Qubits Using Single Electrons Over a Dielectric

John GoodkindPhysics 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.

Vz

1

•The resulting electric field is from an image charge so that

•Very close to the surface, the potential becomes strongly repulsive.

•Application of an electric field yields a first order Stark shift.

0 200 400 600 800 1000-417

-333

-250

-167

-83

0

En

erg

y (G

Hz)

distance above surface (Angstrom)

•In order to apply stark shifts to individual electrons, they must be confined laterally over a micro-electrode.

s u b s t r a t e

g r o u n d p l a n e

i n s u l a t o r

1V V 2

h e l i u m f i l m

v a c u u m

h

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

02 2

22

2

Uz

em

z

For liquid helium =1.05723, =6.95510-3

Thus the Schrodinger equation for the system of interest is that of a 1D hydrogen atom with a modified charge.

For solid Neon = 1.24, = 0.0268

14

1

]/[2

2/30 lzzExpl

, 22

2

2

2

0

l

e

mlU

]2

[211

2

1 22/31

lz

Expzl

zl

, U1 = U0/4

We will use the first two of these eigenstates as qubits.

For liquid heliuml = 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

lm e

2

2

In order to change the state of the qubit we will apply a microwave pulse at frequency, . This adds a time dependent potential to the Hamiltonian.

]cos[]cos[ tVtezE

01 zeER

If U1 - U0, this will cause transitions between states at the “Rabi” frequency.

R 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

10; /2

/1

21 tiUtiU ecetct

lzlzlz 6,82

312,

2

3221211

The the coupled differential equations for the cn(t) are:

The confining force in the plane can be designed so that in plane resonance frequency is comparable to E2 – E1.

+

+

+_

_

_

_

_

_

_

_

-2 -1 0 1 2-8

-4

0

4

E (

kV/c

m)

electrode positions (1 center to center)

Ez at 11.5 nm above surface

Ez at 45.8 nm above surface

-2 -1 0 1 2

-4

0

4

Ex (

kV/c

m)

electrode positions (1 ceneter to center)

Ex at 11.5 nm above surface

-40

0

40

Ex at 45.8 nm above surface

The fields shown are for 1 Volt applied to the electrodes. In practice we can achieve > 20 GHz in plane resonance frequency so that transitions will be induced only between 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

100 200 300 400 500 600

-1

-0.5

0.5

1

100 200 300 400 500 600

-1

-0.5

0.5

1

C1(0)=1 C2(0)=0

On resonance

2/1

Red = real, green= imaginary

Off resonance (10-3)

100 200 300 400 500 600

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

100 200 300 400 500 600

-0.75

-0.5

-0.25

0.25

0.5

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

2212

2

zzd

eV

2122

213

222

212

222

11 zzzz

d

e

d

e

d

zz

d

eV

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.

Logic operations require interactions between qubits and for this case it is due to the Coulomb field of the electrons.

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 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.

E E2 1

Implementation of SWAP gate

In state fields on both qubits the same, the states |10> and |01> are degenerate. The qubits will oscillate between these two states at frequency

183

22

sec10310

d

zeR

If the fields are held in this condition for a time, T, then the system will end up in state

cos[R Tisin[R T]|01>

For RT = /2 this becomes the swap gate.

01

500 1000 1500 2000

0.2

0.4

0.6

0.8

1

Numerical solution for the time dependence of the real and imaginary parts of the two wavefunctions yields:

50 100 150 200

0.2

0.4

0.6

0.8

1

SWAP operation by detuning after ½ cycle. (Wiggles after field shift occur because the basis functions are not Eignefunctions of the Hamiltonian.)

Resonance oscillations

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 timesP.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 ~3103/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-electrodes15 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 HF-water 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 ~ 1 K

• Energy dissipation is negligible

n+Si Porous Si

Au

e-

e-

~

Au

HP33120A SignalGenerator

collector

Electrometer

Vbias

VPS

0 V

VE

VPS

IPS

2 3 4 5 6 7 8 9 10

0.60

0.65

0.70

0.75

0.80

I E/I P

Sx1

04

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.

The frequency dependence of the efficiency of production of emission current, IE .

VPS (V)IPS (mA)

DetectorsSuperconducting 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.

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.

The meander pattern is calculated to give 100% detection efficiency with a misalignment as large as 1 mm. R=5 k.

Meander pattern

Single electron detection with Ti meander

0 100 200 300 400

-1.4

0.0

1.4

2.8

4.2

5.6

6.9

8.3

VPS

= 15 -20 V

N(e-) = 1 - 2R

0 = 3450

R = 3373.3 T = 0.390 KI = 1.8 A

R ()

Relaxation time (ms)

Lower TC by ion implantation in Ti or W.

-100 0 100 200 300 400 500 600 700

3364

3366

3368

3370

3372

3374

3376

VPS

= 15 -20 V

N(e-) = 1 - 2R

0 = 3450

R = 3373.3 T = 0.390 KI = 1.8 A

R

Pulse Count

Resistance transitions due to electron impacts from pulses of ~55 electrons/ mm2 at 1 second intervals.

Raw and filtered data from a resistance change caused by electron impact. The decay is the electronic time constant of the AC coupled signal.

tunnel diode electron source

transition edge detectors

lower plate

micro-structureliquid He film

Schematic of the system