Transcript Probing the Classical-Quantum Transition using Chaos

Probing the Classical-Quantum Transition using Chaos
Chris Amey, Arik Kapulkin, Arjendu Pattanayak
Chaos
Previous Work
Results from the SQUID system
Chaotic dynamics are usually associated
with classical systems. This allows us to use
chaos as a benchmark of how classical a
system is behaving. Our definition of chaos
depends on the divergence of nearby points,
Δs. If nearby points diverge exponentially, then
the system is chaotic.
In previous work, we have shown that there is
non-monotonicity
in
the
classical-quantum
transition of the Duffing two well system, as shown
in figure 3. The classical system has no chaos,
gains structure which was shown to have chaotic
tendencies, and loses the chaos when in the
quantum regime, where we have defined quantum
and classical regimes relative to the size of ћ.
Although there are similarities between the
two systems, SQUIDs are fundamentally
different and in order to change the scale of
the system relative to ћ, different parameters
must be varied. The most promising direction
for non-monotonicity comes from varying the
inductance, L as seen in figure 6.
L = 2*10-10
t
s  Ce 
Log ( s )  t  C
  0  Chaos
L = 3.5*10-10
Figure 3. The Duffing system, the classical limit is on
the left and the quantum limit is on the right, this
suggests non-monotonicity in the transition.
Figure 1. Log of divergence of neighboring points on
the y axis, time on the x axis, this system is chaotic.
Quantum State Diffusion
To
model systems at various sizes, we use
QSD. In traditional quantum mechanics, the
time evolution of a state is dependant on the
Hamiltonian operator. QSD adds a second
operator, the Lindblad to take the environment
into account. In our case, the Lindblad was
generally a combination of the momentum and
position operators. The modified time evolution
equation includes a term which takes
measurement from an outside source into
account (Σdt) and a term which takes random
interactions into account (dξ) in addition to the
standard Hamiltonian term (dt).

i
d   H  dt   L j  L j

j
1 †
1 †
 †
  L j L j  L jL j  L j L j
2
2
j 
 d

  dt

Figure 2. The generic form of the evolution of a
wave state ψ in QSD.
SQUID dynamics
The Duffing system is only a model, there is
no real world system which can be experimented
on. This led our research to superconducting
quantum interference devices (SQUIDs) which
share properties with the Duffing system.
Figure 6. QSD SQUID simulations with varying
inductance values.
Conclusions
Figure 4. A simple rf-SQUID.
The simple SQUID in figure 4 consists of two arms
of superconducting metal with a thin resistive layer
between them at the point k. The Duffing and SQUID
Hamiltonians are very similar mathematically.
In conclusion, we see that there is nonmonotonic behavior associated with variation
of inductance in our SQUID model.
If
inductance is tied to the scale of the system,
then this non-monotonicity extends to the
classical quantum transition.
Acknowledgements
Supervised by Arjendu Pattanayak
Arik Kapulking
pˆ
xˆ

  Cos(xˆ )  I d Sin (d t )   ( xˆpˆ  pˆ xˆ )
2
2 0
References
pˆ
 xˆ xˆ
g

  Cos(t ) xˆ   ( xˆpˆ  pˆ xˆ )
2
4
2 
1
Anthony
Leggett,
in
Percolation,
Localization
Superconductivity (Plenum Press, 1983), pp. 1-41.
2
Ian Percival, Quantum State Diffusion. (Cambridge University
Press, 1998).
3
Ulrich Weiss, Quantum Dissipative Systems, Third ed. (World
Scientific, 2008).
4
A. Kapulkin and A. K. Pattanayak, Physical Review Letters 101
(7), 074101 (2008).
2
j
L = 4*10-10
2
Funded by HHMI
2
2
4
2
Figure 5. Hamiltonian for SQUID (top) and Duffing (bottom), x is
flux or position, p is voltage or momentum, ξ and Γ are frictive
constants, Ω, β and g are scaling parameters (respectively).
and