The a.c. and d.c. Josephson effects in a BEC · Outline What is the a.c. Josephson Effect? History...

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

Presenter�

Presentation Notes�

Must the energy be less than the barrier (CT)? Graph for the two-mode approximation �


Vmicrowaves

h

eV2=ω

Ia.c. Josephson

effect is the voltage

standard


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

Presenter�


Must the energy be less than the barrier (CT)? Graph for the two-mode approximation �

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&


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


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


φh

μΔ−

φωη

μφ

sinJ=

Δ−=

&h

&

ηωμ Ch=Δ


| 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).

Technion Laboratory

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).

BEC Josephson junction

a

5μm

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


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)


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

Presenter�


Delta mu drives thermal atoms�

Effect of the thermal atoms


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−


d.c. Josephson Effect

I

V = 0Tunneling

supercurrent


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

&


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

&


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&


Measured Values

1sec30 −=Jω

1sec9000 −=Δ

=ημωC

CJ ωω << Josephson regime

Thus, pendulum equations


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).

Presenter�


SQUARE**********************************�

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

Compute the Josephson Relations

21

21

21

21

NNNN

+−

≡

−≡−≡Δ

η

φφφμμμ

1μ2μ

111 ,, φψ N 222 ,, φψ N

η&

2

1

22

11

φ

φ

ψ

ψi

i

eN

eN

=

=

φηωη

φη

ηωμφ

sin1

cos1

2

2

−=

−−

Δ−=

J

J

&

h&

ηωμ Ch=Δ

The a.c. and d.c. Josephson effects in a BEC · Outline What is the a.c. Josephson Effect? History...

Documents

Transcript of The a.c. and d.c. Josephson effects in a BEC · Outline What is the a.c. Josephson Effect? History...