Probabilistic vortex crossing criterion for superconducting nanowiresingle-photon detectors

Saman Jahani,1, 2 Li-Ping Yang,1 Adrian Buganza Tepole,3 Joseph C. Bardin,4 Hong X. Tang,5 and ZubinJacob1, a)

1)School of Electrical and Computer Engineering and Birck Nanotechnology Center, Purdue University,West Lafayette, IN 47907 USA.2)Current Address: Moore Laboratory, Department of Electrical Engineering, California Institute of Technology,Pasadena, CA 91125, USA.3)School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA.4)Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst,MA 01003 USA.5)Department of Electrical Engineering, Yale University, New Haven, CT 06511,USA.

Superconducting nanowire single-photon detectors have emerged as a promising technologyfor quantum metrology from the mid-infrared to ultra-violet frequencies. Despite the recentexperimental successes, a predictive model to describe the detection event in these detectors isneeded to optimize the detection metrics. Here, we propose a probabilistic criterion for single-photon detection based on single-vortex (flux quanta) crossing the width of the nanowire.Our model makes a connection between the dark-counts and photon-counts near the detectionthreshold. The finite-difference calculations demonstrate that a change in the bias currentdistribution as a result of the photon absorption significantly increases the probability ofsingle-vortex crossing even if the vortex potential barrier has not vanished completely. Weestimate the instrument response function and show that the timing uncertainty of thisvortex tunneling process corresponds to a fundamental limit in timing jitter of the clickevent. We demonstrate a trade-space between this intrinsic (quantum) timing jitter, quantumefficiency, and dark count rate in TaN, WSi, and NbN superconducting nanowires at differentexperimental conditions. Our detection model can also explain the experimental observationof exponential decrease in the quantum efficiency of SNSPDs at lower energies. This leadsto a pulse-width dependency in the quantum efficiency, and it can be further used as anexperimental test to compare across different detection models.

I. INTRODUCTION

Advancements in quantum technologies strongly de-pends on improvement in the detection of light at thesingle-photon level. This requires near-unity quantumefficiency, sub-picosecond timing uncertainty (timing jit-ter), sub-milihertz dark count rate, large bandwidth, andfast reset time1. Superconducting nanowire single photondetectors (SSPDs or SNSPDs) are highly promising de-tectors in a broad range of frequencies from mid-infraredto ultraviolet2–7 with near unity quantum efficiency8,picosecond-scale timing jitter9–12, fast reset time13, andmilihertz dark count rate14,15. They are composed of athin superconducting nanowire which is biased slightlybelow the superconducting critical current. Photon ab-sorption triggers a phase transition giving rise to gener-ation of a voltage pulse which is measured by a readoutcircuit connected to the nanowire.

Owing to the experimental progress on reducing theamplification noises and the uncertainty of the photonabsorption location, recent breakthrough results haveshown timing jitter below 10 ps9–11,16,17. Hence, the re-sponse function of SNSPDs to a single-photon has ap-proached its intrinsic response limit which only dependson the microscopic mechanism of light-matter interac-tion in nanowires. To further improve the performance

a)Electronic mail: zjacob@purdue.edu

of these detectors, it is required to understand the mi-croscopic mechanism and the trade-space of the photondetection event in these detectors.

Over the past two decades, several important detectionmodels have been proposed to explain the microscopicmechanism of the formation of the first resistive regionin SNSPDs2,3,18–23. In the simplest model, it is assumedthat the energy of the absorbed photon increases the tem-perature at the absorption site leading to the nucleationof a hot-spot which causes the current to be directed tothe sides2,24,25. This may cause the current at the edge tosurpass the critical depairing current causing formationof a normal conducting region across the width. In an-other model, the depletion of superconducting electronsaround the absorption site is responsible for the forma-tion of the resistive region26. Recently, some models havesuggested that the motion of vortices or vortex-antivortexpairs can also induce a phase transition at a lower appliedbias current18,21,23,27–29. Although each of these modelsexplain most of the macroscopic behaviors of SNSPDs,existing models cannot explain or predict the trade-offbetween the quantum efficiency, timing jitter, and darkcounts and their fundamental limits in SNSPDs.

In this paper, we construct a connection between thephoton-induced counts and the dark counts in SNSPDs,which has been recently observed experimentally aroundthe detection current30,31. We propose a probabilisticcriterion for single-photon detection corresponding to thesingle-vortex crossing from one edge of the nanowire to

arX

iv:1

901.

0929

1v3

[ph

ysic

s.in

s-de

t] 3

1 M

ar 2

020

2

a b

xy

0Φ

0Φ

Vortexbarrier

hν

Vortex crossing

Bk T

Figure 1. Superconducting nanowire single photondetectors (SNSPDs). a, When a photon falls on the detec-tor, quasi-particles (QPs) are generated and the bias currentis redistributed. Vortices with magnetic flux quantum of Φ0

are the topological defects of a thin superconductor and arenucleated at the nanowire edge. They can move to the otheredge due to the force exerted by the bias current. b, Beforethe photon absorption, the vortex potential barrier does notallow them to move easily. However, due to the QP multi-plication and current redistribution, the barrier reduces andvortices which are thermally excited can escape the barrierand cross the width of the nanowire. This process generatesa voltage pulse propagating to the two ends of the detector.We provide a probabilistic click definition using this detectionevent.

the other edge. First, we numerically calculate the time-dependent current distribution after the photon absorp-tion and its effect on the vortex potential. We proposethat due to the change in the distribution of the super-conducting electrons, the probability of the vortex cross-ing is significantly enhanced even if the vortex potentialbarrier has not vanished completely.

Then, we define the detection probability based on theprobability of the single-vortex crossing, because the en-ergy released by one vortex moving across the width isenough to induce a phase transition in the superconduct-ing nanowire. We show that the probabilistic behaviorof the single-vortex crossing results in an intrinsic timingjitter on the click event. This intrinsic quantum timingjitter cannot be eliminated even if the geometric positionof photon absorption is known, however, it can be re-duced by engineering the structure and the experimentalconditions at the cost of a degradation of the quantumefficiency and/or an increase in the dark count rate.

Finally, we calculate the quantum efficiency spectrumand show that the quantum efficiency does not suddenlydrop to zero when the photon energy is below a thresh-old. We propose that the response of the detector to thephoton pulse-width can be different for the various detec-tion models. Moreover, the quantum efficiency predictedby our model is strongly dependent on the pulse-width.This effect has not been predicted by the previous detec-tion models. Our work unifies previously known ideas ofvortex crossing phenomenon with the POVM approach ofquantum optics to propose a probabilistic detection crite-rion for SNSPDs. We propose some observable quantitieswhich can be used to experimentally verify the validity ofour probabilistic model. Our model focuses around thedetection threshold (quantum-efficiency≈1) where pho-tons do not have enough energy to form a normal con-ducting hot-spot and the probabilistic behavior of vor-tices is more significant. This is not in contradictionto observations of the vortex/anti-vortex pair unbinding.For higher energies or higher bias currents, the formation

of a hot-spot and, as a result, vortex/anti-vortex pair un-binding might happen before the probabilistic tunnelingof a single-vortex from the edges23,32.

II. DETECTION MECHANISM

Detection mechanism in SNSPDs consists of threesteps: (a) photon absorption and breaking the supercon-ducting electron pairs (known as Cooper pairs) to quasi-particles (QPs) leading to formation of a hot-spot; (b) asa result of the depletion of the Cooper pairs, the super-conducting order parameter is suppressed. This causesthe current density at the absorption location to be re-duced and directed to the sides as illustrated in Fig. 1a;(c) the change in the Cooper pairs and current density re-duces the vortex potential barrier and vortices can moveacross the nanowire and release a measurable voltagepulse (Fig. 1b).

These three steps have been quantitatively described inthe appendix. Our finite-difference calculations of QPsdistribution based on the diffusion model21 for a TaNSNSPD is illustrated in Figs. 2a and 2b at t = 1 ps andt = 5 ps, respectively. We assume a photon with theenergy of hν = 1.5 eV falls at the center of the SNSPDat t = 0. The width and the length of the nanowire areW = 100 nm and L = 1000 nm, respectively. Figures 2cand 2d display the numerical calculation of the currentdensity normalized to the bias current. It is seen thatdue to the hot-spot formation at the center, the currentis directed to the side-walls of the nanowire, and as aresult, the vortex potential barrier is reduced as shownin Fig. 3. If the bias current or the photon energy are highenough, the potential barrier can be vanished completely.

After the single-photon transduction, several processescompete with each other to form the initial normal con-ducting cross-section. Depending on which one occursfirst, different detection models have been proposed. Inhot-spot model, it is assumed that the formation of hot-spot is responsible for the phase transition2,25. Nucle-ation of the hot-spot causes the bias current to be di-rected to the side-walls. If the current density at the edgesurpasses the despairing critical current (Iedge ≥ Ic,dep),it induces a phase transition to the normal conductingstate at the edge and the normal conducting region ex-pands across the width.

In QP model, there is no need to destroy the super-conductivity by surpassing the critical current26. If theCooper pairs are depleted inside a volume with a thick-ness of at least one coherence length (ξ-slab), the phasecoherence is destroyed which results in a phase transition.This requires the number of QPs inside the ξ-slab (Nslab

QP )to exceed the number of the superconducting electrons in-side the ξ-slab: Nslab

QP /Nslabse ≥ 1− Ib/Ic,dep, where Nslab

se

is the initial superconducting-electron number inside theslab before applying the bias current (Ib).

Vortices can also be responsible for the trigger of asingle-photon induced phase transition in SNSPDs. If thephoton transduction causes the vortex potential barrier(Uv) to vanish, vortices move across the width and inducea phase transition20,21. In the next section, we show that

3

t = 1 ps t = 5 ps

t = 1 ps t = 5 ps

W = 100 nm

a b

0.4

0.6

0.8

1

1.2

0.1

0.3

0.5

0.7

c d

Figure 2. a, and b, QP distribution, Cqp(~r, t), normalizedto the initial density of superconducting electrons, nse,0, ina TaN SNSPD at t = 1 ps and t = 5 ps, respectively. It isassumed the photon is absorbed at t = 0. T = 0.6 ◦K. Thephoton energy is hν = 1.5 eV. The nanowire width, length,and thickness are 100 nm, 1000 nm, and 5 nm, respectively.c, and d, Normalized Current density in the y direction att = 1 ps and t = 5 ps, respectively. The current density is nor-malized to the applied bias current. The arrows represent thecurrent density vector, ~j(~r, t). Due to the hot-spot formation,current is redistributed and directed to the side walls.

even if the barrier has not completely vanished and thekinetic energy of the vortices is not enough to surmountthe barrier, there is a considerable probability of single-vortex crossing. This quantum tunneling process causesa new source of timing jitter for the detection event.

III. QUANTUM TIMING JITTER

According to the most accepted quantum measure-ment theory, positive-operator-valued measure (POVM),a quantum detector can be regarded as a black box.Each of its outcomes is represented by a positive Her-mitian operator Πm with non-negative real eigenvalues.The probability that the mth outcome occurs in exper-

-45 -40 -35xv (nm)

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t=0t=1 pst=5 pst=15 ps

0 5 10 15t (ps)

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Uv,

max

a

b

Figure 3. Vortex barrier dynamics a, Vortex potential asa function of the vortex location, xv, around the saddle pointafter the photon absorption for a TaN SNSPD. T = 0.6 ◦K,W = 100 nm, and hν = 1.5 eV. b, Potential barrier peak asa function of time. A change in the Cooper pairs density andcurrent distribution reduce the barrier hight. The potentialhas been normalized to the characteristic vortex energy, ε0.

iments is given by Pm = Tr[ρΠm], where ρ is the ini-tial state of the quantum object to be detected, such asthe state of the incident single-photon pulse. The com-pleteness condition,

∑m Πm = I (I is the identity op-

erator), expresses the fact that the probabilities sum toone:

∑m Pm = 1. For a non-photon-number-resolving

photon detector, there are only two possible outcomes:clicking and non-clicking, characterized by Πc and Πnc

(thus m = c, nc), respectively. The clicking probabil-

ity, Pc = Tr[ρΠc] ≡ P1 + P0, contains two parts: thesingle-photon induced clicking probability, P1, charac-terizing the quantum efficiency of the detector and thedark counting part, P0. Note that we have neglected thenonlinearity in the detector response34,35. Recently, thefigures of merit and time-dependent spectrum of a sin-gle photon in terms of POVMs have been exploited36,37.In the following, we present our microscopic calculationof P1 and P0 based on the single-vortex crossing model.Especially, we introduce the quantum timing jitter in theamplification process, which has not been incorporated

4

NbN

TaN

a

b

0 5 10 15t (ps)

0

100

200

300

400Γ v

(GH

z)WSi

0 5 10 15t (ps)

0

0.1

0.2

0.3

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0.5

P 1

WSi

TaN

NbN

Figure 4. Vortex crossing rate and probability. a,Single-vortex crossing rate as a function of time for NbN, TaN,and WSi SNSPDs. The bias current has been set to achieve asingle photon detection probability of 0.5 for a photon energyof hν = 1.5 eV, which is 0.96ISW, 0.93ISW, and 0.65ISW,for NbN, TaN and WSi SNSPDs, respectively. The reducedtemperature (T/Tc) for TaN, NbN, and WSi at T = 0.6 ◦Kare 0.07, 0.05, and 0.15. W = 100 nm. ISW is defined asthe minimum bias current at which the detector clicks in thetime-bin of the single-photon arrival even if the photon isnot absorbed. Material parameters are derived from ref.33.The enhancement in the vortex crossing rate is as a resultof the suppression of the potential barrier. The probabilityof vortex crossing at the maximum rate is higher. However,there is considerable uncertainty in the vortex crossing timewhich results in a timing jitter in detection event. b, Theevolution of vortex crossing probability as the vortex crossingrate changes. There is a steep change in the probability asthe crossing rate goes up.

into current POVM theory.Even if there is no photon and the bias current is

below the vortex critical current Ic,v, a vortex can bethermally excited and escape the potential barrier sad-dle point to form a normal conducting belt38,39. Thisfalse-count rate is known as dark-count rate which de-teriorates the performance of a single-photon detector1.The time-dependent rate of the vortex crossing can bedescribed as:

Γv(t) = αvIb exp(−Uv,max(t)/kBT ), (1)

where kB is the Boltzmann constant and αv is a con-

10 15t (ps)

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aize

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unts

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trum

ent r

espo

nse

func

tion) t j

1.5h eVυ =

0.75h eVυ =

Ib=0.98ISW

Ib=0.96ISW

Ib=0.94ISW

Ib=0.92ISWLatency

Figure 5. Instrument response function. Estimateddistribution of number of counts registered on the detectoras a function of the delay after the photon absorption in aTaN SNSPD when the incoming single-photon energy is hν =1.5 eV (blue) and hν = 0.75 eV (red); T = 0.6 ◦K and W =100 nm. There is a latency between the photon absorptionand the click registration and the uncertainty in the latencycauses a timing jitter (tj) in the detector.

stant which is measured experimentally38. Uv,max(t) isthe maximum of the potential barrier for vortex cross-ing which changes with time after the single photon ab-sorption event. As a result of the change in the vortexpotential barrier after the photon absorption, the vortexcrossing rate increases exponentially. Figure 4a shows thevortex crossing rate as a function of time after the pho-ton absorption for three different materials. The rate att = 0 corresponds to the dark-count rate38,40–42. How-ever, the rate is enhanced several orders of magnitudewhen the potential barrier reaches its minimum. Thisenhancement during the multiplication and recombina-tion of QPs might be enough to significantly change theprobability of vortex escaping the barrier. It is seen thatthe shape of the vortex crossing rate is different in differ-

5

0.85 0.9 0.95 1Ib/ISW

0

1

2

3

4

5La

tenc

y (p

s)

1.5h eVυ =0.75h eVυ =

Figure 6. The effect of bias current on the latency inTaN SNSPDs. The latency suddenly drops when the quan-tum efficiency approaches unity, and it can be significantlyreduced if the bias current becomes very close to the switch-ing current (ISW ) or the photon energy increases. T = 0.6 ◦Kand W = 100 nm.

ent superconducting materials depending on the numberof QPs generated and how fast they get multiplied, dif-fused across the width, and recombined. Since the cross-ing of vortices occurs independent of the other vortices,the crossing events can be regarded as a Poisson processwith distribution function,

p(nv, t) =nv(t)

nv!e−nv(t), (2)

characterizing the probability of nv vortex crossing thenanowire during the time interval [t0, t]. Here, the time-

dependent function nv(t) =∫ tt0

Γv(t′)dt′ is the mean

number of vortices crossing the nanowire. Hence, we candefine the single-photon detection probability P1 afterthe single-photon absorption (t0 = 0) and before time tas the probability of crossing of at least one vortex as:

P1(t) = 1− p(0, t) = 1− exp

[−∫ t

t0=0

Γv (t′) dt′]. (3)

We have neglected the interaction between vortices dur-ing the vortex crossing.

As seen in Fig. 4b, the detection probability increasesrapidly around the crossing rate maximum. The timederivative of P1(t) is proportional to the single-photoncount rate (also known as instrument response function)measured in experiments9. If the detection efficiency islow, the photon count rate is approximately the same asΓv(t). The rise time of the quantum efficiency is not in-stantaneous due to the finite diffusion speed of the QPsand the hot-spot formation. Hence, there is a fundamen-tal latency and uncertainty between the time of photonabsorption and the quantum vortex tunneling process43.This causes a quantum timing jitter (tj) on photon de-tection event as illustrated in Fig. 5. As shown in Fig. 6,the latency is lower for higher energy photon detectionsince the vortex barrier is suppressed faster. The latency

a

b

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NbN

WSi

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r (ps

)

40 50 60 70 80 90 100Width (nm)

0

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r (ps

)

TaN

NbN

WSi

Figure 7. Timing jitter corresponding to vortex cross-ing. Timing jitter in NbN, TaN, and WSi SNSPDs as a func-tion of (a) temperature (W = 100 nm) and (b) nanowirewidth (T = 0.6 ◦K. hν = 1.5 eV. Decreasing the tempera-ture results in a sharper change in the vortex crossing rate.Hence, the uncertainty of vortex crossing reduces. Reducingthe width of the nanowire causes the QPs to distribute fasteracross the width of the nanowire, and as a result, the potentialbarrier reduces rapidly.

is also reduced if the bias current is increased. There isa sharp drop in the latency when the photon detectionbecomes deterministic.

This type of timing jitter is because of the probabilis-tic tunneling of vortices44,45, which is nonzero even if theabsorption location of the photon is known exactly. Inthe state-of-the-art experiments, the total jitter is dom-inated by the geometric jitter as a result of the uncer-tainty in the position of the transduction event32,46–51.However, even if the geometric jitter is suppressed bydefining the exact longitudinal6,52 and transverse loca-tion of the photon absorption, the quantum jitter cannotbe diminished beyond a limit. However, it can be con-trolled by engineering the structure and controlling theexperimental conditions. Note that to make a quantita-tive comparison between the simulation results and theexisting experimental results, the polarization of incidentphoton and the absorption location in the transverse co-

6

0.8 0.85 0.9 0.95 1Ib/ISW

10-10

10-5

100

P 0 & P

1

P0

P1

tj

0

2

4

6

8

Jitte

r (ps

)

Figure 8. The effect of bias current on TaN SNSPDperformance. a, Timing jitter, dark count probability (P0),quantum efficiency (P1) versus the bias current (Ib). Ib isnormalized to the switching current. Note that ISW is smallerthan Ic,v. Increasing the bias current helps to improve thedetection probability and the timing jitter but at the costof an increase in the dark count probability. T = 0.6 ◦K,W = 100 nm, and hν = 1.5 eV.

ordinate must be considered53. This will be undertakenin a future study.

0 0.5 1 1.5hν (eV)

10-10

10-5

100

P 1

0.6 K1 K

2 K

Figure 9. Single-photon detection probability (quan-tum efficiency) as a function of single-photon energyand temperature in TaN SNSPDs. The bias current isset to have near unity detection probability when the photonenergy is larger than 1 eV. W = 100 nm. For high energy pho-tons, the vortex potential barrier drops to zero. Hence, theclick event which is as a result of the vortex crossing happenscertainly. However, if the photon energy is not high enough tosuppress the potential barrier completely, the vortex crossingevent becomes non-deterministic, and it drops exponentiallyas the photon energy is reduced.

It is seen in Eq. (1) that as the temperature de-creases, the change in the vortex crossing rate becomessharper. This causes the intrinsic timing jitter to re-duce as shown in Fig. 7a which is in agreement withrecent experiments54. Note that the vortex potentialis proportional to the characteristic vortex energy, ε0.Thus, smaller vortex energy in causes a slower change

in the vortex crossing rate, similar to the effect of risingthe temperature. Hence, although the hot-spot forma-tion and relaxation happens faster in WSi due to thesmaller bandgap and faster QP diffusion33, the timingjitter in WSi is comparable with that in TaN because ofthe smaller ε0 in WSi nanowires.

Reducing the width of the nanowire results in a fasterdistribution of the hot-spot across the nanowire width.This leads to a sudden change in the vortex potentialbarrier, and as a result, the timing jitter decreases con-siderably as shown in Fig. 7b. Reducing the width helpsreducing the geometrical timing jitter as well46, however,at the cost of a decrease in the transduction efficiency ofthe device.

Increasing the bias current reduces the vortex poten-tial and increases the vortex crossing rate. This causesnot only an increase in the quantum efficiency (P1)55,56,but also an increase in the dark count probability (P0)as shown in Fig. 8. P0 is defined as the probability of theclick while there is no interaction between the photon andthe detector in the time bin of the photon arrival. ISW

is defined as the minimum bias current which is requiredfor at least one vortex to escape the barrier in the timebin of the photon arrival. Note that ISW is lower thanthe vortex critical current, Ic,v, especially if the temper-ature is not low enough. The bias current has also asignificant impact on the timing jitter corresponding tosingle-vortex crossing. If the detector is biased very closeto the switching current, a small perturbation due to thesingle-photon absorption suppresses the potential barrierand vortex can cross the width. Figure 8 displays the ef-fect of the bias current on the timing jitter as well. Itis seen that the timing jitter drops remarkably when thequantum efficiency approaches unity in agreement withthe recent experimental observations10,57,58. This is be-cause of the significant suppression of the barrier whichleads to the vortex crossing even before the rate reachesits maximum. It is also seen that the slopes of P0 andP1 are identical in non-deterministic region in agreementwith the recent experiments30,31, which confirms the con-nection between the photon counts and the dark countsaround the detection current.

IV. SPECTRAL QUANTUM EFFICIENCY

The non-deterministic behavior of vortices in the casewhen the potential barrier has not vanished completelyallows us to estimate the quantum efficiency probabilityeven for low energy photons. Figure 9 shows the quan-tum efficiency based on the single-vortex crossing modelas a function of the energy of the absorbed single-photonat different temperatures. The bias current is set to havea near unity quantum efficiency when the photon energyis larger than 1 eV. αv in Eq. (1) can be used as a fit-ting parameter to define the actual current with respectto the experimental depairing current. The quantum ef-ficiency approaches P0 when the photon energy goes tozero. As the photon energy goes up, more changes in QPand current distributions are observed. This causes fur-ther suppression of the vortex potential barrier leading

7

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∆ =

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Vortex model

2 4 6 8 101

1.02

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(t)/I b

0 2 4 6 8 10t (ps)

0

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(t)/N

seslab

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Hz)

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0 2 4 6 8 10t (ps)

Increasing

Energy

Increasing

Energy

Increasing

Energy

20 40 60 80 100f (fs)

10-2

10-1

100

P 1

Figure 10. Energy-dependence and pulse-width dependence of the quantum efficiency. a, Normalized QP numbersinside the ξ-slab, b, normalized current density at the edge, and, c, vortex crossing rate for different modes of a photon pulsewith the central energy of hν = 0.75 eV and a pulse-width of τf = 100 fs; T = 0.6 ◦K and W = 100 nm. The insets illustratethe schematic and the detection criteria for each model. It is seen that the number of QPs and current at the edge is not verysensitive to the small change of the photon energy. However, the single-vortex crossing rate is extremely sensitive as a resultof a few percent change in the photon energy. d, Single-photon quantum efficiency versus the pulse-width in vortex model.The quantum efficiency is considerably increased for the very short pulses. This experimental test can verify the validity of ourmodel.

to a higher probability of the vortex crossing. This inturn results in a higher quantum efficiency. Our vortexmodel by itself can explain both constant efficiency athigh energies and exponential decrease of quantum effi-ciency at lower energies seen in experiments59,60. Notethat the photon absorption efficiency is assumed to beone over the entire spectrum. In practice, the absorptionefficiency of a bare nanowire is not very high and doesnot vary significantly at optical frequencies. However,to increase the absorption efficiency, the detector shouldbe placed inside a high-Q cavity61–65 or a low-mode sizewaveguide8,66–70 to enhance the spatial overlap betweenthe optical mode of the incoming photon and the super-conducting electrons of the detector.

Till now, we have assumed in our model that the in-coming photon is a single-mode photon. However, inpractice, the photon has a finite pulse-width and thebandwidth of the photon may affect the performance ofa detector. The response of the detector to a broadbandphoton can be used as an experimental test to compareacross different detection models and verify our theory.In Fig. 10a-10c, we have compared the detection criteriain different models in response to the different modes of

a multi-mode single-photon pulse with central energy ofhν = 0.75 eV and a pulse width of τf = 100 fs. Num-ber of QPs and current at the edge are the main quan-tities to define detection criteria in hot-spot model andQP model21. As shown in Fig. 10a and 10b, the detec-tor performance is not very sensitive when the photonenergy is slightly changed around the central frequencyof the photon. However, as shown in Fig. 10c, a smallperturbation in the photon energy can make a consider-able change in the single-vortex crossing rate since therate exponentially changes with the vortex potential en-ergy. Figure 10d displays the effect of pulse-width on thequantum efficiency in our model. It is seen that there isa remarkable change in the quantum efficiency for ultra-short single-photon pulses. This effect arises due to theexponential tail of the quantum efficiency and clearly dif-ferentiates the proposed vortex model from the existingdetection mechanisms. A controlled experiment can ver-ify whether our model is correct or not.

8

Table I. Experimental tests and observables to verifythe validity of a detection model.

Experimental test Observables

Quantum efficiency spectrumExponential decrease at

low energies

Response function

Latency vs. bias current and

photon energy and

shoulder at threshold

Timing jitter vs. bias current Shoulder at threshold

Broadband single photonPulse-width dependency of

quantum efficiency

V. CONCLUSION

In summary, we have proposed a probabilistic detec-tion criterion in SNSPDs based on a single-vortex mov-ing across the width of the detector. We have shownthat even for a non-vanishing vortex potential barrier,there is a significant enhancement in the rate of the vor-tex crossing after the photon absorption leading to anincrease in the click probability. This non-deterministicprocess insets a considerable intrinsic timing jitter to thedetection event. We have shown the trade-space of thetiming jitter, quantum efficiency, and dark counts for dif-ferent superconducting materials and different nanowirestructures. We have presented the quantum efficiencyspectrum based on our model, which can predict a pulse-shaped dependent quantum efficiency in SNSPDs. Thiseffect is negligible in other proposed models. Our modelcan predict some observables illustrated in Table I whichcan be verified experimentally to confirm or reject ourmodel. It applies specifically to probabilistic (quantum)sources of jitter and further experiments are needed todistinguish such sources from the dominant geometric jit-ter.

Appendix A: Detection mechanism formalism

To find the time-dependent current and QP distribu-tions, we use a modified semi-classical diffusion modelwhich has been originally proposed by Semenov et. al71

and developed by Engel and Schilling21,33.

1. Quasi-particle multiplication

We assume the photon energy (hν) is considerablylarger than the superconducting bandgap (∆), yet notlarge enough to make a phase transition and form a nor-mal conducting core at the position of the photon absorp-tion. Hence, when the photon is absorbed, a hot electronwith a probability density of Ce(~r, t) is created. Since thephoton energy is usually orders of magnitude larger thanthe bandgap, when the hot electron diffuses, it breaksa large number of Cooper pairs (> 100 in the visiblerange) to QPs with a distribution density of Cqp(~r, t).This causes the hot electrons to lose their energy, and

Table II. Material parameters near zero temperatureused in simulations

∆ De Dqp ξ λ τr τqp ς

(eV) (nm2/ps) (nm2/ps) (nm) (nm) (ps) (ps) (%)

TaN 1.3 8.2 60 5.3 520 1000 1.6 25

NbN 2.3 7.1 52 4.3 430 1000 1.6 25

WSi 0.53 10.3 75 8 1400 1000 1.6 25

as a result, the multiplication process slows down with alife-time of τqp due to electron-phonon interaction21:

∂Ce(~r, t)

∂t= De∇2Ce(~r, t), (A1)

∂Cqp(~r, t)

∂t= Dqp∇2Cqp(~r, t)−

Cqp(~r, t)

τr

+ςhν

∆τqp

(nse,0 − Cqp(~r, t)

nse,0

)e−t/τqpCe(~r, t),

(A2)

where De, Dqp, τr, and nse,0 are the hot-electron dif-fusion coefficient, quasi-particle diffusion coefficient, re-combination time, and density of superconducting elec-trons before the photon absorption, respectively. ς isthe QP conversion efficiency which has been assumedconstant. We add the term (nse,0 − Cqp(~r, t)) /nse,0 toinclude the saturation of QP multiplication. We haveignored the electron-phonon and phonon-phonon inter-actions which are considerably slower than the electron-electron interactions21,72. The exact solution of theabove equation in a general form is not easy to derive.Hence, to find the solution numerically, we have useda Finite-Difference Crank-Nicolson method. Since, thehot-electrons diffuse quickly (De � Dqp), to speed-upthe simulations, we have used the analytical solution ofEq. (A1) for the case of an infinite 2D superconductor21.We have assumed a Gaussian distribution for the elec-tron, which is a delta function at t = 0 and the electrondiffuses for t > 0. A grid size of ∆x = ∆y = 1 − 3 nmand a time step of Dqp∆t

/∆x2 = 0.01 is used in our

simulations. Neumann boundary condition for the side-walls and zero-flux at the two ends of the nanowire havebeen considered. The material parameters can be derivedfrom experimental measurements33,38. The parametersthat we have used in this work are listed in Table II.

2. Current redistribution

The current distribution can be calculated by combin-ing superconducting phase coherence condition and con-tinuity equation73:

∇.(~j(~r, t)) = ∇.(

~mnse(~r, t)∇ϕ(~r, t)

)= 0, (A3)

where nse(~r, t) = nse0 − Cqp(~r, t) is the density of su-perconducting electrons after the photon absorption, ϕis the phase of the superconducting order parameter, mand ~ are the electron mass and reduced Planck constant,respectively.

9

3. Single vortex crossing

Vortices and antivortices are the topological defects inthin superconducting films which exist even if there isno applied magnetic field73. Vortices are usually nucle-ated and enter into the nanowire from the edge where thesuperconducting order parameter is suppressed. Londonequation in the presence of a static vortex in a supercon-ducting thin film in xy plane can be written as73,74:

~H(r) + 2πΛ

c∇×~j(r) = zΦ0δ (~r − ~rv) , (A4)

where Λ = 2λ2/d is the Pearl length75, λ is the Londonpenetration depth, d is the film thickness, Φ0 = hc/2eis the magnetic flux quantum due to the presence of a

single-vortex at the position ~rv, ~H is the magnetic field,~j is the current density ignoring the effect of the vortex onthe current, and c is the speed of light in vacuum. Sincethe thickness of the nanowire is significantly smaller thanλ, we have averaged the field and the current in the zdirection. For nanoscale SNSPDs (L� Λ), the first termcan be neglected74, and because of the current continuity(∇.~j = 0), we can write the current density in the form

of a scaler function as ~j(r) = ∇×G(r)z. Thus, eqn. (A4)is reduced to74:

∇2G(r) = − cΦ0

2πΛδ (~r − ~rv) , (A5)

which is equivalent to the 2D Poisson’s equation for acharged particle. For an infinite superconducting filmcase, the interaction energy between vortices and an-tivortices for distances shorter than the Pearl length islogarithmic. This allows Berezinskii-Thouless-Kosterlitz(BKT) transition and the formation of vortex-antivortexpairs below the BKT critical temperature74,76. However,for a thin superconductor with finite width (−W/2 <x < W/2), the long range interaction between vorticesand antivortices is eliminated and single vortices can befound. For a single vortex, eqn. (A5) is reduced to theequation for a charge sandwiched between two parallelgrounded plates. The problem is well-known in electro-statics and can be solved using conformal mapping withz′ = eiπz/W transformation and using image theory74:

G(x, y) =cΦ0

8πΛln

cosh (yπ/W ) + cos ((x+ xv)π/W )

cosh (yπ/W )− cos ((x− xv)π/W ),

(A6)where we have assumed the vortex is placed at x = xvand y = 0. The phase of the order parameter, ϕ, canalso be derived from G since the gradient of ϕ is alsoproportional to the current28:

ϕ (~r, ~rv) = tan−1 cos(πxW

)sinh

(π y−yvW

)sin(πxW

)− cosh

(π y−yvW

)sin(πxvW

) .(A7)

The free energy in presence of a vortex consists of thefield energy and the kinetic energy inside the nanowireand the field energy outside73,74. If we assume the vortexcore radius is ξ and we neglect the core energy of the

vortex, the self-energy of the vortex can be written as20:

U0v (xv) = −Φ0

2cG(|x− xv| → ξ, 0)

=Φ2

0

8π2Λln

(2W

πξcos(πxvW

)), (A8)

If we include the work done by the bias current on asingle vortex due to the Magnus force (dual of the Lorentzforce on a magnetic flux), the total energy of a singlevortex is expressed as28:

Uv(xv)=Φ2

0

8π2Λln

(2W

πξcos(πxvW

))−Φ0

cjy(xv)(xv +

W

2).

(A9)

The Magnus force tries to move the vortex in the di-rection perpendicular to the direction of the applied biascurrent, but it cannot overcome the self-energy of thevortex if the bias current is not high enough. Increasingthe bias current at the edges jy(xv, t) due to the photonabsorption reduces the potential barrier and eases vortexcrossing. This barrier finally turns to zero at the vortexcritical current which is:

Ic,v =cΦ0

4π2 exp(1)ΛξW. (A10)

As seen in eqn. (A7), the phase of the order parameterdepends on the position of the vortex, xv, and the phasedifference at the two ends of a long nanowire (L � W )away from the vortex position can be approximated as:

ϕ(L/2)− ϕ(−L/2) = 2πxv/W. (A11)

Hence, as the vortex moves across the width of thenanowire, it applies a time-dependent phase differencebetween the two terminals of the detector. If the vortexcrosses from one edge at xv = −W/2 to another edge atxv = W/2, it causes a 2π phase-slip at the two ends ofthe nanowire. This phase evolution generates a voltagepulse which can be described by the Josephson effect77:

V (t) =Φ0

2πc

d

dt(ϕ(L/2)− ϕ(−L/2)) =

Φ0

cW

dxvdt

. (A12)

This voltage pulse propagates to the two ends6,78 and isdissipated in the presence of the bias current. If the biascurrent is high enough, the released energy is enough toinduce a phase transition in the nanowire from the super-conducting state to the normal conducting state. Hence,the current Ic,v, which makes the vortex tunneling bar-rier reduce to zero, is also the critical current for phasetransition in a thin superconducting nanowire. This crit-ical current is less than the depairing critical current ina bulk superconductor, Ic,dep,

19,21.Even if the applied bias current is below Ic,v, breaking

of the Cooper pairs and redistribution of the bias cur-rent due to the photon absorption can also change thepotential barrier21:

Uv(xv, t)

ε0=

π

W

∫ xv

ξ−W2

nse(x′, t)

nse,0tan

(πx′

W

)dx′

− 2W

Ic,v exp(1)ξ

∫ xv

−W2

nse(x′, t)

nse,0jy (x′, t) dx′,

(A13)

10

where ε0 = Φ20/8π

2Λ is the characteristic vortex energy.

ACKNOWLEDGEMENT

We thank Sean Molesky, Joseph Maciejko, RudroBiswajs, and Bhaskaran Muralidharan for discussions.This work is supported by DARPA DETECT.

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