Post on 03-Jun-2019
The a.c. and d.c. Josephson effects in a BEC
Technion - Israel Institute of Technology
Jeff Steinhauer
Shahar Levy
Elias Lahoud
Itay Shomroni
Outline
What is the a.c. Josephson Effect?HistoryOur ultra-high resolution BEC systemMeasuring the effectEffects of finite temperatureWhat is the d.c. Josephson Effect?Measurement of this effect
a.c. Josephson Effect
Vmicrowaves
h
eV2=ω
Ia.c. Josephson
effect is the voltage
standard
B. D. Josephson, Phys. Lett. 1, 251 (1962).
S. V. Pereverzev, A. Loshak, S. Backhaus, J. C. Davis, and R. E. Packard, Nature 388, 449 (1997).
He3superfluid
eg Ψ−Ψ eg Ψ+Ψ
Compute the Josephson Relations for atoms
Ψ⎥⎦
⎤⎢⎣
⎡Ψ++∇−=
∂Ψ∂ 22
2
2gV
mti ext
hh
1: Gross-Pitaevskii equation (T = 0)
no interactions
h
ge EE −= 2ω ω depends on the
coupling only
gg E,Ψ
ee E,Ψ
tEie h−
∝Ψ
a.c. Josephson Effect
Vmicrowaves
h
eV2=ω
Ia.c. Josephson
effect is the voltage
standard
B. D. Josephson, Phys. Lett. 1, 251 (1962).
S. V. Pereverzev, A. Loshak, S. Backhaus, J. C. Davis, and R. E. Packard, Nature 388, 449 (1997).
He3superfluid
eg Ψ−Ψ eg Ψ+Ψ
Compute the Josephson Relations
Ψ⎥⎦
⎤⎢⎣
⎡Ψ++∇−=
∂Ψ∂ 22
2
2gV
mti ext
hh
1: Gross-Pitaevskii equation (T = 0)gg E,Ψ
ee E,Ψ)()()()(),( 2211 rtrttr rrr
Φ+Φ=Ψ ψψ2: two-mode approximation
)(1 rrΦ )(2 rrΦ 1222
2111
κψψμψκψψμψ−=−=
&h
&h
ii The Feynman
Lectures
Resulting equations
212211 )(,)( φφ ψψ ii eNteNt ==
3: write ψi as
Josephson Relations
φ
Rigid Pendulum
h
μΔ−
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&
These equations are non-linear
φωη
μφ
sinJ=
Δ−=
&h
&
ηωμ Ch=Δ
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
h
κω 2≡J
12/2 <<η
Assume:
•
• Josephson regime
φη ∝&bulk
Josephson regime
Weak coupling ( )
Phase coherence ( )
Results in:
2 separate condensates ( )
Rigid pendulum equations (like superconducting
case)
CJ ωω <<
h& μφ Δ
−=
2
2⎟⎠⎞
⎜⎝⎛<<
N
J
C
ωω
energyn interactio measures energy coupling measures
C
J
ωω
Previous interferometers
h& μφ Δ
−=
1μ2μ
111 ,, φψ N 222 ,, φψ N
Shin, Y., Saba, M., Pasquini, T. A., Ketterle, W., Pritchard, D. E. & Leanhardt, A. E., Phys. Rev. Lett. 92, 050405 (2004).
Schumm, T., Hofferberth, S., Andersson, L. M., Wildermuth, S., Groth, S., Bar-Joseph ,I . , Schmiedmayer, J. & Krüger, P. Nature Physics 1, 57-62 (2005).
Saba, M., Pasquini, T. A., Sanner, C., Shin, Y., Ketterle, W. & Pritchard, D. E. Science 307, 1945-1948 (2005).
2
2⎟⎠⎞
⎜⎝⎛>>
N
J
C
ωω
η&
Albiez, M., Gati, R., Fölling, J., Hunsmann, S., Cristiani, M. & Oberthaler, M. K. Phys. Rev. Lett. 95, 010402 (2005).
Josephson regime
Continuous readout
Our interferometer
h& μφ Δ
−=
1μ2μ
111 ,, φψ N 222 ,, φψ N
We will observe this relation by the a.c. Josephson effect
η&
a.c. Josephson Effect
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
φ
Rigid Pendulum
h
μΔ−
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&
h
h&
h&
μω
μωη
μφ
Δ=
⎟⎠⎞
⎜⎝⎛ Δ−=
≈Δ
−=
tJ sin
const
a.c. Josephson effect
φh
μΔ−
φωη
μφ
sinJ=
Δ−=
&h
&
ηωμ Ch=Δ
φ
Plasma oscillations
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
φ
Rigid Pendulum
h
μΔ−
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&πφπ <<−
Plasma oscillations
resonance
φωη
μφ
sinJ=
Δ−=
&h
&
ηωμ Ch=Δ
φωη J≈&
JCωωω =
φ
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, PRL 95, 010402 (2005).
F. S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001).
Plasma oscillations
1-D lattice Single BEC Josephson junction
a.c. Josephson Effect
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
φ
Rigid Pendulum
h
μΔ−
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&
h
h&
h&
μω
μωη
μφ
Δ=
⎟⎠⎞
⎜⎝⎛ Δ−=
≈Δ
−=
tJ sin
const
a.c. Josephson effect
φh
μΔ−
φωη
μφ
sinJ=
Δ−=
&h
&
ηωμ Ch=Δ
a.c. Josephson Effect
| F =1, mF = -1 >
h
μω Δ=
| F =2, mF = 1 >
D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, PRL 81, 1543 (1998).
B. P. Anderson and M. A. Kasevich, Science 282, 1686 (1998).
1-D latticeInternal System
Δμ due to asymmetric potential, rather than population difference
Thus, no pendulum equationsηωμ Ch≠Δ
Technion Laboratory
M. Greiner, I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. A 63, 031401(R) (2001).
Ultra high-resolution BEC system
4 mmimaging
potential beam
resolution = 1.2 μm
probe
beam
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
Light barrier
10 µmBECMagnetic trapping
(Zeeman shift)
Laser light sheetkr
Electric dipole potential (Stark shift)
BEC Josephson junction
a
5μm
b
1μm
1/e2 diameter = 1.4 μmS. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
0 0.33 0.66 0.99 1.32 1.65 1.98 2.31 2.64 2.97 3.30 3.63msec
10 μm
2Z1∝μ
11/2
∝μ12
∝μ22
p1
p2
Creating Δμ
→
t (msec)
η
0 2 4 60
0.1
0.2
0.3
0.4
0.5 Δμ = 750 Hz
Δμ = 450 Hz21
21
NNNN
+−
≡η
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
The a.c. Josephson effect
Interferometer calibration
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Δμ/h (kHz)
ω/2π
(kH
z)
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
Effect of the thermal atoms (T > 0)
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&
φωη
μφ
sinJ=
Δ−=
&h
&
ηωμ ch=Δμφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
ηωμ ch=Δ
Damped pendulum
I. Zapata, F. Sols, and A. J. Leggett, Phys. Rev. A 57, R28 (1998).
Pendulum
Δμ drives the thermal atoms
Effect of the thermal atoms
a.c. Josephson effect
φh
μΔ−
Pendulum slows down
Δμ decreases
η decreases
0 50 100 1500
0.05
0.1
0.15
0.2
0.25
t (msec)
η
Effect of the thermal atoms
μφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
ηωμ ch=Δ
Damped pendulum
Macroscopic quantum self-trapping (MQST)Pendulum speed Thermal fraction = 5% (T ≈
0.3 Tc )
Decay of the MQSTThermal fraction = 20% (T ≈
0.5 Tc )
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, PRL 95, 010402 (2005).
MQST
Interferometer relies on the MQST
φh
μΔ−
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
d.c. Josephson Effect
I
V = 0Tunneling
supercurrent
B. D. Josephson, Phys. Lett. 1, 251 (1962).
Can atoms also do this?
image
Applying a Current BiasS. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev. Lett. 84, 4521 (2000).
21
21
NNNN
+−
≡η
is the applied currentequilη&
→
→
0>equilη
0=equilη
Applying a Current Bias
21
21
21
21
NNNN
+−
≡
−≡−≡Δ
η
φφφμμμ
1μ2μ
111 ,, φψ N 222 ,, φψ N
η&
μφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
ηωμ Ch=Δ
μφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
( )equilC ηηωμ −=Δ h
→
Analogous Non-Linear Systems
μφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
( )equilC ηηωμ −=Δ h
μΔ−G
Cωμ h& /Δ−
φω sinJ equilη&
1μ 2μ
U 0φ
φ
equil Jη < ω&
equil Jη > ω&
0=Δ
<
μ
ωη Jequil&
0>Δ
>
μ
ωη Jequil&
→
→
Analogous Non-Linear Systems
μφωη
μφ
Δ−=
Δ−=
GJ sin&h
&
( )equilC ηηωμ −=Δ h
μΔ−G
Cωμ h& /Δ−
φω sinJ equilη&
1μ 2μ
oJequil
Jequil
φωη
ωη
sin=
<
&
&effectJosephsonDC
U 0φ
φ
equil Jη < ω&
equil Jη > ω&
0=Δ
<
μ
ωη Jequil&
0>Δ
>
μ
ωη Jequil&
Δμ - I relation
DC DC JosephsonJosephson effecteffect
Washboard potential
Image
time
η̇eq
uil
rapid variation
Gross-Pitaevskii Equation
U 0φ
φ
equil Jη < ω&
equil Jη > ω&
0=Δ
<
μ
ωη Jequil&
0>Δ
>
μ
ωη Jequil&
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
Measured Values
1sec30 −=Jω
1sec9000 −=Δ
=ημωC
CJ ωω << Josephson regime
Thus, pendulum equations
S. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
Fluctuations
μK 20barrier effective ==B
J
kNωh
nK 100 etemperatur <T
nK 4nsfluctuatio quantum ==B
JC
kωωh
Fluctuations are not expected to play a role
U 0φ
φ
equil Jη < ω&
equil Jη > ω&
BEC SQUID detector of rotation
2sinφωJ1sinφωJ
equilη&
⎟⎟⎠
⎞⎜⎜⎝
⎛ ⋅Ω=
h
rr
&Am
Jequil cos2max ωη
Ω
Superconducting quantum interference device (SQUID)
quantization of circulation
BEC SQUID detector of rotation
rapid variationSimmonds, R. W., Marchenkov, A., Hoskinson, E., Davis, J. C. & Packard, R. E. Nature 412, 55-58 (2001).
He3superfluidS. Levy, E. Lahoud, I. Shomroni, and J. Steinhauer, Nature 449, 579 (2007).
Conclusions
We have made the first observation of the a.c. Josephsoneffect in a single BEC Josephson junction We have made the first observation of the d.c. Josephsoneffect in any atomic system The MQST is seen to be qualitatively altered by the thermal cloudWe have measured the relation between the chemical potential difference and the applied currentThe device is suitable for use in the analog of a SQUID detector This device constitutes a real-time atom interferometer based on the a.c. Josephson effectThis is the first application of our new type of BEC system with ultra high-resolution, capable of applying almost arbitrary potentials and imaging on a tunneling length scale