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section, this exceeds by several orders of magnitude the
required coherence times to perform matter-wave inter-
ferometry using nuclear spins.
An important aspect that could limit the achievable
rotation frequencies is the absorption of photons com-
ing from the laser used to induce the rotation. An ex-
cessive degree of heating of the internal temperature of
the particle will have implications for the sublimation
rate, as discussed in the next section, and for the in- | [
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] |
rate, as discussed in the next section, and for the in-
tensity of the emitted black body radiation, which can
result in excessive decoherence and could be detrimental
for interferometric applications. It is, however, notewor-
thy to mention that in contrast to the optically levitated
setups where rotation has been demonstrated, here we
are proposing a protocol where the nanoparticle is mag-
netically levitated and the laser is used only for a short
time to induce the desired rotations. The short duration | [
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] |
time to induce the desired rotations. The short duration
of the laser pulse together with the high degree of pu-
rity attainable in the production of naphthalene crystals,
which results in low absorption rates, leads us to expect
heating rates below those observed in optical traps with
other materials. In the following, we make a back-of-the-
envelope estimation of the expected heating and conclude
that, while the increase in internal temperature may not | [
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] |
that, while the increase in internal temperature may not
be negligible, it does not impede achieving the required
rotation frequencies.
The rotation frequency νMAS obeys the differential
equation
I ˙νMAS = τopt − τdrag,
(3)
where I is the moment of inertia, τopt is the light-induced
torque, and τdrag is the damping torque due to viscous in-
teraction of the rotating nanoparticle with the surround-
ing gas molecules. τdrag is proportional to νMASP, with | [
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] |
ing gas molecules. τdrag is proportional to νMASP, with
P the pressure of the gas, and the maximal rotation fre-
quency is reached, as soon as τopt = τdrag[14]. At fre-
quencies much lower than this maximum, one can safely
assume that τopt ≫ τdrag, and thus that the rotation
frequency increases linearly in time as νMAS = (τopt/I)t.
For a nanoparticle that is significantly smaller than the
wavelength of the incident light, the optical torque due
to the birefringence of the particle is given by [52] | [
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] |
to the birefringence of the particle is given by [52]
τopt = ϵ0V
2
Re[(χE∗) × E]
+ ϵ0k3V 2
12π
Im[(χ2E∗) × E − (χE∗) × (χE)],
(4)
5
where V is the volume of the particle, and k = 2π/λ
is the wave number of the laser. χ is the orientation-
dependent polarizability tensor, which for an anisotropic,
ellipsoidal particle is diagonal in a frame coinciding with
its principal axes. In that frame, it is determined by the
diagonal elements [60]
χj = 3ϵj − 1
ϵj + 2.
(5)
Here, ϵj = n2 | [
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0.010881366208195686
] |
diagonal elements [60]
χj = 3ϵj − 1
ϵj + 2.
(5)
Here, ϵj = n2
j is the relative permittivity along the jth
principal axis, and E is the possibly complex amplitude
of the electric field E(t) = E exp(−iωt). Notably, for an
isotropic material, the torque is zero. To compute the
torque, the complex contribution to χ will be neglected,
as the absorption of naphthalene in the visible range is
minimal. If the naphthalene is orientated such that the
principal axes of n align with the lab frame, n is given | [
0.4343220591545105,
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] |
principal axes of n align with the lab frame, n is given
by n = diag(1.442, 1.932, 1.775). The largest torque is
achieved with circularly polarized light traveling along
the z axis, namely E = E0/
√
2(1, exp(iπ/2), 0). For a
spherical naphthalene particle of radius r, with moment
of inertia I = 2/5 mr2 and a laser with wavelength λ, we
find
νMAS = a r
λ3 It ,
(6)
where a = 8.15 · 10−11 m4/(Ws2) is a constant, I =
cϵ0|E0|2/2 is the intensity of the laser and t the time | [
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] |
cϵ0|E0|2/2 is the intensity of the laser and t the time
the particle is exposed to the laser light.
We now proceed to estimate the amount of heating
expected as a function of the rotation frequency. If heat
losses due to other mechanisms like black body radiation
are neglected, the heating rate is given by [61]
mcm(T) ˙T = 8π2I r3
λ Im
�ϵ − 1
ϵ + 2
�
,
(7)
where cm(T) is the temperature-dependent specific heat
capacity in units of J/(kgK). This differential equation | [
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] |
capacity in units of J/(kgK). This differential equation
can be solved by the technique of separation of variables.
Plugging eq. (6) in we get
� Tf
Ti
cm(T) dT = 6πνMAS
λ2
a r ρ Im
�ϵ − 1
ϵ + 2
�
.
(8)
Remarkably, the expression is independent of the em-
ployed laser intensity, as the heating rate is proportional
to the laser intensity while the duration of the laser pulse
is inversely proportional to it. We compute specific val-
ues for the expected temperature increase, using data | [
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] |
ues for the expected temperature increase, using data
from Ref. [62] to model the temperature dependence of
the heat capacity of naphthalene. To find ϵ = n2, we
assume that the real part of n is given by the aver-
age over the two axes involved in the rotation, there-
fore Re(n) = 1.687.
On the other hand, we calculate
an upper bound to the imaginary part of n from one of
the main absorption peaks of naphthalene in the optical
range, which is found at 470 nm and shows an absorbance | [
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] |
range, which is found at 470 nm and shows an absorbance
of α = 5 · 10−4 cm−1 [63].
α is related to the imagi-
nary part of the refraction index via Im(n) = αc/(2ω),
where ω is the frequency of the light. We therefore find
Im(n) = 1.75·10−8. In the neighborhood of this peak, the
absorption is expected to be even lower [64]. Naturally, in
experiments, the frequency of the employed laser should
be suitably placed outside any absorption line width of
both naphthalene and pentacene, to minimize the inter- | [
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] |
both naphthalene and pentacene, to minimize the inter-
nal heating of the particle. With these considerations we
use n = 1.687 + i 1.75 · 10−8 as an upper bound.
Putting everything together, we find that, for example,
to achieve a T2 time of 1 s, a spinning frequency of 23 MHz
is required. For a 70 nm particle that was initially pre-
pared at an internal temperature of 5 K this would result
in its temperature increasing up to not more than 11 K
when being accelerated to this frequency.
C.
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] |
when being accelerated to this frequency.
C.
Sublimation
Because naphthalene is a van der Waals crystal, the
question arises as to whether, at the low pressures neces-
sary to conduct matter-wave interferometry experiments,
sublimation events may occur that are detrimental to the
coherence of the experiment. Each time a molecule of
naphthalene evaporates from the crystal, departing with
some momentum ⃗p, it induces a recoil on the levitated
particle, that constitutes a form of noise on its mechanical | [
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particle, that constitutes a form of noise on its mechanical
degrees of freedom. Moreover, each evaporated naphtha-
lene molecule carries into the environment information
on the position and the velocity of the naphthalene crys-
tal, which may also destroy the desired interference pat-
tern. In particular, for situations where the de Broglie
wavelength of the sublimated molecule is shorter than
the spatial superposition of the interferometer, a single | [
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] |
the spatial superposition of the interferometer, a single
molecule loss could be enough to resolve the position of
the naphthalene particle and in turn destroy the interfer-
ence. Thus, we would like to estimate the rate at which
naphthalene molecules detach from the bulk of the crystal
and restrict each of our experimental runs to times for
which the probability of a single naphthalene molecule
evaporating is negligible.
We assume that the rate at
which particles detach from the crystal depends only on | [
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] |
We assume that the rate at
which particles detach from the crystal depends only on
the internal temperature of the crystal, and follow the
argument presented by Tennakone and Peiris to find this
rate [65].
Let us consider a bulk sphere of naphthalene with ra-
dius r, levitated in a completely evacuated container.
Over time, some of the naphthalene will evaporate until
the pressure in the chamber is the same as the vapor pres-
sure pn of naphthalene, at which point, an equilibrium | [
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] |
sure pn of naphthalene, at which point, an equilibrium
is reached between the rate of particles that leave the
crystal and those that are deposited onto it. Following
statistical physics considerations, one can estimate the
rate at which, at a pressure of pn, naphthalene molecules
in the gas collide with the bulk sphere of radius r. This
6
is given by
˙N =
4πr2pn
√2πmNkBT ,
(9)
where mN = 128 a.m.u. = 2.1 · 10−25 kg is the mass of
a single naphthalene molecule and T is its temperature, | [
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] |
a single naphthalene molecule and T is its temperature,
which is also the temperature of the bulk, as the situa-
tion is described at equilibrium. If all incident particles
were to get deposited onto the surface of the naphtha-
lene nanoparticle, eq. (9) would also constitute the rate
at which naphthalene molecules sublimate from the crys-
tal. While this is not true in general, eq. (9) provides an
upper bound to the real rate of sublimation and allows us | [
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] |
upper bound to the real rate of sublimation and allows us
to make a conservative estimation of the available time
before a single molecule escapes (or gets deposited onto)
the crystal, as a function of its temperature and size.
The relation between the vapor pressure of naphtha-
lene and its temperature is found to be well described by
the equation [66]
ln pn
p0
=
�
1 − T0/K
T/K
�
exp
� n
�
i=0
Ai(T/K)i
�
,
(10)
where the parameters Ai are to be experimentally fit, | [
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] |
T/K
�
exp
� n
�
i=0
Ai(T/K)i
�
,
(10)
where the parameters Ai are to be experimentally fit,
and p0 is the vapor pressure at an arbitrarily chosen
temperature T0.
As a reference, we use the values at
room temperature, p0 = 13.08 Pa and T0 = 300 K. As
for the parameters Ai, the values {A0 = 3.272310, A1 =
−2.663498 · 10−4, A2 = −2.929123 · 10−9} for the first
three orders provide a good fit in the temperature range
from 150 K to the triple point at 353.37 K [66]. | [
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] |
from 150 K to the triple point at 353.37 K [66].
With this, we can compute that, for example, at room
temperature the emission rate of naphthalene molecules
from a particle of 70 nm is
˙N ≈ 1010 s−1 which would
limit the duration of a single experimental run to 10−10 s.
This naturally rules out any option of performing in-
terferometry experiments without cooling the internal
temperature of the naphthalene nanoparticle. However,
this rate exhibits a dramatic dependence on tempera- | [
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] |
this rate exhibits a dramatic dependence on tempera-
ture. At 150 K, which is the lowest temperature for which
eq. (10) provides a reliable value, the emission frequency
is ˙N = 40 s−1, extending the available time for each ex-
perimental run to several milliseconds.
Moreover,
for
matter-wave
interferometry
experi-
ments, internal temperatures on the order of a few Kelvin
will be required to reduce noise coming from the emission
of black body radiation, as will be explained in the next | [
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] |
of black body radiation, as will be explained in the next
section. At these temperatures, sublimation rates are ex-
pected to be even lower by orders of magnitude. Thus, we
do not expect sublimation to be a factor limiting the co-
herence of matter-wave interferometry experiments with
naphthalene.
On the other hand, any experiment will consist of a
statistically significant number of runs that allow the re-
construction of the observables of interest. Thus, while | [
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] |
construction of the observables of interest. Thus, while
one could guarantee that on a single experimental run,
no naphthalene molecules are sublimated, it is to be ex-
pected that the total experimental time will extend over
periods where the emission rate of naphthalene particles
is not negligible. This could lead to a drift of the mass
of the particle as the experiment progresses. To under-
stand the extent of this mass variation, we compute the
rate of change of the total mass of a naphthalene particle | [
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] |
rate of change of the total mass of a naphthalene particle
of radius r relative to its initial mass M0. From eq. (9)
we see that, at pressures below the vaporization pressure,
the rate of mass loss is upper bounded by
d
dt
� M
M0
�
= −
�
mN
2πkBT
3pN
ρ r ,
(11)
where ρ = 1.145·103 kg/m3 is the density of naphthalene.
For a particle of 70 nm, at a temperature of 150 K, this
results in a relative mass that varies in time as M/M0 =
1 − 1.8 · 10−10t, which suggests that even on a time scale | [
-0.14171865582466125,
0.15419632196426392,
0.400562584400177,
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] |
1 − 1.8 · 10−10t, which suggests that even on a time scale
of hours the mass of the nanoparticle would not change
beyond its 7th significant digit. We, thus, do not expect
this to affect the experiment.
III.
NAPHTHALENE IN A MAGNETIC FIELD
GRADIENT
This section describes the dynamics of a diamagnetic
particle containing N embedded nuclear spins in the pres-
ence of a magnetic field gradient.
The magnetic field
gradient induces a harmonic confinement of the center of | [
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] |
The magnetic field
gradient induces a harmonic confinement of the center of
mass of the diamagnetic particle and, at the same time,
its interaction with the spin system leads to forces whose
strength and direction depend on the state of the spin
ensemble. These forces can be used to manipulate the
motion of the particle and in particular to expand its
position variance and enhance its sensitivity to exter-
nal perturbations [67]. We introduce a protocol based | [
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] |
nal perturbations [67]. We introduce a protocol based
on microwave pulses that manipulates precisely the state
of the spin ensemble, allowing us to engineer dynamics
that involve a rapid expansion of the center of mass wave
packet and its subsequent reversal. Probing the coher-
ence of such expand-and-recombine dynamics, which can
be done through the measurement of the spin state at
the end of the protocol, allows us to test the linearity of
quantum mechanics at the scales of the employed par- | [
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] |
quantum mechanics at the scales of the employed par-
ticle mass and the reached size of the wave function.
This constitutes an instance of a matter-wave interfer-
ometry protocol, that is, an interferometer that harnesses
the wave character of matter. The novelty with respect
to other matter-wave interferometry protocols based on
spin-dependent forces is the presence of multiple spins,
which boosts the spread of the wave function, resulting
in shorter experimental times and therefore increased ro- | [
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] |
in shorter experimental times and therefore increased ro-
bustness to several sources of noise. We discuss the scal-
ing of the different figures of merit with the number of
spins N. Moreover, we explore the intricacies of the proto-
col, propose a way to measure the spin state through the
position of the nanoparticle, and analyze the resilience of
the protocol against various sources of noise.
7
A.
Advantages of Naphthalene for Stern-Gerlach
type interferometry | [
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] |
7
A.
Advantages of Naphthalene for Stern-Gerlach
type interferometry
Nuclear spins in naphthalene present a promising
pathway for creating macroscopic superpositions of the
nanoparticle in position space.
While current propos-
als for Stern-Gerlach-type macroscopic interferometers
often rely on single electron spins, such as NV centers
in diamond [68, 69], here we argue that the use of nu-
clear spin ensembles presents an intriguing alternative.
Although the gyromagnetic ratio of an electron spin is | [
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] |
Although the gyromagnetic ratio of an electron spin is
roughly three orders of magnitude higher than that of
a nuclear spin, the collective strength of approximately
6 · 107 polarized hydrogen spins in a 70 nm naphthalene
particle exceeds that of an individual electron spin in the
same magnetic field gradient.
Furthermore, the homogeneous distribution of nuclear
spins within naphthalene presents a distinct advantage
over systems reliant on single electron spins.
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] |
over systems reliant on single electron spins.
In tra-
ditional setups, precise positioning of the spin on the
nanoparticle’s center of mass is necessary to prevent un-
desired rotations of the entire particle. However, in naph-
thalene, the evenly dispersed nuclear spins exert a collec-
tive force that acts uniformly on the nanoparticle, avoid-
ing undesired rotational effects.
On the other hand, particles containing embedded
spins with zero-field splitting, such as the NV center in di- | [
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] |
spins with zero-field splitting, such as the NV center in di-
amond, are prone to additional unwanted rotations. For
particles with a zero-field splitting the axis of quantiza-
tion is not determined by the direction of the external
magnetic field but rather by the spin direction that di-
agonalizes the zero-field Hamiltonian. When this quan-
tization axis deviates from the direction of the magnetic
field, it tries to realign, exerting a force that induces a | [
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] |
field, it tries to realign, exerting a force that induces a
torque on the particle. This presents a challenge for ex-
periments leveraging spin-dependent forces for the gen-
eration of spatial superposition.
Matter-wave interfer-
ometry experiments will generally require preparing the
spin in a superposition of states |↑⟩ and |↓⟩. Thus, even
in the case of a perfect alignment of the NV axis with
the magnetic field, one of the superposed states will be | [
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] |
the magnetic field, one of the superposed states will be
aligned parallel to the magnetic field, while the other will
be in an anti-parallel orientation. In this configuration,
any small deviations from this perfect alignment between
the NV axis and the magnetic field will induce torques
dependent on the spin state, which will eventually lead
to entanglement between spin and rotational degrees of
freedom. In contrast, nuclear spin- 1
2 systems lack such | [
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] |
freedom. In contrast, nuclear spin- 1
2 systems lack such
splitting. Any potential zero-field splitting observed in
the nuclear spins of naphthalene arises from dipole-dipole
interactions among nuclear spins, which in our setup are
suppressed by the implementation of magic-angle spin-
ning.
Moreover, the rapid rotation of the nanoparti-
cle contributes to the stabilization of its rotational de-
grees of freedom, akin to magnetic tops, increasing the
robustness of the setup against unwanted torques and | [
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] |
robustness of the setup against unwanted torques and
rotations [70, 71].
B.
Time evolution in a magnetic trap
x
t
xeq,1 xeq,2xeq,3
xeq,4
xeq,0
∆x
κ = 2
κ = 3
κ = 1
κ = 4
κ = 0
ρκ,κ(x, x)
x
χ
FIG. 2. Here, the time evolution of the κ wave packets in
a magnetic field gradient is displayed. The κth wave packet
oscillates around the equilibrium position xeq,κ, as depicted
on the left. After a full period of 2π/Ω, all trajectories meet
again in the center of the trap.
The equilibrium positions | [
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] |
again in the center of the trap.
The equilibrium positions
xeq,κ = (2κ − N)χ are spaced by χ. If the initial spin state
is given by eq. (15), the wave packets corresponding to κ are
in a superposition, as depicted on the right. The Gaussians
are displaced by xeq,κ[1 − cos(Ωt)] and enclosed in an enve-
lope that follows a binomial distribution. The width of the
envelope is given by eq. (22).
The Hamiltonian describing the center-of-mass motion
and spin dynamics of a naphthalene nanoparticle of mass | [
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] |
and spin dynamics of a naphthalene nanoparticle of mass
m containing N nuclear spins, each with gyromagnetic
ratio γp, and embedded in a magnetic field gradient of
the form ⃗B(x) = (B′x+B0)ˆk, with ˆk a unit vector in the
z direction, can be expressed as
ˆH = ˆp2
2m − χV V (B0 + B′ˆx)2
2µ0
+ ℏ
2γp(B0 + B′ˆx)
N
�
n=1
ˆσ(n)
z
,
(12)
where χV = −7.33 · 10−7 is the volume magnetic suscep-
tibility of naphthalene [57], and µ0 = 4π · 10−7 H/m is
the vacuum magnetic permeability. Under the coordinate | [
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the vacuum magnetic permeability. Under the coordinate
transformation x → x − B0
B′ , and a suitable rearranging
of the terms, the Hamiltonian can be rewritten as
ˆH = ˆp2
2m + mΩ2
2
�
ˆx + χ
N
�
n=1
ˆσ(n)
z
�2
− α
� N
�
n=1
ˆσ(n)
z
�2
,
(13)
where we have introduced the parameters
Ω ≡
�
|χV |
ρµ0
B′,
χ ≡
ℏγpµ0
2|χV |V B′
and
α ≡ ℏ2γ2
pµ0
8|χV |V .
(14)
In this form, it becomes clear that the center of mass
oscillates with frequency Ω around an equilibrium po- | [
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] |
oscillates with frequency Ω around an equilibrium po-
sition that is displaced from the origin by an amount
8
χV
ρ in kg/m3 σUTS in MPa H density in 1/m3
-7.33 · 10−7
1145
49
4.3 · 1028
TABLE I. Material constants of naphthalene.
xeq = −χ �N
n=1 ˆσ(n)
z
. This equilibrium position depends
on the state of the spin ensemble.
For clarity, we will presume that the spin ensemble
is initially fully polarized—any implications of imperfect
polarization are discussed in section III D. Hence we can | [
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] |
polarization are discussed in section III D. Hence we can
assume that all spins are initially arranged in their low-
est energy state, aligning uniformly in a single direction.
Subsequently, we apply a π/2-pulse, mapping each spin
into a superposition of spin up and spin down. The pre-
cision with which such a pulse can be delivered to all
the spins in the ensemble is discussed in section III E.
Assuming that the center-of-mass motion is prepared in
a thermal state ˆρth, the resulting state of the combined | [
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] |
a thermal state ˆρth, the resulting state of the combined
system is given by
ˆρ = ˆρth ⊗ 1
2N
N
�
κ,κ′=0
��N
κ
��N
κ′
�
|κ⟩ ⟨κ′| ,
(15)
where |κ⟩ is the κth Dicke state, defined as the linear
combination of all spin states with κ spins up
��N
κ
�
|κ⟩ =
��↑⊗κ↓⊗N−κ�
+ permutations.
(16)
Hamiltonian (13) preserves the Dicke state, as this is an
eigenstate of the operator �N
n=1 σ(n)
z
, and thus, of the
Hamiltonian. Therefore, provided that the initial state | [
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] |
n=1 σ(n)
z
, and thus, of the
Hamiltonian. Therefore, provided that the initial state
is contained in the subspace spanned by the Dicke states,
the state of the system at any time t can be expressed in
terms of Dicke states as
ˆρ(t) = 1
2N
N
�
κ,κ′=0
��N
κ
��N
κ′
�
ˆρκ,κ′(t) ⊗ |κ⟩ ⟨κ′| .
(17)
In the following, we want to find a differential equation
describing the time evolution of the spatial components
of the density matrix ˆρκ,κ′(t). The time evolution of ˆρ(t) | [
-0.0214378722012043,
-0.45224541425704956,
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] |
of the density matrix ˆρκ,κ′(t). The time evolution of ˆρ(t)
is given by von Neumann’s equation ˙ˆρ = − i
ℏ[ ˆH, ˆρ] and
with Ansatz 17, we find
˙ˆρκ,κ′ = − i
ℏ
�
ˆHκˆρκ,κ′ − ˆρκ,κ′ ˆHκ′ + ακ,κ′ ˆρκ,κ′
�
(18)
with
ˆHκ ≡ ˆp2
2m + mΩ2
2
[ˆx + (2κ − N)χ]2
(19)
and
ακ,κ′ ≡ α
�
(2κ′ − N)2 − (2κ − N)2�
,
(20)
and which has the solution
ˆρκ,κ′(t) = e− i
ℏ ˆ
Hκtˆρκ,κ′(0)e
i
ℏ ˆ
Hκ′te− i
ℏ ακ,κ′t.
(21)
The evolution of the initial state in eq. (15) can be
described as follows. At first, the mechanical and spin | [
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] |
described as follows. At first, the mechanical and spin
degrees of freedom are in a product state, however, as
the evolution progresses the initial center-of-mass wave
packet splits into N +1 trajectories, each correlated with
one of the Dicke states, which are parameterized by κ ∈
[0, N]. All trajectories follow a harmonic motion, albeit
each oscillating around a different equilibrium position.
Thus to each Dicke state, we can associate an equilibrium | [
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] |
Thus to each Dicke state, we can associate an equilibrium
position xeq,κ = (2κ−N)χ. It is not difficult to show that
at any given time, one will find the N + 1 wave packets
evenly distributed along dimension x with the nearest-
neighbor spacing given by 2χ sin2(Ωt/2) (see fig. 2 for a
schematic). Nevertheless, not all trajectories are equally
likely, because according to state eq. (15) the probability
for the spin ensemble to be in Dicke state κ is given by
pk =
�N
κ
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] |
for the spin ensemble to be in Dicke state κ is given by
pk =
�N
κ
�
2−N. The variance in the position of the center
of mass at a given time t can be computed to be
⟨∆x⟩2 = ∆x2
th +
�
2χ sin2(Ωt/2)
√
N
�2
,
(22)
where ∆xth is the position variance of the initial thermal
state. eq. (22) already shows the enhancement that can
be achieved in the expansion of the center-of-mass wave
function from a larger number of spins. In comparison
to matter-wave interferometry based on a single electron | [
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-0.28748825192451477
] |
to matter-wave interferometry based on a single electron
spin, such as an NV-center in a diamond nanoparticle,
we observe an enhancement factor of ≈ 1.5 · 10−3√
N in
the variance at a given time t. Here, the factor of 10−3
originates from the ratio between the gyromagnetic ra-
tios of the hydrogen and electronic spins. A naphthalene
nanoparticle with a radius 100 nm would contain approx-
imately 108 nuclear spins, resulting in a 15-fold enhance-
ment of the achievable variance at any given time over | [
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] |
ment of the achievable variance at any given time over
that achievable with a single electron spin. For the pa-
rameters in table II a), one can achieve an expansion of
the wave packet of ⟨∆x⟩2 − ∆x2
th = (566 nm)2 after just
0.5 ms, a position variance that is larger than its radius.
For matter-wave interferometry with spin-dependent
forces, it is often desirable to revert the expansion of
the wavepacket and return the system to its initial state
where spin and mechanical degrees of freedom are decou- | [
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] |
where spin and mechanical degrees of freedom are decou-
pled from each other. Then the coherence of the process
can be tested through a measurement of the spin system,
certifying also the spatial coherence of the superposition
achieved during the protocol.
For the setup described
in this section, this decoupling occurs without additional
intervention at times that are natural multiples of the
trap period, a time at which all the trajectories return
simultaneously to their initial position, see Fig. 2. Nev- | [
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] |
simultaneously to their initial position, see Fig. 2. Nev-
ertheless, with a suitable manipulation of the spin state,
this decoupling can be achieved at times shorter than a
single period of the trap, in this way reducing the influ-
9
κ
?
π
2
π
π
t1
t2
t1
FIG. 3. Protocol for rapid expansion and recombination of the
wave function of the center-of-mass motion in naphthalene.
A hyperpolarized particle is placed in a magnetic gradient,
forming a harmonic trap. Then a π/2 pulse is applied. The | [
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] |
forming a harmonic trap. Then a π/2 pulse is applied. The
spin state is now a superposition of the N + 1 possible Dicke
state. Depending on the spin state, the particle travels along
another trajectory and the wave function expands for a time
t1.
By applying two well-timed π pulses spaced by t2, all
trajectories recombine after another time t1. The trajectories
are denoted by κ ∈ [0, N].
ence of detrimental sources of noise [72]. While this also
reduces the size of the prepared superposition, it allows | [
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reduces the size of the prepared superposition, it allows
us to adjust the protocol to the available coherence time.
First, one lets the wave function spread for a time
t1 < π/Ω—if the expansion time is larger than half of
an oscillation period then the fastest way to decouple
the spin and center-of-mass motion is to leave the sys-
tem complete one full period. At this time, a microwave
π-pulse is applied on the spins, causing them to flip their
state, which for the Dicke states translates into the trans- | [
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] |
state, which for the Dicke states translates into the trans-
formation κ → N − κ. This reverts the force acting on
each of the Dicke states, slowing down the expansion of
the position variance and eventually leading to its com-
pression, see Fig. (3). A second π-pulse is required to
guarantee that all the superposed wavepackets overlap
not only in position but also in momentum. The spac-
ing between the first and second π-pulses depends on the
initial spreading time t1 and is given by
t2(t1) = 2
Ω arcsin
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] |
initial spreading time t1 and is given by
t2(t1) = 2
Ω arcsin
�
sin(Ωt1)
�
5 − 4 cos(Ωt1)
�
.
(23)
Finally, waiting for a time t1 after the second π-pulse, all
trajectories recombine and the position and spin degrees
of freedom are decoupled again. Thus, for such a protocol
the total time is Ttot(t1) = 2t1 + t2(t1).
While at the end of the protocol, all the superposed
trajectories overlap and the spin and motion decouple,
the state of the system is not the same as that at the | [
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] |
the state of the system is not the same as that at the
beginning of the protocol, because each trajectory has
followed a different path in phase space and therefore
acquired a different phase.
The phase picked up by
each trajectory κ, sometimes referred to as the geometric
phase, is given by
ϕκ = (2κ − N)2[(χ/2x0)2 sin(Ωt1) + αTtot/ℏ].
(24)
Note that the phase is the same for trajectories that are
symmetric about the origin, that is, for trajectories κ = 0 | [
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] |
symmetric about the origin, that is, for trajectories κ = 0
and κ = N, for trajectories κ = 1 and κ = N − 1, and so
on. Finally, provided that the whole process is coherent,
at the end of the protocol, the system is found in a state
of the form ˆρ = ˆρth ⊗ |ψf⟩ ⟨ψf| with
|ψf⟩ =
1
2N/2
N
�
κ=0
��N
κ
�
e−iϕκ |κ⟩ .
(25)
Note that, due to the different phases picked up by each
Dicke state, the spin ensemble at the end of the protocol
is in general entangled.
A breakdown of the linearity of quantum mechanics | [
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0.14936712384223938
] |
is in general entangled.
A breakdown of the linearity of quantum mechanics
on the scales at which the protocol is performed would
manifest as a deviation of the final state from that in
eq. (25). Conversely, experimental verification of the final
state in eq. (25) would constitute proof of the validity of
quantum mechanics at such scales. To provide a more
quantitative account of such a possibility, in the next
section, we compute the evolution of the system in the | [
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] |
section, we compute the evolution of the system in the
presence of, arguably, the most widely analyzed model of
wavefunction collapse at macroscopic scales, namely, the
Continuous Spontaneous Localization (CSL) model.
C.
Test of the CSL model
Motivated by the aspiration to provide a unified
framework to describe micro/quantum systems and
macro/classical systems and to solve the so-called mea-
surement problem in quantum mechanics, collapse mod-
els postulate modifications to the Schr¨odinger equa- | [
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] |
els postulate modifications to the Schr¨odinger equa-
tion [73]. These modifications are, typically, in the form
of non-linear and stochastic extensions, whose effects
manifest for systems of large mass and states of wide
spatial delocalization. In contrast, for small masses and
well-localized systems, these modifications are negligible,
and spontaneous collapse models converge to the stan-
dard Schr¨odinger equation. In this sense, they provide
a framework to describe a progressive breakdown of the | [
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] |
a framework to describe a progressive breakdown of the
linearity of quantum mechanics with increasing system
mass. Generally, the effect of these modifications is to in-
duce spontaneous collapses of the wavefunction onto spa-
tially well-localized states. The Continuous Spontaneous
Localization (CSL) model is perhaps the most general
and widely studied of the collapse models. It postulates
a localization of the wavefunction in position space which
depends on two free parameters, λCSL and rCSL. λCSL | [
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] |
depends on two free parameters, λCSL and rCSL. λCSL
has units of s−1 and relates to the frequency of the col-
lapses, while rCSL has units of m and relates to the length
10
r [nm]
N
B′ [T/m] Ttot [ms]
χ[m]
⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) Ω/(4λ) ∆xdis [m] Pflip Ti,max [K]
a)
100
1.8 · 108 2.74 · 105
2
2 · 10−11
0.35
1
1.5 · 10−4
0.1
8
b)
65
5 · 107
9.97 · 105
0.6
2 · 10−11
0.97
1
4.2 · 10−5 0.008
6
c)
45
1.6 · 107
3 · 104
18
2 · 10−9
0.98
10
1.3 · 10−3
0.09
4 | [
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] |
2 · 10−11
0.97
1
4.2 · 10−5 0.008
6
c)
45
1.6 · 107
3 · 104
18
2 · 10−9
0.98
10
1.3 · 10−3
0.09
4
TABLE II. Overview of of some critical specifications of the protocol for different sets of parameters. B′ is chosen in such
a way, that ϕ = 0 and an interference fringe can be resolved. It was assumed that the only noise acting on the state where
spontaneous collapses with λCSL = 10−16 and rCSL = 10−7.
scale of the localization. The dynamics of the CSL model | [
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] |
scale of the localization. The dynamics of the CSL model
at the level of the density matrix, that is, after averaging
over a large number of stochastically occurring collapses,
is given, in coordinate representation, by [73–75]
˙ρ(q′, q′′) = − i
ℏ ⟨q′| [ ˆH, ˆρ] |q′′⟩−ξ
�
1 − e
− (q′−q′′)2
4r2
CSL
�
ρ(q′, q′′),
(26)
where ρ(q′, q′′) = ⟨q′| ˆρ |q′′⟩ with |q⟩ the position eigen-
state, and ξ is a coefficient that depends on the free pa-
rameters λCSL and rCSL, the mass and the shape of the | [
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] |
rameters λCSL and rCSL, the mass and the shape of the
system. For a spherical particle of radius r and mass m,
ξ is given by [76]
ξ = m2
m2
0
λCSLf
�
r
rCSL
�
(27)
with
f (x) = 6
x4
�
1 − 2
x2 +
�
1 + 2
x2
�
e−x2�
,
(28)
and m0 is a reference mass, typically, taken to be the
mass of a nucleon.
Equation 26 indicates that the effect of the collapse
model is to suppress the coherence between different po-
sition eigenstates. The coherence between spatial points | [
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] |
sition eigenstates. The coherence between spatial points
(q′, q′′) that are further away from each other decays
faster than that between closer points.
In the limit
|q′ − q′′| ≪ rCSL, the decay rate of the coherences tends
to ξ(q′ − q′′)2/(4r2
CSL), while between points that are far
away from each other, the decay rate converges to ξ.
1.
Interferometry in the presence of the CSL model
Our goal is to use the interferometric protocol intro-
duced in the previous section to provide bounds for the | [
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] |
duced in the previous section to provide bounds for the
values that the free parameters of the CSL model, λCSL
and rCSL, can take. To that end, we compute the state
of the spin ensemble at the end of the protocol in the
presence of the CSL model and determine how it differs
from that of the noise-free case in eq. (25)
It is noteworthy that the presence of CSL noise does
not distort the trajectories that each Dicke state follows,
but only the coherence between them. Thus, at the end of | [
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0.6828210949897766,
-0.4883047640323639,
0.43729478120803833,
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0.5036121010780334,
0.4636395275592804,
-0.45458680391311646,
-0.40016576647758484,
-0.40416228771209717,
-0.24829623103141785,
0.14551468193531036,
-0.04210794344544411,
0.028743263334035873,
-0.17589794099330902,
-0.3377985656261444,
-0.7935006022453308,
0.23915265500545502,
0.27507075667381287,
-0.39657923579216003,
-0.21097862720489502,
-0.36130431294441223,
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] |
but only the coherence between them. Thus, at the end of
the protocol, all the trajectories overlap, as in the noise-
free case, and the system is in a separable state of the
form ˆρ = ˆρCM⊗ˆρS, where ˆρCM denotes the center-of-mass
state and ˆρS that of the spin ensemble.
Nevertheless,
the loss of coherence between the different trajectories
manifests as a reduction of the purity of the spin state,
which can be computed to be
ˆρS = 1
2N
N
�
κ,κ′=0
��N
κ
��N
κ′
�
e−i(ϕκ−ϕκ′)
e−Λκ,κ′ |κ⟩ ⟨κ′| ,
(29) | [
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] |
ˆρS = 1
2N
N
�
κ,κ′=0
��N
κ
��N
κ′
�
e−i(ϕκ−ϕκ′)
e−Λκ,κ′ |κ⟩ ⟨κ′| ,
(29)
where the relative phase ϕκ was introduced in eq. (24)
and
Λκ,κ′ = ξ
� Ttot
0
(1 − e
−
[Xκ(t′)−Xκ′ (t′)]2
4r2
CSL
) dt′ ,
(30)
as shown in appendix A. Here, Xκ(t) = χ Re[ζκ(t)] is the
classical trajectory of the κth wave packet and is given in
eq. (75). Since ζκ(t) is linear in κ, the decoherence rate
between the κth and κ′th trajectories is proportional to
(κ − κ′)2.
For matter-wave interferometers with a single spin, the | [
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] |
(κ − κ′)2.
For matter-wave interferometers with a single spin, the
standard procedure is to apply a π/2-pulse on the spin at
the end of the protocol, and then measure the magnetic
moment of the spin (1/2)ℏγσz. In the absence of noise,
the pulse returns the spin to the initial state |↑⟩, and the
expectation value of σz is thus found to be one. A loss
of coherence, for example, due to the presence of CSL
noise, is then manifested as a reduction of the expecta- | [
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0.4588254988193512,
0.5878925323486328,
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0.4170473515987396,
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0.1900262087583542,
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0.012505748309195042,
0.5331757664680481,
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0.4126417934894562,
0.02841019630432129,
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0.2745397090911865,
0.3863298296928406,
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0.19522373378276825,
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0.2866215705871582,
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0.30889758467674255,
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0.15066486597061157,
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0.33420616388320923,
0.051702145487070084,
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0.3309546709060669,
0.019367720931768417,
0.31122225522994995,
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0.46987196803092957,
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0.37422263622283936,
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0.23672516644001007,
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0.6220546960830688,
0.7781212329864502,
0.20216049253940582,
0.4892527461051941,
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0.323347270488739,
0.751229465007782,
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0.7792694568634033,
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0.37023672461509705,
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0.3730163872241974,
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0.0977984219789505,
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0.28964564204216003,
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0.01544454600661993,
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0.20697233080863953,
-0.06496582925319672
] |
noise, is then manifested as a reduction of the expecta-
tion value. Thus the expectation value of the magnetic
moment of the spin accounts for the amount of noise ob-
served by the system. The natural extension of such a
protocol for the multi-spin case that we consider here is
the application of a π/2-pulse to each of the spins in the
ensemble and the subsequent measurement of the total
magnetic moment of the nanoparticle
ˆ
M = ℏ
2γp
N
�
n=1
ˆσ(n)
z
,
(31)
with ˆσ(n)
z | [
0.3244142234325409,
-0.2614395022392273,
0.42194169759750366,
-0.14794789254665375,
-0.3680045008659363,
0.11359298974275589,
0.28228676319122314,
-0.7623157501220703,
-0.24236838519573212,
-0.27088281512260437,
0.02303501032292843,
0.5083688497543335,
-0.4119648337364197,
0.3040483891963959,
-0.0315631665289402,
0.2882014513015747,
0.6104459166526794,
-0.04620128497481346,
-0.5469969511032104,
-0.6042426228523254,
0.3296833634376526,
0.16135966777801514,
0.7815871238708496,
0.1723182052373886,
0.07984998822212219,
-0.008493108674883842,
0.6181076765060425,
0.1660643219947815,
0.2789405286312103,
0.15284180641174316,
-0.20917215943336487,
0.5581699013710022,
-0.16030408442020416,
0.08630587160587311,
0.5079208612442017,
0.21195553243160248,
-0.044349946081638336,
0.009736552834510803,
0.12913750112056732,
0.012910744175314903,
0.21741947531700134,
-0.09741713106632233,
0.06755509227514267,
0.33358442783355713,
-0.4696977734565735,
-0.7715095281600952,
-0.7730770111083984,
0.17013782262802124,
-0.32282987236976624,
-0.39353153109550476,
-0.27045053243637085,
0.3731207251548767,
0.361596941947937,
0.5315822958946228,
-0.3084389567375183,
-0.12260349094867706,
-0.5297302603721619,
-0.1702243983745575,
0.46295225620269775,
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-0.33864980936050415,
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0.16857554018497467,
0.3922007083892822,
0.4783751964569092,
0.3031007647514343,
-0.07405294477939606,
-0.37941187620162964,
0.08594278991222382,
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0.014345304109156132,
0.2624537944793701,
0.3372694253921509,
-0.23568710684776306,
0.1111866906285286,
0.25151941180229187,
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0.1263352632522583,
0.40064170956611633,
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0.5472052097320557,
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-0.19858306646347046,
0.7167564034461975,
-0.5876792669296265,
0.36265814304351807,
-0.06936265528202057,
-0.24584801495075226,
0.026167692616581917,
0.7261152863502502,
-0.029306059703230858,
0.7949918508529663,
0.08146268129348755,
-0.6657539010047913,
0.2581420838832855,
0.24231350421905518,
-0.05853686481714249,
-0.04015744850039482,
0.2767849266529083,
-0.16842183470726013,
-0.0754442811012268,
-0.8081474900245667,
-0.38326457142829895,
-0.5027157068252563,
-0.4636712074279785,
-0.016762517392635345,
-0.6008957028388977,
0.3010067343711853,
-0.33807000517845154,
-0.47576814889907837,
0.01116635836660862,
-0.09037183970212936,
0.06340645253658295,
-0.020956117659807205,
0.26546478271484375,
0.49063876271247864,
0.05277876928448677,
1.2988054752349854,
0.04549141600728035,
0.22785475850105286,
0.14443626999855042,
0.013044188730418682,
0.7385843396186829,
0.28217270970344543,
-0.2716078758239746,
0.028423890471458435,
-0.09984582662582397,
1.0093650817871094,
0.1859191358089447,
0.2798706591129303,
-0.2358761429786682,
-0.390390545129776,
-0.08721209317445755,
0.16293126344680786,
-0.3861261010169983,
0.04450904577970505,
-0.22682872414588928,
-0.31303542852401733,
-0.17396238446235657,
0.09521165490150452,
0.22938653826713562,
-0.18548442423343658,
-0.5734565854072571,
-0.20751550793647766,
0.025486793369054794,
0.24721309542655945,
-0.5782690048217773,
0.45083749294281006,
-0.675966739654541,
-0.007659555412828922,
-0.5113822221755981,
-0.7135976552963257,
0.009798414073884487,
0.031224597245454788,
-0.48551297187805176,
0.32829585671424866,
0.11265809834003448,
-0.09812130033969879,
-0.37892237305641174,
-0.3073936998844147,
0.12134622037410736,
-0.6113028526306152,
0.44265028834342957,
0.35216227173805237,
-0.03745318204164505,
-0.28892356157302856,
0.08801154047250748,
-0.26931479573249817,
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0.0740034207701683,
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-0.062129370868206024,
-0.07313337177038193,
0.12896628677845,
0.05995592474937439,
-0.6584420204162598,
-0.2128922939300537,
0.08415783196687698,
-0.2735445201396942,
0.19651934504508972,
0.24697640538215637,
0.3687218129634857,
0.2820468842983246,
-0.5213100910186768,
0.07843365520238876,
-0.11477158963680267,
0.107588030397892,
0.4176006615161896,
-0.8908698558807373,
1.179565668106079,
0.4729771316051483,
0.14727135002613068,
0.7295666337013245,
0.055105701088905334,
-0.13486865162849426,
0.013418824411928654,
-0.1336001306772232,
0.26977160573005676,
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] |
magnetic moment of the nanoparticle
ˆ
M = ℏ
2γp
N
�
n=1
ˆσ(n)
z
,
(31)
with ˆσ(n)
z
the z Pauli operator acting on the nth spin.
However, as we explain below, this will turn out not to
be an optimal strategy.
As shown in appendix B, the expectation value of ˆ
M
after applying a π/2-pulse on all the spins at the end of
the protocol, can be computed to be
⟨ ˆ
M⟩(Ttot)
⟨ ˆ
M⟩(0)
= e−γ(Ttot) cosN−1(4ϕ),
(32)
11
with γ(Ttot) = Λκ,κ±1(Ttot) and ϕ = ϕκ/(2κ − N)2 = | [
0.13525038957595825,
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] |
⟨ ˆ
M⟩(0)
= e−γ(Ttot) cosN−1(4ϕ),
(32)
11
with γ(Ttot) = Λκ,κ±1(Ttot) and ϕ = ϕκ/(2κ − N)2 =
(χ/2x0)2 sin(Ωt1) + αTtot/ℏ. As discussed, Λκ,κ′ is pro-
portional to (κ − κ′)2, and therefore γ(Ttot) is indepen-
dent of κ. Assuming that the distance between nearest
neighbor trajectories is much smaller than rCSL, which
is not a particularly restrictive assumption, γ(Ttot) has a
closed form, although lengthy, solution which is given in
appendix B.
Expression (32) presents two notable drawbacks. | [
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] |
appendix B.
Expression (32) presents two notable drawbacks.
Firstly, it fails to exploit the presence of multiple spins,
resulting in a decay due to CSL noise that remains in-
dependent of the number of spins.
Consequently, the
sensitivity of the protocol mirrors that of the single-
spin scenario, neglecting the potential enhancement from
multiple spins and only leveraging decoherence between
nearest-neighbor trajectories. This limitation stems from
the nature of the operator ˆ
M, and can easily be under- | [
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] |
the nature of the operator ˆ
M, and can easily be under-
stood under the following logic. The measurement of op-
erator ˆ
M after a collective π/2-pulse is equivalent to tak-
ing the expectation value of operator (ℏ/2)γp
�N
n=1 ˆσ(n)
x
over state (29). Given that the ˆσ(n)
x
operators, responsi-
ble for individual spin flips, induce transitions of the form
ˆσ(n)
x
|κ⟩ → {|κ − 1⟩ , |κ + 1⟩}, only coherences between
nearest-neighbor Dicke states contribute to the expecta-
tion value of ˆ
M. | [
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] |
nearest-neighbor Dicke states contribute to the expecta-
tion value of ˆ
M.
Secondly, the presence of the term cosN−1(4ϕ) in
eq. (32) poses another challenge. This term arises from
the relative phase accumulation among different trajec-
tories and restricts the duration of the protocol to times
that guarantee the condition ϕ = nπ, where n is an in-
teger. Any deviations from this condition result in the
rapid vanishing of the expectation value of ˆ
M, regard- | [
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] |
rapid vanishing of the expectation value of ˆ
M, regard-
less of the presence of CSL noise. Given that even the
smallest nanoparticles harbor millions of nuclear spins,
as indicated in Table (II), minor deviations ∆ϕ from nπ
could prove detrimental. A rough estimate suggests that,
since cosN(nπ + ∆ϕ) ≈ 1 − N(∆ϕ)2/2 + · · · , determin-
ing ϕ with a precision below 1/
√
N radians is required.
Nonetheless, with sufficient precision, one can always ad-
just the parameters and vary t1 to meet this criterion. | [
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just the parameters and vary t1 to meet this criterion.
One example would be to set t1 = π/Ω. Then, t2 would
be zero and the total time is Ttot = 2π/Ω. ϕ = nπ then
holds true for
B′ = ℏγ2
p
nV
�
µ3
0ρ
|χV |3 ,
(33)
where n is an integer. In general, the experiment can be
conducted faster by finding the values of t1 that meet the
required conditions for a given set of system parameters.
A more suitable observable for mitigating the impact of
relative phases and harnessing the collective behavior of | [
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0.08371803909540176
] |
relative phases and harnessing the collective behavior of
multiple spins is given by ∝ �N
n=1 ˆσ(n)
x . This observable
induces the transition �N
n=1 ˆσ(n)
x
|κ⟩ = |N − κ⟩ between
Dicke states.
Notably, the two connected states accu-
mulate the same relative phase throughout the protocol,
that is ϕκ = ϕN−κ, effectively eliminating from the ex-
pectation value the dependence on the geometric phase
50
100
150
200
250
300
0.2
0.4
0.6
0.8
N odd
N even
N
�
N
�
n=1
ˆσ(n)
x
�
FIG. 4.
��N
n=1 ˆσ(n)
x
� | [
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] |
50
100
150
200
250
300
0.2
0.4
0.6
0.8
N odd
N even
N
�
N
�
n=1
ˆσ(n)
x
�
FIG. 4.
��N
n=1 ˆσ(n)
x
�
for an arbitrary set of system parame-
ters and protocol duration.
��N
n=1 ˆσ(n)
x
�
is greater for even
values of N and smaller for odd ones.
This discrepancy
arises from the fact that for even N, there exists a trajec-
tory ˜κ = N/2 for which ˜κ = N − ˜κ and thus ΛN/2,N/2 = 0,
a condition not satisfied for odd N. Moreover, this term car-
ries the greatest weight in the sum of eq. (34).
For large | [
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] |
ries the greatest weight in the sum of eq. (34).
For large
N, the difference between odd and even N diminishes as
expected and
��N
n=1 ˆσ(n)
x
�
approaches the bound given by
exp(−ξTtot), which is depicted in red.
ϕ.
Furthermore, the decay rate now relies on the to-
tal number of spins, as contributions to the expectation
value stem not only from nearest-neighbor coherences.
This can be computed as:
� N
�
n=1
ˆσ(n)
x
�
= 1
2N
N
�
κ=0
�N
κ
�
e−Λκ,N−κ .
(34) | [
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] |
This can be computed as:
� N
�
n=1
ˆσ(n)
x
�
= 1
2N
N
�
κ=0
�N
κ
�
e−Λκ,N−κ .
(34)
Figure (4) shows how eq. (34) behaves with increasing N
for an arbitrary set of system parameters and protocol
duration. Notice that as Λκ,κ′ is bounded from below
by ξTtot, the expectation value ⟨�N
n=1 ˆσ(n)
x ⟩ at time Ttot
converges to the value exp(−ξ Ttot) as N increases.
However,
measuring
such
an
observable
presents
greater challenges compared to measuring the magnetic | [
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] |
measuring
such
an
observable
presents
greater challenges compared to measuring the magnetic
moment of the nanoparticle. One way to do so would
be to apply a global π-pulse on the spin ensemble that
is controlled on the state of an ancillary qubit, that is,
Uc = |0⟩⟨0|a⊗�N
n=1 ˆσ(n)
x
+|1⟩⟨1|a⊗I2N , where subscript a
indicates the operators acting on the ancillary qubit, and
I2N is the identity operator on the Hilbert space of the N
spin ensemble. The expectation value of �N
n=1 ˆσ(n)
x
can | [
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] |
spin ensemble. The expectation value of �N
n=1 ˆσ(n)
x
can
then be retrieved by measuring the off-diagonal element
of the density matrix of the ancillary qubit ⟨|0⟩⟨1|⟩a.
Implementing such a controlled gate, however, may
prove challenging in practice as it requires generating
and maintaining coherence between the ancillary qubit
and the N-spin ensemble for the duration of the readout.
One could argue that were such an operation available to
the experimentalist, they could instead use it to prepare | [
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the experimentalist, they could instead use it to prepare
an initial state that is a superposition of the two more
distant Dicke states, |0⟩a ⊗ |κ = 0⟩ + |1⟩a ⊗ |κ = N⟩,
and in this way maximize the achievable position
variance of the interferometer.
Notice, however, that
12
such a strategy would require maintaining the coherence
between the ancilla and the spin ensemble for the
duration of the experiment, while the use of such a gate
only during readout may be significantly less challenging.
2. | [
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] |
only during readout may be significantly less challenging.
2.
Modified protocol
Given the challenges faced by the standard matter-
wave interferometry protocol when attempting to utilize
the collective behavior of multiple spins, as highlighted
above, here, we propose a modified version thereof. This
adaptation aims to effectively leverage the presence of
numerous spins while still relying on the measurement of
the total magnetic moment of the nanoparticle, thus, cir- | [
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] |
the total magnetic moment of the nanoparticle, thus, cir-
cumventing the need for measuring correlations between
the spins.
π
2
π
2
3π
2
|0⟩, |1⟩, |2⟩
κ
κ, α
κ, β
t1 = π
Ω
t2 = 2π
Ω
t1 = π
Ω
FIG. 5. Example for the protocol to test the CSL model with
two spins.
A π/2 is applied in the beginning, bringing all
spins into a superposition of up and down.
Depending on
the spin state, the particle starts oscillating around dκ, the
wave function expands in space. Here, the trajectory the par- | [
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] |
wave function expands in space. Here, the trajectory the par-
ticle takes if all spins are pointing up, in this case κ = 2, is
highlighted. After half a period at t1 = π/Ω, a second π/2
pulse is applied and each trajectory again splits up into N +1
trajectories. Each of the trajectories is denoted by α. For
the κ = 2 branch, the different spin states of the αth trajec-
tory are color-coded. After a full period t2 = 2π/Ω, the αth
trajectories meet again after a time t1 = π/Ω and a third | [
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] |
trajectories meet again after a time t1 = π/Ω and a third
3π/2 pulse is applied. In the unitary case, the particle con-
tinues traveling on the κth trajectory and all paths meet after
another half period in the origin with zero momentum. In
the presence of noise, the coherence between the αth trajec-
tories breaks down. This decoherence is mapped on the spin
state when the αth trajectories meet.
Therefore, there are
also small contributions to Dicke states different from κ after | [
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] |
Therefore, there are
also small contributions to Dicke states different from κ after
the 3π/2 pulse is applied. The N + 1 trajectories on the κth
branch after the 3π/2 pulse are denoted with β.
The modified protocol is depicted in fig. 5 for the mini-
mal case of 2 spins (3 Dicke states). In essence, the intro-
duced modification is the application of additional π/2-
pulses, which has the effect of multiplying the number of
trajectories. The detailed protocol is as follows. Initially, | [
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] |
trajectories. The detailed protocol is as follows. Initially,
a π/2-pulse is applied and the wave packet splits into
N + 1 wave packets each following a different trajectory,
which are numbered consecutively by κ ∈ [0, N]. So far,
the protocol is identical to the standard one. The nov-
elty is introduced after a time t1 = π/Ω, when a second
π/2 pulse is applied. Each of the N + 1 trajectories then
further divides into an additional N+1 trajectories, num- | [
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] |
further divides into an additional N+1 trajectories, num-
bered by α ∈ [0, N], and resulting in a total of (N + 1)2
trajectories that we label (κ, α). In other words, the sec-
ond π/2-pulse performs the transformation (κ) → (κ, α).
After a period t2 = 2π/Ω, the trajectories α generated
by the second π/2-pulse overlap. If another 3π/2-pulse
is applied at this point, these trajectories recombine into
the original N + 1 trajectories, that is, (κ, α) → (κ), see | [
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] |
the original N + 1 trajectories, that is, (κ, α) → (κ), see
fig. 5. Finally, after a time t3 = π/Ω, all trajectories over-
lap, and the spin ensemble is decoupled from the motion
of the nanoparticle, provided that the process is coher-
ent. The total runtime is then given by Ttot = 4π/Ω,
that is, two periods of the oscillation frequency, and the
state of the spin ensemble at the end of the protocol is
given by
ˆρS = 1
2N
N
�
κ,κ′=0
N
�
α,α′=0
N
�
β,β′=0
��N
κ
���N
κ′
�
×
�π
2
�⊗N
κ,α
�3π
2
�⊗N
α,β
�π
2
�⊗N | [
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] |
ˆρS = 1
2N
N
�
κ,κ′=0
N
�
α,α′=0
N
�
β,β′=0
��N
κ
���N
κ′
�
×
�π
2
�⊗N
κ,α
�3π
2
�⊗N
α,β
�π
2
�⊗N
α′,κ′
�3π
2
�⊗N
β′,α′
× e−i(ϕκ,α,β(Ttot)−ϕκ′,α′,β′(Ttot))
× tr{ˆρκ,κ′,α,α′,β,β′(Ttot)} |β⟩ ⟨β′| ,
(35)
where (π/2)⊗N
κ,α and (3π/2)⊗N
κ,α are the matrix elements in
Dicke basis of the π/2- and the 3π/2-pulse, respectively,
whose explicit expression is provided in appendix C. If
the evolution is coherent, we find
tr{ˆρκ,κ′,α,α′,β,β′(Ttot)}
(36)
= tr
�
ˆD
� χ
2x0
ζκ,α,β(Ttot)
�
ˆρ0 ˆD†
� χ
2x0
ζκ′,α′,β′(Ttot)
�� | [
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] |
tr{ˆρκ,κ′,α,α′,β,β′(Ttot)}
(36)
= tr
�
ˆD
� χ
2x0
ζκ,α,β(Ttot)
�
ˆρ0 ˆD†
� χ
2x0
ζκ′,α′,β′(Ttot)
��
,
where ˆD(ζ) = exp
�
ζˆa† − ζ∗ˆa
�
is the displacement oper-
ator and ζκ,α,β(t) are the classical trajectories. These,
together with the geometric phase picked up by each of
them, ϕκ,α,β, are given in appendix D. ˆρ0 is the initial
state of the mechanical degree of freedom. For the pulse
timings described before, ζκ,α,β(Ttot) = 0 and eq. (36)
reduces to 1. On the other hand, the system parame- | [
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-0.17094534635543823,
-0.016654588282108307,
-0.054051924496889114,
-0.006528180558234453,
0.12051555514335632,
-0.2247801274061203,
-0.35360661149024963,
0.0641709640622139,
0.26066017150878906,
-0.03769535571336746,
0.31387439370155334,
0.3754137456417084,
-0.2888261675834656,
-0.36876559257507324,
0.3579563498497009,
-0.2888627052307129,
-0.21623943746089935,
-0.7150943875312805,
-0.782964825630188,
-0.39485087990760803,
0.8936402201652527,
0.03368138149380684,
0.1591639220714569,
0.2178717404603958,
-0.5801137685775757,
-0.26201099157333374,
-0.16113139688968658,
-0.9519741535186768,
-0.25724050402641296,
0.25956666469573975,
-0.6909597516059875,
-0.40571245551109314,
0.2779296636581421,
0.7214077711105347,
0.8493211269378662,
-0.28251174092292786,
-0.5082100629806519,
-0.1434471756219864,
0.004199345596134663,
-0.2277209758758545,
-0.28072863817214966,
-0.2251015305519104,
-0.46544286608695984,
0.10953401774168015,
-0.21805201470851898,
-0.37441831827163696,
0.11604345589876175,
0.4659084379673004,
0.017215974628925323,
-0.28865689039230347,
0.14694656431674957,
-0.5239284634590149,
-0.22751262784004211,
-0.1353137046098709,
-0.3358781337738037,
-0.11480686068534851,
-0.15584763884544373,
-0.4552607834339142,
-0.31741809844970703,
0.4236259460449219,
-5.875315189361572,
-0.21483302116394043,
-0.6422532200813293,
-0.1265764981508255,
-0.07033805549144745,
0.42308366298675537,
0.7147080898284912,
0.15354730188846588,
-0.2573774755001068,
-0.17089484632015228,
0.027687344700098038,
-0.16588589549064636,
-0.13838641345500946,
0.05011773854494095,
-0.13980543613433838,
0.20855829119682312,
0.43241533637046814,
0.23662981390953064,
0.3281896114349365,
0.676839292049408,
0.56865394115448,
-0.321108877658844,
0.005146238021552563,
-0.13558124005794525,
0.3887578248977661,
-0.07468900084495544,
-0.8216606378555298,
0.6165519952774048,
-0.5192491412162781,
-0.20818263292312622,
-0.5010685920715332,
-0.4911161959171295,
-0.1511676013469696,
-0.36535927653312683,
0.7065330743789673,
0.1563788503408432,
0.1522112339735031,
-0.23771296441555023,
0.5501338839530945,
-0.34130534529685974,
-0.19752614200115204,
0.871592104434967,
0.21538794040679932,
-0.30103713274002075,
-0.0615815706551075,
0.06760349869728088,
0.048805978149175644,
0.32844969630241394,
0.11582140624523163,
0.7925422191619873,
-0.7329949140548706,
-0.10391595214605331,
0.4547714293003082,
-0.3760802745819092,
0.03359483927488327,
-0.24825572967529297,
-0.25211840867996216,
0.43856868147850037,
0.0826190859079361,
0.4649116098880768,
-0.045992396771907806,
-0.4017232060432434,
0.18240293860435486,
-0.6419262886047363,
-0.11606812477111816,
0.08798696100711823,
-0.3485725522041321,
-0.08109357953071594,
0.5739777088165283,
0.1486080437898636,
-0.1128186509013176,
-0.39553019404411316,
-0.10084537416696548,
-0.6040631532669067,
0.4065738320350647,
-0.4844691753387451,
0.7273674011230469,
-0.07642519474029541,
-0.14978967607021332,
0.278073251247406,
-0.3420209288597107,
-0.5569533109664917,
-0.1434662938117981,
0.11627204716205597,
0.12704138457775116,
0.033882927149534225,
-0.149953693151474,
-0.6486223936080933,
-0.33335259556770325,
0.11292789876461029,
0.6364375352859497,
0.14759264886379242,
0.3067108988761902,
-0.44573453068733215,
0.014602864161133766,
-0.14049407839775085,
-0.033253178000450134,
-0.29574090242385864,
0.5121750831604004,
-0.0675424337387085,
0.626855194568634,
-0.3052830398082733,
0.34218546748161316,
-0.2377912402153015,
-0.08199922740459442,
0.16412976384162903,
-0.19983510673046112,
0.11035829782485962,
0.5309594869613647,
0.08890990912914276,
-0.14687462151050568,
-0.27457037568092346,
-0.46994027495384216,
-0.0629873126745224,
0.24712225794792175,
-0.4882012605667114,
-0.4351654648780823,
-0.18989531695842743,
0.4912533760070801,
-0.22497348487377167,
0.0263028796762228,
0.2013157606124878,
-0.6722350716590881,
-0.32136547565460205,
0.0815589651465416,
0.14774787425994873,
-0.09127112478017807,
0.5012622475624084,
-0.019113466143608093,
-0.19863207638263702,
-0.03219534456729889,
0.2929978668689728,
-0.3981279730796814,
0.3216160833835602,
0.1684635430574417,
-0.22041454911231995,
-0.011605649255216122,
0.687955915927887,
0.14924173057079315,
-0.33993083238601685,
0.45390990376472473,
0.2842036783695221,
-0.1236257255077362,
-0.6702885627746582,
0.35563743114471436,
0.525968074798584,
0.562510073184967,
-0.271205872297287,
-0.48517677187919617,
0.4052370488643646,
0.8587196469306946,
-0.5181043148040771,
0.28797072172164917,
0.08348371833562851,
-0.3812691867351532,
0.11744154244661331,
-0.13334417343139648,
0.07063660770654678,
0.4216849207878113,
-0.20515082776546478,
0.3433893322944641,
0.5243468284606934,
-0.6790327429771423,
0.0072838058695197105,
0.03848632052540779,
0.07831601798534393,
-0.54377281665802,
-0.5497598648071289,
0.23865659534931183,
0.028965527191758156,
-0.18864645063877106,
-0.3517393171787262,
-0.2207992970943451,
-0.7736961245536804,
0.0932348296046257,
-0.13676047325134277,
-0.12237633019685745,
-0.4525376856327057,
-0.007314897142350674,
-0.3288237750530243,
0.07505850493907928,
0.49142986536026,
-0.23109811544418335,
0.9076794981956482,
0.41989651322364807,
-0.5589280128479004,
-0.34184959530830383,
0.2205134779214859,
-0.1992320716381073,
-0.08548106253147125,
0.7187045216560364,
0.026929061859846115,
0.16146063804626465,
-0.4537956714630127,
0.21603794395923615,
0.12876862287521362,
0.5507713556289673,
1.390921711921692,
-0.4883168041706085,
-0.015818003565073013,
0.18774248659610748,
0.566142201423645,
-0.1498280018568039,
0.656813383102417,
0.343228280544281,
0.28774750232696533,
0.037955448031425476,
-0.10535290092229843,
0.16317105293273926,
-0.06988033652305603,
-0.2161145657300949,
-0.8138566017150879,
0.06920401751995087,
0.1032891497015953,
-0.20268338918685913,
-0.026345623657107353,
0.7035849094390869,
-0.5312256813049316,
-0.41277778148651123,
-0.4626741111278534,
0.40471768379211426,
-0.03629954904317856,
0.512458086013794,
0.26658743619918823,
-0.17396360635757446,
0.1418084055185318,
-0.34887632727622986,
-0.3318568766117096,
0.26207128167152405,
0.20085926353931427,
-0.3855946958065033,
0.08298500627279282,
-0.08120711892843246,
-0.2548188865184784,
-0.5495789647102356,
-0.3248516321182251,
-1.0439351797103882,
0.04779394343495369,
-0.9957833886146545,
0.22681394219398499,
0.01532710250467062,
0.5698662996292114,
0.26184535026550293,
-0.10019996762275696,
-0.3454705774784088,
0.14188437163829803,
0.7366184592247009,
-0.20497320592403412,
0.050960492342710495,
0.6336770057678223,
0.4045066237449646,
0.44058555364608765,
-0.6678820252418518,
-0.2577953040599823,
0.37407344579696655,
0.2621857225894928,
0.3504743278026581,
-0.6043640375137329,
0.41824871301651,
0.7537434697151184,
0.08994434028863907,
-0.006043380592018366,
-0.34122639894485474,
-0.6319652199745178,
-0.4418739974498749,
1.0003987550735474,
-0.4352533221244812,
-0.999387800693512,
0.7938271164894104,
-0.23981377482414246,
0.4291633665561676,
-0.05433566868305206,
0.44774389266967773,
0.17420190572738647,
-0.09164956212043762,
0.3803231418132782,
0.06854452937841415,
0.34989848732948303,
0.005829111207276583,
-0.10375181585550308,
0.011439685709774494,
-0.1543671190738678,
0.062386516481637955,
-0.3057243525981903,
0.30779293179512024,
0.1830969601869583,
0.008121512830257416,
-0.20127537846565247,
-0.21643094718456268,
0.5895297527313232,
-0.1870153397321701,
-0.09732135385274887,
-0.7669450640678406,
0.3033360242843628,
-0.27217650413513184,
0.38177549839019775,
0.26626408100128174,
0.5469037294387817,
-0.2820552885532379,
0.10347983986139297,
-0.3719500005245209,
-0.04836493730545044,
0.66828453540802,
0.10831303149461746,
-0.31133604049682617,
-0.3965357542037964,
0.45699331164360046,
0.2368181198835373,
-0.5180376172065735,
-0.3009047210216522,
0.3697447180747986,
-0.4377770721912384,
-0.14064669609069824,
0.6167858839035034,
-0.007002034690231085,
-0.35675984621047974,
0.9294015765190125,
-0.12945617735385895,
-0.13241133093833923,
0.6596877574920654,
-0.19453606009483337,
0.8270928859710693,
0.5227325558662415,
-0.7451075315475464,
-0.28318071365356445,
0.16916009783744812,
-0.3502151072025299,
0.010449405759572983,
-0.19252575933933258,
0.17319250106811523,
0.574720025062561,
0.3436923325061798,
-0.4792711138725281,
0.46390801668167114,
0.05925360321998596,
0.015854576602578163,
0.2965735197067261,
0.0010744411265477538,
0.1629151701927185,
0.3032248020172119,
0.8288714289665222,
0.8168292045593262,
-0.2972770929336548,
0.17513155937194824,
0.20942671597003937,
0.025591270998120308,
0.44140109419822693,
-0.05871833115816116,
-0.2787310481071472,
-0.12529855966567993,
0.2736221253871918,
-0.3043900430202484,
0.13873961567878723,
-0.6004683375358582,
-0.38258352875709534,
0.28942805528640747,
0.49982649087905884,
0.1191217228770256,
0.26740023493766785,
0.5664540529251099,
-0.013439047150313854,
0.5855582356452942,
0.3198654353618622,
-0.5984383225440979,
-0.5428071022033691,
-0.1255774199962616,
0.6001695394515991,
-0.09513729065656662,
-0.12835314869880676,
-0.2549903988838196,
0.21492530405521393,
0.6139969229698181,
-0.19065260887145996,
0.15399645268917084,
0.2148890495300293,
-0.6929745674133301,
0.35332971811294556,
0.3742029070854187,
-0.37676531076431274,
0.07981180399656296,
-0.024796012789011,
0.052535530179739,
-0.23667104542255402,
-0.6854703426361084,
0.005929804407060146,
-0.5507203936576843,
0.12600333988666534,
-0.35453739762306213,
-0.2125799059867859,
0.1449660360813141,
0.6240074634552002,
-1.012976884841919,
-0.47322967648506165,
-0.019719921052455902,
0.05480079725384712,
-0.06122956797480583,
-0.7344844937324524,
0.4319475293159485,
0.4730859398841858,
0.4597037136554718,
0.3312809467315674,
0.14369627833366394,
0.14848379790782928,
0.45819661021232605,
-0.1269385665655136,
0.2440628856420517,
-0.010567053221166134,
0.699924647808075,
-0.3100201487541199,
-0.23914967477321625,
0.7502803206443787,
0.11767514795064926,
0.007648954633623362,
0.4079214930534363,
-0.18701808154582977,
-0.09508506953716278,
0.28063052892684937,
-0.3156871795654297,
-0.40325409173965454,
-0.47665947675704956,
-0.9927362203598022,
-0.2826889455318451,
0.139569953083992,
0.3219633996486664,
0.022832736372947693,
-0.20370611548423767,
0.6084744930267334,
0.14689022302627563,
-0.4298745393753052,
0.1692299246788025,
-0.16925457119941711,
-0.3418283760547638,
0.09364287555217743,
0.8918283581733704,
-0.4598301649093628,
0.16351056098937988,
0.009449638426303864,
0.4929032027721405,
-0.044743530452251434,
-0.06686323136091232,
0.043702419847249985,
0.1771364063024521,
-0.329643577337265,
0.7545071244239807,
0.3565855324268341,
0.10555054992437363,
-0.5887948274612427,
-0.36216819286346436,
-0.5115921497344971,
0.236406147480011,
0.18145869672298431,
-0.0025971163995563984,
-0.47427091002464294,
-0.09615930169820786,
0.16745400428771973,
-0.3739427626132965,
-0.2673632502555847,
-0.2540077269077301,
-0.373496413230896,
0.15645375847816467
] |
reduces to 1. On the other hand, the system parame-
ters can be chosen to guarantee that ϕκ,α,β = nπ, with
n an integer. For example, for the total runtime of 4π/Ω
described above, this is achieved by choosing
B′ = 2ℏγ2
p
nV
�
µ3
0ρ
|χV |3 .
(37)
13
This is also the case for all examples given in ta-
ble
II.
Under
this
conditions,
we
recover
ˆρS
=
2−N �N
κ,κ′=0
��N
κ
��N
κ′
�
|κ⟩ ⟨κ′|.
Thus, in this noiseless
case, after applying a π/2-pulse and measuring
ˆ
M, we
find ⟨ ˆ
M⟩ (Ttot) = ⟨ ˆ | [
0.0011136780958622694,
0.030021697282791138,
0.3544524013996124,
-0.47621893882751465,
-0.2685237526893616,
0.45534855127334595,
0.5242667198181152,
-1.1363499164581299,
-0.10893727093935013,
-0.4300622344017029,
-0.22893276810646057,
0.6928368806838989,
-0.3403221070766449,
-0.18076014518737793,
0.1119244396686554,
0.38572555780410767,
1.0703096389770508,
-0.020535966381430626,
-0.5794916152954102,
-0.8700456619262695,
0.3686486780643463,
0.31117841601371765,
0.7610349059104919,
0.09683902561664581,
-0.007467355579137802,
0.3256668746471405,
0.3607606887817383,
0.2552545964717865,
0.12643757462501526,
0.14950178563594818,
-0.2510436475276947,
0.5784010887145996,
0.055488187819719315,
0.08478830009698868,
0.4462509751319885,
0.41231268644332886,
0.6211491227149963,
0.012034900486469269,
0.5025274157524109,
0.12129630148410797,
0.11823882162570953,
0.01298457384109497,
0.05707960203289986,
0.280780166387558,
-0.3796525001525879,
-0.5514962077140808,
-1.195724368095398,
0.1020151823759079,
-0.33720511198043823,
-0.43140336871147156,
-0.6387491822242737,
0.015249027870595455,
0.261125385761261,
0.2914856970310211,
-0.3135956823825836,
0.3269493579864502,
-0.6212282776832581,
0.0028332427609711885,
0.4828792214393616,
-0.10508187115192413,
-0.06751032918691635,
-0.4893941879272461,
-0.25736162066459656,
-0.060991071164608,
-0.19443798065185547,
0.6808156967163086,
0.16210544109344482,
0.2559387981891632,
-0.190637469291687,
0.3074357509613037,
0.42422616481781006,
-0.0005213471013121307,
-0.31456494331359863,
0.3612936735153198,
-0.1109607145190239,
0.22344116866588593,
-0.3431740701198578,
-0.2846832573413849,
0.28559181094169617,
0.2365237921476364,
-0.13168881833553314,
-0.2148572951555252,
0.5735797882080078,
-0.6483685970306396,
0.39491531252861023,
-0.08635006844997406,
-0.0342940017580986,
0.5765466690063477,
-0.7332671880722046,
0.1853323131799698,
-0.07477660477161407,
-0.7850000858306885,
0.12188240140676498,
0.5085431933403015,
0.13460876047611237,
0.5129829049110413,
0.028118737041950226,
-0.37996581196784973,
0.7273042798042297,
0.03732132166624069,
-0.15922601521015167,
-0.5031516551971436,
-0.026499081403017044,
-0.10617201775312424,
-0.5360355973243713,
-0.6925235986709595,
-0.5894442200660706,
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] |
case, after applying a π/2-pulse and measuring
ˆ
M, we
find ⟨ ˆ
M⟩ (Ttot) = ⟨ ˆ
M⟩ (0), as expected.
Now, our job is to understand how the expectation
value of
ˆ
M at the end of the protocol deviates from
⟨ ˆ
M⟩ (0) in the presence of CSL noise. When CSL noise
acts on the system the coherences between the different
(κ, α) paths decay at different rates. This decoherence
manifests on ⟨ ˆ
M⟩ in two ways: first, the loss of coherence
between the (κ, α) and (κ ± 1, α′) trajectories results in | [
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