilde{\Phi}_{\delta}(0) \Vert = C \Vert {\Lambda}_{2}^{n} \Vert \eqno(27)$$
where
$$C = \sup_{g:\, g={\Lambda}_2^{n}f}\Vert \tilde{\Phi}_{\delta}(0)g \Vert\quad (\ne 0) \eqno(28)$$
On the other hand, the following property of the norm
$$\Vert A^{\ast}A \Vert = \Vert A {\Vert}^2 \eqno(29)$$
implies
$$\Vert {\Lambda}_{2}^{2^k} \Vert = \Vert {\Lambda}_{2} {\Vert}^{2^k} \eqno(30)$$
for any $k \in {\cal N}$. Consequently
$${\lambda}^{q}(\tilde{\Pi}_{\delta}) = \lim\limits_{k\to\infty}\,{1\over 2^k}\,\log\Vert D_{\epsilon}\tilde{\Pi}_{\delta}^{\epsilon}(t_{2^k}) \Vert \eqno(31)$$
$$ = \lim\limits_{k\to\infty}\,{1\over 2^k}\,\log\Vert{\Lambda}_{2}^{2^k} \Vert = \log\Vert{\Lambda}_{2} \Vert$$
Now, it is clear that for large enough $r$ ($r$ is the effective
constant for kicks) the norm of ${\Lambda}_2$ is larger than 1.
Therefore, we can conclude that for some values of ${\mu}$, $r$
(i.e.~${\chi}$,~${\tau}$,~${\kappa}$)
the quantum characteristic exponent ${\lambda}^{q}$ for quantum variable
$\tilde{\Pi}_{\delta}$ is strictly positive. To comment on this
result let us remark that it is obvious that in polyparametric
cases physics as well as mathematics allow numerous combinations
of stability in certain directions and irregularity in others.
Therefore, we can expect chaotic evolution of our model for
some values of ${\chi}$,~${\tau}$,~${\kappa}$ and regularity for
others and we get a confirmation of such behaviour
(${\lambda}^{q}(\tilde{\Phi}_{\delta})$ can be negative). Moreover,
let us recall that the original Milburn model also has such
a property. Let us finish with the conclusion that
working in the ``pure'' quantum description of a genuine nonlinear
quantum system we are able to speak rigorously about chaotic properties of the
quantum evolution without any (semi)classical limits
and other (semi)classical approximations leading to semiquantal models.
\leftline{\medium{Acknowledgement:}}
We are grateful to Prof.~R.Streater for reading the manuscript
and helpful comments. This work has been partially supported by the grant KBN/1436/2/91.
\bigskip
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\leftline{\medium{References: }}
\noindent
$^1${\bf W.A.Majewski, M.Kuna} On the definition of quantum
characteristic exponents, submitted to {\itmed{Proccedings of
the Workshop on Phase Transitions: Mathematics, Physics,
\break Biology...}}, Prague (1992).
\noindent
$^2${\bf W.A.Majewski, M.Kuna} On quantum characteristic exponents,
SFB 237, Preprint 153, Ruhr-Universit{\"a}t-Bochum(1992)
\noindent
$^3${\bf M.Kuna, W.A.Majewski} On quantum chaos and quantum
characteristic exponents, Rep. Math. Phys. {\bf 33}(1993), to appear
\noindent
$^4${\bf G.J.Milburn}, Phys. Rev. A.{\bf 41}, 6567(1990)
\noindent
$^5${\bf G.J.Milburn, C.A.Holmes}, Phys. Rev. A, {\bf 44}, 4704(1991)
\noindent
$^6${\bf B.Wielinga, G.J.Milburn}, Phys. Rev. A, {\bf 46}, 762(1992)
\noindent
$^7${\bf B.Yurke}, Squeezed light, University of Rochester
and AT{\&}T Bell Laboratories, 1989
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\eject
\bye