Weak values could reveal the hidden effects of quantum interactions

0.15 MB·English

by  Miao Zhang

Weak values could reveal the hidden effects of quantum interactions
Miao Zhang 1
∗
1Department of Physics, Southwest Jiaotong University, Chengdu 610031, China
(Dated: January 23,2015)
Abstract
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Due to the reduced probability of successful post-selection, the weak-value amplification seems to be
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unavailable for the parameter-estimation. Here, we show theoretically that, some effects due to the weak

n
a interactions present only in the properly post-selected sub-ensemble, however are canceled by themselves
J

2 in the total ensemble. From this point of view, the post-selection induced weak value could be one of the
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feasible methods for measuring the weak interaction, since the standard measurement does not work. Ad-
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p ditionally, weemploy the system of trapped ions tosimulate the weakmeasurement andcalculate relevant
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n resultswithoutthefrequently-used weakinteraction approximation.
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∗ [email protected]
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I. INTRODUCTION
The conception of quantum weak measurements was noticed by Aharonov, Albert, and Vaid-
man(AAV)in1988[1]. Theweakmeasurementshavetwokeyfeatures: thecouplingsbetweenthe
quantumsystemsaresufficientlyweak,andtherelevantobservablequantities(act asthepointers)
are measured in the properly post-selected sub-ensemble. The outcome of weak measurements is
theso-calledweak value
hψ |Aˆ|ψ i
f i
A = , (1)
w
hψ |ψ i
f i
where|ψ i,|ψ i,and Aˆare theinitialstate, post-selectedstate,and theobservableoperatorofthe
i f
measuredsystem,respectively. ComparingtotheusualquantummeasurementA = hψ |Aˆ|ψ i,the
i i
weakvalueA couldgeneratesomeinterestingresults[1–5],asthepost-selection|ψ iisinduced.
w f
Recently, the weak value has been employed to study the foundational questions of quantum
mechanics, e.g., the Hardy’s paradox [6], the Leggett-Garg inequality [7], the Heisenberg’s un-
certainty relation [8], the Cheshire Cat [9], and the superluminal velocities [10]. Much attentions
are also paid on thepractical applicationsof the weak value, e.g., detecting tinyvalues of a quan-
tity [11]. Considering an observable quantity depends on g|A | (with g being a small parame-
w
ter) [12], then the weak value A can significantly change the pointer with |hψ |ψ i| → 0, or
w f i
say amplifyingthe weak signal of g [13–19]. However, serval recent papers claim this amplifica-
tion offering no fundamental metrological advantage, due to the necessarily reduced probability
ofsuccessfulpost-selection[20–23], i.e.,|hψ |ψ i|2 → 0.
f i
In this work, we recommend an useful weak value, which could reveal the hidden effects of
the weak quantum interactions. Here the hidden effects mean that, the g-induced changes of
someobservablequantitiescan notbedetected bytheusualquantummeasurement(withthetotal
ensemble). However,thesechangescanbeconditionallymeasuredbytheweakvaluetechnique. If
thesespecial observablequantitiesare thebetterones forexperimentallydetecting,then theweak
value technique could be considered as an effective candidate for measuring the small parameter
g of weak interaction. Moreover, we prove the relevant results by a simple example which is
calculated without the frequently-used weak interaction approximation, and employ the powerful
experimental-systemoftrapped ionstosimulatetheweak measurements.
This work is organized as follows. In Sec. II, we show that the usual measurement can not
detect some special effects of the weak quantum interactions. In Sec. III, we show the condi-
tionalmeasurementscouldrevealthesehiddeneffects,andcalculateanexamplewithouttheusual
2
weak interaction approximation. In Sec. IV, we employ the trapped ions to simulate the weak
measurement,and finallygiveaconclusioninSec. V.
II. THEVANISHEDEFFECTSOFONE-ORDERg
Weconsiderthestandardvon NeumannmeasurementHamiltonian
Hˆ = ~g AˆPˆ. (2)
0
Where,~isthePlanckconstantdividedby2π,AˆandPˆ arerespectivelytheoperatorsofthequbit
and the pointer (with the coupling frequency g between them). After the above interaction, the
0
final stateofthetotalsystemcan beformallywrittenas
|ψ i = e igAˆPˆ|S i|φ i. (3)
f − 0 0
Where, g = g t (with the interaction duration t) describes the coupling strength between the
0
pointer and the qubit, |S i and |φ i are respectively the initial states of the qubit and the pointer.
0 0
Consideringtheinteractionissufficientlyweak, i.e.,g → 0,thenthefinal state(3)can beapprox-
imatelywrittenas
|ψ i ≈ |S i|φ i−igAˆPˆ|S i|φ i. (4)
f 0 0 0 0
by neglectingthehighorderofO(g2).
Directly,theprobabilityoffindingthepointerstate|xireads
2
I = |hx|ψ i|
f
≈ |hx|φ i|2 −ig hφ |xihx|Pˆ|φ ihS |Aˆ|S i−c.c
0 (cid:16) 0 0 0 0 (cid:17)
(5)
= I 1−ig(P hS |Aˆ|S i−c.c)
0h w 0 0 i
= I 1+2gIm(P hS |Aˆ|S i)
0h w 0 0 i
2
withI = |hx|φ i| and
0 0
hx|Pˆ|φ i
0
P = . (6)
w
hx|φ i
0
3
Above, c.c is theconjugatecomplex numberof the first term in the bracket, and the high order of
O(g2) hasbeen neglected again. Thisapproximationshowsthat: when
P hS |Aˆ|S i = Real, (7)
w 0 0
thereisnoeffectduetotheone-orderg. Thus,wesaythat: theobservableI isinsensitivetothepa-
rameterg,andtheweak-valueamplificationmaybeuseful(whenI isfavorableforexperimentally
detecting).
Worth of note that, if |xi is the state of atoms at position x, the observable I can be regarded
as the “lightness” of atoms beam. This is similar to the light intensity, which is proportional
to |E~(x)|2 of the Maxwell field E~(x). Specially, we consider the two-level transition operator
Aˆ = σˆ = |eihg|+|gihe| and the initial state |S i = α|gi+β|ei of the qubit. Here, α and β are
x 0
thenormalizedcoefficients of thetwo orthogonalstates |giand |ei. When P = Real, theEq. (5)
w
can berewrittenas
I = I [1−igP (α β +β α−c.c)]
0 w ∗ ∗
= I [1+2gP Im(α β)−2gP Im(α β)]
0 w ∗ w ∗
(8)
= I +I −I
0 g g
= I
0
with I = 2gP Im(α β)I . This result shows that, under the condition of (7), the g-induced
g w ∗ 0
effects areaccurately canceled by themselves.
III. THEWEAKMEASUREMENTS
With the post-selected state |S i of the qubit, the probability of finding the pointer state |xi
f
reads:
2
I = |hx|hS |ψ i|
s f f
2
ˆ ˆ
≈ hS |S ihx|φ i−ighS |A|S ihx|P|φ i
(cid:12) f 0 0 f 0 0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) (9)
2
≈ I ×|hS |S i| ×[1−igP (A −c.c)]
0 f 0 w w
2
= I ×|hS |S i| ×[1+2gP Im(A )]
0 f 0 w w
withtheweak value
hS |Aˆ|S i
f 0
A = . (10)
w
hS |S i
f 0
4
FIG. 1: (Color online) Sketch for the observable I0 = |hx|φ0i|2 affected by the one-order g. Under the
situation of Eq. (7), there is no effect of g with the usual measurement (the dark line). However, in the
properly post-selected sub-ensemble (theredorblueline),theg-induced effect(i.e.,I )appears.
g
The Eq. (9), with A 6= Real, means that the effect due to the one-order g presents now in the
w
post-selected sub-ensemble of |S i. On the other hand, within the remaining sub-ensemble of
f
|Sfi,wehavethesimilarresult: theprobabilityoffinding|xi reads
2
Is = I × hSf|S i ×[1+2gP Im(Aw)] (11)
0 0 w
(cid:12) (cid:12)
(cid:12) (cid:12)
withthecorrespondingweak value
hSf|Aˆ|S i
Aw = 0 . (12)
hSf|S i
0
Considering|S i = |giand |Sfi = |ei,theEq.(9)and (11)can befurtherwrittenas
f
I = I × |α|2 +2gP Im(α β) , (13)
s 0 w ∗
(cid:2) (cid:3)
Is = I × |β|2−2gP Im(α β) (14)
0 w ∗
(cid:2) (cid:3)
and thenthetotalprobabilityoffindingthepointerstate|xireads
I +Is = I ×[1+2gP Im(α β)−2gP Im(α β)]
s 0 w ∗ w ∗
= I +I −I (15)
0 g g
= I
0
This result is same to that of Eq. (8): the g-induced effects vanished from the usual measure-
ment with the total ensemble. However, the Eq. (13) and (14) show that, the g-induced changing
of observable I presents magically in the properly post-selected sub-ensemble (see the dramatic
0
Fig. 1). Moreover, it is easily to calculate the maximal Im(α β) = 1/2 (according to the normal-
∗
izedcondition|α|2+|β|2 = 1). Thus,theEq.(13)and(14)reducefurthertoI = I ×(1 +gP )
s 0 2 w
and Is = I ×(1 −gP ),respectively.
0 2 w
5
WenowcalculateanexamplewithouttheweakinteractionapproximationpreformedinEq.(4).
ConsideringPˆ = aˆ aˆ and Aˆ = σˆ , theHamiltonian(2)reads
† x
Hˆ = ~g aˆ aˆσˆ , (16)
0 † x
with aˆ and aˆ being respectively the creation and annihilation operators in the Fock state presen-
†
tation [24]. Without of generality, we write the initial state of the pointer as |φ i = c |ni,
0 n n
P
i.e., a superposed Fock state with the occupancy number n ≥ 0 and the normalized coefficients
|c |2 = 1. Consequently,theinitialstateofthetotalsystemreads
n n
P
|ψ i = |φ i(α|gi+β|ei) = c (α|gi+β|ei)|ni (17)
0 0 X n
n
and thefinal statereads
|ψ i = e igaˆ†aˆσˆx|ψ i = c (αe igaˆ†aˆσˆx|gi|ni+βe igaˆ†aˆσˆx|ei|ni)
f − 0 n n − −
P
= c αe ignσˆx|gi+βe ignσˆx|ei |ni
Pn n(cid:0) − − (cid:1) (18)
= c {α[cos(gn)|gi−isin(gn)|ei]+β[cos(gn)|ei−isin(gn)|gi]}|ni
n n
P
= c (η |gi+η |ei)|ni
n n gn en
P
withη = αcos(gn)−iβsin(gn)and η = βcos(gn)−iαsin(gn). Immediately,wehave
gn en
|η |2 = |α|2cos2(gn)+|β|2sin2(gn)−icos(gn)sin(gn)(βα −αβ )
 gn ∗ ∗

|η |2 = |β|2cos2(gn)+|α|2sin2(gn)+icos(gn)sin(gn)(βα −αβ ) (19)
 en ∗ ∗
2 2
|η | +|η | = 1
 gn en



and thefollowingresults.
(I)Fortheusual measurementofFockstate|mi,wehave
|hm|ψ i|2 = |c |2|(η |gi+η |ei)|2 = I (20)
f m gm em m
with|c |2 = I .
m m
(II)Fortheconditionalmeasurement,wehave
|hm|hg|ψ i|2 = I |η |2 ≈ I [|α|2 +2gmIm(α β)], (21)
f m gm m ∗
|hm|he|ψ i|2 = I |η |2 ≈ I [|β|2−2gmIm(α β)], (22)
f m em m ∗
Here, weconsidergm → 0andneglectthehighorderofO(g2). Thus,theresultsare sametothat
2 2 2
ofEq.(13)and (14)withP = m, and |hm|hg|ψ i| +|hm|he|ψ i| = |hm|ψ i| = I .
w f f f m
6
FIG.2: (Color online) Sketch for the considered model to implement the weak measurement with trapped
ions. Here, twoions are confined in a single harmonic potential and manipulated by the laser pulses. Two
internal levels ofion 1act asthe quantum system (qubit), thecenter-of-mass (CM)motion of the twoions
servesasthepointer. Thelaser1isapplied torealizethedesirable weakinteraction (16)betweenthequbit
and pointer, and the laser 2 couples the pointer (i.e., the CM mode) to the atomic levels of ion 2 for final
readout(bytheelectron shelvingmethod)[28].
IV. SIMULATINGWEAKMEASUREMENTINTHEION-TRAP
To simulate the above weak measurement, we consider two ions trapped in a linear Paul
trap [25–28], as it showing dramatically in Fig. 2. The two internal atomic states of ion 1 is
encoded as the qubit. By the so-called sideband excitation [26, 27], this qubit can be coupled
to the center-of-mass (CM) motion of the two vibrational ions, and consequently the desirable
Hamiltonian of (16) could be realized. The ion 2, manipulated by the lase beam 2, is used to ex-
perimentallymeasuretheCMmode. UndertheLamb-Dickeapproximationandtherotatingwave
approximation, the first-red-sideband excitation (generated by the laser pulse 1) is described by
thewell-knownHamiltonian[28]
Hˆ = ~Ω eiδtaˆτˆ +e iδtaˆ τˆ . (23)
i 0 + − †
(cid:0) −(cid:1)
Here, aˆ and aˆ are respectively the boson creation and annihilation operators of the CM mode,
†
τˆ = | ↑ih↓ | and τˆ = | ↓ih↑ | are the two-level flip operators of ion 1 (with the internal
+
−
states | ↓i and | ↑i), and Ω is the Rabi frequency of the sideband coupling, and δ is the relating
0
detuning (decided by the frequencies of the applied laser beams). When δ = 0, the Hamiltonian
(23)describesastandardJaynes-Cummings(JC)couplingbetweentheinternalandexternalstates
oftrapped ions.
Under the large detuning: Ω ≪ δ, the time-evolution operator of Hamiltonian (23) can be
7
approximatelywritten as:
Uˆ(t) = 1+(cid:0)−~i(cid:1)R0tHˆi(t1)dt1 +(cid:0)−~i(cid:1)2R0tHˆi(t1)R0t1 Hˆi(t2)dt2dt1 +··· ≈ e−~iHˆet, (24)
with the effective Hamiltonian Hˆ = (~Ω2/δ)[aˆ aˆ(| ↑ih↑ | − | ↓ih↓ |) + | ↑ih↑ |] [29]. In the
e 0 †
interactionpicturedefinedbyUˆ = exp(−ig t| ↑ih↑ |),thisHamiltoniancanbefurtherwrittenas
R 0
Hˆ = ~g aˆ aˆτˆ (25)
e′ 0 † z
with τˆ = | ↑ih↑ |−| ↓ih↓ | and g = Ω2/δ. We define |ei = 1 (| ↑i+| ↓i) and |gi = 1 (| ↑
z 0 0 √2 √2
i−| ↓i) being the qubit of ion 1. These states satisfy the orthogonal condition he|gi = 0 and the
normalized condition he|ei = hg|gi = 1. As τˆ |ei = |gi = σˆ |ei and τˆ |gi = |ei = σˆ |gi, the
z x z x
Hamiltonian(25)canbewrittenasthedesirableformHˆ = ~g aˆ aˆσˆ ofEq.(16)andconsequently
0 † x
get thesameresultsto(18).
Experimentally, the internal atomic states of the trapped ions can be well measured by serval
methods such as the electron shelving technique [28]. This method can be briefly described as
follows. Applying a strong laser resonantly drives the transition between the state | ↑i and an
auxiliary atomic state |auxi. If the ion is in state | ↑i, the laser-induced resonance fluorescence
canbequicklydetected,whereasthereisnosignalasthetransition| ↓i ⇔ |auxiislargedetuning.
Here, the post-selection (i.e., the detection of state |ei or |gi) needs additional operations, since
these states are the superposition ones of | ↑i and | ↓i. We apply a single-qubit operation to the
ion 1, i.e., Uˆ = exp(−iHˆ t/~) with theHamiltonian Hˆ = ~Ω (e iθσˆ +eiθσˆ ). By setting the
s s s s − +
−
parameters Ω t = π/4 and θ = −π/2, we have Uˆ |ei = | ↑i and Uˆ |gi = −| ↓i. Consequently,
s s s
wehave
|ψ i = Uˆ |ψ i = c η Uˆ |gi+η Uˆ |ei |ni = c (η | ↑i−η | ↓i)|ni (26)
f′ s f X n(cid:16) gn s en s (cid:17) X n en gn
n n
and thesameresultsto(20)∼ (22), i.e.,
2
hm|ψ i = I ,
(cid:12) f′ (cid:12) m
(cid:12)hm|h↓ |ψ(cid:12) i 2 ≈ I [|α|2+2gmIm(α β)], (27)
(cid:12) f′ (cid:12) m ∗
(cid:12)hm|h↑ |ψ i(cid:12)2 ≈ I [|β|2−2gmIm(α β)].
(cid:12) f′ (cid:12) m ∗

(cid:12) (cid:12)
As that demonstrated in many experiments, the external motional states of trapped ions can
bemappedto theirinternal statesby thelasermanipulations[26–28], and measured consequently
by the electron shelving technique. For simplicity, we suppose that the CM mode is prepared
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1
initially in the superposition state |φ i = c |ni = c |0i + c |1i (with the normalized
0 n=0 n 0 1
P
coefficients |c |2 + |c |2 = 1), and the internal state of ion 2 is initially in | ↓ i. We apply a
0 1 r
first-red-sideband laser pulse to implement the operation Uˆ = exp(−iHˆ t/~) on ion 2. Here,
r r
Hˆ = ~Ω aˆτˆ +aˆ τˆ is theJC couplingbetween the CM modeand the internal states ofion
r r 2+ † 2
(cid:0) −(cid:1)
2, with the two-level flip operators τˆ = | ↑ ih↓ | and τˆ = | ↓ ih↑ |. When Ω t = π/2, this
2+ r r 2 r r r
−
operation generates the logic: |0i| ↓ i → |0i| ↓ i, |1i| ↓ i → (−i)|0i| ↑ i. Consequently, we
r r r r
have
2 2
h↑ |ψ i = hm|ψ i = I ,
(cid:12) r f′′ (cid:12) (cid:12) f′ (cid:12) m
(cid:12)h↑ |h↓ |ψ(cid:12) i 2(cid:12)= hm|h(cid:12)↓ |ψ i 2 ≈ I [|α|2 +2gmIm(α β)], (28)
(cid:12) r f′′ (cid:12) (cid:12) f′ (cid:12) m ∗
(cid:12)h↑ |h↑ |ψ i(cid:12)2 = (cid:12)hm|h↑ |ψ i(cid:12)2 ≈ I [|β|2 −2gmIm(α β)].
(cid:12) r f′′ (cid:12) (cid:12) f′ (cid:12) m ∗

(cid:12) (cid:12) (cid:12) (cid:12)
with |ψ i = Uˆ |ψ i| ↓ i and m = 1. These results are same to that of Eq. (27). Worth of
f′′ r f′ r
note that, the above operations (i.e., Uˆ , Uˆ , and the preparation and readout of the initial and
s r
final states) are that frequently used in the system of laser-manipulated trapped ions. Thus, the
above simulation of weak measurements could be experimentally feasible. Indeed, the quantum
entanglement with eight ions has been successfully demonstrated (need more operations) in this
powerfulexperimentalplatform[30].
V. CONCLUSION
In this study, we present a very simple version to explain the usefulness of weak values. We
showedthat theweak valueswork well underthecertain conditions,e.g., Eq. (7). Withthisequa-
tion,someeffectsoftheweakinteractionspresentonlyintheproperlypost-selectedsub-ensemble,
howeverarecanceledbythemselvesinthetotalensemble. Fromthispointofview,theweakvalue
can be regarded as one of the effective methods for measuring the weak interaction, as the usual
measurement does not work. We presented an example to exactly calculate the relevant results.
These results prove that the frequently-used weak interaction approximation is effective indeed.
Additionally, we showed that the famous experimental-system of trapped ions could also be uti-
lized to simulate the weak measurements. Finally, we hope this paper could help understanding
theweak valuesandmovingtherelevantresearches forward.
Acknowledgements: ThisworkwassupportedbytheNationalNaturalScienceFoundationof
ChinaGrantsNo. 11204249.
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