Fate of photon blockade in the deep strong coupling regime

Alexandre Le Boite,1, ∗ Myung-Joong Hwang,1, ∗ Hyunchul Nha,2, 3 and Martin B. Plenio1

1Insitut fur Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universitat Ulm, D-89069 Ulm, Germany2Department of Physics, Texas A&M University at Qatar, Education City, P.O.Box 23874,Doha, Qatar

3School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea

We investigate the photon emission properties of the driven-dissipative Rabi model in the so-called ultrastrong and deep strong coupling regimes, where the atom-cavity coupling rate g becomescomparable or larger than the cavity frequency ωc. By solving numerically the master equation in thedressed-state basis, we compute the output field correlation functions in the steady-state for a widerange of coupling rates. We find that, as the atom-cavity coupling strength increases, the systemundergoes multiple transitions in the photon statistics. In particular, a first sharp anti-bunchingto bunching transition, occurring at g ∼ 0.45ωc, leading to the breakdown of the photon blockadedue to the counter-rotating terms, is shown to be the consequence of a parity shift in the energyspectrum. A subsequent revival of the photon blockade and the emergence of the quasi-coherentstatistics, for even larger coupling rates, are attributed to an interplay between the nonlinearity inthe energy spectrum and the transition rates between the dressed states.

PACS numbers: 42.50.Ar, 03.67.Lx, 42.50.Pq, 85.25.-j

I. INTRODUCTION

Cavity quantum electrodynamics (cavity QED) hasproved to be a powerful platform both for testing funda-mental laws of quantum physics [1, 2] and implementingquantum information protocols [3, 4]. A strong atom-cavity interaction, larger than any dissipation rates, canbe readily achieved in cavity QED setups [5–7], givingrise to interesting nonlinear effects induced by effectivephoton-photon interactions. One of the most famous isthe photon blockade effect [9], where the presence of a sin-gle photon inside the cavity is sufficient to inhibit the ab-sorption of another photon. The photon blockade resultsin a highly nonclassical emission of photons from the cav-ity, characterized by a strong antibunching. The photonblockade has been investigated theoretically and experi-mentally in a large variety of systems, such as atomic cav-ity QED [10, 11], semiconductor nanostructures [12, 13]and superconducting circuits [14–16].

The effective photon-photon interaction in a cavityQED system is well captured, in the strong couplingregime, by the Jaynes-Cummings (JC) model. Its energyspectrum has a nonlinear ladder structure consisting ofa doublet of energy eigenstates with the same numberof excitation n. This nonlinearity is responsible for thephoton blockade. More precisely, the avoided crossingbetween two states forming a doublet shows a character-istic scaling of g

√n, where g is the atom-cavity coupling

strength [17]. Therefore, the JC model predicts that, fora fixed number of excitations, the nonlinearity of the en-ergy spectrum increases with the coupling strength andthe photon-blockade effect gets more pronounced.

Meanwhile, recent experimental progress in tailoringthe light-matter interaction has made it possible to

∗These authors contributed equally to this work.

achieve a coupling strength that is comparable or evenlarger than the cavity frequency [18–25]. In this so-calledultrastrong coupling regime, the rotating-wave approxi-mation on which the Jaynes-Cummings model is based isno longer valid and thus the counter-rotating terms can-not be neglected. A description of the cavity QED sys-tem based on the Rabi model, whose spectrum and eigen-states have been a subject of intensive studies [23, 26–30],is therefore necessary. In order to investigate output pho-ton correlations, which are experimentally relevant formany setups, several works have also considered driven-dissipative scenarios [31–35]. They have shown that inthe ultrastrong coupling regime, the usual quantum op-tical master equation and input-output relations led tounphysical predictions and had to be modified.

In this context, the question of the fate of the photon-blockade effect in the ultrastrong coupling regime natu-rally arises. It was shown in Ref. [34] that there is noqualitative change in the output-photon statistics whenthe coupling strength is of the order of 10% of the cav-ity frequency, except when a bias on the qubit, whichbreaks explicitly the parity symmetry of the Rabi model,is introduced. In the latter case, an output-photon statis-tics that is significantly different from the standard sce-nario based on the JC model is predicted. The effectof counter-rotating terms on the output-photon statis-tics and the photon blockade when the coupling strengthis of the same order of the cavity frequency [25, 36–38],however, has remained largely unexplored.

In this paper, we study the photon emission proper-ties of a cavity QED system in the ultrastrong couplingregime for a wide range of atom-cavity coupling strengthg, up to several multiples of the cavity frequency ωc. Tothis end, we investigate the steady-state properties of theRabi model in a driven-dissipative setting. We considera weak excitation limit where the driving field intensityis smaller than the dissipation rates of the cavity andthe atom. By solving the master equation in the dressed

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state basis, we compute the output field correlation func-tions in the steady-state and show that, as the couplingstrength increases, the counter-rotating terms lead to abreakdown and revival of the photon blockade, followedby the emergence of a quasi-coherent photon statistics.We explain the multiple transitions in the photon statis-tics due to the counter-rotating terms, by analyzing theenergy spectrum of the Rabi Hamiltonian and the prop-erties of the dressed states. In particular, the first break-down of the photon blockade, evidenced by a sharp anti-bunching to bunching transition in the photon statistics,occurs at g ∼ 0.45ωc and it is attributed to the parityshift occurring in the energy spectrum, which induces acascaded emission of photons.

The paper is organized as follows: the theoreticalframework is presented in Sec. II. Section III is de-voted to the main numerical results and key observa-tions. More detailed discussions on the three differentphases of photon emission are divided into three subsec-tions: the break-down and revival of the photon-blockadeeffect (Secs. III A and III B respectively) and the tran-sition to a coherent regime (III C). We conclude in Sec.IV.

II. THEORETICAL FRAMEWORK

We consider the Rabi Hamiltonian, describing a singlecavity mode coupled to a two-level atom,

Hr = ωca†a+ ωaσ+σ− − g(a+ a†)σx, (1)

where we have introduced the photon annihilation op-erator a, and the Pauli matrices σx, σy (with σ± =12 (σx ± iσy)). Here, ωc is the cavity frequency, ωa theatomic transition frequency, and g the atom-cavity cou-pling strength. In the following we will focus on a reso-nant case, i.e., ωc = ωa. An important property of theRabi Hamiltonian is that the parity of the total numberof excitations, Π = exp[iπ(a†a + σ+σ−)], is a conservedquantity. It is therefore convenient to label the eigen-states of the system using the notation |Ψ±j 〉 for the jth

eigenstate (j = 0, 1, ..) of the ± parity subspace and E±jfor the corresponding energy. For example, the groundstate of Hr is |Ψ+

0 〉, which is the lowest energy state ofthe + parity subspace; the first excited state of Hr, whichcorresponds to the lowest energy state of the − paritysubspace, is |Ψ−0 〉.

We focus in this paper on a driven-dissipative scenariowhere the cavity is externally driven by a monochro-matic field and both the cavity and the atom are coupledto their environments, leading to dissipation. The totaltime-dependent Hamiltonian of the system is

H(t) = Hr + F cos(ωdt)(a+ a†), (2)

with F the intensity of the driving field and ωd its fre-quency. The dynamics of this open quantum system is

governed by a master equation of the form,

∂tρ = i[ρ,H(t)] + Laρ+ Lσρ, (3)

where the term Laρ + Lσρ describes the dissipation ofthe system excitations into the electromagnetic environ-ment. In the standard quantum optical master equa-tion, the quantum jump operators appearing in the termLaρ+ Lσρ are given by the annihilation operator of thecavity a and the atom σ−; the validity of this approachrelies on the fact that the atom and the cavity can beregarded as being independently coupled to their ownenvironment. In the ultrastrong coupling regime, thisassumption is no longer valid and the dissipation shouldbe described as quantum jumps among the eigenstates ofthe total atom-cavity Hamiltonian [33]. A correct masterequation, taking fully into account the coherent couplingbetween the cavity and the atom, can therefore be ex-pressed in the dressed-state basis {|Ψp

j 〉} with p = ±, in

which the Hamiltonian (without driving) is diagonal. Inthis basis, the dissipative part reads,

Laρ+ Lσρ =∑p=±

∑k,j

Θ(∆ppjk)(

Γppjk +Kppjk

)D[|Ψp

j 〉〈Ψpk|],

(4)

where Θ(x) is a step function, i.e., Θ(x) = 0 for x ≤ 0 andΘ(x) = 1 for x > 1, and p = −p. We have also introducedthe following notation, D[O] = OρO†− 1

2 (ρO†O+O†Oρ).

The quantities Γppjk and Kppjk denote the rates of transition

from a dressed-state |Ψpk〉 to |Ψp

j 〉 due to the atomic andcavity decay, respectively; the transition rates are definedas

Γppjk = γ∆ppjk

ωc|〈Ψp

j |(a− a†)|Ψp

k〉|2, (5)

Kppjk = κ

∆ppjk

ωc|〈Ψp

j |(σ− − σ+)|Ψpk〉|

2, (6)

where ∆ppjk = Epk −E

pj is the transition frequency and γ,

κ are respectively the cavity and the atom decay rates.Note that the transition between states belonging to thesame parity space is forbidden because both operatorsa− a† and σ− − σ+ change the parity of the state.

In the following, we will be interested in the long timedynamics of Eq. (3). Except for a sufficiently small g,Eq (3) generally does not have a particular rotating-frame where the equation becomes time-independent.Therefore, the solution has a residual oscillation at thedriving frequency ωd even in the t → ∞ limit. Thesteady-state properties are then obtained by averagingthe solution over several driving periods, which corre-sponds to a time integrated measurement in an actualexperiment [34].

3

g/ω0

10-1

100

ωd/ω

0

0.5

1

1.5

×10-3

0.5

1

1.5

2

2.5

3(a)

g/ω0

10-1

100

ωd/ω

0

0.5

1

1.5

0

0.5

1

1.5

2(b)

FIG. 1: (color online). (a) Intensity Iout as a function of g/ωc,

and ωd/ωc. (b) Second order correlation function g(2)(0). Toimprove readability, the color scale for the autocorrelationfunction saturates at g(2)(0) = 2. The dissipation and drivingparameters are γ/ωc = 10−2 and F/ωc = 10−3.

III. RESULTS

Our aim is to study the statistics of the photons emit-ted from the cavity as a function of the coupling strengthg in the weak excitation limit (F/γ, F/κ � 1). Whenthe atom-cavity coupling strength is sufficiently small,the output field is proportional to the intracavity fielda, and the correlation functions of the intracavity fieldare therefore the relevant observables for characterizingthe emitted light. This is not the case in the ultrastrongcoupling regime and the standard input-output relationis modified [31, 32, 34]. In this case, as shown in Ref. [34],

the output field is proportional to an operator X+, de-fined in the dressed-state basis as:

X+ =∑p=±

∑k,j

Θ(∆ppjk)∆pp

jk|Ψpj 〉〈Ψ

pj |i(a

† − a)|Ψpk〉〈Ψ

pk|.

(7)The second-order correlation function then reads

g(2)(0) =〈X−X−X+X+〉〈X−X+〉2

, (8)

and the intensity of the emitted photons is proportionalto Iout = 〈X−X+〉. A strong photon antibunching indi-cated by g(2)(0)� 1 will be the signature of the photon-blockade effect.

Figure 1 shows a phase diagram for Iout and g(2)(0)in the steady-state as a function of the coupling strength

g/ω0

1 2 3

I ou

t

10-4

10-3

10-2

γ/ωc = 10

-2

γ/ωc = 10

-3

(a)

g/ω0

1 2 3

g(2

) (0)

10-4

10-2

100

102

γ/ωc = 10

-2

γ/ωc = 10

-2

(b)

FIG. 2: (color online). (a) Intensity Iout and (b) Second or-

der correlation function g(2)(0) as a function of g/ωc, for anexternal driving resonant with the transition |Ψ+

0 〉 → |Ψ−1 〉.

Results are shown for two values of the dissipation rate,γ/ωc = 10−2 (solid blue line) and γ/ωc = 10−3 (dashed redlines). In both cases F/γ = 10−1.

and the driving frequency. They are obtained by solv-ing Eq. (3) numerically for a fixed driving strength anddissipation rates, F/γ = 10−1 and γ/ωc = κ/ωc = 10−2.The two dark lines in Fig. 1 (a) correspond to a reso-nant driving of the two transitions |Ψ+

0 〉 → |Ψ−0 〉 and|Ψ+

0 〉 → |Ψ−1 〉. According to the standard photon block-

ade scenario, a strong anti-bunching occurs for both driv-ing frequencies. This persists even when one enters theultrastrong coupling regime (g/ωc ≈ 10−1), where it iswell-known that the rotating-wave approximation breaksdown [34]. In this range of the coupling strength, thecounter-rotating terms make the photon-blockade effectmore pronounced as evidenced by the decreasing g(2)(0).

Interestingly, however, as the coupling strength in-creases further, entering the so-called deep strong cou-pling regime (g/ωc ≈ 1), the photon statistics of the firsttransition and the second transition become drasticallydifferent. Regarding the first transition |Ψ+

0 〉 → |Ψ−0 〉,

the emitted light remains strongly antibunched. Thisfeature can be easily understood from the fact that, inthe deep strong coupling regime, the lowest two eigen-states of the Rabi model |Ψ+

0 〉 and |Ψ−0 〉 becomes pro-gressively closer in energy as g increases, eventually form-ing an effective two-level system that is separated fromthe higher energy states by the cavity frequency ωc(forg/ωc & 1) [23]. The nonlinearity for the first transi-tion therefore increases monotonically as a function ofg, and hence the photon blockade effect becomes morepronounced. It is important to note however that the in-tensity Iout of the emitted photon goes to zero for the firsttransition for g/ωc & 1, which eventually sets a practicallimit for the observation of the photon blockade effect bydriving this transition.

Richer and more interesting properties of the emittedphotons are observed when driving the second transition.The most prominent feature is that the dip in g(2)(0) dis-appears for 0.45 . g/ωc . 1. The dip briefly reappearsfor a larger value of g before the emitted light eventu-

4

ally becomes coherent g(2)(0) = 1. For a more preciseunderstanding, we present in Fig. 2 a cut of the phasediagrams of Fig. 1 following the second transition.

Overall, in terms of statistics of the emitted photons,there appears three distinctive phases in the deep strongcoupling regime: (a) For 0.45 . g . 1, there occursa sharp transition from antibuncing to bunching leadingto the breakdown of the photon blockade (b) For 1 . g .2.5, another bunching-to-antibunching transition leads tothe revival of the photon blockade. (c) For g & 2.5,g(2)(0) converges to 1, that is the emitted light becomescoherent. This indicates that the statistics of emittedphotons reverts back to that of a linear cavity for couplingstrengths that are multiple times larger than the cavityfrequency. The mechanisms giving rise to these threedifferent kinds of photon statistics are analyzed in moredetails in the remainder of this section.

A. Breakdown of the photon blockade(0.45 . g/ωc . 1)

Generally speaking, two main conditions need to befulfilled in order to observe a photon antibunching whendriving the second transition |Ψ+

0 〉 → |Ψ−1 〉: i) anhar-

monicity of the spectrum, ii) no cascaded emission ofphotons. In order to understand the sharp transition tothe photon bunching in this range of coupling strength,we investigate the structure and the selection rules of theenergy spectrum of the Rabi model, which is presentedon Fig. 3 (a). One key element is the change in parityfor the second excited state, indicated by a vertical line.It occurs for a certain coupling strength gc ≈ 0.45ωc thatcoincides with the value where the sharp transition oc-curs in Fig. 2 (b). For g < gc, |Ψ−1 〉 is the second excitedstate, which means that both the first and second ex-cited states belong to the - parity subspace. In this case,the only decay channel available is the direct transition|Ψ−1 〉 → |Ψ

+0 〉. For g > gc, however, |Ψ−1 〉 is the third

excited state and the second excited state now belongsto the + parity subspace. An important consequence isthat, above gc, two decay channels are available: the firstone is simply the direct transition |Ψ−1 〉 → |Ψ

+0 〉 that re-

sults in the emission of a single photon, and the secondone is |Ψ−1 〉 → |Ψ

+1 〉 → |Ψ

−0 〉 → |Ψ

+0 〉.

More insight on the photon statistics can be gainedby investigating the transition rates between the dif-ferent dressed-states χ−+

ij = Γ−+ij + K−+

ij . Note thatthese transition rates depend on the energy eigenvaluesand eigenstates through Eq. (5) and they are propo-tional to the dissipation rates γ and κ. In Fig. 3 (a),the rates for all the transitions involved when pumpingthe second transition are plotted. Since the transitionrate χ−+

10 is much larger than χ−+11 , the direct transi-

tion channel, |Ψ−1 〉 → |Ψ+0 〉, is the dominant one. But

whenever the system relaxes through the second chan-nel, |Ψ−1 〉 → |Ψ

+1 〉 → |Ψ

−0 〉 → |Ψ

+0 〉, it triggers a cascaded

photon emission. Hence, this second channel introduces

10−1 100

g/ω0

0

1

2

3

E± j

[ω0]

10−1 100

g/ω0

1

2

χppij

[γ]

1 1.5 2 2.5 3g/ω0

10−4

10−3

10−2

10−1

η

(c)

FIG. 3: (color online) (a) Energy spectrum of the Rabi Hamil-tonian (without driving). Black dotted lines indicate energylevels with an even number of excitations while red solid linescorrespond to an odd number of excitations. The parity shiftof the second excited state is denoted by an horizontal line.Blue arrows show the available decay channels when drivingthe second transition |Ψ+

0 〉 → |Ψ−1 〉. (b) Transition rates be-

tween the different dressed states, χ+−00 (yellow squares), χ+−

11

(blue circles), χ−+01 (inverted green triangles) and χ+−

01 (redtriangles), as a function of the coupling strength. (c) Anhar-monicity of the Rabi Hamiltonian, measured by the quantity∆+−

21 − ∆−+10 . The intersection with the vertical line shows

the value of g for which the anharmonicity is equal to γ (here,γ/ωc = 10−2).

fluctuations in the photon statistics and is responsiblefor the strong photon bunching. The value of g for whichg(2)(0) is the highest (see Fig. 2 (b)) corresponds indeedto the value of g where the transition rate χ−+

11 is thehighest.

For a fixed value of g, the effect of the change in thelevel structure can be also seen by looking at Iout andg(2)(0) as a function of the pump frequency ωd. An exam-ple of such a plot for g/ωc = 0.75 is shown in Fig. 4. Theother parameters are the same as in Fig. 2. The two dipsin g(2)(0) correspond to the two transitions |Ψ+

0 〉 → |Ψ−0 〉

and |Ψ+0 〉 → |Ψ

−1 〉, and they match the two peaks in the

photon intensity. In terms of photon statistics, the con-trast between the first and the second transition is par-ticularly striking. Driving resonantly the first transitionresults in near-perfect antibunching, while a resonant ex-citation of the second transition leads to a very strongbunching (g(2)(0) & 10). Note that the photon bunch-ing is even more pronounced when the external laser isoff-resonant. The intensity of the photon is however ex-tremely low in this case. A similar “superbunching” be-

5

ωd/ω

0

0.5 1 1.5

I ou

t

10-10

10-8

10-6

10-4

10-2(a)

ωd/ω

0

0.5 1 1.5

g(2

) (0)

10-5

100

105

1010

γ/ω0= 10

-2

γ/ω0= 10

-3

(b)

FIG. 4: (color online). (a) Intensity Iout and (b) g(2)(0) as afunction of the driving frequency ωd, for g/ωc = 0.75. F andγ are the same as in Fig.(2).

havior for weak and non-resonant excitation has also beenpredicted in the Jaynes-Cummings Hamiltonian [39] andfor the Kerr Hamiltonian [40].

B. Revival of the photon blockade (1 . g/ωc . 2.5)

Fig. 3 (b) shows that for g/ωc & 1, the transitionrate χ+−

11 decreases and goes eventually to zero. In otherwords, the first decay process of the cascaded decay chan-nel closes. As a result, the direct transition becomesagain the only available decay channel, and the fluctu-ations in the photon statistics start to diminish. Theclosing of the second decay channel is related to the factthat the transition amplitudes χppij are proportional to the

energy spacing ∆ppij . Indeed, for g > 1, all the transition

frequencies ∆ppii , and in particular ∆−+

11 , decreases expo-nentially with g/ωc [38]. This is clearly visible on theenergy spectrum presented in Fig. 3 (a); for large valuesof g/ωc, the energy spectrum can be seen as a collectionof doublets of quasi-degenerate eigenstates {|Ψ−j 〉, |Ψ

+j 〉}.

At the same time, the spacing between the doublets ofquasi-degenerate states becomes constant (equal to thecavity frequency ωc) when g increases, suppressing theanharmonicity of the spectrum. The resulting photonstatistics in this regime thus depends on an interplay be-tween the closing of the second decay channel and thevanishing anharmonicity. More specifically, a revival ofthe photon blockade effect can occur if the second decaychannel is closed while a sufficiently large anharmonicitystill remains in the energy spectrum. Fig. 3 (c), where theanharmonicity η = ∆+−

21 −∆−+10 is compared to the dissi-

pation rate γ as a function of g, shows that this is indeedthe case for 1 . g/ωc . 2.5, providing a clear explana-tion for the revival of the photon blockade. Note that thismechanism also explain why the region for which photonantibunching is recovered becomes larger for smaller val-ues of γ; see the red dotted line in Fig. 2 (b).

C. Reversion to non-interacting photons(g/ωc & 2.5)

The photon correlation in this regime converges tog(2)(0) = 1. It is interesting to observe that a verystrong atom-cavity coupling leads to a poissonian statis-tics, which is characteristic of photons emitted from alinear cavity. To understand this rather counter-intuitiveobservation, it is illuminating to consider the limitingcase of g/ωc → ∞ [38]. In this limit, the energy spec-trum is that of an displaced harmonic oscillator andtherefore it is harmonic with a characteristic frequencyof ωc. The doubly degenerate j-th eigenstates are given

by |j,±g/ωc〉|±〉 where |j, α〉 = eαa†−α∗a|j〉, with α a

complex number, is the j-th displaced Fock state, andσx|±〉 = ±|±〉. For large but finite values of g/ωc, theterm ωaσ+σ− lifts the degeneracy between |j,±g/ωc〉|±〉,leading to an exponentially supressed energy splitting,

〈n,−g/ωc|n, g/ωc〉 ∝ e−2g2/ω2cLn(4g2/ω2

c ), where Ln isthe nth Laguerre polynomial.

If the exponentially suppressed anharmonicity of thespectrum for g/ωc � 1 is smaller than the dissipationrate γ, the system will essentially behave as a linear sys-tem, and the emitted light will show a Poissonian statis-tics. Fig. 3 (c) clearly shows that the anharmonicityη = ∆+−

21 − ∆−+10 becomes comparable to the dissipa-

tion rate γ at g ∼ 2, and it becomes smaller for largervalues of g. This is consistent with the range of couplingstrength where the reversion to non-interacting photonsoccurs, i.e., g/ωc & 2.5. We emphasize that the mech-anism behind the breakdown of the photon blockade inthis regime differs fundamentally from the one reportedin section III A. In the latter case it occurs due to a par-ity shift in the spectrum and gc does not depend on γ orany other external parameter (see Fig. 2 (b)).

IV. CONCLUSION

In this paper, we have investigated the output photonstatistics of a cavity coupled to a two-level atom inthe ultrastrong and deep strong coupling regimes. Wehave shown that, when the second available transition isdriven in the weak excitation limit, the photon-blockadeeffect observed in the Jaynes-Cummings regime disap-pears for 0.45 . g . 1, which we attributed to a parityshift in the energy spectrum of the Rabi Hamiltonianthat induces a cascaded emission of photons. We havealso found that for 1 . g/ωc . 2.5 the second-orderautocorrelation function exhibits an oscillatory behavior,leading to a revival of the photon blockade in this rangeof parameters. This revival, characterized by a strongphoton antibunching, takes place when the cascadeddecay channel closes while the energy spectrum remainsanharmonic. For a even larger coupling strength, theanharmonicity of the spectrum becomes smaller thanthe dissipation rates, and as a consequence, we observedthat the emitted light is coherent. For the dissipation

6

rates considered here, this occurs for g/ωc & 2.5. Theregime of parameters considered here is reachable withcurrent technologies and the different regimes of photonemission presented in this paper could therefore beobserved in circuit QED experiments.

This work was supported by the EU IntegratingProject SIQS, the EU STREPs DIADEMS and EQUAM,the ERC Synergy Grant BioQ and Alexander von Hum-boldt Professorship as well as the DFG via the SFBTRR/21 and SPP 1601.

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