Transcript Small Josephson Junctions in Resonant Cavities

H2.005
Small Josephson Junctions in
Resonant Cavities
David G. Stroud, Ohio State Univ.
Collaborators: W. A. Al-Saidi,
Ivan Tornes, E. Almaas
Work supported by NSF DMR01-04987
Motivation: to develop models for controllable two-level
superconducting systems for possible use in quantum
computation
What is a Josephson Junction?
Current I
I
I
i
s
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s
A Josephson junction is a weak link between
two strong superconductors
Small junction: typically region between
superconductors is insulating (i); junction
underdamped
Hamiltonian of a small Josephson
junction
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Let  be the phase difference across a junction
Let n be the difference in the number of Cooper
pairs on the two superconducting islands
Then the Hamiltonian of the junction is
H  (1 / 2)Un  J cos
2

where U=(2e) /C is the charging energy and J is
the Josephson coupling energy.
2
(C=capacitance)
Commutation relations
[n,  ]  i
Schrodinger equation is
H  E
where
H  (U / 2) /   J cos
2
2
Wave function which solves this equation is that
of a particle of mass  2 / U moving in a cosine
potential
Voltage-biased Josephson junction

If junction is voltage-biased,the kinetic
energy term in the Hamiltonian becomes
Un / 2  U (n  n) / 2
2
2
If Josephson energy vanishes, energy eigenvalues
are just
En  U (n  n) / 2
2
With n = 0, 1, 2,…..
Schematic of voltage-biased
junction
Cg=gate capacitance;
Vg=gate voltage
Cg
C J , EJ
Vg
Junction
Hamiltonian is
H  EC ( n 
EC
C gVg
2e
( 2e) 2

2(C g  C J )
) 2  E J cos
Schrodinger equation for voltagebiased small junction
H  E
H  (U / 2)(i /   n) 2  J cos
Solve to obtain eigenvalues and wave functions as
functions of n
Experiment: junction can be placed in macroscopic
superposition of two distinct eigenstates [e. g. Bouchiat et
al, Phys. Scr. T76, 165 (1998); Nakamura et al, PRL 79, 2328
(1997); Nature 398, 786 (1999); Zorin et al, cond-mat/0105211]
Single-Mode Resonant cavity
Cavity fields
E,B
Hamiltonian of Resonant
Cavity Mode is

H  (a a  1 / 2)
Here  is the cavity frequency, a and a+ are photon
annihilation/creation operators. Energies are (m+1/2)  
Cavity-junction interaction
In presence of a vector potential, phase
difference in expression for Josephson
energy is replaced by a gauge-invariant
phase difference:
cos  cos[  g (a  a )]

Here g represents the junction-cavity coupling

strength, and a  a is related to the cavity
vector potential in the junction
Strength of junction-cavity coupling
c (2 )
2
g 
[
E
(
x
)

ds
]
2

 0
2
3
2
Note: line integral taken across
junction
E (x) 

=
Normalized cavity electric field
Cavity frequency
 0  Flux quantum ( = hc/2e)
Thus, maximum coupling occurs when E-field is
concentrated in as small a volume as possible
Schematic illustration of geometry for a voltage-biased
Josephson junction in a microcavity
C J , EJ
I
Supercon
S
Supercon
S
I
I
Vg
Cg
S
Microcavity
Josephson junction is two superconductors
(S and S) separated by insulating region
Solution of combined junctioncavity problem
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Compute Hamiltonian matrix in product basis
|km> of junction and cavity states (k=0,1,2,…=
junction states; m=0,1,2…= cavity states)
Diagonalize to obtain eigenvalues and
eigenfunctions as functions of g, offset voltage,
cavity frequency, etc.
At certain values of offset voltage, states of
cavity and junction are highly entangled – i. e.,
cannot be written as products of cavity and
junction states.
Energy level diagram for junctioncavity system

n
Lowest eigenvalues E/U of the junction-cavity system, plotted
as a function of n for the following parameters:   /U=0.3,
J/U=0.7, and several values of g as indicated. Two arrows
indicate pairs of degenerate states [Al-Saidi and Stroud,
PRB65, 014512 (2002)].
Time-dependent behavior of
Josephson-cavity system, n =0.26
[Obtained by solving time-dep’t Schrodinger equation starting from
|k=0,m=1>]
n
Left: energy in junction (solid line) and in cavity (dashed line)
versus time. Right: probability that junction is in first excited
state (full line) and that the cavity is in m=1 (one-photon)
state (dashed line). Calculation includes ALL the quantum
states of junction and cavity. Cavity oscillates between m =0
and m = 1.
Josephson-cavity system,
n
= 0.06
Same as previous picture, except that offset voltage is
tuned so that junction oscillates between ground and
first excited state, and cavity oscillates between m=0
and m=2 states.
Two-level approximation
Calculate Rabi frequency including only the two
nearly degenerate states.
 For n = 0.26, these are |k=0;m=1> and
|k=1;m=0>. For n = 0.06, they are |k=0,m=2>
and |k=2;m=0>.
 For the first case, 2-state approx. gives a Rabi
frequency within 1% of exact calculation. For
second case, it gives a Rabi frequency within
20% of exact calc.

”Collapse and revival”
[coined by J. H. Eberley et al, PRL 44, p. 1323 (1980)]
Full curve: Time-dependent probability that junction in first excited state,
given resonator initially in a “coherent state” (coherent superposition of
different number states) and junction in ground state. Dashed curve:
junction treated in two-level approximation, but all photon states included.
[Al-Saidi and Stroud, Phys. Rev. B65, 014512 (2002).]
Jaynes-Cummings Model
[E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963)]
Hamiltonian for a two-state system coupled to a
single mode of a quantized electromagnetic field
(or any single mode oscillator):
H  ( E2  E1 )S z  a  a   (a  S  aS )
E2  E1 
S , S , S z
Energy splitting of two lowest levels
are components of spin-1/2 operators
Our numerical results using full Hamiltonian for junctioncavity system very close to Jaynes-Cummings results
including only two lowest junction levels.
Remarks about single junction
coupled to single-mode cavity
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Quantum behavior closely resembles that
of two-state system coupled to cavity
mode, at certain values of offset voltage
Junction does NOT have to be a chargingenergy dominated*; charging and
Josephson energy can be comparable
*Regime studied by Shnirman et al, PRL79, 2371 (1997) and by
Buisson and Hekking, cond-mat/0008275.
Several Junctions in a Resonant
Cavity
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Is it possible to couple several Josephson
junctions to the SAME resonant cavity?
Yes! The mathematical model is very similar to
one used to treat several identical two-level
atoms placed in the same resonant cavity: the
Dicke model [R. H. Dicke, 1954]
The models would be identical if the junctions
were truly two-level systems, but this is true
only approximately
Model Hamiltonian for SeveralJunction Problem
H  Hcavity   H j  Hint
Hcavity
Hj
Hint
=
cavity Hamiltonian
=
Hamiltonian for jth junction
=Hamiltonian
interaction
for junction-cavity
Dicke model

The Hamiltonian for the Dicke model is
H  [(a a  1 / 2)   S z ]   ( a S   aS )

j

j

a
Here a and
are the annihilation and creation
*
operators for the photon mode, S’s are components
of the spin-1/2 operators, and  = strength of
coupling between the two-level system and the
cavity mode. Junction-cavity system differs from the
Dicke model because junctions are non-identical and
non 2-level systems.
j
Rabi oscillations in a system of
several junctions in a cavity
This Figure plots the time-dependent “inversion” S(t) (sum of probabilities
that each junction is in its excited state) for systems of 2 and 4 junctions
in a resonant cavity, given that initially exactly one junction is excited.
Rabi frequency is approximately proportional to square root of number of
junctions. [Calculated using full Hamiltonian] (Al-Saidi and Stroud, Phys.
Rev. B65, 224512 (2002).)
S(t) for coherent initial state, one
and two junctions
Time-dependent “inversion” S(t) (=sum of probabilities that each junction
is in its excited state) for (a) N = 1 and (b) N = 2 junctions, given that
resonator is initially in a “coherent” state (eigenstate of annihilation
operator), and junctions in their ground state. Solid lines: full numerical
solution; dashed lines, two-level approximation. The two junctions are
assumed identical.
Remarks about system of several
junctions in a cavity
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Two-level (Dicke) approximation works quite well.
Influence of higher levels can be incorporated into an effective
dipole-dipole interaction between the “qubits”:
H '   ( S( j ) S( k )  S( j ) S( k ) )
j


k
If junctions are non-identical, there are still Rabi oscillations
(frequency proportional to square root of junction number) if
junctions slightly different. For larger differences, each junction
couples independently to the cavity.
In “classical limit” (many photons), dynamical equations predict “selfinduced resonant steps” in the IV characteristics (Almaas and Stroud,
2002) in agreement with expt (Barbara et al, PRL, 1999).
Long Josephson “ring” with crossed
static and cavity magnetic fields
H
Vortex in ring behaves
like Cooper pair in small
junction
II
I


Cavity
magnetic
field H’
v
Vortex in ring is like a
magnetic moment
interacting with magnetic
fields
Qubit from vortex coupled to cavity
magnetic field
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Magnetic moment of vortex couples to applied static
magnetic field +magnetic field of cavity.
When current is applied across the ring, vortex is driven
around the ring. Applied static magnetic field produces
cosine potential.*
With magnetic field of cavity mode, quantum
Hamiltonian of system is formally identical (for small
vortex velocities) to that of small junction
Hence, should be able to make qubit analogous to that
produced by small junction coupled to a cavity mode
(but via magnetic, not electric, fields.
Quantum mechanics in case of static field already seen in experiments,
and calculated by Wallraff et al (Nature vol.425, p. 6954 2003).
Hamiltonian for vortex-cavity
system

Hamiltonian is (for slow vortex)
H=
p /(2mv )  H cos  H ' sin(a  a )  a a
2

 I /(2e)
Form of Hamiltonian is identical to that of small
junction coupled to electric field of cavity

Summary
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Have computed quantum states of states of
Josephson junctions coupled to single-mode
cavities.
Junctions may behave like two-level systems
which can be strongly entangled with cavity
states.
Several junctions in a cavity, can behave like
groups of controllable two-level systems which
can be entangled via the cavity.
Work in progress: (i) Other geometries for two
level systems; (ii) coupling via magnetic fields of
cavity modes; (iii) calculation of dissipation.