Cusps in the quench dynamics of a Bloch state

0.97 MB·

by  J. M. Zhang

epl draft
Cusps in the quench dynamics of a Bloch state
J. M. Zhang1,2 and Hua-Tong Yang3
1 Fujian Provincial Key Laboratory of Quantum Manipulation and New Energy Materials, College of Physics and
Energy, Fujian Normal University, Fuzhou 350007, China
2 FujianProvincialCollaborativeInnovationCenterforOptoelectronicSemiconductorsandEfficientDevices,Xiamen,
361005, China
63 School of Physics, Northeast Normal University, Changchun 130024, China
1
0
2
n PACS 03.65.-W–QuantumMechanics
u PACS 02.30.Rz–Integral Equations
J
Abstract –We report some nonsmooth dynamics of a Bloch state in a one-dimensional tight
3
binding model with the periodic boundary condition. After a sudden change of the potential
2
of an arbitrary site, quantities like the survival probability of the particle in the initial Bloch

state show cusps periodically, with the period being the Heisenberg time associated with the
]
h energyspectrum. Thisphenomenonisanonperturbative counterpartofthenonsmooth dynamics
p observed previously (Zhang and Haque, arXiv:1404.4280) in a periodically driven tight binding
- model. Underlyingthecuspsisan exactly solvablemodel, which consists of equally spaced levels
t
n extendingfrom −∞ to+∞, between which two arbitrary levels are coupled to each other bythe
a same strength.
u
q
[

4
v
9 Introduction. – In quantum mechanics, there exist fact that the curves of some physical quantities like the
6two parallel themes [1]. One is about the static proper- probability of finding the particle in the initial state, are
5tiesofasystem,namelytheeigenstatesandeigenvaluesof structured. Specifically, they show cusps periodically in
3
the Hamiltonian. The other is about the dynamics of the time.
0
.system, namely how the wavefunction or the expectation The cusps here are somehow similar to the cusps ob-
1
values of variousphysical quantities evolve. While for the
0 served previously in Ref. [5], which are also in the tight
former, there exist many theorems which give us a good
6 binding model setting (the cusps there were observedear-
1pictureofthewavefunctionsinmanycases;forthelatter, lier in quantum optics settings [6,7] but were not fully
:the relevant mathematics is far less developed, and hence
v accountedfor). The only difference in the scenariois that
we often have little intuition. Actually, the dynamics of a
i there the defect potential is modulated sinusoidally in-
Xsystemcanbe verysurprising [2]. This is the case evenin
stead of being held fixed. However, the crucial difference
rthesingle-particlecase,asdemonstratedbythecelebrated
a is that, there the cusps (called kinks) are a perturbative
phenomena of dynamical localization[3] and coherentde-
effect and survive only in the weak driving limit, while
struction of tunneling [4].
heretheyareagenericnonperturbative effectandthusare
Here we report some unexpected dynamics in the set- very robust.
ting of the one-dimensional tight binding model, which is
Inthefollowing,weshallfirstdescribethe phenomenon
arguably the simplest model in solid state physics. It is
by presenting the numerical observations. Then we will
about a very simple scenario. Take a tight binding model
identify the essential features of the underlying Hamilto-
with periodic boundary condition and put a particle in
nian, from which we define an idealized model. The phe-
some eigenstate, i.e., a Bloch state with some momen-
nomenonisthenaccountedforbysolvingthe dynamicsof
tum. Then suddenly quench it by changing the potential
the ideal model analytically. Finally, we discuss its physi-
of an arbitrary site. The rough picture is that the parti-
cal implications.
cle will be reflected by the newly introduced barrier, and
the particle will perform Rabi oscillation between the ini-
tial Bloch state and its time-reversed counterpart. How- Periodically appearing cusps. – The Hamiltonian
ever, exact numerical simulation reveals the unexpected ofanN-sitetightbindingmodelwiththeperiodicbound-
p-1
J. M. Zhang1,2 Hua-Tong Yang3
1 1 1
(a) (b) (c)
0.8 0.8 0.8
Pr0.6 0.6 0.6
&
Pi0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
0 500 1000 1500 2000 0 500 1000 1500 0 500 1000
t t t
Fig. 1: (Color online) Time evolution of the probability of finding the particle in the initial Bloch state |kii (Pi, solid lines)
and in the momentum-reversed Bloch state |−kii (Pr, dotted lines). Note that Pi+Pr 6= 1 in general as other Bloch states
are occupied too, but Pi+Pr = 1 to a good accuracy when the cusps show up. The values of the parameters (N,ki,U) are
(401,80,1.5), (301,75,2), and (201,50,12) in (a), (b), and (c), respectively.
0.55 The most prominent feature of the curves is the cusps.
In each panel of Fig. 1, the cusps are equally spaced in
0.5 time. They appearsimultaneouslyinthe curvesofP and
i
Pr Pr. Sometimes, the cusp in one of the two curves is not
&0.45 so clearly visible, but the corresponding one in the other
Pi curveiswellshaped. Ofallofthethreepanels,panel(b)is
0.4 especiallyregular. Notonlythecuspsappearperiodically,
bothcurvesaresimplyperiodic. Moreover,whenthecusps
0.35 happen, P =0.5 or 1.
430 440 450 460 470 i,r
t One might wonder whether the cusps are cusps in the
mathematical sense as the system is a finite one. Indeed,
Fig. 2: (Color online) Details of thecusps enclosed by the red theyarenot. InFig.2,thecuspsenclosedbytheredboxin
boxinFig.1(b). Thecuspsaresmoothedonasmalltimescale.
Fig. 1(b) are magnified—They are quite smooth. Hence,
the cusps appear to be cusps only on a relatively large
time scale. However, as we shall see below, behind the
ary condition is (~=1 throughout this paper) roundedcuspshereisanexactlysolvablemodelconsisting
of infinitely many levels, where the cusps are cusps in the
N−1
Hˆ = (l l+1 + l+1 l ). (1) rigorous sense.
0
− | ih | | ih | It is worthy to emphasize the essential difference be-
l=0
X tween the cusps here and those observed previously in
Here l is the Wannier state on site l. The eigenstates Ref. [5]. There, it is a first order perturbative effect. The
| i
are the well-known Bloch states k defined as l k = cusps exist only in the weak driving limit, or specifically,
| i h | i
exp(iql)/√N. Here k is an integer defined up to an in- only when the survival probability P is close to unity,
i
tegral multiple of N and q = 2πk/N is the associated and between the cusps the survival probability is a linear
wave vector. function of time. In contrast, here apparently the cusps
Nowconsidersuchascenario. Initiallythe particleisin arestillverysharpevenwhenP constantlydropstozero.
i
someBlochstate ki . Thenattimet=0,thepotentialon Moreover, the functional form of the curves between the
| i
somesite j issuddenly changedtoU. Thatis,weaddthe cuspsis neither linearnorexponential,but asweshallsee
termHˆ1 =U j j totheHamiltonian(1). Becauseofthe below, quadratic.
| ih |
periodicboundarycondition,wecanassumej =0without
Explanation by a truncated and linearized
lossofgenerality. Inthe ensuingnontrivialdynamics,two
model. – To accountforthe cuspsin Fig.1, weneed to
quantitiesofparticularinterestarethesurvivalprobability
have a close survey of the structures of the un-perturbed
and the reflection probability
HamiltonianHˆ andtheperturbationHˆ . Figure3shows
0 1
Pi(t)=|h+ki|Ψ(t)i|2, Pr(t)=|h−ki|Ψ(t)i|2, (2) tthuerbdaitsiponersHˆioncroeulaptleiosnt,wεo(qa)rb=itr−a2rycoBslqo,chofsHtˆa0t.esTwhiethpearn-
whichare, respectively,the probability of finding the par- 1
equal amplitude
ticleintheinitialBlochstateandthemomentum-reversed
Bloch state. Both quantities can be easily calculated nu- g = k Hˆ k =U/N, (3)
1 1 2
mericallyasinFig.1. Thereweshowthenumericalresults h | | i
of P and P as functions of time. regardless of the values of k and k .
i r 1 2
p-2
Cusps in the quench dynamics of a Bloch state
ε Im(ψ )
0
(0,1)
−π −qi qi π q
(1,0)
θ
Re(ψ )
0
Fig.3: (Coloronline)Dispersionrelationε(q)=−2cosqofthe
tight binding model (1). The parameter qi =2πki/N denotes
thewavevectorof theinitial Bloch state. The dotted straight Fig.4: (Coloronline)Ageneric(redsolidlines)trajectoryofψ0
lines are local linear approximations to the dispersion curve. on the complex plane according to Eq. (15a). It is analogous
OnlytheBlochstatesinsidethecirclesparticipatesignificantly to the bouncing of a classical ball inside a circular billiard.
inthedynamicsandthusareretainedinthetruncatedHamil- The green dashed closed trajectory corresponds to the case of
tonian. θ=π/2.
Acrucialfactrevealedbynumerics isthatinthe evolu- does the probability P of finding the systemin the initial
i
tionofthe wavefunction,essentiallyonlythosefew Bloch level R evolve in time? We have the decomposition
0
states with wave vectors q ≃ ±qi contribute significantly |Ψ(0)i| =i(|A−0i+|A+0i)/√2. Since |A−0i is an eigenstate
to the wave function. Now since locally the dispersion of Hˆ, we see that at an arbitrary time later, the wave
curve ε(q) can be approximated by a straight line (it is function has the form
especially the case at q = π/2 where ε′′(q) = 0), we are
± M
led to truncate and linearize the model. 1 1
Ψ(t) = A− + ψ (t)A+ . (6)
Of all the Bloch states, we retain only two groups cen- | i √2| 0i √2 n | ni
n=−M
teredat k . Eachgroupconsistsof2M+1(the condi- X
i
|± i
tionsonM will be discussedlater)stateswith wavenum- Bytheinitialcondition,ψ (0)=δ . TheprobabilitiesP
n n,0 i
bers symmetrically distributed around ki or −ki. Let us and Pr, which are our primary concern, can be expressed
now refer to them as Rn and Ln , where R and as
{| i} {| i}
L mean right-going and left-going, respectively, and n
1 1
ranges from M to M. By choice, R k +n and P = 1+ψ 2, P = 1 ψ 2. (7)
− | ni ≡ | i i i 4| 0| r 4| − 0|
L k n . After linearizing the dispersion curve
n i
| i ≡ |− − i
at qi, the energy of the degenerate states Rn and Ln Hence, the aim is to calculate ψ0(t).
± | i | i
is n∆, with Projecting the time-dependent Schr¨odinger equation
i∂ Ψ(t) = Hˆ Ψ(t) onto A† , we get i∂ ψ = n∆ψ +
∆=(4πsinq )/N. (4) ∂t| i | i | ni ∂t n n
i 2gS, where S is a collective, auxiliary quantity defined as
Here we have chosen the energy of k as the zero of
energy. Again, the perturbation Hˆ1 c|o±uplieis two arbitrary S(t) M ψm(t). (8)
states in the retained set of states with equal amplitude ≡
m=−M
X
g =U/N.
The point is that it is independent of n. Introducing the
Thistruncatedandlinearizedmodelcanbepartiallydi-
agonalized by introducing a new basis as A± =(R so-calledHeisenbergtime T =2π/∆,weseethatforfixed
L )/√2. Referringto the originalHamilt|onniainHˆ|,nthie±y M,withrespecttothe reducedtimet/T,the dynamicsof
n 0
| i thetruncatedmodelisdeterminedbytheratiog/∆alone.
are even- and odd-parity states with respect to the de-
fected site, respectively. It is easyto see that A− , which The quantity ψn can be solved formally as
areeigenstatesofthe originalHamiltonianHˆ0|wnitiheigen- t
values n∆, are also eigenstates of the total Hamiltonian ψ (t)=e−in∆tδ i2g dτe−in∆(t−τ)S(τ). (9)
n n,0
Hˆ =Hˆ0+Hˆ1 with the same eigenvalues. In the yet to be − Z0
HdˆiagaonndalHiˆzedarseubspace of {|A+ni}, the matrix elements of Plugging this into (8), we get an integral equation of S,
0 1
t M
hA+n1|Hˆ0|A+n2i=n1∆δn1,n2, hA+n1|Hˆ1|A+n2i=2g, (5) S(t)=1−i2gZ0 dτ n=−Me−in∆(t−τ)!S(τ). (10)
for arbitrary n . X
1,2
Now the scenario is like this. Initially the system is Heretoproceedanalyticallyfurther,weusesomefactver-
in the state Ψ(0) = k = R . The problem is, how ified by numerics (see Figs. 5 and 6). Numerically, it is
i 0
| i | i | i
p-3
J. M. Zhang1,2 Hua-Tong Yang3
1 1
(a) (b)
m(S) 0 ψm()0 0
I I
−1 −1
2 1
3 3
0 2 0 2
1 1
Re(S) −2 0 t/T Re(ψ0) −1 0 t/T
Fig.5: (Color online)Timeevolution(bluesolidlines) of(a)theauxiliaryquantityS and(b)ψ0 forM =10andg/∆=0.125.
In each panel, thered line indicates theanalytical predictions of (15a) or (15b). Compare (b) with Fig. 4.
observed that as M , the trajectories of ψn (in par- 1 (a)M=5
→ ∞
tlaicruglearM, ψ, 0it)icsolnevgeitrigmeaqtueictoklyr.eplTahceertehfoerefi,niftoerssuumffimciaetnitolny 2ψ|00.5
|
in the bracket by an infinite one. That is, 0
1
(b)M=10
t +∞
S(t) 1 i2g dτ e−in∆(t−τ) S(τ). (11) 2ψ|00.5
≃ − Z0 n=−∞ ! |
X 0
1
Using the Poisson summation formula [8], the kernel (c)M=20
n+=∞−∞e−in∆(t−τ) can be rewritten as T n+=∞−∞δ(t − 2ψ|00.5
τ nT). Substituting this new form of the kernel into |
P− P
(11), we get S(t)=1 igTS(t), for 0<t<T, by noting 0
∞ − 0 0.5 1 1.5 2 2.5 3
that dtδ(t)=1/2. We then solve for 0<t<T, t/T
0
R S(t)= 1 , (12) Fig. 6: (Color online) Time evolution of |ψ0|2 for finite values
1+igT of M, with g/∆ = 0.5. In each panel, the blue solid line
indicatesthenumericalexactvaluewhilethegreendashedline
which is a constant. Substituting this into (9), we get
theanalyticalformula(15a),whichcorrespondstotheM →∞
limit. In each period, thelatter is a parabola.
i2gt 1 i2g(t T/2)
ψ (t)=1 = − − , (13)
0
− 1+igT 1+igT
which is linear in t. We note that as t T−, (1+ ψ0 2)/2. Hence, generally Pi+Pr < 1 as ψ0 2 < 1,
| | | |
→ butatt=rT,when ψ 2 =1,wehaveP +P =1,which
0 i r
| |
1 igT is satisfied to a good accuracy in Fig. 1.
ψ (t) − =e−iθ (14)
0
→ 1+igT A peculiar feature of (12) and (15b) is that S is not
continuous at t = rT. For example, by the definition
forsomeθ ( π,π). Thatis,afteroneperiod,ψ returns
to its initia∈l v−alue, except for a phase accumulat0ed. This (8), S(t = 0) = ψ0(t = 0) = 1, however by (12),
S(t = 0+) = 1. This should be an artifact of our treat-
complete revival means, ψ (t+T) = ψ (t)e−iθ for all n. 6
n n ment involving the M limit. To see how this dif-
We thus have for t = rT +s, with r being a nonnegative → ∞
ficulty is solved for finite M, we demonstrate the typical
integer and s [0,T),
∈ time evolution behavior of S with M = 10 in Fig. 5(a).
We see that in the interval of rT <t <(r+1)T, S oscil-
1 i2g(s T/2)
ψ0(t) = − − e−irθ, (15a) latesrapidlyaroundtheconstantvaluepredictedby(15b),
1+igT
and at about t= rT, the orbit of S quickly transits from
1
S(t) = e−irθ. (15b) aroundoneconstantvaluetoaroundthenext. Alongwith
1+igT
the time evolution of S in Fig. 5(a), we show in Fig. 5(b)
In Fig. 4, the trajectory of ψ0 on the complex plane is the time evolution of ψ0. We see that the numerical ex-
illustrated. It bounces inside the unit circle elastically act value of ψ0 follows the analytical prediction of (15a)
like a ball. Hence, its kinematics has the regularity of closely, with much smaller oscillation amplitude than S.
the irrational rotation [9]. We see that ψ 2 is a periodic This is reasonable in view of (9), where S appears in the
0
function of time t. At t = rT, it return|s to| unity and in- integral and thus its oscillation is averagedout.
between it is a quadratic function of t. By (7), P +P = Further evidences demonstrating that the simple for-
i r
p-4
Cusps in the quench dynamics of a Bloch state
1 1 1
(a) (b) (c)
0.8 0.8 0.8
Pr0.6 0.6 0.6
&
Pi0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
0 500 1000 1500 2000 0 500 1000 1500 0 500 1000
t t t
Fig. 7: (Color online) Comparison between the numerical exact values of Pi,r and the analytical predictions. The panels
correspondtothoseinFig.1onetooneandinorder. Theanalyticalcurvesaresolid (respectively,dotted)ifthecorresponding
numerical curves are dotted (respectively, solid). The dotted lines are hardly visible, which proves that the numerical and
analytical results agree with each other very well.
mula(15a)isagoodapproximationforfiniteM (Anyway, 0.2 (a)
thereareonlyafinitenumberoflevelsintheoriginaltight
1
binding model) arepresentedin Fig.6. Therewe see that =0.1
n
P
even for M = 5, the formula (15a) captures the behavior
of ψ 2onthescaleofT verywell,andasM increases,the
| 0| 0
curve converges to that predicted by (15a) very quickly. 1
(b)
Having verified that (15a) is reliable even for finite lev-
els, we now apply the theory to the original problem.
R0.5
P
There we have ∆ = 4πsinq /N and g = U/N. Using
i
(7) and (15a), we can calculate P in Fig. 1 analytically.
i,r
The results are presented in Fig. 7 together with those 0
0 5 10 15 20
numerical data in Fig. 1. We see that the analytical ap- t/T
proximation and the numerical exact results agree very
well. We can also understand the regularity of Fig. 1(b) Fig.8: (Coloronline)(a)PopulationontheBlochstate|ki+1i
now. For U = 2 and q = π/2, gT = 1 (regardless of the and (b) total population on those right-moving Bloch states.
i
valueof N)andhence θ =π/2andthe trajectoryofψ is Ineachpanel,thereareactually twocurves,thebluesolidone
0
the closed one in Fig. 4. By (7), it results in the regular for the numerical exact result and the green dotted one for
the analytical approximation [Eqs. (17) and (20)]. But they
behavior of P in Fig. 1(b).
i,r
coincidewell. ThescenarioisasinFig.1(c). In(b),thetipsof
Another regularity in all panels of Fig. 1 and Fig. 7 is
thecuspsarelocatedonthereddashedcurveof(1+cosωt)/2
that,by Fig.4, the cusps ofP are locatedonthe curves
i,r with ω=θ/T.
of (1 cosωt)/2 respectively, with ω = θ/T. This fact is
±
inaccordwiththeroughpicturethatinthelongterm,the
particle performs Rabi oscillation between the two Bloch realisticmodel,asfarasthequantitiesPi,r areconcerned.
states k . But since ω = 2g, we see that, because of
|± ii 6 Populations on other Bloch states. – So far, we
couplingtootherBlochstates,the oscillationfrequencyis
have focused on ψ , the quantity relevant for calculating
0
not simply determined by the direct coupling g between
the populations onthe Blochstates k . But the exact
i
the two. From the point of view of quantum chaos, the |± i
solution above allows us also to calculate all other ψ ,
n
system in question has a very regular dynamics.
whicharerelatedtothepopulationsonotherBlochstates.
Finally, we are prepared to discuss the conditions on
By (9), similar to (15a), we have (t=rT +s)
M. On the one hand, M should be large enough so that
the M limit of the dynamics of ψ has almost been 2g
achieve→d. ∞On the other hand, M should0 be small enough ψn(t)= n∆(1+igT)(e−in∆s−1)e−irθ, (16)
so that the linearizationapproximationis valid. For fixed
forn=0. By(6),theanalyticpredictionofthepopulation
values of U and qi, as the ratio g/∆=U/4πsinqi is inde- 6
on the Bloch states (k +n) is
pendentofN,thelowerboundsetbythefirstconditionis |± i i
N-independent. On the contrary,the upper bound set by 1 4g2sin2(n∆t/2)
the second condition is obviously linearly proportional to Pn(t)= 4|ψn(t)|2 = (1+g2T2)n2∆2. (17)
N. Therefore, for arbitrary U and q , there will be room
i
for M if N is sufficiently large. When it is the case, the This is expected to be a good approximation for small n.
ideal model with M = is a good approximation of the Indeed, as shown in Fig. 8(a), Eq. (17) agrees with the
∞
p-5
J. M. Zhang1,2 Hua-Tong Yang3
numerical exact result for P very well for the scenario in a single-particle model. They are different on the one
1
Fig.1(c). Thecrucialfeatureof(17)isthattheamplitude hand from those reported in Refs. [5–7] in that they are
of P shrinks as 1/n2. This is in line with the numerical deeply nonperturbative, andonthe other handfromthose
n
findingthatonlythoseBlochstatesinthevicinityof k inRefs.[12–14]inthatthefunctionsin-betweenthecusps
i
|± i
are significantly populated. It also explains why the M = are not exponential but quadratic or linear. The different
limitisrelevantfortheoriginalmodel,whichhasonlya behaviors stem from the different hamiltonian structures.
∞
finitenumberoflevelsandagloballynonlinearspectrum— In Refs. [12–14], the model has the level-band structure,
The n-large states are only negligibly populated both in namely, an extra level couples to a band, and there is no
the ideal model and in the realistic model. coupling between the levels inside the band. In contrast,
Another quantity of interest is the total population on here the ideal model consists of only a band, and two ar-
those Bloch states moving to the right, i.e., bitrary levels in the band are coupled.
Although we do not believe the phenomenon reported
[N/2]
here is universal, we do think it provides a good exam-
P (t)= k Ψ(t) 2. (18)
R |h | i| pledemonstratingthatthe dynamicsofamodel,eventhe
kX=1 simplest one, can be very surprising. The point is that,
Here the lower and upper bounds of summation actually thoroughunderstandingofthestaticpropertiesofamodel
do not need to touch the band edges, as the contribution (as we do for the model in question) does not imply thor-
isprimarilyfromthevicinityofk . Bythecorrespondence ough understanding of its dynamical properties. There is
i
k +n R for small n, P should can be approxi- a big gap from the former to the latter in many cases.
i n R
| i ↔ | i
mated by In view of the intensive study of nonequilibrium dynam-
ics of many-body systems nowadays [15], it is worthy to
1
R Ψ(t) 2 = 1+ψ 2+ ψ 2 . (19) emphasize that, actually the dynamics of single-body or
n 0 n
|h | i| 4 | | | | few-bodysystemsisalreadycomplexenoughandfarfrom
n∈Z (cid:18) n6=0 (cid:19)
X X
being fully understood.
Using (15a), (17), and the equality sin2nα/n2α2 =
n∈Z Acknowledgments.– Thisworkissupportedbythe
π/α, for 0 < α < π [5,10], it is straightforward to reduce
P FujianProvincialScienceFoundationundergrantnumber
(19) to
2016J05004.
rθ 1 gs
cos2 sin r+ θ . (20)
2 − 2 1+g2T2 REFERENCES
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[14] W. Zhu and D. L. Zhou,arXiv:1507.02390.
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[15] A.Polkovnikov,K.Sengupta,A.Silva,and M.Vengalat-
The cusps are reminiscent of the quantum dynamical tore, Rev.Mod. Phys. 83, 863 (2011).
phase transition discovered by Heyl et al. [11], but in
p-6