Quantum reflection of S-wave unstable states

•Tunneling times Definitions, interpretations, density of states•Delay times in quantum collisions S-wave threshold singularity•Quantum reflection relation to s-wave scattering•Applications

N. G. Kelkar Dept. Fisica, Universidad de los Andes, Bogota, Colombia

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 barrierand 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 dimensionMartin 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 matrixF. 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 interactionThe 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, suchthat a typical element was

and the average time delay for a particle injected in the channel

average collision time beginning in the channel

Indeed, in a phase shift formulation of the S-matrix

In the presence of inelasticities,

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 3DWith 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 thep + 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 stateclose to thresholdT. Mersmann et al., Phys. Rev. Lett. 98, 242301 (2007)

The photoproduction of η - mesic 3He was investigated using the TAPScalorimeter 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 reflectiono 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.