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
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