# Direct Observation of Ultrafast Singlet Exciton Fission in Three Dimensions

Arjun Ashoka  Cavendish Laboratory, University of Cambridge, J. J. Thomson
Avenue, Cambridge CB3 0HE, United Kingdom  Nicolas Gauriot  Cavendish
Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE,
United Kingdom  Aswathy V. Girija  Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Nipun
Sawhney  Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom  Alexander J. Sneyd  Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United
Kingdom  Kenji Watanabe  Research Center for Functional Materials, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi
Taniguchi  International Center for Materials Nanoarchitectonics, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Jooyoung
Sung  Department of Emerging Materials Science, DGIST, Daegu, 42988 Republic
of Korea  Christoph Schnedermann  Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom  Akshay Rao
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge
CB3 0HE, United Kingdom<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present quantitative ultrafast interferometric pump-probe microscopy
capable of tracking of photoexcitations with sub 10 nm spatial precision in
three dimensions with 15 fs temporal resolution, through retrieval of the full
transient photoinduced complex refractive index. We use this methodology to
study the spatiotemporal dynamics of the quantum coherent photophysical
process of ultrafast singlet exciton fission. Measurements on microcrystalline
pentacene films grown on glass (SiO2) and boron nitride (hBN) reveal a 25 nm,
70 fs expansion of the joint-density-of-states along the crystal a,c-axes
accompanied by a 6 nm, 115 fs change in the exciton density along the crystal
b-axis. We propose that photogenerated singlet excitons expand along the
direction of maximal orbital $\pi$-overlap in the crystal a,c-plane to form
correlated triplet pairs, which subsequently electronically decouples into
free triplets along the crystal b-axis due to molecular sliding motion of
neighbouring pentacene molecules. This methodology opens a new route for the
study of three dimensional transport on ultrafast timescales.

##  I Introduction

Elucidating the three dimensional transport of excitations in condensed matter
is key to advancements in our understanding and utilisation of functional
materials ranging from novel quantum systems to next-generation optoelectronic
materials AndersonOriginal ; QHE ; Blumberger ; PRLTroisi . Of particular
interest are quantum coherent processes in heterogeneous, disordered systems
which require both ultrafast time resolution and local measurements to study
and understand their transport characteristics STLInGaN ;
zong2021unconventional ; NatPhyCoherence . Singlet exciton fission is a widely
studied example of such a process that has gained relevance in the fields of
photovoltaics and quantum computing. Here a photo generated singlet exciton
converts to an electronically and spin entangled pair of triplets at nearly
half the singlet energy on ultrafast timescales. The correlated triplet pair
then separates into individual triplet excitons through the loss of electronic
and spin coherence Miyata_2019 ; Chan2013 ; Rao2017 ; Smyser2020 . Numerous
ultrafast studies have probed the coherent dynamics of the singlet fission
process, however direct observation of the real space three dimensional
spatial dynamics of ultrafast singlet fission remains an outstanding goal
Miyata_2019 ; Rao2017 .

The field of optical pump-probe microscopy, which extends standard pump-probe
spectroscopy to a microscope geometry, is well suited to study coherent
transport processes in condensed matter systems, as it provides direct
visualisation of the transport of excitations Sung2019 ; EIs ; Delor2019 .
Conventionally, optical pump-probe microscopy has used point-scanning
methodologies which provide a two-dimensional picture of the transport process
with 10s of nm resolution and 100 fs time resolution. In contrast, reports of
widefield optical pump-probe microscopy have demonstrated that interferometric
contrast could provide a way to visualise three dimensional transport
Delor2019 . However, while the use of phenomenological Gaussian point-spread-
functions to describe the measured images can yield sub-10 nm lateral
precision, it cannot quantify out-of-plane transport Sung2019 ; Pandya ; Sneyd
; Delor2019 ; Schnedermann2019 . Moreover, the measured images naturally arise
from changes in the full complex refractive index $n+ik$, which cannot be
retrieved using such phenomenological Gaussian point-spread-functions, which
in turns rules out quantitative measurement of the transient joint-density-of-
states (JDOS).

Here we introduce quantitative interferometric ultrafast pump probe
microscopy. Using a first-principles analytic optical model, we show it is
possible to fully describe these interferometric pump-probe images and
quantify changes in the full transient complex refractive index $n(t)+ik(t)$
and retrieve both lateral and out-of-plane transport with sub-10 nm precision,
with 15 fs time resolution (see Supplemental Material 6 for localisation
precision and Supplemental Material 4 demonstration of time resolution). We
report a full three dimensional picture of ultrafast singlet fission in
polycrystalline pentacene films, suggesting that the photoexcited singlet
exciton expands along the direction of maximal orbital $\pi$-overlap
(a,c-axes) to form correlated triplet pairs, which subsequently electronically
decouples into free triplets along the crystal b-axes due to lattice modes
that drive the intermolecular sliding motion of neighbouring pentacene
molecules.

Figure 1: Experimental setup and optical model. a) A schematic of the
propagation of a time delayed weakly interacting probe through a locally
perturbed three dimensional sample imaged on a camera. b) Transient
transmission and extracted transient complex refractive index spectra,
measured bands indicated in grey. c) Radially averaged differential
transmission images measured at different planes through a film of pentacene
150 fs after photoexcitation. The radial averages are mirrored about the x=0
plane for clarity. Dashed lines indicate the interferometrically enhanced
planes chosen for measurement. d) Microscope image of the microcrystalline
domains of the measured pentacene film on hBN and SiO2 substrates, with
labelled crystal geometries relative to the substrate. (Scale bar = 20 $\mu m$
)

##  II Experimental Setup and Optical Model

Our experimental setup (Fig 1a) is based on a transmission widefield pump-
probe microscope equipped with an oil immersion objective (numerical aperture
= 1.1). A pump pulse (560 nm, 13 fs) is focused onto the sample by the
objective to produce a near diffraction-limited local photoexcitation
(Supplemental Material 1). After a variable time delay, a counter-propagating
widefield probe pulse (750 nm, 7 fs, $\sim$20 $\mu$m FWHM) is transmitted
through the sample and imaged onto an emCCD detector. The effect of the pump
pulse is to photoinduce a three-dimensional spatially varying, local complex
refractive index change, $\Delta\tilde{n}=\Delta n+i\Delta k$ (Fig 1a). This
index change weakly perturbs the time-delayed plane-wave probe pulse incident
on the sample, leading to local changes in its phase and amplitude
SheikBahae1990 . The large background unperturbed probe field interferes with
this perturbed probe field to form a spatial interference pattern along the
propagation direction. The objective and imaging lens then image this spatial
interference pattern combined with the attenuation of the probe on a camera.

Our optical model is based on the treatment of the diffraction-limited optical
perturbation produced by the pump pulse in a thin film semiconductor as a well
defined local complex refractive index perturbation, similar to a gold
nanoparticle or polystyrene bead (Fig 1a). Before the pump pulse arrives, when
the system is in the ground state (pump-off), the polarisation $P$ measured by
the probe is given by, $P=\epsilon_{0}\chi^{(1)}E$. Using $D=\epsilon
E=\epsilon_{0}E+P$, the ground state dielectric function $\epsilon_{off}$ is
therefore given by, $\epsilon_{off}=\epsilon_{0}(1+\chi^{(1)})$. After the
arrival of the pump pulse, the system is in the excited state (pump-on), the
polarisation $P$ measured by the probe is given by,
$P=\epsilon_{0}(\chi^{(1)}E+\chi^{(3)}E_{pu}E_{pu}E)$, where $E_{pu}$ is the
pump electric field. Similarly, the excited state dielectric function
$\epsilon_{on}$ is therefore given by,
$\epsilon_{on}=\epsilon_{0}(1+\chi^{(1)}+\chi^{(3)}E_{pu}E_{pu})$. As the
probe pulse is always temporally separated from the pump, time-ordering allows
us to treat the pump-on and pump-off probe signals as measures of the
photoexcited and ground state dielectric functions (or complex refractive
index) of the material, respectively. The overall perturbation to the
dielectric function can therefore be written as,

$\epsilon_{on}-\epsilon_{off}=\Delta\epsilon=\epsilon_{0}\chi^{(3)}E_{pu}^{2},$
(1)

where $\chi^{(3)}$ is the third order non-linear susceptibility,
$\epsilon_{0}$ is the permittivity of free space and $E_{pu}$ is the pump
electric field Wang1994 ; Conforti . In order to link this to the refractive
index, we use the fact that as $\tilde{n}=\sqrt{\epsilon}$,
$\Delta\tilde{n}=\frac{1}{2\sqrt{\epsilon}}\Delta\epsilon$ for small
perturbations. As the time integrated intensity of the pump absorbed by the
material is given by $I_{pu}=\frac{cn_{0}\epsilon_{0}}{2}E_{pu}^{2}$, the
photoinduced refractive index change can be written as,

$\Delta\tilde{n}(r,z)=\frac{1}{2{n_{0}}}\chi^{(3)}\frac{2}{cn_{0}}I_{pu}(r,z)$
(2)

which importantly shows that upon photoexcitation, $\Delta\tilde{n}$ is
proportional to the intensity of the pump pulse.

As the intensity at the back-focal-plane of the objective is a TEM00 mode,
exploiting radial symmetry to average over polarisation effects, the pump is a
focused Gaussian beam attenuated through the depth of the sample,

$I_{pu}(r,z)=I_{0}\left[{\frac{\sigma_{0}}{\sigma(z)}}\right]^{2}\exp\left[{\frac{-r^{2}}{2\sigma(z)^{2}}}-\frac{\omega_{pu}\alpha
z}{c}\right]$ (3)

where $\sigma(z)=\sigma_{0}{\sqrt{1+{\left({\frac{z}{z_{R}}}\right)}^{2}}}$,
$z_{R}$ being the Rayleigh range, $\omega_{pu}$ is the pump frequency and
$\alpha$ is the dimensionless, thickness corrected extinction coefficient for
the pump. For thin film samples of thickness less than $z_{R}$, and ignoring
in-plane anisotropy, $\sigma$ can be approximated to be constant
($\mathcal{O}\left(z/z_{R}\right)^{2}$) through the thickness of the sample.
We can therefore approximate the photoinduced refractive index change as,

$\Delta\tilde{n}(r)=(\Delta n_{0}+i\Delta
k_{0})\exp\left[\frac{-r^{2}}{2\sigma_{0}^{2}}-\frac{\omega_{pu}\alpha
z}{c}\right]$ (4)

over the sample length, where we have absorbed the constants and $\chi^{(3)}$
into the transient optical constants $\Delta n_{0}+i\Delta k_{0}$
$=\frac{1}{2\tilde{n_{0}}}\chi^{(3)}\frac{2}{cn_{0}}I_{0}$. We compute the
near-field electric field by multiplying the incident field with the laterally
varying Fresnel complex transmission coefficients of a Gaussian disc situated
about the photoexcitation’s principal plane $z_{c}$,

$E^{nf}(r,z_{c})=E_{i}[(1-r_{1}(r)][1-r_{2}(r))]t(r),$ (5)

where $t=e^{iL\tilde{n}\kappa}$ captures the attenuation of $E_{i}$ through
the sample, $r_{1}=\frac{n_{air}-\tilde{n}}{n_{air}+\tilde{n}}$ captures the
reflection at the air-sample interface and
$r_{2}=\frac{\tilde{n}-n_{s}}{\tilde{n}+n_{s}}$, captures the substrate-sample
interface. As the microscope system we study is an oil-immersion system, the
substrate-oil interface just before the objective is near index matched and
not taken into consideration. Here $n_{s}$ is the substrate refractive index,
$L$ is the sample thickness and $\kappa$ is the probe wavevector. The
amplitude and phase of $E^{nf}$ therefore encodes three dimensional spatial
information of photoexcited carrier transport through $\Delta\tilde{n}(r)$ and
$z_{c}$.

The probe electric field at the objective’s input aperture can be approximated
as the Fourier transform of the near-field electric field,
$E^{ff}(k_{r},z_{c})=\mathfrak{F}(E^{nf}(r,z_{c}))$, where $k_{r}$ is the
spatial frequency of the probe in the radial direction and $\mathfrak{F}$ is
the Fourier transform. To study the spatial interference of the probe in
different planes in the object space where the field is interferometrically
enhanced, we calculate the far-field electric field of the plane $\Delta
z=z_{0}-z_{c}$ by propagating the plane wave decomposition an extra distance
$\Delta z$, $E^{ff}(k_{r},\Delta z)=e^{i\Delta
z\sqrt{\kappa^{2}-k_{r}^{2}}}E^{ff}(k_{r},z_{c})$.

To calculate the image on the camera by the objective-imaging lens system, we
compute an inverse Fourier transform after filtering the high spatial
frequencies as the objective’s NA specifies, yielding,

$E^{im}(r^{\prime},\Delta z,n,k)=\int_{-\kappa NA}^{\kappa
NA}E^{ff}(k_{r},\Delta z)e^{-ik_{r}r^{\prime}}dk_{r}$ (6)

which is a simplified version of the Richards-Wolf integral RichardsandWolf1 ;
BornWolf:1999:Book ; Gibson1989 .

We calculate the widefield normalised differential transmitted image of a
photoexcited refractive index change (at a given probe wavelength),
$\Delta\tilde{n}(r)=\Delta n(r)+i\Delta k(r)$ centred at $z_{c}=z_{0}-\Delta
z$ on a spatially constant, static background $\tilde{n}_{0}(r)=n_{0}+ik_{0}$
centered about $z_{0}$,

$\frac{\Delta T}{T}(r^{\prime},z_{0}-\Delta
z,\Delta\tilde{n})=\frac{|E^{im}(r^{\prime},\Delta
z,\tilde{n_{0}}+\Delta\tilde{n}(r))|^{2}}{|E^{im}(r^{\prime},z_{0},\tilde{n_{0}})|^{2}}-1.$
(7)

Experimental control over the image plane $z_{0}$ is achieved by translating
the imaging lens in the infinity space of the objective to relay different
conjugate planes to the camera RFiSCAT . The effective axial distance
accessible by moving the imaging lens by a distance $z_{0}^{\prime}$ is given
by the axial magnification of the imaging system (Supplemental Material 7).
Ensuring that neither the imaging lens nor the objective move during a pump-
probe measurement fixes $z_{0}$, allowing us to track changes in $z_{c}$,
$\sigma$ and transient refractive index $\Delta\tilde{n}$ as a function of
pump-probe delay.

Our model therefore characterises the normalised differential transmitted
images in terms of the physical parameters $\Delta\tilde{n}(r,\sigma,z_{0})$,
the probe wavelength and the sample thickness. We benchmark our model against
Finite-Difference-Time-Domain (FDTD) calculations and find excellent
agreement, demonstrating that our near-field approximations and the
computational challenging aberrations are systematically cancelled in the
differential far-field images (Supplemental Material 2 and 3). Critically,
this enables us to extract the three dimensional transport and $\Delta
n_{0}+i\Delta k_{0}$ through fitting the measured data set, which would be
prohibitively computationally challenging using purely FDTD methods.

##  III Results

Figure 2: Triplet decoupling dynamics at 740 nm and 250 $\mu$J cm-2. a)
Measured radially averaged $\Delta T/T$ map and spatially averaged signal
kinetic. b) Fit of the optical model to the radial average at 13, 70 and 1000
fs. c) Schematic of the measured pentacene film on hBN and SiO2 substrates,
with labelled crystal geometries relative to the substrate. d,e)Transient
optical constants display the expected signs based on Fig 1b and show an
ultrafast rise followed by a constant JDOS. f) Transport in the a,c-crystal
plane measured in-plane on SiO2 (blue) and along the a-crystal direction
measured out-of-plane on hBN (orange) is absent. g) Transport in the
b,c-crystal plane measured in-plane on hBN (orange) and along the b-crystal
direction measured out-of-plane on SiO2 (blue) displays a few nm 115 fs change
in the exciton density (dashed line is fits to an exponential).

The process of ultrafast singlet fission is considered to occur in three
steps. First, the photogenerated singlet $S_{1}$ converts to an electronically
and spin entangled triplet pair state $TT$. Second, through the loss of
electronic correlation the $TT$ state converts to a spatially separated
triplet pair $T...T$. Finally, through the loss of spin correlation, the
spatially separated spin entangled triplet pair $T...T$ separates into two
uncorrelated triplets $T+T$ Miyata_2019 . As spin coherence is typically lost
on longer timescales than the electronic coherence in polyacenes and as we
probe the electronic states through their optical transitions, we solely focus
on the loss of electronic correlation from $TT$ to $T...T$ and make no comment
on the spin correlation Miyata_2019 . We study the dynamics of singlet exciton
fission in two well-explored spectral bands of the archetype organic
semiconductor pentacene: the photobleaching band at 670 nm and the
photoinduced absorption band at 740 nm (Fig 1b) Wilson2011_Pc . It has been
established that the 670 nm band primarily tracks the ground-state bleach and
hence contains the JDOS of both the photoexcited singlet $S_{1}$ and resulting
entangled $TT$ and uncorrelated $T...T$ triplet pairs Wilson2011_Pc ;
Lochbrunner2007 . The 740 nm feature however, is not present immediately after
photoexcitation and tracks the JDOS of only the entangled $TT$ and separated
$T...T$ triplet pair through transitions to higher lying triplet states. We
describe the transitions in these systems through their JDOS rather than as
single energetic transitions as in extended thin film systems, the molecules
are not isolated and the density of states cannot be treated as one-
dimensional.

Analysis of the transient transmission spectra based on a Kramers-Kronig
differential dielectric function predicts $\Delta n>0$ and $\Delta k<0$ at 670
nm and $\Delta n<0$ and $\Delta k>0$ at 740 nm (Fig 1b) 2021extracting . We
demonstrate our interferometic sensitivity to the photoexcitation by measuring
the differential transmitted images 150 fs after photoexcitation at both
wavelengths through several image planes (Fig 1c), demonstrating three
dimensional point-spread-functions that are reminiscent of those reported in
state-of-the-art static interferometric scattering microscopy iscatAxialLong .
We access different image planes by translating the imaging lens in the
infinity space of our microscope to relay different planes from the image
space onto the camera, similar to the concept of remote focusing RFiSCAT . The
degree of interferometic constrast determines our ability to resolve out-of-
plane carrier transport and accurately retrieve the transient refractive
indices. We therefore study spectral bands with a non-zero real refractive
index change and a suitable imaging plane within the axial focus (Fig 1c,
dashed lines) (Supplemental Material 7) Unser_Limits ; Wang1994 .

In order to unravel the three dimensional dynamics of singlet fission in
microcrystalline pentacene films, we study pentacene films evaporated on SiO2
and on hexagonal boron nitride (hBN). On SiO2 pentacene crystallises with its
b-axis near-perpendicular to the substrate, whereas on hBN pentacene
crystallises with the b-axis in-plane (Fig 1d and Supplemental Material 8)
Amsterdam2020 . This enables us to validate any measured out-of-plane
transport against in-plane transport in the orthogonal orientation as well as
confirm the direction of the out-of-plane transport. We begin by studying the
uncongested photoinduced absorption band at 740 nm to establish the spatial
dynamics of the triplet excitons and use this information to study the more
complicated dynamics of the ground state bleach band at 670 nm.

###  III.1 Loss of Triplet Pair Electronic Correlation

Figure 3: Singlet fission dynamics at 670 nm and 250 $\mu$J cm-2. a) Measured
radially averaged $\Delta T/T$ map and spatially averaged signal kinetic. b)
Fit of the optical model to the radial average at 13, 70 and 1000 fs. c)
Schematic of the measured pentacene film on hBN and SiO2 substrates, with
labelled crystal geometries relative to the substrate. d,e) Transient optical
constants display the expected signs based on Fig 1b along with a sub 1-ps
feature associated with the singlet fission process and subsequent constant
JDOS. f) Transport in the a,c-crystal plane measured in-plane on SiO2 (blue)
and along the a-crystal direction measured out-of-plane on hBN (orange)
displays a 25 nm 70 fs expansion which can be resolved in $\sigma_{a,c}$
(dashed line is exponential fit). g) Transport in the b,c-crystal plane
measured in-plane on hBN (orange) and along the b-crystal direction measured
out-of-plane on SiO2 (blue) displays a 30 nm 180 fs expansion (dashed lines
are exponential fit).

We study the $TT$ and $T...T$ exciton dynamics in the 740 nm photoinduced
absorption (PIA) band (on both SiO2 and hBN) at 250 $\mu$J cm-2 where singlet
fission is the dominant photophysical process (Fig 2) TTA_Poletayev . At these
densities, there is on average one photoexcitation per 100 pentacene
molecules. As shown in Fig 2a, there are spatiotemporal changes to the signal
during the first 500 fs. Our model is able to capture the measured
differential transmission image in this regime (Fig 2b) (see Supplemental
Material 5 for details on the fitting procedure). A PIA band is expected to
result from an increased JDOS, as the triplets are formed via the fission
process and as the JDOS $\propto$ Im($\tilde{\epsilon}$), where
$\tilde{\epsilon}$ is the dielectric function and
$\tilde{\epsilon}=\tilde{n}^{2}$, $\Delta k>0$ is expected at 740 nm. We
retrieve the expected signs of $\Delta n$ and $\Delta k$ (Fig 1b) and find an
ultrafast rise due to formation of triplets and a subsequently constant JDOS
(Fig 2d,e). Transport of the photoexcitation in the a,c-crystal plane measured
through $\sigma_{a,c}$ on SiO2 and $\Delta z_{a}$ on hBN (Fig 2c,f) is absent
to within our noise floor of 4 nm peak-to-peak. However, transport along the
b-crystal direction measured through $\Delta z_{b}$ on SiO2 displays clear few
nm transport with a timescale of 115 $\pm$ 11 fs (Fig 2g). When measured in
the rotated crystal orientation on hBN, this transport appears as the 6 nm
change in the exciton density in the b,c-crystal plane measured through
$\sigma_{b,c}$ (Fig 2g).

The 6 nm, 115 fs change in the exciton density along the crystal b-axis at 740
nm must correspond to a photophysical process relating exclusively to the
triplet population. This process must be distinct from the formation of the
$TT$ state which would be correlated with the rise of this spectral feature.
Further, the 115 fs timescale is distinct from previously reported $TT$
formation timescales in pentacene Wilson2011_Pc . Hence this process must be
related to the loss of electronic correlation in the triplet pair. Recent
reports suggest that a 1 THz (1 ps period) lattice vibration in pentacene
crystals associated with sliding motion of neighbouring pentacene molecules
along the crystal b-axis modulates the $\pi$-overlap and therefore the
J-coupling between adjacent pentacene molecules, resulting in triplet pair
separation from $TT\rightarrow T...T$ SciAdv1Thz . Our measured 6 nm change in
the spatial triplet exciton density along the same crystal b-axis over 500 fs
could be related to a lattice distortion arising from this 1 THz mode which
changes the local excitonic density along this axis. A full 1 ps oscillation
period of the 1 THz sliding mode is not needed to decouple the $TT$ state to
the $T...T$ state, and as we show below, free triplets are formed within 200
fs of photoexcitation. We additionally note that the triplet exciton in
pentacene is known to be polarised along the b-axis which further suggests a
strong change in polarisability due to this mode PRB_Pc . Further theoretical
investigations of the response of the excitonic wavefunction to phonon modes
along this axis in the vibronic and transient delocalisation framework are
called for Bakulin_2015 ; Alvertis2020 ; Giannini2022 .

###  III.2 Singlet to Correlated Triplet Transition

Figure 4: Fast triplet annihilation dynamics at 740 nm and 750 $\mu$J cm-2.a)
Measured radially averaged $\Delta T/T$ map and spatially averaged signal
kinetic. b) Fit of the optical model to the radial average at 45, 70 and 1000
fs.c) Schematic of the measured pentacene film on hBN and SiO2 substrates,
with labelled crystal geometries relative to the substrate. d,e) Transient
optical constants display the expected signs based on Fig 1b and the JDOS
decays due to triplet-triplet annihilation expected at these fluences. f,g)
Apparent transport due to the density dependent TTA cross-section is seen on
both SiO2 and hBN and appears to be isotropic.

We study the spatio-temporal dynamics the $S_{1}$ exciton via the 670 nm
photobleaching band in the same fluence regime of 250 $\mu$J cm-2 (Fig 3). As
shown in Fig 3a, there are spatiotemporal changes to the signal during the
first 500 fs. Our model captures the measured differential transmission image
and we are able to retrieve the correct signs of the $\Delta\tilde{n}$ based
on Fig 1b (Fig 3b,d,e) 2021extracting . We recall that the 670 nm band is
congested due to overlapping spectral features of the $S_{1}$, $TT$, $T...T$
and different oscillator strengths which makes a quantitative interpretation
of $\Delta\tilde{n}(t)$ at 670 nm challenging. Transport of the
photoexcitations in the a,c-crystal plane of pentacene is tracked through
$\sigma_{a,c}$ on SiO2 and $\Delta z_{a}$ on hBN (Fig 3c). We observe a 25 nm
expansion in $\sigma_{a,c}$ with a timescale of 72 $\pm$ 13 fs, but no
measurable correlated transport in $\Delta z_{a}$ (Fig 3f). We note that the
high localisation precision is significantly diminished on this spectral band
as the required interferometic contrast can only be gained through
substantially defocussing ($|\Delta z|>500$ nm at 670 nm compared to $|\Delta
z|<50$ nm at 740 nm), where the remote focussing model begins to fail (see
Supplemental Material 7).

As the triplet ($TT$, $T...T$) excitons studied at 740 nm and display no
transport in the a,c-crystal plane, any transport of in the a,c-crystal plane
measured at the 670 nm ground state bleach necessarily arises from the $S_{1}$
exciton. The 70 fs timescale matches previously reported timescales of singlet
fission that are sensitive to the $S_{1}\rightarrow TT$ transition Rao2017 ;
Wilson2011_Pc . The presence of this feature in the crystal-a,c plane is
consistent with the fact that the J-coupling between pentacene molecules
required for the $S_{1}\rightarrow TT$ transition is maximal in direction of
maximal $\pi$-overlap, i.e., the crystal-a,c plane Zaykov2019  (Fig 1). This
suggests that the $S_{1}\rightarrow TT$ transition occurs in the crystal-a,c
plane which results in a spatial expansion of the exciton density potentially
due to the formation of charge transfer (CT) states yielding a velocity
$\mathcal{O}(10^{5})$ m s-1. Theoretical calculations of singlet fission in
solid pentacene using ab-inito Green’s function methods predict a singlet
exciton bandwidth of 100 meV over the unit cell, which yield an estimate of a
group velocity of $\mathcal{O}(10^{4})$ m s-1, suggesting that this transport
phenomena cannot be rationalised as typical coherent transport of the $S_{1}$
state PRLSivan . The slower timescale of the expansion along the b-crystal
direction measured in the b,c-crystal plane of pentacene is tracked through
$\sigma_{b,c}$ on hBN is, however, difficult to interpret due to the
overlapping transport of the triplets at 740 nm along the same direction (Fig
3g).

###  III.3 Triplet-Triplet Annihilation

Lastly, at high excitation densities, triplet-triplet annihilation (TTA) can
dominate photo-physics of pentacene and influence subsequent triplet transport
and decay pathways. To study the effects of TTA in the ultrafast regime, we
study the 740 nm band in the high fluence regime at 750 $\mu$J cm-2 (Fig 4)
TTA_Poletayev . At these densities, there is on average one photoexcitation
per 30 pentacene molecules and the spatiotemporal dynamics show a decay over
the first picosecond (Fig 4a). Even in the high density limit, our model
satisfactorily captures the measured data demonstrating robustness to the
magnitude of the external $\tilde{n}$ perturbation (Fig 4b). We find that both
$\Delta n$ and $\Delta k$ decay as a function of time as would be expected for
a reduction in the JDOS of triplets through a TTA process (Fig 4d,e).

As the exciton-exciton annihilation cross-section scales with the exciton
density, the decay rate varies spatially over the photoexcitation profile.
With time, this leads to a flat-topped exciton density profile, which when
approximated as a Gaussian in Eqn. 4, results in an apparent expansion
EEA_Deng2020 . We resolve the apparent expansion from TTA in the a,c-crystal
plane through both measured in-plane through $\sigma_{a,c}$ on SiO2 and out-
of-plane through $\Delta z_{a}$ on hBN (Fig 4c,f). We resolve a similar
expected apparent expansion in the b-crystal direction, suggesting that the
TTA process is isotropic (Fig 4g). The fast timescale for the onset of TTA
suggests that such free triplets are formed within 200 fs in pentacene as TTA
typically occurs for spatially separated triplets, i.e. $T...T$. This fast
triplet annihilation process is must be to higher lying triplet states and not
back to the singlet manifold as the latter process is endothermic and would be
expected to occur on longer timescales. The fast sub-200 fs transport along
the b-crystal axis seen in the low-fluence regime at 250 $\mu$J cm-2 also
appears to be present in the high density regime, suggesting that this feature
is related to the singlet fission process.

##  IV Conclusion

To summarise, we have utilised interferometric pump-probe microscopy to reveal
the three dimensional picture of singlet fission and triplet annihilation in
microcrystalline pentacene films with sub 10 nm spatial precision and 15 fs
temporal resolution. Our results suggest that the photoexcited singlet exciton
expands along the direction of maximal orbital $\pi$-overlap in the crystal
a,c-plane to form correlated triplet pairs, which subsequently decouple into
free triplets along the crystal b-axis due to molecular sliding motion of
neighbouring pentacene molecules. The fast formation of free triplets in
pentacene results in the fast onset of isotropic triplet annihilation dynamics
at high excitation densities, critical to applications of singlet fission that
utilise triplets. Our technique is not limited to studying excitons in
pentacene through optical pump-probe techniques on femtosecond timescales, but
can be applied for other pump-probe schemes involving electron or X-ray
sources over any experimentally accessible timescale Tatiana . Going forward
our approach will enable direct insights into the transport of excitations in
a range of condensed matter systems over a variety of timescales.

##  V Methods

###  V.1 Pump-probe microscope setup

A pump pulse (560 nm, 13 fs) is focused onto the sample by the objective to
produce a near diffraction-limited local photoexcitation. After a variable
time delay, a counter-propagating widefield probe pulse (750 nm, 7 fs,
$\sim$20 $\mu$m FWHM) is transmitted through the sample and imaged onto an
emCCD detector. Widefield probe images in the presence and absence of the pump
excitation are recorded by chopping the pump pulse at 80 Hz. The pump and
probe pulses are derived from a Yb:KGW amplifier (1030 nm, 5 W, 200 kHz,
LightConversion) via white-light-continuum generation and subsequent spectral
filtering and compression with chirped mirrors (Supplemental Material 1)
Sung2019 ; Schnedermann2019 .

###  V.2 Fabrication of the hbN - Pentacene Sample

100 nm hBN flakes were exfoliated from bulk crystals on glass substrate. 100
nm on pentacene was then evaporated on the sample. Thin films of pentacene
were prepared by thermal evaporation in an ultrahigh vacuum environment of
$10^{(}-8)$ mbar at a constant evaporation rate of 0.02 nm/s. The rate was
monitored using a calibrated quartz crystal microbalance and the deposition
was stopped once the desired film thickness of 100 nm was obtained. The
deposition was done onto 1 mm-thick quartz substrates and 0.2 mm-thick glass
coverslips that were cleaned by sequential sonication in acetone and isopropyl
alcohol.

##  VI Acknowledgements

A.A acknowledges funding from the Gates Cambridge Trust and the as well as
support from the Winton Programme for the Physics of Sustainability. A.V.G
acknowledges funding from the European Research Council Studentship and
Trinity-Henry Barlow Scholarship. C.S. acknowledges financial support by the
Royal Commission of the Exhibition of 1851. We acknowledge financial support
from the EPSRC and the Winton Program for the Physics of Sustainability. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement no. 758826).

##  VII Competing interests

The authors declare that there are no competing financial interests.

##  VIII Data and materials availability

The data that support the plots within this paper and other findings of this
study are available at the University of Cambridge Repository
(https://doi.org/XXXXX).

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# Distinguishability dynamics and quantum speed limits in the Bloch ball

T.J. Volkoff<EMAIL_ADDRESS>K.B. Whaley  Dept. of Chemistry, UC Berkeley,
Berkeley, CA 94720

###### Abstract

We derive a necessary and sufficient condition on a time-independent
Hamiltonian and arbitrary initial state of a two-level quantum system which
guarantees that the initial state will time evolve to another state from which
it can be distinguished with an a priori established minimal error probability
$\delta\in[0,1/2]$. The time required for the initial state to evolve to such
a probabilistically distinguishable state is compared to previously
established quantum speed limits giving lower bounds on the time required for
evolution between nonorthogonal states. We use the explicit distinguishability
time to provide a succinct derivation of the quantum brachistochrone, i.e.,
the Hamiltonian generating shortest time evolution between two states having
equal magnitude Bloch vectors. As a consequence of the distinguishability
condition, we find that if a given unitary does not evolve a given pure state
to an orthogonal state, then there exist mixed states in the Bloch ball that
time evolve faster than the given pure state and that reach a greater level of
distinguishability under the given unitary. We show that such mixed states
exist even in the case of nonunitary evolution by considering a two-level
system resonantly coupled to a single mode electromagnetic field.

##  I Introduction

The minimal length of time required for a given quantum state to evolve to an
orthogonal state under unitary time evolution provides an ultimate bound for
the processing speed of a quantum computer, regardless of the physical
substrate used for the quantum information processing Lloyd (2000). Orthogonal
states also form a valuable resource for quantum communication and for the
efficiency of quantum algorithms Kitaev et al. (1999). However, in practice,
perfectly orthogonal states are not always achievable; for this reason, it is
not surprising that the problem of optimally distinguishing elements of a set
of nonorthogonal quantum states continues to be subject of active research
(see, e.g., Refs.Bergou et al. (2004); Samsonov (2009)). Methods for
generation and manipulation of nonorthogonal states are vital for high
precision control of quantum dynamics and for optimal covariant quantum state
estimation Holevo (1982). The unavoidability of nonorthogonal initial and
final states of realistic quantum dynamics has led to the study of generalized
quantum speed limits, i.e., lower bounds on the minimal time required for an
initial state to evolve into a state which is imperfectly distinguishable from
the initial state Volkoff and Whaley (2014); Giovannetti et al. (2003).

In this paper we consider the more general question of determining a simple
necessary and sufficient condition on a given time-independent Hamiltonian $H$
and initial qubit state $\rho$ indicating that the unitary time evolution
generated by $H$ produces a state that is probabilistically distinguishable
(with maximal success probability $1-\delta$) from $\rho$. The proof of such a
condition requires a notion of distinguishability time, which we briefly
define. In order to operationally define a distinguishability time
$\tau_{\delta}(\rho,\mathcal{E}_{t},\Delta)$, representing the minimal time
required for an initial quantum state $\rho$ to time evolve to a state from
which it is probabilistically distinguishable with error $\delta$, one must
specify in addition to a parametrized quantum dynamical map $\mathcal{E}_{t}$,
a discrimination prodecure $\Delta$ that maps the pair
$(\rho,\mathcal{E}_{t}(\rho))$, consisting of initial state $\rho$ and its
time evolved counterpart $\mathcal{E}_{t}(\rho)$, to the interval $[0,1/2]$.
This interval represents the probability of unsuccessful discrimination
events. Then one can define

$\tau_{\delta}(\rho,\mathcal{E}_{t},\Delta):=\min\\{t|\Delta(\mathcal{E}_{t}\rho,\rho)=\delta\\}.$
(1)

Importantly, in order for $\tau_{\delta}(\rho,\mathcal{E}_{t},\Delta)$ to be
well-defined, there must exist a time $t>0$ such that
$\Delta(\rho,\mathcal{E}_{t}\rho)=\delta$. The calculation of
$\tau_{\delta}(\rho,\mathcal{E}_{t},\Delta)$ for a given time-evolution and
discrimination procedure may require solving for the full path
$\\{\mathcal{E}_{t}\rho|t\geq t_{0}\\}$ and optimizing over a set of quantum
measurements determined by the decision procedure $\Delta$. The general
solution of this poses a considerable challenge. However, determination of a
range of precisions $\delta$ in $[0,1/2]$ such that
$\tau_{\delta}(\rho,\mathcal{E}_{t},\Delta)$ is meaningful can be obtained
from, e.g., an upper bound on $\Delta(\rho,\mathcal{E}_{t}\rho)$ for all $t$.
In addition, it is useful to derive tight lower bounds on $\tau_{\delta}$ that
reveal the physical properties of the state and the dynamics that lead to the
true value of the distinguishability time.

The present definition of $\tau_{\delta}$ is easily generalizable to the case
of optimal multistate distinguishability dynamics Samsonov (2009) and to maps
$\mathcal{E}_{t}$ parametrized by an arbitrary smooth manifold instead of a
line. The discrimination procedure $\Delta$ implicitly depends on a set of
available measurements and postprocessing of measurement results. We define a
symmetric relation on the set of quantum states termed
“$(1-\delta)$-distinguishability” which will be useful in the discussion that
follows by the following relationship: $\rho$ and $\sigma$ are
$(1-\delta)$-distinguishable if and only if $\Delta(\rho,\sigma)=\delta$.

In the present work, we focus primarily on the case in which $\mathcal{E}_{t}$
is unitary evolution generated by a time-independent Hamiltonian,
$\\{\mathcal{E}_{t}(\rho)|t\geq 0\\}$ is a path in the quantum state space of
a two-level system, and $\Delta$ is the minimal error probability for binary
quantum state discrimination Helstrom (1976). Not all Hamiltonians generate a
unitary time-evolution taking a given initial state to an orthogonal state or
even to a $(1-\delta)$-distinguishable state for a given value of $\delta$
(results along these lines for various finite dimensional Hilbert spaces can
be found in Ref. Morley-Short et al. (2014)). Our main theorem completely
determines the relationship between distinguishability of states on a unitary
path in the Bloch ball and the statistical geometry of that path. Given a
Hamiltonian and a distinguishability parameter $\delta\in[0,1/2]$, the theorem
allows qubit states to be split into two classes: 1) those initial states
that, under unitary evolution generated by $H$, become
$(1-\delta)$-distinguishable from the starting point, and 2) those initial
states that do not. For the first class, the theorem allows one to introduce a
partial order on the states based on the minimal time required to get to
$(1-\delta)$-distinguishability, i.e., the quantum speed limits of their
respective orbits in the Bloch ball. As a consequence, we determine the subset
of the Bloch ball that reaches $(1-\delta)$-distinguishable states faster than
a given pure state under the evolution generated by an arbitrary time-
independent qubit Hamiltonian.

A brief outline of the paper is as follows: in Section II, we review two lower
bounds on the distinguishability time when $\Delta$ is taken to be the minimum
error binary quantum state discrimination procedure and compare these bounds
to the true unitary distinguishability time of a qubit. Section III contains
the main theorem providing a necessary and sufficient condition for evolution
to a $(1-\delta)$-distinguishable state and an explicit expression for the
distinguishability time of a two-level system. We use the main theorem to
provide a short solution to the problem of the quantum brachistochrone in
Section IV. In Section V, we extend the analysis to non-unitary qubit dynamics
and show that a set of mixed states evolving to $(1-\delta)$-distinguishable
states faster than a given pure state exists even in the presence of amplitude
damping due to coupling to a monochromatic electromagnetic field. We also use
this example to illustrate the role of the initial field state in controlling
the minimal distinguishability time, showing how this can be tailored to speed
up arrival to a target level of distinguishability.

##  II Distinguishability time

Rigorous notions of uncertainty tradeoffs between measurements of energy and
time have been developed in terms of the orthogonalization time, i.e., the
minimum time required for an initial quantum state to evolve under the action
of a given (unitary or nonunitary) quantum dynamical map to a state from which
it is completely distinguishable Dodonov and Dodonov (2015); Giovannetti et
al. (2003); Deffner and Lutz (2013); Taddei et al. (2013). In traditional
approaches to time-energy uncertainty, a decay time or half-life of a quantum
system scales inversely with the root mean square energy fluctuations of the
system. These approaches were made mathematically rigorous by the derivation
of a lower bound on the pure state orthogonalization time which scales
inversely with the variance of the generator of evolution (we call this bound
MT⟂ after the seminal work of Mandelstam and Tamm Mandelstam and Tamm (1945)
which was put on a geometric footing by Aharanov and Anandan Anandan and
Aharonov (1990)).

The orthogonalization time can also be bounded below by a function of the
expected value of the generator of evolution (we call such a bound ML⟂ after
Margolus and Levitin Margolus and Levitin (1996)). An important difference
between the ML⟂ and MT⟂ bounds is that the former is a kinematic bound,
resulting from a linear approximation to the fidelity of the initial state and
the time-evolved state, while the latter was derived by consideration of
geodesics of an appropriate metric on quantum state space.

Recently, the ML⟂ and MT⟂ bounds have been generalized to bounds on the
distinguishability time of Eq.(1) for general quantum states evolving under
unitary maps Volkoff and Whaley (2014). When the unitary path is generated by
$H=H^{\dagger}$ and
$\Delta(\rho,\sigma):=p_{\text{err}}(\rho,\sigma)=1/2-1/4\|\rho-\sigma\|_{1}$,
where $p_{\text{err}}(\rho,\sigma)$ is the minimal error probability for
binary state discrimination of quantum states $\rho$ and $\sigma$ Helstrom
(1976), the time $t$ required for $\rho$ to reach a
$(1-\delta)$-distinguishable state is bounded below by the following
distinguishability times:

$\displaystyle\tau_{\delta}^{MT}$ $\displaystyle=$
$\displaystyle{2\sin^{-1}(1-2\delta)\over\sqrt{\mathcal{F}(\rho,H)}}$ (2)
$\displaystyle\tau_{\delta}^{ML}$ $\displaystyle=$
$\displaystyle{\pi\hbar(1-\sqrt{1-(1-2\delta)^{2}})\over 2(\text{tr}(\rho
H)-E_{0})}$ (3)

where $E_{0}$ is the least eigenvalue of $H$ and $\mathcal{F}(\rho,H)$ is the
quantum Fisher information on the unitary path containing $\rho$ and generated
by $H$ (see Ref.Helstrom (1976) or Section III for a definition). Clearly, the
unified distinguishability bound satisfies $\lim_{\delta\rightarrow
0}\max\\{\tau_{\delta}^{MT},\tau_{\delta}^{ML}\\}=\max\\{\text{MT}_{\perp},\text{ML}_{\perp}\\}$.
For a two-level system with time-evolution generated by
$H=\hbar\omega_{0}\vec{n}\cdot\vec{\sigma}$, we will see in Section III that
the states $c_{1}|{0}\rangle_{\vec{n}}+c_{2}|{1}\rangle_{\vec{n}}$ with
$|c_{1}|=|c_{2}|=1/\sqrt{2}$ are the only ones that saturate
$\tau_{\delta}^{MT}$; not surprisingly, these are also the only states
saturating the ML⟂ and MT⟂ bounds Horesh and Mann (1998); Margolus and Levitin
(1996).

For a two-level system in a state
$\rho(\vec{r})=\mathbb{I}+\vec{r}\cdot\vec{\sigma}/2$ evolving by the
Hamiltonian $H=\hbar\omega_{0}(\vec{n}\cdot\vec{\sigma}+\mathbb{I})$ (the
identity is added so that $H$ has positive semidefinite spectrum), application
of Eq.(3) yields

$\tau_{\delta}^{ML}={\pi(1-\sqrt{1-(1-2\delta)^{2}})\over
2\omega_{0}(\vec{n}\cdot\vec{r}+1)}.$ (4)

This bound is not consistent with the rotational symmetry of the dynamics. For
example, the state with Bloch vector $\rho(-\vec{r})$ gives a different bound
than that for $\rho$. The bound $\tau_{\delta}^{ML}$ remains valid if
$\vec{n}\cdot\vec{r}$ is replaced by $|\vec{n}\cdot\vec{r}|$. However, our
explicit calculation of $\mathcal{F}(\rho,H)$ in Section III, combined with
the fact that $\sin^{-1}(1-2\delta)\geq(\pi/2)(1-\sqrt{1-(1-2\delta)^{2}})$
for $\delta\in[0,1/2]$, leads to the conclusion that for $\delta\in(0,1/2)$,
the Mandelstam-Tamm bound $\tau_{\delta}^{MT}$ is greater than the Margolus-
Levitin bound $\tau_{\delta}^{ML}$. Hence we will focus here on
$\tau_{\delta}^{MT}$ as the lower bound on the distinguishability time for the
two-level system. Similar to ML⟂, we expect $\tau_{\delta}^{ML}$ to be useful
for analysis of the distinguishability dynamics of systems with constant
energy but large quantum Fisher information, which for pure states is
equivalent to large fluctuations of the energy. One potentially important
application of the quantity $\tau_{\delta}^{ML}$ is thus for analysis of
distinguishability dynamics in incompressible liquids with large heat capacity
near thermal phase transitions.

##  III Single qubit distinguishability dynamics

Before we state the main theorem, we make note of some of its corollaries for
pure states which are well-known. Consider a Hamiltonian
$H=\hbar\omega_{0}\vec{n}\cdot\vec{\sigma}$ (where $\|\vec{n}\|=1$ and
$\vec{\sigma}:=(\sigma_{x},\sigma_{y},\sigma_{z})$) and an initial state
$|{\psi}\rangle$. $H$ has operator norm $\hbar\omega_{0}$ and we will set
$\omega_{0}=1$ for convenience. By acting on $|{\psi}\rangle$ with the time-
evolution operator $U(t):=e^{-iHt/\hbar}$ to produce $|{\psi(t)}\rangle$, one
finds that a time $t$ such that $\langle\psi|\psi(t)\rangle=0$ exists if and
only if the Bloch vector $\vec{r}$ representing $|{\psi}\rangle$ on the Bloch
sphere is orthogonal to $\vec{n}$. These states are superpositions
$|{\phi(\varphi)}\rangle={1\over\sqrt{2}}(|{0}\rangle_{\vec{n}}+e^{i\varphi}|{1}\rangle_{\vec{n}})$
of the lowest and highest energy states ($|{1}\rangle_{\vec{n}}$ and
$|{0}\rangle_{\vec{n}}$, respectively) of $H$. Such superpositions define a
great circle of states on the Bloch sphere having Bloch vector orthogonal to
$\vec{n}$. A measurement of the observable $H$ in a state
$|{\phi(\varphi)}\rangle$ has variance $1$, the largest possible value for all
pure states. For any $\varphi$, the state $U(t)|{\phi(\varphi)}\rangle$ is
orthogonal to $|{\phi(\varphi)}\rangle$ when $t=\pi/2$.

On the other hand, there are no completely distinguishable mixed states (i.e.,
nontrivial convex combinations of pure states) in the Bloch ball. Furthermore,
if the initial state is mixed, it cannot be distinguished completely from any
pure state. Mathematically, these facts follow from the fact that mixed states
of the Bloch ball have rank two and so they cannot have support which is
disjoint from the support of any other state of the Bloch ball. Hence, when
the initial state is mixed, there is no hope to achieve $1$-distinguishability
through any type of evolution, unitary or nonunitary. However, the evolution
may still result in a $(1-\delta)$-distinguishability of initial and final
states for some $\delta>0$. Here, we consider unitary evolutions which result
in $(1-\delta)$-distinguishability between the initial and final states
instead of perfect distinguishability (i.e., $1$-distinguishability) and
derive the set of quantum states which evolve to
$(1-\delta)$-distinguishability faster than a given pure state. In the finite
dimensional case considered here, the condition of
$(1-\delta)$-distinguishability of two qubit states is made easier by the fact
that the trace norm $\|\cdot\|_{1}$ appearing in the expression
$p_{\text{err}}(\rho,\sigma)$ can be calculated as the sum of the absolute
values of the eigenvalues of $\rho-\sigma$. For the statement of the main
theorem, we again take $H=\hbar\omega_{0}\vec{n}\cdot\vec{\sigma}$ and
$\omega_{0}=1$. The proof is made easier by the use of a simple lemma.

Lemma Let $\rho_{0}={\mathbb{I}\over 2}+{\vec{r}_{0}\cdot\vec{\sigma}\over 2}$
be an initial quantum state on the unitary path
$\rho_{t}=e^{-iHt/\hbar}\rho_{0}e^{iHt/\hbar}$ generated by $H$. Then the
quantum Fisher information on this path satisfies

$\mathcal{F}(\rho_{t},H)=4\|\vec{n}\times\vec{r}_{0}\|^{2}.$ (5)

for all $t\geq 0$.

Proof By definition, $\mathcal{F}(\rho_{t},H):=\text{tr}(L^{2}\rho_{t})$,
where $L=L^{\dagger}$ is the symmetric logarithmic derivative operator, i.e.,
the unique observable satisfying ${d\rho_{t}\over dt}={1\over
2}(L\rho_{t}+\rho_{t}L)$ for all $t$. $L$ depends on the state $\rho_{t}$
through its Bloch vector $\vec{r}_{t}$ and also on the vector $\vec{n}$
defining $H$. The von Neumann equation gives ${d\rho_{t}\over
dt}=-i/\hbar[H,\rho_{t}]$, so that $L$ must satisfy ${1\over
2}[L,\rho_{t}]_{+}=-i/\hbar[H,\rho_{t}]=(\vec{n}\times\vec{r}_{t})\cdot\vec{\sigma}$
for all $t$. Writing $L=\vec{v}_{t}\cdot\sigma$ and solving for $\vec{v}_{t}$
results in $\vec{v}_{t}=2(\vec{n}\times\vec{r}_{t})$. Taking the variance of
the resulting operator $L=2(\vec{n}\times\vec{r}_{t})\cdot\vec{\sigma}$ in the
state $\rho_{t}$ and using that fact that $\text{tr}(\rho L)=0$ gives the
result $\mathcal{F}(\rho_{t},H)=4\|\vec{n}\times\vec{r}_{t}\|^{2}$. The Bloch
vector $\vec{r}_{t}$ satisfies a quantum equation of motion that is the same
as the classical equation of motion for a magnetic moment in a constant
magnetic field which gives rise to Larmor precession, and so
$\|\vec{n}\times\vec{r}_{t}\|=\|\vec{n}\times\vec{r}_{0}\|$ for all $t$. In
this classical analogy, the constant value of the norm corresponds to
conservation of angular momentum. $\square$

Because the observable $L$ has units of $[t]^{-1}$, $\mathcal{F}(\rho,H)$ has
units of $[t]^{-2}$. The geometric relationships among the symmetric
logarithmic derivative, the Hamiltonian, and the state $\rho$ can be seen in
Figure 1a). More general expressions for the symmetric logarithmic derivative
and quantum Fisher information for a qubit have been obtained previously Jiang
(2014), but for the case of unitary evolution the vectorial expression is
exceptionally useful. In the context of time-dependent quantum magnetometry
with a collection of qubits, Eq.(5) reproduces the relevant quantum Fisher
information appearing in the quantum Cramér-Rao bound Liu et al. (2015).
Clearly, when $\|\vec{r}\|=1$, i.e., when $\rho$ is pure, the quantum Fisher
information takes the well known value $\mathcal{F}(\rho,H)=4\text{tr}((\Delta
H)^{2})\rho)/\hbar^{2}$ Braunstein and Caves (1994). We now state and prove
the main theorem.

Theorem 1 There exists a $t\geq 0$ such that an initial quantum state
$\rho={1\over 2}({\mathbb{I}}+{\vec{r}\cdot\vec{\sigma}})$ evolves to a state
$U(t)\rho U(t)^{\dagger}$ satisfying $p_{\text{err}}(\rho,U(t)\rho
U(t)^{\dagger})=\delta$ if and only if:

$2(1-2\delta)\leq\sqrt{\mathcal{F}(\rho,H)}.$ (6)

where $\|\vec{a}\|=(\vec{a}\cdot\vec{a})^{1/2}$ is the Euclidean norm of
$\vec{a}\in\mathbb{R}^{3}$.

Proof It follows from the algebra of the Pauli matrices that

$\rho-U(t)\rho
U(t)^{\dagger}=\sin^{2}(t)(\vec{r}-(\vec{r}\cdot\vec{n})\vec{n})\cdot\vec{\sigma}+\sin
t\cos t(\vec{r}\times\vec{n})\cdot\vec{\sigma}.$ (7)

We derive the conditions on $\vec{r}$ which guarantee the existence of $t$
such that $p_{\text{err,min}}(\rho,U(t)\rho U(t)^{\dagger})=\delta$ is
satisfied. Evaluating the trace norm of Eq.(7) allows one to write the
following expression for $p_{\text{err}}(\rho,U(t)\rho U(t)^{\dagger})$:

${1\over 2}-{1\over 2}\|\sin^{2}t(\vec{r}-(\vec{r}\cdot\vec{n})\vec{n})+\sin
t\cos t(\vec{r}\times\vec{n})\|.$ (8)

The expression in (8) is equal to $\delta$ if and only if

${1-2\delta\over\sqrt{\|\vec{r}\|^{2}-(\vec{r}\cdot\vec{n})^{2}}}=|\sin t|.$
(9)

A value of $t$ satisfying the above equation exists if and only if
${1-2\delta\over\sqrt{\|\vec{r}\|^{2}-(\vec{r}\cdot\vec{n})^{2}}}\leq 1$.
Using the Lemma, we have
$\sqrt{\mathcal{F}(\rho,H)}=2\|\vec{n}\times\vec{r}\|=2\sqrt{\|\vec{r}\|^{2}-(\vec{r}\cdot\vec{n})^{2}}$,
which was required. $\square$

An immediate corollary of Theorem 1 is that pure qubit states are the only
states for which there exists an orthogonalizing unitary evolution. In
addition, it is clear from expanding the vector norm in Eq.(8) that the
minimal error for distinguishing $\rho$ and $\rho(t)$ occurs at $t=\pi/2$ (in
units of $\omega_{0}^{-1}$). It follows from Eq.(9) that if $\rho$ saturates
the inequality (6), then $p_{\text{err}}(\rho,U(\pi/2)\rho
U(\pi/2)^{\dagger})=\delta$, i.e., $\rho(t)$ is $(1-\delta)$-distinguishable
from $\rho$ at time $t=\pi/2$.

Having derived the necessary and sufficient condition for a given qubit state
to reach a $(1-\delta)$-distinguishable state under a given unitary evolution,
we are equipped to find the set of states that evolve to
$(1-\delta)$-distinguishable states in less time than the given state. In
particular, it follows as a corollary of Theorem 1 that there are mixed states
that evolve more quickly to $(1-\delta)$-distinguishable states than certain
pure states, as long as $\delta>0$. To make this clear, we take
$H=\hbar\omega_{0}\sigma_{z}$ without loss of generality and first note that
the maximal quantum Fisher information of all paths in the Bloch ball
generated by $H$ is achieved for the pure states
$|{0}\rangle+e^{i\varphi}|{1}\rangle/\sqrt{2}$ ($\varphi\in[0,2\pi)$); these
are the “fastest” time-evolving states, reaching $(1-\delta)$-distinguishable
states in time $\sin^{-1}(1-2\delta)\leq\pi/2$. Now, consider a pure state
$|{\psi}\rangle$ with Bloch vector given by angular parameters
$(\theta_{\psi},\varphi_{\psi})$ on the Bloch sphere with
$\theta_{\psi}=\sin^{-1}(1-2\delta)$ (Fig. 2). Then Eq.(9) implies that
$\tau_{\delta}(|{\psi}\rangle,e^{-i\omega_{0}t\sigma_{z}},p_{\text{err}})=\pi/2$.
It follows from Theorem 1 that the set $S$ of states defined by
$S=\\{\rho|\sqrt{\mathcal{F}(\rho,H)}/2\geq\sin\theta_{\psi}=1-2\delta\\}$
reach $(1-\delta)$-distinguishable states and do so in a time
$t\in[\sin^{1}(1-2\delta),\pi/2]$, i.e., in a time less than or equal to the
$(1-\delta)$-distinguishability time of $|{\psi}\rangle$. The set of states
satisfying this condition lie in the spherical ring illustrated in Fig. 2.
Hence, to find the set of states evolving faster than a given state $\rho$,
one must only look for the states $\sigma$ such that
$\mathcal{F}(\sigma,H)\geq\mathcal{F}(\rho,H)$.

Figure 1: The magnetic field vector $\vec{n}$, the Bloch vector $\vec{r}$, and
the direction vector $2(\vec{n}\times\vec{r})$ of the symmetric logarithmic
derivative plotted relative to the 2-sphere. The square root of the quantum
Fisher information is equal to the operator norm $\|L\|$ of $L$ which is twice
the shaded area.  Figure 2: Given a pure quantum state $|{\psi}\rangle$ with
Bloch vector corresponding to polar angle $\theta_{\psi}$ and Hamiltonian
$H=\hbar\omega_{0}\sigma_{z}$, the shaded region containing $\rho$ represents
those states with a shorter distinguishability time for all values of
$\delta$.

##  IV Distinguishability on minimal time paths

As an example of utility of the notion of distinguishability time, we provide
a simple solution of the problem of the quantum brachistochrone, i.e., given
two states $\rho_{1}$, $\rho_{2}$ with equal Bloch vector magnitudes, to
identify a Hamiltonian $H_{*}$ generating a unitary path such that $\rho_{1}$
evolves to $\rho_{2}$ in the shortest possible time. Previous approaches to
this problem make use of a variational principle on projective Hilbert space
Carlini et al. (2006) or saturating an upper bound on the energy fluctuations
in a 2-dimensional Hilbert subspace Mostafazadeh (2009). Our solution solves
the brachistochrone for two mixed states with equal Bloch vector magnitudes
and the pure state brachistochrone follows as a corollary when the Bloch
vectors are on the Bloch sphere.

Theorem 2 Let $\rho_{1}$, $\rho_{2}$ be quantum states of a two-level system
and have Bloch vectors $\vec{r_{1}}$, $\vec{r_{2}}$, respectively, with
$\|\vec{r_{1}}\|=\|\vec{r_{2}}\|$. Then the Hamiltonian $H_{*}$ having
operator norm $\hbar\omega_{0}$ which takes $\rho_{1}$ to $\rho_{2}$ in the
minimal time is given by

$H_{*}=\hbar\omega_{0}{\vec{r_{1}}\times\vec{r_{2}}\over\|\vec{r_{1}}\times\vec{r_{2}}\|}\cdot\vec{\sigma}.$
(10)

Proof The assumptions of the theorem come only from the observation that
unitary dynamics of a mixed state with Bloch vector magnitude $R$ take place
on a 2-dimensional sphere of radius $R$. To prove the Theorem, let
$\omega_{0}=1$ and consider the ansatz $H_{*}=\hbar\vec{q}\cdot\vec{\sigma}$
with $\|\vec{q}\|=1$. We assume that there exists $t$ such that
$\rho_{2}=e^{-iH_{*}t/\hbar}\rho_{1}e^{iH_{*}t/\hbar}$. Then the time evolved
Bloch vector of $\rho_{1}$ satisfies

$\cos(2t)\vec{r_{1}}-\sin(2t)(\vec{r_{1}}\times\vec{q})+2\sin^{2}(t)(\vec{r_{1}}\cdot\vec{q})\vec{q}=\vec{r_{2}}.$
(11)

Taking the Euclidean dot product of Eq.(11) with $\vec{r_{1}}$ gives the
equation

$\sin^{2}t={\|\vec{r_{1}}\|^{2}-\vec{r_{1}}\cdot\vec{r_{2}}\over
2\sin^{2}\theta_{1}\|\vec{r_{1}}\|^{2}}$ (12)

where $\theta_{1}$ is defined as the angle between $\vec{r_{1}}$ and
$\vec{q}$. The time $t$ is minimal when $\theta_{1}=\pi/2$; we will call this
time $T$. If $\theta_{1}$ takes this value, then it is clear from Eq.(11) that
$\vec{q}\cdot\vec{r_{2}}=0$. Up to a factor of $\pm 1$, this specifies
$\vec{q}={\vec{r_{1}}\times\vec{r_{2}}\over\|\vec{r_{1}}\times\vec{r_{2}}\|}$.
To finish the proof, we show that taking $H_{*}=\vec{q}\cdot\vec{\sigma}$ with
$\vec{q}={\vec{r_{1}}\times\vec{r_{2}}\over\|\vec{r_{1}}\times\vec{r_{2}}\|}$
(i.e., taking the “+” sign) causes $\rho_{1}$ to unitarily evolve to
$\rho_{2}$ in the minimal possible time allowed by Eq.(9). Since the minimal
error probability for distinguishing $\rho_{1}$ and $\rho_{2}$ is
$\delta={1/2}-{1/4}\|\vec{r_{1}}-\vec{r_{2}}\|$ and using
$\|\vec{r_{1}}\|=\|\vec{r_{2}}\|$, we have
$2(1-2\delta)=\|\vec{r_{1}}-\vec{r_{2}}\|=\sqrt{2(\|\vec{r_{1}}\|^{2}-\vec{r_{1}}\cdot\vec{r_{2}})}$.
For $\theta_{1}=\pi/2$, $\mathcal{F}(\rho_{1},H_{*})=4\|\vec{r}_{1}\|$. Hence,
Eq.(12) shows that $\rho_{1}$ evolves to $\rho_{2}$ on the path generated by
$H_{*}$ in time

$T=\sin^{-1}{\|\vec{r}_{1}-\vec{r}_{2}\|\over\sqrt{\mathcal{F}(\rho_{1},H_{*})}}=\sin^{-1}{(1-2\delta)\over\sqrt{\mathcal{F}(\rho_{1},H_{*})}}.$
(13)

Therefore, according to Eq.(9), $H_{*}$ gives the shortest time unitary
evolution taking $\rho_{1}$ to $\rho_{2}$. $\square$

The form of Eq.(10) could have been anticipated by noting that the goal is to
identify the unitary path connecting $\rho_{1}$ and $\rho_{2}$ that has
greatest quantum Fisher information (i.e., greatest Bures line element
Braunstein and Caves (1994)) and then noting that the Bures line element is
just a Euclidean line element on Bloch vectors Hubner (1992). Hence, one seeks
a great circle on the sphere of radius $\|\vec{r_{1}}\|$ connecting $\rho_{1}$
and $\rho_{2}$ in the Bloch ball. The quantum Fisher information on the path
$e^{-itH_{*}/\hbar}\rho_{1}e^{itH_{*}/\hbar}$ is

$\mathcal{F}(\rho_{1},H_{*})={4(1-\|r_{1}\|^{2}\cos^{2}(\varphi_{12}))\over\sin\varphi_{12}}$
(14)

where $\varphi_{12}\in[0,\pi]$ is the angle between $\vec{r_{1}}$ and
$\vec{r_{2}}$.

To obtain the pure state brachistochrone, consider two linearly independent
pure states $|{\psi_{1}}\rangle$, $|{\psi_{2}}\rangle$ with
$\langle\psi_{1}|\psi_{2}\rangle=:z\in\mathbb{R}$ to span a Hilbert subspace,
form the orthonormal basis with elements

$|{e_{\pm}}\rangle=|{\psi_{1}}\rangle\pm|{\psi_{2}}\rangle/\sqrt{2+2z}.$ (15)

Then the state $|{\psi_{1}}\rangle\langle{\psi_{1}}|$
($|{\psi_{2}}\rangle\langle{\psi_{2}}|$) has Bloch vector
$\vec{n_{1}}=(\sqrt{1-z^{2}},0,z)$ ($\vec{n_{1}}=(-\sqrt{1-z^{2}},0,z)$) and,
from Eq.(10), one has $H_{*}=-\hbar\omega_{0}\sigma_{y}$, where
$\sigma_{y}=-i|{e_{+}}\rangle\langle{e_{-}}|+\text{h.c.}$. Rewriting
$-\sigma_{y}$ gives the quantum brachistochrone for two pure states:

$H_{*}={-i\hbar\omega_{0}\over\sqrt{1-z^{2}}}\left(|{\psi_{1}}\rangle\langle{\psi_{2}}|-|{\psi_{2}}\rangle\langle{\psi_{1}}|\right).$
(16)

Note also that under brachistochrone evolution to a pure state
$|{\psi_{2}}\rangle$, an initial pure state $|{\psi_{1}}\rangle$
orthogonalizes to the state $\propto z|{\psi_{1}}\rangle-|{\psi_{2}}\rangle$
after a time $\pi/2$.

##  V Cavity qubit distinguishability

In Theorem 1, we have established the necessary and sufficient condition for
an arbitrary qubit state to reach a $(1-\delta)$-distinguishable state under
unitary evolution in the Bloch ball. The restriction to unitary evolution is
made here primarily on account of the relevance of such maps for quantum
computations, where Stinespring’s dilation theorem Stinespring (1955) allows
one to consider any completely positive, trace-preserving map as a unitary map
on an amplified system consisting of the original system and some ancilla
qubits. However, within current physical realizations of quantum computers, it
is not a trivial task to simply append the needed ancilla qubits and carry out
the unitary map; it may therefore often be more efficient to focus efforts on
engineering dissipative quantum evolutions that optimally take initial states
to the desired final states in the Bloch ball.

With this in mind, we now consider non-unitary distinguishability dynamics by
introducing a resonant interaction of the qubit with a single-mode
electromagnetic field, e.g., in a cavity quantum electrodynamics (CQED)
experiment. We shall analyze the distinguishability of the initial qubit state
from its reduced time evolved counterpart. In the case of unitary evolution,
the only way to decrease the distinguishability time for a given initial qubit
state is to increase the energy scale $\hbar\omega_{0}$ of the qubit.
Intuitively, one expects that a more energetic quantum state moves faster in
Hilbert space and this is verified by the inverse scaling of the ML⟂ and MT⟂
bounds with the operator norm of the Hamiltonian. However, for a qubit
resonantly coupled to a lossless single-mode electromagnetic field, coherent
control of the initial field state can allow the qubit distinguishability time
to be tuned for any given value of the qubit energy splitting.

Figure 3: Minimal error probability for distinguishing an initial pure state
in the $xz$-plane of the Bloch sphere from
$\text{tr}_{E}(U(t)(\rho_{S}\otimes|{\psi}\rangle\langle{\psi}|)U^{\dagger}(t))$
for $t\in[0,100\omega_{0}^{-1}]$ with $|{\psi}\rangle=|{\alpha}\rangle$ and a)
$\alpha=3$ b) $\alpha=1$. c) Reduced distinguishability dynamics of qubit
initially in $\rho_{S}(0)=|{0}\rangle\langle{0}|$ coupled to various initial
field states: $|{\psi}\rangle=|{\alpha}\rangle$ (black, dashed);
$|{\psi}\rangle=|{e_{0}}\rangle\langle{e_{0}}|$ given by Eq.(20) (black,
solid); $|{\psi}\rangle\propto|{\alpha}\rangle+|{-\alpha}\rangle$ (blue);
$|{\psi}\rangle\propto|{\alpha}\rangle-|{-\alpha}\rangle$ (red). $\alpha=3$
for all curves.

We calculate the quantity $p_{\text{err}}(\rho_{S}(t),\rho_{S}(0))$ where
$\rho_{S}(0)$ is the initial qubit state, $|{\psi}\rangle$ is the initial
(pure) state of the field, and
$\rho_{S}(t)=\text{tr}_{E}(V(t)(\rho_{S}(0)\otimes|{\psi}\rangle\langle{\psi}|)V(t)^{\dagger})$
is the reduced time-evolved state of the qubit (Fig.3). The unitary $V(t)$ is
the time-evolution operator generated by the Jaynes-Cummings Hamiltonian
Klimov and Chumakov (2009)

$H_{JC}:=\hbar\omega_{0}(a^{\dagger}a+{\sigma_{z}\over
2})+g(a^{\dagger}\sigma_{-}+a\sigma_{+})$ (17)

at field frequency $\omega_{0}=1$, zero detuning, and qubit-field coupling
$g=\omega_{0}/20$. We parametrize the initial qubit state by

$\rho_{S}(0)={\mathbb{I}\over 2}+{\|\vec{r}\|\over
2}({\cos(\theta)}\sigma_{x}+{\sin(\theta)}\sigma_{z})$ (18)

for ease of visualization. The reduced time-evolved qubit state $\rho_{S}(t)$
is calculated with the Kraus representation of the reduced dynamics defined in
the Fock state representation,

$\rho_{S}(t)=\sum_{n=0}^{100}E_{n}(t)\rho_{S}(0)E_{n}(t)^{\dagger},$ (19)

where $E_{n}(t):=\langle n|V(t)|\psi\rangle$ is a bounded operator on the
qubit Hilbert space. We employ $N=100$ Kraus operators, corresponding to
contributions from field photon Fock states up to $N=100$ and vary the initial
product state of qubit and field as described below.

Figs. 3a) and 3b) show the time dependence of the minimal error probability
for distinguishing the time-evolved reduced qubit state from the initial qubit
state when the initial field state is a coherent state with variable
intensity, i.e., $|{\psi}\rangle=|{\alpha}\rangle$, $\alpha\in\mathbb{C}$. Two
effects of the field intensity $|\alpha|^{2}$ on the minimal error of optimal
discrimination between $\rho_{S}(0)$ and $\rho_{S}(t)$ are evident here: i)
increasing the field energy causes large amplitude oscillations in the
distinguishability with respect to time, and ii) the timescale of revivals of
distinguishability is extended by increasing the field energy (Fig. 3a)). In
the frame corotating with the qubit, the timescale of the fast oscillations of
$p_{\text{err}}$ is governed by the quantum Rabi frequency
$f_{n}:=g\sqrt{n}/\hbar$ associated to each Fock component of the initial
field state $|{\psi}\rangle$. In the limit of zero detuning, the relevant
timescale of revivals of the distinguishability for an initially coherent
field is $\mathcal{O}(\hbar|\alpha|/g)$.

The reduced distinguishability dynamics of a two-level quantum system in an
arbitrary environment are simplified by the fact that a two-level system can
only undergo two dynamical processes: 1) emission/absorption (i.e., losing or
gaining energy), and 2) dephasing (i.e., gaining or losing quantum coherence
in a chosen basis). The form of the qubit-field interaction term in the
Jaynes-Cummings model, Eq.(17), suggests that the timescales associated to
oscillations, collapses, and revivals of distinguishability should depend on
specific features of the photon number distribution of the initial field state
$|{\psi}\rangle$. For example, starting with an initially excited qubit state
$\rho_{S}(0)=|{0}\rangle\langle{0}|$, the off-diagonal terms of
$\rho_{S}(0)-\rho_{S}(t)$ depend linearly on the product $\langle n\pm
1|\psi\rangle\overline{\langle n|\psi\rangle}$ and its complex conjugate. By
preparing the field in an even or odd coherent state
$|{\psi}\rangle\propto|{\alpha}\rangle\pm|{-\alpha}\rangle$, these dephasing
contributions to the distinguishability vanish. This is illustrated in Fig. 4,
where the solid red and blue traces reveal that at a constant intensity
$|\alpha|^{2}=9$, the qubit exhibits distinguishability revivals of greater
amplitude and at shorter times in the even/odd coherent fields than in the
coherent field.

Figure 4: Reduced distinguishability dynamics of qubit initially in
$\rho_{S}(0)=|{0}\rangle\langle{0}|$ coupled to various initial field states:
(black, dashed) $|{\psi}\rangle=|{\alpha}\rangle$; (black, solid)
$|{\psi}\rangle=|{e_{0}}\rangle\langle{e_{0}}|$ given by Eq.(20); (blue)
$|{\psi}\rangle\propto|{\alpha}\rangle+|{-\alpha}\rangle$; (red)
$|{\psi}\rangle\propto|{\alpha}\rangle-|{-\alpha}\rangle$ ($\alpha=3$ for all
curves).

The damping (diagonal) contribution to $\rho_{S}(0)-\rho_{S}(t)$ can also be
tuned in order to change the timescale of distinguishability revivals.
Consider the initial field state $|{e_{0}}\rangle$ given by

$|{e_{0}}\rangle:={|{\alpha}\rangle+|{-\alpha}\rangle+|{i\alpha}\rangle+|{-i\alpha}\rangle\over
2e^{-|\alpha|^{2}/2}\sqrt{2\cosh|\alpha|^{2}+2\cos|\alpha|^{2}}}$ (20)

which exhibits a photon number distribution having support on $n$ such that
$n=0\text{ mod }4$ Volkoff (2015). Like the even and odd coherent states, this
field state does not provide a dephasing contribution to
$\rho_{S}(0)-\rho_{S}(t)$. However, for this state, only the Kraus operations
$E_{m}(t)$ with $m\in\\{1,3\\}\text{ mod }4$ contribute to the dynamics. In
the solid black trace of Fig. 3c), this feature is seen to prevent the
complete destructive interference of quantum Rabi frequencies from occurring
at short timescales.

Note that in this work, we do not consider the reduced distinguishability
dynamics of the field, but focus solely on the qubit dynamics. Analysis of the
photon statistics, field entropy, and other physical characteristics of the
reduced state of the field were analyzed in Ref.Buz̆ek et al. (1992).

An analog of Theorem 1 for the reduced qubit distinguishability dynamics would
require an analytical relation between the time-dependent quantity
$p_{\text{err}}(\rho_{S}(0),\rho_{S}(t))$ and the time-dependent quantum
Fisher information of the qubit. In Fig. 5, we demonstrate instead by example
that given a pure state with Bloch vector not orthogonal to the $z$-axis,
there are mixed states that evolve to a $(1-\delta)$-distinguishable state
faster than the given pure state even in the presence of nonunitary dynamics.
As expected from the linear field-qubit coupling in the Hamiltonian of
Eq.(17), the quantum state of the field plays an important role. For $\delta$
in the range $\sim 0.15$ to $0.50$, the mixed states on the $x$-axis with
Bloch vector magnitude $9/10$ and initial field states $|{\alpha}\rangle$ and
$|{e_{0}}\rangle$ (red and black dashed lines, respectively) are seen to reach
their $(1-\delta)$-distinguishable states faster than the pure state at polar
angle $3\pi/8$ in the $xz$-plane with initial field state $|{\alpha}\rangle$
(black solid line). For smaller, but still intermediate, vales of $\delta$ the
pure state evolves faster than the mixed state with initial field
$|{e_{0}}\rangle$ but slower than the mixed state with initial field
$|{\alpha}\rangle$. For small $\delta$, the pure state is faster than both
mixed states.

This result is relevant to realistic quantum computations because it shows
that given a known fidelity loss $\delta>0$, i.e., given knowledge of the
error rate of the unitary gates, of the environment of the computer, etc., one
is able to operate the computer optimally using mixed states. For example,
imperfectly prepared qubit states could thereby be used to increase the
processing speed of a lossy quantum computer. This strategy not only speeds up
the quantum computer, but also reduces the change in entropy associated with
any quantum computation.

Figure 5: Minimal error probability for distinguishing various initial states
$\rho_{S}(0)$ of a qubit in the $xz$-plane of the Bloch ball from their
respective reduced, time-evolved states $\rho_{S}(t)$ given initial field
state $|{\psi}\rangle$. (Black solid) $\rho_{S}(0)$ having polar angle
$3\pi/8$ and $\|\vec{r}\|=1$, $|{\psi}\rangle=|{\alpha}\rangle$; (red)
$\rho_{S}(0)$ having polar angle $\pi/2$ and $\|\vec{r}\|=9/10$,
$|{\psi}\rangle=|{\alpha}\rangle$; (black dashed) $\rho_{S}(0)$ having polar
angle $\pi/2$ and $\|\vec{r}\|=9/10$, $|{\psi}\rangle=|{e_{0}}\rangle$.
$\alpha=3$ in all cases.

##  VI Conclusion

We have introduced the formal notion of distinguishability time as a
generalization of the orthogonalization time appearing in rigorous definitions
of the quantum speed limit and solved for the distinguishability time of
unitary evolution of a two-level system (Eq.(9)) in the case that quantum
binary distinguishability is used as the discrimination procedure. In
particular, by bounding the quantum Fisher information from below, we have
derived a necessary and sufficient condition on a quantum state $\rho$ and
Hamiltonian $H$ such that $\rho$ will evolve in time $t\geq 0$ to a state
$e^{-iHt}\rho e^{iHt}$ from which it is distinguishable with maximal success
probability $1-\delta$ for any $\delta\in[0,1/2]$. As a corollary of this
condition, we determined the set of quantum states that evolve to
$(1-\delta)$-distinguishable states faster than a given quantum state under
unitary time-evolution. The formalism developed in Section III was used in a
simple proof of the quantum brachistochrone for a two-level system.

We have extended the distinguishability analysis of time-evolved states to the
reduced Liouville-von Neumann dynamics of a two-level system resonantly
coupled, via the Jaynes-Cummings interaction, to a single-mode electromagnetic
field. The timescale of revivals of qubit state distinguishability under non-
unitary dynamics induced by the quantum state of the field was found to depend
on both the strength of the field and the photon number statistics. In the
context of CQED-based quantum operations, these results suggest the
possibility of engineering the field state and the field energy to tune the
distinguishability dynamics. Calculation of the reduced distinguishability
dynamics of a qubit under more general qubit-field interactions will be made
in future work.

###### Acknowledgements.

This work was supported by National Science Foundation Grant No. CHE-1213141.

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