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between the (κ, α) and (κ ± 1, α′) trajectories results in
the decoherence between the κth and κ±1th trajectories.
This enhances the loss of coherence between neighboring
trajectories at the end of the protocol, which is picked up
by the expectation value of ˆ
M. Second, the loss of coher-
ence between trajectories (κ, α) and (κ, α′) prevents the
successful completion of the transformation (κ, α) → (κ)
after the third 3π/2-pulse. The spin state does not return | [
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] |
after the third 3π/2-pulse. The spin state does not return
perfectly to the κth Dicke state but to a superposition of
Dicke states, inducing a transformation of trajectories of
the form (κ, α) → (κ, β). Therefore, not all trajectories
overlap at the end of the protocol and, thus, the spin and
mechanical degrees of freedom do not become separable.
This, however, is not detrimental as it enhances the de-
coherence due to the CSL noise on the spin ensemble, | [
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] |
coherence due to the CSL noise on the spin ensemble,
making it more easily detectable. The full expression of
the state of the spin ensemble at the end of the protocol
in the presence of CSL noise is given in appendix D.
Unfortunately, ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) has no closed form
expression and needs to be evaluated numerically. More-
over, its numerical evaluation becomes increasingly hard
with a growing number of spins N.
Our simulations
are limited to values of N ≈ 200.
In fig. 6 we plot
the value ⟨ ˆ | [
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] |
Our simulations
are limited to values of N ≈ 200.
In fig. 6 we plot
the value ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) as a function of (χ/rCSL)N.
Even though the numerical results clearly suggest that
⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) is a function of (χ/rCSL)N, we have
been unable to prove this analytically. To explore the full
range of the function we combine curves corresponding to
different values of χ where for each curve N ranges from
10 to 250. It is evident that the expectation value of ˆ
M | [
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] |
10 to 250. It is evident that the expectation value of ˆ
M
now depends on the number of spins as desired, and can,
thus, outperform the sensitivity of a single spin interfer-
ometer. It is noteworthy to mention, that this observable
is lower bounded by ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) ≥ exp(−ξ Ttot),
with the bound corresponding to the case where the co-
herence between any pair of Dicke states decays at the
maximum decay rate, that is, at ξ. We observe that as
(χ/rCSL)N grows the magnetic moment of the nanopar- | [
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-0.48969539999961853,
0.11287616938352585,
-0.16501985490322113,
0.7128654718399048,
-0.1946205049753189
] |
(χ/rCSL)N grows the magnetic moment of the nanopar-
ticle approaches this limit. This lower bound can be used
to restrict the values that the free parameters of the CSL
model can take. For instance, if a given experiment mea-
sures at the end of the protocol a magnetization ⟨ ˆ
M⟩exp,
an upper bound on ξ is given by
ξ <
ln ⟨ ˆ
M⟩exp
Ttot
,
(38)
0.2
0.4
0.6
0.8
1
χ = 10−10 m
χ = 10−9 m
χ = 10−8 m
χ = 10−7 m
10−3
10−2
10−1
100
101
102
0.97
0.98
0.99
1
χ
rCSL N
⟨ ˆ
Mtot⟩/⟨ ˆ
M0⟩ | [
0.07572811841964722,
0.06975597143173218,
0.923015832901001,
-0.07829110324382782,
-0.017548009753227234,
0.0532057099044323,
0.19994215667247772,
-0.769538402557373,
-0.026695016771554947,
-0.04472927376627922,
-0.2634999752044678,
0.3382877707481384,
-0.0203678160905838,
0.35329583287239075,
-0.045468930155038834,
0.964031457901001,
0.8219567537307739,
0.3600991368293762,
-0.7694270014762878,
-1.0362820625305176,
0.2381768822669983,
-0.09124674648046494,
0.5932002067565918,
0.12829799950122833,
0.10537704825401306,
0.18353943526744843,
-0.046860624104738235,
-0.2740591764450073,
0.45999646186828613,
0.40051963925361633,
-0.15038765966892242,
0.5493414998054504,
0.013694512657821178,
-0.07060110569000244,
0.4912683665752411,
0.06904784590005875,
0.22046655416488647,
0.1640157699584961,
0.27947694063186646,
-0.3157402276992798,
0.35303208231925964,
-0.16004440188407898,
0.2070772349834442,
0.5331171751022339,
-0.4608526825904846,
-0.6456962823867798,
-1.0405999422073364,
0.08588527888059616,
-0.038533467799425125,
-0.32961195707321167,
-0.11644046753644943,
0.16241247951984406,
0.22197698056697845,
0.16460372507572174,
-0.6089929342269897,
0.2716171443462372,
-0.8475351929664612,
-0.20665735006332397,
0.7100131511688232,
-0.157578244805336,
0.04138454794883728,
-0.08439286053180695,
-0.3139614462852478,
0.29953688383102417,
0.0806097611784935,
0.7326218485832214,
0.06745661050081253,
0.12499619275331497,
-0.5238030552864075,
0.15340650081634521,
0.6958602070808411,
0.27376219630241394,
-0.11231966316699982,
0.2339990884065628,
0.21244904398918152,
0.5499140024185181,
-0.23077505826950073,
-0.4377067983150482,
0.4442257583141327,
0.3948443531990051,
0.06985633075237274,
-0.11235744506120682,
0.4226757287979126,
-0.24167890846729279,
0.24500596523284912,
0.522795557975769,
0.18186400830745697,
0.712401270866394,
-0.44064003229141235,
0.36399027705192566,
0.07440342009067535,
-0.34609735012054443,
0.2555137872695923,
0.249003604054451,
0.30856841802597046,
0.48969730734825134,
0.05870082601904869,
-0.5181225538253784,
0.13984237611293793,
0.1893988847732544,
-0.13063105940818787,
0.02465084008872509,
-0.45596256852149963,
-0.09011353552341461,
-0.5295339226722717,
-0.4527569115161896,
-0.5465278625488281,
-0.5330780744552612,
-0.2258591204881668,
-0.10556600987911224,
-0.855032742023468,
0.1879967749118805,
-0.5732637047767639,
-0.31970125436782837,
-0.1632850468158722,
-0.6216824054718018,
0.04219330474734306,
-0.4979945421218872,
0.2656644284725189,
0.6773185729980469,
-0.40117350220680237,
1.075486183166504,
-0.056226786226034164,
0.35915160179138184,
0.459045946598053,
0.2728588879108429,
0.6206185817718506,
0.9127232432365417,
-0.41447708010673523,
0.3380351960659027,
0.07634084671735764,
1.052160620689392,
0.32557833194732666,
-0.024375248700380325,
-0.17790193855762482,
-0.4921156167984009,
-0.08129531890153885,
0.11663667857646942,
-0.6647352576255798,
0.1673978865146637,
-0.01430788915604353,
-0.4794836938381195,
-0.1545337736606598,
0.09876558184623718,
0.08323659747838974,
-0.35413676500320435,
-0.4498569667339325,
0.03998946771025658,
-0.1094239354133606,
0.13329778611660004,
-0.055305663496255875,
0.3546350300312042,
-0.8838059902191162,
-0.15684685111045837,
-0.2951824367046356,
-0.4248601198196411,
-0.3466818928718567,
0.028464891016483307,
-0.3611465394496918,
0.04551597312092781,
-0.12094752490520477,
0.16547130048274994,
-0.1881902515888214,
-0.12289445847272873,
0.4498038589954376,
-0.33396610617637634,
0.774806022644043,
-0.17193514108657837,
-0.28998443484306335,
0.5070543885231018,
-0.08360940217971802,
-0.06426998972892761,
-0.32754379510879517,
-0.06309041380882263,
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0.21338596940040588,
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0.3754039406776428,
0.2136365920305252,
-0.025301439687609673,
-0.636293351650238,
0.02802163176238537,
-0.21272075176239014,
-0.38220342993736267,
0.5449245572090149,
0.4879341125488281,
0.5829545259475708,
0.36349186301231384,
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0.12584982812404633,
-0.3775940537452698,
-0.025747964158654213,
0.6097701787948608,
-1.058565616607666,
1.6517930030822754,
0.14454776048660278,
-0.09981895983219147,
0.4560058116912842,
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-0.016859598457813263,
-0.26053884625434875,
0.36569443345069885,
0.34415435791015625,
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-0.7363743782043457,
0.4563431739807129,
0.2749018669128418,
-0.06500190496444702,
-0.7035040855407715,
-0.17146079242229462,
0.48696771264076233,
-0.07235372811555862,
-0.0676225945353508,
-0.3658677935600281,
-0.3027077615261078,
-0.42932188510894775,
-0.012881708331406116,
0.06621444225311279,
0.3284185230731964,
0.16120103001594543,
0.8914036750793457,
-0.6598446369171143,
0.33511918783187866,
0.5865874290466309,
-0.16723918914794922,
0.744384229183197,
0.7099334001541138,
0.3986806273460388,
0.49082058668136597,
-0.23211444914340973,
-0.45567622780799866,
-0.608704686164856,
0.1939186155796051,
0.11785483360290527,
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0.40017083287239075,
0.13680200278759003,
0.07352292537689209,
0.615768551826477,
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0.16457109153270721,
0.7083909511566162,
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0.6354352831840515,
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0.07867912203073502,
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0.17950986325740814,
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0.22218438982963562,
0.02409275248646736,
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1.0110477209091187,
0.32050302624702454,
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0.09969140589237213,
-0.4845636785030365,
-0.09555203467607498,
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0.007915779016911983,
0.5354799628257751,
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0.08753161132335663,
0.7416182160377502,
0.7276981472969055,
0.1893603652715683,
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0.5166209936141968,
0.24313345551490784,
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0.035998743027448654,
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0.01319909282028675,
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0.580088198184967,
0.4399780333042145,
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0.07779703289270401,
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0.4480586647987366,
0.37029775977134705,
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] |
χ = 10−8 m
χ = 10−7 m
10−3
10−2
10−1
100
101
102
0.97
0.98
0.99
1
χ
rCSL N
⟨ ˆ
Mtot⟩/⟨ ˆ
M0⟩
FIG. 6. Illustrates of the behavior of ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) for
the parameters in table II a) (upper plot) and table II b)
(lower figure) and for fixed χ. Due to limited computation
times, ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) could not be evaluated numerically
for N ≈ 108 as required, but only for N < 250. Therefore,
additionally to varying N, χ was varied as this is equivalent | [
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] |
additionally to varying N, χ was varied as this is equivalent
to varying the gyromagnetic ratio of the spins. In different
colors, one sees ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) for different, fixed χ and
N ∈ [10, 250]. As one can see, the plots strongly indicate that
⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) is a function of (χ/rCSL)N. Furthermore,
for (χ/rCSL)N > 102, it approaches its lower limit given by
exp(−ξTtot) as indicated by the red, dashed line. For all pa-
rameters in table II, (χ/rCSL)N ≫ 102. Therefore we can as- | [
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] |
rameters in table II, (χ/rCSL)N ≫ 102. Therefore we can as-
sume that ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) is very close to its lower bound.
as any larger value of ξ would result in stronger decoher-
ence and therefore a lower value of ⟨ ˆ
M⟩ at the end of the
protocol. As ξ depends on rCSL and λCSL, they can be
bounded by measuring ⟨ ˆ
M⟩exp.
In fig. 7, the parameter space of the CSL model is
shown, marking the areas already excluded by various
other experiments. In the same figure, we show the areas | [
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] |
other experiments. In the same figure, we show the areas
that could, in principle, be excluded by our protocol if
operated with the set of parameters in table II (b). Based
on this analysis, the parameters proposed by GRW could
be potentially excluded. Comparable bounds could be
put with the other two parameter sets in table II. How-
ever, achieving complete exclusion of the parameter space
would likely necessitate significantly longer experimental
times or heavier particles. | [
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] |
would likely necessitate significantly longer experimental
times or heavier particles.
The question arises as to whether the scheme can be
14
Adler
GRW
FIG. 7. Exclusion plot of the parameter space of the CSL
model, with λCSL in the vertical axis and rCSL in the hor-
izontal one.
Various experimentally excluded areas are de-
picted: the green area corresponds to cantilever-based ex-
periments [77], the blue region represents experiments con-
ducted with the gravitational wave detectors LIGO and LISA | [
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] |
ducted with the gravitational wave detectors LIGO and LISA
Pathfinder [78–80], the brown bounds are derived from spon-
taneous X-ray emission tests [81, 82], while the gray area is
theoretically excluded [83]. The values for λCSL and rCSL pro-
posed by Adler [84] and by Ghirardi, Rimini and Weber [85]
are highlighted in black. Additionally, we show in red, the
region that could, in principle, be excluded with pentacene-
doped naphthalene when selecting parameters as in table II | [
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] |
doped naphthalene when selecting parameters as in table II
b). Similar exclusion lines would be applicable for the other
two sets of parameters.
further enhanced by adding additional π/2 and 3π/2
pulses. This would, on the one hand, increase the du-
ration of the interferometric protocol as well as the max-
imal width of the delocalization of the wave function,
both of which would increase the sensitivity to possible
collapses of the wave function. However, as error sources | [
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] |
collapses of the wave function. However, as error sources
due to the increased number of pulses would also increase
it is difficult to assess whether further improvements can
be obtained and we leave the detailed analysis to future
work. In any case, we do not expect a significant im-
provement for the parameter regime investigated in this
work, as ⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0) already approaches its lower
bound with the scheme presented here.
D.
Measurement schemes | [
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] |
M⟩(0) already approaches its lower
bound with the scheme presented here.
D.
Measurement schemes
We now move on to the question of the measurement
of ⟨ ˆ
M⟩. One possibility could be to leverage the pres-
ence of the photo-excited triplet state of the embedded
pentacene molecule and its magnetic interaction with the
nuclear spin ensemble. In particular, the polarization of
the nuclear spin ensemble would have two notable effects
on the pentacene molecule. On the one hand, it would | [
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] |
on the pentacene molecule. On the one hand, it would
shift the energy splitting between the magnetic triplet
states |+1⟩ and |−1⟩. On the other hand, the polariza-
tion of the spin-lattice could be transferred to the triplet
state of the pentacene molecule, provided that this is pre-
pared in a suitable initial state. Both of these effects are,
in principle, measurable and proportional to the polar-
ization of the spin ensemble. However, they are bound to | [
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] |
ization of the spin ensemble. However, they are bound to
be more sensitive to the state of the spins near the pen-
tacene molecule, while underestimating the contribution
from spins sitting further from it. We thus turn to alter-
native measurement schemes that can be sensitive to the
global nuclear spin polarization. In particular, we pro-
pose to exploit the spin-motion coupling of the nanopar-
ticle to map expectation values of the spin ensemble onto
measurements of the center-of-mass position. | [
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] |
measurements of the center-of-mass position.
The Hamiltonian of our system is of the form
ˆH = ℏΩˆa†ˆa + ℏλ ˆZ(ˆa + ˆa†),
(39)
where ˆZ = �N
n=1 σ(n)
z
is the observable of interest and
λ = (γpB′/2)
�
ℏ/2mΩ. ⟨ ˆZ⟩ is directly related to ⟨ ˆ
M⟩
by ⟨ ˆ
M⟩ = ℏ/2γ⟨ ˆZ⟩.
For the particle diameters below
200 nm that we consider, it is safe to assume that the
magnetic field gradient is constant across the particle.
Consequently, a flip of any of the nuclear spins yields the | [
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] |
Consequently, a flip of any of the nuclear spins yields the
same momentum kick which, in turn, can then be mea-
sured by the displacement.
The evolution under such
a Hamiltonian has an exact solution, which in the in-
teraction picture w.r.t.
H0 = ℏΩa†a, is given by the
unitary-evolution operator
ˆU = eλ ˆ
Z[α(t)ˆa†−α∗(t)ˆa]ei λ2
Ω ˆ
Z2[t− sin(Ωt)
Ω
],
(40)
where α(t) = (1 − eiΩt)/Ω.
To see how such an evolution maps the operator ˆZ onto
the position, we consider the time evolution of the dimen- | [
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] |
the position, we consider the time evolution of the dimen-
sionless position operator ˆX = ˆa + ˆa† in the Heisenberg
picture
ˆX(t) = ˆU † ˆX(0) ˆU = ˆX(0) + 4λ ˆZ
Ω
sin2(Ωt/2).
(41)
In this form, it becomes evident that, for any state of
the spin-motion composite system, the expectation value
of the ˆZ operator can be retrieved from the expectation
value of the position operator as
⟨ ˆZ⟩ =
Ω
4λ sin2(Ωt/2)[ ⟨ ˆX⟩ (t) − ⟨ ˆX⟩ (0)].
(42)
Here, the coefficient Ω/(4λ) captures the required preci- | [
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] |
(42)
Here, the coefficient Ω/(4λ) captures the required preci-
sion in the position measurement.
While this will, in
general, depend on the parameters of the system, we
find that the position measurement precision required to
make meaningful measurements of the spin polarization
is not that great. For example, for the case of a particle
15
of radius r = 100 nm and a magnetic field gradient of
B′ = 105 T/m, we find that Ω/(4λ) = 0.6. This suggests
that a position displacement of two times the ground | [
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] |
that a position displacement of two times the ground
state width, which in this example is x0 = 6 · 10−11 m,
would provide enough sensitivity to distinguish a change
in ⟨ ˆ
M⟩ of 1 unit, corresponding to the change induced
by a single nuclear spin flip. This is quite a remarkable
measurement precision, taking into account that opti-
cal position measurements of levitated nanoparticles with
≈ 10−10 m resolution have been demonstrated [86] and
as low as 10−12 m with charged particles [87]. For an | [
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] |
as low as 10−12 m with charged particles [87]. For an
initially fully polarized particle, ⟨ ˆZ⟩ = N. The change
in the displacement of a naphthalene crystal which is ini-
tially fully polarized is then given by
∆xdis = x0
4λ
Ω
�
1 − ⟨ ˆ
M⟩(Ttot)
⟨ ˆ
M⟩(0)
�
.
(43)
In table II, Ω/(4λ) as well as ∆xdis for three different
examples are given.
The presence of the pentacene molecule presents an
additional method to read out the position of the naph-
thalene particle.
By probing the Zeemann splitting of | [
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] |
thalene particle.
By probing the Zeemann splitting of
the photo-excited triplet state of pentacene, which de-
pends on the position of the particle due to the presence
of the magnetic field gradient, information on the posi-
tion of the particle can be retrieved. Such techniques are
standard in magnetometry with NV centers [88].
The lifetime of the photo-excited triplet state limits the
resolution of the measurement, as this determines the line
width. In naphthalene, the lifetime of the excited triplet | [
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] |
width. In naphthalene, the lifetime of the excited triplet
states is in the order of τ = 50 µs [35], corresponding
to a line width ∆ωpent = τ −1 = (2π)3.1 kHz. Interac-
tions with the surrounding nuclear spins would in gen-
eral broaden this line width, but as magic angle spinning
suppresses dipole-dipole interactions, this broadening is
also suppressed in our setup. To resolve the change in
displacement due to the CSL model rapidly, this should
ideally induce a larger Zeemann shift than ∆ωpent. | [
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ideally induce a larger Zeemann shift than ∆ωpent.
For the example discussed above, with a magnetic field
gradient of B′ = 105 T/m, we find the following relation
between position displacement ∆x and the shift of the
transition frequency ∆ω in SI units, ∆ω ≈ 1.8 · 1013∆x.
This suggests that a displacement of the order of the zero-
point motion of the nanoparticle, which in this case, for
a particle of r = 100 nm, is x0 = 7 · 10−11 m, induces a
frequency shift on the order of kHz, and corresponds to a | [
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] |
frequency shift on the order of kHz, and corresponds to a
magnetic field change on the order of µT. Hence, already
displacements a few times the zero-point motion could,
in principle, be resolved with this method. To see the
effect of the CSL model, for the numbers as in table II,
displacements on the order of 10−5 m and larger would
have to be resolved. This results in Zeeman shifts in the
order of MHz and higher, which is well above the limit
imposed by the line width. | [
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] |
order of MHz and higher, which is well above the limit
imposed by the line width.
In reality, the nuclear spins in the nanoparticle will
not be fully polarized.
When the nuclear spins of the
nanoparticle are polarized by a percentage p, the initial
r
ωMAS
x0
d
⃗B = (B′(x) + B0)⃗ez
x
z
y
FIG. 8.
Here, a spherical nanoparticle with radius r in a
magnetic field gradient is depicted. The particle rotates with
ωMAS around an axis that is tided with respect to the mag- | [
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] |
ωMAS around an axis that is tided with respect to the mag-
netic field. A spin in this particle will see an oscillating mag-
netic field ⃗Beff = {B0 + B′[x0 + d sin(ωMASt)]}⃗ez. x0 is the
x coordinate of the point on the rotation axes closest to the
spin and d is the distance between the spin and this point
along the x-axis.
magnetization and consequently the final magnetization
too will be reduced by a factor of p. In table II, a few
examples with different sets of parameters for an initially | [
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] |
examples with different sets of parameters for an initially
50% polarized particle are given.
E.
Pulses
The protocol suggested in the lines above operates un-
der the assumption that the particle is undergoing magic
angle spinning. Due to this rotation, the magnetic field
observed by the nuclear spins will in general be time-
dependent, and thus, it prompts the question of whether
this affects our ability to deliver with sufficient precision
the microwave pulses required for the protocol. In this | [
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] |
the microwave pulses required for the protocol. In this
section, we look into answering this question.
The particle is sitting in a magnetic gradient field
B(x) = B0 + B′x along the z-direction and is rotat-
ing with frequency ωMAS around a rotation axis that
is tilted with respect to the magnetic field by an angle
ϕMAS = arccos
�
1/
√
3
�
≈ 54.74°. In this arrangement, a
microwave pulse is applied to the system with frequency
ωM and Rabi frequency ΩM. The Hamiltonian describing | [
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ωM and Rabi frequency ΩM. The Hamiltonian describing
such a configuration for a given spin i is provided by
ˆH =ℏ
2γp{B0 + B′[x0 + d sin(ωMASt)]}ˆσ(i)
z
+ ℏ
2ΩM cos(ωMt)ˆσ(i)
x .
(44)
Here, x0 is the x coordinate of the point on the rotation
axes closest to the spin and d is the distance between
the spin and this point along the x-axis, as depicted in
fig. 8.
We define the parameters ΩMAS = γpB′d and
ω0 = γp(B0 + B′x0). Notice that the latter corresponds
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] |
We define the parameters ΩMAS = γpB′d and
ω0 = γp(B0 + B′x0). Notice that the latter corresponds
16
to the Larmor frequency of a spin sitting on the rotation
axis at x0.
To elucidate the effect of the magic angle
spinning on our ability to perform the desired spin rota-
tion, we move to an interaction picture with respect to
H0 = (ℏ/2)ΩMAS sin(ωMASt)ˆσ(i)
z , resulting in the Hamil-
tonian
ˆHint =ℏ
2ω0ˆσz + ℏ
2ΩM cos(ωMt)
�
e−i ΩMAS
ωMAS cos(ωMASt)σ(i)
−
+ ei ΩMAS
ωMAS cos(ωMASt)σ(i)
+
�
.
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] |
2ω0ˆσz + ℏ
2ΩM cos(ωMt)
�
e−i ΩMAS
ωMAS cos(ωMASt)σ(i)
−
+ ei ΩMAS
ωMAS cos(ωMASt)σ(i)
+
�
.
(45)
We identify that the required regime to suppress the ef-
fect of magic angle spinning is given by ωMAS ≫ ΩMAS,
where we recover the Hamiltonian corresponding to a
static spin with energy splitting ω0 that is driven at fre-
quency ωM, that is ˆHint = ℏ/2(ω0ˆσz + ΩM cos(ωM)ˆσx).
Thus, in this regime, all spins in the body of the naph-
thalene nanoparticle behave as if they were sitting on | [
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] |
thalene nanoparticle behave as if they were sitting on
the rotation axis, with each spin taking the position on
the axis that is the closest to its actual position in the
crystal. Since d cannot be larger than the radius of the
particle, assuming a spherical particle of radius r, an up-
per bound on ΩMAS is given by ΩMAS,max = γpB′r. For
the parameters given in table II a), we find ΩMAS,max ⪅
(2π)1.2 MHz, which is several orders of magnitude below
the rotation frequencies on the order of hundreds of MHz | [
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] |
the rotation frequencies on the order of hundreds of MHz
considered in our protocol to attain efficient magic angle
spinning. We, thus, do not expect the rotation of the
particle to impose limitations on our ability to deliver
the required spin rotation pulses.
Nevertheless, the rotation axis extends in the direction
of the magnetic field gradient, and thus, not all spins on
the axis will have the same energy splitting. Notice, that
the rotation axis could be arranged such that it is con- | [
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] |
the rotation axis could be arranged such that it is con-
tained in the y − z plane therefore giving all the spin
on the axis the same x coordinate.
This would, how-
ever, not solve the problem, as any magnetic field ex-
hibiting a gradient along the x direction, would have to
exhibit also a gradient along the y direction, to satisfy
Maxwell equations. For a particle with a 70 nm radius
that sits in a magnetic field gradient of B′ = 5·104 T/m,
will experience a variation of the magnetic field from one | [
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] |
will experience a variation of the magnetic field from one
end of the axis to the other of ∆B = 7 · 10−4 T, as-
suming that the rotation axis is fully contained in the
x − z plane.
Thus, the energy splittings of the spins
sitting on the axis are contained in a frequency band
of width ∆ωparticle = (2π)30 kHz.
As radio frequency
pulses on hydrogen can achieve Rabi frequencies above
100 kHz≫ ∆ωparticle, composite pulse techniques (see e.g.
Ref. [89, 90]) offer a potential path to compensate for | [
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] |
Ref. [89, 90]) offer a potential path to compensate for
detunings and Rabi frequency errors that are up to ap-
proximately half of the Rabi frequency.
During the protocol, the delocalization of the center-of-
mass wavepacket will further broaden the resonance peak
of the spin ensemble. In the modified protocol, the pulses
are applied while the position state’s width is given by
eq. (22). When neglecting the position variance of the ini-
tial state, the width is given by ⟨∆x⟩ ≈ 2χ
√
N, where χ | [
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] |
tial state, the width is given by ⟨∆x⟩ ≈ 2χ
√
N, where χ
represents the spacing between the equilibrium positions
around which the different trajectories oscillate, and is
defined in eq. (14). The frequency band associated with
a given position width is ∆ωwp = γp⟨∆x⟩B′. As χ is
inversely proportional to B′, ∆ωwp is independent of the
magnetic field gradient, and only depends on system con-
stants and the size of the particle. In particular, one finds
∆ωwp = (2π)2r−3/2·10−4. For a particle with r = 45 nm, | [
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] |
∆ωwp = (2π)2r−3/2·10−4. For a particle with r = 45 nm,
the spectrum width is approximately ∆ωwp ≈ 20 MHz.
We would like to point out that composite pulses alone
will not suffice to compensate for such detunings. Other
potential approaches might involve the use of chirped
pulses, which can implement broadband rotations [91].
Modified schemes of the protocol that momentarily turn
off the magnetic-field gradient during the duration of the
pulses could also provide a viable means to efficiently | [
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] |
pulses could also provide a viable means to efficiently
apply the required spin rotations to the entire spin en-
semble. While this aspect is crucial to the protocol, the
detailed design of such pulses is considered beyond the
scope of this work and will be presented elsewhere.
IV.
POSSIBLE NOISE SOURCES
In this section, we analyze some of the most relevant
sources of noise acting on the interferometer without be-
ing exhaustive. A more complete analysis will be required
for concrete experimental setups. | [
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] |
ing exhaustive. A more complete analysis will be required
for concrete experimental setups.
A.
Coherence time of the nuclear spins
As already discussed, for the T1 time of nuclear spins in
naphthalene, with approximately 920 h at 25 K [34–38],
and the short timescales of the proposed interferometry
protocols (< 1 ms), even for the large number of nuclear
spins in the considered nanoparticles, N = 106 −108, the
probability that at least one of the spins in the ensem- | [
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] |
probability that at least one of the spins in the ensem-
ble flips during the protocol is small. However, as in the
proposed experiment with its rapid rotation, the T1 time
may turn out to be shortened somewhat and it is there-
fore pertinent to analyze the effect of stochastic spin flips
on the performance of the protocol.
When a spin flips, it changes the equilibrium position
around which each state of the spin ensemble is oscillat-
ing due to the presence of the magnetic field gradient. | [
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] |
ing due to the presence of the magnetic field gradient.
For a joint eigenstate of all the individual σ(i)
z
operators,
e.g. |↑↑↓↑ · · ·⟩, the equilibrium position will jump to the
right or the left depending on whether the spin involved
in the rotation undergoes a spin flip from |↓⟩ to |↑⟩ or the
opposite. Since the trajectories of the interferometer are
associated with Dicke states, and Dickes states are su-
perpositions of spin states with the same number of spin | [
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] |
perpositions of spin states with the same number of spin
up, we expect that all trajectories, except for κ = 0 and
κ = N, will split into an incoherent superposition of two
trajectories in the event of a spin flip. As a consequence,
trajectories do not recombine in a single point, as would
17
Flip
|κ⟩ |κ − 1⟩ |κ + 1⟩
FIG. 9. An example of the simplest case of the interferometric
protocol without π pulses involves the trajectories recombin-
ing at the center after a total time of Ttot = 2π/Ω. If a T1 | [
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] |
ing at the center after a total time of Ttot = 2π/Ω. If a T1
flip occurs during the protocol, for all states except κ = 0
and κ = N, the κth Dicke state transitions into an incoher-
ent superposition of |κ + 1⟩ and |κ − 1⟩. Consequently, the
trajectories split into an incoherent superposition of two tra-
jectories oscillating around the κ+1th and κ−1th equilibrium
points, respectively. Thus, the trajectories will recombine at
two different points. In phase space, the maximum distance | [
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] |
two different points. In phase space, the maximum distance
between these two points is given by |∆ζ| = 4χ/x0, as shown
in appendix E.
be the case in the noise-free scenario. Instead, two re-
combination points emerge, one to the left and one to
the right of the original recombination point, such that
half of the trajectories recombine on one of these points
and half on the other. In fig. 9, this process is depicted for
a six-spin case. In appendix E, an analytical derivation | [
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] |
a six-spin case. In appendix E, an analytical derivation
of this phenomenon is provided. For each additional spin
flip, the number of recombination points is doubled, such
that if two spin flips occur then trajectories recombine in
4 different points, and so on. The result of this process is
that at the end of the protocol spins and motion will not
end up in a product state. This contributes to the loss of
purity of the spin ensemble upon tracing the mechanical | [
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] |
purity of the spin ensemble upon tracing the mechanical
degrees of freedom, and is, therefore, not desirable.
While an exhaustive numerical simulation of this form
of noise is unattainable for any realistic number of spins,
it is useful to calculate the spatial spread of the recombi-
nation points to understand how much decoherence can
be expected from such stochastic spin flips.
If the re-
combination points are spread over a length scale that is
below the zero-point motion of the nanoparticle, then one | [
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] |
below the zero-point motion of the nanoparticle, then one
can expect the spin and motion to be almost separable,
and the induced decoherence to be small.
In general,
the distance between the recombination points will de-
pend on the parameters of the system and the instant at
which the spin-flip occurs. For the protocol described in
this work, the maximal distance between the two recom-
bination points in dimensionless position space is given
by
|∆ζ| = 4χ
x0
,
(46) | [
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] |
bination points in dimensionless position space is given
by
|∆ζ| = 4χ
x0
,
(46)
corresponding to the case where the flip happens while
the wave packet is maximally expanded. χ is the spacing
between the spin-dependent equilibrium points around
which the trajectories oscillate and is given in eq. (14).
For the parameters determined by eq. (37), χ/x0 will be
in the order of one.
Moreover, T1 flips could be even
more destructive if the flip happens in between the π/2 | [
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] |
Moreover, T1 flips could be even
more destructive if the flip happens in between the π/2
and 3π/2 pulses in the modified protocol, as the fanned-
out trajectories will not recombine in one point anymore.
Therefore even a single flip has to be avoided.
Assuming that the spin-flips occur independently from
each other, at a rate given by N/T1, and follow a Poisson
distribution, the probability that during an experimental
run of duration Ttot, at least one spin flips is given by
Pflip = 1 − exp
�
−Ttot
T1
· N | [
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] |
run of duration Ttot, at least one spin flips is given by
Pflip = 1 − exp
�
−Ttot
T1
· N
�
.
(47)
With the parameters in table II a), the run time of the
protocol to test the CSL model is given by Ttot = 4π/Ω =
2 ms. The total number of nuclear spins is N = 1.8 · 108.
We then find that the probability for at least one spin
to flip is 10 %. By increasing the magnetic field gradient,
the protocol can be conducted faster and this probability
reduced. For the parameters in table II b), Ttot is given | [
0.6138695478439331,
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0.5217064023017883,
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] |
reduced. For the parameters in table II b), Ttot is given
by 0.6 ms, and the total number of spins is 5 · 107. For
these values, the probability for one or more spin flips is
reduced to 0.8 %. These values might be further reduced
if the experiment is conducted with the bulk temperature
of the naphthalene nanoparticle below 25 K, as required
to avoid other forms of noise such as that coming from
black body radiation.
Regarding the T2 time of the nuclear spins, we refer the | [
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] |
black body radiation.
Regarding the T2 time of the nuclear spins, we refer the
reader to section II B, where we describe the use of magic
angle spinning. By rotating the particle sufficiently fast
around an axis tilted by the magic angle with respect to
the magnetic field, T2 times of hundreds of milliseconds or
even seconds can be achieved. For example, if the particle
is rotated at a frequency of 23 MHz, a T2 time of 1 s can
be expected. To check to what extent T2 processes can be | [
0.7656110525131226,
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] |
be expected. To check to what extent T2 processes can be
tolerated, we describe them using the Lindblad equation
ˆρ = − i
ℏ[ ˆH, ˆρ] + Γ
2
N
�
n=1
�
ˆσ(n)
z
ˆρˆσ(n)
z
− ˆρ
�
(48)
where Γ = T −1
2
. Since the Hamiltonian in eq. (13) com-
mutes with the Lindbladian, the two processes can be
described independently. Pure dephasing can also be de-
scribed using Kraus operators
ˆ
M0 = e− Γ
2 t1 ,
ˆ
M1 =
�
1 − eΓt 1 + ˆσz
2
,
ˆ
M2 =
�
1 − eΓt 1 − ˆσz
2
.
(49)
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] |
ˆ
M0 = e− Γ
2 t1 ,
ˆ
M1 =
�
1 − eΓt 1 + ˆσz
2
,
ˆ
M2 =
�
1 − eΓt 1 − ˆσz
2
.
(49)
18
Pure dephasing of N spins can therefore be written as
S[ˆρ, t] =
2
�
i1,...,iN=0
Mi1 ⊗...⊗MiN ˆρM †
i1 ⊗...⊗M †
iN . (50)
Bringing everything together, we find for the full-time
evolution under eq. (48)
ˆρ(t + ∆t) = Mdp[ˆρ, ∆t] = ˆU(∆t)S[ˆρ, ∆t] ˆU †(∆t)
(51)
where ˆU is given by exp
�
−(i/ℏ) ˆH ∆t
�
.
Nevertheless, the π/2 and 3π/2 pulses do not commute
with S[ˆρ, t]. Therefore, the full-time evolution of the pro- | [
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-0.008499592542648315,
-0.08105023950338364,
-0.22268523275852203,
0.25363093614578247,
0.13178014755249023,
-0.6410510540008545,
-0.11023561656475067,
0.7654633522033691,
0.15617668628692627,
-0.7321198582649231,
0.1508578658103943,
0.1667482554912567,
-0.5282130837440491,
0.5929829478263855,
0.281838595867157,
0.10029518604278564,
-0.03236209973692894,
-0.17804986238479614,
0.5234435200691223,
-0.24705877900123596,
-0.13947056233882904,
-0.1289352923631668,
0.4897836446762085,
0.11198169738054276,
-0.4535663425922394,
-0.10146132111549377,
-0.10884087532758713,
-0.5351589322090149,
0.44297757744789124,
0.6627059578895569,
0.21202416718006134,
0.4421360492706299,
0.010318191722035408,
0.4862646758556366,
0.28059104084968567,
0.007730628829449415,
-0.2877216041088104,
0.3930455446243286,
-0.3559805750846863,
0.6468249559402466,
0.024187222123146057,
0.1924004703760147,
-0.8039793968200684,
-0.15805234014987946,
0.1017778217792511,
-0.5354958176612854,
-0.009492812678217888,
0.26519161462783813,
-0.038605958223342896,
0.1460302174091339,
0.01664568856358528,
-0.694977343082428,
0.4040088653564453,
0.7924823760986328,
-0.297541081905365,
-0.28536462783813477,
0.2256554812192917,
0.4469294548034668,
-0.06508717685937881,
-0.085227370262146,
0.3680446147918701,
-0.225803405046463,
-0.5237128734588623,
-0.620800256729126,
-0.33127281069755554,
-0.08327905088663101,
0.5697113275527954,
-0.07752341032028198,
0.2255798727273941,
0.29653751850128174,
0.2330331653356552,
-0.6652082204818726,
-0.23715436458587646,
0.1435781568288803,
-0.4198477864265442,
0.04316602274775505,
0.31113627552986145,
-0.0898120179772377,
-0.19110751152038574,
0.6637893915176392,
0.16250012814998627,
0.009573319926857948,
-0.11201569437980652,
-0.28247955441474915,
0.2264719307422638,
0.46017467975616455,
-0.18612082302570343,
-0.13878598809242249,
0.33360785245895386,
0.30366024374961853,
-0.2649674415588379,
0.6780958771705627,
-0.2827529013156891,
0.06457804143428802,
0.33464208245277405,
-0.05258525535464287,
-0.09470906108617783,
0.2738981246948242,
-0.4061656892299652,
0.30246666073799133,
0.5857598781585693,
-0.9817172884941101,
0.3966153860092163,
0.09471999853849411,
-0.0218118354678154,
-0.4491068124771118,
-0.16029813885688782,
0.3178546726703644,
0.29214465618133545,
-0.018002305179834366,
-0.40597981214523315,
-0.03299729898571968,
-0.48614922165870667,
0.2926120162010193,
-0.11742811650037766,
-0.25303179025650024,
-0.33016324043273926,
-0.18789824843406677,
-0.43845489621162415,
-0.026054637506604195,
-0.12608443200588226,
-0.314596951007843,
1.0023142099380493,
1.0120242834091187,
-0.4691374599933624,
-0.3748471140861511,
-0.39194026589393616,
-0.386187881231308,
-0.28730499744415283,
0.612902045249939,
-0.08363877981901169,
0.45234864950180054,
-0.1364573836326599,
0.15330974757671356,
-0.0004820618196390569,
0.24309612810611725,
1.6967698335647583,
-0.11851667612791061,
-0.02498582936823368,
-0.015034358017146587,
0.7514023780822754,
-0.5850829482078552,
0.6613934636116028,
0.3807125687599182,
-0.1875462681055069,
-0.18701809644699097,
0.11711771041154861,
0.09050076454877853,
-0.11444278061389923,
0.04602032154798508,
-0.47756820917129517,
-0.2230747640132904,
-0.018412720412015915,
0.04672849550843239,
0.37359240651130676,
0.1367400884628296,
-0.4396566152572632,
-0.2868867516517639,
-0.5052727460861206,
0.32695266604423523,
-0.7659665942192078,
0.2156437188386917,
-0.19615189731121063,
-0.0981874018907547,
0.2088620364665985,
0.09932834655046463,
0.02676267921924591,
0.315887451171875,
0.3548767566680908,
0.22571834921836853,
-0.07498445361852646,
-0.23311293125152588,
-0.01893852837383747,
-0.2911435663700104,
-0.33151379227638245,
-0.5800544023513794,
0.22611340880393982,
-0.5043169856071472,
0.14374691247940063,
-0.030271926894783974,
-0.15256339311599731,
0.14645802974700928,
-0.43789348006248474,
-0.04653014987707138,
0.16603299975395203,
0.38715267181396484,
-0.49786612391471863,
-0.18576297163963318,
0.6485833525657654,
0.43019041419029236,
-0.08582229167222977,
-0.05391398072242737,
-0.4967317581176758,
0.6523565649986267,
0.031111285090446472,
0.07372156530618668,
-0.6258260011672974,
0.07558787614107132,
0.5984199643135071,
0.05461699515581131,
-0.04316168278455734,
-0.6355790495872498,
-0.39980214834213257,
-0.3873852491378784,
0.024343330413103104,
-0.24619656801223755,
-0.3694743514060974,
0.6360547542572021,
0.1126970499753952,
0.0010278940899297595,
-0.09442951530218124,
0.10632852464914322,
0.7915685772895813,
-0.5015511512756348,
0.42245304584503174,
-0.4639967083930969,
0.3785083293914795,
-0.2793946862220764,
-0.014607277698814869,
-0.28584763407707214,
-0.42043107748031616,
0.2405216246843338,
-0.07096241414546967,
0.2424757331609726,
-0.11167993396520615,
0.00953588541597128,
-0.1632898598909378,
-0.08014532923698425,
0.9037954211235046,
-0.14223098754882812,
-0.11992581188678741,
-1.1406323909759521,
-0.6988750696182251,
-0.07209818065166473,
0.7419552206993103,
0.3624197840690613,
0.26778918504714966,
-0.565710723400116,
-0.2281615436077118,
-0.26572322845458984,
-0.22350944578647614,
0.3875565528869629,
-0.02154291793704033,
-0.6021589636802673,
0.16492971777915955,
0.4887062907218933,
-0.2573678493499756,
-0.6295685768127441,
-0.19522035121917725,
0.48023858666419983,
0.043352801352739334,
-0.11232128739356995,
0.28940248489379883,
-0.39606383442878723,
-0.23232117295265198,
0.4503631293773651,
-0.12561164796352386,
-0.35183462500572205,
0.33385294675827026,
0.0780438482761383,
0.7789688110351562,
0.3701260983943939,
-0.502838671207428,
-0.612507700920105,
0.46308979392051697,
-0.030377021059393883,
-0.19180068373680115,
-0.31974080204963684,
0.28379154205322266,
0.39415857195854187,
0.08570890873670578,
-0.5191026926040649,
0.47962284088134766,
0.38111960887908936,
-0.16897839307785034,
0.08186949044466019,
-0.06737104058265686,
0.21855053305625916,
0.17542977631092072,
0.7311527132987976,
1.087404489517212,
-0.13357609510421753,
-0.16431058943271637,
0.17662522196769714,
-0.01966417022049427,
0.322229266166687,
-0.42846667766571045,
-0.347373902797699,
0.11860261112451553,
0.22188648581504822,
-0.4829600155353546,
-0.5435145497322083,
-0.3630949854850769,
-0.4941410720348358,
-0.020478559657931328,
0.37645813822746277,
-0.605827808380127,
-0.18352243304252625,
0.4756665527820587,
0.234008327126503,
0.29690247774124146,
-0.14742407202720642,
-0.2680390179157257,
-0.4070967733860016,
-0.19470764696598053,
0.5785186290740967,
0.16070368885993958,
-0.021248195320367813,
-0.16543154418468475,
0.23174206912517548,
0.35058125853538513,
0.1489814668893814,
0.24806420505046844,
0.4730006158351898,
-0.34937816858291626,
0.10598618537187576,
0.025165773928165436,
-0.4717239439487457,
0.13607744872570038,
-0.1888059675693512,
0.1696157157421112,
-0.48782822489738464,
-0.5890302062034607,
-0.18190915882587433,
-0.14601942896842957,
0.19403313100337982,
-0.2251720279455185,
-0.40523427724838257,
0.5149919986724854,
0.7120198011398315,
-0.846912682056427,
-0.2382219135761261,
0.12568674981594086,
0.2889672517776489,
-0.02308633364737034,
-0.25742194056510925,
0.9045810103416443,
0.7282450199127197,
0.44677990674972534,
0.6647772192955017,
0.13574664294719696,
0.40558922290802,
0.9264290928840637,
-0.4869292378425598,
0.18936601281166077,
-0.644154965877533,
0.11678357422351837,
-0.4059310555458069,
-0.10749869793653488,
0.7521300911903381,
0.10642063617706299,
-0.11855174601078033,
0.11862034350633621,
0.0016448107780888677,
0.18588487803936005,
0.1474062204360962,
-0.3807087242603302,
-0.4594138562679291,
-0.24302105605602264,
-0.6540755033493042,
-0.8603841066360474,
0.1177411675453186,
0.5574927926063538,
0.16719993948936462,
-0.6267185807228088,
0.024989183992147446,
0.1781306266784668,
-0.17310678958892822,
0.42209622263908386,
-0.06946690380573273,
-0.21830180287361145,
0.047163985669612885,
0.7440404891967773,
-0.5281400680541992,
-0.2362712323665619,
0.1743043065071106,
0.47528213262557983,
-0.07245761901140213,
-0.4913650155067444,
-0.48380887508392334,
0.008816108107566833,
-0.17061764001846313,
0.5401788949966431,
0.3673768937587738,
0.12572088837623596,
-0.1890045702457428,
-0.25475266575813293,
-0.1603088080883026,
-0.13976150751113892,
0.009336451068520546,
0.0026951709296554327,
-0.35781610012054443,
0.38064154982566833,
0.16843198239803314,
-0.061441365629434586,
-0.17170703411102295,
-0.594298779964447,
0.022588742896914482,
-0.03969627246260643
] |
with S[ˆρ, t]. Therefore, the full-time evolution of the pro-
tocol is given by
ˆρ
� π
Ω
�
= Mdp
�
ˆρ0, π
Ω
�
,
ˆρ
�3π
Ω
�
= Mdp
�
U ⊗N
π/2 ˆρ
� π
Ω
�
U †⊗N
π/2 , 2π
Ω
�
,
ˆρ
�4π
Ω = Ttot
�
= Mdp
�
U ⊗N
3π/2ˆρ
�3π
Ω
�
U †⊗N
3π/2 , π
Ω
�
,
(52)
where ˆρ0 is as in eq. (15). The average magnetization
is then given by ⟨ ˆ
M(Ttot)⟩ = tr{ ˆ
MU ⊗N
π/2 ˆρ(Ttot)U †⊗N
π/2 }
While we did not find an analytical expression for
⟨ ˆ
M(Ttot)⟩, we evaluated it explicitly for N ∈ [2, 20]. In
all cases, we found
⟨ ˆ
M(Ttot)⟩ | [
-0.2589249014854431,
-0.6621953845024109,
0.8453370928764343,
-0.31764522194862366,
-0.20760558545589447,
0.24428293108940125,
-0.3056623935699463,
-0.47907790541648865,
-0.2505910098552704,
-0.2975872755050659,
0.04946689307689667,
0.5025426745414734,
-0.29979419708251953,
-0.11968105286359787,
0.2690819203853607,
0.8787722587585449,
0.6993861794471741,
-0.01407951582223177,
-0.8529972434043884,
-0.7474624514579773,
0.3381599485874176,
-0.42409709095954895,
0.8686242699623108,
0.5266059637069702,
0.35541635751724243,
-0.45880424976348877,
0.5577325224876404,
0.0969751849770546,
0.2913954555988312,
0.4014325439929962,
-0.1564498394727707,
0.529668390750885,
-0.01957627572119236,
0.07505521178245544,
0.31742915511131287,
-0.06987035274505615,
0.6040804982185364,
0.07765975594520569,
0.16407757997512817,
-0.051934100687503815,
0.46891841292381287,
0.07548465579748154,
-0.16637416183948517,
0.7386708855628967,
-0.064883753657341,
-0.2761854827404022,
-1.411362648010254,
0.35318830609321594,
-0.237162247300148,
-0.39672037959098816,
-0.07919550687074661,
-0.038949932903051376,
0.09764561802148819,
0.09372596442699432,
-0.08301851898431778,
0.5200822949409485,
-0.7010678648948669,
-0.00818302296102047,
0.6113762855529785,
-0.21316811442375183,
0.2845316231250763,
0.43158161640167236,
-0.047143254429101944,
0.5210981965065002,
-0.138994961977005,
0.6429944634437561,
-0.11424494534730911,
0.40454378724098206,
-0.3339521288871765,
-0.09717883914709091,
0.22336305677890778,
0.33931732177734375,
-0.057585492730140686,
0.2294796258211136,
-0.1345532089471817,
0.20957590639591217,
-0.5197163820266724,
-0.2642327845096588,
0.35179653763771057,
-0.0817168653011322,
-0.1559731811285019,
-0.006143629550933838,
0.4121745824813843,
-0.20732204616069794,
0.46612560749053955,
0.44858062267303467,
-0.19570551812648773,
0.5861350297927856,
-0.19834548234939575,
0.2855033874511719,
0.18997375667095184,
-0.24534089863300323,
-0.0712977796792984,
0.2491249144077301,
0.5388668179512024,
-0.15271782875061035,
-0.5056570172309875,
-0.4997774660587311,
0.09938608855009079,
0.039735786616802216,
-0.5390809774398804,
0.06166331097483635,
-0.4215087890625,
-0.16135449707508087,
0.10402659326791763,
-0.6617349982261658,
-0.4217069149017334,
-0.509448230266571,
-0.24131642282009125,
0.07020644098520279,
-1.1789608001708984,
-0.051277898252010345,
-0.14963793754577637,
-1.0202510356903076,
-0.003991574514657259,
-0.22421234846115112,
0.10770314931869507,
-0.1649901568889618,
0.06276106834411621,
0.5735304355621338,
-0.4529443085193634,
0.5420191884040833,
-0.38493606448173523,
0.1513601541519165,
-0.0409100167453289,
-0.08085016161203384,
0.30581626296043396,
0.6810702681541443,
-0.7701843976974487,
-0.030811581760644913,
0.18648043274879456,
0.7364690899848938,
0.2778990864753723,
0.3366105556488037,
-0.03523517772555351,
-1.2522035837173462,
-0.11550354957580566,
-0.06575991213321686,
-0.4799734652042389,
-0.21759314835071564,
-0.2880817949771881,
-0.19122862815856934,
-0.1579599827528,
-0.2557927966117859,
0.23691140115261078,
-0.4689633846282959,
-0.21705341339111328,
0.016271285712718964,
-0.2929708659648895,
-0.002626174595206976,
-0.0938468649983406,
0.12687163054943085,
-0.44582459330558777,
0.1625654697418213,
-0.7254945039749146,
-0.14464852213859558,
0.40005746483802795,
0.3102060854434967,
-0.6208171248435974,
-0.19356496632099152,
0.2678317725658417,
-0.19788973033428192,
0.10769584774971008,
-0.4483151137828827,
0.4063642919063568,
-0.4585508108139038,
0.7841198444366455,
0.01924295164644718,
0.11199046671390533,
0.29600173234939575,
-0.29357966780662537,
-0.07417523860931396,
-0.11960061639547348,
0.13384273648262024,
-0.3090345561504364,
0.05474140867590904,
-0.2179102599620819,
-0.031476058065891266,
0.15748436748981476,
-0.15121135115623474,
-0.7182620763778687,
0.004157021176069975,
-0.08487236499786377,
0.09430999308824539,
0.5706533789634705,
0.411834180355072,
0.4666922688484192,
-0.06399545818567276,
0.44970569014549255,
0.15390247106552124,
0.20474529266357422,
-0.14848282933235168,
0.07175633311271667,
-0.47464457154273987,
1.6622495651245117,
0.22252722084522247,
-0.8940290808677673,
0.25410130620002747,
0.2513275742530823,
0.427324503660202,
-0.5751466751098633,
-0.09870892018079758,
0.2609541118144989,
-0.18392734229564667,
-0.18807540833950043,
-0.04330891743302345,
0.06731896102428436,
-0.03935473412275314,
0.13638819754123688,
0.1147066056728363,
0.5602230429649353,
-0.5130151510238647,
0.11615215986967087,
-0.44267791509628296,
0.0729639083147049,
-0.35722723603248596,
-0.07419873774051666,
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0.02933073416352272,
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] |
⟨ ˆ
M(Ttot)⟩, we evaluated it explicitly for N ∈ [2, 20]. In
all cases, we found
⟨ ˆ
M(Ttot)⟩
⟨ ˆ
M(0)⟩
= 1
2
�
e−4ΓTtot − e
−4ΓTtot−64( 1
2 +¯n)
�
χ
x0
�2
+ e−2ΓTtot + e
−2ΓTtot−64( 1
2 +¯n)
�
χ
x0
�2�
,
(53)
where ¯n is the mean occupation number of the initial
thermal position state.
For our choice of parameters,
χ/x0 is on the order of one. Also, in general, ¯n can be
very large, so we assume (1/2 + ¯n)(χ/x0)2 ≫ 1. Under
this assumption, ⟨ ˆ
M(Ttot)⟩/⟨ ˆ
M(0)⟩ becomes
⟨ ˆ
M(Ttot)⟩
⟨ ˆ
M(0)⟩
= 1
2
� | [
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0.4505951404571533,
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0.07907043397426605,
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0.2563016712665558,
0.056283287703990936,
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0.4706282317638397
] |
this assumption, ⟨ ˆ
M(Ttot)⟩/⟨ ˆ
M(0)⟩ becomes
⟨ ˆ
M(Ttot)⟩
⟨ ˆ
M(0)⟩
= 1
2
�
e−4ΓTtot + e−2ΓTtot�
(54)
which
also
constitutes
a
lower
bound
on
⟨ ˆ
M(Ttot)⟩/⟨ ˆ
M(0)⟩. For Ttot = 1 ms and Γ−1 = T2 = 1 s,
we find ⟨ ˆ
M(Ttot)⟩/⟨ ˆ
M(0)⟩ ≈ 0.997.
B.
Collisions with air molecules
Since no experiment can be conducted in a perfect vac-
uum, there will always be some gas molecules that might
collide with the nanoparticle. For a spherical particle in
an ideal gas containing gas particles of mass mg, temper- | [
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] |
an ideal gas containing gas particles of mass mg, temper-
ature Tg, and pressure P, the collision frequency is given
by
γg = 8πPr2
mgvg
,
(55)
where vg =
�
8kBTg/(πmg) is the mean velocity of the
gas molecules.
We assume cryogenic temperatures of
Tg = 8 K, low pressure of P = 5·10−10 Pa, and a gas com-
position that is mainly dominated by H2 molecules with
mg = 2 a.m.u. Notice that gas pressures as low as 10−16
Pa have been demonstrated [92]. We are interested in the | [
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] |
Pa have been demonstrated [92]. We are interested in the
probability of finding no collision, as, in general, a single
collision would be enough to destroy the visibility of the
interferometer. For this set of parameters, a particle of
radius r = 100 nm and an experimental run of duration
Ttot = 2 ms, the probability of having no collisions with
air molecules is exp(−γg Ttot/3) ≈ 0.9. Therefore, the
conditions to guarantee no collisions with air molecules,
while demanding, are, in principle, achievable.
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] |
while demanding, are, in principle, achievable.
C.
Black body radiation
Another possible noise source is induced by thermal
photons that are either scattered or emitted by the
nanoparticle.
Just as the noise induced by the CSL
model, the noise due to black body radiation is well de-
scribed by a master equation of the form [93]
˙ρ(q′, q′′) = − i
ℏ ⟨q′| [ ˆH, ˆρ] |q′′⟩−ξbb
�
1 − e
− (q′−q′′)2
4r2
bb
ρ(q′, q′′)
�
.
(56)
Here, rbb is called the localization distance, and ξbb rep- | [
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] |
− (q′−q′′)2
4r2
bb
ρ(q′, q′′)
�
.
(56)
Here, rbb is called the localization distance, and ξbb rep-
resents the localization strength. Both parameters de-
pend on the specific decoherence process. The localiza-
tion distance of thermal photons is related to the thermal
wavelength by rbb = λth/2[94]. Therefore, for thermal
photons, we find rbb = π2/3ℏc(2kBTb/e), where Tb/e is
the temperature of the bulk and environment, respec-
tively. For scattered photons, the localization strength is
given by [93, 94] | [
0.3670288026332855,
-0.01663895696401596,
0.353075236082077,
-0.4142170548439026,
-0.4116359055042267,
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0.287970632314682,
-0.7544063329696655,
0.07912745326757431,
-0.11223778873682022,
-0.16384314000606537,
0.13152441382408142,
-0.2182895690202713,
0.46708181500434875,
0.27610376477241516,
1.0330430269241333,
0.8911635279655457,
0.13102667033672333,
-0.5888270735740662,
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0.6211268901824951,
0.28671523928642273,
0.5303826928138733,
0.40934839844703674,
0.11340388655662537,
0.3469497859477997,
0.6157460808753967,
0.0070381928235292435,
0.11593212932348251,
0.004626225680112839,
-0.5696352124214172,
0.3691682517528534,
0.14771205186843872,
0.3323756456375122,
0.8666933178901672,
0.2389170229434967,
0.2783651649951935,
-0.03967781737446785,
0.16190671920776367,
-0.07178932428359985,
0.06910381466150284,
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0.42077264189720154,
0.3899490237236023,
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0.3599630296230316,
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0.11400621384382248,
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0.20570310950279236,
0.47591468691825867,
-0.3865905702114105,
0.26072627305984497,
-0.6778504252433777,
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] |
tively. For scattered photons, the localization strength is
given by [93, 94]
ξbb,sc
4r2
bb
= 8! 8ζ(9)c r6
9π
�kBTe
ℏc
�9
Re
�ϵ − 1
ϵ + 2
�2
.
(57)
Here, ζ(x) is the Riemann function.
For the absorp-
tion and emission of thermal photons, the localization
strength is given by
ξbb,em/ab
4r2
bb
= 16π5c r3
189
�kBTb/e
ℏc
�6
Im
�ϵ − 1
ϵ + 2
�
.
(58)
Here, ϵ is the dielectric constant of naphthalene in the
spectrum range of the thermal photons. Due to the lack | [
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] |
spectrum range of the thermal photons. Due to the lack
of available data, we assumed a conservative value of
ϵ = 1.1 + i0.15. For rbb ≫ (q′ − q′′), the noise term ap-
proaches ξbb,sc(q′−q′′)2/(4r2
bb), while for rbb ≪ (q′−q′′),
it saturates to ξbb,sc. In any case, both limits are upper
bounds for all q′, q′′. At 4 K, we find rbb ≈ 0.5 mm which,
in general, is neither significantly larger nor smaller than
the delocalization of the particle in our interferometric | [
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] |
the delocalization of the particle in our interferometric
protocol. Nonetheless, we will consider ξbb,sc as an up-
per bound for this analysis. To ensure that the noise in-
duced by possible spontaneous collapses is greater than
19
the noise induced by interactions with thermal photons,
one must ensure ξbb = ξbb,sc + ξbb,em + ξbb,ab < ξ, where
ξ is the localization strength of the CSL model as in
eq. (26).
For a particle of r = 100 nm and at cryo-
genic temperatures, the noise due to emission and absorp- | [
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] |
For a particle of r = 100 nm and at cryo-
genic temperatures, the noise due to emission and absorp-
tion dominates over the noise induced by the scattering
of thermal photons, as one can easily convince oneself.
Thus, it is justified to only ensure that ξbb,em + ξbb,ab =
2ξbb,em/ab < ξ. In table II, the maximum internal or bulk
temperature that can be tolerated to ensure ξbb < ξ for
different parameters is shown.
Note, that ξbb already
represents an upper limit for the noise induced by ther- | [
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] |
Note, that ξbb already
represents an upper limit for the noise induced by ther-
mal photons; hence, it may be possible to tolerate slightly
higher temperatures.
V.
CONCLUSION
The significant polarization capabilities of naphtha-
lene, coupled with its levitation potential, can open new
avenues in matter-wave interferometry and NMR ap-
plications. In particular, our investigation underscores
the promise of pentacene-doped naphthalene in over-
coming several limitations inherent in Stern-Gerlach-type | [
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] |
coming several limitations inherent in Stern-Gerlach-type
matter-wave interferometers relying on materials hosting
color centers.
On the one hand, the transient nature
of the photo-excited triplet state used for polarization
ensures the disappearance of the associated decoherence
channels post-polarization, in contrast to the constant
presence of NV centers in diamonds. On the other hand,
the remarkably prolonged polarization lifetime in naph-
thalene suggests minimal magnetic impurities, removing | [
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] |
thalene suggests minimal magnetic impurities, removing
a major source of decoherence present in diamond-based
interferometers. Moreover, the homogeneous distribution
of nuclear spins, and the absence of a preferential axis
of magnetization, mitigates the emergence of undesirable
torques, which are challenging to compensate for in other
types of interferometers and typically lead to a loss of in-
terferometric feasibility. Building upon these advantages, | [
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] |
terferometric feasibility. Building upon these advantages,
we have developed a multi-spin matter-wave interferome-
try protocol and assessed its capacity to establish bounds
for the free parameters of collapse models. Importantly,
our noise analysis indicates that our interferometer can
improve upon existing bounds using state-of-the-art or
near-term technology.
Besides the applications in matter-wave interferome-
try, we believe that levitated naphthalene may be used | [
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] |
try, we believe that levitated naphthalene may be used
to push the boundaries of NMR applications, by im-
plementing magic angle spinning at unparalleled rota-
tional frequencies, and thus reach unprecedented nuclear
spin coherence times.
Moreover, our proposal for de-
tecting spin ensemble polarization with exceptional ac-
curacy through position displacement measurements fur-
ther augments the catalog of potential NMR applica-
tions made possible by levitation.
It is pertinent to | [
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] |
tions made possible by levitation.
It is pertinent to
note that pentacene-doped naphthalene serves as just
one member of a broader family of optically polariz-
able materials, to which much of our analysis is trans-
ferable.
Examples would be p-dibromobenzene doped
with p-dichlorobenzene [95] or p-terphenyl doped with
pentacene [96].
Finally, it is noteworthy to mention that our analysis
relies on properties of naphthalene, such as its refractive
index or heat capacity, that were measured in bulk sam- | [
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] |
index or heat capacity, that were measured in bulk sam-
ples and in restricted ranges of temperature and pressure
conditions. Thus, their behavior outside these regimes,
such as in nanoscale particles, remains incompletely un-
derstood. While this naturally limits the extent of our
conclusions, it also opens an unprecedented opportunity
to explore these material properties in a levitated ar-
rangement, going beyond the already charted regimes.
Taken together, these factors make a compelling case | [
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] |
Taken together, these factors make a compelling case
for the exploration of naphthalene in the context of levi-
tation and promise exciting advancements for the quan-
tum applications of levitated optomechanical platforms.
VI.
ACKNOWLEDGMENTS
We thank Benjamin Stickler for insightful discussions
about levitated systems.
Furthermore, we would like
to thank Tim R. Eichhorn, Markus Schw¨orer and Jens
Pflaum for fruitful discussions about naphthalene. This
work was supported by the ERC Synergy grant Hy- | [
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] |
work was supported by the ERC Synergy grant Hy-
perQ (Grant No. 856432) and the QuantERA Project
LEMAQUME (Grant No. 500314265).
20
APPENDIX
A.
Decoherence between κth and κ′thave packet
In this section, we want to find an expression for
tr{ˆρκ,κ′(Ttot)} in the presence of the CSL model. Using
eq. (26), together with eq. (18), the master equation gov-
erning the time evolution of ρκ,κ′(q′, q′′) = ⟨q′| ˆρκ,κ′ |q′′⟩
is given by
⟨q′| ˙ˆρκ,κ′ |q′′⟩ = − i
ℏ ⟨q′| ˆHκˆρκ,κ′ − ˆρκ,κ′ ˆHκ′
+ ακ,κ′ ˆρκ,κ′ |q′′⟩
− ξ | [
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] |
is given by
⟨q′| ˙ˆρκ,κ′ |q′′⟩ = − i
ℏ ⟨q′| ˆHκˆρκ,κ′ − ˆρκ,κ′ ˆHκ′
+ ακ,κ′ ˆρκ,κ′ |q′′⟩
− ξ
�
1 − e
− (q′−q′′)2
4r2
CSL
�
ρκ,κ′(q′, q′′). (59)
It can be easily shown that
e
− (q′−q′′)2
4r2
CSL ρ(q′, q′′) =
1
√πrCSL
⟨q′|
� ∞
−∞
dx e
− (ˆq−x)2
2r2
CSL ˆρe
− (ˆq−x)2
2r2
CSL |q′′⟩ .
(60)
We define
T[ˆρ] ≡
1
√πrCSL
� ∞
−∞
dx e
− (ˆq−x)2
2r2
CSL ˆρe
− (ˆq−x)2
2r2
CSL |q′′⟩
(61)
and write the master equation as
⟨q′| ˙ˆρκ,κ′ |q′′⟩ = − i
ℏ ⟨q′| ˆHκˆρκ,κ′ − ˆρκ,κ′ ˆHκ′ + ακ,κ′ ˆρκ,κ′
− ξ (ˆρ − T[ˆρ]) |q′′⟩ .
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ℏ ⟨q′| ˆHκˆρκ,κ′ − ˆρκ,κ′ ˆHκ′ + ακ,κ′ ˆρκ,κ′
− ξ (ˆρ − T[ˆρ]) |q′′⟩ .
(62)
First, we go into the rotating picture to eliminate the
unitary time evolution by defining
ˆρS
κ,κ′ = ˆU †
κ(t)ˆρκ,κ′ ˆUκ′(t)e
i
ℏ ακ,κ′ .
(63)
With
ˆUκ(t) = e− i
ℏ ˆ
Hkt.
(64)
The equation governing the evolution of ˆρS
κ,κ′ is then
given by
˙ˆρS
κ,κ′ = −ξ
�
ˆρS
κ,κ′ − T S[ˆρS
κ,κ′]
�
with
T S[ˆρS
κ,κ′] = ˆU †
κ(t)T[ˆρκ,κ′] ˆUκ′(t).
T S[ˆρS
κ,κ′] can be further simplified by using the unitary
nature of ˆUκ(t). With ˆqκ ≡ ˆU † | [
0.04626281559467316,
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0.017490204423666,
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0.3100008964538574,
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0.3713745176792145,
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0.17097489535808563,
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0.2123454064130783,
0.3243076801300049,
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0.18721024692058563,
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] |
T S[ˆρS
κ,κ′] can be further simplified by using the unitary
nature of ˆUκ(t). With ˆqκ ≡ ˆU †
κ(t)ˆq ˆUκ(t) we find
T S[ˆρS
κ,κ′] =
1
√πrCSL
×
� ∞
−∞
dx e
− (ˆqκ−x)2
2r2
CSL ˆρS
κ,κ′e
−
(ˆqκ′ −x)2
2r2
CSL .
(65)
To evaluate ˆqκ, one first has to find an expression for
the time evolution of the κth wave packet. We realize that
we can write ˆHκ in terms of the ladder operators ˆa, ˆa†
defined by ˆq = x0(ˆa† + ˆa) and ˆp = ip0(ˆa† − ˆa). x0 and p0
are given by x0 =
�
ℏ/(2mΩ) and p0 =
�
ℏmΩ/2. By | [
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] |
are given by x0 =
�
ℏ/(2mΩ) and p0 =
�
ℏmΩ/2. By
using the properties of the displacement operator ˆD(ζ) =
exp
�
ζˆa† − ζ∗ˆa
�
, we can write
e− i
ℏ ˆ
Hκt = eiΩtˆa†ˆa ˆD
� χ
2x0
(2κ − N)
�
e−iΩtˆa†ˆa.
(66)
Each π pulse flips all spins, therefore κ goes to N − κ.
We can use the properties of the displacement operator
to find the full-time evolution along the κth trajectory,
which can be written as
ˆUκ(t) = e
−i
�
χ
2x0
�2ϕκ(t) ˆD
� χ
2x0
ζκ(t)
�
e−iΩtˆa†ˆa.
(67) | [
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] |
which can be written as
ˆUκ(t) = e
−i
�
χ
2x0
�2ϕκ(t) ˆD
� χ
2x0
ζκ(t)
�
e−iΩtˆa†ˆa.
(67)
ζκ(t) represents the classical phase space trajectory of
the κth wave packet. The position of the center of this
wave packet is given by Xκ(t) = χ Re[ζκ(t)].
ϕκ(t) =
ϕ′
κ+(2κ−N)2αTtot/ℏ as well as ζκ(t) are given in eq. (75)
and eq. (76).
By using the properties of the displacement operator
and the Baker–Campbell–Hausdorff formula, ˆqκ can be
written as
ˆU †
κ(t)ˆq ˆUκ(t) = cos(Ωt)ˆq + sin(Ωt)ˆpx0
p0
+ Xκ(t).
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] |
written as
ˆU †
κ(t)ˆq ˆUκ(t) = cos(Ωt)ˆq + sin(Ωt)ˆpx0
p0
+ Xκ(t).
(68)
As we are interested in tr(ˆρκ,κ′), we use the linearity
and the cyclic properties of the trace and [ˆqκ, ˆqκ′] = 0 to
find
tr
�
T S[ˆρS]
�
=
1
√πrCSL
tr
�� ∞
−∞
dx e
(ˆqκ−x)2−(ˆqκ′ −x)2
2r2
CSL
ˆρS
�
= tr
�
e
−
(ˆqκ−ˆqκ′ )2
4r2
CSL
ˆρS
�
.
(69)
When plugging eq. (68) into eq. (69), all terms involving
ˆq and ˆp cancel out and one finds
∂ tr
�
ˆρS
κ,κ′
�
∂t
= −ξ
�
1 − e
−
(Xκ(t)−Xκ′ (t))2
4r2
CSL
�
tr
�
ˆρS
κ,κ′
�
. (70) | [
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0.15076807141304016,
-0.42179593443870544,
-0.809640645980835,
-0.6313796639442444,
0.322509765625,
0.2752819061279297,
0.2763829827308655,
-0.32329079508781433,
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0.07297156006097794,
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] |
∂ tr
�
ˆρS
κ,κ′
�
∂t
= −ξ
�
1 − e
−
(Xκ(t)−Xκ′ (t))2
4r2
CSL
�
tr
�
ˆρS
κ,κ′
�
. (70)
This differential equation can be easily solved by
tr
�
ˆρS
κ,κ′(Ttot)
�
=e−Λκ,κ′ tr
�
ˆρS
κ,κ′(0)
�
(71)
with
Λκ,κ′ = ξ
� Ttot
0
(1 − e
−
[Xκ(t′)−Xκ′ (t′)]2
4r2
CSL
) dt′.
(72)
The protocol is designed such that the κth and the
κ′ th wave packets overlap, meaning that ζκ(Ttot) =
ζκ′(Ttot).
Hence,
ˆUκ(Ttot) and
ˆUκ′(Ttot) are the
same up to a phase.
Therefore, tr
�
ˆρS
κ,κ′(Ttot)
�
=
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] |
Hence,
ˆUκ(Ttot) and
ˆUκ′(Ttot) are the
same up to a phase.
Therefore, tr
�
ˆρS
κ,κ′(Ttot)
�
=
21
tr{ˆρκ,κ′(Ttot)} exp(−i∆ϕκ,κ′(Ttot)) holds true, where we
defined ∆ϕκ,κ′ ≡ ϕκ(Ttot) − ϕκ′(Ttot).
At t = 0, the
wave packets did not split up yet. Therefore, we know
tr{ˆρκ,κ′(0)} = tr{ˆρ(0)} = 1. tr{ˆρκ,κ′(Ttot)} is then given
by
tr{ˆρκ,κ′} =e−i∆ϕκ,κ′ e−Λκ,κ′ .
(73)
If (Xκ(t′) − Xκ′(t′)) ≪ rCSL, Λκ,κ′ simplifies to
Λκ,κ′ ≈ (Ttot)
ξ
4r2
CSL
� Ttot
0
(Xκ(t′) − Xκ′(t′))2 dt′.
(74)
ζκ = (2κ − N)
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0.06664334982633591,
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1.079659104347229,
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0.9573602676391602,
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0.4674266278743744,
0.4820394814014435,
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0.2825787663459778,
0.37250950932502747,
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0.28863558173179626,
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0.3544803559780121,
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0.5227011442184448,
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0.01898798905313015,
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0.18631292879581451,
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0.2617155909538269,
0.8486563563346863,
1.1896849870681763,
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0.13541008532047272,
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0.47225984930992126,
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0.26437675952911377,
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0.6713009476661682,
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0.40304622054100037
] |
Λκ,κ′ ≈ (Ttot)
ξ
4r2
CSL
� Ttot
0
(Xκ(t′) − Xκ′(t′))2 dt′.
(74)
ζκ = (2κ − N)
1 − e−iΩt
0 ≤ t < t1
e−iΩ(t1+t) − 2e−it1Ω + 1
t1 ≤ t < t1 + t2
2e−iΩ(t1+t2) − e−iΩ(t1+t2+t) − 2e−it1Ω + 1
t1 + t2 ≤ t < Ttot
(75)
and
ϕ′
κ(t) = − (2κ − N)2
sin(tΩ)
0 ≤ t < t1
2 [sin(tΩ) + sin(t2Ω)] − sin(Ω(t1 + t))
t1 ≤ t < t1 + t2
2 sin(tΩ) + 2 sin(t1Ω) + 4 sin(t2Ω)−
2
�
sin((t + t2)Ω) + sin((t1 + t2)Ω)
�
+ sin((t + t1 + t2)Ω)
t1 + t2 ≤ t < Ttot
.
(76)
B.
Expectation value of ⟨ ˆ
M⟩ | [
-0.10038726776838303,
-0.3822261691093445,
0.46330925822257996,
-0.2954566478729248,
0.29494354128837585,
0.658586859703064,
0.10465237498283386,
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0.11659836769104004,
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0.06579675525426865,
0.7031288743019104,
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0.29144659638404846,
0.3257179260253906,
0.6143365502357483,
0.005496019497513771,
-0.4749353229999542,
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0.4732587933540344,
0.5427406430244446,
0.42029833793640137,
-0.15799078345298767,
-0.2001412808895111,
0.1402357965707779,
0.04014305770397186,
-0.29224249720573425,
0.4364158809185028,
0.1809227615594864,
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0.5897071361541748,
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0.023947283625602722,
0.5709405541419983,
-0.09916874766349792,
0.31148985028266907,
0.1523001790046692,
-0.036481574177742004,
0.038538698107004166,
0.28520795702934265,
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0.3415682017803192,
0.4392879605293274,
-0.5133766531944275,
-0.7623208165168762,
-1.9541367292404175,
-0.5624335408210754,
0.26749712228775024,
-0.9673682451248169,
0.2346741110086441,
0.1626093089580536,
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0.12307669222354889,
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0.35843637585639954,
-0.5317354202270508,
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0.7550477981567383,
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0.2239852398633957,
-0.04663636162877083,
0.3464464843273163,
0.3654663860797882,
-0.02839537523686886,
1.018876075744629,
0.11377183347940445,
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0.2174157351255417,
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0.45852744579315186,
0.3899463713169098,
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0.8050702810287476,
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0.18601450324058533,
0.6601475477218628,
-0.288038969039917,
0.14628297090530396,
0.08400941640138626,
-0.02613590471446514,
-0.6327893733978271,
0.5007005929946899,
0.30288979411125183,
0.04185774549841881,
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-0.5232228636741638,
0.31482309103012085,
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0.3144835829734802,
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0.5874261856079102,
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0.27445676922798157,
0.4356888234615326,
-0.13209283351898193,
0.9354529976844788,
0.6416602730751038,
-0.11159408837556839,
0.09079111367464066,
-0.509606122970581,
-0.06843322515487671,
-0.597452700138092,
-0.33739569783210754,
0.11473089456558228,
0.31873565912246704,
0.32467105984687805,
-0.031735148280858994,
-0.3206287920475006,
0.46141913533210754,
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0.3049379289150238,
0.24886010587215424,
-0.27592504024505615,
0.21446745097637177,
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0.31920790672302246,
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0.7655367255210876,
-0.002548854099586606,
-0.7015973925590515,
0.32744351029396057,
0.25001761317253113,
-0.34774988889694214,
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0.30291566252708435,
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0.16609971225261688,
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0.7084178924560547,
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0.5860387086868286,
-0.6364007592201233,
0.21927669644355774,
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] |
�
+ sin((t + t1 + t2)Ω)
t1 + t2 ≤ t < Ttot
.
(76)
B.
Expectation value of ⟨ ˆ
M⟩
We want to find the expectation value of ⟨ ˆ
M⟩(Ttot) =
tr
�
ˆU ⊗N
π/2 ˆρs(Ttot) ˆU †⊗N
π/2
ˆ
M
�
, where ˆUπ/2 is a π/2 pulse act-
ing on the spins and ˆρS is given by eq. (29). Due to the
cyclic properties of the trace and since ˆUπ/2ˆσz ˆU †
π/2 = ˆσx,
⟨ ˆ
M⟩(Ttot) is given by
⟨ ˆ
M⟩(Ttot) = ℏγp
2N+1
N
�
κ,κ′=0
��N
κ
��N
κ′
�
× e−i∆ϕκ,κ′ e−Λκ,κ′ tr
�
ˆX |κ⟩ ⟨κ′|
�
.
(77)
Here, ˆX = �N
n=1 ˆσ(n)
x | [
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] |
N
�
κ,κ′=0
��N
κ
��N
κ′
�
× e−i∆ϕκ,κ′ e−Λκ,κ′ tr
�
ˆX |κ⟩ ⟨κ′|
�
.
(77)
Here, ˆX = �N
n=1 ˆσ(n)
x
is the sum over all ˆσx operators.
To find an expression for ˆX |κ⟩, one can to use combi-
natorics and the relation ˆσx |↑ / ↓⟩ = |↓ / ↑⟩. It can be
shown that
��N
κ
�
ˆX |κ⟩ =
�
[N − (κ − 1)]
�� N
κ − 1
�
|κ − 1⟩
+ (κ + 1)
�� N
κ + 1
�
|κ + 1⟩
�
.
(78)
Since Dicke states are orthonormal, we find that for
⟨ ˆ
M⟩(Ttot) only terms like κ′ = κ ± 1 contribute to the | [
-0.1673905849456787,
-0.03840114176273346,
0.24165260791778564,
-0.29129835963249207,
-0.1799950748682022,
0.020786814391613007,
0.09258174896240234,
-0.6627572774887085,
-0.11047250777482986,
-0.19875989854335785,
-0.38198792934417725,
0.3234064280986786,
-0.2481520026922226,
0.30059900879859924,
0.3647695779800415,
0.7321146726608276,
0.7796259522438049,
0.3229062557220459,
-0.5093879103660583,
-0.5579942464828491,
0.233847975730896,
0.1914242058992386,
0.5281378626823425,
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] |
⟨ ˆ
M⟩(Ttot) only terms like κ′ = κ ± 1 contribute to the
double sum. Furthermore, Xκ(t) is linear in κ, therefore
only the nearest neighbors’ decoherences contribute and
Xκ(t′) − Xκ±1(t′) is independent of κ and can be taken
out of the sum. The left sum has a closed form solution
that is given by
⟨ ˆ
M⟩(Ttot) = ⟨ ˆ
M⟩(0)e−γ cosN−1(4ϕ)
(79)
with
ϕ = (χ/2x0)2 sin(Ωt1) + αTtot/ℏ,
γ = Λκ,κ±1.
(80)
If Xκ,κ±1(t′) ≪ rCSL at all times γ simplifies to γ =
ξ/(2r2
CSL)
� Ttot
0 | [
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] |
γ = Λκ,κ±1.
(80)
If Xκ,κ±1(t′) ≪ rCSL at all times γ simplifies to γ =
ξ/(2r2
CSL)
� Ttot
0
Xκ,κ±1(t′)2 dt′, which has a closed form
solution given by
γ =
ξ
4r2
CSL
χ2
4Ω
�
14(2t1 + t2)Ω − 8(t1 + t2)Ω cos(t1Ω)
+ 8t1Ω(−2 cos(t2Ω) + cos((t1 + t2)Ω)) + 8 sin(t1Ω)
− 11 sin(2t1Ω) + 4 sin(3t1Ω) − sin(4t1Ω)
+ 24 sin(2(t1 + t2)Ω) − 4 sin(3(t1 + t2)Ω)
+ 3 sin(4(t1 + t2)Ω) − 8 sin((2t1 + t2)Ω)
+ 5 sin(2(2t1 + t2)Ω) − 16 sin((t1 + 2t2)Ω)
− 4 sin(2(t1 + 2t2)Ω) − 12 sin(3t1Ω + 2t2Ω) | [
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] |
+ 5 sin(2(2t1 + t2)Ω) − 16 sin((t1 + 2t2)Ω)
− 4 sin(2(t1 + 2t2)Ω) − 12 sin(3t1Ω + 2t2Ω)
+ 8 sin(2t1Ω + 3t2Ω) − 8 sin(4t1Ω + 3t2Ω)
+ 4 sin(5t1Ω + 3t2Ω) + 4 sin(3t1Ω + 4t2Ω)
− 4 sin(5t1Ω + 4t2Ω) + sin(6t1Ω + 4t2Ω)
�
.
(81)
C.
π/2 pulse in Dicke basis
To
write
down
the
explicit
expression
for
⟨ ˆ
M⟩(Ttot)/⟨ ˆ
M⟩(0)
for
the
modified
protocol,
the
22
matrix elements of a π/2 and 3π/2 pulse in the Dicke
basis are needed.
To do that, we first look at the closely related | [
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] |
basis are needed.
To do that, we first look at the closely related
Hadamard gate, i.e., we want to know what H⊗N |κ⟩
looks like on the Dicke basis. To do so, let us look at
the eigenvectors of the Hadamard gate H on the spin ba-
sis. In this section, ket vectors in the spin basis will be
denoted by |·⟩S
H⊗N |k⟩S =
1
2N/2
�
a∈{0,1}
(−1)a·k |a⟩S
(82)
where ”1” corresponds to ”↑” and ”0” corresponds to
”↓”. By definition, the Dicke state |κ⟩ can be written in
the spin basis as
��N
κ
�
|κ⟩ =
�
k∈π(1, ..., 1 | [
0.028100138530135155,
0.01814035139977932,
0.2581527829170227,
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0.1694590449333191,
0.21619006991386414,
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0.13227418065071106,
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0.40228763222694397,
0.13029341399669647,
0.37997785210609436,
0.9974185228347778,
0.14002090692520142,
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0.0809587612748146,
0.04402182251214981,
0.41837865114212036,
0.1030178964138031,
0.10708964616060257,
0.2240072339773178,
0.36296120285987854,
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0.6961504220962524,
0.24079228937625885,
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0.30788207054138184,
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1.1103603839874268,
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-0.35028794407844543,
-0.07367412000894547,
-0.1824103742837906
] |
the spin basis as
��N
κ
�
|κ⟩ =
�
k∈π(1, ..., 1
� �� �
κ
,0, ..., 0
� �� �
N−κ
)
|k⟩S =
�
k∈π(k0)
|k⟩S
(83)
where π is the permutation operator and k0 is the vector
k0 = {1}⊗κ{0}⊗(N−κ). Hence, H⊗N |κ⟩ can be written
as
H⊗N |κ⟩ =
1
2
N
2
��N
κ
�
�
a∈{0,1}
�
k∈π(k0)
(−1)k·a |a⟩S .
(84)
Lets look at the vectors of the form |a0⟩S
=
��{1}⊗a0{0}⊗(N−a0)�
and find the weighting factor of
|a0⟩S i.e.
�
k∈π(k0)(−1)k·a0.
More explicitly, we want
to find an expression for
�
k∈π(k0)
(−1)k·a0 =
�
k∈π(k0)
(−1)
�a0 | [
-0.08564084023237228,
-0.4034724831581116,
0.40165048837661743,
0.02612253651022911,
-0.3948405981063843,
-0.12979662418365479,
-0.15590664744377136,
-0.6642627120018005,
-0.3263033628463745,
0.0953894630074501,
-0.1843661665916443,
-0.02081996202468872,
-0.22269298136234283,
0.5370545387268066,
-0.02236873283982277,
0.4898957908153534,
0.6552688479423523,
0.19635379314422607,
-0.9186683297157288,
-0.810900092124939,
0.30868077278137207,
0.0030717302579432726,
0.5195838809013367,
0.0042908634059131145,
0.16869589686393738,
0.1265057623386383,
0.20691107213497162,
-0.16091153025627136,
0.7078295350074768,
0.29608047008514404,
-0.31877392530441284,
0.48007309436798096,
0.2611846327781677,
0.06354117393493652,
0.3785143196582794,
-0.2071562558412552,
0.5053179860115051,
0.22647437453269958,
0.3050421178340912,
-0.5585533380508423,
0.4492926299571991,
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0.10965877771377563,
0.601092517375946,
-0.2138795256614685,
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0.34165501594543457,
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0.026547206565737724,
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0.1529836505651474,
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0.4026692509651184,
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0.15436822175979614,
0.0043032243847846985,
0.1528078317642212,
0.18595683574676514,
0.4480017423629761,
0.35268089175224304,
0.7086390256881714,
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0.42619311809539795,
-0.3796299993991852,
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0.007653628475964069,
0.22504009306430817,
-0.028536159545183182,
-0.038498640060424805,
0.07497057318687439,
0.14268353581428528,
0.050122715532779694,
-0.39505475759506226,
0.19259220361709595,
-0.33613133430480957,
-0.7351366877555847,
0.01571769081056118,
0.1584637612104416,
-0.6436631083488464,
0.4238061010837555,
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0.6441498398780823,
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0.658102810382843,
-0.02779604308307171,
0.10521408915519714,
0.18899081647396088,
0.3496623635292053,
0.3258901536464691,
0.5335964560508728,
-0.0831179991364479,
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0.8203986883163452,
0.056792162358760834,
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0.060836583375930786,
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0.11874733865261078,
-0.24983948469161987,
-1.055335283279419,
-0.29468610882759094,
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-0.4184134304523468,
0.058956313878297806,
-0.7687317132949829,
0.2395312637090683,
-0.7431944012641907,
0.9103243350982666,
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0.23037098348140717,
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0.4207437038421631,
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0.17107991874217987,
0.14541789889335632,
0.26262497901916504,
-0.8279107213020325,
1.2910888195037842,
0.6400433778762817,
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0.27748391032218933,
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0.3104701638221741,
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0.13304734230041504,
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0.000008212216016545426,
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0.25141116976737976,
1.3192118406295776,
-0.3210655748844147,
0.1875472515821457,
0.580542266368866,
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1.0155497789382935,
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0.08704204857349396,
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] |
More explicitly, we want
to find an expression for
�
k∈π(k0)
(−1)k·a0 =
�
k∈π(k0)
(−1)
�a0
i=1 ki .
(85)
To evaluate this expression, let us look at fig. 10. We
sum over all permutations of k0 that contain κ ones in
total. For a given permutation that contains ν ones in
the first a0 entries, there are κ−ν ones in the last N −a0
entries. There are
�a0
ν
��N−a0
κ−ν
�
possibilities to distribute
the ν and N − ν ones that way. Now, ν goes from zero
to κ. Hence, we find
�
k∈π(k0)
(−1)
�a0
i=1 ki =
κ
�
ν=0
(−1)ν | [
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0.06967157870531082,
-0.05078825727105141,
0.7428486347198486,
0.03482949733734131,
-0.08236851543188095,
-0.015778882429003716,
0.061455074697732925,
0.04129375144839287,
0.08954493701457977,
0.30617496371269226,
0.02906782552599907,
-0.15887218713760376,
-0.3770117163658142,
0.377352237701416,
-0.33241403102874756,
0.004750449676066637,
0.33163174986839294,
0.7664357423782349,
0.01161737646907568,
-0.2897171080112457,
0.015195619314908981,
-0.6809267401695251,
-0.41578665375709534,
0.31599661707878113,
-0.4693360924720764,
0.14956822991371155,
-0.17225246131420135,
0.42169424891471863,
-0.22195450961589813,
0.3941175043582916,
0.8120981454849243,
-0.3967457413673401,
-0.22379979491233826,
0.03175777196884155,
0.5528181195259094,
-0.5813955068588257,
0.5584248900413513,
0.31804290413856506,
-0.36957547068595886,
-0.06057523936033249,
-0.2712167203426361,
0.3313964605331421,
0.1302344650030136,
-0.19090914726257324,
-0.5242081880569458,
-0.5900850296020508,
-0.05013998597860336,
0.06134883314371109,
0.3784538209438324,
0.5602986812591553,
-0.11176769435405731,
-0.1001233458518982,
-0.8015580177307129,
0.3415737450122833,
-0.5220945477485657,
0.2226250171661377,
0.009217179380357265,
-0.2514787018299103,
0.37520378828048706,
0.1573507934808731,
-0.5102109313011169,
0.04987439513206482,
0.23749110102653503,
0.21173913776874542,
-0.06568621844053268,
-0.08455558121204376,
0.12427695095539093,
-0.11618591099977493,
-0.7323256134986877,
-0.6525289416313171,
0.04449436441063881,
-0.8979603052139282,
0.06934158504009247,
-0.13100530207157135,
0.2026229351758957,
0.29003602266311646,
-0.730366051197052,
-0.6036065220832825,
-0.34834691882133484,
0.6202921271324158,
-0.24942676723003387,
-0.49975845217704773,
0.07686204463243484,
-0.293866366147995,
-0.3840070366859436,
0.0736284926533699,
-0.7268108129501343,
0.7088908553123474,
0.23624195158481598,
-0.1313610076904297,
-0.1058083102107048,
0.2187349498271942,
0.3971528112888336,
-0.23203007876873016,
-0.23862439393997192,
-0.743270754814148,
-0.24655641615390778,
-0.18816682696342468,
0.5772724747657776,
-0.3739384114742279,
-0.17249685525894165,
0.12487594783306122,
-0.4721188247203827,
-0.13377051055431366,
0.1733183115720749,
0.28705570101737976,
0.501538872718811,
0.3253636956214905,
0.21594499051570892,
-0.28722476959228516,
0.3468678891658783,
-0.12327674776315689,
0.05245356261730194,
0.0021186950616538525,
-0.6024126410484314,
-0.1457419991493225,
-0.1599593609571457,
0.6058148741722107,
-0.08853735774755478,
-0.08639427274465561,
0.0316607728600502,
0.006634721998125315,
0.4223374128341675,
-0.34616991877555847,
-0.09389589726924896,
-0.91866534948349,
-0.22549660503864288,
0.24669218063354492,
0.6702881455421448,
0.36546552181243896,
0.3395845293998718,
-0.218095600605011,
-0.29505667090415955,
-0.22263102233409882,
-0.1415768265724182,
0.015303104184567928,
-0.10927963256835938,
-0.14312775433063507,
0.29461681842803955,
0.5195949673652649,
0.14292041957378387,
-0.39074912667274475,
-0.03556317090988159,
0.5584390163421631,
0.010794613510370255,
0.22916889190673828,
0.0430377721786499,
-0.5630776286125183,
-0.04623747617006302,
0.7770266532897949,
-0.38561415672302246,
0.1593502163887024,
0.31263604760169983,
0.16011515259742737,
0.12697969377040863,
0.6989979147911072,
-0.39997267723083496,
-1.0470718145370483,
0.0982092097401619,
-0.15925751626491547,
0.11889499425888062,
-0.1505075842142105,
0.2764127552509308,
0.7411983013153076,
0.5471293926239014,
-0.7205620408058167,
-0.12263259291648865,
0.01577935926616192,
0.4765138626098633,
-0.4755454957485199,
0.0886942595243454,
-0.07431759685277939,
0.43781670928001404,
1.0197070837020874,
1.1037473678588867,
-0.2105427235364914,
-0.33569225668907166,
0.3193761110305786,
0.16921785473823547,
-0.18469035625457764,
-0.5656905770301819,
-0.012516300193965435,
0.261761337518692,
-0.027561217546463013,
-0.0945683941245079,
-0.6066816449165344,
-0.26086369156837463,
-0.5670064687728882,
0.24827523529529572,
-0.013516662642359734,
-0.7814080715179443,
0.050829265266656876,
0.18846160173416138,
0.2487003654241562,
0.29190540313720703,
-0.08211353421211243,
-0.1382438987493515,
-0.40982785820961,
0.10217662155628204,
0.3043445646762848,
-0.21857692301273346,
-0.24605277180671692,
-0.17583175003528595,
-0.12777353823184967,
0.19368810951709747,
0.30774450302124023,
0.21544432640075684,
0.28448617458343506,
-0.23263686895370483,
0.08683411777019501,
0.34573692083358765,
-0.5782256126403809,
-0.13152262568473816,
0.19995182752609253,
0.3829050064086914,
-0.3669015169143677,
-0.5589993000030518,
0.03405563160777092,
0.1724408119916916,
0.25091296434402466,
-0.29722127318382263,
-0.2026810646057129,
0.2878919839859009,
0.5695906281471252,
-0.6210747361183167,
-0.3992140591144562,
0.059089917689561844,
0.1645079404115677,
0.019598571583628654,
-0.4722612500190735,
0.9747207164764404,
1.127508282661438,
-0.24233411252498627,
0.2766837477684021,
-0.348314106464386,
0.3713829219341278,
0.7855715751647949,
-0.36990171670913696,
0.5032832622528076,
-0.3498308062553406,
0.17403769493103027,
-0.48615357279777527,
-0.07160557806491852,
0.8655295372009277,
0.4072144627571106,
-0.04407655820250511,
-0.2038455456495285,
-0.24980565905570984,
-0.3333991467952728,
0.4045104682445526,
-0.057194631546735764,
-0.26124003529548645,
-0.5358196496963501,
-0.6405865550041199,
-0.6861129999160767,
0.6237093806266785,
0.41626185178756714,
0.14807462692260742,
-0.4730800688266754,
0.5752736926078796,
-0.2191259264945984,
0.3057551681995392,
0.3617445230484009,
-0.32302987575531006,
0.025694163516163826,
-0.30257463455200195,
0.7310556769371033,
-0.688974916934967,
0.5493069291114807,
0.6935023069381714,
0.6658041477203369,
0.35557326674461365,
-0.1648823469877243,
-0.7224075794219971,
0.13851898908615112,
-0.17809021472930908,
0.9377267956733704,
0.4234997630119324,
0.6206601858139038,
-0.19101271033287048,
-0.6677922010421753,
-0.6210933327674866,
0.5017691850662231,
-0.13338793814182281,
0.15676820278167725,
0.05505358800292015,
0.016186479479074478,
0.02870704047381878,
0.3169216513633728,
0.22482310235500336,
-0.44010859727859497,
-0.18488486111164093,
0.4025314152240753
] |
to κ. Hence, we find
�
k∈π(k0)
(−1)
�a0
i=1 ki =
κ
�
ν=0
(−1)ν
�a0
ν
��N − a0
κ − ν
�
.
(86)
Since we sum over all permutations of k0, the weighting
factor of all |a⟩S with the same number of spins up will
be the same. We can summarize all |a⟩S with with same
numbers of spin up in the Dicke state
��N
α
�
|α⟩ and find
H⊗N |κ⟩ =
1
2
N
2
N
�
α=0
�
�
�
�
�N
α
�
�N
κ
�
κ
�
ν=0
(−1)ν
�α
ν
��N − α
κ − ν
�
|α⟩ .
(87)
a0
N − a0
ν
”1”
κ − ν
”1”
FIG. 10. There are
�a0
ν
�
possibilities to place ν ones in the | [
0.021883618086576462,
-0.08438129723072052,
0.5060803294181824,
-0.10697640478610992,
-0.37986302375793457,
0.15010429918766022,
0.19589175283908844,
-0.7374684810638428,
-0.52227383852005,
-0.11843295395374298,
-0.29772040247917175,
0.21394583582878113,
-0.0572650283575058,
0.4798142611980438,
0.2225574553012848,
0.6368994116783142,
0.7131454348564148,
0.2630235254764557,
-1.0679038763046265,
-0.8853219151496887,
0.43866774439811707,
0.006281251087784767,
0.6967921257019043,
-0.08909255266189575,
0.10534476488828659,
0.01535122748464346,
0.19141636788845062,
-0.3611760139465332,
0.36073213815689087,
0.2705850899219513,
-0.48181819915771484,
0.5184654593467712,
0.06573636084794998,
-0.02238263003528118,
0.32259541749954224,
-0.34287703037261963,
0.5360628366470337,
0.14981390535831451,
0.3430822789669037,
-0.5738264918327332,
0.3693941831588745,
-0.05600808933377266,
0.3627113103866577,
0.5332841277122498,
-0.40718090534210205,
-0.3943376839160919,
-1.123911738395691,
-0.017447752878069878,
0.06710921972990036,
-0.7586260437965393,
0.18232473731040955,
-0.13853716850280762,
0.21596123278141022,
0.21113547682762146,
-0.06856708228588104,
0.049266304820775986,
-0.6647030115127563,
-0.10037906467914581,
0.728196382522583,
0.0817737802863121,
0.2314419001340866,
-0.0321250855922699,
0.19116856157779694,
0.43164974451065063,
0.41883793473243713,
1.18000066280365,
-0.04640451446175575,
0.07226163893938065,
-0.45943403244018555,
0.023390423506498337,
0.1500200778245926,
-0.15190231800079346,
0.009210691787302494,
-0.0727802962064743,
0.09177131205797195,
0.22781455516815186,
-0.2073000967502594,
-0.5475136637687683,
0.44875720143318176,
-0.07876752316951752,
-0.3825507164001465,
-0.2882601022720337,
0.43490034341812134,
-0.37922343611717224,
0.36630508303642273,
0.2901739776134491,
-0.1505483239889145,
0.5760102272033691,
-0.6673256158828735,
0.6817933320999146,
0.024475757032632828,
-0.04219648614525795,
0.19865202903747559,
0.19009345769882202,
0.19035738706588745,
0.43634673953056335,
-0.10512441396713257,
-0.25877654552459717,
0.5477849841117859,
0.1347716897726059,
-0.0731804296374321,
0.05443722754716873,
-0.18807925283908844,
-0.1881023496389389,
-0.3467750549316406,
-0.7926703691482544,
-0.2661954462528229,
-0.34225398302078247,
-0.01697225123643875,
-0.015588415786623955,
-0.923875093460083,
0.541612446308136,
-0.33263278007507324,
-0.46299460530281067,
-0.09417632222175598,
0.18564027547836304,
0.2043377161026001,
-0.3720034062862396,
0.10946120321750641,
0.37070903182029724,
-0.28023624420166016,
0.5502614378929138,
-0.005253050476312637,
0.7913601994514465,
0.39936983585357666,
0.11063000559806824,
0.4424196481704712,
0.6838057637214661,
-0.3809359669685364,
-0.006728494539856911,
0.06444235891103745,
0.7403873801231384,
0.1788591593503952,
0.15124402940273285,
-0.0842808336019516,
-0.7551209330558777,
-0.3519056737422943,
0.05761679261922836,
-0.6017610430717468,
0.10621979087591171,
-0.05916944891214371,
-0.040941495448350906,
-0.40000325441360474,
-0.03031918965280056,
0.15223649144172668,
-0.2590824067592621,
-0.23611882328987122,
0.12139955908060074,
0.003298999508842826,
0.3914446532726288,
0.005441689398139715,
0.26460564136505127,
-0.573005199432373,
0.22181130945682526,
-0.24273726344108582,
-0.6198456883430481,
-0.040595315396785736,
0.17304687201976776,
0.0217384472489357,
0.05758688598871231,
-0.40537527203559875,
-0.14558890461921692,
-0.004730158485472202,
-0.5025355815887451,
0.48878222703933716,
-0.7368504405021667,
0.6040969491004944,
-0.4041561484336853,
-0.11569596827030182,
0.16852499544620514,
-0.6471434235572815,
-0.10729435086250305,
-0.3710394501686096,
0.17944905161857605,
-0.4309893846511841,
-0.4844318628311157,
0.18143847584724426,
0.27943119406700134,
0.22161640226840973,
-0.27023863792419434,
-0.773201048374176,
-0.41645780205726624,
-0.2480027675628662,
-0.4150165021419525,
0.308820515871048,
0.5792630314826965,
0.15110459923744202,
0.23814113438129425,
0.27429261803627014,
0.5033653378486633,
0.16088327765464783,
0.034516140818595886,
0.40771394968032837,
-0.7634834051132202,
1.2324990034103394,
0.35997137427330017,
-0.3875772953033447,
-0.015125500969588757,
-0.1536291092634201,
0.1510349065065384,
-0.13465391099452972,
0.2048896998167038,
0.22568058967590332,
0.36355873942375183,
-0.15487638115882874,
0.30610391497612,
0.04556562006473541,
0.0342034287750721,
-0.4289892613887787,
0.5081547498703003,
0.1923685520887375,
-0.08944385498762131,
0.24284528195858002,
-0.4993908107280731,
-0.017687667161226273,
-0.13900993764400482,
0.18683801591396332,
0.27687257528305054,
0.3904944062232971,
-0.05097970366477966,
0.8508192896842957,
-0.27066975831985474,
0.4490475654602051,
0.43234649300575256,
0.17647120356559753,
0.8840854167938232,
0.504741370677948,
0.10057640820741653,
0.14642393589019775,
-0.27866074442863464,
-0.20972830057144165,
-0.19747096300125122,
0.2662259340286255,
0.1885620504617691,
0.01245411578565836,
0.776040256023407,
-0.05552869662642479,
-0.3680804669857025,
0.26247358322143555,
-0.6825963258743286,
0.12840360403060913,
0.32711708545684814,
-0.47095969319343567,
0.6936869621276855,
0.11215031147003174,
0.06369460374116898,
-0.4126017689704895,
0.076134093105793,
-0.5238155126571655,
-0.2136574685573578,
-0.044667817652225494,
-0.2827766239643097,
0.19768065214157104,
-0.8570489287376404,
-0.14222648739814758,
0.15888723731040955,
-0.08098102360963821,
-0.27864712476730347,
0.1727861911058426,
0.4565623998641968,
-0.24704283475875854,
-0.2644752860069275,
-0.024366531521081924,
-0.3582478165626526,
-0.6945393681526184,
-0.12030066549777985,
-1.088560700416565,
-0.20830881595611572,
0.8058279156684875,
0.2998945415019989,
0.3953026235103607,
-0.26131704449653625,
-0.9196158051490784,
-0.33168381452560425,
-0.3326820731163025,
-0.7136567831039429,
-0.39990752935409546,
0.5633894205093384,
-0.6042097210884094,
-0.20265698432922363,
0.1372544765472412,
0.9241288900375366,
0.5890304446220398,
0.16463908553123474,
-0.29201140999794006,
-0.19960592687129974,
-0.165877565741539,
-0.9789459109306335,
-0.2380579262971878,
-0.8498010039329529,
-0.18441987037658691,
0.5764843821525574,
-0.4061756134033203,
-0.8187143802642822,
0.12919656932353973,
0.2917027175426483,
-0.5780164003372192,
-0.3746284246444702,
0.08944164961576462,
-0.030159980058670044,
-0.4738609492778778,
0.2746025621891022,
-0.3075001835823059,
0.13822340965270996,
0.010937086306512356,
-0.5099779367446899,
-0.6182175874710083,
0.303225040435791,
-5.817797660827637,
-0.06600914150476456,
-0.5962985754013062,
0.3907136023044586,
-0.14261464774608612,
0.13306638598442078,
0.6022952198982239,
-0.27968162298202515,
-0.11571963876485825,
0.04788532853126526,
-0.05354154482483864,
-0.02623293548822403,
-0.28896135091781616,
0.6312447190284729,
-0.005635509733110666,
0.4728877544403076,
0.4152218997478485,
0.276800274848938,
0.3999726474285126,
-0.009430717676877975,
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] |
(87)
a0
N − a0
ν
”1”
κ − ν
”1”
FIG. 10. There are
�a0
ν
�
possibilities to place ν ones in the
yellow entries of k′ and
�N−a0
κ−ν
�
possibilities to place κ − ν
ones in the green entries.
The N-particle Hadamard gate in Dicke basis is therefore
given by
H⊗N
κ,α =
1
2
N
2
�
�
�
�
�N
α
�
�N
κ
�
κ
�
ν=0
(−1)ν
�α
ν
��N − α
κ − ν
�
.
(88)
What is left is to retrieve the π/2 pulse from the
Hadamard gate. We can do that by applying the X op-
erator on the Hadamard gate as one can check easily for | [
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] |
erator on the Hadamard gate as one can check easily for
the one spin case
XH =
1
√
2
�
0 1
1 0
� �
1
1
1 −1
�
=
1
√
2
�
1 −1
1
1
�
=
�π
2
�
.
(89)
A X gate flips |↑⟩S to |↓⟩S and the other way around.
Therefore, X⊗N acts on a Dicke state like X⊗N |α⟩ =
|N − α⟩. We find with a few arithmetic transformations
U ⊗N
π/2 |κ⟩ = X⊗NH⊗N |κ⟩
=
1
2
N
2
N
�
α=0
�
�
�
�
�N
α
�
�N
κ
�
κ
�
ν=0
(−1)ν
�N − α
ν
��
α
κ − ν
�
|α⟩ .
(90)
Therefore, the matrix elements of U ⊗N
π/2 |κ⟩ in Dicke basis
are
�π
2
�⊗N
κ,α = ⟨α| U ⊗N | [
-0.38008642196655273,
0.008626803755760193,
0.2688549757003784,
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0.16193436086177826,
0.2042362093925476,
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0.6340362429618835,
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-0.48443037271499634,
0.08564411848783493,
0.0478803925216198,
-0.1627492904663086,
0.24530799686908722,
-0.09518378973007202,
-0.039760179817676544,
0.048625487834215164,
-0.35387328267097473,
-0.17718344926834106,
0.11282423883676529,
0.17215028405189514,
-0.11795864254236221,
-0.2829556167125702,
0.5053317546844482,
0.46850115060806274,
0.026037026196718216,
0.39096158742904663,
0.3920465409755707,
-0.12934863567352295,
0.3464653491973877,
0.10815960168838501,
0.20043273270130157,
-0.6382877230644226,
-0.29247143864631653,
-0.6181359887123108,
0.6030771732330322,
1.1213265657424927,
-0.12077097594738007,
-0.032392509281635284,
-0.2566153407096863,
-0.352748841047287,
-0.558281660079956,
0.28904426097869873,
-0.3324814736843109,
-0.30504271388053894,
-0.5241652131080627,
-0.3591761887073517,
-0.1535574346780777,
0.22113284468650818,
-0.2050924152135849,
0.25237226486206055,
-0.10190720111131668,
0.7770653963088989,
0.03464822471141815,
0.20433369278907776,
0.35919132828712463,
-0.005745601840317249,
-0.5172222256660461,
-0.7566752433776855,
0.32457247376441956,
-0.39057961106300354,
0.2655986547470093,
0.5139502882957458,
0.6773083209991455,
0.6209458708763123,
-0.21175096929073334,
0.19895689189434052,
0.024687889963388443,
-0.017054351046681404,
0.4328569173812866,
0.18866848945617676,
0.3050774931907654,
-0.5086116790771484,
-0.47632622718811035,
-0.19417844712734222,
0.7089654803276062,
0.2014547884464264,
-0.24476325511932373,
-0.16992056369781494,
0.1893930584192276,
0.14777369797229767,
-0.044519372284412384,
-0.3111327886581421,
-0.6066898107528687,
-0.05909160524606705,
-0.08105335384607315
] |
(90)
Therefore, the matrix elements of U ⊗N
π/2 |κ⟩ in Dicke basis
are
�π
2
�⊗N
κ,α = ⟨α| U ⊗N
π/2 |κ⟩
=
1
2
N
2
�
�
�
�
�N
α
�
�N
κ
�
κ
�
ν=0
(−1)ν
�N − α
ν
��
α
κ − ν
�
.
(91)
Furthermore, a 3π/2 pulse is given by U3π/2
=
UπUπ/2. One can easily convince oneself, that Uπ |α⟩ =
(−1)α+1 |N − α⟩. Therefore, we find
�3π
2
�⊗N
κ,α
= (−1)α+1
2
N
2
�
�
�
�
�N
α
�
�N
κ
�
κ
�
ν=0
(−1)ν
�α
ν
��N − α
κ − ν
�
.
(92)
23
D.
Expectation value of ⟨ ˆ
M⟩, modified protocol | [
0.2866465151309967,
-0.005593091249465942,
0.4445228576660156,
-0.3356648087501526,
-0.24669839441776276,
0.17091652750968933,
0.24483169615268707,
-0.6695827841758728,
-0.152933731675148,
0.2683704197406769,
0.0526445209980011,
0.2746499478816986,
-0.2491256296634674,
-0.05108194425702095,
0.14974331855773926,
0.6746340990066528,
0.8012048006057739,
-0.032393552362918854,
-0.4322662949562073,
-0.9647294282913208,
0.3124634027481079,
0.13513140380382538,
0.9408944249153137,
0.27687790989875793,
0.029534701257944107,
0.408336877822876,
0.533701479434967,
-0.36135345697402954,
-0.24800781905651093,
0.345755010843277,
-0.24473729729652405,
0.4402136206626892,
0.2051829844713211,
-0.22539067268371582,
0.16130715608596802,
0.5781758427619934,
0.49156439304351807,
0.24434839189052582,
-0.08899807929992676,
-0.41524580121040344,
0.5905553102493286,
-0.3099575638771057,
-0.12954847514629364,
0.45707646012306213,
-0.08721034973859787,
-0.07594361156225204,
-1.4744782447814941,
0.11280452460050583,
-0.16452732682228088,
-0.6850884556770325,
0.12846338748931885,
-0.15852664411067963,
0.17426742613315582,
0.23063597083091736,
-0.1709294468164444,
0.0016594668850302696,
-0.4034329354763031,
-0.19066469371318817,
0.507886528968811,
0.022209996357560158,
0.2777709662914276,
-0.008044672198593616,
0.0004214923828840256,
0.3544595539569855,
0.2148827761411667,
0.6964172720909119,
0.028963185846805573,
0.6488878130912781,
-0.29693013429641724,
-0.33332717418670654,
0.28579550981521606,
0.002397290663793683,
0.007954270578920841,
-0.07769548147916794,
-0.09274902939796448,
0.19575147330760956,
-0.48674309253692627,
-0.18496201932430267,
0.037974391132593155,
-0.25988540053367615,
-0.07832397520542145,
-0.2669931650161743,
0.6090157628059387,
-0.11265900731086731,
0.18838873505592346,
-0.4110763370990753,
0.08798424899578094,
0.5355571508407593,
-0.20044416189193726,
0.4604824483394623,
-0.45902058482170105,
-0.5021235346794128,
0.30636194348335266,
0.2726459801197052,
0.05406545102596283,
0.23388876020908356,
-1.001638412475586,
0.06325656175613403,
0.5407959222793579,
0.05512122064828873,
0.027653275057673454,
-0.17177674174308777,
-0.20650221407413483,
-0.0675438642501831,
0.013842032290995121,
-0.7105752825737,
-0.2234867364168167,
-0.3180907666683197,
-0.4089968800544739,
0.157468780875206,
-0.8591539263725281,
0.5243247747421265,
0.02571878582239151,
-0.6053099632263184,
0.09788412600755692,
0.13123798370361328,
0.23202013969421387,
0.025614582002162933,
-0.14653053879737854,
0.5461632609367371,
-0.06586182117462158,
0.7000699639320374,
-0.07231227308511734,
0.23168526589870453,
0.3756122887134552,
-0.10916410386562347,
0.48072242736816406,
0.8693367838859558,
-0.6110658049583435,
-0.1585649698972702,
-0.26394763588905334,
0.7860264778137207,
-0.1204025074839592,
0.41781124472618103,
0.018560457974672318,
-1.154322624206543,
-0.2087695300579071,
-0.25381261110305786,
-0.6184425950050354,
0.025032512843608856,
-0.017484189942479134,
0.44808098673820496,
-0.3584001958370209,
-0.14033478498458862,
0.20823106169700623,
-0.18371443450450897,
0.42511075735092163,
0.07393745332956314,
0.13332675397396088,
0.04337010905146599,
0.13742844760417938,
0.24884861707687378,
-0.31004491448402405,
0.04922439530491829,
-0.4473721385002136,
-0.3520267903804779,
-0.12080689519643784,
0.01633126102387905,
-0.12146341800689697,
0.4449346363544464,
-0.15462282299995422,
-0.0691317617893219,
0.1179228127002716,
0.06721869111061096,
0.613568902015686,
-0.5701988935470581,
0.8795044422149658,
-0.6343657374382019,
0.4569135904312134,
-0.0011697968002408743,
-0.5380827188491821,
-0.10579197108745575,
-0.4751637578010559,
0.4839380979537964,
-0.12018854171037674,
-0.3594772517681122,
0.3771759569644928,
0.3532351851463318,
-0.13768763840198517,
-0.39491620659828186,
-0.7341665029525757,
-0.27031272649765015,
-0.3404650092124939,
-0.04438341036438942,
0.5530679225921631,
0.3714965283870697,
0.5967170000076294,
0.3660469353199005,
0.28468555212020874,
0.4832729995250702,
0.42015326023101807,
0.16594816744327545,
0.13534501194953918,
-0.5987586975097656,
1.0892359018325806,
-0.04073171317577362,
-0.562628448009491,
-0.09859415888786316,
0.11433286964893341,
0.5763015747070312,
0.26196393370628357,
-0.04429422318935394,
0.31400299072265625,
0.165958434343338,
0.053077977150678635,
0.17079497873783112,
0.5700134634971619,
0.024330921471118927,
-0.26835715770721436,
0.1597823202610016,
0.4946441948413849,
-0.00641603721305728,
-0.11183782666921616,
-0.23818962275981903,
-0.3175491392612457,
0.16586552560329437,
0.39592066407203674,
0.22344115376472473,
0.27402764558792114,
0.09602051228284836,
0.6033463478088379,
-0.052453283220529556,
-0.0011921996483579278,
0.7946353554725647,
-0.060029298067092896,
1.0272977352142334,
0.33144012093544006,
-0.04280174896121025,
0.03625694662332535,
-0.5531646609306335,
-0.324126660823822,
-0.00012191560381324962,
-0.20099516212940216,
0.6105672717094421,
0.3758781850337982,
0.46681469678878784,
-0.10828784853219986,
-0.39113906025886536,
0.35617610812187195,
-0.2873220145702362,
0.8583504557609558,
0.24400590360164642,
-0.759506106376648,
0.39579299092292786,
0.40988868474960327,
-0.06545121222734451,
-0.102430060505867,
0.2630294859409332,
-0.22698207199573517,
-0.44362232089042664,
0.30396586656570435,
-0.2861878573894501,
0.10260097682476044,
-0.6744839549064636,
-0.15609082579612732,
-0.16390575468540192,
-0.05051264539361,
-0.272506445646286,
0.4849171042442322,
0.17503197491168976,
-0.18782185018062592,
-0.3261292278766632,
0.3915637135505676,
-0.22096742689609528,
-0.678447425365448,
0.15367190539836884,
-1.298056960105896,
0.20612125098705292,
0.4448131322860718,
0.04652810096740723,
0.29370495676994324,
0.1055191308259964,
-0.9794723987579346,
-0.060378603637218475,
-0.5133771300315857,
-0.767756998538971,
-0.3067565858364105,
0.3417837917804718,
-0.5483552813529968,
-0.4051770567893982,
-0.027604782953858376,
0.9856254458427429,
0.45629653334617615,
0.10692308098077774,
-0.9987328052520752,
-0.3659313917160034,
-0.2562395930290222,
-0.782037615776062,
-0.6624614000320435,
-0.6484920382499695,
-0.32223260402679443,
0.37078943848609924,
-0.35399192571640015,
-0.6024250388145447,
0.15457148849964142,
0.21996226906776428,
-0.3783212900161743,
-0.2159547209739685,
0.3825989365577698,
-0.07730961591005325,
-0.18704067170619965,
-0.061983320862054825,
-0.17071257531642914,
0.17134559154510498,
-0.1228918805718422,
-0.7449907064437866,
-0.61330246925354,
0.2604188323020935,
-5.722812175750732,
-0.00739864120259881,
-0.2484581172466278,
0.2407868206501007,
-0.22278648614883423,
0.2690011262893677,
0.7165658473968506,
-0.10595934092998505,
0.36960989236831665,
0.012168314307928085,
-0.5599867105484009,
-0.14369750022888184,
-0.11127953976392746,
0.41865667700767517,
0.11924823373556137,
-0.05127432197332382,
0.40342164039611816,
-0.059393975883722305,
0.2546083629131317,
0.3786989450454712,
0.6890339255332947,
-0.21497169137001038,
0.0655323937535286,
-0.07403421401977539,
0.40513545274734497,
0.005194723140448332,
0.00036534364335238934,
0.7957817316055298,
-0.04918263480067253,
-0.7348374724388123,
-0.28037363290786743,
-0.21755951642990112,
0.23003624379634857,
-0.19964733719825745,
0.41158798336982727,
0.14170324802398682,
0.9011462926864624,
-0.14146248996257782,
0.21598105132579803,
-0.9637041091918945,
0.442449688911438,
0.5546969175338745,
-0.12580816447734833,
-0.5630555152893066,
0.3103622794151306,
0.19101278483867645,
-0.11989742517471313,
-0.006610646843910217,
-0.4286870062351227,
0.5302513241767883,
0.08685839921236038,
-0.2842164635658264,
-0.2648887038230896,
-0.1414058804512024,
-0.6222501993179321,
-0.5363147854804993,
-0.32981500029563904,
0.16089756786823273,
-0.24399757385253906,
0.7251719832420349,
-0.05434086173772812,
-0.03935960307717323,
0.3474423289299011,
-0.5323868989944458,
-0.13459590077400208,
0.12475291639566422,
-0.7110012173652649,
-0.08027362078428268,
0.48158568143844604,
0.20206347107887268,
-0.47848066687583923,
0.09440679103136063,
0.18431183695793152,
-0.7146288156509399,
0.607677698135376,
-0.021699700504541397,
0.11300932615995407,
-0.10954410582780838,
-0.5315067768096924,
0.08730330318212509,
-0.15426011383533478,
-0.2637118697166443,
-0.1017758697271347,
0.7074690461158752,
0.41730767488479614,
-0.05502571910619736,
-0.11857260018587112,
0.03582703694701195,
-0.2746686041355133,
0.26218608021736145,
0.8572520017623901,
0.08215406537055969,
0.35162943601608276,
0.3098251223564148,
0.06289903074502945,
0.426483154296875,
0.07553393393754959,
-0.4113316535949707,
0.6540897488594055,
-0.2565114200115204,
0.5973928570747375,
-0.22310808300971985,
0.0328865572810173,
-0.8939815759658813,
0.29546213150024414,
0.16188639402389526,
-0.46905139088630676,
0.04673605039715767,
0.4805031716823578,
0.08608219027519226,
-0.17300471663475037,
-0.2127988487482071,
-0.8112006783485413,
-0.12668892741203308,
0.5904785394668579,
-0.3269340693950653,
-0.48511725664138794,
-0.3311615586280823,
0.7020730376243591,
-0.058701012283563614,
-0.4837392568588257,
0.547771692276001,
-0.09627999365329742,
-0.5515262484550476,
-0.07292364537715912,
-0.36171120405197144,
0.010352501645684242,
0.6132826209068298,
-0.5760926008224487,
0.2170284539461136,
0.4614652097225189,
0.38819676637649536,
-0.28093427419662476,
-0.3508165776729584,
0.12686079740524292,
-0.3373934328556061,
-0.30975040793418884,
0.18332719802856445,
0.21101704239845276,
0.04228467866778374,
0.4832683503627777,
-0.01585557498037815,
0.1561838686466217,
-0.4008939266204834,
0.3958653211593628,
0.5701019763946533,
0.2722960114479065,
-0.33261874318122864,
-0.46614307165145874,
0.055945709347724915,
-0.0024893786758184433,
-0.5015376210212708,
0.4693043828010559,
-0.16453219950199127,
0.03979998081922531,
0.40375515818595886,
0.26082342863082886,
0.2715364396572113,
0.17853058874607086,
-0.2770005464553833,
0.4230136275291443,
-0.11099904030561447,
-0.3014717996120453,
0.26849961280822754,
0.2789042592048645,
-0.098885178565979,
-0.7366007566452026,
-0.40738677978515625,
0.1455453634262085,
0.5693008899688721,
0.04036886990070343,
-0.42685768008232117,
-0.5355495810508728,
0.12451332807540894,
0.3957081139087677,
0.0386090986430645,
-0.21480149030685425,
-0.10040390491485596,
-0.18992626667022705,
-0.5339853763580322,
0.255403071641922,
-0.14838555455207825,
-0.13458524644374847,
1.1842702627182007,
1.1270265579223633,
-0.3463183343410492,
-0.7535281181335449,
0.06535058468580246,
-0.9997507929801941,
0.08844703435897827,
0.7655715346336365,
-0.11806084960699081,
0.3067677617073059,
-0.26080912351608276,
0.0590011402964592,
-0.38003915548324585,
0.30318912863731384,
1.0109608173370361,
-0.5838276147842407,
-0.3028523325920105,
-0.29939737915992737,
0.5725309252738953,
-0.050369225442409515,
0.6242983341217041,
-0.06722963601350784,
-0.18523138761520386,
0.04122169315814972,
-0.10749568790197372,
-0.12577363848686218,
-0.36918917298316956,
0.12249086052179337,
-0.5510380864143372,
-0.5359240174293518,
0.30885860323905945,
-0.371425062417984,
0.7195296883583069,
0.22015301883220673,
-0.26782873272895813,
-0.49576786160469055,
-0.754319965839386,
0.6870970129966736,
-0.8096950054168701,
0.21114204823970795,
0.434653639793396,
-0.2717139720916748,
0.1953689157962799,
0.09162762761116028,
-0.5644282698631287,
-0.26995500922203064,
0.22217459976673126,
-0.015659451484680176,
0.11862626671791077,
-0.24209098517894745,
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] |
κ
�
κ
�
ν=0
(−1)ν
�α
ν
��N − α
κ − ν
�
.
(92)
23
D.
Expectation value of ⟨ ˆ
M⟩, modified protocol
For the protocol as depicted in fig. 5, the classical tra-
jectories of the (κ, α β)th path is given by
ζκ,α,β(t) =
˜κ
�
1 − e−iΩt�
0 ≤ t < π
Ω
˜α
�
1 + e−iΩt�
− 2˜κe−iΩt
π
Ω ≤ t < 3π
Ω
˜β
�
1 + e−iΩt�
− 2˜κe−iΩt
3π
Ω ≤ t < 4π
Ω
.
(93)
with ˜κ = 2κ − N, ˜α = 2α − N and ˜β = 2β − N. One
can check easily, that for t = Ttot = 4π/Ω and β = κ
all trajectories meet in the origin with zero momentum. | [
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