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Selim Jochim, UniversitΓ€t Heidelberg One, two, three, many Creating quantum systems one atom at a time

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Page 1: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Selim Jochim, UniversitΓ€t Heidelberg

One, two, three, many

Creating quantum systems one atom at a time

Page 2: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Selim Jochim, UniversitΓ€t Heidelberg

One, two, three, many

Creating quantum systems one atom at a time

Page 3: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Selim Jochim, UniversitΓ€t Heidelberg

One, two, three, many

Creating quantum systems one atom at a time

Page 4: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two trapped interacting particles …

interacting singlet

Ground state of the Helium atom:

No analytic solution available, we learn how

to apply powerful numerical techniques:

Hartree Fock method.

Page 5: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Many body limit

Define quantities like the Fermi energy,

density, pressure ….

… apply local density approximation …

But when are such approximations

justified?

This is an ancient problem!

Page 6: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Sorites Paradox

How many grains make a heap?

β€’ 1 grain of sand does not make a heap.

β€’ If 1 grain does not make a heap then 2 grains of sand do not.

β€’ If 2 grains do not make a heap then 3 grains do not.

β€’ …

β€’ If 9,999 grains do not make a heap then 10,000 do not.

From Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/sorites-paradox/

http://commons.wikimedia.org/wiki/File%3ASossusvlei_Dune_Namib_Desert_Namibia_Luca_Galuzzi_2004.JPG

http://plato.stanford.edu/entries/sorites-paradox/
Page 7: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Ultracold neutral atoms

Bose Einstein condensates of large samples of atoms: Macroscopic wave

function: Number of particles is so large that a constant density of atoms is

observed in experiments:

http://jila.colorado.edu/bec/images/bec.png

Removing one single atom does not make a difference!

Measure: 𝑛 𝒓 = ⟨ Ψ† 𝒓 Ξ¨ 𝒓 ⟩

Page 8: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Our approach

Reduce the complexity of a system as much as possible

until only the essential parts remain!

In most physical systems:

Range of interaction

significantly complicates the description

Page 9: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Ultracold atoms are an ideal tool …

The interactions between ultracold atoms can be effectively pointlike

(contact interaction)

van der Waals interaction: range of π‘Ÿπ‘£π‘‘π‘Š ∼ 1nm

In the experiments we have:

β€’ extremely low density (interparticle spacing ~ 1Β΅m)

β€’ extremely low momentum, such that πœ†π‘‘π΅ =β„Ž

2πœ‹π‘šπ‘˜π‘‡β‰« π‘Ÿπ‘£π‘‘π‘Š

Page 10: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

β€’ extremely low momentum, such that πœ†π‘‘π΅ =β„Ž

2πœ‹π‘šπ‘˜π‘‡β‰« π‘Ÿπ‘£π‘‘π‘Š

(This is the opposite limit desired in collision experiments:

shorter wavelength enhances resolution)

Here:

β€’ If πœ†π‘‘π΅ is sufficiently large, all the information about internal structure of the

atom is hidden in a single quantity, the scattering length 𝒂

β€’ We can even tune the scattering length to any desired value by simply

applying a magnetic field (Feshbach resonances).

Ultracold atoms are an ideal tool …

Page 11: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

The 6Li atom

11

0 200 400 600 800 1000 1200-5

0

5

Scatt

eri

ng

len

gth

(1000 a

0)

Magnetic field (G)

S=1/2, I=1

β†’ half-integer total angular momentum

β†’ 6Li is a fermion

F=3/2

En

erg

y

magnetic field [G]

F=1/2

|>|>

mI= 0

mI= 1

Tuning interactions: Feshbach resonance in 6Li

a↑↓

6Li ground state

NO interaction between identical particles

Page 12: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

C. Regal et al., Nature 424, 47-50 (2003)

Two-body system: Tune the binding

energy of a weakly bound molecule:

Tunability of ultracold systems

600 800 1000 1200-10

-5

0

5

10

Scatt

eri

ng

len

gth

(1000 a

0)

Magnetic field (G)

Binding energy:

Size (>> range of interaction):

Feshbach resonance: Magnetic-

field dependence of s-wave

scattering length

/( ) r ar e

2

2BEma

We have prepared such molecules up to ~1Β΅m in size!

G. ZΓΌrn et al., PRL 110, 135301 (2013)

Page 13: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

This talk

Page 14: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Outline

β€’ How do we prepare our samples?

β€’ How many particles do we need to form a heap?

β€’ Controlling the motion of two particles in a double well

Page 15: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

A picture from the lab …

109 laser cooled atoms at ~1mK

Thomas Lompe ( MIT)

Matthias Kohnen

( PTB Braunschweig) Friedhelm Serwane

( UC S. Barbara) Timo

Ottenstein

( Dioptic)

Page 16: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

A container for ultracold atoms

This might still work

for liquid nitrogen ….

We need to isolate the atoms from the environment:

… here we use the focus of a laser beam:

Optical dipole trap depth: π‘ˆ ∝ 𝐼(𝒓)

Page 17: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Evaporative cooling

It just works the same:

For our cup of coffee …

… and for our cold atoms:

Cool from ~1mK down to

below 1Β΅K

Just reduce the trap depth, i.e. laser power

Page 18: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Ultracold gas of fermions

About 50000 atoms @ 250nK, TF~1Β΅K

~100Β΅m

Absorption imaging of ultracold clouds:

resonant laser beam

CCD camera

Page 19: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Single atom detection

CCD

1-10 atoms can be distinguished with

high fidelity > 99%

one atom in a MOT

1/e-lifetime: 250s

Exposure time 0.5s

Fluorescence normalized to atom number

Page 20: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Towards a finite gas …

The challenge:

achieve ℏω ≫ π‘˜π‘‡

Page 21: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Creating a finite gas of fermions

β€’ superimpose microtrap (~1.8 Β΅m waist)

p0= 0.9999

β€’ 2-component mixture in reservoir

E

n

1

Fermi-Dirac dist.

~100Β΅m

Page 22: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Creating a finite gas of fermions

β€’ switch off reservoir

p0= 0.9999

+ magnetic field gradient in

axial direction

Page 23: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Spilling the atoms ….

β€’ We can control the atom number with exceptional precision!

β€’ Note aspect ratio 1:10: 1-D situation

β€’ So far: Interactions tuned to zero …

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

2% 2%

co

un

ts

fluorescence signal

96%

5 6 7 8 9 100

20

40

60

80

100

120

140

6.5%5%

88.5%

co

un

ts

fluorescence signal

F. Serwane et al., Science 332, 336 (2011)

Page 24: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Letβ€˜s study the interacting

system!

Page 25: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Precise energy measurements

β€žbareβ€œ RF – transition

RF – transition with interaction

RF photon+ Ξ”E

RF photon

Radio Frequency spectroscopy

Page 26: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Measure the interaction energy

vary the number of majority particles:

0,0 0,5 1,0 1,5 2,0 2,5 3,00,0

0,2

0,4

0,6

interaction shift [kHz]

Tra

nsfe

rre

d f

ractio

n

0 1 2 3 4 5

0,0

0,5

1,0

1,5

2,0

E

[Δ§

ll]

N

Page 27: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Measure the interaction energy

vary the number of majority particles:

0,0 0,5 1,0 1,5 2,0 2,5 3,00,0

0,2

0,4

0,6

interaction shift [kHz]

Tra

nsfe

rre

d f

ractio

n

0 1 2 3 4 5

0

1

2

g

=0.25

g

=0.8

g

=2.0

E

[Δ§

ll]

N

Page 28: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

10 8 6 4 2 0

0,07

0,1

0,4

0,7

1

kF/g

E/E

F

lnteraction energy in dimensionless units

Analytic solution of the two particle problem

T.Busch et al., Found. Phys. 28, 549 (1998)

Analytic solution for an infinite number of majority particles

J. McGuire, J. Math. Phys. 6,432 (1965)

(local density approximation)

noninteracting strongly repulsive

A. Wenz et al., Science 342, 457 (2013)

Page 29: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

0 1 2 3 4 5

0

1

2

g

=0.25

g

=0.8

g

=2.0

E

[Δ§

ll]

N

10 8 6 4 2 0

0,07

0,1

0,4

0,7

1

kF/g

E/E

F

lnteraction energy in dimensionless units

N↑=1

N↑=2N↑=3N↑=4N↑=5

Analytic solution of the two particle problem

T.Busch et al., Found. Phys. 28, 549 (1998)

Analytic solution for an infinite number of majority particles

J. McGuire, J. Math. Phys. 6,432 (1965)

(local density approximation)

noninteracting strongly repulsive

A. Wenz et al., Science 342, 457 (2013)

Page 30: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

10 8 6 4 2 0

-0,01

0,00

0,01

0,02

0,03

0,04

0,05

-E

/EF

-kF/g

0,07

0,1

0,4

0,7

1

E/E

F

lnteraction energy in dimensionless units

N↑=1

N↑=2N↑=3N↑=4N↑=5

2 particles

N particles

3 particles

S.E. Gharashi et al.,

PRA 86, 042702 (2012)

A. Wenz et al., Science 342, 457 (2013)

G.E. Astrakharchik and

I. Brouzos, arxiv:1303.7007

Page 31: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Approaching the many body limit…

… with very few particles (in a one-dimensional system)

A. Wenz et al., Science 342, 457 (2013)

Interesting things to look at:

β€’ Polaron physics in various dimensions

β€’ The Kondo problem

β€’ Andersonβ€˜s orthogonality catastrophe

Page 32: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Realize multiple wells …

….. with similar fidelity and control?

Basic building blocks of matter!

Page 33: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

The setup

Light intensity distribution

Page 34: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

A tunable double well

J

initial spatial wave function:

spin wave function (stationary)

Page 35: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

β€’ Interactions switched off:

A tunable double well

J

0 25 50 750

1

2

Ato

m n

um

ber

in w

ell

|R>

Time (ms)

well |π‘…βŸ©well |𝐿⟩

switch off left

well before

counting

atoms

Page 36: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two interacting atoms

U

J

0 25 50 750

1

2

Ato

m n

um

be

r in

we

ll |R

>

Time (ms)

c)

well |π‘…βŸ©well |𝐿⟩

Interaction leads to entanglement:

Page 37: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two strongly interacting atoms

U

J

well |π‘…βŸ©well |𝐿⟩

U

J

well |π‘…βŸ©well |𝐿⟩

In a balanced double well, they can only tunnel together!

Page 38: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two strongly interacting atoms

U

J

well |π‘…βŸ©well |𝐿⟩

U

J

well |π‘…βŸ©well |𝐿⟩

We can compensate for the interaction energy by applying a tilt!

Page 39: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two strongly interacting atoms

β€’ Observe number statistics in the right well (time averaged)

-15 -10 -5 0

0,0

0,5

1,0

2 atoms

Pro

ba

bili

ty in

we

ll |R

>

tilt [J]B=740G

two atoms

Page 40: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two strongly interacting atoms

β€’ Observe number statistics in the right well (time averaged)

-15 -10 -5 0

0,0

0,5

1,0

1 atom

2 atoms

Pro

ba

bili

ty in

we

ll |R

>

tilt [J]B=740G

U

two atoms

one atom

Page 41: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Two strongly interacting atoms

β€’ Observe number statistics in the right well (time averaged)

-15 -10 -5 0

0,0

0,5

1,0

0 atoms

1 atom

2 atoms

Pro

ba

bili

ty in

we

ll |R

>

tilt [J]B=740G

two atoms

one atom

no atomsU

Page 42: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Preparing stationary states

β€’ If we ramp on the second well slowly enough, the system will remain in its

ground state:

Page 43: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Eigenstates of a symmetric DW

1

2(| ⟩𝐿𝐿 + | βŸ©π‘…π‘… )

1

2(| βŸ©πΏπ‘… + | βŸ©π‘…πΏ )

1

2(| ⟩𝐿𝐿 + | βŸ©πΏπ‘… + | βŸ©π‘…πΏ + | βŸ©π‘…π‘… )

Page 44: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Preparing stationary states

β€’ Number statistics for the balanced case depending on the interaction

strength:

-2 0 2 4 6 8 100,0

0,5

1,0

Pro

ba

bili

ty

Interaction U (J)

Single occupancy

Page 45: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Preparing stationary states

β€’ Number statistics for the balanced case depending on the interaction

strength:

-2 0 2 4 6 8 100,0

0,5

1,0

Pro

ba

bili

ty

Interaction U (J)

Single occupancy

Double occupancy

Page 46: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Measuring energies

13

Trap modulation spectroscopy

Page 47: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Measuring energies

First-order

tunneling ~ 2𝐽 Second-order

tunneling ~4 𝐽

π‘ˆ

2

13

Trap modulation spectroscopy

Page 48: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Measuring energies

First-order

tunneling ~ 2𝐽 Second-order

tunneling ~4 𝐽

π‘ˆ

2

Super exchange energy!

responsible for spin ordering in the many body ground state

13

Page 49: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

How to go to a many-body system?

β€’ Inspired by a top-down approach: D. Greif et al., Science 340, 1307-1310 (2013) (ETH ZΓΌrich)

β€’ Dimerize a lattice filled with spin-1/2 fermions to observe spin correlations

Page 50: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Outlook

Combination of multiple double wells

β€’ Preparation of ground states

in separated double wells

β€’ Combination to larger system

Can this process be done adiabatically ?

Can it be extended to larger systems ?

14

Page 51: One, two, three, many Creating quantum systems one atom at ...Β Β· β€’ extremely low momentum, such that πœ†π‘‘π΅= β„Ž 2πœ‹π‘šπ‘˜π‘‡ β‰«π‘Ÿπ‘£π‘‘π‘Š (This is the opposite

Vincent KlinkhamerGerhard ZΓΌrn

Thank you for your attention!

Funding:

AndreWenz

Thomas Lompe(-> MIT)

DhruvKedar

SelimJochim


Opposite Word

Gated RNN & Sequence Generationyvchen/f106-adl/doc/171030...Β Β· 2017. 10. 30.Β Β· Recurrent Neural Network β€’Given function f: β„Žβ€², = β„Ž, h0 f h1 y1 x1 f h2 y2 x2 f h3 y3 x3

A ROCKET IS A DEVICE THAT SENDS GAS IN ONE DIRECTION TO MOVE IN THE OPPOSITE DIRECTION

Square Roots & Pythagorean Theorem. The opposite of + is - The opposite of is The opposite of x 2 is Inverse Operations…

€¦  · Web viewConcept(s): Opposite signs of numbers indicate locations on opposite sides of 0 on the number line. The opposite of the opposite of a number is the number itself

Opposite adjectives2

CS224W: Machine Learning with Graphs Jure Leskovec, Hongyu ...Β Β· Β‘Edges in KG are represented as triples(β„Ž,$,%) Β§head(β„Ž)has relation $ with tail(%). Β‘ Key Idea: Β§Model entities

Quadrilateral Proofs Page 4-5. (a) One pair of opposite side both parallel and congruent (b) Both pairs of opposite sides congruent (c) Both pairs of

First evidence of Bβ†’β„Ž K and Recent Results on X c and Y ...Β Β· M. Beneke, et al. NPB 811 155 (2009) BESIII, PRD 99, 072008 (2019) Belle set the upper limit on ℬ(Bβ†’β„Ž c K+)

Opposite At

!=1 $=24 β„Ž= 5 =38876 ft-kip

DERIVATIVES: RATES OF CHANGE … · INSTANTANEOUS RATE OF CHANGE Now, suppose that we define β„Žas the difference between and . This implies that = +β„Ž Here, note that as β„Žβ†’0,

Footings - seismicconsolidation.com€¦ · The value of distortion normally allowed in design is 1/3300 for π‘™π‘™β„β„Žβ‰€3.0 in sandy soils, 1/2500 for π‘™π‘™β„β„Žβ‰€3.0 in

Fiat-Shamir: from Practice to TheoryΒ Β· βˆ€PPT 𝐴, Pr β„Žβ†π» π‘₯←𝐴(β„Ž) T,β„Ž T∈ = H ... Need enc. with KDM security for bounded functions. Known [BHHI10,BGK11,A11]

Opposite adjectives

Think opposite

PATH FOLLOWING FOR THE F18-HARV AIRCRAFTΒ Β· 2017-03-10Β Β· structure πΎπ‘€π‘Ž β„Ž( ), and after that one can apply root locus by changing the sign, cause the resulted transfer

Sample Design Summary: ESS Round 10...2. PROB2 | β„Ž=1 3. PROB3 | β„Ž= 7 𝐡 β„Ž 4. PROB4 | β„Ž= 1 𝑁𝑙 β„Ž Statistic Description π‘β„Ž Population 15+ size of β„Ž-th stratum

JUST LIFETIME INCOME (JuLI)Β Β· Fund Fact Sheet [(1+ β„Ž π‘Ž π‘Ž β„Ž 6 π‘Ž )

Opposite Day

Discrete Time Convolution - University of Kansasperrins/class/F14_360/lab/...Β Β· 2014-10-16Β Β· Suppose we wanted their discrete time convolution: = βˆ—β„Ž = β„Ž βˆ’ ∞ π‘š=βˆ’βˆž

Drain Source GateΒ Β· 2016. 12. 10.Β Β· Zener-Klein tunneling Zener-Klein Tunneling, Pauli blocking: 𝐹𝜎 = 4 2 β„Ž Þ ΓŸπ‘πΎ 4πœ‹ = . ; βˆ’β„Ž= Þ𝐹 πœ‹2 ℏ ( βˆ’ ) 𝒛

Impacts of Autonomous Vehicles on Parking and …...2019/05/17 Β Β· Regular Vehicle Parking Autonomous Vehicle Parking 21/35 Parking options Home o 𝐢𝐢 β„Ž = 2π‘₯π‘₯ β„Ž 𝑐𝑐

Opposite Opinions

Assignment 4 6 sem B - BGUDesign of Algorithms – Assignment 4 2016 sem B 2. , run β„Ž ( 1 ,… , β„Ž)and update the results in the field ΧͺΧ•Χ Χ•Χ›Χ  Χ¨Χ‘Χ‘Χ”

ΰΈͺΰΈΈΰΈ”ΰΈ’ΰΈ­ΰΈ”ΰΈœΰΈΉΰΉ‰ΰΈšΰΈ£ΰΈ΄ΰΈ«ΰΈ²ΰΈ£ Executive ...€¦ · Title: ΰΈͺΰΈΈΰΈ”ΰΈ’ΰΈ­ΰΈ”ΰΈœΰΈΉΰΉ‰ΰΈšΰΈ£ΰΈ΄ΰΈ«ΰΈ²ΰΈ£ Executive Excellence Author: ζΈŽγ„ŽβˆŽδ€Žβ¨ŽΰΌŽα€Žδ°€αΈŽβŒŽβ¬Žβ„Žε°¨ΰΈ£ΰΈ΅

NeurΓ‘lis HΓ‘lΓ³k - EΓΆtvΓΆs LorΓ‘nd Universitypospet.web.elte.hu/NNFP.pdfΒ Β· 2016-12-17Β Β· Automatikus differenciΓ‘lΓ‘s ∘ βˆ˜β„Žβ€²= β€²βˆ˜ βˆ˜β„Ž β€²βˆ˜β„Žβ„Žβ€² HΓ‘trafelΓ©

Two opposite parts of one whole. - Pine Creek High SchoolΒ Β· Two opposite parts of one whole. A dichotomy is an idea or classification split in two. When you point out a dichotomy,

CHAPTER 6mrjaffe.net/GeomRef/unit6solutions.pdfΒ Β· Opposite angles are congruent. If one angle increases by xΒ°, the opposite angle also increases by xΒ°. Consecutive angles are supplementary

Adjective Opposite

One and Two Step Equations Using Opposite Operations

PowerPoint Presentation - Carnegie Mellon Universitygdicaro/15381/slides/381-F18...β„“we have to look where β„“=0 β„“=βˆ’ 2 =1 π‘š βˆ’β„Ž 𝝓 =𝟎 β„Ž ; = 0+ 1πœ™1 + 2πœ™2 +β‹―+

librarykvnewcanttblog.files.wordpress.comΒ Β· Web viewIf an exterior angle of a triangle is 1300 and one of the interior opposite angle is 600. Find the other interior opposite angle

Calculus - Weebly1 Calculus Chapter 4A Approximation using the derivative It’s an algebraic take on the limit rule !"#=lim!#+β„Žβˆ’!(#) β„Ž ** that β€˜formula’ is familiar to you

Chapter 1 Q1.19)Β Β· 2 Μ‚=2.114+0.0388 At X=30 Μ‚β„Ž=2.114+0.0388 (30)=3.278863 when the entrance test score increases by one point, the mean response increase by 0.038827. Q1.20)