Download Quantum reflection and dwell times of

Quantum reflection of
S-wave unstable states
N. G. Kelkar
Dept. Fisica, Universidad de los Andes, Bogota, Colombia
•Tunneling times
Definitions, interpretations, density of states
•Delay times in quantum collisions
S-wave threshold singularity
•Quantum reflection
relation to s-wave scattering
•Applications
How long does it take a particle to
tunnel through a barrier?
•
•
Phase time, extrapolated phase time
1. Based on a wave packet approach
2. Arrival of the incident and departure of the reflected or
transmitted wave packets
3. Interference between incident and reflected waves not
considered
E. H. Hauge and J. Stovneng, Rev. Mod. Phys. 61, 917 (1989)
Dwell time
Average time spent by a particle in a given region of space
1. Average over reflection and transmission in one dimension
2. Scattering channels in three dimensions
Concept introduced by F. T. Smith, Phys. Rev. 118, 349 (1960)
One dimensional tunneling: M. Büttiker, Phys. Rev. 27, 6178 (1983)
•
Larmor time
1. Spin precession in a weak magnetic field
2. Baz and Rybachenko gave the theoretical formalism
If the electrons have a direction of polarization perpendicular to the
direction of the field, the time spent in the field region was
proportional to the expectation value of a spin component
Larmor time = phase time + oscillating terms
A. J. Baz´, Sov. J. Nucl. Phys. 4, 182 (1967); 5, 161 (1967)
V. F. Rybachenko, Sov. J. Nucl. Phys. 5, 635 (1967)
• Traversal time, Büttiker Landauer time
• Complex times
• Group delay time
M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).
H. G. Winful, Phys. Rev. Lett. 91, 260401 (2003)
• The Dwell time and asymptotic phase time provide reliable and
complementary information on time aspects of a tunneling process
One dimensional treatment of dwell and
phase times
Reflection amplitude:
Transmission amplitude:
For a “sharp” wave packet (transmitted)
Follow the peak
Phase time :
: arrival at the barrier
:departure (due to transmission)
Similarly, reflection phase derivative :
Weighted sum (defined as group delay by Winful)
The dwell time was defined by Büttiker as,
Where N – number of particles within the barrier
and j – incident flux given as
Does not distinguish if the particles got reflected or transmitted
Büttiker: The extent to which the spin undergoes a Larmor precession is
determined by the dwell time of a particle in the barrier.
Hauge: the above statement cannot always be true
V. S. Olkhovsky and E. Recami, Phys. Rep. 214, 339 (1992); H. Winful,
Phys. Rep. 436, 1 (2006) … controversies with Hartmann effect
THE PHASE AND DWELL TIME CONNECTION
Hauge review, Winful (2003), Büttiker (1983) in one dimension
Martin Ph. A. Acta Physica Austrica Suppl 23, 157 (1981)
In the review of Hauge, it was shown for a rectangular barrier of height
and width d, for the opaque case
The last term – self-interference term – due to overlap of the incident
and the reflected waves in front of the barrier
Phase time becomes singular near threshold
DENSITY OF STATES
Relation between dwell time and the density of states was given in
3D by G. Iannaccone, Phys. Rev. B 51, R4727 (1995)
density of states in a region Ω, N- number of incoming
channels
For the relation in 1D, the number of channels reduce to two, since
there are only left and right incoming waves and for a symmetric
potential, the dwell time relation to density of states is
V. Gasparian and M. Pollak, Phys. Rev. B 47, 2038 (1993).
Delay times in 3D scattering
An intuitive picture:
Time delay in a resonant (R) scattering process is much
larger than in a non-resonant (NR) scattering process
A+BA+B
E.P. Wigner, Phys. Rev. 98, 145 (1955)
A simple wave packet description of a collision implies a delay time of the
magnitude
Simple picture: Wave packet with superposition of two frequencies:
Following the peak in the two terms, for the first term
and the second term
Implying that the interaction has delayed the particle by a time
TIME DELAY CAN BE NEGATIVE !
Wigner put a limit from the principle of causality
a- radius of the scattering centre
Close to resonances – time delay should large and positive
Connection to density of states – Beth - Uhlenbeck formula
density of states with interaction
density of states without interaction
THE PHASE AND DWELL TIME DELAY CONNECTION
Subtracting the density of states without interaction from both sides of
i.e.,
and
Near threshold singularity also present in Wigner’s time delay:
however, only for s-waves. With the scattering phase shift,
, we get,
and
For l = 0, Wigner’s time delay blows up near threshold.
Smith’s lifetime matrix
F. T. Smith, Phys. Rev. 118, 349 (1960)
Collision time: The limit as R  ∞ of the difference between the time
the particles spend within a distance R of each other (with interaction)
and the time they would have spent there without interaction
The matrix elements of the collision matrix Q in terms of the scattering
matrix S were given as
is the velocity in the
channel and
is an element of S
Based on the collision time idea, Smith defined a delay time matrix, such
that a typical element was
and the average time delay for a particle injected in the
average collision time beginning in the
Indeed, in a phase shift formulation of the S-matrix
In the presence of inelasticities,
channel
channel
In a transition matrix (T-matrix) formulation of the S-matrix
with
for s-wave scattering
Is there a way to relate the s-wave threshold singularity in Wigner
and Smith’s time delay relations and the threshold singularity in the
phase time delay in tunneling in one dimension?
YES – reflection in 1D can be viewed as a back scattering in 3D
With there being no angle dependence of the scattering amplitude
in the case of s-waves, the s-wave 3D motion can be viewed as a
1D motion in the radial coordinate r
QUANTUM REFLECTION
•
•
•
•
Reflection of a particle in a classically allowed
region where there is no classical turning point
This can happen above potential barriers
M. V. Berry and K. E. Mount, Rep. Prog. Phys. 35, 315 (1972)
At the edge of attractive potential tails
H. Friedrich et al., Phys. Rev. A 65, 032902 (2002)
Recent interest: experiments with ultra cold atoms
F. Shimizu, Phys. Rev. Lett. 86, 987 (2001); T. Pasquini et al., Phys.
Rev. Lett. 93, 223201 (2004); ibid, 97, 093201 (2006); H. Oberst,
Phys. Rev. A 74, 052901 (2005)
Importance: reduces the probability that the atoms come close
enough to be influenced by short ranged forces.
In atom-surface interaction, it can prevent particles from
sticking or being inelastically scattered
For quantum reflection to occur the semiclassical WKB approximation
must be violated. Essential condition for WKB is that,
the De Broglie wavelength, and the parameter
when WKB is valid.
BADLANDS are regions where this condition is not satisfied
Larger the badlands  higher the quantum reflection
As the energy E  0,
WKB becomes less reliable  quantum reflection dominates
For a one-dimensional transmission-reflection problem at positive energies,
the asymptotic wave function with incidence from the left is
where
and
are the transmission and reflection amplitudes
and the S-matrix is given as,
and
due to time reversal invariance and
for symmetric potentials
F. Arnecke, H. Friedrich and J. Madroñero, Phys. Rev. A74, 062702 (2006);
H. Friedrich and A. Jurisch, Phys. Rev. Lett. 92, 103202 (2004); H. Friedrich
and J. Trost, Phys. Rep. 397, 359 (2004); U. Kuhl et al., Phys. Rev. Lett. 94,
144101 (2005); W. O. Amrein and Ph. Jacquet, Phys. Rev. A75, 022106 (2007)
Getting back to the delay time matrix of Smith,
and using the R and T as channels,
For quantum reflection at low energies
In terms of the T-matrix then,
Recall the phase time delay expression
Replacing
Comparing with the time delay expression of Smith,
└-------------------------------------------------------------┘
↓
 Dwell time delay
THE THRESHOLD SINGULARITY
The self-interference term near threshold,
Where aR is the real part of the scattering length and the dwell time delay
such that the phase time delay
Smoothly vanishes near threshold and emerges as the right
definition for the density of states of metastable states at all
energies.
Applications
1. Eta - Mesic Nuclei

η – pseudoscalar meson, mass ~ 547 MeV, η-N threshold lies
close to the S11 resonance N*(1535)

η-N interaction is attractive near threshold

Searches for exotic quasibound (metastable) states of η
mesons and nuclei


Search via peaks in the dwell time delay distributions in the
elastic scattering of η + A  η + A
Specifically : η-3He and η-4He
The t-matrix for η-nucleus scattering is written using the finite rank
approximation of few body equations
N. G. Kelkar, K. P. Khemchandani and B. K. Jain,
J. Phys. G 32, 1157 (2006); J. Phys. G 32, L19 (2006).
N. G. Kelkar, Phys. Rev. Lett. 99, 210403 (2007).
Experimental evidences (direct or indirect) of
eta-mesic nuclei
Indirect evidence from η meson producing reactions like the
p + d  3He + η
The very rapid rise of the total cross section to its maximum value within
0.5 MeV from threshold hints toward the existence of an eta-mesic state
close to threshold
T. Mersmann et al., Phys. Rev. Lett. 98, 242301 (2007)
The photoproduction of η - mesic 3He was investigated using the TAPS
calorimeter at the Mainz Microtron accelerator facility MAMI.
A binding energy of (-4.4 ± 4.2 ) MeV and a width of (25.6 ± 6.1 MeV)
is deduced for the quasibound η - mesic state in 3He.
M. Pfeiffer et al., Phys. Rev. Lett. 92, 252001 (2004)
Applications …
2. Near threshold scalar mesons
– the σ

Dwell time delay – density of states of a resonance

Fourier transform of density of states  survival probability

Critical times for the transition from the exponential to
the non – exponential decay law
(to be discussed in the talk of M. Nowakowski)
SUMMARY
o Phase time = Dwell time + Self-interference
in tunneling
o Identifying times with density of states with interaction, the density
of states without interaction is subtracted to obtain a relation
between delay times
o Phase time delay  Wigner’s time delay and the singularity in
s-wave Wigner’s time delay corresponds to the self-interference
term due to quantum reflection
o Dwell time delay can be evaluated, goes smoothly to zero near
threshold and gives the correct behaviour of the density of states of
a metastable state at all energies.