6th International Schoolon Spintronics and Quantum Information Technology(SPINTECH6)
8/2 (2011) Matsue

Quantum Control and Quantum Simulation
with Ultracold Atomswith Spin

Kyoto University, JSTY. Takahashi

Outline of LectureI)

State-of-the-art of ultracold atom technologyII) Quantum simulation of Hubbard model with ultracold atoms
in optical latticeII-1) quantum magnetism and superfluidity
metal-Mott insulator transition, anti-ferromagnetic orderII-2) adiabatic cooling of atoms with spins
Pomeranchuk cooling
III) Quantum control of atomic spin ensemble
III-1) quantum description of collective atomic spincoherent spin state, squeezed spin state
III-2) quantum description of polarization of lightquantum Stokes operators
III-3) quantum interface and control
QND measurement of spin, quantum feedback control

Outline of LectureI)

State-of-the-art of ultracold atom technology

typical experimental setup

Laser Cooling and Trapping“optical trap”
anti-Helmholtz coils

Vint   p  E

U pot (r )  

E (r ) 2

CCD

2

Six laser beams
for laser cooling

500 mm“magnetic trap”• Number: 107• Density: 1011/cm310mm• Temperature: 10µK“Magneto-optical Trap”

Vint   m  BEvaporative Cooling“remove (=evaporate) energetic atoms in a trap”“atoms remained in a trap have lower average energy（＝lower temperature）“T~100 nK

1nK 以下の温度まで冷却可！

Atomic Gases Reach the Quantum Regime !
“Boson versus Fermion”“Bose-Einsten Condensation”“Fermi Degeneracy”
6Li and 7Li

87Rb

Momentum Distribution

Spatial Distribution[E. Cornell et al, (1995)][R. Hulet et al, (2000)]

Feshbach Resonance:
ability to tune an inter-atomic interactionas   l / kCollision is in Quantum Regime !
It is described by s-wave scattering length as 0  4 f 0  4 as
22

Coupling between “Open Channel” and “Closed Channel”

PotentialControl of Interaction(as)B
as ( B)  abg (1 )B  B0

Molecular State

F2+F2

Two Atoms-C6/R3

F1+F1[C. Regal and D. Jin, PRL90, 230404(2003)]

Realization of Atomic BCS
TBCS  0.3TF exp( 
2k F a s)(k F ) 2
E F  k BTF 2m 3N  3  ω
1T < TBCS

T~TBCST > TBCSB ~ BresB ~> Bres

B >> Bres

D. Jin GroupOptical Lattice
Vo ( x)  Vo sin 2 (k L x)

λ/2
Atoms in optical latticeelectrons in crystalline lattice

Hubbard Model:H  t  c c j  U  ni nii , j 
iti , ji-th

j-thi

U

New Technique: Single Site Observation[WS. Bakr, I. Gillen, A. Peng, S. Folling, and M. Greiner, Nature 462(426), 74-77(2009)]
Fluorescence Imaging

87Rb

New Technique: Single Site Manipulation[C. Ewitenberg et al, Nature 471, 319(2011)]

Outline of LectureII) Quantum simulation of Hubbard model with ultracold atoms
in optical latticeII-1) quantum magnetism and superfluidity
metal-Mott insulator transition, anti-ferromagnetic orderII-2) adiabatic cooling of atoms with spins
Pomeranchuk coolingFermions in a 3D optical latticeH   J  c c j  U  ni , ni ,    i ni
i , j iii“Fermi-Hubbard Model”
λlattice
Tunability ofparameters

luck of defects

λlatticeλlatticeluck of impurities

λlattice/2

Phase Diagram of
Repulsive Fermi-Hubbard ModelSpin UPSpin DOWN

Mott Insulator

Metal

Anti-Ferro
Magnetism

U/J[T. Esslinger, Annu. Rev. Condens. Matter Phys. 2010. 1:129-152,
R. Micnas, J. Ranninger, S. Roaszkiewicw, Rev. Mod. Phys. 62, 113(1990)]

One of the Goals：
quantum simulation of high-Tc super-conductivity
theoryexperiment

AFSC
SC

holeelectron(carrier doping)holex(carrier doping)[in T. Moriya and K. Ueda, Rep. Prog.Phys.66(2003)1299]

electronCurrent Status of
Quantum Simulation of Fermi Hubbard Model:“Formation of (paramagnetic) Mott insulator”[R. Jördens et al., Nature 455, 204 (2008)] [U. Schneider, et al., Science 322,1520(2008)]“Suppression of Doubly Occupied Sites
by Strong Repulsion”

Current Status ofQuantum Simulation of Fermi Hubbard Model:“Formation of (paramagnetic) Mott insulator”[R. Jördens et al., PRL 104, 180401 (2010)]

Quantum Monte-Carlo
Mean-Field
Theoryexperiment

You Need Cooling for Quantum Magnetism(paramagnetic)
Mott Insulator

Anti-Ferro Magnetic OrderCoolings  k B ln(2)Spin Degree of Freedom is Cool
Pomeranchuk Cooling[Pomeranchuk, (1950)]

Discovery of Superfluid 3He by Osheroff, Lee, Richardson

Initial state: Spin-depolarized
and also with degeneracy:

Final state: Spin depolarized:

Adiabatic changes ~ kB𝜋2T/TF
liquid 3Heatoms in traps ~kBln(N)solid 3He atoms in Mott Insulator“entropy flows from motional degrees of freedom to spin,
which results in the cooling of the system”

Outline of LectureIII) Quantum control of atomic spin ensemble
III-1) quantum description of collective atomic spin
coherent spin state, squeezed spin state
III-2) quantum description of polarization of lightquantum Stokes operators
III-3) quantum interface and control
QND measurement of spin, quantum feedback control

collective atomic spin

J X   j x(i ) , J Y   j y(i ) , J Z   j z(i )yix
NAzix

NAyx

J X J Y  N Az[Jx , Jy]=i Jz

JJ

iy

JZJ2yx

Coherent Spin StateSqueezed Spin StateJ X  J Y  J / 2

J X  J / 2 , J Y  J / 2

Quantum Interface“Memory”

QND-interaction: “Communication”
Hint=  SxJx
Atomic Spins: Interface
Light Polarization:
Collective Spin Operators

Quantum Stokes Operators

J X   jx(i ) , J Y   j y(i ) , J Z   jz(i )Sx =(a+ † a+ - a- † a- )/2,Sy =(a+ † a- - a- † a+ )/2i,
Sz = (a+ † a- + a- † a+ )/2,iii[ J  , J m ]  i m J

[S , S m ]  i m  S
 , m ,  x , y , z

 , m ,  x , y , z

j(1) j(2) j(3)j(N)

SzzySySx

“Faraday Rotation” as QND interaction
YT et al., PRA 60, 4974, (1999);
A. Kuzmich et al., Europhys. Lett. 42, 481(1998)

polarimeter“Hint= α SxJx”

JX“Sy”

polarimeter

linear polarized offresonant probe laser
lightprobe laser

detectordetector+

“Quantum State Preparation”
Projection of the Result of QND Measurement
Jx(0)
Sy(0)

Jx(t)= Jx(0)Hint= α SxJx

QND pulseSy(t) :Measurement

yx

yyyx

polarization
detectionxx

Now, 5 groups in the world have successfully created Squeezed Spin States

Typical Spin-QND Measurement:
Scheme and Result
Joint probability distribution

OpticalPump.

12.5 µs“QND”“Verifying”“The squeezed spin state is
created CONDITIONALLY”

Implementing Quantum Feedback Control
Joint probability distribution

OpticalPump.

“QND”“Verifying”

Experimental Setup for
Quantum Feedback Control

Implementing Quantum Feedback Control
Joint probability distribution

OpticalPump.

“QND”“Verifying”-1.4dB
Reduction

Maxwell Demon“Classical”

Reduction of (Thermodynamic) Entropy without Work

“Quantum”

Reduction of Shannon Entropy without Work

Outline of LectureI)

State-of-the-art of ultracold atom technologyII) Quantum simulation of Hubbard model with ultracold atoms
in optical latticeII-1) quantum magnetism and superfluidity
metal-Mott insulator transition, anti-ferromagnetic orderII-2) adiabatic cooling of atoms with spinsdemagnetization cooling, Pomeranchuk cooling
III) Quantum control of atomic spin ensemble
III-1) quantum description of collective atomic spin
coherent spin state, squeezed spin state
III-2) quantum description of polarization of lightquantum Stokes operators
III-3) quantum interface and control
QND measurement of spin, quantum feedback control

Quantum Magnetism via Quantum Feedback ?
(paramagnetic)
Mott InsulatorAnti-Ferro Magnetic Order

Coolings  k B ln(2)Measurement & Feedback Control
With Single Atom Level

Thank you very much for attention

16 AugustMount Daimonji at Kyoto