Vitaly Shumeiko

Dept of Microtechnology and NanoscienceChalmers University of Technology, Göteborg Sweden

Zeno regime in Macroscopic Quantum Tunneling

ESF Conference, Obergurgl, 6-9 June 2010

Background

Aim: possibilities to slowdown quantum decay (MQT) of non-dissipative state of current biased Josephson junctions by means of fast temporal manipulations (Zeno regime)

Similar effect has been experimentally investigated with atoms trappedin optical lattice, PRL 87, 040402, 2001

Dynamical control of MQT in Josephson junction has been theoreticallystudied, PRL 92, 200403, 2004

Here we revisit this problem using different technique

Discussions: G. Kurizki, D. Dasari, A. Ustinov

Macroscopic Quantum Tunneling

eV2Δ

IS

S

JJ

MQT = tunnel switching from non-dissipative to dissipative current branch

Quantum Tunneling

P(t) = exp (- Γt )

Ψ(t) = exp (- iHt ) |0>

Free evolution

ΔU

- lnP

t

Periodic watching: tm << 1/ΔU

P(t) ≈ exp (- Γzeno t )

Quantum Zeno effect

B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, 756 (1977)P Facchi and S Pascazio, J. Phys. A: Math. Theor. 41 (2008) 493001

Quantum Tunneling

ΔU

Watching!

Ψ(tm) = exp (- iHtm ) |0> → |0> Projective measurement

Γzeno = (<H2> - <H>2 ) tm

- lnP

t1/ΔU

Zeno regime

JJ switching DOES NOT exhibit Zeno effect !

∂tφ = 2eV = 4Δ

After escape, “particle” accelerates till threshold velocity, when single particle tunneling channels opens;Then JJ switches to dissipative branch = measurementBefore switching event – unitary evolution

eV2Δ

I

tm ~ Δ/ωp2 >> 1/ ωp

What is a measurement time?

JJ is a meter itself

MQT: what is measured?

MQT: how to get Zeno regime?

static

dynamic

x x

ΔU

E = k2

t2 >> 1/ΔU

E0

completelyopen

MQT: periodic modulation

closed

ΔU

t1 << 1 /ΔU

E = k2

open closed open

|k1> |k2>

|k’>

|0>

|k’>

|k’>

openopen closed

Correction ~ (1 / ΔU t2) b

= Zeno effect !

destructive interferencet2 >> 1/ΔU

Conclusion

To achieve Zeno regime one has to open well for (short) time intervals, t1<< 1/ΔU, then close for (long) time intervalst2 >> 1/ΔU

The system measures itself. It gradually performs projection on bound state during time >> 1/ΔU

Evolution is purely unitary!

Rapid modulation: t2 << 1 / ΔU

Decay from modulated well = decay from effective static well(Kapitza regime)

ΔU

E

ΔU

E

Ueff

E0 < Ueff E0 > Ueff

Stay Go

C00

E’0

Ueff =ΔUt1 /(t1+t2)

Ueff

SUMMARY

Studied: decay of a quantum state in quantum well into continuumunder rapid modulation of the barrier transparency (instant opening-closing)

Found: two distinctly different regimes: “incoherent” (Zeno) and coherent. In both cases state evolution is purely unitary.

Incoherent regime: well is kept closed during time longer than inverse level frequency. In this case, the leaking state is effectively projected on the original bound state (self-measurement) leading to the Zeno effect – substantial suppression of the decay rate. Coherent regime: manipulation cycle (open-close) is shorter than inverse level frequency. A finite fraction of the state stays in the well at t = ∞, for ratio of open-close durations being smaller than certain critical value.