Collectively induced many-vortices topology via rotatory Dicke quantum phase transition

3.8 MB·

by  Priyam Das

Collectively induced many-vortices topology
via rotatory Dicke quantum phase transition
Priyam Das,1 Mehmet Emre Tasgin,1,∗ and O¨zgu¨r E. Mu¨stecaplıo˘glu2
1Institute of Nuclear Science, Hacettepe University, Ankara - 06800, Turkey
2Department of Physics, Ko¸c University, Sarıyer, I˙stanbul, 34450, Turkey
We examine the superradiance of a Bose-Einstein condensate pumped with a Laguerre-Gaussian
laser of high winding number, e.g., (cid:96)=7. The laser beam transfers its orbital angular momentum
(OAM) to the condensate at once due to the collectivity of the superradiance. An (cid:96)-fold rotational
6 symmetric structure emerges with the take place of rotatory superradiance. (cid:96) number of single-
1 charge vortices appear at the arms of this structure. Even though the pump and the condensate
0 profilesinitiallyhavecylindricalsymmetry,weobservethatitisbrokento(cid:96)-foldrotationalsymmetry
2 duringthesuperradiance. Breakingofthecylindricalsymmetryintothe(cid:96)-foldsymmetryandOAM
transfer to the BEC become possible after the same critical pump strength. Reorganization of the
r
p condensateresemblestheorderingintheexperimentbyEsslingerandcolleagues[Nature,264,1301
A (2010)]. We show that the critical point for the onset of the reorganization follows the form of the
Dicke quantum phase transition.

8
2 PACSnumbers: 42.50.Ct42.50.Nn05.30.Rt

]
s
I. INTRODUCTION to it [12]. Experiments on the transfer of OAM to BECs
a
g via optical pumping [13–16] make us understand that
- Superradiance(SR)isthecollectiveemissionofanen- there exist a critical threshold for breaking the collective
t
n semble of atoms into an initially unoccupied mode [1, 2]. behavior of N-particle entanglement of a BEC [17]. A
a Atoms interact with the common electromagnetic field BEC can be made partially rotate, if the recoil energy
qu within the extent of the pump’s coherence length, and ((cid:126)ωR),asingleatomgains,exceedsthemeanatom-atom
. are excited to many-body states. Vacuum fluctuations interactionenergy(Uint/N)whichisthesituationforthe
at inanemptyphotonmodestimulatethedecay(emission) optical frequencies. Therefore, regular optical pumping
m of atoms to this mode [3]. Hence, SR is often called cannot transfer BEC to a rotatory state collectively.
as the spontaneous collective emission of an ensemble. Weusethecollectivity[10]ofrotatorySR[17]toopti-
-
d This makes SR strongly directional [4]. Probability of callyinducemanyvorticesinaBEC,atasingletime. We
n the ensemble to spontaneously decay to an empty mode pump the BEC with a Lauguerre-Gaussian (LG) beam
o isproportionaltotheexponentialoftheensemblelength of higher modes (upto (cid:96) = 7), see Fig. 1. We observe
c
[ along this direction [4]. that the (cid:96)-fold rotational symmetric density structures
SR is a quantum phase transition (PT) [5–7], which emerge, Fig. 5, even though the pump and BEC ini-

2 can be treated with the Dicke model [1]. Above a crit- tially have cylindrical symmetric intensities. Interest-
v
ical pump (or effective atom-photon coupling) strength, ingly, single-charge vortices emerge at the arms of the
3
phase transition results in the macroscopic occupation (cid:96)-foldsymmetricstructures,seeFig.5. Wedeterminethe
1
4 of the electronic excited states. The collectivity, which spatial form of the superradiant scattered pulse, F(x,y),
6 SR induces, leads to bipartite entanglement of particles evolving with the equations of motion (EOM). Form of
0 [8], photon-atom entanglement [8], single-mode nonclas- thescatteredSRfieldisfreeofconstraints,e.g.,nomode
1. sicality [9] but more importantly, to N-particle (collec- expansion is used.
0 tive) entanglement [10] of particles via spin-squeezing Reorganizationofthecondensateinto(cid:96)-foldrotational
6 [11]. Spin-squeezing is the collective entanglement of all symmetric structures, resembles the take place of spa-
1 (N) of the ensemble atoms, where the measurement of tial ordering in normal SR, explored by Esslinger and
v: a single atom influences all the remaining ones instan- colleagues [7]. In Refs. [6, 7, 18, 19], the appearance
i taneously. In a Bose-Einstein condensate (BEC), such a of spatial ordering arises due to the collective transfer
X
collectiveentanglementisshowntobeinducedpurelyby of linear momentum via normal SR. Here, (cid:96)-fold rota-
r the atom-atom collisions [11]. For the superradiant scat- tional ordering (and quantized vorticity) takes place due
a
tering from a BEC, this preexisting collectivity enables to the excess OAM transfer to the BEC via rotatory SR.
an earlier induction of the phase transition since BEC is Breakingofthecylindricalsymmetryintothe(cid:96)-foldsym-
already in the symmetric superradiant Dicke states [1]. metry and OAM transfer to the BEC become possible
When a BEC is stirred, it does not rotate until a total after the same critical pump strength. Quantized ro-
of N(cid:126) orbital angular momentum (OAM) is transferred tations (single-charge vortices) also emerge in an (cid:96)-fold
rotational ordered form. We observe that rotatory SR
emergeswithoutaprecedingnormalSR,unlikeRef.[17],
when the damping is small.
∗ Correspondingauthor: [email protected] Onset(dynamics)ofasuperradiantspontaneousdecay
2
cake BEC would consist of radial edge-fire modes, in the
xy-plane shown in Fig. 1. End-fire modes are emitted
BEC through the edge, along the s=xxˆ+yyˆdirections since
theBECistightlyconfinedalongthez-axis. Whenthex-
y profile of the scattered radiation has a smaller winding
number compared to the pump winding, OAM is trans-
ferred to the BEC. If the scattering is collective, that is
superradiant,OAMistransferredtotheBECcollectively
at a single time, i.e. via eNt/τ emission behavior [4].
ℓ-mode The second quantized effective Hamiltonian, Hˆ, of the
LG laser system can be written as the sum
Hˆ =Hˆ +Hˆ +Hˆ +Hˆ +Hˆ(2) (2)
FIG. 1. A pancake shaped BEC is illuminated with a strong 0 col f af af
LG laser of winding number (cid:96). Profile of the laser, Eq. (1)
oftheenergyoftheexternal(motional)statesoftheBEC
carries(cid:96)(cid:126)amountofOAMperphoton. Aboveacriticalpump
strength η > η BEC superradiates in the x-y plane due to (cid:90)
c
the confinement along the z-axis [23]. Hˆ0 = d3rΨˆ†(r)HˆgΨˆ(r), (3)
interatomic collision Hamiltonian
(cid:90)
can be demonstrated only within the second-quantized Hˆ =g d3rΨˆ†(r)Ψˆ†(r)Ψˆ(r)Ψˆ(r), (4)
scheme [4, 20], since it is triggered by vacuum fluctua- col s
tions. NeverthelessPTmanifestsitselfalsoinsemiclassi-
the total electromagnetic energy of the scattered field
caltreatmentsasinstabilities[18]onwhichexperimental
results [7] can be mapped consistently. (cid:90)
Weshowthat(i)angularmomentumtransferredtothe Hˆf =(cid:126)ωe d3rΦˆ†(r)Φˆ(r), (5)
BEC, (cid:104)Lˆ (cid:105), becomes nonzero only after a critical pump
z
strength η > η , see Fig. 6, which is accompanied by couplingoftheatomswiththepump(off-diagonalterm)
c
the breaking of the cylindrical symmetry into the (cid:96)-fold (cid:90)
rotational symmetry, see Fig. 7. (ii) Temporal width of Hˆ =(cid:126)g d3rΨˆ†(r)Φˆ†(r)Φ (r)Ψˆ(r)+H.c., (6)
af a L
the scattered pulse peak is inversely proportional to the
number of atoms [21, 22], see Fig. 3, which is character- and the interaction of the scattered field with the atoms
istic to SR. For η sufficiently larger than ηc, scattered (diagonal term)
field intensity displays in-phase scattering, I ∼ N2, be-
(cid:90)
havior. (iii) Transition is in the Dicke form such that Hˆ(2) =2(cid:126)g d3rΨˆ†(r)Φˆ†(r)Φˆ(r)Ψˆ(r), (7)
af a
critical pump strength (η ) depends on the frequency of
c
the end-fire mode (∆) and diagonal atom-field coupling where we make parametric pump approximation for the
(U0) in the form suggested by Nagy et al. [7, 18], see LG laser. Here, Hˆg = −(cid:126)2∇2/2m+V(r) is the first-
Fig. 8. quantized Hamiltonian for the BEC with V(r) is the
trap potential. Φˆ(r) is the field operator for the su-
perradiantly scattered pulse, whose spatial behavior is
II. HAMILTONIAN determined by the EOM. Ψˆ(r) is the field operator for
the BEC. g is the effective strength for the atom-field
a
We examine the dynamics of a pancake BEC [23], see interaction [24, 25].
Fig. 1, which is illuminated with a strong LG laser of Due to the symmetry of the system and the form of
optical frequency ωL along the symmetry axis z. LG the pump (1), we make separation of variables in the
pump has a spatial profile BEC and the field operators, Ψˆ(r) = fˆ(x,y)Zˆ(z) and
ΦL(r)=EL (r/wL)(cid:96)e−r2/2wL2ei(cid:96)φeikLz (1) Φˆ(r) = Fˆ(x,y)Zf(z). This reduces the computational
efforts. We obtain the time evolution for each operator
andcarriesan(cid:96)(cid:126)amountofOAMperphoton. Detuning using the Heisenberg EOM, e.g., i(cid:126)Fˆ˙(x,y)=[Fˆ,H] (see
enablestheadiabaticeliminationoftheelectronicexcited
the Appendix). Since we are interested in the field am-
state [24, 25]. w is the radial width and k is the wave-
L L plitudes, we replace all of the operators by c-numbers,
vector of the LG pump. Electric field amplitude of the
e.g. fˆ(x,y)→f(x,y).
laser is E = α ε where |α |2 is the number of pump
L L L L We scale the BEC wave-function Ψ(r) and the scat-
photons and ε = ((cid:126)ω /(cid:15) V)1/2 with V the volume of √
L L 0
tered field Φ(r) with N [18]. Parameters η =
the laser cavity. √
Similar to the end-fire modes emitted out of a pen- ga N|αL| and U0 =gaN control the strength of the off-
cil shaped condensate [4], scattered radiation off a pan- diagonal(Hˆ )anddiagonal(Hˆ(2))atom-fieldcouplings,
af af
3
respectively. Off-diagonal coupling favors the macro-
scopic occupation the scattered field [5] while energy of
the diagonal coupling is minimized with a microscopic
scattered field. Hence, the two works against each other.
Ifg isnegative(positive),collisiontermsupports(works
s
against) the SR transition [6, 18].
Tightly confined BEC does not superradiate along the
z-direction. SR field profile along the z-direction non-
vanish only in the condensate, which has an extent of
(cid:39)10nm that is much smaller than the wavelength of the
radiation. Therefore, we neglect Z (z) profile in Φ(r).
f
Additionally, again due to the tight confinement of the
FIG. 2. (a) OAM transfer to the BEC follows the (b) scat-
BECalongthez-direction,vorticespointingalongthex-
tering peak (rotatory SR) at t (cid:39) 1.45/ω . Reorganization
R
y directions are energetically unfavorable. The scattered of the BEC profile, from azimuthal symmetry to (cid:96)-fold ro-
light (end-fire mode) propagates almost in the x-y plane tational symmetry, with (cid:96) quantized vortices, emerges close
due to the strong directionality of the SR [4, 26], thus before the intensity peak.
has no propagation component along the z-direction.
We note that, in obtaining the EOM we do not make
any mode expansion in the BEC, Ψ(r), or the scattered
SR field, Fˆ(x,y), profiles. We aim to observe the free
(without constraint) evolution of the scattered field and
the BEC through the EOM. This is because, during the
time evolution, BEC and the scattered field may attain
unexpected profiles, which can maximize/minimize the
interaction terms.
We examine the behavior of the system both in the
presence and the absence of damping. Following the
mean field theory of free space superradiance from an
ensembleofatoms,weintroduceaphenomenologicallin- FIG. 3. The temporal widths of the pulse peaks are pro-
ear loss term into the field dynamical equation in our portional to 1/N, as suggested by Mandel & Wolf [21]. We
Maxwell-Bloch type equations [19, 27, 28], in Eq. (A1). observe that FWHM is ∆τ=0.0048 for N =N0; ∆τ= 0.0023
Presence of damping causes the 3-fold ordering and the for N = 2N0 and ∆τ=0.0012 for N = 4N0. This is a be-
OAM transfer to take place at higher pump intensities, havior characteristics to superradiant scattering√. We remind
that N appears in the parameters as η = g N|α | and
as should be expected. a L
U =g N. (Weshiftedthethreecurvestothesamepeakpo-
0 a
sitionandnormalizedthemaxintensityto1andnodamping
is assumed.)
III. VORTICES WITH (cid:96)-FOLD ROTATIONAL
SYMMETRY
We time evolve F(x,y) and Ψ(r) = f(x,y)Z(z) using metryinitially. Thewindingnumberofthelaseristrans-
theEOMsgivenintheAppendix. InFig.2,macroscopic formed to the 3-fold rotational symmetry. Three single-
OAM transfer to the BEC follows the emission peak (at chargevorticesappearatthearmsofthe3-foldrotational
t (cid:39) 1.45/ω ). In Fig. 2(a), the (cid:104)Lˆ (cid:105) can have values symmetric density profile. By enclosing small spatial re-
R z
different than pump LG photon’s OAM due to collec- gions around the three vortex positions, we calculate the
tive nature ofthesuperradiant scattering. The temporal expectation value of the Lˆz operator and find that they
width of the peak (∆τ), in Fig. 2(b), follows the super- carry a single charge. In Fig. 4(d) we observe the expan-
radiant form suggested by the Refs. [21, 22]. When we sion ofthe three vortices when the laserfield is lifted off.
increasethenumberofatomsbyntimes,weobservethat Remaining OAM (mtot−(cid:96))(cid:126), per atom, is distributed to
temporal width of the peak shrinks to ∆τ/n, see Fig. 3. the body of the BEC, where m = (cid:104)Ψ(r)|Lˆ |Ψ(r)(cid:105) is
tot z
Additionally, the peak intensity follows the form ∝ N2 the total OAM of the BEC.
form. Hence, the scattering displays the superradiant In our simulations, (cid:96)-fold rotational symmetric struc-
character. turesemergeforthepumpingwithaLGlaserofwinding
In Fig. 4, we depict the time evolution of the spa- number (cid:96), see Fig. 5. Single-charge vortices emerge at
tial profile for both the BEC and the scattered field. In theendsof(cid:96)-foldrotationalsymmetricstructures. When
the close proximity of the scattering peak, Fig. 2(b), az- the OAM transferred to BEC is smaller than the (cid:96)-fold
imuthal symmetric BEC profile reorganizes to 3-fold ro- symmetry, m <(cid:96), (cid:96)-fold symmetric structures emerge
tot
tational symmetry, Fig. 4(b). This happens even though again. However, in this case no single-charge vortex ap-
both BEC and LG pump intensity have cylindrical sym- pear at the arms. The whole OAM is distributed to the
4
(a) (b)
(a) (b)
(c) (d)
(c) (d)
(d)
(e) (f)
FIG. 4. Time evolution of the profiles (corresponding to Fig.
2) of the BEC density |ψ(x,y)| and the scattered field inten-
sity|F(x,y)|forpumpingwithan(cid:96)=3modeLGlaserabove
the critical pump strength η > η . 3-fold rotational sym-
c
metric structures appear (b) both in the BEC and scattered FIG. 5. (cid:96)-fold rotational symmetric structures emerge for
field, even though both have cylindrical symmetry initially. pumping with LG laser, profile (1), of winding number (cid:96). (cid:96)
(c)Single-chargevorticesappearattheendsofthestructure number of single-charge vortices appear at the ends of (cid:96)-fold
suchthat(d)expansionofthevorticescanbeobservedwhen structures above a critical pump strength of the (cid:96)-mode LG
the laser is turned off at t(cid:39)1.51. Quantized vortices appear laser. (η values are 50, 60, 70, 120, 120, 150 and the times
suddenly when mtot > 3. [In (c) and (d), scattered field are profilesbelongtoaret=2.2,1.4,1.56,0.39,0.23,0.15,respec-
verysharp,sothatgraphslookasiftheyarefadedduetothe tively, and even a higher value of damping is chosen, κ=3.)
scaling of the colormap.]
bodyoftheBEC.Noquantizedvorticitycanbeidentified
in the body of the BEC.
Wenotethat,indifferencetovortexcreationviastimu-
latedemission[29,30],amountsofOAMwhicharelarger
than the one for the pump can be transferred to the
BEC via SR. There are two reasons. First, since SR
is a collective process, the atoms can absorb and emit
multi-photons collectively. Hence, more than one pho-
ton per atom can contribute to the collective emission.
Second, in our case, the spatial profile of the BEC side-
mode in which atoms can recoil is not constrained, since FIG.6. OAMtransfertotheBECcanbeachievedonlyafter
a stimulating laser do not impose the spatial profile of acriticalpumpstrengthη>η ,whichdependsontheactual
c
the mode [29, 30]. values of end-fire mode frequency (∆), diagonal atom-field
coupling (U ), damping (κ) and atom-atom collisions (g ).
0 s
IV. CRITICAL PUMP STRENGTH
In Fig. 6, we observe that OAM transfer to the BEC In Fig. 6, we observe a gradual increase in the OAM
canbeachievedonlyafteracriticalpumpstrengthη >η which is managed to be transferred to the BEC. Note
c
depending on the actual values of the diagonal coupling that: whenwesayOAMtransfertotheBEC,forthecase
(U0), collision (gs) and damping of the scattered field of non-integer (cid:104)Lˆz(cid:105) values, we mean that BEC can have
(κ) [18]. (cid:96)-fold rotational ordering in the density can this transient OAM values in real time dynamics only
appear only after η > η , where OAM transfer to the for a short time unlike quantized vorticity. We underline
c
BEC becomes possible [31]. The 3-fold ordering in the that, for η < ηc, BEC do not support OAM even for a
BEC profile becomes visible for η >η , see Fig. 7. short time.
c
In the absence of damping, κ = 0, we clearly observe Despite the gradual increase in (cid:104)Lˆ (cid:105), the visibility of
z
the staircase of plateaus at (cid:104)Lˆ (cid:105) = 1 and (cid:104)Lˆ (cid:105) = 2 hori- the (cid:96)-fold rotational ordering appears quite sharply just
z z
zontal lines. This resistance against the the OAM trans- after η > η (cid:39) 2.25. In Fig. 7, we observe that between
c
fer is due to the rotationless nature —unless multiplies η=1.50-2.25, visibility of the 3-fold ordering remains al-
of N(cid:126) OAM is gained [12]— of the BEC. When we in- most constant, while for η > η (cid:39) 2.25 3-fold ordering
c
troduce damping, these plateaus become unobservable. appears and remains almost constant between η = 2.75-
5
3.50. Therefore, onset of the OAM transfer to the BEC (cid:96) number of single-charge vortices appear at the arms of
is accompanied by a 3-fold rotational symmetric order- the (cid:96)-fold symmetric structures.
ing [31]. In the time evolution, these structures appear after a
Eventhoughthecomplexityofthefunctionsdetermin- sharp scattering peak, Fig. 2(b). The temporal width of
ing the superradiant emission and the recoiled BEC pre- this peak follows the superradiant characteristics intro-
vents us from establishing a simple mapping, presented duced in Refs. [21, 22], see Fig. 3.
in Ref.s [6, 7, 18], in Fig. 8 we show that η follows a
c Inthispaper,wedonottraptheSRscatteredpulsein
form similar to the one predicted by Nagy et al. [18]. In
a cavity in the x-y directions. However, if such a cavity
Fig. 8, we plot log∆-logη and log∆-logη . We observe
c c wouldbepresentlinearordering[7]androtatoryordering
the ηc ∼ ∆1/2 for U0 = gs = 0 and ηc ∼ U01/2 behaviors wouldbeexpectedtoemergetogether—possiblyvortices
predicted by Ref. [18]. According to Ref. [18], η ∼U1/2 distributed in a different pattern among the linearly or-
c 0
behavior should be observed for the large values of U dered structure.
0
since ∆=1 can be neglected in this regime.
V. SUMMARY ACKNOWLEDGEMENT
Insummary,weobservethatBECreorganizesto(cid:96)-fold TheauthorsthankL.YouandL.Dengformanyuseful
rotational symmetry above a critical pump strength η . discussions and acknowledge the financial support from
c
For η < η , OAM transfer to the BEC does not appear. TUBITAK Projects Grant No. 112T927 and 114F170.
c
When η > η , OAM transfer [31] and (cid:96)-fold rotational O¨.E.M. acknowledge support from TUBITAK Project
c
symmetry show up mutually. For excess OAM transfer, Grant No. 112T974.
Appendix A: Equations of Motion
Equations of motions transforms to the following in the scaled form.
dF(x,y) (cid:90) (cid:90)
i =(−iκ+∆)F(x,y)+η|f(x,y)|2F (x,y) |Z(z)|2dz+2U F(x,y)|f(x,y)|2 |Z(z)|2dz (A1)
dt L 0
df(x,y) (cid:90)
i =η(F (x,y)F∗(x,y)+F∗(x,y)F(x,y))f(x,y) |Z(z)|2dz
dt L L
(cid:90) (cid:90)
+2U f(x,y)|F(x,y)|2 |Z(z)|2dz+g¯ |f(x,y)|2f(x,y) |Z(z)|4dz
0 s
−1(cid:2)(cid:0)∂2+∂2(cid:1)f(x,y)(cid:3)(cid:90) |Z(z)|2dz− 1f(x,y)(cid:90) Z∗(z)∂2Z(z)dz
2 x y 2 z
1 (cid:90) 1(cid:18)ω (cid:19)2 (cid:90)
+ (x2+y2)f(x,y) |Z(z)|2dz+ z f(x,y) z2|Z(z)|2dz, (A2)
2 2 ω
r
idZ(z) =ηZ(z)(cid:90) (cid:0)|f(x,y)|2F (x,y)F∗(x,y)+|f(x,y)|2F∗(x,y)F(x,y)(cid:1)d2r
dt L L
(cid:90) (cid:90)
+2U Z(z) |f(x,y)|2|F(x,y)|2d2r+g¯ |Z(z)|2Z(z) |f(x,y)|4d2r
0 s
−1Z(z)(cid:90) f∗(x,y)(cid:0)∂2+∂2(cid:1)f(x,y)d2r− 1∂2Z(z)(cid:90) |f(x,y)|2d2r+ 1Z(z)(cid:90) (x2+y2)|f(x,y)|2d2r
2 x y 2 z 2
1(cid:18)ω (cid:19)2 (cid:90)
+ z z2Z(z) |f(x,y)|2d2r. (A3)
2 ω
r
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FIG. 7. Visibility of the 3-fold rotational ordering in the BEC profile (|F(x,y)|) for different η values. Profiles correspond to
solidlineinFig.6. Foreachη value,weperformthetimeevolutionandcapturetheBECprofilewhere3-foldorderingismost
observable. From η=η (cid:39)2.25 to η=2.5, the visibility ofthe 3-fold symmetry jumps. Only after η>η =2.25, OAM transfer
c c
to BEC can be achieved. For η<η , BEC cannot handle OAM even for a small time. We note that, 3-fold ordering does not
c
change much in the three graphics η=1.5, 1.75 and 2.0; but changes apparently at η=2.25 and 2.5. Visibility also does not
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[31] Itisworthnotingthatsuddenjumpsinorderparameters
appearwhentheorderparameterisplottedwithrespect