Qubits Using Single Electrons
Over a Dielectric
John Goodkind
Physics Department
University of California, San Diego
Collaborators:
Manyam Pilla, Xinchang Zhang, Alex Syshchenko,
Brian Naberhuis
Mrk Dykman, MSU, Arnold Dahm CWRU
Work supported by ARO and NSF
A clean physical system for
qubits
• The Hamiltonian for the system is well known and
simple enough to be used to calculate the parameters
required for logic operations
• The interactions with the outside are also well known so
that relaxation times can be estimated.
• Swap and CNOT gates are easily implemented as a
consequence of the interactions between qubits. The
swap gate can be turned on and off by detuning specific
qubits with the Stark shift. Refocusing can be used in
exactly the same manner as for nmr qubits.
•electron, in vacuum, above the surface of a dielectric, is attracted
to that surface by the induced polarization.
•The resulting electric field is from an image charge so that
1
V
z
•Very close to the surface, the potential becomes strongly repulsive.
•Application of an electric field yields a first order Stark shift.
Energy (GHz)
0
-83
-167
-250
-333
-417
0
200
400
600
800
1000
distance above surface (Angstrom)
•In order to apply stark shifts to individual electrons, they must
be confined laterally over a micro-electrode.
va cuum
h
he luim fm
il
g round p al ne
ni su al to r
sub s tra te
V1
V2
d
Some properties of the system
•qubits are coupled to electric fields so that they can be
manipulated by easily obtainable DC and microwave
fields
•easy to detect 1 electron (compared to 1 spin)
•qubits are coupled by Coulomb force
•qubits are in vacuum, weakly interacting with rest of
universe at operating T = 10 mK
•It is fabricated by standard lithographic techniques
Thus the Schrodinger equation for the system of interest is that of
a 1D hydrogen atom with a modified charge.

 2 2m  e 2

 2 
 U   0
2
z
  z

 1

4  1
For liquid helium =1.05723, =6.95510-3
For solid Neon  = 1.24, = 0.0268
We will use the first two of these eigenstates as qubits.
e2

 0  3 / 2 zExp[  z / l ] ,
U0 
2
2l
2 ml
l


 1  1 31/ 2  z  1 z 2  Exp[  z ] ,
U 1 = U 0/4
2l 
2l
2l 
2
 2
2
l
me2
For liquid helium
l = 7.64 nm, U1 – U0 ~ 118 GHz
Confirmed by experimental measurements.
For solid Neon
l =1.98 nm, U2 – U1 = 2.1 THz.
At this frequency optical techniques can be used
In order to change the state of the qubit we will apply a
microwave pulse at frequency, W. This adds a time
dependent potential to the Hamiltonian.
ezE cos[Wt ]  V cos[Wt ]
If W  U1 - U0, this will cause transitions between states at
the “Rabi” frequency.
WR 
eE 1 z 0

WR  1 GHz per V/cm
The time dependent problem can be solved by expanding
the time dependent wave function in terms of the
eigenstates of the time independent Hamiltonian and then
using the interaction representation.
We write the expansion as
3
z11  a
 ; t 2 c1
z 22  6a
t e
iU1t / 
0  c2e
iU2t / 
1
The the coupled differential equations for the cn(t) are:
32
z12  2
a
81
dc1
i
 Ee cosWt  z11c1  z12e it c2
dt
dc2
i
 Ee cosWt  z 21e it c1  z 22c2
dt




3
31
z11  l , z12  2 l , z 22  6l
2
82
The confining force in the plane can be designed so that in
plane resonance frequency is comparable to E2 – E1.
Ez at 11.5 nm above surface
Ez at 45.8 nm above surface
_
_
+
_
Ex (kV/cm)
E (kV/cm)
4
0
Ex at 11.5 nm above surface
Ex at 45.8 nm above surface
_
4
40
0
0
-4
-4
+
-8
-2
-1
0
1
_
2
electrode positions (1  center to center)
_
-40
-2
-1
0
1
2
electrode positions (1  ceneter to center)
The fields shown are for 1 Volt applied to the electrodes. In
practice we can achieve > 20 GHz in plane resonance
+ will be induced only between
frequency so that transitions
_
_
the same in plane states.
Ez must be reduced to  400 V/cm over the electrodes. Can
be done by superimposing a uniform field.
One qubit electron gates
On resonance
1
1
C1(0)=1
0.5
100
200
300
400
0.5
500
600
C2(0)=0
100
1/ 2
-0.5
-0.5
-1
-1
Red = real, green= imaginary
200
300
400
500
600
Off resonance (10-3)
1
0.75
0.5
0.5
0.25
0.25
100
-0.25
-0.5
-0.75
200
300
400
500
100
600
-0.25
-0.5
-0.75
200
300
400
500
600
Detuning by Stark shift
Potentials applied to the electrodes shift the energy level
spacing as described above (about 1.4 GHz for 100 Volt/cm).
Thus, if we use fixed microwave frequency individual
electrons will be tuned into resonance by adjusting the DC
potentials. This provides the ability to address individual
qubits.
The interaction between qubits
Logic operations require interactions between qubits and for
this case it is due to the Coulomb field of the electrons.
V
e2
d 2  z1  z 2 
2
d = 500 nm and <(z1 – z2)> between the first excited state
and the ground state for helium is 34.4 nm. Expand and
keep only first order terms.
2

e  1  z1  z 2   e 2
e2 2
V
1 
 3 z1  z 22  2 z1 z 2
 
d  2  d   d 2d
2


The last term leads to a SWAP operation when two qubits are in
resonance.
This interaction also leads to an electric field from one electron
on the other when the two are in different states (different
elevations above the surface). This shifts the resonance
frequencies so that if E2  E1  W for qubit #1 when qubit #2
is in the ground state then #1 will not respond to the microwave
radiation when #2 is in the excited state. This provides the
CNOT operation.
Implementation of SWAP gate
In state 01 fields on both qubits the same, the states |10> and
|01> are degenerate.
The qubits will oscillate between these two states at frequency
2
WR 
e 0 z1
d 3
2
 3  108 sec 1
If the fields are held in this condition for a time, T, then the
system will end up in state
cos[WR T|10>isin[WR T]|01>
For WRT = p/2 this becomes the swap gate.
Numerical solution for the time dependence of the real and
imaginary parts of the two wavefunctions yields:
Resonance oscillations
SWAP operation by detuning after ½ cycle. (Wiggles after
field shift occur because the basis functions are not
Eignefunctions of the Hamiltonian.)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
500
1000
1500
2000
50
100
150
200
Limiting the interactions in a
many qubit system
For more than two qubits, this system like many others, has
no simple way to turn off the interaction. The swap operation
can be turned off by detuning, but the first terms in the
expansion are always present.
•The magnitude and range of the interaction is limited due to
screening by the ground plain (1/r5)
•The interaction can be switched by a factor of 10 by
fabricating a metallic wall on the faces of the square qubit
array and connecting to to ground to turn off the interaction.
•Refocusing can be used to effectively eliminate the
interaction
Relaxation times
P.M.Platzman, M.I. Dykman, Science 284, 1967 (1999)
M.I. Dykman, P.M.Platzman, P. Seddighrad Submitted for
publication August, 02 (calculations including horizontal
confinement)
•Coupling to the many electron modes: ignorable for h/d 0.5
•Coupling to surface displacement
1 ripplon decay neglibile,
2 ripplon ~3103/sec for 0< E<300 V/cm
Phonons, surface displacement ~103 to 104/sec
modulation of dielectric constant 104 to6 104
•Dephasing due to ripplon scattering 0.7 102 at T=10 mK
•Dephasing from Johnson noise from 1K into the electrode
 5 102/sec
•Ripplon induced sideband absorption, limits usable Rabi
frequency (no quantitative estimate in this article)
The ripplons are overdamped in 3He and do not exist in solid
Neon. No ripplon induced sidebands. No superfluid film
covering everything.
Decay due to coupling to phonons varies as vs-3 so that it will
become negligible in solid Neon.
Hardware
Need:
•Micro-electrode structure
•Low T low energy electron source
•Means to read out the states of the electrons
•Microwave source
Micro-electrodes
15 mm square chip.
Ground plane will cover
only the lead wires of the
left and right triangles.
Thickness gauge
capacitors
Location of
microelectrodes and
leads to be written by
ebeam lithography
•Original pattern of two rows of posts separation = 500 nm
•Ebeam lithography is independent of the photolithography lead
pattern and can be changed from the original 2 row pattern.
Electron source
Porous silicon diode prepared by anodization in HFwater ethanol solution of n+ doped Si, 30mA/cm2,
illuminated with 30 mW/cm2, 5 m RTO, 0.5 m
anneal in N2 (to be published).
• Yields as a few as 50 electrons per pulse with
energies less than 10 eV.
• Operating parameters are independent of T below ~
1K
• Energy dissipation is negligible
Porous Si
n+Si
Au
0V
Au
0.80
Vbias
0.75
e-
4
collector
HP33120A Signal
Generator
Electrometer
~
IE/IPSx10
e-
VPS
0.70
0.65
IPS
VPS
0.60
VE
2
0.0
0.0
nA)
3.0
2.0
0.5
1.0
IPS, 30
(mA)
30 mA/30mW/cm
sec
5min RTO 0.5min N2 Anneal
200 Hz, 4.2 K
1.5
a
IPS (A)
2
30 mA/30mW/cm , 30 sec
5min RTO 0.5min N2 Anneal
0.5
2.0
2
5
6
7
8
9
10
The frequency dependence of the
efficiency of production of emission
current, IE .
2.0
1.6
1.2
0.8
0.4
0.0
20
A)
IE (nA)
1.0
- 150 Hz, 300 mK
- 250 Hz, 75 mK
IE = -0.015 + 0.762 IPS
4
1/f (msec)
Schematic of experimental setup to measure
properties of the PS diode. Individual pulses from
the HP33120A of the form shown are applied. The
frequency setting of the generator determines the
slope of the sawtooth.
1.5
3
60
40
T = 300 mK, f = 150 Hz
-5 V /8.34
IPS = 4.46 10 e
A
PS
a
30
40
50
V (V)
60
T = 4.2 K, f = PS
150 Hz
-3
VPS/9.5
70
80
90
Detectors
•





Superconducting microbolometers
Tests with short wire, 1  by 40 by
150 nm thick Aluminum (shown without
ground plane)
Three TC’s corresponding to transitions
of the three different lead widths
Operated only close to TC = 1.2K
Resistance pulse height is fixed, width
(duration) depends on energy
Total number of electrons in area of detector less than one per
pulse so that only a fraction of pulses cause change in resistance
Field configuration was incorrect for 100% efficiency (alignment
too critical). New meander pattern described below.
Meander pattern
The meander pattern is calculated to give 100% detection
efficiency with a misalignment as large as 1 mm. R=5 kW.
1.5 cm square chip with 1 mm square
opening in the ground plane at the
center.
Holes through the chip between
meander patterns to allow
electrons to pass from the
source to the microelectrodes.
Edges of the meander pattern
and the ground plane.
Single electron detection with Ti
meander
Raw and filtered data from a resistance
change caused by electron impact. The
decay is the electronic time constant of the
AC coupled signal.
Resistance transitions due to electron
impacts from pulses of ~55 electrons/
mm2 at 1 second intervals.
8.3
3376
VPS= 15 -20 V
6.9
3374
-
N(e ) = 1 - 2
R0 = 3450 W
R = 3373.3 W
T = 0.390 K
I = 1.8 A
5.6
VPS= 15 -20 V
-
N(e ) = 1 - 2
R0 = 3450 W
R = 3373.3 W
T = 0.390 K
I = 1.8 A
3370
RW
R (W)
4.2
3372
2.8
3368
1.4
0.0
3366
-1.4
3364
0
100
200
300
400
Relaxation time (ms)
Lower TC by ion implantation in Ti or W.
-100
0
100
200
300
400
Pulse Count
500
600
700
Schematic of the system
tunnel diode electron source
transition edge detectors
lower plate
liquid He film
micro-structure