# Storage and retrieval of heralded telecommunication-wavelength photons using
a solid-state waveguide quantum memory

Mohsen Falamarzi Askarani  Institute for Quantum Science and Technology, and
Department of Physics & Astronomy, University of Calgary, 2500 University
Drive NW, Calgary, Alberta, T2N 1N4, Canada  Marcel.li Grimau Pugibert
Institute for Quantum Science and Technology, and Department of Physics &
Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta,
T2N 1N4, Canada  current address: University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland  Thomas Lutz  Institute for Quantum Science and
Technology, and Department of Physics & Astronomy, University of Calgary, 2500
University Drive NW, Calgary, Alberta, T2N 1N4, Canada  current address: ETH
Zürich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland  Varun B. Verma  National
Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305,
USA  Matthew D. Shaw  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA  Sae Woo Nam
National Institute of Standards and Technology, 325 Broadway, Boulder,
Colorado 80305, USA  Neil Sinclair  Institute for Quantum Science and
Technology, and Department of Physics & Astronomy, University of Calgary, 2500
University Drive NW, Calgary, Alberta, T2N 1N4, Canada  current address:
California Institute of Technology, 1200 East California Boulevard, Pasadena,
California 91125, USA  Daniel Oblak  Institute for Quantum Science and
Technology, and Department of Physics & Astronomy, University of Calgary, 2500
University Drive NW, Calgary, Alberta, T2N 1N4, Canada  Wolfgang Tittel
Institute for Quantum Science and Technology, and Department of Physics &
Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta,
T2N 1N4, Canada  current address: QuTech, Delft University of Technology, 2600
GA Delft, The Netherlands

###### Abstract

Large-scale quantum networks will employ telecommunication-wavelength photons
to exchange quantum information between remote measurement, storage, and
processing nodes via fibre-optic channels. Quantum memories compatible with
telecommunication-wavelength photons are a key element towards building such a
quantum network. Here, we demonstrate the storage and retrieval of heralded
1532 nm-wavelength photons using a solid-state waveguide quantum memory. The
heralded photons are derived from a photon-pair source that is based on
parametric down-conversion, and our quantum memory is based on a 6 GHz-
bandwidth atomic frequency comb prepared using an inhomogeneously broadened
absorption line of a cryogenically-cooled erbium-doped lithium niobate
waveguide. Using persistent spectral hole burning under varying magnetic
fields, we determine that the memory is enabled by population transfer into
niobium and lithium nuclear spin levels. Despite limited storage time and
efficiency, our demonstration represents an important step towards quantum
networks that operate in the telecommunication band and the development of on-
chip quantum technology using industry-standard crystals.

Many efforts towards future quantum networks kimble2008quantum  have focused
on employing photons at wavelengths in the C band (1530-1565 nm) due to the
possibility of low-loss transmission using existing fiber-optic
telecommunication infrastructure. Local nodes in a quantum network are
envisioned to process information using (optical) chips that are comprised of
(elementary) quantum computers kimble2008quantum , while the synchronization
of information is enabled by quantum memories that store and retrieve quantum
information lvovsky2009optical . The latter underpins the operation of quantum
repeaters, which promise to transmit quantum information through lossy
channels or over intercontinental distances sangouard2011quantum .

Efforts to realize long-distance quantum communication have been bolstered by
the significant progress of researchers to develop of quantum-optical
technology 2009photonicQtechnology . However, much of this work is not
compatible with C-band photons without the use of optical frequency conversion
tanzilli2005photonic . To avoid the conversion step, efforts have focused
towards quantum technology that operates in the C-band, which has resulted in
the development of efficient photon detectors natarajan2012SNSPD  and single-
photon sources Quantumdots . Nonetheless, a C-band quantum memory has turned
out to be particularly challenging component to realize and in particular so
if it is to be integrated into a solid-state platform such that it is easy to
integrate with other photonics components.

Significant progress towards quantum memories has been made in the last
decade, with most focus on the development of optical, microwave or radio-
frequency to matter interfaces bussieres2013prospective . One promising
approach to implement quantum memory is based on cryogenically-cooled rare-
earth-ion-doped crystals as they often feature long optical and spin coherence
times, as well as suitable energy-level structures Charles2011REI .
Achievements with rare-earth-ion-doped crystals include broadband storage of
entangled photons saglamyurek2011broadband ; clausen2011quantumcrystal ,
teleportation into quantum memories bussieres2014teleportationintomemory ,
storage in nano-fabricated structureszhong2017on-chipstorage , and storage
assisted by impedance-matched cavities sabooni2013cavity ; afzelius2014cavity
. Furthermore, rare-earth-ion-based quantum memory and signal processing has
been demonstrated using industry-standard waveguides, such as titanium-
indiffused lithium niobate (Ti4+:LiNbO3) saglamyurek2011broadband ;
saglamyurek2014manipulation ; sinclair2017properties , which are promising for
efficient on-chip information processing.

Er-doped crystals are unique in that they offer a ground-to-excited level
transition in the C band with a coherence time that is the longest of any
optical transition in a solid bottger2003erbium . This property, found in Er-
doped yttrium-orthosilicate (Er:Y2SiO5), prompted the realization of a photon-
echo memory protocol for coherent light at the single-photon level
2010YSiOmemorynoise , but not for non-classical (single photon) light. A
disadvantage of most Er-doped materials, including Er:Y2SiO5, is the
relatively long lifetime of the excited level (up to $\approx$11 ms) as
compared to that of sub-levels of the ground level ($\approx$130 ms), which
makes optical pumping to the latter inefficient. This poses a significant
obstacle to achieve a high-efficiency in photon-echo based quantum memory
protocols, which rely on the optical pumping of sub-ensembles to shelving
levels such as the electronic sub-levels of the ground state. This obstacle is
partially overcome in Er-doped fibers, which due to their amorphous (rather
than crystalline) structure feature reduced spin-spin interaction strengths
compared to crystals and, as a result, extended ground-level lifetimes hole-
erbium . This feature allowed a recent demonstration of quantum memory for
non-classical and entangled states of light at telecommunication wavelengths
(saglamyurek2015quantum, ). Despite the appeal of Er-doped fibres for all-
fiber implementations, decoherence arising from their amorphous structure
hole-erbium ; Coherence-fiber  currently restricts the resulting memories’
storage times to less than one hundred nanoseconds, though improvements at
ultra-low temperatures may be possible saglamyurek2016multiplexed . Thus, the
demonstration of a telecommunication-wavelength quantum memory for non-
classical light using a solid-state crystal, in particular a crystalline
waveguide that can be integrated with telecommunication-industry devices (such
as modulators that are realized with Ti4+:LiNbO3 waveguides), is an important
and yet-to-be achieved goal.

Here we demonstrate a quantum memory for heralded single photons at 1532.05 nm
wavelength using an erbium- and titanium-indiffused lithium-niobate waveguide
(Er3+:Ti4+:LiNbO3) (see Methods for fabrication details) cooled to 0.6 K. By
spectrally-selective optical pumping to shelving levels, which we identify as
arising from the (superhyperfine) coupling of 7Li and 93Nb nuclear spins in
LiNbO3 to an external magnetic field, we prepare a 6 GHz-bandwidth Atomic
Frequency Comb (AFC) quantum memory. Heralded single-photons generated from a
spontaneous parametric down conversion (SPDC) based photon pair-source are
stored and retrieved from the AFC. The quantum nature of the recalled heralded
photons is verified by their cross-correlation value with the heralding
photon.

## Experiment

Our quantum memory is based on the AFC protocol, which is well suited to the
spectroscopic properties of rare-earth ion doped materials and has underpinned
much progress towards efficient and broadband quantum memories liu2006sREIs-
book . For example, storage times as long as milliseconds AFC2015ms ,
efficiencies as high as 56% sabooni2013cavity , storage bandwidths of several
GHz saglamyurek2011broadband  and fidelities up to 99.9% Guo2012realization
have been achieved. An AFC is comprised of a series of spectral absorption
lines that are equidistantly detuned by $\Delta$ (i.e. a ’comb’ of absorption
features) unitefficiency1 . An AFC can be prepared on the inhomogeneously
broadened absorption line of rare-earth ions by spectrally-selective optical
pumping into long-lived sub-levels (often hyperfine ground-levels). After
preparation, an incident photon is absorbed by the comb to generate a
collective atomic excitation that is described by

$\left|\Psi\right>_{A}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}c_{j}e^{i2\pi\delta_{j}t}e^{-ikz_{j}}\left|g_{1},...e_{j},...g_{N}\right>,$
(1)

where $N$ is the number of ions and $\delta_{j}$ is the detuning of the $j$th
atom’s transition frequency with respect to the frequency of the incoming
photon. The longitudinal position of the $j$th ion is denoted by $z_{j}$, and
the coefficient $c_{j}$ is related to its excitation probability. After
absorption, each term in Eq. (1) accumulates a different phase according to
its detuning $\delta_{j} = m_{j}\Delta$, where $m_{j}$ is an integer.
Consequently, all terms coherently rephase at a time $\tau = 1/\Delta$,
yielding collective re-emission of the photon in its original quantum state.
Under certain conditions the retrieval process can in principle reach unit
efficiency unitefficiency1 ; unitefficiency2 ; unitefficiency3 . The
experimental performance of the AFC quantum memory hinges on the spectroscopic
properties of the rare-earth-ion doped crystal used. For example, the level
structure and population dynamics influence the storage efficiency, operating
wavelength as well as bandwidth lvovsky2009optical ; Tittel2010reviewQM ;
bussieres2013prospective .

Figure 1: a) Experimental timing sequence for AFC preparation as described in
the Methods. ”$\cdot\cdot\cdot$” denotes repetition of the sequence. b)
Experimental set-up as described in the main text and Methods consists of two
main parts. Top half, which is for optical pumping for spectral hole burning
measurements and AFC generation consists of a continuous wave laser (CW
laser), phase modulator (PM), accousto-optic modulator (AOM), polarization
controller (PC), optical switches, a circulator, and a photo-detector (PD)
which is used to record the optical depth spectra shown in Figs. 2a and 3.
Bottom half, which is for heralded single photon generation and detection
consists of a pulsed pump laser, second harmonic generation (SHG), spontaneous
parametric down conversion (SPDC), Fabry-perot cavity (FP), superconducting
nanowire single-photon detectors (SNSPD), time to digital converter (TDC).

There have been only a few spectroscopic studies of the low-temperature
properties of the ${}^{4}I_{15/2}$ $\leftrightarrow$ ${}^{4}I_{13/2}$ telecom-
wavelength transition in our material of choice, Er3+-doped LiNbO3. Ahdiye2009
; Charlesppt ; thiel2010LNbulk ; Charles2011REI ; thiel2012LNcrystalprocessing
; lutz2017powder . The Er ion is a so-called Kramers ion and the electronic
spin degeneracy is lifted by the Zeeman effect in an applied magnetic field.
The result is a split doublet in both the excited and ground level of the
${}^{4}I_{15/2}$ $\leftrightarrow$ ${}^{4}I_{13/2}$ transition liu2006sREIs-
book . A ground-level electronic Zeeman splitting of 1.6 MHz/G was measured
for a 0.005%-doped Er-doped lithium niobate bulk crystal (Er3+:LiNbO3, not
containing titanium) in magnetic field parallel to the c-axis of the crystal
Charlesppt . In addition, rare-earth ions can couple to nuclear spins of
crystal lattice ions (superhyperfine coupling)liu2006sREIs-book . For
Er3+:LiNbO3, the electron spin (1/2) of erbium is known to couple to nearby
lithium ($I=3/2$) and niobium ($I=9/2$) spins, and possibly other impurities
Charlesppt . A study of Er3+:LiNbO3 at temperatures as low as 1.6 K using
light polarized orthogonal to the crystal’s c-axis (i.e. E$\perp$c) revealed a
180 GHz-wide inhomogeneously-broadened line, a 2 ms population lifetime of the
${}^{4}I_{13/2}$ level, and an optical coherence lifetime of up to 117 $\mu$s
at a 50 kG field oriented parallel to the c-axis (B$\parallel$c)
thiel2010LNbulk ; Charles2011REI . The same transition of Er3+ was studied in
a Ti4+:LiNbO3 waveguide at 3 K with E$\perp$c-polarized light to reveal a 250
GHz-wide inhomogneously broadened line, a population lifetime matching that of
the bulk material, and a coherence lifetime of 18.2 $\mu$s in a 4 kG field
oriented B$\parallel$c Ahdiye2009 . Regardless, of the slightly degraded
performance111We expect that at a lower temperature, optimized magnetic field,
and with consideration of instantaneous spectral diffusion liu2006sREIs-book ,
the coherence lifetime in the waveguide will approach that of the bulk,
similarly to what was observed for Tm3+:Ti4+:LiNbO3 sinclair2017properties .
of the waveguide compared to bulk Er3+:LiNbO3 under these conditions, the
coherence properties suffice for up to $\mu$s-long storage times.

For our Er3+:Ti4+:LiNbO3 waveguide to be suitable for the AFC protocol we
additionally need to verify the existence shelving levels with a lifetime
significantly exceeding that of the excited state. To this end we perform
spectral hole burning to determine the structure and population dynamics of
the hyperfine levels (i.e. electronic Zeeman and superhyperfine levels) hole-
erbium  using the setup sketched in Fig. 1 with the crystal at 0.6 K. Fig. 2a
depicts the resulting absorption spectrum (at a magnetic field of 19 kG),
which displays a rich side- and anti-hole structure, which persists up to a
time delay of 15 minutes. (Here and henceforth, the light polarization and the
magnetic field orientation are set perpendicular and parallel to the c-axis of
the crystal, respectively, i.e. E$\perp$c and B$\parallel$c.) As described in
the Methods, the side-hole detunings are determined by the excited-level
hyperfine structure, whereas the detunings of the anti-holes are given by the
energy-level differences between the excited- and ground-level hyperfine
structure Thierryside-antiholes .

Figure 2: a) Optical depth over a 100 MHZ spectral range centered at 1532.05
nm in a 19 kG magnetic field showing evidence of spectral hole burning. Side
holes due to the interaction of the Er3+ electronic spin with 93Nb, 6Li, and
7Li nuclear spins of LiNbO3 are labeled accordingly. The apparent side-holes
at $\pm$50 MHz are due to imperfect frequency sweep. b) Detuning of 7Li and
93Nb side-holes as function of magnetic field. Side-holes for 6Li are resolved
only at the highest magnetic field. Uncertainty bars are calculated from the
between the negative and positive detuned side-holes.

To determine the origin of the substructure, hole burning is repeated at
different magnetic fields, and the detuning of the central frequencies of each
side-hole is measured after a time delay of 30 ms. We plot the field-dependent
detuning in Fig.2b for three pairs of distinguishable side-holes. A fit to
their field dependence gives values of 1.15$\pm$0.01, 1.721$\pm$0.006, and
0.572$\pm$0.002 kHz/G, respectively. This agrees well with the superhyperfine
coupling of Er3+ to 93Nb, 7Li, and 6Li nuclear spins in the host crystal as
previously measured in Er3+:LiNbO3 1991nuclearspinsplitting ; thiel2010LNbulk
. At fields below 10 kG the superhyperfine splitting is not resolvable, but we
still observe long-lived spectral holes. These are likely due to optical
pumping to long lived electronic Zeeman levels of Er3+. However, no side or
anti-hole structure is discernible and thus no Zeeman level splitting can be
determined.

To gain a better understanding of the population dynamics we perform time-
resolved spectral hole burning using magnetic fields of 0.35, 0.6, 0.8 and 19
kG. Based on earlier measurements of the spectral diffusion in Er3+:LiNbO3
thiel2010LNbulk  we expect that the few MHz linewidth will dominate over any
spectral diffusion in our material. For the largest field, we monitor the
depth of the central hole after a variable time delay between $\sim$30 s and
$\sim$20 min and estimate the $1/e$ hole lifetime to be 10 min. This agrees
well with that observed using Er3+:LiNbO3 under similar conditions Charlesppt
, where the population trapping mechanism was attributed to 7Li nuclear spins
i.e. a superhyperfine level. For the three lower fields, we vary the time
delay from 20 ms to 4.5 s and record the depth of the central hole feature.
The hole-decay features two distinct exponential decays. The $1/e$ hole
lifetime for the faster decay is field-independent at $\sim$60 ms, while for
the slower second decay it is of 1.0, 1.36, and 2.44 s for the three field
values, respectively. The relative weights of all decays do not vary with
field (see upcoming manuscript upcoming ). We conjecture that the fast decay
arises from relaxation of Er electronic Zeeman levels or superhyperfine
levels. The former is supported by the short lifetimes of Zeeman levels that
have been observed in many Er-doped crystals at these magnetic fields,
including Er3+:LiNbO3. As for the latter, even though no clear superhyperfine
structure is visible, the levels may be buried within the laser linewidth-
broadened hole and thus residual optical pumping into superhyperfine levels
could be possible. We attribute the long, magnetic-field dependent decay to
relaxation of Zeeman levels of Er ions that belong to different magnetic
classes, in the sense that their electronic Zeeman splitting is significantly
different from neighboring Er ions. As a result of this electronic spin
detuning, the spin flip-flop relaxation mechanism, which is the dominant one
at these fields, is suppressed. Hence, relatively long Zeeman level lifetimes
can be achieved similar to that in Er-doped fibers hole-erbium . Such spin-
inhomogeneous broadening will be manifested by very broad anti-holes, which is
corroborated by our inability to identify any anti-holes in the hole burning
spectra. Further support for our interpretation comes from studies of
Tm3+-doped LiNbO3 bulk crystal and waveguide sinclair2018 ;
thiel2012LNcrystalprocessing , in which significant anti-hole broadening
caused by a distribution of Tm nuclear spin splittings was measured. This was
attributed to the magnetic disorder of LiNbO3 (e.g. due to Li+-Nb+5 anti-
sites, as well as Nb+5 and O vacancies). Further hole burning studies are
necessary to shed light on these conjectures.

Based on our hole burning measurements, it should be possible to realize the
AFC by exploiting the long-lived population trapping mechanisms at either low
or high fields. Nevertheless, we find that the AFCs generated at low fields
are marred by a large absorption background when the bandwitdh is increased
beyond about a GHz upcoming . Hence, the high-field configuration with
trapping in superhyperfine levels turns out to be advantageous. We perform the
optical pumping such that the comb periodicity somewhat coincides with the
excited-level splitting caused by interactions with 7Li and 93Nb, which for
the chosen magnetic field of 16.5 kG, are 28 MHz and 19 MHz, respectively.
Based on this we chose an AFC storage time of $\tau=48~{}ns$. A 200 MHz-
bandwidth section of our 6 GHz-bandwidth AFC structure is shown in Fig. 3.
From the trace we expect an efficiency close to 1$\%$. However, limited laser
intensity at large detunings leads to a non-uniform AFC and an averaged
efficiency of around 0.3%. The absorption background is due to imperfect hole
burning. The finesse – the ratio of tooth width $\gamma$ to tooth spacing
$\Delta$ – is two.

Figure 3: A 200 MHz-bandwidth section of our 6 GHz-bandwidth AFC.

## Results and discussion

To demonstrate that our waveguide can be used as a quantum memory, we create
and store (absorb) heralded telecommunication-wavelength photons of about 6
GHz bandwith, which are then re-emitted after a pre-programmed storage time of
48 ns. The experimental setup is outlined in Fig. 1 and the components
described in detail in the Methods. Fig. 4 depicts the histogram (red colour)
of heralded detection events after the memory. The large peak $t=0$
corresponds to photons that are directly transmitted through our waveguide and
thus not absorbed by the AFC structure. At $t=48$ ns the peak corresponding to
stored and retrieved photons appears (in Fig. 4 it is magnified by a factor of
20 and coloured blue). The sequence of detection peaks separated by 12.5 ns
are due to uncorrelated photons generated during different cycles of the 80
MHz repetition rate SPDC pump laser (often referred-to as accidental
coincidences). Finally, the large number of smaller peaks in the histogram are
due to spurious temporal modes of the pump laser – a consequence of imperfect
alignment of the laser cavity – causing further uncorrelated photon-pair
production events in the SPDC crystal.

Figure 4: Quantum memory for a heralded telecommunication-wavelength photons.
The histogram depicts the time-resolved signal-idler coincidence detection
rate. The time axis is defined relative to the detection of the transmitted
idler photons. The photons that are emitted by the AFC (colored in blue,
magnified 20$\times$) are measured after a storage time of 48 ns. The sequence
of coincidence peaks that repeat at a rate of 80 MHz is due to repetitive
pumping of the SPDC source (explained in the main text).

In order to assess the non-classical nature of the photons recalled from our
quantum memory, we measure the second-order cross-correlation function g(2)
extracted from the coincidence histogram (Fig. 4). This function is defined as
$g^{(2)}=\frac{P_{SI}}{P_{S}\times P_{I}}$. Here $P_{SI}$ corresponds to the
probability of detecting a coincidence between the signal and idler photons,
and $P_{S}$ ($P_{I}$) is the probability of detecting the signal (idler)
photon. To prove that quantum correlations between the two photons are
present, g(2) must be greater than 2 tapster1998photon . First, bypassing the
quantum memory, we determine g(2) of our photon pair source (without storage)
to be around 20. It is limited by multi-pair emissions caused by the high pump
intensity tapster1998photon . After one photon has been stored in our quantum
memory, we measure a g(2) of 7.1$\pm$4, demonstrating that non-classical
correlations between the members of photon pairs are still present after
storage.

The memory efficiency is limited by several factors compared to the ideal
efficiency of 54% for forward recall unitefficiency1 , including the remaining
absorption background, low finesse ($\Delta/\gamma$) of the AFC (where
$\Delta$ is the tooth spacing and $\gamma$ the linewidth of each tooth) and,
if $\gamma$ were reduced to increase the finesse and approach the maximum
efficiency, insufficient optical depth. The limited optical depth and the lack
of phase matching for backwards recall (in which case the efficiency can reach
100%) can be remedied with an impedance-matched cavity unitefficiency2 .
However, this solution is not beneficial without also reducing the remaining
background loss, which we attribute, in our demonstration, to the complexity
of the superhyperfine level structure and the possibility of laser-induced
enhancement of spin relaxation thiel2012LNcrystalprocessing ; sinclair2018 ,
an effect that has also been observed, and mitigated for the creation of AFCs
in Tm3+:Ti4+:LiNbO3 sinclair2017properties ; sinclair2018 . In particular, the
presence of (broad) anti-holes within the AFC troughs, which is possible for a
host with considerable disorder such as LiNbO3 thiel2012LNcrystalprocessing ,
may not allow complete removal of the absorption background. The finesse of
our AFC is currently limited to two by our optical pumping method
unitefficiency1 . It is ultimately limited by the homogeneous linewidth of the
used transition.

We believe that further characterization of the atomic level structure and
dynamics of the material, in particular through detailed hole burning
measurements (e.g. with varying magnetic field strength and orientation), as
well as optimization of the AFC preparation steps (e.g. using back-pumping or
spin-mixing methods), the background absorption can be significantly reduced.
To this end, the origin of the aforementioned long-lived holes at low magnetic
fields should be studied further upcoming . Furthermore, wavelength-dependent
measurements may allow the identification of more favorable (super- or
Zeeman-) hyperfine levels due to different magnetic sub-classes of ions.
Furthermore, a lower sample temperature and concentration could allow for more
efficient spectral hole burning in addition to suppressing decoherence
mechanisms (e.g. spin flip-flops).

Note that our current quantum memory implementation does not allow photons to
be retrieved on demand, which requires in an additional step the reversible,
coherent, mapping to a third long-lived level (AFC_spinwave, ). Nonetheless,
applications such as quantum repeaters may utilize memories of fixed storage
times provided that spectral or spatial multiplexing is utilized
NeilPLRmultiplexing .

## Conclusion

We have demonstrated the storage and retrieval of heralded 1532 nm
telecommunication-wavelength photons using a cryogenically-cooled Er-doped
lithium niobate waveguide. The non-classical nature of the storage process is
demonstrated by a measurement of the cross-correlation between the heralding
and the retrieved photons. We employed an AFC quantum memory that is based on
optical pumping of atomic population into superhyperfine ground levels, and we
have detailed the limitations as well as possible improvements of our memory,
in particular its limited storage efficiency. Our work is a step towards
developing on-chip quantum technology in the C band using standard materials
that are employed by the telecommunication industry, and will benefit quantum
networks.

METHODS

## Er-doped lithium niobate waveguide

To fabricate the Er3+:Ti4+:LiNbO3 waveguide, z-cut congruent LiNbO3 is Er-
doped over a length of 10 mm by indiffusion of a vacuum-deposited (electron-
beam evaporated) 8 nm-thick Er layer at 1130 ∘C for 150 h in an Ar-atmosphere.
This step is followed by a post-treatment in O (1 h) to get a full re-
oxidization of the crystal. Er substitutes for Li when incorporated into the
LiNbO3 lattice. The indiffusion results in a 3.6 $\times$ 1019 cm-3 near-
surface concentration and a Gaussian concentration profile that features a 1/e
penetration depth of 8.2 $\mu$m baumann1996LiNbOfabrication . Next, the
waveguide is created by indiffusion of Ti. To do this, a 98 nm-thick titanium
layer is deposited on the Er-doped surface of the LiNbO3 substrate using
electron beam evaporation. From this layer, 7 $\mu$m-wide Ti strips are
defined by photo-lithography and chemical etching, and subsequently in-
diffused at 1060 ∘C for 8.55 h. This process leads to a single-mode waveguide
with a 4.5 $\times$ 3 $\mu$m full-width-at-half-maximum intensity distribution
for transverse magnetic-polarization. Note that an analogous procedure, which
is detailed in Ref. saglamyurek2011broadband , was used to fabricate the
similar Tm3+:Ti4+:LiNbO3 waveguide.

In the experiments the Er3+:Ti4+:LiNbO3 waveguide is mounted in an adiabatic
diamagnetic refrigerator (ADR) cryostat kept below 0.6 K and the in and out
coupling of light is achieved by fiber butt-coupling. The overall in-out
coupling transmission of our cryogenic setup is 20$\%$ (measured at a
wavelength of 1532.05 nm).

## Spectral hole burning and AFC preparation

Spectral hole burning is performed by shining continuous wave (CW) light at
zero detuning $\Delta$ and subsequently recording the optical depth (OD)
profile in a relevant spectral region centred on $\Delta=0$. The setup for
this is depicted in the top half of Fig. 1b. First a continuous-wave laser is
switched to the crystal – by using an acousto-optic modulator (AOM) – in order
to optically-pump a few MHz-wide frequency band of the inhomogeneous
absorption line (at $\Delta=0$ corresponding to 1532.05 nm) until
transparency, i.e. a spectral hole is created. The OD profile is recorded
after a given delay by switching on the AOM and sweeping the frequency of the
light impinging on the crystal over 100 MHz using serrodyne phase-modulation.
The transmitted light is then detected on a photo-detector. A magnetic field
oriented B$\parallel$c of up to 20 kG can be applied using a solenoid magnet.

The AFC is generated in a similar fashion except measures have to be taken to
store and absorb single photons. First step is optical pumping for 300 ms as
shown in Fig. 1a. This involves the 1532.05 nm-wavelength CW optical pumping
light is frequency-swept over a 6 GHz bandwidth by serrodyne phase-modulation
while the intensity modulated using an AOM. In the OD spectrum this procedure
creates features of high (low pump power intensity) and low (high pump power
intensity) absorption, as is shown in Fig. 3 for a magnetic field of
B$\parallel$c=16.5 kG . Following this is a 30 ms wait time in order to avoid
the impact of noise-inducing spontaneously emitted photons from the
${}^{4}I_{13/2}$ excited-level. Finally, during a 200 ms storage and retrieval
duration heralded single photons are stored and retrieved in the waveguide
memory. The optical switches and circulator are employed to suppress any stray
otpical pump light during the storage and retrieval of the heralded single
photons (see Fig. 1a).

## Spectral hole structure

The CW light incident on the waveguides, results in the transfer of atomic
population to the ${}^{4}I_{13/2}$ excited level, which subsequently decays
and is redistributed among the hyperfine levels of the ${}^{4}I_{15/2}$ ground
state. The resulting absorption profile allows the determination of the number
of sub-levels. Furthermore, the detuning of the various features from the
center of the (main) spectral hole can be used to extract the coupling
strength of the involved spins and to identify population-trapping mechanisms.
In addition, time-resolved measurements of the absorption profile can give
insight into population dynamics. Regions of increased (decreased)
transparency that are detuned from the burning frequency are referred to as
side- (anti-) holes.

## SPDC-based heralded single photon source

Photon pairs are generated using SPDC, a process in which a pump photon is
converted into a pair of photons (whose total energy and momentum equal that
of the pump photon) by using a second-order nonlinearity tanzilli2001SPDC . As
shown in Fig. 1, our production of photon pairs begins with a mode-locked
laser that emits 6 ps-duration pulses at a rate of 80 MHz at a wavelength of
1047 nm. These pulses are frequency-doubled to 523.5 nm by means of second
harmonic generation using a periodically-poled lithium niobate (PPLN) crystal
and are then used to pump a second PPLN crystal that performs SPDC to yield
frequency-correlated photon pairs of 795 nm (signal) and 1532 nm (idler)
wavelength. After passing through an interference filter that removes the
remaining 523 nm light, a dichroic mirror spatially separates the photons from
each pair. Finally, the bandwidth of each photon is set by filtering, using a
12- (6-) GHz bandwidth fiber-Bragg grating (Fabry-Perot filter) for the 1532
(795) nm photon. The 1532 nm idler photon is coupled to the waveguide and then
stored. After retrieval, it is detected by a superconducting nanowire single-
photon detector (SNSPD) fmarsiliSNSPD . The heralding 795 nm signal photon is
directly detected by the SNSPD. All detection events are recorded by a time-
to-digital converter.

## Acknowledgments

The authors thank Wolfgang Sohler, Mathew George, and Raimund Ricken for
providing the Er3+:Ti4+:LiNbO3 waveguide, Francesco Marsili for assistance
with development of the SNSPDs, Jakob H. Davidson for help with aligning the
waveguide, and Gustavo Amaral and Erhan Saglamyurek for useful discussions.
This work is funded through Alberta Innovates Technology Futures (AITF), the
National Science and Engineering Research Council of Canada (NSERC), and the
Defense Advanced Research Projects Agency (DARPA) Quiness program. W.T.
acknowledges funding as a Senior Fellow of the Canadian Institute for Advanced
Research (CIFAR), and V.B.V. and S.W.N. acknowledge partial funding for
detector development from the Defense Advanced Research Projects Agency
(DARPA) Information in a Photon (InPho) program. Part of the detector research
was carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space
Administration (NASA).

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# Partial and full tunneling processes across potential barriers

Philip Caesar Flores<EMAIL_ADDRESS>Max-Born-Institute, Max-Born Straße
2A, 12489 Berlin, Germany  Dean Alvin L. Pablico<EMAIL_ADDRESS>Eric A.
Galapon<EMAIL_ADDRESS>Theoretical Physics Group, National Institute of
Physics
University of the Philippines Diliman, 1101 Quezon City, Philippines

###### Abstract

We introduce the concept of partial-tunneling and full-tunneling processes to
explain the seemingly contradictory non-zero and vanishing tunneling times
often reported in the literature. Our analysis starts by considering the
traversal time of a quantum particle through a potential barrier, including
both above and below-barrier traversals, using the theory of time-of-arrival
operators. We then show that there are three traversal processes corresponding
to non-tunneling, full-tunneling, and partial tunneling. The distinction
between the three depends on the support of the incident wavepacket’s energy
distribution in relation to the shape of the barrier. Non-tunneling happens
when the energy distribution of the quantum particle lies above the maximum of
the potential barrier. Otherwise, full-tunneling process occurs when the
energy distribution of the particle is below the minimum of the potential
barrier. For this process, the obtained traversal time is interpreted as the
tunneling time. Finally, the partial-tunneling process occurs when the energy
distribution lies between the minimum and maximum of the potential barrier.
This signifies that the quantum particle tunneled only through some portions
of the potential barrier. We argue that the duration for a partial-tunneling
process should not be interpreted as the tunneling time but instead as a
partial traversal time to differentiate it from the full-tunneling process. We
then show that a full-tunneling process is always instantaneous, while a
partial-tunneling process takes a non-zero amount of time. We are then led to
the hypothesis that experimentally measured non-zero and vanishing tunneling
times correspond to partial and full-tunneling processes, respectively.

The time it takes for a quantum particle to tunnel through a potential barrier
has always eluded physicists since the advent of quantum mechanics [1, 2]. A
number of tunneling time definitions have been offered in the literature [3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13], e.g., Wigner phase time [3], Büttiker-
Landauer time [4], Larmor time [5, 6, 7], Pollak-Miller time [8], dwell time
[9]; but a consensus on whether quantum tunneling is instantaneous or not is
yet to be reached [14, 15, 16, 17, 18, 19, 20, 21]. The development of
ultraprecise techniques in strong-field physics has been expected to close the
debate once and for all, but the existence of contradictory experimental
results only further divided the physics community [22, 23, 24, 25, 26, 27,
28, 29]. This diversity poses a challenge to theoretical treatments that only
predict either zero or non-zero tunneling times, imploring a formalism that
could accommodate both seemingly contradictory results. In this Letter, we
offer such a formalism using the theory of time-of-arrival (TOA) operators
[30].

We start our analysis by investigating the expected quantum traversal time
across a contiguous barrier system, i.e., $V(q)=V_{1}$ for $-a<q<-l$ and
$V(q)=V_{2}$ for $-l<q<-b$, where $V_{1}<V_{2}$. We define the traversal time
as the amount of time a particle traverses through the barrier region,
including both above-barrier and below-barrier traversals. We choose a
contiguous potential barrier, instead of the usual square barrier, as it
naturally showcases three possible traversal processes: (i) non-tunneling,
(ii) full-tunneling, (iii) and partial-tunneling. The distinction between
these three processes is shown in Fig. (1). Non-tunneling or classical above-
barrier traversal happens when the energy distribution $E$ of the quantum
particle lies entirely above $V_{2}$. Meanwhile, full-tunneling process occurs
when the energy distribution of the particle is entirely below $V_{1}$. The
partial-tunneling process occurs when the energy distribution that lies
between $V_{1}$ and $V_{2}$. This physically suggests that the quantum
particle traversed above $V_{1}$ and tunneled through $V_{2}$, not the entire
barrier region. The single square barrier can only describe the first two
processes. We will show later that full-tunneling is always instantaneous
while partial-tunneling and non-tunneling takes a non-zero amount of time. A
double square barrier system has also been considered in Ref. [31] but their
analysis is focused on the generalized Hartman effect, which is different to
the current paper’s objectives.

Figure 1: Three traversal processes involving a quantum particle with incident
energy $E$ traversing through a contiguous square barrier system.

With clear distinctions between the three traversal processes, we now
determine the corresponding quantum traversal time. We work under the
assumption that there exists a TOA-operator $\hat{\mathrm{T}}$ corresponding
to an arrival at some specific point in the configuration space for a given
interaction potential $V(q)$. The traversal time can be extracted by
determining the arrival time difference between two identical wave packets,
the first one encounters a potential barrier while the other traverses freely
without obstruction [32, 31].

Our measurement scheme follows the prescription of Refs. [32, 31, 33, 34, 35]
and shown in Fig. (2). We place a detector $D_{T}$ at the far right of the
barrier system to announce the arrival of the particle at the origin $q=0$.
Likewise, a similar detector $D_{R}$ is placed at the far left of $D_{T}$. A
localized wave packet $\psi(q)$ is launched in between the potential $V(q)$
and detector $D_{R}$ towards the origin at time $t=0.$ An arrival time is
measured when $D_{T}$ clicks while no measurement is recorded when $D_{R}$
clicks. An average barrier TOA, $\bar{\tau}_{B}$, can be obtained when the
same experiment is repeated over a large number of trials, with the same
initial state for every measurement. A similar experiment is then performed in
the absence of the potential barrier. The average free TOA, $\bar{\tau}_{F}$,
is also measured with the detector $D_{T}$. The barrier traversal time is then
extracted from the difference
$\Delta\bar{\tau}=\bar{\tau}_{F}-\bar{\tau}_{B}$.

The above measurement scheme essentially coincides with the tunneling delay
time used by Steinberg, Kwiat, and Chiao in their seminal single-photon
tunneling time experiment [36]. They employed a two-photon source in which
pairs of photons are emitted simultaneously. One particle traverses a tunnel
barrier while the other twin particle encounters no barrier. Their operational
definition of tunneling time is extracted from the comparison between the TOA
of the two conjugate particles. The difference, however, with our measurement
scheme is that what they measured is $-\Delta\bar{\tau}$.

Figure 2: Measurement scheme for the expected quantum traversal time.

Within the theory of TOA-operators, the measured average value
$\bar{\tau}_{B}$ can be obtained as the expectation value of some barrier TOA-
operator $\hat{\mathrm{T}}_{B}$ for a given state $\psi$, that is,
$\bar{\tau}_{B}=\langle\psi|\hat{\mathrm{T}}_{B}|\psi\rangle$. Similarly, the
average free time-of-arrival appears as
$\bar{\tau}_{F}=\langle\psi|\hat{\mathrm{T}}_{F}|\psi\rangle,$ where
$\hat{\mathrm{T}}_{F}$ is the corresponding free-particle TOA-operator. The
arrival time difference then assumes the form

$\Delta\bar{\tau}=\langle\psi|\hat{\mathrm{T}}_{F}|\psi\rangle-\langle\psi|\hat{\mathrm{T}}_{B}|\psi\rangle.$
(1)

We highlight that $\Delta\bar{\tau}$ is not yet the barrier traversal time
itself, but the latter can be obtained from it.

Using the rigged Hilbert space formulation of quantum mechanics, a TOA-
operator has the general form

$(\hat{\mathrm{T}}\psi)(q)=\dfrac{\mu}{i\hbar}\int_{-\infty}^{\infty}dq^{\prime}\,T(q,q^{\prime})\,\mathrm{sgn}(q-q^{\prime})\,\psi(q^{\prime}),$
(2)

in coordinate representation [30]. The factor $\mathrm{sgn}z$ is the signum
function, $\mu$ is the mass of the incident particle, and $T(q,q^{\prime})$ is
referred to as the time kernel factor (TKF). The construction of the integral
operator $\hat{\mathrm{T}}$ then translates to the construction of
$T(q,q^{\prime})$ for a given interaction potential $V(q)$. A closed-form
expression for $T(q,q^{\prime})$ can be obtained by performing Weyl-
quantization on the classical TOA at the origin $\mathcal{T}_{C}(q,p)$ given
by

$\mathcal{T}_{C}(q,p)=-\mathrm{sgn}(p)\sqrt{\frac{\mu}{2}}\int_{0}^{q}\frac{dq^{\prime}}{\sqrt{H(q,p)-V(q^{\prime})}}.$
(3)

The quantization, however, should be restricted to the trajectories that pass
through the arrival point and coincides with the first arrival of the particle
[30]. This gives the following Weyl-quantized TKF,

$\tilde{T}(\eta,\zeta)=\dfrac{1}{2}\int_{0}^{\eta}ds\,{{}_{0}}F_{1}\left[;1;\dfrac{\mu}{2\hbar^{2}}(V(\eta)-V(s))\,\zeta^{2}\right],$
(4)

where, $T(q,q^{\prime})=\tilde{T}(\eta,\zeta)$, $\eta=(q+q^{\prime})/2$,
$\zeta=q-q^{\prime}$, and ${{}_{0}}F_{1}(;a;z)$ is a particular hypergeometric
function [30]. Equation (2), together with Eq. (4), define a TOA-operator that
satisfies Hermiticity, time-reversal symmetry, and Dirac’s correspondence
between classical and quantum observables.

Now, in the absence of a potential barrier, substitution of $V(q)=0$ into Eq.
(4) yields the free TKF $\tilde{T}_{F}(\eta,\zeta)=\eta/2$ or equivalently
$T_{F}(q,q^{\prime})=(q+q^{\prime})/4$. Substitution of $T_{F}(q,q^{\prime})$
into Eq. (2) gives the free TOA-operator $\hat{\mathrm{T}}_{F}$ which is
equivalent to $(\hat{\mathrm{T}}_{AB}\psi)(q)=\int dq^{\prime}\langle
q|\hat{\mathrm{T}}_{AB}|q^{\prime}\rangle\psi(q^{\prime})$, where
$\hat{\mathrm{T}}_{AB}=-(\mu/2)(\hat{\mathrm{q}}\hat{\mathrm{p}}^{-1}+\hat{\mathrm{p}}^{-1}\hat{\mathrm{q}})$
is the well-known Aharonov-Bohm free time operator [37]. Note that
$\hat{\mathrm{T}}_{F}$ is canonically conjugate with the free Hamiltonian
$\hat{\mathrm{H}}_{F}=\hat{\mathrm{p}}^{2}/2\mu.$

On the other hand, the barrier TOA-operator $\hat{\mathrm{T}}_{B}$ is
constructed by solving for the barrier TKF
$T_{B}(q,q^{\prime})=\tilde{T}_{B}(\eta,\zeta)$. This is done by dividing the
$s$ integral in Eq. (4) into four non-overlapping regions separated by the
edges of the two distinct barriers. The barrier TKF
$\tilde{T}_{B}(\eta,\zeta)$ then have four pieces corresponding to the four
regions described in Fig. (2) and are given by

$\displaystyle\tilde{T}_{B,I}(\eta,\zeta)=$ $\displaystyle\frac{\eta}{2},$
(5a) $\displaystyle\tilde{T}_{B,II}(\eta,\zeta)=$
$\displaystyle\frac{\eta+b}{2}-\frac{w_{1}}{2}I_{0}(\kappa_{1}|\zeta|),$ (5b)
$\displaystyle\tilde{T}_{B,III}(\eta,\zeta)=$
$\displaystyle\frac{\eta+b+w_{1}}{2}-\frac{b}{2}I_{0}(\kappa_{2}|\zeta|)$ (5c)
$\displaystyle-\frac{w_{1}}{2}{{}_{0}F_{1}}\left(;1;\frac{\kappa^{2}_{21}}{4}\zeta^{2}\right),$
(5d) $\displaystyle\tilde{T}_{B,IV}(\eta,\zeta)=$
$\displaystyle\frac{\eta+L}{2}-\sum_{n=1,2}\frac{w_{n}}{2}J_{0}(\kappa_{n}|\zeta|),$
(5e)

where $\kappa_{n}=\sqrt{2\mu V_{n}}/\hbar$ for $n=1,2$,
$\kappa^{2}_{2,1}=2\mu(V_{2}-V_{1})/\hbar^{2},$ and $w_{n}$ are the widths of
each barrier [31]. The above results have been derived using the identities
${}_{0}F_{1}(;1,x)=I_{0}(2\sqrt{x})$ for $x>0$ and
${}_{0}F_{1}(;1,x)=J_{0}(2\sqrt{x})$ for $x<0$, where $I_{0}(z)$ and
$J_{0}(z)$ are specific Bessel functions. Substitution of Eq. (5) into Eq. (2)
gives the barrier TOA-operator $\hat{\mathrm{T}}_{B}$. It can be shown that
$\hat{\mathrm{T}}_{B}$ with the TKF $\tilde{T}_{B,r}(\eta,\zeta)$ for
$r=I,II,III,IV$ is canonically conjugate with the Hamiltonians in the
respective regions, $r$ [38].

Having constructed the operators $\hat{\mathrm{T}}_{B}$ and
$\hat{\mathrm{T}}_{F}$, we now evaluate the arrival time difference
$\Delta\bar{\tau}$ in Eq. (1). We assume an incident wave packet of the form
$\psi(q)=e^{ik_{o}q/\hbar}\varphi(q)$ centered at $q=q_{o}$ with a mean
momentum expectation value $p_{o}=\hbar k_{o}$. We also impose the support of
$\varphi(q)$ to lie entirely to the left of the barrier system. The latter
assumption suggests that there is a zero probability that the quantum particle
is already within the barrier region or at the transmission side at the
initial time $t=0$. Under this condition, our barrier TOA-operator
$\hat{\mathrm{T}}_{B}$ depends only on the piece
$\tilde{T}_{B,IV}(\eta,\zeta)$.

In the $(\eta,\zeta)$ coordinates, the arrival time difference can be
rewritten in the form

$\Delta\bar{\tau}=-\frac{2\mu}{\hbar}\,\mathrm{Im}\,\int_{0}^{\infty}d\zeta\,\Phi(\zeta)\,e^{ik_{0}\zeta}\,\left[\tilde{T}_{F}(\eta)-\tilde{T}_{B,IV}(\eta,\zeta)\right],$
(6)

where,
$\Phi(\zeta)=\int_{-\infty}^{\infty}d\eta\,\bar{\varphi}(\eta-\zeta/2)\,\varphi(\eta+\zeta/2)$,
and $\mathrm{Im}(z)$ denotes the imaginary part of the integral. Evaluation of
Eq. (6) leads to

$\displaystyle\Delta\bar{\tau}=$
$\displaystyle\dfrac{L}{v_{0}}\mathrm{Im}\,Q^{*}\,-\,\sum_{n=1}^{2}\,\dfrac{w_{n}}{v_{0}}\,\mathrm{Im}\,R_{n}^{*},$
(7a) $\displaystyle Q^{*}=$ $\displaystyle
ko\int_{0}^{\infty}d\zeta\Phi(\zeta)\,e^{ik_{0}\zeta},$ (7b) $\displaystyle
R_{n}^{*}=$ $\displaystyle
ko\int_{0}^{\infty}d\zeta\Phi(\zeta)\,J_{0}(\kappa_{n}\zeta)\,e^{ik_{0}\zeta}.$
(7c)

We determine the physical significance of $\Delta\bar{\tau}$ by taking its
classical limit. This is done by considering the high energy limit
$k_{0}\to\infty$ for fixed $\kappa_{n}.$ One then finds the factor
$\mathrm{Im}\,Q^{*}\sim 1$ while $\mathrm{Im}\,R_{n}^{*}\sim v_{0}/v_{n}$,
where $v_{n}$ is the particle’s speed on top of the barrier with width
$w_{n}$. Since $\mathrm{Im}\,R_{n}^{*}$ depends on the ratio between the
speeds $v_{0}$ and $v_{n}$, we can interpret it as the index of refraction of
the $n$th barrier. Hence, we find the relation $\Delta\bar{\tau}\sim
L/v_{0}-\,\sum_{n=1}^{2}\,w_{n}/v_{n}.$ The first term is simply the free
classical TOA across the region of length $L$. On the other hand, the second
term is identified as the classical barrier traversal time.

The above classical limits suggest that the expected quantum traversal time
across the barrier region can be extracted from the second term of Eq. (7a).
Evaluation of $R_{n}^{*}$ using the Fourier transform of the full incident
wave function $\psi(q)$, i.e.,
$\tilde{\psi}(k)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}e^{-ikq}\psi(q)$, the
second term leads to

$\bar{\tau}_{B}=\sum_{n=1}^{2}\dfrac{w_{n}}{\nu_{0}}\int_{\kappa_{n}}^{\infty}dk\dfrac{|\tilde{\psi}(k)|^{2}-|\tilde{\psi}(-k)|^{2}}{\sqrt{k^{2}-\kappa_{n}^{2}}}.$
(8)

Equation (8) clearly shows the contribution of the positive and negative
momentum components of the incident wave function $\tilde{\psi}(k)$. Within
our framework, $\bar{\tau}_{B}$ can be interpreted as the dwell time in the
barrier region which is the total average time that our incident particle
spends in the barrier, regardless of transmission (where the positive
components dominate) or reflection.

The measurable quantum traversal time $\bar{\tau}_{trav}$ at the transmission
channel then comes from the positive momentum components. Hence, we can now
define the barrier traversal time as

$\bar{\tau}_{trav}=\,\sum_{n=1}^{2}\,\dfrac{\mu\,w_{n}}{\hbar\,k_{0}}\,\int_{\kappa_{n}}^{\infty}dk\,\dfrac{|\tilde{\psi}(k)|^{2}}{\sqrt{k^{2}-\kappa_{n}^{2}}}.$
(9)

To extract the time durations for the three tunneling processes, we rewrite
Eq. (9) as

$\displaystyle\bar{\tau}_{\text{trav}}=$
$\displaystyle\dfrac{L}{\nu_{o}}(R_{\text{part}}+R_{\text{non}}),$ (10a)
$\displaystyle R_{\text{part}}=$
$\displaystyle\dfrac{w_{1}k_{o}}{L}\int_{\kappa_{1}}^{\kappa_{2}}dk\dfrac{|\tilde{\psi}(k)|^{2}}{\sqrt{k^{2}-\kappa_{1}^{2}}},$
(10b) $\displaystyle R_{\text{non}}=$
$\displaystyle\dfrac{k_{o}}{L}\sum_{n=1}^{2}w_{n}\int_{\kappa_{2}}^{\infty}dk\dfrac{|\tilde{\psi}(k)|^{2}}{\sqrt{k^{2}-\kappa_{n}^{2}}}.$
(10c)

Both Eqs. (9) and (10a) clearly suggest that only the energy components
satisfying $k>\kappa_{1}$ contribute to any measurable quantum traversal time,
in keeping with the results of Refs. [32, 31]. In addition, we can extract
three specific traversal time regimes from Eq. (10a) depending on the support
of the distribution $|\tilde{\psi}(k)|^{2}$. These regimes coincide with our
definition of (i) non-tunneling or classical above-barrier traversal, (ii)
full-tunneling, and (iii) partial-tunneling processes.

The non-tunneling regime occurs when the distribution $|\tilde{\psi}(k)|^{2}$
has a compact support in $k>\kappa_{2}$ such that the full energy distribution
of the incident wave packet lies above the two potential barrier heights. The
quantity $\bar{\tau}_{\text{non}}=(L/\nu_{o})R_{\text{non}}$ from Eq. (10c)
then gives the expected above-barrier quantum traversal time and can be
rewritten as

$\bar{\tau}_{\text{non}}=\,\sum_{n=1}^{2}\,\dfrac{\mu w_{n}}{\hbar
k_{0}}\,\int_{\kappa_{2}}^{\infty}dk\,|\tilde{\psi}(k)|^{2}\,\tau(k),$ (11)

where $\tau(k)=\hbar\sqrt{k^{2}-\kappa_{n}^{2}}/\mu$ are just the classical
traversal times on top of entire barrier system.

The full-tunneling regime occurs when the support of $|\tilde{\psi}(k)|^{2}$
lies below $\kappa_{1}$, so that Eq. (9), or equivalently Eq. (10a), results
to a vanishing traversal time. Since all the energy components are below the
barrier heights, the vanishing traversal time is interpreted as instantaneous
tunneling time, that is,

$\bar{\tau}_{\text{tun}}=0.$ (12)

Last, the partial-tunneling regime occurs when the momentum distribution
$|\tilde{\psi}(k)|^{2}$ has components that lie between
$\kappa_{1}<k<\kappa_{2}$. These components that lie between
$\kappa_{1}<k<\kappa_{2}$ corresponds to a particle that traverses above
$V_{1}$ and tunnels through $V_{2}$. Thus, the particle did not tunnel through
the entire barrier system, hence, the name partial-tunneling. The quantity
$\bar{\tau}_{\text{part}}=(L/\nu_{o})R_{\text{part}}$ is now interpreted as a
“partial-traversal time” since Eq. (10b) indicates that the measured value
originates from the momentum components that traversed above the barrier
$V_{1}$.

We wish to highlight that the resolution of the quantum tunneling time problem
starts when everyone agrees that the term tunneling time be used to describe a
full-tunneling process. When an incident particle only partially tunnels
through a barrier region, one should avoid using the term tunneling time. For
our case, we used the term partial-traversal time. Within our framework, we
conclude that quantum tunneling, whenever it happens, is always instantaneous.

The above results hold in general. We can model an arbitrary potential barrier
$V(q)$ as a system of composite square barriers with varying heights and
widths, i.e., $V(q)=\sum_{n=1}^{\infty}V_{n}$ each having a width $w_{n}$. In
the continuous limit, $w_{n}\rightarrow 0$, the barrier traversal time is

$\displaystyle\bar{\tau}_{\text{trav}}=$
$\displaystyle\frac{L}{\nu_{o}}(R_{\text{part}}+R_{\text{non}}),$ (13a)
$\displaystyle R_{\text{part}}=$
$\displaystyle\dfrac{k_{o}}{L}\int_{b}^{a}dx\int_{\kappa(x)}^{\kappa_{\text{max}}}dk\dfrac{|\tilde{\psi}(k)|^{2}}{\sqrt{k^{2}-\kappa(x)^{2}}},$
(13b) $\displaystyle R_{\text{non}}=$
$\displaystyle\dfrac{k_{o}}{L}\int_{b}^{a}dx\int_{\kappa_{\text{max}}}^{\infty}dk\dfrac{|\tilde{\psi}(k)|^{2}}{\sqrt{k^{2}-\kappa(x)^{2}}},$
(13c)

where $\kappa_{\text{max}}=\sqrt{2\mu V_{\text{max}}}/\hbar$ indicates the
maximum value of the the barrier height and $\kappa(x)=\sqrt{2\mu
V(x)/\hbar^{2}}$.

Notice the distinction between traversal time and tunneling time. For the
general case when a wave packet has both above, below, and in-between barrier
energy components, the traversal time through a potential barrier is the sum
of the vanishing tunneling time $\bar{\tau}_{\text{tun}}$, non-zero partial
traversal time $\bar{\tau}_{\text{part}}$, and above-barrier traversal time
$\bar{\tau}_{\text{non}}$, i.e.,

$\bar{\tau}_{\text{trav}}=\bar{\tau}_{\text{tun}}+\bar{\tau}_{\text{part}}+\bar{\tau}_{\text{non}}=\bar{\tau}_{\text{part}}+\bar{\tau}_{\text{non}}.$
(14)

If one does not properly differentiate partial and full-tunneling processes,
the contribution of the partial traversal time may be mistakenly identified as
the tunneling time especially if one only considers tunneling with respect to
$\kappa_{\text{max}}$.

For a square potential barrier system, we can simply take $V(x)=V_{0}$ so that
$\bar{\tau}_{\text{part}}=\bar{\tau}_{\text{tun}}=0$. Quantum tunneling for
this case is only described by a full-tunneling process, and we recover the
predictions of Ref. [32] that only above barrier energy components contribute
to the barrier traversal time. For smooth barriers with compact support, Eq.
(13b) indicates that the partial traversal time
$\bar{\tau}_{\text{part}}=(L/\nu_{o})R_{\text{part}}$ will always be non-zero
since the $\kappa_{\text{min}}=0$, provided that a segment of the support of
$\tilde{\psi}(k)$ is below $\kappa_{\text{max}}$. Thus, quantum tunneling for
this case is only described by a partial-tunneling process.

The main advantage of our treatment in comparison with other tunneling time
definitions is its simplicity and generality. Specifically, other tunneling
time definitions such as the Büttiker-Landauer time, Larmor time, and Pollak-
Miller time involves calculating the transmission amplitude for propagating
through the barrier, which will require solving the Schödinger equation.
However, our treatment only requires information on the incident wavepacket
and the interaction potential, which allows it to be applicable in any system
without the need of any further calculations. Of course, our current analysis
is anchored on the assumption that the incident wavepacket does not initially
‘leak’ into the barrier. In addition, our treatment encompasses both non-zero
and zero traversal times depending on the initial state of the particle and
the shape of the potential barrier system. This is not obtained in other
approaches so it makes sense why they cannot explain the seemingly
contradictory reports in tunneling time experiments.

Following our results, we conclude that the non-zero tunneling time reported
in Ref. [36] is due to a partial-tunneling process. Specifically, the tunnel
barrier used was a multilayer dielectric mirror that has an $(HL)^{5}H$
structure, where $H$ represents titanium oxide with a refractive index of
$n_{H}=2.22$ while $L$ represents fused silica with a refractive index of
$n_{L}=1.41$. This setup is an optical analogue of a system of contiguous
square barriers with heights $V_{H}$ and $V_{L}$, where $V_{L}<V_{H}$, and can
be considered as a potential barrier with jump discontinuities. Since there
was no way that the momentum distribution of the incident photon can be
controlled such that all the momentum components are below $V_{L}$, then the
photon exhibits partial tunneling resulting to a non-zero partial traversal
time.

Figure 3: Potential curves for a Helium atom in the presence of an external
electric field $E$ with $Z_{eff}=1.6875$, $I_{p}=0.90357\mathrm{au}$. In
strong-field physics, an electron with bound state energy $-I_{p}$ tunnels
through the an effective potential barrier $V_{eff}=-Z_{eff}/q-Eq$ shown in
the red line. The above model is based on the analysis of Ref. [39].

We also argue that $\bar{\tau}_{\text{part}}$ may explain any upper bound in
the tunneling time of attoclock experiments which reported instantaneous
tunneling time [24, 25, 26, 22, 27]. In particular, theoretical modelling of
quantum tunneling in strong field-physics based on Simpleman’s model considers
an electron which starts from a bound state and tunnels through the barrier to
become a continuum state. The atomic Coulomb potential $V_{a}(q)=-Z_{eff}/q$
is distorted by an external electric field $E$ to create an effective
potential barrier $V_{eff}(q)=-Z_{eff}/q-Eq$ as shown in Fig. (3), where
$Z_{eff}$ is the effective nuclear charge [39]. Semiclassical analysis of
tunneling time considers the electron traversing through the barrier
$V_{eff}(q)$ of length $d$ with point $d$ being considered as the classical
tunnel exit. The corresponding tunneling time is called the Keldysh time.

It has been argued that the Keldysh time may substantially exceed the actual
tunneling time [40]. In our formalism, this process is exactly a partial-
tunneling process since the particle has an above-barrier energy component in
the region between the turning point $d_{+}$ and classical exit $d$. Assuming
there is a zero probability that the particle is already in the barrier
region, the non-zero Keldysh time corresponds to our non-vanishing partial-
traversal time and the reason why the former exceeds the actual tunneling time
is due to the contribution of the above-barrier energy components of the
incident particle. Hence, the Keldysh time should not be interpreted as the
tunneling time. Another way to show a substantial non-vanishing traversal time
is when the support of the particle’s initial state extends into the barrier
region which follows from the result of Ref. [31].

On the other hand, quantum tunneling based on Kullie’s model considers
tunneling where a particle enters the barrier region at point $d_{-}$ and
exits at point $d_{+}$ [41]. It is found that tunneling time decreases as a
function of increasing field strength $E$ [41, 40]. In our framework, this can
be explained by the decrease in the contribution of the above-barrier energy
components of the incident particle. For sufficiently strong electric field,
all energy components are exactly below the effective potential barrier so
that we get an instantaneous quantum tunneling.

It is often mentioned that the tunneling time using attoclock ionization
technique gives seemingly contradictory results. Some experiments suggest an
instantaneous quantum tunneling while some find the opposite result. One may
argue that our model does not exactly coincide with attoclock experiments.
However this is not entirely true since the character of these experiments
still reduce to our current model. That is, a quantum particle travels through
a potential barrier and is detected at the other side. The time it takes to
traverse the barrier is then measured. Following our results, we are led to
the hypothesis that experimentally measured non-zero tunneling times
correspond to partial-tunneling processes while vanishing tunneling times
correspond to full-tunneling processes.

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