Controlling and Measuring
Heat Transport in Ion Traps
April 22nd 2015
Alejandro Bermudez Martin Bruderer
Martin B. Plenio
Heat Transport on the Nanoscale
Carbon Nanotubes
Molecular Nanojunctions
Ultracold Atomic Gases
Quantum Dots
& Tunnel Junctions
Ion Crystals
Overview
Physics in Ion Traps
Experimental Tools
Heat Transport
Quantum Dots
Fourier’s Law
Trapped Ion Set-up
•
Singly-charged ions (Be+, Ca+, Mg+, …)
•
Coulomb interaction between ions
•
Single ions & ion crystals
•
Control over motional & internal states (QIP)
H.C. Nägerl, W. Bechter, J. Eschner, F. Schmidt-Kaler & R. Blatt, Appl. Phys. B 66, 603 (1998)
J. Eschner, G. Morigi, F. Schmidt-Kaler & R. Blatt, J. Opt. Soc. Am. B 20, 1003 (2003)
ted Ising
spins
with
Examples - Experiments
y
spin having
unit magnetic
andset
wePlanck’s
have setconstant,
Planck’s constant,
, having
pin
unit magnetic
moment,moment,
and we have
3
For
three
define
; as
J1,2the
5nearestJ2,3 as the nearest,.equalh,toequal
one. to
Forone.
three
spins,
wespins,
definewe
J1 ;
J1,2 5J1J2,3
neighbour
interaction
J1,3next-nearest-neighbour
as the next-nearest-neighbour
s
eighbour
interaction
and J2 ;and
J1,3 Jas
2 ;the
(i)
(i)
interaction
(Fig.
sa the
denote
Pauliof
matrices
of the ith spin.
n
nteraction
(Fig. 1b),
and1b),
sa and
denote
Pauli the
matrices
the ith spin.
We initialize
each
spin parallel
to atransverse
strong transverse
field and then
n initialize
We
each spin
parallel
to a strong
field and then
Observing
Frustration
in
Spin
Systems
on Heat Transport
diabatically
lower
fieldthe
relative
the Ising
couplings
so that theso that Experiments
adiabatically
lower
field
relative
to the
Ising
the
e
2 the
3 to
3 couplings
1
2
Freericks
, G.-D. Lin , L.-M. Duan & C. Monroe
e
d
a
a
b
b J
2
J2
n
AF
AF
4 MHz and n3 5 3.674 MHz. Off-resonance laser beams
s
3
1
2
3
1
2
ye illuminate the ions, driving stimulated
Raman
transitions
the
forces in
e spin states
AFand also imparting Jspin-dependent
J
1
1
J1
J1 quantum
ction.
As discussed in Methods, this
allows
simu10–12
n
cthe Zigzag
Ising Hamiltonian with
Tilt a transverse field CM
c
n
Zigzag X
TiltX
CM
(i) (j)
(i)
e
H~
Ji,j sx sx zBy
sy
ð1Þ
- 3
ivj III
i
IV 3
II
I
t
J1
IV
III
II
I
fis an effective
J2
J1
uniform transverse magnetic
field
with each
s
J2
• Internal states = effective spins
? ?
J (kHz)
J (kHz)
ng unit magnetic moment, and we have set Planck’s constant,
0
oe one.
For• three
spins, we
J1 ;modes
J1,2 5 J2,3 as the nearestInteractions
viadefine
collective
0
er interaction and J2 ; J1,3 as the next-nearest-neighbour
ground states
• Determine
a
(i)
n (Fig. 1b), and sa denote the Pauli matrices of the ith spin.
ol
–3 each spin parallel to a strong transverse field and then
ylize
M. Ramm et al., New J. Phys. 16, 063062 (2014)
K. Kim et al.,
590
–3
–2 Nature 465,
–1 (2010)
0
~
µ
ally
to the
e lower
Islamfield
et al.,relative
Science 340,
583 Ising
(2013) couplings so that the
–3R. the
A. Bermudez et al., PRL 111, 040601 (2013)
–3
–2 New J. Phys.~ 14,–1093042 (2012) 0
ngure 1 | Frustrated
A. Bermudez
et al.,
Ising spins.
a, Simplest case
µ of spin frustration, with
d
hree antiferromagnetic spins on a triangle. b, Image of three trapped atomic
1
bspins.
| Frustrated
a, intensified
Simplest case
of spin frustration, with
Yb1 Figure
ions in 1the
experiment,Ising
taken
with an
charge-coupled-
this Otto engine
runs in- finite
time and has thus
Examples
Proposals
Verifying the Quantum Jarzynski Equality
Single Ion Heat Engine (Otto Cycle)
work on
system
!(0)
F =
he
W/kT
!(⌧ )
kT lnhe
i=
Z
W/kT
i
dW p(W )e
W/kT
1 over
(color
Control
initial online).
& final states
•FIG.
(a) Energy-frequency
diagram of the
• Control over trapping
cycle
ion. and
The
continuous
line
Resolve
singlefor
Fockthe
statesradial mode of the
cooling
of single ions
•Otto
• Heating
thedone
ideal
process, while •the
dots motion
showofthe
of
Determine work
on system
Generate
axialresults
mode
•represents
the
Monte Carlo simulations. (b) The pictograms illustrate the
G. Huber, F. Schmidt-Kaler, S. Deffner & E. Lutz, PRL 101, 070403 (2008)
R. Dorner,
S.R. Clark, L. Heaney,
R. Fazio, J. Goold
& V. Vedral,
PRL 110,for
230601the
(2013) radial state.
four
individual
strokes
of Energy-frequency
the
cycle
1
(color
online).
(a)
diagram of the
lor FIG.
online).
(a)
Energy-frequency
diagram
of
the
O. Abah, …, F. Schmdt-Kaler, … & E. Lutz, PRL 109, 203006 (2012)
(c)
the
Paul
trap:
theThe
rf electrodes
cycle mode
for of
theof
radial
mode
ofcontinuous
the
ion.
continuoushave
line
for Otto
the Geometry
radial
thetapered
ion.
The
line
What are the basic tools?
Motional and Internal States
• Motional and internal states are decoupled
• Manipulate internal states via laser coupling
|f i
|ni
|2i
|1i
|0i
1
V (x) = m! 2 x2
2
motional states
(harmonic trap)
1
p (|gi + |ei)
2
⌦⇠I
|ei
|gi
internal states
(electron configuration)
laser
!L = !eg
Optical Pumping - State Preparation
• Resonant driving of transition |ei to |f i
• Radiative decay into |ei and |gi
|ni
|2i
|1i
|0i
1
V (x) = m! 2 x2
2
motional states
(harmonic trap)
|f i
!ef
|ei
|gi
internal states
(electron configuration)
Resonance Fluorescence - Read Out
• Resonant driving of transition |ei to |f i
• See that level |ei is populated
|ni
|2i
|1i
|0i
1
V (x) = m! 2 x2
2
motional states
(harmonic trap)
|f i
!ef
|ei
|gi
internal states
(electron configuration)
Resonance Fluorescence - Read Out
• Resonant driving of transition |ei to |f i
• See that level |ei is populated
|ni
|2i
|1i
|0i
1
V (x) = m! 2 x2
2
motional states
(harmonic trap)
|f i
!ef
|ei
|gi
internal states
(electron configuration)
State-dependent Potential
• AC-Stark shift from far-detuned driving
• Position-dependent intensity ⌦(x)
|ei
{
|ei
|ei
⌦2 (x)
Ve (x) =
2
!e > !
⌦(x)
|gi
|gi
|gi
!g < !
Vg (x) =
⌦2 (x)
2
Couple Motional and Internal States
• Drive transitions that change HO state
• Requires narrow HO states
|0, ei
|1, ei
carrier
|2, ei
blue sideband
red sideband
|0, gi
|1, gi
|2, gi
···
Spin-Mode Coupling
• Internal states act as spins
|ei ⌘ | "i |gi ⌘ | #i
• Spin-spin interactions via common mode)
|0, ei
a†
+a
|2, ei
|1, ei
a†
+
+
+a
+
|0, gi
|1, gi
= |eihg|
= |gihe|
|2, gi
Hint ⇠
†
(a
+
a
)
x
Sideband Cooling
• Combine coherent driving & spontaneous decay
• Control temperature down to hni ~ 0.05
• Doppler cooling hni ~ 3 - 10
|0, ei
|0, gi
|1, ei
|1, gi
|2, ei
|2, gi
hni '
✓ ◆2
!
⌧1
Temperature Measurement
• Thermal population of motional state hni
• Start in |gi and observe population in
|0, ei
|0, gi
|1, ei
|1, gi
|ei after fixed time
|2, ei
|2, gi
Pered
hni
=
blue
Pe
1 + hni
Temperature Measurement
• Thermal population of motional state hni
• Start in |gi and observe population in
|0, ei
|0, gi
|1, ei
|1, gi
|ei after fixed time
|2, ei
|2, gi
Pered
hni
=
blue
Pe
1 + hni
Collective Modes
Hamiltonian for motional ion states
H=
X ⇣ p2
i
⌘ e2 X
1
1
i
2
+ ri ! ri +
2m 2
2
|ri rj |
i6=j
Axial modes
Radial modes
harmonic oscillator modes ~ collective modes.
Fourier’s Law
Fourier’s Law
J. Fourier, Théorie Analytique de la Chaleur (1822)
Fourier’s Law
J. Fourier, Théorie Analytique de la Chaleur (1822)
Fourier’s Law
• How does Fourier’s law emerge from microscopic laws?
• When do temperature gradients occur?
• Observe Fourier’s law on nanoscales?
Z. Rieder, J. L. Lebowitz & E. Lieb, J. Math. Phys. 8, 1073 (1967)
M. Michel, G. Mahler & J. Gemmer PRL 95, 180602 (2005)
A. Asadian, D. Manzano, M. Tiersch & H. J. Briegel, Phys. Rev. E 87, 012109 (2013)
Trapped-Ion Crystal Toolbox
Functionalities of ions
•
Full control over internal states
•
Separation of time scales
•
Different atomic species
•
Bulk ions
•
Heat reservoir ions
•
Multi purpose ions
Vibron Hopping Model
Hamiltonian for motional ion states
H=
X ⇣ p2
i
⌘ e2 X
1
1
i
2
+ ri ! ri +
2m 2
2
|ri rj |
i6=j
Tight-binding model for vibron hopping
H=
X
i
!i a†i ai +
X
Jij a†i aj + h.c.
i6=j
! ⇠ 1MHz
J ⇠ 10kHz
• Small radial oscillations above ground state (vibrons)
• Coupling from dipole-dipole interaction J ⇠ 1/d3
• Heat transport by vibron hopping
D. Porras & J. I. Cirac, PRL 93, 263602 (2004)
chain of weakly coupled
harmonic oscillators
Heat Reservoirs
Tight-binding model for vibron hopping
H=
X
!i a†i ai
+
i
X
Jij a†i aj + h.c.
i6=j
n
¯L
n
¯R
Continuous heating/cooling of edge ions
L ↵ ⇢↵ =
↵
2
(¯
n↵ + 1)(2a⇢↵ a†
• Effective cooling rate
{a† a, ⇢↵ }) +
↵
2
n
¯ ↵ (a $ a† )
↵ = {L, R}
↵
• Reservoirs at constant temperature
n
¯↵
• Cooling much faster than hopping
↵
Jij
Spin-Vibron Coupling
Tight-binding model for vibron hopping
H=
X
!i a†i ai
i
+
X
Jij a†i aj + h.c.
i6=j
n
¯L
n
¯R
Couple internal states to vibrons
HiSV
1
= (A +
2
z
i ) cos(⌫t
1
A cos(⌫t)a†i ai
2
1
z †
=
i ai ai
2
)a†i ai
(1)
HiSV =
⊳ Photon-assisted tunneling
(2)
HiSV
⊳ Probing & Disorder
Ballistic Transport
Vibron occupations
Jij and project
Assume ↵
th
dynamics onto state ⇢th
⌦
⇢
⌦
⇢
bulk
L
R
hni iss =
hI vib iss =
¯L
Ln
+
L+
L R
L
+
R
¯R
Rn
R
(¯
nL
n
¯R)
•
Ballistic transport of vibrons
•
Anomalous heat transport
Spin-Induced Disorder
Strong spin-vibron coupling
HiSV
1
=
2
Spins in superposition
1
p
|+ii =
(| "ii + | #ii )
2
z †
i ai ai
Random binary alloy (RBA) model (bosonic)
H
RBA
=
X
"i a†i ai
+
i
X
Jij a†i aj + h.c.
i6=j
Binary diagonal disorder
"i 2 {!i
1
2
, !i +
1
2
}
Exploit “quantum parallelism”
B. Velicky, S. Kirkpatrick & H. Ehrenreich, Phys. Rev. 175, 747 (1968)
B. Paredes, F. Verstraete & J. I. Cirac, PRL 95, 140501 (2005)
Fourier’s Law
Ballistic
Disorder
Homogeneous
Inhomogeneous
Clear signature of Fourier’s law
Dephasing from Electrodes
Dynamic fluctuations from noisy electrodes
Voltage
Tight-binding model with dephasing noise
H =
X
"i (t)a†i ai
i
+
X
Jij a†i aj + h.c.
i6=j
Markovian noise with correlation length ⇠c
"i (t) = !i + !(t)
Description as Lindblad operator
L ⇢=
X
i,j
e
dij /⇠c
(2ni ⇢nj
{nj ni , ⇢})
Fourier’s Law
Ballistic
Dephasing
Disorder
Homogeneous
Inhomogeneous
Thermal Quantum Dot
Single site & thermal leads
| # · · · #i| i| " · · · "i
!# ⌧ !
!"
Photon-assisted tunneling
HiSV
H
1
= A(1 +
2
PAT
=
X
!
=A
†
z
)
cos(⌫t)a
i
i ai
JiPAT a†i a
+ h.c
i
Full control of heat current through TQD
Single-spin heat switch
| i = | "i
| i = | #i
Nonequilibrium Schrödinger’s Cat
Single site & thermal leads
| # · · · #i| i| " · · · "i
!# ⌧ !
!"
Photon-assisted tunneling
HiSV
H
1
= A(1 +
2
PAT
=
X
!
=A
†
z
)
cos(⌫t)a
i
i ai
JiPAT a†i a
+ h.c
i
Full control of heat current through TQD
Single-spin heat switch
| i = | "i
| i = | #i
1
| i = p (| "i + | #i)
2
Verify Fluctuation Theorems
Single site & thermal leads
| # · · · #i| i| " · · · "i
!# ⌧ !
!"
!
Verify fluctuation theorems for bosons (switch on)
p(N, t)
lim
= e!  N (
t!1 p( N, t)
D
S)
N = number of vibrons
Use Ramsey probe to measure current
M. Esposito, U. Harbola & S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009)
Non-Invasive Ramsey Probe
|↑〉
Operator couples weakly to spin
HiSV =
|↓〉
π/2 pulse
x
ii
1
hOi iss t e
= cos
S(!) =
π/2 pulse
z
i
Spin evolution
h
free induction
1
Oi
2
Z
2
S(0)t
1
0
dt hhOi (t)Oi (0)iiss e
⊳ Oscillations with frequency
⊳ Damping by fluctuations
i!t
O
δO²
Measure occupations and thermal currents
-1
MB and D. Jaksch, New J. Phys. 8, 87 (2006)
G. B. Lesovik, F. Hassler & G. Blatter, PRL 96, 106801 (2006)
Summary
Implement
thermal
reservoirs
Measure currents
& local temperatures
Control & manipulate
heat flow
Analytical Results
Local vibron hopping and dephasing
Vibron occupations
Vibron current
Long-range coupling 1/d³ probably not essential
A. Asadian, D. Manzano, M. Tiersch & H. J. Briegel, Phys. Rev. E 87, 012109 (2013)
Zig-zag Crossover
Reduce radial trapping
for zig-zag crossover
linear
zig-zag
zag-zig
• Anomalous transport for linear chain
• Zig-zag results in temperature gradient and suppressed heat current
• Nonlinearity & coupling of axial and radial modes
K. Pyka et al., Nat. Commun. 4, 2291 (2013)
A. Ruiz, D. Alonso, M. B. Plenio & A. del Campo, Phys. Rev. B 89, 214305 (2014)
N. Freitas, E. Martinez & J. P. Paz, Preprint arXiv:1312.6644 (2013)
Experimental Status
• Heat transport in chain with up to 37 ions
• Measure vibron occupation via cooling time
• Coupling
10kHz stronger than cooling rate (need
↵
Jij )
M. Ramm, T. Pruttivarasin & H. Häffner, New J. Phys. 16, 063062 (2014)
T. Pruttivarasin, M. Ramm, I. Talukdar, A. Kreuter & H. Häffner, New J. Phys. 13 075012, (2011)