Download Quantum interference in the classically forbidden region: A parametric oscillator

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PHYSICAL REVIEW A 76, 010102共R兲 共2007兲
Quantum interference in the classically forbidden region: A parametric oscillator
1
M. Marthaler1,2 and M. I. Dykman1
Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
2
Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures (CFN),
Universität Karlsruhe, D-76128 Karlsruhe, Germany
共Received 20 March 2007; published 20 July 2007兲
We study tunneling between period-2 states of a parametrically modulated oscillator. The tunneling matrix
element is shown to oscillate with the varying frequency of the modulating field. The effect is due to spatial
oscillations of the wave function and the related interference in the classically forbidden region. The oscillations emerge already in the ground state of the oscillator Hamiltonian in the rotating frame.
DOI: 10.1103/PhysRevA.76.010102
PACS number共s兲: 03.65.Xp, 74.50.⫹r, 05.45.⫺a, 05.60.Gg
Nonlinear micro- and mesoscopic vibrational systems
have attracted much interest in recent years. In such systems
damping is often weak, and even a comparatively small resonant field can lead to bistability, i.e., to coexistence of forced
vibrations with different phases and/or amplitudes. Quantum
and classical fluctuations cause transitions between coexisting vibrational states. The transitions are not described by the
conventional theory of metastable decay, because the states
are periodic in time and the systems lack detailed balance.
Experimentally, classical transition rates have been studied
for such diverse vibrational systems as modulated trapped
electrons 关1兴, Josephson junctions 关2兴, nano- and micromechanical oscillators 关3–5兴, and trapped atoms 关6兴, and the
results are in agreement with theory 关7,8兴.
Currently much experimental effort is being put into
reaching the quantum regime 关9,10兴. In this regime tunneling
between coexisting classically stable periodic states should
become important, for weak dissipation. It was first studied
for a resonantly driven oscillator, where a semiclassical
analysis 关11兴 made it possible to find the tunneling exponent
in a broad parameter range 关12兴.
Tunneling is particularly interesting for a parametrically
modulated oscillator. Here, the coexisting classical periodic
states have period 2␶F, where ␶F is the modulation period.
Such period-2 states are identical except that the vibrations
are shifted in phase by ␲. Therefore the corresponding quantum states 共Floquet states兲 are degenerate. Tunneling should
lift this degeneracy, as for a particle in a symmetric static
double-well potential. Earlier the tunneling matrix element
was found 关13兴 for modulation at exactly twice the oscillator
eigenfrequency ␻0. Recently the tunneling exponent was obtained in a general case where the modulation frequency
␻F = 2␲ / ␶F is close to 2␻0 关14兴.
In this paper we show that tunneling between period-2
states of a parametrically modulated oscillator displays unexpected features. We find that the tunneling matrix element
oscillates with varying ␻F − 2␻0, periodically passing
through zero. These oscillations are accompanied by and are
due to spatial oscillations of the wave function in the classically forbidden region.
For resonant modulation, 兩␻F − 2␻0兩 Ⰶ ␻F, and for a small
amplitude of the modulating field F the oscillator dynamics
is well described by the rotating wave approximation 共RWA兲
关15兴. The scaled RWA Hamiltonian ĝ as a function of the
1050-2947/2007/76共1兲/010102共4兲
oscillator coordinate Q and momentum P in the rotating
frame is independent of time. In a broad parameter range it
has a symmetric double-well form shown in Fig. 1. The
minima correspond to the classical period-2 states, in the
presence of weak dissipation. Respectively, of utmost interest are tunneling transitions between the lowest single-well
quantum states of ĝ.
A simple model of a nonlinear oscillator that describes
many experimental systems, cf. Refs. 关1–6兴, is a Duffing
oscillator. The Hamiltonian of a parametrically modulated
Duffing oscillator has the form
1
1
1
H0 = p2 + 共␻20 + F cos ␻Ft兲q2 + ␥q4 .
2
2
4
共1兲
For ␻F close to 2␻0 and for comparatively small F,
1
2
␦ ␻ = ␻ F − ␻ 0,
兩 ␦ ␻ 兩 Ⰶ ␻ 0,
F Ⰶ ␻20 ,
共2兲
even where the oscillator becomes bistable its nonlinearity
remains relatively small, 兩␥具q2典兩 Ⰶ ␻20. For concreteness we
set ␥ ⬎ 0; the results for ␥ ⬍ 0 can be obtained by replacing
␦␻ → −␦␻ in the final expressions.
0.6
g
0.0
0.5
-0.6
-2.0
0.0
-1.0
P
0.0
Q
1.0
-0.5
2.0
FIG. 1. 共Color online兲 The scaled effective Hamiltonian of the
oscillator in the rotating frame g共Q , P兲, Eq. 共5兲, for ␮ = 0.5. The
minima of g共Q , P兲 correspond to the period-2 vibrations. The eigenvalues of ĝ give scaled oscillator quasienergies.
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©2007 The American Physical Society
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M. MARTHALER AND M. I. DYKMAN
To describe a weakly nonlinear oscillator it is convenient
to make a canonical transformation from q and p to the
slowly varying coordinate Q and momentum P,
U†qU = Cpar关P cos共␻Ft/2兲 − Q sin共␻Ft/2兲兴,
U† pU = − Cpar␻F关P sin共␻Ft/2兲 + Q cos共␻Ft/2兲兴/2,
共3兲
where Cpar = 共2F / 3␥兲1/2 and
关P,Q兴 = − i␭,
␭ = 3␥ប/F␻F .
共4兲
The dimensionless parameter ␭ plays the role of ប in the
quantum dynamics in the rotating frame 关14兴.
The transformed oscillator Hamiltonian has the form
共F2 / 6␥兲ĝ, where ĝ ⬅ g共Q , P兲,
1
1
1
g共Q, P兲 = 共P2 + Q2兲2 + 共1 − ␮兲P2 − 共1 + ␮兲Q2
4
2
2
␺±共Q兲 =
1
冑2 关␺l共Q兲 ± ␺l共− Q兲兴,
共7兲
where ␺l共Q兲 is the “single-well” wave function of the left
well of g共Q , P兲 in Fig. 1. It is maximal at the bottom of the
well Ql0 = −共1 + ␮兲1/2 and decays away from the well. To the
leading order in ␭, the corresponding lowest eigenvalue of ĝ
is gmin + gq, where gmin = −共1 + ␮兲2 / 4 is the minimum of
g共Q , P兲 and gq = ␭共␮ + 1兲1/2 is the zero-point energy.
The wave function ␺l共Q兲 is particularly simple for ␮ ⬍ 0.
In the classically forbidden region between the wells, 兩Q兩
⬍ 兩Ql0兩, it has the form
␺l = C关− i⳵ Pg兴−1/2 exp关iS0共Q兲/␭兴,
共8兲
where S0共Q兲 is given by the equation g共Q , ⳵QS0兲 = gmin + gq,
S0共Q兲 =
共5兲
冕
Q
P−共Q⬘兲dQ⬘ ,
Ql0+Lq
˜ 兲1/2兴1/2 ,
P±共Q兲 = i关1 + Q2 − ␮ ± 2共Q2 − ␮
关we use here a more conventional notation g共Q , P兲 instead of
g共P , Q兲 used in Ref. 关14兴兴. The terms ⬀exp共±in␻Ft兲 with n
艌 1 in ĝ have been disregarded.
The time-independent operator ĝ is the scaled oscillator
Hamiltonian in the rotating frame. Its eigenvalues multiplied
by F2 / 6␥ give oscillator quasienergies, or Floquet eigenvalues. Formally, ĝ is a Hamiltonian of an auxiliary stationary
system with variables Q , P, and the eigenvalues of ĝ give the
energies of this system. The operator ĝ depends on one parameter
We keep in S0 only the contribution from the branch
P−共Q兲, because P−共Q兲 is zero on the boundary of the classically forbidden range Ql0 + Lq. For −␮ Ⰷ ␭ and 兩Ql0 + Lq兩
⬎ 兩Q兩 the action S0共Q兲 is purely imaginary. The wave function ␺l共Q兲 monotonically decays with increasing Q.
The prefactor in the wave function 共8兲 is determined by
the complex classical speed of the oscillator
␮ = 2␻F␦␻/F.
˜ 兲1/2 .
⳵ Pg = 2P−共Q兲共Q2 − ␮
˜ = ␮ − g q,
␮
共6兲
For ␮ ⬎ −1, g共Q , P兲 has two minima located at P = 0, Q
= ± 共␮ + 1兲1/2. For ␮ 艋 1 the minima are separated by a saddle
at P = Q = 0, as shown in Fig. 1. When friction is taken into
account, the minima become stable states of period-2 vibrations. The function g共Q , P兲 is symmetric as a consequence of
the time translation symmetry: the change 共P , Q兲 → 共−P ,
−Q兲 corresponds to shifting time in Eq. 共3兲 by the modulation period ␶F.
We assume the effective Planck constant ␭ to be the small
parameter of the theory, ␭ Ⰶ 1. Then the low-lying eigenvalues of ĝ form doublets. Splitting of the doublets is due
to tunneling between the wells of g共Q , P兲. Since g共Q , P兲
= g共−Q , −P兲 is symmetric, the problem of level splitting
seems to be similar to the standard problem of level splitting
in a double-well potential 关16兴. As in this latter case, we will
analyze it in the WKB approximation.
The major distinction of the present problem comes from
the difference between the structure of g共Q , P兲 and the
Hamiltonian considered in Ref. 关16兴. The momentum
P共Q ; g兲 as given by equation g共Q , P兲 = g has four branches,
with both real and imaginary parts in the classically forbidden region of Q. This leads to new features of tunneling and
requires a modification of the method 关16兴.
We will consider splitting ␦g of the two lowest eigenvalues of ĝ. Because of the symmetry, the corresponding wave
functions ␺±共Q兲 are
1/2
−1/4
Lq = ␭/g1/2
.
q ⬅ ␭ 共␮ + 1兲
共9兲
共10兲
The normalization constant C in Eq. 共8兲,
C = 关共␮ + 1兲/␲兴1/4 exp共− 1/4兲,
共11兲
is obtained by matching, in the range Lq Ⰶ Q − Ql0 Ⰶ 兩Ql0兩, Eq.
共8兲 to the tail of the Gaussian peak of ␺l共Q兲, which is centered at Ql0.
We are most interested in the parameter range ␮ Ⰷ ␭
where tunneling displays unusual behavior. For such ␮ the
˜.
momentum P−共Q兲 becomes complex in the range 兩Q兩 ⬍ ␮
This means that the decay of the wave function is accompanied by oscillations. To correctly describe them we had to
keep corrections ⬀gq in Eq. 共9兲.
We first rewrite Eq. 共9兲 in the form
冋
˜ 兲1/2 −
P−共Q兲 ⬇ i 1 − 共Q2 − ␮
gq/2
˜ 兲1/2
1 − 共Q2 − ␮
册
.
共12兲
Equation 共12兲 applies for Q − Ql0 Ⰷ Lq. It is seen that P−共Q兲
has two branching points inside the classically forbidden re˜ 1/2. The WKB
gion. The closest to Ql0 is the point Qbr = −␮
˜ 1/2. The wave
approximation breaks down for small Q + ␮
function in this region can be shown to be proportional to
˜ 1/2兲共2␮
˜ 1/2 / ␭2兲1/3). Therefore ␺l osAiry function Ai(−共Q + ␮
˜ 1/2.
cillates with Q for positive Q + ␮
In contrast to the standard WKB theory of the turning
point, the prefactor in ␺l contains two factors that experience
˜ 1/2, see Eqs. 共8兲 and 共10兲. The full solution in
branching at −␮
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QUANTUM INTERFERENCE IN THE CLASSICALLY…
FIG. 2. 共Color online兲 The wave function of the ground state in
the left well ␺l共Q兲 in the oscillation region for ␭ = 0.09 and ␮
= 0.5. The solid line shows explicit expressions 共13兲–共15兲, the
dashed line shows numerical results. Inset: ␺l共Q兲 near its second
zero with higher resolution.
˜ 1/2
the oscillation region can be obtained by going around −␮
in the complex plane following the prescription 关16兴. For
˜ 1/2 this gives
␭2/3 Ⰶ Q + ␮
FIG. 3. 共Color online兲 Scaled matrix element of tunneling between period-2 states as a function of the scaled detuning of the
modulation frequency from twice the oscillator eigenfrequency. The
solid lines show explicit expression 共16兲, the dashed lines show the
result of numerical calculations. Inset: a higher-resolution plot of
兩␦g兩 / 2 vs ␮ near the zero of ␦g at ␮ = 6␭. The data refer to ␭
= 0.09.
␦g = − ␭2兵2共1 − ␮兲␺l共0兲␺l⬘共0兲 − ␭2关␺l共0兲␺l⵮共0兲
+ ␺l⬘共0兲␺l⬙共0兲兴其
␺l ⬇ 2C兩⳵ Pg兩−1/2 exp关− Im S0共Q兲/␭兴cos ⌽共Q兲,
or, with account taken of Eq. 共13兲,
⌽共Q兲 = ⌽1共Q兲 + ⌽2共Q兲.
共13兲
␦g =
The term Im S0共Q兲 in the amplitude of the wave function
共13兲 is determined by Eq. 共9兲. The phase ⌽共Q兲 has two
terms. The term ⌽1共Q兲 comes from the exponential factor in
the WKB wave function 共8兲,
⌽1共Q兲 = ␭
−1
冕
2⌽1共0兲 = ␲共␮␭−1 − 1兲/2
Re P−共Q兲dQ,
共14兲
where Re P−共Q兲 is given by Eq. 共12兲 in which we set 共Q2
˜ 兲1/2 → i共␮
˜ − Q2兲1/2; therefore Re P−共Q兲 ⬎ 0. It is simple to
−␮
write ⌽1 and Im S0共Q兲 in explicit form.
The term ⌽2共Q兲 in Eq. 共13兲 comes from the prefactor in
␺l共Q兲, Eq. 共8兲,
⌽2共Q兲 ⬇
A = 共␮ + 1兲1/2 + ␮ ln兵␮−1/2关1 + 共␮ + 1兲1/2兴其,
Q
˜ 1/2
−␮
冉
16␭1/2共␮ + 1兲5/4 −A/␭
e
cos关2⌽1共0兲兴,
共␲␮兲1/2
1
␮ − Q2
arcsin
2
1 + ␮ − Q2
冊
1/2
−
␲
.
4
共15兲
Decay and oscillations of the wave function described by
Eq. 共13兲 are compared in Fig. 2 with the results of a numerical solution of the Schrödinger equation ĝ␺ = g␺. The leftwell wave function was obtained numerically as a sum of the
two lowest-eigenvalues solutions, cf. Eq. 共7兲. In this calculation the basis of 120 oscillator Fock states was used. A good
agreement between analytical and numerical results is seen
already for not too small ␭ = 0.09.
The above solution allows us to find the tunnel splitting
␦g = g− − g+ of the symmetric and antisymmetric states 共7兲.
Following the standard approach for a symmetric doublewell potential 关16兴 we multiply the Schrödinger equations for
the involved states ĝ␺l = gl␺l and ĝ␺± = g±␺± by ␺*± and ␺*l ,
respectively, integrate over Q from −⬁ to 0, and subtract the
results. This gives
共␮ Ⰷ ␭兲.
共16兲
Clearly, ␦g may be positive or negative, that is, the symmetric state may have a lower or higher quasienergy than the
antisymmetric state.
The dimensional splitting 共F2 / 6␥兲兩␦g兩 gives twice the
matrix element of tunneling between period-2 states of
the oscillator. This matrix element has an exponential factor
exp共−A / ␭兲 关14兴. In addition, it contains a factor oscillating
as a function of the scaled frequency detuning ␮ / ␭
= 6␻2F共␻F − 2␻0兲 / 3␥ប. The oscillation period is ⌬共␮ / ␭兲 = 4.
These oscillations are shown in Fig. 3.
The oscillations of ␦g result from the wave function oscillations in the classically forbidden region. This can be
seen from the analysis of ␺l共Q兲 near the positive-Q boundary
˜ 1/2. The wave function for Q
of the oscillation region, Q = ␮
˜ 1/2 Ⰷ ␭ is a combination of the WKB waves with imagi−␮
˜ 兲1/2兴. The coefficients in
nary momenta P±共Q兲 ⬇ i关1 ± 共Q2 − ␮
this combination can be found in a standard way 关16兴. They
˜ 1/2兲. Only the wave with
are determined by the phase ⌽共␮
P−共Q兲 contributes to the tunneling amplitude, since P+ re˜ 1/2兲
mains imaginary in the right well of g共Q , P兲. For ⌽共␮
= 共4n − 3兲␲ / 4 this wave has zero amplitude, leading to ␦g
˜ 1/2兲 = 2⌽1共0兲 − ␲ / 4, we immediately
= 0. By noting that ⌽共␮
obtain from Eq. 共16兲 that ␦g = 0 for ␮ = 2n␭ with integer n, in
agreement with Fig. 3.
The occurrence of spatial oscillations of the ground state
wave function of the scaled Hamiltonian ĝ does not contra-
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PHYSICAL REVIEW A 76, 010102共R兲 共2007兲
M. MARTHALER AND M. I. DYKMAN
dict the oscillation theorem, because ĝ is not a sum of the
kinetic and potential energies and is quartic in P. The motion
described by the Hamiltonian g共Q , P兲 is classically integrable. Respectively, the quantum problem is different from
dynamical tunneling in classically chaotic systems 关17–19兴;
the effect we discuss has not been considered for such systems, to the best of our knowledge.
The effect is also qualitatively different from photonassisted or suppressed tunneling in systems with stationary
double-well potentials: our oscillator has a single-well potential, the bistability is a consequence of resonant modulation,
and the Hamiltonian ĝ is independent of time. On the other
hand, there is a remote similarity between the oscillations of
the tunneling matrix element for period-2 states and for electron states in a double-well potential in a quantizing magnetic field 关20兴. However, not only is the physics different,
but our approach is also different from that in Ref. 关20兴; in
particular, it makes it possible to find ␦g analytically. The
approach can be extended also to a resonantly driven Duffing
oscillator, where the RWA Hamiltonian has a structure similar to Eq. 共5兲 关7,12兴.
Tunnel splitting can be observed by preparing the system
in one of the period-2 states and by studying interstate oscillations, cf. Refs. 关18,19兴. This requires that the tunneling rate
共␦␻ / 2␮␭兲兩␦g兩 exceed ␻F / 4Q, where Q is the oscillator
quality factor. The splitting sharply increases with increasing
␭. It will be shown separately that for comparatively large ␭
共but still for 兩␦g兩 Ⰶ gq兲 the RWA applies and relaxation remains small provided ␦g2 Ⰷ C␭ / Q with C␭ ⱗ 1. Our RWA
numerical results indicate that ␦g still oscillates with ␮ for
␭ = 0.25− 0.3 and is well described by Eq. 共16兲 for ␮ ⲏ 2␭.
The local peak of 兩␦g兩 for ␭ = 0.3 and the characteristic extremum of d2␦g / d␮2 for ␭ = 0.25 occur where 兩␦g兩 ⬇ 0.01.
Such ␦g may be large enough for detecting the effect in
modulated Josephson junctions where Q = 2360 has been
reached in the range of bistability 关10兴.
In conclusion, we used the WKB approximation to study
the wave functions of the period-2 states of a parametrically
modulated oscillator. We showed that these wave functions
can display spatial oscillations in the classically forbidden
region, in the rotating frame. These oscillations lead to oscillations of the matrix element of tunneling between the
period-2 states with the varying frequency of the modulating
field.
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We are grateful to V. N. Smelyanskiy and F. Wilhelm for
stimulating interactions. This research was supported in part
by the NSF through Grant No. PHY-0555346.
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