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all trajectories meet in the origin with zero momentum.
The phase each trajectory accumulates at time t is given
by ϕκ,α,β(t) = ϕ′
κ,α,β + αTtot/ℏ with
ϕ′
κ,α,β(t) =
˜κ2 sin(Ωt)
0 ≤ t < π
Ω
�
2˜α˜κ − ˜α2�
sin(Ωt)
π
Ω ≤ t < 3π
Ω
�
2˜β˜κ − ˜β2�
sin(Ωt)
3π
Ω ≤ t < 4π
Ω
. (94)
As one can easily check, the phase each trajectory accu-
mulates for all κ, α, β is a multiple of π at t = Ttot =
4π/Ω and if the parameters are chosen according to
eq. (37). For an initial state as in eq. (15) and | [
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] |
eq. (37). For an initial state as in eq. (15) and
ˆρκ,κ′,α,α′,β,β′ = ˆUκ,α,β ˆρth ˆU †
κ′,α′,β′,
(95)
the final spin state is given by
ˆρS = 1
2N
N
�
κ,κ′=0
N
�
α,α′=0
N
�
β,β′=0
��N
κ
���N
κ′
�
×
�π
2
�⊗N
κ,α
�3π
2
�⊗N
α,β
�π
2
�⊗N
α′,κ′
�3π
2
�⊗N
β′,α′
× tr{ˆρκ,κ′,α,α′,β,β′(Ttot)} |β⟩ ⟨β| .
(96)
For the trajectories that recombine, the trace over
ˆρκ,κ′,α,α′,β,β′(Ttot) is given by
e−ξ
� T
0 [1−e
−[Xκ,α,β (t′)−Xκ′,α′,β′ (t′)]2/(4r2
CSL)] dt′,
(97)
where Xκ,α,β(t) is given by Xκ,α,β(t) = χ Re[ζκ, α, β(t)]. | [
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] |
CSL)] dt′,
(97)
where Xκ,α,β(t) is given by Xκ,α,β(t) = χ Re[ζκ, α, β(t)].
If the trajectories do not recombine, the trace over
ˆρκ,κ′,α,α′,β,β′(Ttot) goes to zero mainly because the wave
packets do not perfectly overlap. The overlap between
two thermal states, where one is displaced by ζi and the
other by ζj, is given by tr
�
ˆD(ζi)ˆρth ˆD†(ζj)
�
. To find an
expression for this overlap, we expand ˆρth in coherent
states
ˆρth =
�
dα2
π
e− |α|2
¯n
¯n
|α⟩ ⟨α|
(98) | [
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states
ˆρth =
�
dα2
π
e− |α|2
¯n
¯n
|α⟩ ⟨α|
(98)
where ¯n is the occupation number. To evaluate the trace,
we use the definition of a coherent state |α⟩ = ˆD(α) |0⟩
and the properties of the displacement operator.
One
then finds
tr
�
ˆD(ζi) |α⟩ ⟨α| ˆD†(ζj)
�
= eϕe∆ζα∗−∆ζ∗αe
−|∆ζ|2
2
(99)
with ∆ζ = ζi −ζj and ϕ = (ζ∗
j ζi −ζjζ∗
i )/2. All that’s left
is solving the integral to find
tr
�
ˆD(ζi)ˆρth ˆD†(ζj)
�
= eϕ−|∆ζ|2( 1
2 +¯n).
(100)
As the overlap goes with exp(−¯n), it goes to zero very | [
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] |
�
= eϕ−|∆ζ|2( 1
2 +¯n).
(100)
As the overlap goes with exp(−¯n), it goes to zero very
quickly for large occupation numbers.
E.
Spin Flips
The T1 time gives a time scale, at which a nuclear spin
statistically flips. In this subsection, we want to analyze
how such a flip affects the time evolution of the levitated
nanoparticle. For that we split the time evolution into
the time evolution before and after the spin flip. If the
spins are initially fully polarized, the state is given by
ˆρ(0) = 1
2N
N
�
κ=0
��N | [
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] |
spins are initially fully polarized, the state is given by
ˆρ(0) = 1
2N
N
�
κ=0
��N
κ
�
|κ⟩ ⊗ ˆρx.
(101)
Until time τ, the nanoparticle evolves undisturbed ac-
cording to the given protocol. The time evolution of the
κth branch is then given by
ˆUκ(τ) = e−iϕκ(τ) ˆD
� χ
2x0
ζκ(τ)
�
e−iτˆa†ˆa.
(102)
If one spin flips, a Dicke state transforms like
��N
κ
�
|κ⟩N →
��N − 1
κ − 1
�
|0⟩1 ⊗ |κ − 1⟩N−1
+
��N − 1
κ + 1
�
|1⟩1 ⊗ |κ + 1⟩N−1 .
(103)
Therefore, each branch in turn splits into an incoherent | [
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0.3926486372947693,
-0.8504429459571838,
-0.7490622401237488,
0.5518931746482849,
-0.4207073450088501,
-0.1343413144350052,
0.039270877838134766,
0.8402237296104431,
0.3553035259246826,
-0.2115289270877838,
0.6984860897064209,
-0.5932115316390991,
-0.004183194134384394,
-0.3707452714443207,
-0.17071112990379333,
-0.38568583130836487,
-0.34232375025749207,
0.191229909658432,
-0.1628188192844391,
0.4571453928947449,
-0.2579643130302429,
-0.1070786714553833,
-0.1907634139060974,
-0.10713086277246475,
0.4133152663707733,
-0.28045332431793213,
-0.09868965297937393,
-0.4350615441799164,
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0.07695416361093521,
0.5633119344711304,
0.2903272211551666,
0.44112861156463623,
-0.381828635931015,
0.05190584808588028,
-0.6261988878250122,
-0.01576055958867073,
0.3272465169429779,
-0.40462765097618103,
-0.33949148654937744,
0.28072696924209595,
0.4176003038883209,
0.19798946380615234,
-0.5013166069984436,
-0.06221778690814972,
0.5562595725059509,
0.4233356714248657,
-0.38435253500938416,
0.0934249609708786,
-0.5828997492790222,
-0.5999672412872314,
0.4867401421070099,
-0.32404395937919617,
-0.0884174257516861,
0.34740033745765686,
0.22171486914157867,
0.8850647211074829,
0.24198882281780243,
-0.29215797781944275,
-0.7236025333404541,
0.29514241218566895,
-0.07562949508428574,
-0.16410285234451294,
-0.7161365151405334,
-0.016883043572306633,
0.5030736327171326,
0.1798761636018753,
-0.5284065008163452,
0.37760448455810547,
0.14401091635227203,
0.33776235580444336,
0.026009976863861084,
-0.020458031445741653,
-0.22334566712379456,
0.510159969329834,
0.6175114512443542,
1.0613563060760498,
-0.7175177931785583,
-0.27879977226257324,
-0.1103668287396431,
0.15083928406238556,
0.7072283625602722,
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0.5185467004776001,
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0.19683589041233063,
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0.3550056517124176,
0.03602062910795212,
0.05908830836415291,
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0.19491179287433624,
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0.01974942721426487,
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0.17362800240516663,
0.6045083403587341,
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0.24091166257858276,
0.09653698652982712,
0.26688292622566223,
-0.6328612565994263,
-0.621271550655365,
0.07205628603696823,
0.31266042590141296,
0.1551753282546997,
-0.08028474450111389,
-0.1911236196756363,
0.43617555499076843,
0.6536049246788025,
-0.6351171731948853,
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0.32658520340919495,
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0.16177783906459808,
-0.24302566051483154,
0.9642619490623474,
0.42447468638420105,
0.3188574016094208,
0.1275777965784073,
0.38632717728614807,
0.06374052166938782,
1.0188584327697754,
-0.40027645230293274,
0.3676406443119049,
-0.659545361995697,
0.1788802593946457,
-0.7178484201431274,
-0.06560062617063522,
0.5424351692199707,
0.44892677664756775,
-0.16139163076877594,
0.1321869194507599,
0.20938020944595337,
-0.5986576080322266,
0.6008913516998291,
-0.33022281527519226,
-0.4988905191421509,
-0.01032947190105915,
-0.7334057688713074,
-0.791898250579834,
0.40191981196403503,
0.6589234471321106,
0.04681871831417084,
-0.23945088684558868,
0.5023869872093201,
0.08601441234350204,
-0.344535768032074,
0.15699392557144165,
-0.2347966730594635,
-0.5412766933441162,
-0.3106033504009247,
0.5359671115875244,
-0.4199904799461365,
0.0701628178358078,
-0.2525532841682434,
0.738549530506134,
0.31602731347084045,
-0.29967060685157776,
-0.16539950668811798,
0.19705025851726532,
-0.23134271800518036,
0.38753002882003784,
0.36820417642593384,
0.14058859646320343,
0.200896754860878,
-0.21890807151794434,
-0.26939132809638977,
0.18559400737285614,
-0.03775959834456444,
-0.18111838400363922,
-0.418464332818985,
0.05776926502585411,
-0.19618070125579834,
-0.40027663111686707,
-0.0507974699139595,
-0.1736951321363449,
0.1948157399892807,
-0.16076016426086426
] |
��N − 1
κ + 1
�
|1⟩1 ⊗ |κ + 1⟩N−1 .
(103)
Therefore, each branch in turn splits into an incoherent
superposition of two branches. One with κ − 1 spins up
and one with κ + 1 spins up. The time evolution for the
full protocol along the κth trajectory is therefore given
by
ˆU ±
κ (Ttot, τ) =e−iϕκ±1(Ttot−τ) ˆD
� χ
2x0
ζκ±1(Ttot − τ)
�
×e−i(Ttot−τ)ˆa†ˆa
×e−iϕκ(τ) ˆD
� χ
2x0
ζκ(τ)
�
e−iτˆa†ˆa.
(104)
As ζκ is linear in (2κ − N), we can write it as ζκ(t) =
(2κ − N)˜ζ(t). ζκ±1 can therefore be rewritten as ζκ±1 = | [
-0.08150320500135422,
-0.18830016255378723,
0.4411090910434723,
-0.2644558250904083,
-0.307790070772171,
0.04501819983124733,
0.38255205750465393,
-0.7669654488563538,
-0.11826034635305405,
-0.10205177217721939,
-0.10200528055429459,
-0.11744371056556702,
-0.14217011630535126,
0.34915992617607117,
0.06511381268501282,
0.3763132095336914,
0.9572122097015381,
0.16383780539035797,
-0.7710721492767334,
-0.963828980922699,
0.4095652997493744,
0.5408325791358948,
0.20041978359222412,
-0.05993104353547096,
-0.16847790777683258,
0.22203761339187622,
0.04217154160141945,
-0.43763092160224915,
0.5386742353439331,
0.07690990716218948,
-0.25113436579704285,
0.18166902661323547,
0.1809786856174469,
-0.16968297958374023,
0.5346691012382507,
-0.046306949108839035,
0.3949989378452301,
-0.045496318489313126,
0.47285374999046326,
-0.41140738129615784,
-0.0949675664305687,
-0.1072685495018959,
0.07691562175750732,
0.495034396648407,
-0.14578044414520264,
-0.4840558171272278,
-1.193211555480957,
0.11263041198253632,
0.17956310510635376,
-0.5530056357383728,
-0.25339940190315247,
-0.15446802973747253,
0.2560461163520813,
0.23988361656665802,
-0.2656763792037964,
0.38907790184020996,
-0.6981338858604431,
-0.30070629715919495,
0.6479308009147644,
0.04163449630141258,
-0.0009209309355355799,
-0.10170336812734604,
-0.018435180187225342,
0.4917322099208832,
-0.01617683656513691,
0.8959107995033264,
0.14874672889709473,
0.344707727432251,
-0.48147469758987427,
-0.025429753586649895,
0.44077619910240173,
-0.02206648699939251,
-0.12120800465345383,
0.15007352828979492,
0.15771791338920593,
0.3363334834575653,
-0.4014260470867157,
-0.19752560555934906,
0.2881839871406555,
0.08190412819385529,
-0.48301661014556885,
-0.2390976995229721,
0.00034620321821421385,
-0.30291420221328735,
0.591538667678833,
0.1951909065246582,
0.12423030287027359,
0.47300148010253906,
-0.7227609753608704,
0.5248075127601624,
-0.36832112073898315,
-0.38057029247283936,
0.12418454140424728,
0.26239103078842163,
-0.04848160967230797,
0.547601580619812,
-0.24778808653354645,
-0.33015334606170654,
0.6195017695426941,
0.14270278811454773,
0.021786123514175415,
-0.13885796070098877,
-0.3364560902118683,
-0.3561125695705414,
-0.24359287321567535,
-0.75971919298172,
-0.53569495677948,
-0.6217559576034546,
0.023616109043359756,
-0.11786231398582458,
-0.7734261751174927,
0.05208009108901024,
-0.2966017425060272,
-0.1832740157842636,
-0.10445412993431091,
0.3180030584335327,
-0.3025343120098114,
-0.7350318431854248,
0.11881063133478165,
0.1854846328496933,
-0.1542707234621048,
0.9470212459564209,
0.20702427625656128,
0.37547263503074646,
0.5742624402046204,
0.26308175921440125,
0.8150374293327332,
0.6249460577964783,
-0.42146921157836914,
-0.06787338107824326,
-0.05271852761507034,
0.9451877474784851,
0.1508101224899292,
0.06589889526367188,
-0.12492197751998901,
-0.5602965354919434,
-0.18371909856796265,
0.35374715924263,
-0.645156741142273,
0.23624034225940704,
0.1461484283208847,
-0.10419690608978271,
0.08047796040773392,
0.0019046711968258023,
0.25823551416397095,
0.14294417202472687,
-0.41191989183425903,
-0.08837009221315384,
-0.03851187229156494,
0.07882801443338394,
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0.18505795300006866,
-0.4301071763038635,
0.20911426842212677,
-0.4904147982597351,
-0.32540446519851685,
-0.33000996708869934,
0.536916196346283,
-0.4516337811946869,
0.11452995240688324,
-0.03739280253648758,
-0.17769111692905426,
0.05687011778354645,
-0.2662647068500519,
0.45071175694465637,
-0.19659185409545898,
0.8053215742111206,
0.08249428868293762,
-0.05829853564500809,
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-0.3326667845249176,
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0.08184991776943207,
0.23753926157951355,
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-0.8285101652145386,
-0.08782795071601868,
-0.4189956486225128,
-0.6064596772193909,
0.04024677351117134,
0.559880793094635,
0.050760071724653244,
0.40974292159080505,
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0.5537672638893127,
-0.16989389061927795,
0.41179370880126953,
0.41177091002464294,
-0.8549562692642212,
1.2910140752792358,
0.035344693809747696,
-0.25961780548095703,
-0.02181403338909149,
-0.12429384142160416,
0.10889144986867905,
-0.31605276465415955,
0.30891409516334534,
0.16194815933704376,
0.01942266710102558,
-0.14288794994354248,
0.43149110674858093,
0.11816371232271194,
-0.17084455490112305,
-0.2866901755332947,
-0.06184890866279602,
0.10849033296108246,
-0.3178754448890686,
-0.0362735390663147,
0.09755226224660873,
-0.11775700747966766,
-0.09214900434017181,
0.008899844251573086,
0.024060193449258804,
0.46791714429855347,
0.35264110565185547,
0.7224453687667847,
-0.149997279047966,
0.49638232588768005,
0.3349005877971649,
0.10716500133275986,
0.9597125053405762,
0.6893048882484436,
0.40095624327659607,
0.4872182309627533,
-0.4002968668937683,
-0.33311691880226135,
-0.5316331386566162,
0.2944244146347046,
0.1987159103155136,
0.09704703092575073,
0.1325571984052658,
-0.020226886495947838,
-0.3387894034385681,
0.515893816947937,
-0.3165934085845947,
0.17338263988494873,
0.7534463405609131,
-0.5615197420120239,
0.5972330570220947,
0.022509809583425522,
0.14797617495059967,
-0.2603500783443451,
-0.057611752301454544,
-0.30241817235946655,
-0.22914929687976837,
-0.36726850271224976,
0.2491295039653778,
0.10899803042411804,
-0.9999310970306396,
-0.3831291198730469,
-0.2421470284461975,
0.03128768876194954,
-0.13953909277915955,
0.03786235675215721,
0.24258078634738922,
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0.13103803992271423,
-0.13090239465236664,
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-1.3247653245925903,
-0.48499029874801636,
1.0258148908615112,
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-0.6315315365791321,
0.03751618415117264,
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] |
(2κ − N)˜ζ(t). ζκ±1 can therefore be rewritten as ζκ±1 =
ζκ(t) ± 2˜ζ. The effect of the spin-flip can be separated
24
from the undisturbed time evolution, by rewriting the
displacement operator like
ˆD
� χ
2x0
ζκ±1(Ttot − τ)
�
= ˆD
�
± χ
x0
˜ζ(Ttot − τ)
�
× ˆD
� χ
2x0
ζκ(Ttot − τ)
�
.
(105)
We therefore find
ˆU ±
κ (Ttot, τ) =e−i ˜ϕκ(Ttot,τ) ˆD
�
± χ
x0
˜ζ(Ttot − τ)
�
×e−iϕκ(Ttot) ˆD
� χ
2x0
ζκ(Ttot)
�
e−iTtotˆa†ˆa.
(106)
The second line describes the unperturbed evolution, | [
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] |
� χ
2x0
ζκ(Ttot)
�
e−iTtotˆa†ˆa.
(106)
The second line describes the unperturbed evolution,
while the first line describes the effect of the flip. The
flip induces an additional phase ˜ϕκ(Ttot, τ) = ϕκ±1(Ttot−
τ) − ϕκ(Ttot − τ). The protocol is designed such that,
if the initial state is a thermal state, all trajectories
recombine in the center.
The displacement operator
in the first line indicates that, when a flip happens,
the trajectories either recombine in ˆD( χ
x0 ˜ζ(Ttot − τ))
and or in
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] |
the trajectories either recombine in ˆD( χ
x0 ˜ζ(Ttot − τ))
and or in
ˆD(− χ
x0 ˜ζ(Ttot − τ)), each with a probabil-
ity of 50 %.
The distance in phase space is given by
|∆ζ(τ)| = 2χ/x0|˜ζ(Ttot − τ)|. For the first protocol de-
scribed here, |˜ζ(Ttot −τ)| is bounded by two. This bound
is realized, if the flip happens while the wave function is
maximally spread at τ = π/Ω. The maximal displace-
ment is therefore given by |∆ζ(τ)| < 4χ/x0.
[1] K. G. Libbrecht and E. D. Black, Toward quantum- | [
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0.309251993894577,
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0.7535110712051392,
0.01565096341073513,
0.11256493628025055,
-0.05201324075460434,
-0.2592352330684662,
-0.9110783934593201,
0.4201430082321167,
0.06444419175386429,
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0.37164363265037537
] |
ment is therefore given by |∆ζ(τ)| < 4χ/x0.
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] |
milled from polycrystalline 12c diamond, Physical Re-
view Applied 20, 044045 (2023).
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] |
states, Nuclear Instruments and Methods in Physics Re-
search Section A 678, 91 (2012).
[43] T. R. Eichhorn, A. J. Parker, F. Josten, C. M¨uller,
J. Scheuer, J. M. Steiner, M. Gierse, J. Handwerker,
M. Keim, S. Lucas, M. U. Qureshi, A. Marshall, A. Sal-
hov, Y. Quan, J. Binder, K. D. Jahnke, P. Neumann,
S. Knecht, J. W. Blanchard, M. B. Plenio, F. Jelezko,
L. Emsley, C. C. Vassiliou, P. Hautle, and I. Schwartz,
Hyperpolarized solution-state nmr spectroscopy with op- | [
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] |
K. Bhimani, E. Blalock, B. Bos, M. Busch, M. Buuck,
T. Caldwell, Y.-D. Chan, C. Christofferson, P.-H. Chu,
M. Clark, C. Cuesta, J. Detwiler, Y. Efremenko, H. Ejiri,
S. Elliott, G. Giovanetti, M. Green, J. Gruszko, I. Guinn,
V. Guiseppe, C. Haufe, R. Henning, D. Hervas Aguilar,
E. Hoppe, A. Hostiuc, I. Kim, R. Kouzes, T. Lannen V.,
A. Li, A. Lopez, J. L´opez-Casta˜no, E. Martin, R. Martin,
R. Massarczyk, S. Meijer, T. Oli, G. Othman, L. Paudel,
W. Pettus, A. Poon, D. Radford, A. Reine, K. Rielage, | [
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] |
tion by microwave-induced optical nuclear polarization
at 77 k, Physical Review Letters 84, 171 (2000). | [
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