Quantum Zeno switch for single-photon coherent transport

0.34 MB·English

by  Lan Zhou

Quantum Zeno switch for single-photon coherent transport
Lan Zhou,1,2 S. Yang,3 Yu-xi Liu,2,4 C. P. Sun,3,2 and Franco Nori2,4,5
1Department of Physics, Hunan Normal University, Changsha 410081, China
2Advanced Science Institute, The Institute of Physical and Chemical Research (RIKEN), Wako-shi 351-0198, Japan
3Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing, 100080, China
4CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan
5Center for Theoretical Physics, Physics Department, Center for the Study of Complex Systems,
The University of Michigan, Ann Arbor, Michigan 48109-1040, USA.
Using a dynamical quantum Zeno effect, we propose a general approach to control the coupling
between a two-level system (TLS) and its surroundings, by modulating the energy level spacing
of the TLS with a high frequency signal. We show that the TLS–surroundings interaction can
be turned on or off when the ratio between the amplitude and the frequency of the modulating
0
field is adjusted to be a zero of a Bessel function. The quantum Zeno effect of the TLS can also be
1
observedbythevanishingofthephotonreflectionatthesezeros. Basedontheseresults,wepropose
0
2 a quantum switch to control the transport of a single photon in a 1D waveguide. Our analytical
results agree well with numerical results using Floquet theory.
n
a
PACSnumbers: 03.65.Xp,32.80.Qk,42.50.Dv,03.67.Lx
J

4
I. INTRODUCTION
] e
h
p ThequantumZenoeffect(QZE)[1]wasoncedescribed W = w+W cos(vt) v- 4x
- asthequantumversionoftheexpression“awatchedpot a a
t g
n never boils”. Namely, an unstable particle, if observed
a (a) (b)
constantly, will not decay [2–6]. This effect has been
u k
explained (e.g., Refs. [1, 7, 8]) in terms of wave-packet
q - p 0 p
[ collapse: onceaquantummeasurementisperformed,the
superposed quantum state is reduced to a single one of
3 FIG. 1: (color online) (a) Schematic diagram of a quantum
the allowed eigenstates of the measured observable.
v switch: thecoupledresonatorwaveguideiscoupledtoaTLS
However, many authors (e.g., Refs. [9–15]) explained
1 with an energy level spacing modulated by Ωa(t). The Rabi
5 the QZE [16–18] without invoking the wave function col- oscillationcanappearintheeffectivemulti-bandstructureof
1 lapse. This dynamical QZE does not involve quantum frequencyshownin(b),withboundstates(inred)inthegap.
2 states collapsing in one eigenstate, and thus evolves in
. time. Indeed, Asher Peres [9] argued that the decay of
2
1 an unstable quantum system can be slowed down and
8 even halted by its continual interaction with an exter- A discrete coordinate scattering approach [21] was de-
0 nal system, without measurement or observation. This veloped to study a quantum switch for the coherent
v: dynamical QZEissometimescalled“bang-bang”control transportofasinglephotonalonga1Dcoupled-resonator
i (e.g.,inRefs.[13–15]). Here,weapplythedynamicQZE waveguide (CRW). That quantum switch [21] can be re-
X to realize a quantum switch. alized by changing either the energy level spacing or
r The QZE can be used as a dynamical mechanism to the coupling constants between the waveguide and a
a
controller. Physical systems proposed to realize single-
suppresstheenvironment-induceddecoherenceofaquan-
photon switches include: (i) Superconducting transmis-
tum system [7, 15]. This implies that the coupling be-
sion line resonators coupled to a superconducting charge
tween the system and its environment can be controlled
qubit [21]; (ii) A TLS coupled to a photonic crystal
viatheQZE.Here,weproposeaquantum“Zenoswitch”
“defect-cavity” waveguide [20]. For the case (i), the con-
to turn on or off the quantum coherent transport of par-
trollability can be realized by changing the energy level
ticles along a one-dimensional (1D) continuum.
spacing of the charge qubit. However, it seems difficult
A quantum switch [19–21] is regarded as an active de-
to control well the photon transport in (ii), because the
vice to control the transport of particles or the transfer
couplings and TLS parameters are fixed once the sample
of quantum states. It plays a similar role to the gate
is fabricated.
voltage, which turns on or off the transmission of elec-
trons, in conventional electronic circuits. The quantum Inthispaper, forasingle-photonpropagatingina1D
version of the switch means that the operations involve CRW, a dynamical QZE switch is proposed with a tun-
asinglequantum. Anexampleisthesingle-photon tran- able effective coupling to a TLS. This coupling is con-
sistor [22–25], where the transport of a single photon in trolled by an applied frequency-modulated electromag-
a 1D waveguide can be controlled by a two-level system neticfield. Thephotoncanbetransmittedeitherdirectly
(TLS) externally driven by a classical field. through the continuum, or indirectly, via a discrete en-
2
ergy level provided by the TLS. These two channels in- Foraveryfastmodulationoffrequencyν (i.e.,∆/ν 0)
∼
terfere with each other. Their destructive quantum in- and a large Ω, we have
terference prevents the photon transport while the con-
structive one leads to a total transmission [28]. The HI J0(Ω/ν).
∝
dynamic Zeno effect of the TLS could be actively con-
The values of the decoupling points (i.e., Ω/ν =
trolled by a frequency modulation, to switch between ∼
2.40,5.52,...) are just the zeros of J (x). Equation (5)
the two quantum interference channels. When the dy- 0
clearlyshowsthattheeffectiveinteractionvanishes when
namicZenoeffectslowsdownthetransitionsintheTLS,
theratioΩ/ν(=amplitude/frequencyofthedrivingfield)
the photon transport appears only when the TLS is in
is a zero of the Bessel function J (x), therefore de-
itsgroundstate,nodestructivequantuminterferenceoc- [∆/ν]
coupling S from the TLS. At these zeros, the dynamical
curs. Then, effectively, the TLS and the continuum are
QZE occurs; namely, the TLS initially prepared in its
decoupled.
excited state will remain there and will not decay to the
ground state.
The above arguments to realize the QZE dynamics
II. DECOUPLING MECHANISM USING
does not depends on the concrete form of the quantum
DYNAMICAL QZE.
system S. Thus its based idea can be generally used for
the various decoupling schemes.
WegenerallyconsideraquantumsystemSwithachar-
Notice that the Bessel functions J (x) look roughly
n
acterizedfrequencyω andfreeHamiltonianH ,coupled
c e like oscillating sine or cosine functions, and have an infi-
to a TLS, with its ground g and excited e states, and
nite number of zeros. The decays of the Bessel functions
energy level spacing ω (w|ithi ~=1). We |asisume that a
a are proportional to 1/√x. Except asymptotically large
periodically modulated field is applied to TLS so that
x, the roots of the Bessel functions are not periodic.
ω Ω (t)=ω +Ωcos(νt), (1)
a a a
→
III. SINGLE PHOTON QUANTUM SWITCH IN
where Ω is the amplitude of the modulation with fre- A 1D WAVEGUIDE.
quency ν. Hamiltonian
Based on the above dynamical QZE mechanism, we
H =Ω (t)e e +Ge g +G∗ g e +H (2)
a | ih | | ih | | ih | e now propose a quantum device, which behaves as a
switchtocontroltheincidentphotontransportinaCRW
describes the quantum system S and the TLS, as well as
made of a periodic array of identical coupled resonators.
the coupling between them. The coupling coefficient G
The main difference between this device and the one in
depends on S-variables so that
Ref. [21] is that here the TLS energy level spacing is
eiHetGe−iHet =Ge−iωct. (3) now modulated by a periodic field, allowing us to use
the remarkable dynamic QZE in a very unusual man-
ner. The 1D CRW is schematically shown in Fig. 1(a),
In the interaction picture, the coupling between S and
where a TLS is embedded in one of the resonators and
the TLS is modeled by the Hamiltonian
is modulated with the external periodic forcing of am-
HI =G +∞ Jn Ω ei(nν−∆)t e g +H.c., (4) (pjlit=ude Ω,and,fre)qubeencthyeνc,rei.aet.ionωaop→eraΩtoar(to).f tLheetjath†j
(cid:18)ν(cid:19) | ih | −∞ ··· ∞
n=X−∞ single mode cavity, all with the same frequency ω. The
Hamiltonian of the CRW reads
where the detuning ∆ = ω ω . In Eq. (4), we have
c a
used [28] the Fourier-Bessel se−ries identity, H =ω a†a ξ a†a +H.c. , (6)
C cXj j j − Xj (cid:16) j j+1 (cid:17)
eixsinγ = J (x)einγ,
n
withtheinter-cavitycouplingconstantξ,whichdescribes
Xn
the photon moving from one cavity to another. For con-
with the nth Bessel function J (x) of the first kind. venience, we take the 0th cavity as the coordinate-axis
n
For a fast modulation with large ν, the lowest fre- origin and also assume that the TLS is located in this
quency terms in Eq. (4) dominate the dynamical evo- 0th cavity. Under the rotating wave approximation, the
lution. Thesetermsaredeterminedbythenear-resonant interaction between the 0th cavity field and the TLS is
and resonant condition (nν ∆) 0; here n = [∆/ν] is described by a Jaynes-Cummings Hamiltonian
− ≃
theintegernearestto∆/ν. Tothelowestfrequencyterm
of the index n=[∆/ν] , Eq. (4) is approximated as HI =Ωa(t)|eihe|+g(cid:16)a†0|gihe|+|eihg|a0(cid:17), (7)
Ω with coupling strength g and modulated transition fre-
H GJ e(i[∆/ν]ν−∆)t e g +H.c.. (5)
I ≈ [∆/ν](cid:18)ν(cid:19) | ih | quency Ωa of the TLS. By employing the Fourier trans-
3
formation the high frequency modulation Ω (t) leads to the decou-
a
pling between the TLS and the CRW, and thus photons
1
a = eikja , (8) in the CRW will propagate freely without feeling the in-
j k
√N Xk fluence from the localized TLS. Therefore, this modu-
lated TLS acts as a quantum “Zeno switch”.
the second term in Eq. (7) and H can be rewritten in
C
k-space. In the rotating frame with respect to H +
C
Ω (t) e e, the interaction Hamiltonian reads IV. FLOQUET-THEORY-BASED NUMERICAL
a
| ih |
SIMULATION OF THE DYNAMICAL QZE
g Ω
H (t)= J a†σ ei(∆−ǫk−nυ)t+H.c. ,
II √NXk;n n(cid:18)ν(cid:19)h k − i Let us further study the QZE decoupling mechanism
(9) and the Zeno switch by numerically solving the time-
where we have used the Fourier-Bessel series identity. dependent Hamiltonian HC+HI. Note the total Hamil-
Here, the dispersion relation ǫ = 2ξcosk describes an tonian is periodic in time, with period 2π/ν. Using Flo-
k
energy band of width 4ξ (the lattice constant l is as- quet theory, we define
sumed to be unity), σ = g e. The H (t) in Eq. (9)
− II
| ih | H =H +H i∂ (11)
effectively describes a multi-band Rabi oscillation. F C I− t
In this work we are mainly interested in the investiga-
as in, e.g., Ref. [29]. Each eigenstate f (t) , with eigen-
tionofthetransmissionandlocalizationofsinglephoton | n i
value ε , of H can be used to define an eigenstate
in high-frequency regimes, corresponding to ∆,ξ ν. n F
≪
For small ξ, compared with the modulation frequency ν,
F =exp( iε t) f (12)
n n n
the gaps are large and there is no energy band overlap | i − | i
[thebandstructureisillustratedinFig.1(b)]. Otherwise of H with zero eigenvalue. Then the superposition of
F
thereexistsacomplexquantumdynamicswithquantum these eigenstates
chaos. Inthehigh-frequencyregime,thefastmodulation
in Ωa(t) implies the resonance condition n = 0. There- Φ(t) = cn Fn (13)
fore, the dynamic of the CRW+TLS is described by the | i | i
Xn
effective Hamiltonian
determines the general solution of the time-dependent
Ω
H =H +ω e e +gJ a† g e + e g a . Schr¨odinger equation corresponding to HC +HI. With
C a| ih | 0(cid:18)ν(cid:19)(cid:16) 0| ih | | ih | 0(cid:17) an initial state Φ(0) , the coefficients cn are given by
| i
In the one excitation subspace, the wavefunction at ar- c = f (0) Φ(0) .
n n
bitrary time h | i
Our task now is to diagonalize H in a discrete Hilbert
F
φ(t) = u (t)e−iωat 1 g +u (t)e−iωat 0e space with the spatio-temporal basis vectors
j j e
| i | i | i
Xj
j;m = j m , (14)
t t
| i | i⊗| i
is a superposition of the photon at the jth cavity with
atom in the ground state and no photon in all cavities where mt is the discrete temporal coordinate of the time
withatomintheexcitedstate. Theequationsfortheex- t, satisfying
cited state amplitude u (t) and the amplitudes u (t) of
e j
singlephotonstatesderivedfromtheSchroedingerequa- t mt =exp(iνmtt)/√T, (15)
h | i
tion with Hamiltonian H are given by
with a large T; j is the eigenstate of the discrete space
| i
iu˙j(t)=∆uj(t)−ξ[uj−1(t)+uj+1(t)]+Gue(t)δ(j100.a) tcroioxrdelienmateen.tsOonftHhiFstaimreei-nvdareypienngdbenatsisof{t|jim,me.ti},thema-
For the dynamic QZE in this paper, the initial state
iu˙ (t) Gu (t) (10b)
e ≃ 0 Φ(0) = 0,e;nt = 0 evolves to Φ(t) . To keep the
| i | i | i
numerical accuracy and save computing time we divide
where the overdot indicates the derivative with respect
thetotaltimetintoN smalltimeintervalsτ,t=Nτ. In
to time. These amplitude equations (10) show that the
eachinterval,thesystemisevolvedfrommτ to(m+1)τ.
interaction between the single photon and the TLS is
The final state of this interval becomes the initial state
characterized by the effective coupling constant
of the next interval. For a sufficiently small value of τ,
G=gJ (Ω/ν). an accurate wave function can be obtained, even when
0
not too many m are used to represent H [29].
t F
{| i}
For zeros of zeros of J0(Ω/ν) , i.e., Figure2(a) shows the probability Pe(t) for the TLS to
be in the excited state. When Ω/ν is tuned to the vicin-
Ω=2.40ν,5.52ν,... ity of the zeros of J (x) (e.g., Ω/ν = 2.40), the initially
∼ 0
4
1.0 the acceleration (anti-Zeno effect).
Following the analysis in Ref.[31], we calculate the
0.8
P (t)e0.6 W /n = 0.0 probability
0.4 W /n = 1.0 Pe(t)=exp[ R(t)Q(t)] (16)
0.2 W /n = 2.4 −
W /n = 3.5 to reflects the decay law (for detailed caculation see the
0.0
0 5 10 15 20 appedix), where
(a) t J(W /n)
0 t
1.0 1.0 Q(t)= dτ V (τ)2 (17)
Z | |
0
P (t )ef0.5 0.5 is the effective interaction time with

0.0 0.0 Ω
V (t)=exp i sin(υt) (18)
(cid:20)− υ (cid:21)
-0.5 -0.5
0 2 4 6 8 10 here. In the frequency domain, the effective decay rate
(b) W /n
∞
R(t)= Φ˜(ω+∆)F (ω)dω (19)
FIG.2: (coloronline)(a)Timedependenceoftheprobability t
Z
Pe(t)fortheTLSatitsexcitedstatewithdifferentΩ/ν,with −∞
parameters: ∆ = 0, ξ = 1.0, g = 0.25, ν = 10 and L = is the overlap of the normalized spectral modulation in-
41 resonators; t is in units of 1/ξ. Note that the excited tensity
state does not decay (i.e., QZE) when Ω/ν =2.4. (b) Pe(tf)
(winesolidline)versusΩ/ν isplottedforacertaintimet = t
f
20 using the same data as in (a). Obviously, the points for Ft(ω)=(cid:12)Z dτV (τ)e−iωτ(cid:12)/Q(t), (20)
Pe(tf) = 1 (corresponding to no decay for the TLS) agree (cid:12) 0 (cid:12)
(cid:12) (cid:12)
well with the zeros (orange vertical arrows) of J0(Ω/ν) (blue and the reservoir co(cid:12)upling spectrum Φ(cid:12)˜(ω), which is the
dashed line).
Fourier transformation of the memory function or reser-
voir response function
excited TLS does not decay. Otherwise (e.g., Ω/ν = 1.0 J2
and 3.5), the probability P (t) in the excited state de- Φ(t)= ei2ξtcosk. (21)
e N
creases with time. Figure 2(b) shows how P (t ) varies Xk
e f
with the ratio Ω/ν at the instant t = 20 for J (Ω/ν).
f 0 Obviously, V (t) is periodic with the period υ−1, the
It clearly demonstrates that the zeros of J (Ω/ν) corre-
0 Bessel function J (A/υ) of the first kind becomes the
spond to P (t )=1, i.e., the TLS does not decay. Thus n
e f Fourier components of V (t). At time t much larger than
the numerical results agree well with our analytical re-
the period υ−1, the decay rate reads
sults on the dynamical QZE.
+∞ Ω ∞ (ω ∆ nυ)t
R(t)=t J2 Φ˜(ω)sinc2 − − dω,
n(cid:18)υ(cid:19)Z 2
V. QUANTUM ZENO DYNAMICS n=X−∞ −∞
(22)
where sincx = sinx/x. Figure 2 roughly shows the de-
In this section we give the above mentioned dynamic
pendence of the decay rate on the ratio of the driving
QZEamorestandardapproachincurrentliteratures[31],
amplitude to the driving frequency. As we mentioned
which confirms the suitability for using the idea of QZE.
before, we are interested in the high-frequency regime
Within the subspace spanned by e0 and gj , if the
| i | i with parameters ∆,ξ ν. Since the width of the reser-
TLS is prepared initially in the excited state, the evolu- ≪
voir spectrum is proportional to ξ, the relation of these
tionoftheTLS-CRWsystemwillleadtoasuperposition
parametersimplythatthemodulationfrequencyismuch
of states, corresponding to the TLS initially excited and
greater than the inverse correlation time of the contin-
nophotonsintheCRWandtheTLSinthegroundstate uum. Therefore, Φ˜(ω)doesnotchangesignificantlyover
and one photon in the CRW. The CRW forms a contin-
the spectral intervals ν−1. In this case, we can make the
uum. Therefore,theinteractionprocessbetweentheTLS
approximation
andtheCRWcanbedescribedintermsofadiscretelevel
coupled to a continuum, and will lead to a dissolution of Ω (ω ∆)t ∞
the discrete state over an interval of width R, which is R(t)=tJ02(cid:18)υ(cid:19)sinc2 −2 Z Φ˜(ω)dω. (23)
−∞
the decay rate of the discrete state to the continuum. A
universal mechanism of quantum-mechanical decay con- The above equation describes that the state decays into
trol is usually based on periodic coherent pulses. This all the channels of the reservoir. Since the effective de-
control can yield either the inhibition (Zeno effect) or cay rate is averaged over all decay channels [7], we work
5
in the QZE regime. Obviously, the decay rate without gives a band of width 4ξ with its minimum lies at k =0.
modulation is suppressed by a factor J2(A/υ) when a Withintheband,thescatteringwavefunctionisassumed
0
periodical modulation is applied. When t is much larger as
than both υ−1 and an effective correlation time of the
U (j)=exp(ikj)+rexp( ikj), (27)
reservoir, one obtains k
−
for j <0, and
Ω
R(t)=J2 Φ˜(∆) (24)
0 (cid:18)υ(cid:19) Uk(j)=sexp(ikj),
forj >0, withtheright-andleft-movingwavesexp(ikj)
i.e. the extension of the golden rule rate to the case of
and exp( ikj). The boundary conditions
a time-dependent coupling [31]. Therefore, the atom re-
−
mainsinitsinitialstatewhentheratiobetweenthemod- U 0+ =U 0− (28a)
k k
ulation amplitude and the frequency meets the zeroes of
[V(E )+∆ E ]U(cid:0) (0(cid:1))=ξ[U(cid:0) ( (cid:1)1)+U (1)] (28b)
the Bessel function. We also note that when A = 0, the k k k k k
− −
0th Bessel function J (0) = 1, which means the decay
0 resultinthereflectionamplituder =s 1andtransmis-
rate R(t)=Φ˜(∆) without any modulation. −
sion amplitude
ω ω
k a
s= − . (29)
ω ω +ig2J2(Ω/ν)/v
VI. SCATTERED AND LOCALIZED PHOTONS k− a 0 g
WITH DYNAMICAL QZE
Here,
We now study the photon localization and the mea- vg =2ξsink
surement on the QZE via photon scattering. The pur-
is the group velocity and
pose for this study is twofold: (i) Utilize the QZE-based
mechanism to turn on or off the transmission of photons ω =ω 2ξcosk
k c
intheCRW,sothatanidealsingle-photontransistorcan −
be realized; (ii) Witness the dynamical QZE in the TLS is the incident energy of the single photon.
by measuring the single photon scattering. We further Equation(29)indicatesthatwhenωaisinsidetheband
considertherelationbetweenphotonlocalizationandthe (ωk = ωc 2ξcosk), a dip down to zero will occur in
−
appearance of dynamic QZE. the transmission line-shape s2. This zero transmission
| |
Solutions to Eqs. (10) can be found in the form of is a quantum interference (Fano) effect characterized by
either localized states around the location of the TLS or certain discrete energy states interacting with the con-
as a superposition of extended propagating Bloch waves tinuum. The width of the transmission line-shape is de-
incident,reflectedandtransmittedbytheTLSembedded termined by the ratio of the effective coupling strength
in the CRW. It was done by first Fourier transforming G = gJ0(Ω/ν) to group velocity vg. Obviously, the
the equations of motion of u (t) and u (t) in Eq. (10) to decoupling of the TLS-photon interaction occurs when
j e
obtain their Fourier transforms Uk(j) and Ue(j) J0(Ω/ν) vanishes. Consequently, the transmission dip
disappears, i.e., the single photon goes through the TLS
E U (j)=∆U (j) ξ[U (j 1)+U (j+1)]+GU δ scattererwithoutabsorptionoremission. Inotherwords,
k k k k k e j0
− − by adjusting the amplitude and frequency of the modulat-
E U (j)=GU (j)δ . (25)
k e k j0 ing field Ω (t), the TLS spontaneous emission and stim-
a
ulated transition are suppressed. Measuring the photon
Eliminating the amplitude U (j) in the above equation,
e transmission or reflection would show the occurrence of
we obtain the discrete-coordinate scattering equation
the dynamic QZE.
TheCRWwiththeTLSembeddedincanalsosupport
[V(E )+∆ E ]U (j)=ξ[U (j 1)+U (j+1)], (26)
k k k k k
− − localized modes. We now discuss how to observe the
dynamicQZEfromthelocalizationofphotons,described
where
by the photonic bound state in the CRW. The bound
V(E )=g2J2(Ω/ν)/E stateisassumedtohavethefollowingsolutionswitheven
k 0 k parity for the eigenvalue equation (26):
is a resonant potential resulting from the second-order
U (j)=Cexp( κj),j >0)
κ
transition process due to the coupling with the TLS. It −
U (j)=Cexp(κj),j <0), (30)
behaves as an infinite δ-potential on the resonance E = κ
k
0. In the absence of the TLS, a solution of Eqs. (10) has
which is localized around the 0th site where the TLS is
the form U (j)=exp(ikj), where k is the wave number
k embedded. Here, the imaginary wave vector κ labels the
of the Bloch waves. Therefore,
energy
E =∆ 2ξcosk E =∆ 2ξcoshκ (31)
k κ
− ±
6
|U (j)|2 from its surroundings. We apply this QZE-based de-
10 k coupling mechanism to realize a quantum switch for a
1
8 photon. The dynamic QZE switch studied here allows
0.8 to control the single-photon transport in a CRW by us-
6 0.6 ing a TLS which is periodically modulated in time. Our
Wn /4 0.4 analytical results agree well with our numerical results
0.2 using Floquet theory. This proposal should be realizable
2
experimentally [30], because the quantum Zeno switch
0
0 only depends on the ratio between the amplitude Ω and
-500 -250 0 250 500
the frequency ν of the external modulation. This QZE
j
photon switch might be useful to quantum information
science, for the coherent control of single photons.
FIG. 3: (color online) Contour plot of the square |Uκ(j)|2 We note that if one puts atoms in a MOT, with the
of the bound-state wave function amplitude versus the ratio
atomicfrequencymodulatedbytheACStarkeffect, and
Ω/ν and the space coordinate j. The parameters here are
observes the light scattering, complete reflection won’t
the same as in Fig. 2. The horizontal dashed lines are the
occur at the correct ratio of modulation strength to fre-
zeros of J0(x), corresponding to extended (along j) photon
wavefunctions quency, which just like the one discussed here. However,
due to the linear dispersion of the reservoir which atoms
in a MOT interacts with, photon localization can not be
of a localized photon within the energy gap. The con- observed.
tinuity of U (j) at the boundary j = 0, i.e. Eq.(28)
κ Asweknowthat,inreality,allquantumsystemsinter-
determines the existence condition of the bound state
act with the environment, which results in the inelastic
Ω scattering of a single photon. The inelastic scattering of
g2J02(cid:18)ν(cid:19)=2ξ(ξsinh2κ±∆sinhκ). (32) photonswouldreducethetransmissionofphotonsaswell
as the quantum switching efficiency. The decoherence or
where the lower sign gives the living condition for the dissipation either influences the free propagation of the
boundstatewhichliesupperthebandwithenergyE = single photon, or broadens the width of the line shape
κ
∆+2ξcoshκ. On the resonance ∆=0, the width of the at the resonance, according to its contributions to the
bound state is determined by scattering process. It is known that photon propagates
freely in these resonators, all the dissipation factors of
sinh2κ= g2J02(Ω/ν), (33) the resonators have effect on the free propagation of the
2ξ2 single photon. However, the decay of the two-level sys-
tem influences the scattering process, since the energy of
which can be adjusted by the ratio Ω/ν through J (x).
0 photon is not conservative before and after its interact-
At the zeros of this Bessel function, κ = 0 and the pho-
ingwiththetwo-levelsystem,therefore,thewidthofthe
ton is delocalized. When off resonance, the energies of
lineshape is broadened.
the two bound states are asymmetric with respect to the
This work is supported by New Century Excel-
center of the band. In Fig. 3, we plot the single-photon
lent Talents in University (NCET-08-0682), NSFC
distribution U (j)2 as a function of the discrete coordi-
| κ | No.10935010,No.10474104,andNo.10704023,NFRPC
nate j, and the ratio Ω/ν. It shows that: (1) The wave
No. 2006CB921205, and 2007CB925204, and Scientific
packet of the single-photon spreads along the CRW as
Research Fund of Hunan Provincial Education Depart-
Ω/ν increases, because
mentNo.09B063. FNacknowledgespartialsupportfrom
J x−1/2; theNSA,LPS,ARO,NSFgrantNo.EIA-0130383,JSPS-
0 ∼ RFBR 06-02-91200, and the JSPS-CTC program. We
(2)Thewidthofthelocalizedphotonstateoscillatesfol- thank S. Ashhab for useful discussions.
lowingJ (Ω/ν);(3)Duetothismodulation-inducedZeno
0
effect, the wave packet will be extended at the zeros of
Appendix A: Quantum Zeno Dynamics of Two Level
J (x); (4) The full-spreading of the photon wave packet
0 System in An Artificial Bath
does not occur as a periodic function of Ω/ν because the
roots of J (x) are not periodic. The above investigation
0
indicates that the dynamic Zeno effect can be character- In this appendix we follows the references [31] to dis-
ized by the delocalization of photons. cuss the quantum zeno dynamics of the two level system
in the coupled-resonator waveguide (CRW), which be-
have as an artificial bath.
VII. CONCLUSION To this end, we make a Fourier transformation
1
Based on the dynamical QZE induced by a high- aˆ = eikjaˆ (A1)
j k
frequency modulation, we show how to decouple a TLS √N Xk
7
for Hamiltonian H +H in Eqs.(6) and (7). In the sub- ercase is given by
C I
space supported by the complete basis e0 , gk , the
Hamiltonian can be rewritten as {| i | i} u =U e−i(ω−2ξcosk)t,u =U e−i[ωat+Aυ sin(υt)] (A5)
k k e e
J
H = [ωk k k + kg 0e +H.c.]+Ωa(t) e e The above differential equations about Uk and Ue give
Xk | ih | √N | ih | | ih | an exact integro-differential equation for Ue(t)
(A2)
whereω =ω 2ξcosk. Therefore,thestateatarbitary t
k c− U˙ = ǫ∗(t) dτU (τ)ǫ(τ)Φ(t τ)ei∆(t−τ) (A6)
time e − Z e −
0
|φ(t)i= uj(t)|1jgi+ue(t)|0ei (A3) where the detuning ∆=ωa−ωc. Here
Xj
J2
Φ(t)= ei2ξtcosk (A7)
lies within the subspace spanned by 0e and kg . Then
N
weobtaintheequationsforamplitude|sui and|u ibysub- Xk
k e
stituting the state φ(t) into the Schr¨odinger equation
| i is the memory function or call reservoir response func-
tion, and the modulation function reads
i∂tUk = J Uee−i(ωa−ωk)te−iAυ sin(υt)
√N ǫ(t)=e−iAυ sin(υt). (A8)
i∂tUe = J Ukei(ωa−ωk)teiAυ sin(υt) (A4)
√N Intheweakcouplinglimit,theapproximationU (τ)
Xk e ≈
U (t)canbemadeontheright-handsideoftheequation
e
wheretherelationbetweenthecapitalletterandthelow- (A6). Then we have
t t
U (t)=exp dt ǫ∗(t ) dt ǫ(t )Φ(t t )ei∆(t1−t2)Θ(t t ) (A9)
e 1 1 2 2 1 2 1 2
(cid:20)−Z Z − − (cid:21)
0 0
The Fourier transformation of Φ(t)ei∆tΘ(t) is obtained The effective decay rate
J2 ∞
Φ˜(ω+∆)/2= δ(ω+∆+2ξcosk), R(t)= Φ˜(ω+∆)F (ω)dω (A12)
2N t
Xk Z−∞
which is expressed in terms of the coupling spectrum
is written as the overlap of the reservoir coupling spec-
density Φ˜(ω). The coupling spectrum is defined as the
trum and normalized spectral modulation intensity.
Fouriertransformationofthereservoirresponsefunction
Ifthestates k belongstoaspectrallydenseband,the
1 ∞ CRW forms a|‘reiservoir” spectral distribution, Φ˜(ω) =
Φ˜(ω)= Φ(t)eiωtdt. (A10)
2π Z J2ρ(ω), with spectral density given by
−∞
Further introducing the the effective interaction time π
ρ(ω) = δ(ω+2ξcosk)dk
t Z
Q(t)=Z0 dτ|ǫ(τ)|2 (A11) = 1−π∞ e−iωxJ (2ξx)dx
0
4Z
and defining the normalized spectral modulation inten- −∞
sity 0 2ξ < υ ω
| − |
t t = ∞ 2ξ =|υ−ω|
F (ω) = Q−1(t) dt ǫ∗(t )e−iωt1 dt ǫ(t )eiωt2  1/2 2ξ > υ ω
t Z 1 1 Z 2 2 √4ξ2−ω2 | − |
0 0
t 2 
= Q−1(t) dt′ǫ(t′)eiωt′ , R(t)inEq.(A12)wasobtainedbyKofmanandKurizki
(cid:12)(cid:12)Z0 (cid:12)(cid:12) forverygeneralcasestodiscussthedynamicsofZenoand
(cid:12) (cid:12) anti-Zeno effects in a heat bath. We only demonstrate
the norm of amplitu(cid:12)de reads (cid:12)
theiruniversalapproachwithoursetupforcoherentcon-
Q(t) ∞ trol of single photon transfer. The detailed calculation
U (t) =exp Φ˜(ω+∆)F (ω)dω
| e | (cid:20)− 2 Z t (cid:21) for the decay rate begins with the effective interaction
−∞
8
time Q(t)=t and the modulation spectrum The above equation shows that the decay rate is deter-
mined by: 1) the parameters of the driving field, i.e.
+∞ Ω ratio of modulation strength Ω to frequency υ. 2) the
F (ω) = J2 sinc2ω +
t n(cid:18)υ(cid:19) n overlapthereservoircouplingspectrumandthemodula-
n=X−∞ tionspectrum. Thereforethewidthandthecenterofthe
+∞ Ω Ω spectrum are important factors. Obviously, the width of
tJn1(cid:18)υ(cid:19)Jn2(cid:18)υ(cid:19)sinc2ωn1sinc2ωn2 Φ˜(ω) is less than 4ξ, the center of Φ˜(ω) lies in ωMG =0.
nX16=n2 The width of Ft(ω) is t−1, ∆ is its centra. This follow-
ing results can be obtained: 1). If ratio of modulation
where ω = (ω nυ)t/2 and sincx = sinx/x. It yeilds
n − strength Ω to frequency υ satisfy J0(Ω/υ)=0, then the
the effective decay rate
effective decay rate R(t) = 0. 2). For sufficiently long
times,i.e. tismuchlargerthanbothυ−1 andaneffective
tJ2 +∞ A sin(∆+2ξcosk+nυ)t 2 correlation time (4ξ)−1, the decay rate
R(t)= J 2 .
(cid:12) n (cid:12)
Xk N (cid:12)(cid:12)n=X−∞ (cid:18)υ(cid:19)(∆+2ξcosk+nυ)t/2(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
When no modulation is applied, i.e. A=0 and t Ω
→∞ R(t)=J2 Φ˜(∆), (A13)
0 (cid:18)υ(cid:19)
R(t)=2πΦ˜(∆)
which is the extension of the golden rule rate to the case
3). When time t υ−1, with υ ∆,2ξ, the normalized
ofatime-dependentcoupling. Inthispaper,weareinter-
∼ ≫
spectral modulation intensity is a small varying function
estedinaregimewithparameters∆,ξ ν,whichmeans
≪ over the interval 4ξ, therefore one can make the approx-
that the modulation frequency is much greater than the
imation
inverse correlation time of the continuum. The effective
decay rate reads
tJ2 Ω sin(∆+2ξcosk)t 2 R(t)=tJ2 Ω sinc2(ω−∆)t ∞ Φ˜(ω)dω.
R(t)=Xk N (cid:12)(cid:12)(cid:12)J0(cid:18)υ(cid:19)(∆+2ξco2sk)t/2(cid:12)(cid:12)(cid:12) 0 (cid:18)υ(cid:19) 2 Z−∞
Ω(cid:12)(cid:12) ∞ sin(ω−∆)t 2(cid:12)(cid:12)
=tJ02(cid:18)υ(cid:19)Z−∞Φ˜(ω)(cid:12)(cid:12)(cid:12)(ω−∆2)t/2(cid:12)(cid:12)(cid:12) dω Schinacnenetlhse, tehffeecqtuivaentduemcaZyernaoteeffisecatvegreangeerdalloyveorccaullrsd.ecay
(cid:12) (cid:12)
(cid:12) (cid:12)
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