Spatial adiabatic passage and Josephson effect for

Bose-Einstein condensate in a double-well trap

BLTP, JINR, Dubna, Moscow region, Russia

V.O. Nesterenko

SAP, Okinawa, Japan, 25-27.05.2016

A.N. Novikov (BLTP, JINR, Dubna, Moscow region, Russia)

E. Suraud (LPQ, Universite Paul Sabatier, Toulouse, France)

Trapped BEC:

- Unique in the precision and flexibility in control and manipulations.

Unique laboratory for investigation of various quantum scenarios.

- Crossover of BEC with other areas (nano, superfluidity, Josephson effect, qauge theories, quantum control,

quantum informatics, topological systems, quantum turbulence, nuclear physics,

astrophysics,..)

- Analogy between:

Josephson effect in superconductors (SJJ)

and BEC in a double-well trap (BJJ)

Motivation

V.O.N, A.N. Novikov, and E. Suraud, Laser Phys., 24 125501 (2014)

- The main point to be studied:

non-linear impact of the interaction between BEC atoms

Method:

- Gross-Pitaevskii equation,

- three-mode approximation

- control of barrier penetrabilities

Results:

- detrimental role of the non-linear interaction effect

- passage beyond STIRAP

- generation of topological (Berry phase) phase

1

2

3

(1)

(3)

(2)

Circular well configuration

Dynamics of BEC in M-well trap: model

System of GPE equations of M-component BEC

2

1 1

ˆ[ | | ] ( )( )( 1)M M

k jk k kj kjkj j

j j

h tit

g t

with the order parameter:

kjg - interaction

( )kj t - coupling

( ) exp{ }( ) ( )k kk t N it t

( )kN t

( )k t

- normalized

population

- phase Equations for phases and populations:

1

( ) ( ),M

k k

k

t r

1

sin( )M

k j k k

j

jkjN N Nt

t

1 1

1cos(

2( )) ( )

M

kj

j

k j k jk

j k

k

M

j

j tN

NN

Et

t

The key parameter regulating

interaction-coupling ratio

( ) ( )kj kjt K t 2

2

( )( ) exp{ }

2

kj

kj

t tt

/ 2 , 2k kE E K Kt t

2

kj

kj

U N

K

kj kjU g

Scaled dimensionless time

1

( ) 1M

k

k

N t

A. Smerzi et al, PRL, 79, 4950 (1997)

V.I. Yukalov et al, PRA, 56, 4845 (1997)

E.M. Graefe et al, PRA, 73, 013617 (2006)

Circular 3-step STIRAP transport of BEC

as a function of interaction -- well 1

-- well 2

-- well 3

1

2

3

(1)

(3)

(2)

circular configuration

Couplings

iΩ

Populations

iN

2

UN

K

time time

STIRAP:

- is complete at , distorted at , and breaks down

at lager interaction

- in general the interaction is detrimental for the passage

- the passage is not fully adiabatic even without interaction

This is natural since the intermediate well anyway has to get some

temporary population.

Λ< 0.5Λ=0

D – detuning

STIRAP vs non-adiabatic transport

STIRAP intuitive order

of pulses

Non-adiabatic transport in the TWT is also rather effective.

2

2

( )( ) exp{ }

2

kj

kj

t tt

d – time interval between

the pump and stokes pulses

STIRAP-generation of phases

in 3-well system

0 50 100 150 200

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

Ph

ase

s

Time

dynamical

geometric

total

The problem: to develop the transport scenario where

0dyn

tot geom

tot dyn geom

0

Im ( ( ), ( ))T

dyn dt t t arg( (0), ( ))geom T

R. Balakrishnan, M. Mehta,

EPJD, 33, 437 (2005).

unconventional

geometric phase

geom dyn

S.-L. Zhu and Z.D. Wang,

PRL 91, 187902 (2003)

Λ=0.2

Method:

- Gross-Pitaevskii equation,

- two-mode approximation (TMA)

- control of barrier penetrability

and depth detuning (two parameters)

- generalization of Landau-Zener and

Rosen-Zener transport protocols

Results:

- Interaction between BEC atoms favors

the transport !

-new possibilities while using two control

parameters (no adiabatic limit)

Interaction between BEC atoms is detrimental for adiabatic

STIRAP transport in TWT.

What about the interaction impact for SAP in DWT?

|1> |2>

Δ(t) < 0

( )t

Transport

LZ vs LZ+RZ transport of BEC

|1> |2>

Δ(t) < 0

( )t0

( ) ( )t K t 2

2( ) exp{ }

2

tt

Gaussian coupling:

1

2

1E ,

2

1E ,

2

t

t

2

UN

K

Linear case (no interaction):

2

2LZP e

21 ( )LZP P N t

0

Linear case: no difference

Nonlinear case ?

Adiabatic transfer for

LZ + RZ LZ

1 2E ( )( E )) (tt t t

LZ+RZ:

two control parameters

(detuning and penetrability)

2

4

Nonlinearity impact is asymmetric:

- favorable for repulsive BEC

- harmful for attractive BEC

The repulsive interaction increases

the chemical potential and thus

penetrability of the barrier. This favors

the passage.

Attractive interaction has the opposite

effect.

The reasons of the effects

Windows with P=0 at small detuning

rate . No adiabatic lmit!

Two control parameters: penetrability

and detuning .

We lose the adiabatic limit if not both

control parameters support the

adiabaticity:

e.g. small (slow) and large

LZ LZ + RZ

2( )P N t2

UN

K

Physics behind the nonlinear effect

N.V. Vitanov et al

Adv. Atom. Mol. Opt. Phys.

46, 55 (2001)

2 2 3 / 21[ ( ) ( ) ( ) ( )] [ ( ) ( )]

2t t t t t t

The repulsive interaction increases the chemical potential and thus

the barrier penetrability

Then the adiabatic condition

adopts higher process rates and .

As a result, the repulsive interaction strongly favors (speeds up)

the SAP in a double-well trap

Physics behind the nonlinear effect

N.V. Vitanov et al

Adv. Atom. Mol. Opt. Phys.

46, 55 (2001)

2 2 3 / 21[ ( ) ( ) ( ) ( )] [ ( ) ( )]

2t t t t t t

The repulsive interaction increases the chemical potential and thus

the barrier penetrability

Then the adiabatic condition

adopts higher process rates and .

As a result, the repulsive interaction strongly favors (speeds up)

the SAP in a double-well trap

That time we have not yet recognized that our SAP is actually

the dc Josephson effect.

SAP vs dc/ac Josephson effect

V.O. N., A.N. Novikov, and E. Suraud,

”Transport of the repulsive Bose-Einstein condensate in a double-well trap:

interaction impact and relation to Josephson effect”,

Laser Phys., 24, 125501 (2014).

Modification of the model:

- production of SAP by the barrier shift

- 3D TD-GPE for the total order parameter - no two-mode approximation, etc

- both weak and strong coupling

- experimental parameters and schemes

Only in our study

- the full set of principle values was scrutinized.

- the approximate similarity of Josephson dc/ac in SJJ and BJJ was shown

(I, , )

Many previous studies including early ones

F. Dalfovo, L. Pitaevskii and S. Stringari,

PRA 54, 4213 (1996)

A.Smerzi, S. Fantoni, S.Giovanazzi, and

S.R. Shenoy, PRL 79, 4950 (1998)

S. Giovanazi, A. Smerzi and S. Fantoni

PRL 84, 4521 (2000)

Experimental observation of Josephson oscillations and self-trapping

M. Albiez, et al, Phys. Rev. Lett., 95 010402 (2005)

R. Gati, M.Albietz, et al, Appl. Phys. B82, 207 (2006)

2 25HzJO 2 80HzMQST

0.50cz 1 2N Nz

N

2 2 21( ) cos

2x b

xV m x x V

d

R. Gati and M.K. Oberthaler, JPB, 40, R61 (2007)

Barrier shift as suitable

control parameter.

2 25Hz 2 23Hz

2 78Hz 2 72Hz

JO

MQST

Exper. Theor.

Good agreement with Heidelberg

experiment for N=1000!

3D TD-GPE results:

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

N=1000

zz

zz

t [ms]

/

/

N=2000

/

N=5000

/

t [ms]

N=10000

JO Population imballance Phase difference

- JO survives, MQST converges to JO

- important test for our model

0 20 400

2

4

6

8

0 20 400

2

4

6

8

0 20 400

2

4

6

8

0 20 400

2

4

6

8

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

0 20 40-1

0

1

/

N=1000

/

t [ms]

N=10000

/

N=2000

/

N=5000

zz

t [ms]

zz

MQST

V.O.N., A.N. Novikov, and E. Suraud, JPB, 45, 225303 (2012)

|1> |2>

Δ(t) < 0

( )t

Weakly coupled BECs

SC SC =

The current in BEC is generated by the barrier shift:

- slow (adiabatic) shift stationary Josephson effect

- rapid shift non-stationary Josephson effect

( )cv v

( )cv v

S. Giovanazzi, A. Smerzi, and S. Fantoni,

PRL 84 4521 (2000)

Stationary effect, - small supercurrent

- no voltage V=0

- constant phase difference

determines the current magnitude

- constant direct current (dc)

- slow adiabatic passage

Josephson effect

0I I

Tunneling transfer of Cooper pairs of

electrons via a thin dielectric layer

separating two superconductors

Fully determined by phase difference.

Non-stationary effect - large resistive current

- non-zero voltage V>0

- sum of super and normal currents

- alternating current (ac)

- emission of photons

- fast non-adiabatic passage

|1> |2>

Δ(t) < 0

( )t

SJJ -- Superconductor Josephson Junction

BJJ – Bose Josephson Junction

s 0I =I sin( )

2eV

h

Josephson

equations

0I I

s nI I I

Analogy between SJJ and BJJ:

- 2 superconductors 2 superfluid BECs

- thin dielectric potential barrier

- in both cases: tunneling of bosons, weak coupling

- initiation by: current barrier shift

R L

sI I

- phase difference

0I - critical current

Model

22 2( , ) [ ( ) | ( , ) | ] ( , )

2exti r t V r g r t r t

t m

Time-dependent 3D

Gross-Pitaevskii

equation for

order parameter

confinement

+ barrier

BEC interaction,

nonlinearity

2 2 2 2 2 2 2 00

0

( ( ))( ) ( ) cos ( )

2ext x y z

x x tmV r x y z V

q

HO confinement barrier

3 2| (r,t)| =Ndr

0 ( )

2

LN = | (r,t)|

x t

dydz dx

0

2

( )

N = | (r,t)|R

x t

dydz dx

3

j

3

Im (r,t)

=arctan ,

Re (r,t)

j

j

dr

dr

2( , ) | (x,y,z,t)|x t dydz

population imbalance ,j L R L

s

Nz=

zI =-

2

RN

N

R L

h

current

phase difference

chemical potential

difference

22 2( , ) [ ( ) | ( , ) | ] ( , )

2exti r t V r g r t r t

t m

2

2

2 2 1 sin

cos2 21

sz I K z

z NUK z

z

In two-mode approximation (TMA),

the GPE is reduced to the system

of equations similar to Josephson

equations:

s 0I =I sin( )

h

Josephson

equations

s

20

zI =-

2

I=I 1 sinz

Exact current:

Approximate

TMA current:

Similarity between GPE equations of motion and Josephson equations

small z

K –barrier penetrability

U – interaction between BEC atoms

0I K

Tunneling transport induced by the barrier shift: Potential: HO confinement + barrier (element of a periodic optical lattice)

Velocity profiles:

2( ) cos ( )2

m

tV t V

T

0V const1)

2)

t 0 T

0V

t 0 T

V(t)

Asymmetric stationary initial state Asymmetric final state Symmetric initermed. state

t

N1(0)=800, N2(0)=200

z(0) = [N1(0)-N2(0)]/N = 0.6

t=0 t=T/2 t=T

Sharp changes at the beginning and end

of the evolution result in strong undesirable

dipole oscillations

S. Giovanazzi, A. Smerzi, and S. Fantoni,

PRL 84 4521 (2000)

Initial barrier shifts for z(0)=0.6 ( )

d= 3 nm - no inter.

d=500 nm - with inter.

- trap parameters and initialization

of the process like in Heisenberg

experiment

-initial equilibrium asymmetric state

- barrier position as a control parameter

- transfer during the time T to get the

inverse population

- weak coupling

-- the chemical potential is always

below the barrier top tunneling

-interaction:

- more coupling

- larger initial barrier shift

- the similar technique was used in

Trap-density configuration M. Albiez, et al,

PRL 95 010402 (2005)

V.O.N., A.N. Novikov, and E. Suraud,

J. Phys. B 45 225303 (2012)

1000 atoms 87Rb

800, 200L RN N

0

0

2 78

2 66

2 90

420

5.2

x

y

z

Hz

Hz

Hz

V hHz

q m

ideal BEC

v~ nm/s

repulsive BEC

v~ m/s

soft barrier

velocity

Population transfer in ideal and repulsive BEC

Interaction allows robust transfer three orders of magnitude faster than in ideal BEC

Impact of interaction between BEC atoms

Interaction:

- allows robust transfer three orders

of magnitude faster,

- provides a wide plateau for velocities

with a complete population transfer

- At some critical velocity

the adiabatic passage fails.

0 [ / ]sv m s

0 [ / ]sv m s

Does this process correspond to ac/dc Josephson effects?

Does the transport critical velocity correspond to Josephson critical current?

-z(T)P=-

z(0)

For complete inversion:

z(0)=0.6

z(T)=-0.6

22 /s

cv m s

Phase difference and current :

ideal BEC

soft barrier velocity

(adiabatic case)

- behavior of the current is driven by

- the remaining at the end of the process (geometric phase?)

- unlike the familiar dc, is not constant and is not zero (though small),

which is explained by using time-dependent velocity of the barrier/

- more complicated picture than in the paper of Giovanazzi

s 0I =I sin( )

h

Stationary Josephson effect (dc)

repulsive BEC, soft velocity profile 0sI I -

-

- =const

0

Up to this small variation, BEC transport can be associated with dc (stationary)

Josephson effect.

The variation of and current is caused by using time-dependent velocity of the

barrier.

The transport:

-

- ~ 1-4 Hz not zero but small

- rad changes but not much 1

h

0I I

- exact current I(t) from GPE

.. approximate current i(t) from TMA-GPE

slow

medium

fast

Non-stationary Josephson effect (ac) 0sI I

th

0

The transport:

- at a critical velocity ~ 12-13

the transport is transformed into

high-frequency modulated oscillations

~ 10-100 Hz becomes large

- changes linearly with time

- a slight modulation is caused by

dipole oscillations caused by the

rapid barrier velocity

h

/m s

So we obviously get the dc-ac transfer.

The Josephson ac should not be confused

with MQST where also

Indeed in ac case the oscillations are around

z=0 while in MQST

t

0z

Conclusions

- Approximate analogy between dc/ac in SJJ and BJJ is confirmed

- important role of nonlinearity in BJJ:

the critical current is increased by 3 orders of magnitude,

- need in soft velocity time-profile

- Analysis of SJJ-BJJ analogy in terms of currents, chemical potentials and

phase differences.

For weak coupling in DWT, there is the analogy between:

- space adiabatic passage dc Josephson effect

(LZ, RZ, …)

- break of adiabatic passage transfer to ac Josephson effect

- critical SAP velocity critical Josephson current

Then SAP in DWT (weak coupling) is always driven by the phase difference

and fulfills Josephson equation s 0I =I sin( )

Levy’s experiment for dc/ac Josephson effect in BEC (1) S. Levy et al,

Nature 449, 579 (2007)

2 224

2 26

r

z

Hz

Hz

87Rb510 atoms of h

s 0I =I sin( )

ac:

Change of the system from asymmetric

equilibrium to symmetric non-equilibrium

to produce

dc: current at 0

Adiabatic increase of the current

to avoid dipole oscillations

ac

dc

MQST

adiabatic

current

t

h

equilz , I notation:

SQUIDs (Superconducting QUantum Interference Device)

SQUID atomic SQUID C.A. SACKETT,

“An atomic SQUID”

Nature, 505, 166 (2014)

DC SQUID (two JJ)

- measurement of very small magnetic

fields up to 185 10 Tesla

0

2a b

0

hc

e

-magnetic flux quantum

(very small value)

Thus high sensitivity of SQUID

,2 2

a s b s

I II I I I

SAP-Josephson perspectives (1) : atomic SQUIDs C. Ryu, P.W. Blackburn, A. A. Blinova, and M. G. Boshier (Los Alamos),

“Experimental Realization of Josephson Junctions for an Atom SQUID”

PRL, 111, 205301 (2013)

- toroidal trap with two JJ, strong coupling

painted potential

- two barriers move toward each other

- slow (dc) and rapid (ac) barrier shifts

- measurements of BEC density to see

if the system is able to adapt barrier shifts

- dc: no density change

- ac: regions of low and high density in two

toroidal sector

-rotation sensing

(instead of measurement of magnetic fields

in superconductor SQUIDs) **Superconductor SQUIDS are widely used for precise measurements of weak magnetic fields

SAP-Josephson perspectives (2): soBEC

Reviews:

J. Dalibard, F. Gerbier, G. Juzeli˜unas, and P. ˙.Ohberg, Rev. Mod. Phys. 83, 1523 (2011).

V. Galitski and I. B. Spielman, Nature 494, 49 (2013).

P.-G. Wang, J. Zhang, Front. Phys., 9, 598 (2014)

Experiment:

Y.-J. Lin, KJ. Jimenez-Garcia and I.B. Spielman,

Nature 471, 83 (2011)

- engineering synthetic magnetic fields and spin-orbit couplings (SOC)

- artificial gauge potentials

- crossover with spintronics, topological insulators, Majorana fermions, …

- artificial SOC: using Raman-dressed pseudo-spin states of BEC atoms to

transfer light momentum to the atom motion

To our knowledge, there are no yet studies on dc/ac Josephson in

SOC BEC. This is in our nearest plans.

Thank you for the attention