Realizing Topological Transition in a Non-Hermitian Quantum Walk with Circuit QED

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by  Yizhou Huang

Realizing Topological Transition in a Non-Hermitian Quantum Walk with Circuit
QED
Yizhou Huang,1 Zhang-qi Yin,1, and W. L. Yang2
∗
1Center for Quantum Information, Institute for Interdisciplinary
Information Sciences, Tsinghua University, Beijing 100084, China
2State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
We extend the non-Hermitian one-dimension quantum walk model [PRL 102, 065703 (2009)] by
taking dephasing effect into account. We prove that the feature of topological transition does not
change even when dephasing between the sites within units is present. The potential experimental
6 observationofourtheoreticalresultsinthecircuitQEDsystemconsistingofsuperconductingqubit
1 coupled to a superconducting resonator mode is discussed and numerically simulated. The results
0 clearly showa topological transition in quantumwalk, and display therobustnessof such asystem
2 tothedecayanddephasingofqubits. Wealsodiscusshowtoextendthismodeltohigherdimension
in the circuit QED system.
r
p
A PACSnumbers:

6
I. INTRODUCTION Hamiltonian,as donein [10]. Later,the same modelwas

] extended to multi-dimensional systems [11].
h
p As a quantum analog of the well-known classical ran- AsinFig. 1,thequantumwalkcouldberealizedonan
- dom walk, quantum random walk serves as a fascinating 1Dbipartiteindexedlatticewheredecaysites(blue)and
t
n framework for various quantum information processes, non-decaysites (white) appear in turn. The strengths of
a such as basic search[1], universal quantum computation hopping betweensites are characterizedby v (within the
u [2],quantummeasurement[3]etc. Apartfromitsnumer- same unit) and v′ (between neighboring units), and due
q ous applications, quantum random walk itself also dis- to that, a random “walker” starting from a non-decay
[
plays new traits different from its classical counterpart, site of unit m0 may end up decaying from the system

2 suchasthefastspreadingofthewavefunctioncompared fromanotherunitmwithprobabilityPm. Giventhe rel-
v to a classical random walk [4], which was used for ex- ative strengths of v and v′, the average displacement of
74 plaining the high efficiency photosynthetic energy trans- a “walker” ( mm·Pm−m0) before it decays is quan-
7 fer assisted by environment [5–8]. Ref. [9] considered an tized as an iPnteger (0 for v′ < v and −1 for v′ > v) no
1 one-dimensional(1D) quantum randomwalk ona bipar- matter what the decay co-efficient γ >0 is. As shown in
0 tite lattice, whereatopologicaltransitionwasfoundand Ref. [12], such system displayed Parity-Time Symmetry
. experimental implementation in quantum dots or cav- [13–15]. These differentdisplacements correspondto un-
1
ity QED (CQED) was briefly discussed. In it, the en- broken and broken Parity-Time Symmetry regimes and
0
6 vironment effect was included by using a non-Hermitian consist of a topological transition. In this work we will
1 show that such transition is robust even in the presence
: of qubit dephase. Note that this idea can be experi-
v
mentally realized in a CQED setting, where a qubit is
i
X coupled to one resonator mode [16, 17]. Specifically, the
r v v two states of the qubit (e , g ) represent two sites of
a the same unit, where the|hiigh|eri-energy qubit state (e )
v’ | i
has decay rate γ. The cavity photon number state n
| i
represents the n-th unit. Here we suppose that the cav-
ity decay κ is much less than qubit decay γ and is thus
unit m unit m+1
negligible. Such a system could be an atom in a cavity
coupled to one cavity mode or similarly, as CQED sys-
decay decay
tems wherea superconductingJosephsonjunctionwhich
actsasaqubit,iscoupledtoalumpedLCoscillatorinan
FIG. 1: (color online) Non-Hermitian quantum walk model.
A particle can hop between the nearest sites with strength v superconducting electric circuit. Recently, circuit QED
orv′ dependingonwhetherthathopcrossesunitboundaries. system attracts a lot of attention, as it could be easily
When the particle is on the blue sites, it would decay with scaledupandcontrolled[18]. We willsticktothe circuit
rate γ. QED system setup from now on.
Ontheotherhand,therewereexperimentsrealizingbi-
partite non-Hermitian quantum random walk in optical
waveguide[19]withsatisfyingresults,whichinessenceis
∗Electronicaddress: [email protected] a classical simulation of a quantum effect. However, the
2
non-Hermitian quantum walk in circuit QED is a fully jumpapproachin[21]andputthe qubitdecayterminto
quantumexperiment. AnothermeritofcircuitQEDsys- theHamiltonian,whichyieldsaneffectivenon-Hermitian
tem is thatextensionto higher dimensionquantumwalk Hamiltonian
is relatively easy, as a high-dimensional quantum walk
1 1iγ
can be implemented by coupling the qubit to more than H =g σ+a˜+σ a˜ +Ω/2(σ++σ )+ ∆ǫσz e e .
− † −
one resonatormode. We will show that novel features of 2 −2 2 | ih |
(cid:0) (cid:1) (3)
a quantum walk emerges with the growth of dimension-
Weputsubscript“org”todistinguishtheoriginalHamil-
ality.
tonianH fromthisHamiltonianthatwewillworkwith
In this paper, we first extend the analytical theory in org
from now on.
[9] to accommodate the existence of dephase, and show
Ideally, if the system is initialized at photon number
that the qubit dephase, one major source of decoher-
state N (which means the “walker”starts at exact site
ence in normal circuit QED systems, won’t affect the | i
N),andstaysclosetoN duringthewholeevolutionpro-
topological structure experimentally observed. Numeri-
cess, then we can use the approximation
cal calculations are carried out for different parameters
of the system, and different factors affecting the results
a˜ √N n 1 n ,a˜ √N n n 1 (4)
are considered and discussed, including the preparation ≈ | − ih | † ≈ | ih − |
X X
of the system, the decay rate of the qubit, the detun-
ing of the system and the effect of dephasing. We will to simplify the operators. The conditions on which this
show that such quantum walk realization is feasible in approximation holds will be discussed later. Then, as-
circuitQED setting, andgoodtopologicaltransitioncan suming ∆ǫ 0, we have
≈
be observed with modest requirement on the system pa-
rameters afore-mentioned. v =Ω/2,v′ =g√N. (5)
Thus, when v < v (Ω/2 > √N) the average photon
′ g
II. THE CIRCUIT QED MODEL numberuponmeasurementshouldbeN (nochange),and
when v < v (Ω/2 < √N) the average photon number
′ g
We consider a system where a two-level qubit is cou-
upon measurement should be N 1 (change of 1).
pledtoaresonatorfield,withexternalmicrowavedriving. Due to the existence of deca−y term iγ e−e the
Suchasystemcanbe expressedinstandardcircuitQED −2 | ih |
Hamiltonian is not Hermitian. However, it satisfies the
Hamiltonian
condition of Parity-Time symmetry, that is, after un-
Horg =g σ+a˜+σ−a˜† +Ω/2(σ++σ−)+ 21∆ǫσz, (1) (dpˆergoingpˆ,piarityi()xˆtr→ans−foxˆr,mpˆat→ion−, tpˆh)eannedwtHimaemriletvoenrisaanl
(cid:0) (cid:1) → − → −
H is only a constant away from the old Hamiltonian
where σz,σ arePaulioperatorsforthe qubit, g charac- ′
±
(H = H +cI). And recent work [12] has proven that
terizesthecouplingstrength,anda˜(a˜ )arelowering(ris- ′
ing)operatorsontheresonatormode†inarotatingframe, when we have Ωg/2 > √N the eigenvalues of the Hamil-
∆ǫis(real)detuningofthesystem,consistsoftheenergy tonianare totally real,while Ω/2 <√N there’s a pair of
g
differencesbetweenthe twostatesofthe qubitminusthe complex energy eigenvalues, which correspond to unbro-
minimum energygapfor the resonator,Ω is the strength
ken and broken PT symmetries respectively.
of external drive on the atom.
Furthermore, we consider decay of the excited state
of the qubit, together with qubit dephase. We use γ to
III. ANALYTICAL THEORY
characterize qubit decay strength, and d as the dephase
factor of the qubit. The effects of such two terms on the
Theanalyticaltheoryforthistopologicaltransitionhas
system can be added as Lindblad operators as
beenexplainedindetailbefore[9]. Here,wewillimprove
ρ˙(t)= i[H ,ρ(t)]+d[σzρ(t)σz ρ(t)] theiranalyticaltheorytoincludethequbitdephase. The
org
− − (2)
+γ σ ρ(t)σ++σ+σ ρ(t)+σ σ+ρ(t) , Lindblad master equation of the system is [22]
− − −
In usual ex(cid:2)perimental setup [20], the strength(cid:3)of cav- ρ˙(t)= i[H,ρ(t)]+d[σzρ(t)σz ρ(t)], (6)
− −
ity decay is at the magnitude of 1 kHz or less, which
is significantly smaller than other∼parameters in the sys- where d characterizes dephase rate for qubit.
tem, such as the coupling strength g which could reach If the system is pure, one could use ψnA to denote
∼Th1e0re0fMoreH,zwaendignqourbeitthdeeccaayviγtyisdeincaytheeffoercdtse.r oBfeMsidHezs., tthhee parmespelnitcuedeofadtep|ghias⊗e |tnhie adnednsiψtynBmaattr|iexi o⊗f |tnhie. syIsn-
the external drive Ω is easily tunable to fulfill the topo- tem is mixed. We can similarly use density matrix as
logical transition conditions. ρAn1A,n2(0) = ρBn1A,n2(0) = ρAn1B,n2(0) = 0, ρBn1B,n2(0) =
Since we are only interested in the state of the system δ δ , where ρAA is the matrix entry correspond-
n1,0 n2,0 n1,n2
when the qubit first decays, we followed the quantum ingto g g n n andsocanbeinterpretedforAB,
1 2
| ih |⊗| ih |
3
BA and BB. We have Given the way we conduct Fourier Transfer, we have
ρk,k′ = ρ k′, k, which means θk,k′ = θ k′, k. Thus we
∆n = n ∞γρBB(t)dt . (7) have − − − −
h i ·(cid:18)Z n,n (cid:19)
Xn 0 dk dk
p ∂ ∂ θ (t)= p ∂ ∂ θ (t)
k t 1 k,k k t 2 k, k
I 2π −I 2π − −
Then we translate this to momentum space by having
dk
= p ∂ ∂ θ (t). (16)
ρC,C′ = 1 dk dk eikne ik′n′ρC,C′ (8) I 2π k t 2 k,k
n,n′ (2π)2 I I ′ − k,k′
Here we also use the fact that p is even function in
k
k. Given the fact that ∂ θ (t) +∂ θ (t) = 0, that
where C,C A,B , the Hamiltonian becomes separa- 1 k,k 2 k,k
′
∈{ } integration yields zero, and we have ∆n =I .
ble as for a 2 2 subspace of g/n k we have h i 0
× | i⊗| i Given that the system eventually decays completely,
p (t )=0, we have
H = ǫA vk (9) k →∞
k (cid:18) vk∗ ǫB −iγ/2(cid:19) dk ∂θk′,k(0)
∆n = . (17)
with v = Ω +g√Ne ik and ǫ ǫ =∆ǫ. h i −I 2π ∂k′ (cid:12)(cid:12)k′=k
k 2 − B− A (cid:12)
Using integration by part, we have that Since ρ(0) is pure and diagonal in t(cid:12)he basis of σz, for
ρ(ǫ) consider up to the first order of ǫ 0+, one could
nρBn,Bn = (2πi)2 I dkI dk′ei(k−k′)n∂1ρBk,Bk′. (10) fiaffnedctthρa(tǫ)tthoetdheepfihrassteotredremr. Td(hσuzsρ,(bty)σt→zre−atρin(tg))ρ(dǫo)eassn’at
pure state (which is similar to the case in Ref. [9]), we
Summing over n and integrate over time, we have
have
∆n =iγ ∞dt dk∂ ρBB, (11) ∆n = dk∂arg(−ivk∗). (18)
h i Z I 2π 1 k,k h i −I 2π ∂k
0
where we use ∂1ρBk,Bk as a shorthand of ∂∂k′ρBk′B,k|k′=k. tThheeintotepgorlaotgiiocnalosftrucivturedoiesstnh’tatc,oinftaΩi/n2t>hega√xiNs,ortihgeinn
abNiliotwy,thoantetchaensduebfisnyestpemk(to)f:=moρmAk,eAkn+tuρmBk,Bkk haasstnh’tedpercoaby- and ∆n = 0; if Ω−/2k∗< g√N, then the integration of
h i
tdiellcotimmpeots,itwiointhas∂tρpBkk,Bk=′(t−)=γρuBk,kBk,k.′(At)ls·oe,iθok,nk′e(tc)a.nWuistehptholaatr, −−oriiivΩgk∗i/n2i,stahannudsawn∆tiitn-hclor=ackdwiu1is.segc√onNt,ouwrhoifchaccoirnctlaeincesntthereedaxaits
one can write h i −
∆n = iγ ∞dt dk eiθk,k(t)∂1uk.k(t) IV. NUMERICAL RESULTS AND
h i 2π Z0 I (cid:16) DISCUSSIONS
+u (t) i eiθk,k(t) ∂ θ (t) .
k,k 1 k,k
· · · (cid:17)
(12) We use a simple numericalintegrationtechnique todo
the simulation. A MAXN = 320[23] (which is the to-
Considering the fact that θ (t) = 0 as the di- tal dimension of the Hilbert space under consideration)
k,k
agonal terms of a density matrix are real, and dimensional complex vector V (matrix, if dephase is in
∂ u (t) = ∂ u (t), one can find that the first term consideration) is used to store the state of the system.
1 k,k 2 k,k
eiθk,k(t)∂1uk.k(t) induces an integration of a closed con- In the absence of dephase, a matrix U = e−i·H·t is com-
tour, and is thus zero. We can reach puted and U V simply yields the new state vector V.
·
InthepresenceofqubitdephaseweuseLindbladmaster
1 ∞ equation(6). Integrationiscarriedoutalongthewayun-
∆n = dt dk(∂ p (t) ∂ θ (t)). (13)
t k 1 k,k tiltheamplitudeofV converges[24]. Thetimeintervalt
h i 2π Z I ·
0
shrinks by half each time and Richardson Extrapolation
Defining is carried out until the final result converges.
A first glimpse of the results are provided in Fig. 2
dk with N =100 (which is large, as proposed in [9] to meet
I0 =I 2π (pk∂1θk,k(t)|∞0 ), (14) theassumptionof(4)). Goodtopologicaltransitionisob-
servedif the system is initialized in a Fock state (black).
we have through integration by part SincelargenumberFockstatesareusuallyhardtopre-
pare,weconsideredanalternative: coherentstate(Pois-
∞ dk sondistribution), which the authors of [9] have also cho-
∆n =I p ∂ ∂ θ (t). (15)
h i 0−Z I 2π k t 1 k,k sen by a classical driving. However, the simulation (red
0
4
A. Different Decay Factors and Energy Level
Differences
100.0 In this section we consider the effect of qubit decay
factors γ and detuning ∆ǫ on this experiment.
Intheory[9], qubitdecayfactorγ onlyaffectsthe (ex-
pected) evolutiontime, not the topologicaltransition,so
Æ æ99.5 the expected result should be the same for different γ’s.
Start with Fock State We run a simulation for different initial N’s with identi-
N Start with
Poisson Distribution cal parameters like g and ∆ǫ for different decay factors
Incoherent Hopping γ’s, with result in Fig. 4. We observe that the qubit de-
99.0
0 10 20
/ (2g)
16
FIG.2: (coloronline)ResultsofabasicsimulationforN=100 128
with different starting distributions (Poisson / Fock) of the 6
resonator, and comparison withclassical incoherenthopping.
TheindependentvariableΩ/2gcapturestherelativestrengths
ofinterandintraunithoppinginthequantumrandomwalk,
and N is the average resonator energy upon qubit decay. Æ æ 5
h i
The other parameters are g = 1 (set as a benchmark), γ = = 0.4
−4
4,∆ǫ=10 while Ω varies. N = 1.5
= 4
4 = 100
0 2 4 6
line in Fig. 2) turns out to be quite similar in the case / (2g)
(a)
of non-coherent hopping (classical walker) despite large
initial N. This shows that, contrary to the Ref. [9], the
Fock states are indeed necessary for testing topological
111
transition. 110
109
101
Considering the fact that high energy Fock states are
hard to come by, we relax the condition of N >> 1 and
examine some cases with small Fock numbers. An ex-
treme case of N =1 is displayed in Fig. 3(a) which still Æ æ100
preserves the basic traits of a topological transition de-
= 0.4
spitethesmallinitialFocknumber. Furtherinvestigation N = 1.5
= 4
= 100
99
0 10 20
Average Photon Number 01Upon Qubit Decay..50 SFtaorct kw Sithta te Photon Number Change --100on Average...050 NNNNN ===== 12951050 FqNtiuIvGbe=i.stt14dr0:ee0nc.a(gcytTohhlfsoaercotfinooirndnlteγiepn’rseen)aundnRdedneietsnru tv/(l(rat baa(sr))2iuaoignbnfili)tetsiihamΩol/upN2lpagit=nicogan5pintawuntirhdtehes(qbtduh)iaffeinnertrietuelinamat-l
0.0
0 2 4 6 -2 -1 0 1 2 random walk, and N is the average resonator energy upon
((a) ) / g ( )( b/ g) qubit decay. The hothier parameters are g = 1,∆ǫ = 10−4
while Ω varies. One can observe that if qubit decay factor
γ is small, the curve significantly deviates from theoretical
FIG. 3: (color online) Results of simulation for small initial
deduction,while for large γ’s, thecurvesconverge.
N’s and comparison. The parameters in both figures are
(same as Fig. 2) g = 1,γ = 4,∆ǫ = 10−4 while Ω varies.
(a) Results of simulation with N =1. (b) Displaying change cayfactorγ cannotbetoosmall(atleastγ >gshouldbe
in “decay energy” with respect to Ω/2/g √N satisfied), otherwise the photon number upon measure-
−
mentwouldbetoolarge. Thisisnecessaryforbothsmall
and large N’s. We found that γ = 4 (4 times the cou-
into the problem yields Fig. 3(b), which shows that the plingstrength)isroughlywherequbitdecayfactorstarts
curve for different N’s overlap. to negatively affect the observed topological transition,
5
we will stick to this value from now on. approximation (4) plays a major role here, just like in
Thediscrepancybetweenanalyticaltheoryandnumer- Fig. 4, as small qubit decay γ yields wilder amplitude
ical results lies in the assumption of equation (4), which span to overturn approximation (4).
is only valid if the system stays close to photon num-
ber N throughout the whole evolution. If decay factor
γ is small, the quantum walker can walk far away from B. Dephasing Effects
theinitialsite,overturningthatassumption. Specifically,
since we have
Dephasing characterizes one major effect of noise on
circuit QED systems, making a quantum system less
a˜= √n n 1 n ,a˜ = √n n n 1 , (19)
†
| − ih | | ih − | “quantum” but more “classical”. Here, we consider de-
Xn Xn
phaseofthequbitduetoexternalfieldfluctuation,which
the strengths of inter-site hoppings grow stronger as the is the major source of dephase in a circuit QED sys-
site number(orenergylevelinoursimulation)increases, tem, and can be written in Lindblad master equation
which explains why for small γ’s the curves deviate sig- (6), where d characterizes the strength of such dephase.
nificantly to higher energy levels. Such dephase keeps the diagonal elements of ρ, but in
Luckily, big γ’s does not present an experimental dif- addition to other standard evolution the non-diagonal
ficulty in reality, as big qubit decays are usually easy to elements of it shrinks exponentially by e dt. Here, we
− ·
generate. runanumericalsimulationwherequbitdecayrateγ and
And Fig. 5 shows the effect of detuning on this exper- otherparameterslikeg,∆ǫarekeptidenticalfordifferent
iment. Ideally,we want∆ǫ=0,whichis anexactmatch qubit dephase factor d’s.
The results are in Fig. 6, which shows that the topo-
6
Æ æ 5 5
N
4 Æ æ
Incoherent Hoppings
N d = 0
5 d = 8
4 d=100
= 0
0 2 4 6
Æ æ = 1
/ (2g)
N = 10
FIG. 6: (color online) Results of Simulation with different
4 Dephase Factor d’s For the Qubit under initial N = 5. The
0 4 8 independent variable Ω/2g captures the relative strengths of
/ (2g) inter and intra unit hopping in the quantum random walk,
and N is the average resonator energy upon qubit decay.
FIG. 5: (color online) Results of Simulation with different Thehpariameters are (same as Fig. 2)g =1,γ =4,∆ǫ=10−4
System Detuning∆ǫ’s underinitial N =5, with (upper) un-
while Ω varies.
der qubit decay γ = 4 and (lower) γ = 20. Notice that the
difference between ∆ǫ = 1 and ∆ǫ = 0 in (lower) is so tiny
logicaltransitiongrowslessdistinctasthequbitdephase
that one can barely distinguish them in the plot. The inde-
pendentvariableΩ/2g capturestherelativestrengthsofinter factordgrows. Yetevenwhenthedephaseofthequbitis
andintraunithoppinginthequantumrandomwalk,and N verylarge(d=100)comparedwiththeotherelementsof
h i
is the average resonator energy upon qubit decay. The other the system Ω,γ,g < 10, good topological transition can
parameter is g=1 while Ω varies. stillbe observed,as comparedto the classicalincoherent
hopping.
tothequantumrandomwalkscenario. Simulationshows This numericalresults agreewith analyticaltheory we
that as long as the detuning of the system is kept small put forward earlier in this paper, that this topological
(Fig. 5 shows that∆ǫ < g,Ω suffices), it won’t have a structure is protected from the effect of dephase. No
tangible effect on the result of this experiment. matterhowmuchdephasewehaveinthesystem,aslong
It’sworthwhiletonoticethatintheory,neither∆ǫnor as such dephase comes from external field fluctuation,
γ should affect the experimental results. Yet in Fig. 5, andcanbetreatedasMarkovianinthetimescaleofother
onecanobservethatforafixed∆ǫ=10,increasingqubit operations in the system, it won’t affect experimental
decay γ yields better topological transition. We believe results.
6
V. 2-DIMENSIONAL QUANTUM WALK, AND A. Analytical Theory for Higher Dimensional
ITS REALIZATION IN A CIRCUIT QED SYSTEM Random Walk
The theory for higher dimensional case without de-
As an extension to what’s previous discussed, we have
phasehasbeenexplainedin[11],whichisasimpleexten-
also investigated 2-dimensional quantum walk. In this
sion of the one-dimensional case, and so is the following
case eachunit still consists of a decaying site and a non-
section considering dephase.
decaying site, however,a “walker”couldmake inter-unit
Supposethedimensionisd. Asintheone-dimensional
hopping in two dimensions, with their strengths charac-
case, one can still use ρCC′ to denote the state of the
terized by v′ and v′′ each. The strength of hopping be- system, where n and n na,nr′e d-dimensional vectors and
tweentwositesofthesameunitisstillv. Fig. 7provides ′
C,C A,B . The same Fourier Translation into mo-
a sketch of the model. ′
mentu∈m{space}yields that for subspace k,
v’’ v’’ Hk =(cid:18)AǫAk∗ ǫB Aik~γ/2(cid:19) (21)
−
unit m,m+1 unit m+1,m+1 with Ak = Ω2 + dα=1g(α)√Ne−ikα. Defining pk(t) in
v v’ v the same way as P
pk =ρAk,Ak+ρBk,Bk (22)
decay decay and with ∂tpk = −γψkB,Bk(t) and integrate by part, one
can reach
v’’ v’’ h∆nαi=I (2dπd−)d1k1 (cid:26)iγZ ∞dtI d2kπα∂kα,1ρBk,Bk(cid:27),
− 0
(23)
unit m,m unit m+1,m
whiletheshorthand∂ meanstakingpartialderivative
kα,1
v v’ v only on the α-th component of k, and only on the first
k in the density matrix. It’s not hard to see that the
expressioninsidethebraceisexactlyasequation(11)and
thus is either 0 or 1, no matter what dephasing factors
decay decay we have.
Now, if one come back to the definition of Ak =
v’’ v’’
Ω2+ dα=1g(α)√Ne−ikα,onewaytounderstandequation
FIG.7: (coloronline)Theorysetupoftwodimensionalquan- (23)Pistofirstfixd 1angleskβ=α,thenseewhetherthe
tum walk. The system is similar to the one in Fig. 1, ex- integrationofkα fro−m0to2πwo6 uldcasetheangleofAk
ceptwithonemoredimension. Threecolors(red,orangeand to also shift 2π, and integration over those d 1 angles.
−
green) are used to distinguish different kindsof hopping(red During the integrationof the other d 1 angles,we may
for intra-site, orange and green for inter-site on two differ- observe new topological structures, li−ke the middle area
ent dimensions). The dashed lines are used to indicate the
of Fig. VB, with some old traits still remaining,like the
boundaries of each site.
two sides of Fig. VB.
For the 2-dimensional case that is relatively easy to
simulate,assuming g >g , the results are
The benefits of a circuit QED system is that it’s rel- 1 2
atively easy to accommodate this change by adding a
1 0 v >v+v
′ ′′
second resonator mode coupled to the qubit. Like be- −
∆n = 1+ θ1 , ∆n = θ2 v v v ,
fore, if one uses e/g n1 n2 to identify the state h 1i − π h 2i −π | − ′|≤ ′′
| i⊗| i⊗| i 0 0 v <v v
of the system, then the Hamiltonian of the system is ′ ′′
−
  (24)
H = g1 σ+a˜1+σ−a˜1† +g2 σ+a˜2+σ−a˜2† with cosθ1 = N(g12Ω−√g2N2)−g2Ω2/4,cosθ2 = N(g12Ω−√gN22)g+2Ω2/4,
+Ω(cid:0)/2(σ++σ )+(cid:1) 1∆ǫσ(cid:0)z 1iγ e e (cid:1), (20) v = Ω/2,v′ = √Ng1,v′′ = √Ng2 in this system. One
− 2 − 2 2 | ih | can see a detailed theory deduction in Appendix A.
whereσ’sarestillPaulioperatorsonthequbit,anda˜ ’s
1/2
are lowering/ rising operatorson the first or second res- B. Numerical Results
onator field in a rotating frame, ∆ǫ is (real) detuning of
thesystem,andγ characterizesthedecaystrengthofthe We run numerical simulations for the 2-dimensional
excited qubit state. case in a similar manner as the 1-dcase, except that the
7
tensor product of two vectors is stored. We truncated
photonnumberatMAXN=20withhindsightknowledge
from the 1-dimensional simulations. From Figure VB 0.0
onecanseethattheresultsofthesimulationpreservethe
basic properties ofa 2-dtransition. It also showsthat in
the 2-dimensional case, a more stringent requirement is
Æ Næ
put onto the system as a much bigger decay factor γ is
-0.5
necessarytopreservethetopologicalnatureofthesystem
compared to the one-dimensional case. Theory
d = 0
d = 1
-1.0 d = 8
d = 100
0.00
0.0
-0.05
Æ Næ Æ Næ
-0.5 -0.10
-0.15
-1.0
0.00 0.25 0.50 0.75 1.00
0.05
v / (v + v’ + v’’)
0.00 FIG. 9: (color online) 2D Hoppings: Results of simulation
with different dephase factor d’s for the qubit under initial
-0.05 N = 5, and comparison with the theoretical (ideal) case
Æ Næ Theory for a 2-dimensional random walk in (upper) the first res-
-0.10
= 25 onator mode, and (lower) the second resonator mode. The
-0.15 = 5 coupling′ st′r′engths are set to g1 = 2,g2 = 1 to satisfy con-
strain v/v = 2. The independent variable is calculated by
0.00 0.25 0.50 0.75 1.00 v/(v+v′+v′′) := Ω/(Ω+2√N(g1+g2)) to fit in the form
v / (v + v’ + v’’) of equation (24). ∆N := N N is the average resonator
h i h i−
energy change upon qubit decay. The other parameters are
FIG. 8: (color online) 2D Hoppings: Results of simula- −4
γ = 25,∆ǫ = 10 while Ω varies. From the fact that the
tion with different qubit decay factor γ’s under initial N =
curvesfordifferentdephasefactor d’salmost overlaponecan
5, and comparison with the theoretical (ideal) case for a
seethat qubitdephasefactor doesn’t affect thesimulated re-
2-dimensional random walk in (upper) the first resonator
sult.
mode, and (lower) the second resonator mode. The cou-
pling strengths are set to g1 = 2,g2 = 1 to satisfy con-
′ ′′
strain v/v = 2. The independent variable is calculated by
v/(v+v′+v′′) := Ω/(Ω+2√N(g1+g2)) to fit in the form VI. CONCLUSION
−4
of equation (24). The other parameter is ∆ǫ = 10 while
Ω varies. From the fact that a similar qubit decay (γ = 5) As a key element of this experimental realization, a
as 1d hoppings leads to large deviation to the ideal topolog- measurement of the photon number in the cavity needs
ical structure one can observe that a larger qubit decay is tobecarriedouttheinstancethatthequbitdecays. One
necessary for 2d experiments. way to implement such scheme is to prepare a low-decay
qubit that’s coupledto another resonatormode that has
much decay. By continuous measurement of that other
resonator mode [25], which provides much of the qubit
We have also numerically simulatedthe effect of qubit
decay,one canimmediately sense the decay ofthe qubit,
dephase onthe final result. Using a similar method with
and thus carried out a energy measurement in the main
section IV.B, we simulated equation (6) exactly and dis-
resonator mode to retrieve photon number.
playedtheresultsinFig. VB,whichshowsthatdifferent
In summary, we considered the quantum walk on an
qubit dephase factor d’s doesn’t affect our result.
one-dimensional bipartite indexed lattice, where each
Due to the fact that analytical theory for different di- unit has one decay site and one non-decaysite. We have
mensions followsimilar integrationovercontours,we ex- proved the topological transition in this non-Hermitian
pect that in higher dimensional case, the topological ef- systemisnotaffectedbyboththedecayandthedephas-
fect would still be protected from qubit dephase. ing between decay and non-decay sites within units. We
8
proposed a circuit QED implementation where a qubit depend on qubit dephase d, and as long as detuning ∆ǫ
is coupled to one resonator mode. We have shown that of the system is kept smaller than the coupling strength
quantumrandomwalkcanbe realizedwithouttoo much g, and qubit decay is kept big (none of which are strin-
modification to the standard JC Hamiltonian, and that gent), topological transition would be observable.
topological transition could be observed to some extent
even with small photon number N. We found that there We would like to thank L. Sun for helpfull discus-
was no topological transition if the systems starts in a sions. ThisworkisfundedbytheNBRPC(973Program)
coherentstate,whichiscontrarytoRef. [9]. Ournumer- 2011CBA00300(2011CBA00301),NNSFCNO.61435007,
ical results show that the topological structure doesn’t NO. 11574353,NO. 11274351and NO.11474177.
[1] M.Szegedy,QuantumSpeed-UpofMarkovChainBased [15] LChang,XJiang,SHua,CYang,JWen,LJiang,GLi,
Algorithms,inProceedings of the 45th IEEESymposium GWang,MXiao,Parity-timesymmetryandvariableop-
on Foundations of Computer Science (IEEE, New York, tical isolation in active-passive-coupled microresonators,
2004), pp.32-41. Nature Photonics 8, 524 (2014).
[2] A. M. Childs, Universal computation by quantum walk, [16] HerbertWalther,BenjaminTHVarcoe,Berthold-Georg
Phys.Rev.Lett. 102, 180501 (2009). Englert and Thomas Becker, Cavityquantumelectrody-
[3] Z. H. Bian, J. Li, H. Qin, X. Zhan, R. Zhang, namics, Rep.Prog. Phys. 69, 1325 (2006).
B. C. Sanders, and P. Xue, Realization of Single- [17] Zhang-qi Yin and Fu-li Li, Multiatom and resonant in-
QubitPositive-Operator-ValuedMeasurementviaaOne- teraction scheme for quantum state transfer and logical
DimensionalPhotonicQuantumWalks,Phys.Rev.Lett. gates between two remote cavities via an optical fiber,
114, 203602 (2015). Phys. Rev.A 75, 012324 (2007).
[4] A.Ambainis,E.Bach,A.Nayak,A.Vishwanath,J.Wa- [18] A.Blais, J. Gambetta, A.Wallraff, D.I.Schuster,S.M.
trous, One-dimensional quantum random walk, in Pro- Girvin, M. H. Devoret, and R. J. Schoelkopf, Quantum-
ceedings of the 33rd Annual AMC Symposium onTheory information processing with circuit quantum electrody-
of Computing (ACM Press, New York,2001), pp.37-49. namics, Phys. Rev.A 75, 032329 (2007).
[5] Gregory S. Engel et al, Evidence for wavelike energy [19] J.M.Zeuner,M.C.Rechtsman,Y.Plotnik,Observation
transfer through quantum coherence in photosynthetic ofatopologicaltransitioninthebulkofanon-Hermitian
systems. Nature(London) 446, 782 (2007). system, Phys.Rev.Lett. 115, 040402 (2015).
[6] M. Mohseni, et al, Environment-assisted quantumwalks [20] M. Reagor, H. Paik, G. Catelani, L. Sun, C. Axline, E.
in photosynthetic energy transfer, J. Chem. Phys. 129 Holland, I. M. Pop, N. A. Masluk, T. Brecht, L. Frun-
174106 (2008). zio, M. H. Devoret, L. Glazman, and R. J. Schoelkopf,
[7] Qing Ai, Tzu-Chi Yen, Bih-Yaw Jin, and Yuan-Chung Reaching10mssinglephotonlifetimesforsuperconduct-
Cheng, Clustered Geometries Exploiting Quantum Co- ing aluminum cavities, Appl. Phys. Lett. 102, 192604
herence Effects for Efficient Energy Transfer in Light (2013).
Harvesting, J. Phys. Chem. Lett. 4, 2577 (2013). [21] Plenio M. B, Knight P. L. The quantumjump approach
[8] Qing Ai, Yuan-Jia Fan, Bih-Yaw Jin and Yuan-Chung todissipativedynamicsinquantum.Rev.Mod.Phys.70,
Cheng, An efficient quantum jump method for coherent 101 (1998).
energy transfer dynamics in photosynthetic systems un- [22] H. M. Wiseman and G. J. Milburn, Quantum Measure-
dertheinfluenceoflaserfields,NewJ.Phys.16,053033 ment and Control. (Cambridge University Press, Cam-
(2014). bridge, England, 2009).
[9] M.S.RudnerandL.S.Levitov,TopologicalTransitionin [23] Although with hindsight we only need the dimension of
a Non-Hermitian Quantum Walk, Phys. Rev.Lett. 102, Hilbert space for the resonator to be 2N, which means
065703 (2009). MAXN 4N is simply enough. Choosing such a large
∼
[10] CarlM.Bender,MakingSenseofNon-HermitianHamil- numberistomakesurethatwedon’tneedtochangethe
tonians, Rept.Prog. Phys. 70, 947 (2007). configurationofthesystemfordifferentinitialFockstate
[11] M.S.RudnerandL.S.Levitov,Phasetransitionsindis- numberslike N =1 and N =100.
sipativequantumtransport and mesoscopic nuclearspin [24] Due to the existence of “dark state” for finite systems,
pumping,Phys. Rev.B 82, 155418 (2010). that amplitude won’t go to exactly zero.
[12] S. Longshi, Convective and absolute PT-symmetry [25] L. Sun,A.Petrenko,Z. Leghtas, B. Vlastakis, G. Kirch-
breaking in tight-binding lattices, Phys. Rev. A 88, mair,K.M.Sliwa, A.Narla,M.Hatridge,S.Shankar,J.
052102 (2013). Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and
[13] C. M. Bender and S. Boettcher, Real Spectra in Non- R.J.Schoelkopf,TrackingPhotonJumpswithRepeated
Hermitian Hamiltonians Having PT Symmetry, Phys. QuantumNon-Demolition ParityMeasurements,Nature
Rev.Lett. 80, 5243 (1998). (London) 511, 444 (2014).
[14] C. E Ru¨ter et al, Observation of parity-time symmetry
in optics, NaturePhysics 6, 192 (2010).