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molecular localization, time-dependent Schrodinger equation
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% This is a LaTeX file containing 2 PostScript figures
%
% Dr. Andrea Sacchetti
% Department of Mathematics,
% University of Modena,
% Via Campi 213/B, I-41100, Modena, Italy
%
% email: Sacchetti@unimo.it
%

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\documentstyle [preprint,aps]{revtex}
\begin{document}

\draft

\title {Molecular localization induced by collisions}

\author
{Vincenzo Grecchi}

\address
{Dipartimento di Matematica, Universit\`a di Bologna, \\
Piazza di Porta S. Donato 5, I--40127 Bologna, Italy,\\
email: Grecchi@dm.unibo.it.}

\author
{Andrea Sacchetti}

\address
{Dipartimento di Matematica, Universit\`a di Modena,\\
Via Campi 213/B, I--41100 Modena, Italy,\\
email: Sacchetti@unimo.it}

\date {\today}

\maketitle

    \begin {abstract}
    {\bf Abstract:}
    We consider a periodically driven double well potential as a dynamical
    model for molecular localization induced by collisions. \ If the
    frequency of the collisions is high enough, we have effective
    localization, due to the larger instability of one of the metastable
    states. \ We show also that the red shift effect of the inversion line
    can be given by this model.
    \end{abstract}

\pacs {PACS: 03.65.-w, 33.80.Be, 73.40.Gk, 82.90.+j}

% 03.65.-w, Quantum mechanics
% 33.80.Be, Level crossings and optical pumping
% 73.40.Gk, Tunneling
% 82.90.+j, other topics in physical chemistry

    In this letter we return on the old problem of the explanation of the
    {\it molecular localization} (hereafter ML) hypothesis, successfully
    used in chemistry as the {\it molecular structure} concept, in the
    rigorous quantum mechanics (hereafter QM) framework \cite {W}. \ QM
    predicts that stationary states have a distribution probability
    satisfying the same symmetrical properties of the Hamiltonian; in
    contrast, ML hypothesis considers molecular states having a shape that
    drastically breaks the symmetry of the Hamiltonian. \ The qualitative
    explanation of this apparent contradiction is simple: the molecule is
    not an isolated system, and there are no stationary states \cite {A}. \
    The problem is the understanding of the basic effect that makes a
    simple molecular system to have some macroscopic properties.

    In particular, we consider the case of the ammonia molecule, where the
    model for the motion of the nitrogen atom is given by means of a
    double well potential with large internal barrier; we expect to
    observe the pyramidal shape, i.e. the nitrogen atom in one well, with
    probability $1/2$ \cite {W}. \ Experiments show that the {\it
    inversion line}, related to the splitting of the ground state, vanishes
    (red shift) in the limit of large pressure of the gas and we have the
    pyramidal shape with full probability. \ The argument of the vanishing
    of the splitting (hereafter VS) is theoretically relevant; indeed,
    localization of a double well stable state, by means of a small static
    perturbation, is accompanied by the growing of the splitting \cite
    {JL}. \ Recently, in a joint work, A Martinez and us showed that the ML
    and VS are both obtained in the case of metastable states \cite
    {GMS}.

    In this paper the instability in the double-well model is generated in
    a dynamical way by means of a periodically time-dependent external
    perturbation. \ The time-dependent perturbation simulates the dynamical
    influence of the environment on the ammonia molecule, that is the
    collisions with the other molecules of the gas where the collision
    frequency is related to the gas pressure. \ The periodicity is not a
    natural condition, but is the simplest one and turns out to be
    compatible with the expected results, at least for a frequency value
    not too large; in fact, for very high frequencies one should take into
    account non periodic effects due to the occurrence of less frequent but
    stronger collisions. \ The results give the generality of the ML for
    high frequency and in the simultaneous limit of small perturbation and
    large barrier between the wells; in particular we have taken the
    strength of the perturbation proportional to the square root of the
    splitting. \ Moreover, by means of a small static
    perturbation term, which would partially restore the broken symmetry,
    we make the two Floquet resonances to (anti-)cross. \ In such a case we
    obtain the gradual vanishing of the splitting (red shift of the
    inversion line) at a critical pressure; at this value of the
    pressure (frequency), we have that the lifetime of the system is of
    the same order of the beating period.

    In order to have explicit results we consider an elementary double well
    model given by a one-dimensional Schr\"odinger operator $H$ with a
    real-valued symmetric double well potential $V$ such that $\lim_{|x|\to
    \infty} V(x)=0$. \ The discrete spectrum of $H$ is given by two
    negative non-degenerate eigenvalues (energy levels) $\lambda_1
    <\lambda_2$ and the continuous spectrum by the real-positive axis. \
    The difference between the two levels gives the small splitting, due to
    the large barrier between the wells, defined as $\omega =2\delta$,
    $\delta= (\lambda_2 -\lambda_1)/2 \hbar$. \ The asymmetrical
    time-dependent perturbation $W$, that simultaneously introduces the
    breaking of the symmetry and the instability of the system, is given by
    means of a potential periodic in time on one of the wells and vanishing
    on the other well; that is $W(x,t) = \epsilon v(\mu t) f(x)$ where
    $\epsilon$ is a small real and positive parameter, $v(t)$ is a
    periodic (kick-type) function with period $2\pi $ and $f(x)$ is a
    real-valued function with compact support contained in one well.

    By means of new techniques for the analysis of non-autonomous
    Hamiltonian systems we compute here the Floquet spectrum of the
    time-dependent Schr\"odinger operator $H+W$ with the rigorous control
    of the error \cite {SW}. \ To this end let the solution of the
    time-dependent Schr\"odinger equation $i\dot \phi = (H+W)\phi$ be
    written as
%
\begin {equation}
\phi (t,x) = a_+ (t) \psi_+ (x) + a_- (t) \psi_- (x) + \phi_c (t,x)
, \label {l2}
\end {equation}
%
    where $\psi_{\pm}= (\psi_1 \pm \psi_2)/{\sqrt {2}}$ are the {\it
    single-well} states, $\psi_{1,2}$ are the eigenvectors of $H$
    associated to the eigenvalues $\lambda_{1,2}$ and $\phi_c=P_c \phi$
    where $P_c$ denotes the projection operator on the eigenspace
    associated to the continuous spectrum of $H$, that is $\langle \phi_c
    (t,\cdot ) ,\psi_{\pm} \rangle =0$ for any $t$; $\langle \cdot ,\cdot
    \rangle$ denotes the usual scalar product on the Hilbert space $L^2$. \
    The substitution of $\phi$ by (\ref {l2}) leads to the following system
    of equations
%
\begin {equation}
\left \{
\begin {array}{lcl}
i \dot a_+
&=& \Omega a_+ - \delta a_-  +
\langle \psi_+ ,W\phi \rangle \\
i \dot a_- &=& -\delta a_+ + \Omega a_- +
\langle \psi_- ,W\phi \rangle \\
i \dot \phi_c &=& H \phi_c +P_c W \phi
\end {array}
\right. \label {l3}
\end {equation}
%
    where $\Omega = (\lambda_1 + \lambda_2)/2$. \ Let $\tau = \delta t$ be
    a {\it slow time} (in this way the beating period becomes $2\pi$), $F$
    be the $2\times 2$ matrix with elements $F_{\pm ,\pm} = \langle
    \psi_\pm , f \psi_\pm \rangle$, $I=s_1$ be the first Pauli matrix, $A
    (\tau )$ be the column matrix with elements $A_\pm (\tau ) = a_\pm (t)
    e^{i \Omega t }$ and $R_c$ be the column matrix with elements
    $R_{c,\pm} = \frac {1}{\delta} e^{i\Omega t} \langle \psi_\pm ,W \phi_c
    \rangle$, then the first two equations of the above system can be
    written as
%
\begin {equation}
i A'(\tau ) = \left [ - I + ({\epsilon}/{\delta}) v\left (
{\mu}\tau /{\delta} \right ) F \right ] A(\tau ) + R_c. \label {l3bis}
\end {equation}

    We remark that in the limit of large barrier between the two wells then
    $\delta \ll 1$ and the single-well states $\psi_\pm$ are asymptotically
    given by $\psi_0 (\pm x)$, where $\psi_0 (x)$ is localized in one well,
    for instance the right one \cite {LL}. \ If, as we can assume without
    lack of generality, $f(x)$ has compact support contained in the right
    well, then it follows that $F_{+,+}\sim f_0 = \langle \psi_0, f \psi_0
    \rangle$ and the other terms of $F$ can be neglected in the large
    barrier limit.

    Now, it is a matter of integration by parts and use of the third
    differential equation of the system (\ref {l3}), to obtain that the
    remainder term in equation (\ref {l3bis}) can be arranged as
%
\begin {eqnarray*}
R_c=({\epsilon^2}/{\delta}) U A + R,
\end {eqnarray*}
%
    where $U$ is a $2\times 2$ matrix independent of $\tau $ and $R$ is a
    remainder term obtained collecting the other contributes. \ As above,
    in the large barrier limit, we have that
%
\begin{eqnarray*}
    U_{+,+} \sim u_0= -4\sum_{n\not= 0} |c_n|^2 \Lambda (\Omega + n\mu ),
\end{eqnarray*}
%
    where $c_n$ are the Fourier coefficients of $v(t)$, and
%
\begin{eqnarray*}
\Lambda (E) =  \langle f \psi_0 ,
[H - (E  +i0)]^{-1} P_c f  \psi_0 \rangle ,
\end{eqnarray*}
%
    the other terms of $U$ can be neglected; we remark that $\Im u_0 (\mu )
    \le 0$ for any $\mu$ and, generically, $\Im u_0 (\mu )<0$ provided that
    $\Omega + n\mu >0$ for some $n$ such that $c_n \not= 0$. \ If we denote
    $v=v_p+v_0$, where $v_0$ is the mean value of $v(t)$ in one period, and
%
\begin {eqnarray*}
M= - I + ( {\epsilon v_0}/{\delta}) F + ( {\epsilon^2}/{\delta} ) U,
\end {eqnarray*}
%
   then (\ref {l3bis}) takes the form
%
\begin{eqnarray*}
i A'(\tau ) = \left [ M + ({\epsilon}/{\delta}) v_p \left ( {\mu}\tau
/{\delta} \right ) F \right ] A (\tau )+  R,
\end {eqnarray*}
%
    which has solution
%
\begin{eqnarray*}
A(\tau ) = e^{ -i (\epsilon /\delta )\int_0^{\tau} v_p(\mu s/\delta ) ds F }
\left [e^{-i M\tau }A(0) + R_A(\tau ) \right ],
\end{eqnarray*}
%
    where $e^{ -i (\epsilon /\delta ) \int_0^{\tau} v_p(\mu s/\delta ) ds F
    } = 1 + O (\epsilon /\mu )$, for any $\tau$ fixed, since $v_p$ has mean
    value zero. \ In order to bound the elements $R_{A,\pm }$ of the
    remainder term $R_A(\tau )$, in the simultaneous limit of small
    perturbation and large barrier, we assume the following time-decay
    behavior for some $s>1$: $e^{-i H t} P_c \sim O(t^{-s} )$ as specified
    in \cite {M}. \ Then we can prove that for $\epsilon$ small enough and
    for any $\tau$ fixed
%
\begin {eqnarray}
|R_{A,\pm} (\tau )| \le ({\epsilon^3}/{\delta}) C \max |A_\pm (0)|, \label
{l8bis}
\end {eqnarray}
%
    for some positive constant $C$.

    Hence, the beating effect between the two wells is completely described
    by means of the eigenvalues $\ell_{1,2}$ of the matrix $M$; indeed,
    from (\ref {l8bis}) we obtain that
%
\begin {eqnarray*}
A_\pm (\tau ) \sim c_{1,\pm} e^{-i\ell_1 \tau } + c_{2,\pm }e^{-i\ell_2
\tau },
\end {eqnarray*}
%
    for any $\tau$ fixed and for some $c_{1,\pm}$ and $c_{2,\pm}$. \ In the
    large barrier limit, the leading term of these eigenvalues is given by
    $\ell_{1,2} \sim \left [ u \pm \sqrt {1 +u^2} \right ]$, where
    $u=(u_0k+z)/2$ and where we define $k=\epsilon^2/\delta$ and
    $z=\epsilon v_0 f_0 /\delta$. \ In the following we assume $v_0$ small
    enough (in particular $|v_0| \le \epsilon$); in the opposite case, it
    follows that the main effect of the perturbation is a drastic
    destruction of the symmetry and the time-dependent term does not affect
    the dynamics in a substantial way.

    Now, in order to give a more explicit description, we consider the
    one-dimensional double-well model with piece-wise constant static
    potential $V(x)=V_1 \chi_{[0,x_1]}(|x|) - V_2 \chi_{(x_1,x_1+a)}(|x|)$,
    where $\chi_I (x)$ is equal to $1$ for any $x\in I$ and $0$ otherwise,
    $a,x_1,V_1$ and $V_2$ are positive constants such that the discrete
    spectrum of $H$ consists of only two eigenvalues; for instance, $a=5$,
    $V_2=0.25$, $x_1=4$ and $V_1$ large enough such that $\Omega =-0.06932$
    and $\delta \ll 1$. \ For what concerns the perturbation we take $f(x)
    = \chi_{(x_1,x_1+a)}(x)$ and $v_p(t)$ with Fourier coefficients
    $c_n=c_{-n}=\frac {\sin (n \pi /N) }{2n/N} (1-(-1)^n)$, $n=1,\ldots ,
    N$, $c_j=c_0=0$, $|j|>N$, we fix $N=12$. \ In order to compute the
    function $\Lambda (E)$, in the large barrier limit, we can use the
    Dirichlet operator $H_D$ on $L^2(0,+\infty)$ with potential equal to
    $-V_2$ on the interval $(0,a)$ and zero otherwise. \ From this and from
    the fact that $f$ and $\psi_0$ are real-valued functions it follows
    that
%
\begin {eqnarray*}
&&\Lambda (E) \sim \left \langle f \psi_0
,[H_{D} - (E+i0) ]^{-1} f \psi_0 \right \rangle \\
&& =  - \frac
{2}{w} \int_0^a f(x) \psi_0 (x) u_1 (x) dx \int_x^a f(y) \psi_0 (y)
u_2 (y) dy
\end {eqnarray*}
%
    where $u_1$ and $u_2$ are the solutions of the equation $[H_{D}-
    (E+i0)]u=0$ such that $u_1 (0)=0$ and $u_2 (+\infty )=0$, and $w$
    denotes the Wronskian \cite {K}.

    In the case $k \ll 1$, then $\ell_{1,2} \sim \pm 1$ and we still have
    the unperturbed beating effect. \ In contrast, for $v_0=0$ and $k$
    fixed and large enough, i.e. $k\ge k_0$ for some $k_0$ but such that
    $\epsilon k \ll 1$ in order to use (\ref {l8bis}), we obtain the wanted
    result: that is the beating effect gradually disappears for the
    frequency $\mu$ increasing and larger than a threshold value, in fact
    we take into account only a finite number of Fourier harmonics of
    $v_p$. \ Indeed, $\Im \ell_1 \sim 0 $ and $\Im \ell_2 \sim k \Im u_0
    (\mu) < 0$ for $\mu$ not too small (see Figure \ref {fig1}a), then we
    obtain the destruction of the beating effect because the right
    single-well state rapidly decays, actually it is a metastable state
    with lifetime shorter than the beating period. \ In order to consider
    the contribution given by the static term $v_0$, we define the minimum
    splitting value defined for any $\mu$ as $\Delta (\mu )= \min_z |\Re
    (\ell_2 -\ell_1 )|$, where $k\ge k_0$ is fixed as above. \ Then we have
    the vanishing of $\Delta (\mu )$ for any $\mu > \mu^\star$, as appears
    in Figure \ref {fig1}b, where $\mu^\star$ is the critical frequency for
    which we have an exact crossing of the two eigenvalues $\ell_{1,2}$
    (see Figure \ref {fig2}b).

    In Figures \ref {fig2}a--c we plot the imaginary and real parts of the
    eigenvalues $\ell_{1,2}$ for three different values of the frequency. \
    For $\mu$ equal to the critical value $\mu^\star$ we have the exact
    crossing (Figure \ref {fig2}b); in such a case we observe a damped
    beating effect with beating period much greater than the unperturbed
    one. \ For $\mu$ slightly larger than the critical value $\mu^\star$
    we have the vanishing of the real part of the splitting and the
    imaginary part of $\ell_{1,2}$ is substantially unaffected (Figure \ref
    {fig2}c); in such a case we have the destruction of beating effect and
    the localization result. \ In contrast, for $\mu < \mu^\star$ we have
    the avoided crossing of the real part of the eigenvalues $\ell_{1,2}$
    (Figure \ref {fig2}a) and we found, at the avoided crossing point, the
    unperturbed splitting value, i.e. $\Delta (\mu )$ close to $2$; in
    such a way we have that the static term $v_0$ would restore the broken
    symmetry due to $v_p$.

    We close by underlining that the study of the dynamics of a two-level
    system under an external time-dependent perturbation is interesting by
    itself and it appears in many other fields; from the study of spin
    systems coupled with an environmental sea \cite {NHC} to the study of
    quantum transport of electrons in heterostructures modulated by
    external fields \cite {MFDQNR}, to name but a few. \ We also recall
    previous discussions about the possible destruction of the beating (or
    tunneling) effect \cite {GDJH} in the different limit $\delta \ll \mu
    \ll \epsilon$, that is for the external frequency going to zero
    together with the perturbation strength, and for $v(t)=\cos (t)$, for
    particular values of the ratio $\epsilon /\mu$ proportional to the
    zeros of the zero-th Bessel function.

\bigskip

    {\bf Conclusions:} In the present research we have studied a
    periodically driven double well model in order to understand the large
    validity of the molecular structure concept. \ On the basis of our results it is
    possible that molecular collisions determine (at a critical pressure) a
    critical instability on a generic beating state which becomes a more
    stable localized state (definitely stabilized by the polarization of
    the environment \cite {A} \cite {JL}). \ Moreover, partially restoring
    the symmetry by means of a small static perturbation, we make the two
    Floquet resonances to (anti-)cross, so that the splitting vanishes at
    the critical pressure (red shift of the inversion line). \ The
    extension to non periodic perturbations, expected to improve the
    results for large pressure, will be discussed in a following research.

 \bigskip

    {\bf Acknowledgements:} This work is partially supported by the Italian
    MURST and CNR-GNFM. \ VG is supported by the INFN and by the University
    of Bologna (funds for selected research topics); AS is supported by the
    program "Progetti di ricerca orientata" of the University of Modena.

\bigskip

\begin{figure}
    \caption {In figure (a) we plot the imaginary part of the eigenvalues
    $\ell_1$ (broken line) and $\ell_2$ (full line) for $z=0$ and $k=50$
    fixed and $\mu \in (0.01,0.016)$, dots line represents the reference
    value $-2$. \ Occurrence of cusps, for the values of $\mu =\Omega /n $
    for some $n $ such that $c_n \not= 0$, is a consequence of the fact
    that the function $\Lambda (E)$ is an analytic function with branch
    point at $E=0$ [12]. \ In figure (b) we plot the minimum
    splitting value $\Delta (\mu)$, showing the vanishing for any $\mu$
    larger than the critical value $\mu^\star$. \ We see that the imaginary
    part of $\ell_{2}$ is near the value $-2$ at $\mu=\mu^\star$.
    }
    \label {fig1}
\end{figure}

\medskip

\begin{figure}
    \caption {We plot the real (full line) and imaginary part (broken line)
    of the eigenvalues $\ell_{1,2}$ for $k=50$ fixed. \ For the critical
    value $\mu = \mu^\star \approx 0.01536$ (figure (b)) we have an exact
    crossing; vanishing of the real part of the splitting and avoided
    crossing of the real part of the eigenvalues are observed,
    respectively, for $\mu >\mu^\star$ ($\mu \approx 0.156$ in figure (c))
    and $\mu <\mu^\star$ ($\mu \approx 0.15$ in figure (a)).}
    \label {fig2}
\end{figure}

\begin{references}

    \bibitem {W} This problem has been subject of a large interest since
    the early years of quantum mechanics: Hund F 1927, Z. Phys. {\bf 43},
    803; for a review see the papers by Woolley R G 1976, Adv. Phys. {\bf
    25}, 27-52, and Wightman A S 1995, Nuovo Cimento B {\bf 110}, 751-769.

    \bibitem {A} Anderson P W 1949, Phys. Rev. {\bf 75}, 1450-1450,
    Margenau H 1949, Phys. Rev. {\bf 76}, 1423-1429.

    \bibitem {JL} Claviere P, Jona Lasinio G 1986, Phys. Rev. A {\bf 33},
    2245-2253, and Grecchi V, Martinez A 1995, Commun. Math. Phys. {\bf 166}, 533-548.

    \bibitem {GMS} Grecchi V, Martinez A, Sacchetti A 1996, J. Phys. A:
    Math. and Gen. {\bf 29}, 4561-4587.

    \bibitem {SW} The mathematical details are leaved to a forthcoming
    paper by us, where we make use of technical results given by E Mourre
    1981, Commun. Math. Phys. {\bf 78}, 391-409, Perry P, Sigal I M, Simon
    B 1981, Ann. Math. {114}, 519-567 and in particular Soffer A, Weinstein
    M I 1998 J. Stat. Phys. {\bf 93}, 359-391.

    \bibitem {LL} \S 50 in Landau L, Lifsits E M 1958 {\it Quantum
    mechanics}, Oxford: Pergamon Press.

    \bibitem {M} This condition holds under some very general assumption on
    $V$, mainly we request that the potential rapidly decreases at infinity
    and that the threshold energy $E=0$ is neither an eigenvalue nor a
    resonance, for more details see Theorem 7.6 in Murata M 1982, J. Func.
    An. {\bf 49}, 10-56, and Jensen A, Kato T 1979, Duke Math. J. {\bf 46},
    583-611.

    \bibitem {K} See \S III.6 in Kato T 1976 {\it Perturbation theory for
    linear operators,} New-York: Springer-Verlag.

    \bibitem {NHC} Nagels B, Hermans L J F, Chapovsky P L 1997, Phys. Rev.
    Lett. {\bf 79}, 3097-3100, and Shao J, H\"anggi P 1998, Phys. Rev.
    Lett. {\bf 81 }, 5710-5713.

    \bibitem {MFDQNR} Madison K W, Fischer M C, Diener R B, Qian Niu,
    Raizen M G 1998, Phys. Rev. Lett. {\bf 81}, 5093-5096.

    \bibitem {GDJH} The destruction of tunneling for a two-level model has
    been proved for the first time by Grossman F, Dittrich T, Jung P,
    H\"anggi P 1991, Phys. Rev. Lett. {\bf 67}, 516-519; see also the
    papers by Kayanuma Y 1994, Phys. Rev. A {\bf 50}, 843-845, Tsukada N,
    Gotoda M, Isu T, Nunoshita M, Nishino T 1996, Jpn. J. Appl. Phys. {\bf
    35}, L1490-L1493, and Schatzer L, Weigert S 1998, Phys. Rev. A {\bf
    57}, 68-78. \ In a semi-classical regime this problem has been also
    considered by Berry M V 1984, J. Phys. A: Math. and Gen. {\bf 17},
    1225-1233.

    \bibitem {N} Occurrence of cusps as threshold effect is discussed in a
    more general framework in \S 17.2 by Newton R G 1982 {\it Scattering
    theory of waves and particles}, New-York: Springer-Verlag (2nd ed.).

    \end {references}

\end {document}

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