Table Of ContentNoise-Free Measurement of Harmonic Oscillators with Instantaneous Interactions
M. Franc¸a Santos,1,∗ G. Giedke,2,† and E. Solano2,3,4,‡
1Dep. de F´ısica, Universidade Federal de Minas Gerais, Belo Horizonte, 30161-970, MG, Brazil
2Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
3Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany
4Secci´on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´olica del Peru´, Apartado Postal 1761, Lima, Peru
(Dated: February 1, 2008)
Wepresent a method of measuring thequantumstate of a harmonic oscillator through instanta-
neousprobe-systemselectiveinteractionsoftheJaynes-Cummingstype. Weprovethatthisscheme
is robust to general decoherence mechanisms, allowing the possibility of measuring fast-decaying
systems in the weak-coupling regime. This method could be applied to different setups: motional
states of trapped ions, microwave fields in cavity/circuit QED, and even intra-cavity optical fields.
7
0 PACSnumbers: numbers: 03.65.Wj,42.50.Dv,42.50.Vk
0
2
Measuring the quantum state of a harmonic oscilla- placements, the associated Wigner and Q functions at
n
tor, or, equivalently, its associated Wigner function [1], any point in phase space with arbitrarily small influence
a
J isafundamentaltaskofquantumphysics. Somepropos- of decoherence. From this data the full Wigner func-
8 als allow a direct measurement, like propagating optical tion can be reconstructed by a simple fit as in [6] or by
fields tested with homodyning techniques [2]. Others, more sophisticated techniques [16], e.g., involving maxi-
2 duetoproblemsofaccessibility,requireindirectmeasure- mum likelihood estimation [17] and taking into account
v
ment schemes via interaction with a probe. This is the the imperfections of the measurement process [2].
4
case of microwave fields in 3D cavities [3], circuit cavity Typically, a selective interaction can be built when a
0
0 QED with superconducting qubits [4], or the motion of three-level probe, driven by a classical and a quantized
6 trappedions[5]. Differentastheyare,knowntechniques field, is reduced to two metastable states after adiabatic
0 share a common problem: the noisy action of decoher- elimination of the third level, allowed by a large detun-
6 ence due to the probe-system finite interaction times. ing ∆. The associated Hamiltonian reads [12, 13]
0
h/ miRcreocwenatvley,ctahveityrecQoEnDstru(CctQioEnDo)f waaWs siguncecresfsufunlcltyiorneailn- Hˆ = ~Ω21 g g +~Ω22aˆ†aˆe e
p eff ∆| ih | ∆ | ih |
ized [6] with the aid of a dispersive probe-system inter-
- Ω Ω
nt action[7]. Unfortunately,dispersivecouplingisknownto +~ 1 2 g eaˆ†+ e g aˆ . (1)
a beslowandtherequiredinteractiontimeallowsdecoher- ∆ (cid:0)| ih | | ih | (cid:1)
u ence processes to disturb the measurement. In Ref. [8], Here, g , e are the (metastable) ground and excited
q a resonantmethod was proposed,but the Wigner recon- states{o|ftihe| tiw}o-levelprobe, aˆ,aˆ† aretheharmonicos-
v: struction depends on the possibility of monitoring a few cillatorannihilationandcreat{iono}perators,respectively,
Xi Rabicycles,addinguptolongprobeobservationperiods. and Ω Ω1Ω2/∆ is the effective JC coupling strength.
In the case of trapped ions, the measurement techniques The firs≡t and second terms on the r.h.s. of Eq. (1) are
r
a arequitesimilarandarecentexperiment[9]madeuseof AC-Stark shifts associatedwith eachof the probe levels,
numericalintegrationoverseveralRabi cycles to achieve the second one depending on the number of oscillator
the goal. These long interaction times are particularly excitations. This means that any effort at tuning the
harmful in the case of fast-decaying systems. For exam- JC coupling to resonancewillsucceed only for a selected
ple, state reconstruction of intracavity optical fields has JC doublet : g N , e N 1 , leaving all other
N
not been experimentally attempted, to our knowledge, doublets, forHwhic{h|ni|=iN| ,i|slig−htlyi}or completely off-
due to their weak coupling with atomic probes. resonance. It can be sho6 wn that, under proper tuning of
In this paper, we present a method to measure the the excitation fields, the condition Ω Ω √N assures
2 1
≫
quantumstateofaharmonicoscillatorthroughinstanta- neat selectivity in the JC model, where resonant Rabi
neous probe-system interactions [10, 11], preventing de- oscillations will happen only inside the subspace . In
N
H
coherence from disturbing the measurement. The har- this case, Eq. (1) turns into the selective Hamiltonian
monic oscillator is allowed to interact with a two-level
probeforanarbitrarilyshorttimeviaaselectiveinterac- HN=~√NΩ g e N N 1 + e g N 1 N ,
| ih |⊗| ih − | | ih |⊗| − ih |
tion[12,13,14]intheJaynes-Cummings(JC)model[15]. (cid:0) (cid:1) (2)
b
The information is then collected from the second time
derivative of the probe population at zero interaction describing the flip-flop interaction of two effective spin-
time. The scheme permits to measure the population 1/2systems, g , e and N 1 , N . Specificimple-
{| i | i} {| − i | i}
field distribution and, with the support of coherent dis- mentationshavebeenproposedinmicrowaveCQED[12]
2
andintrappedions[13],butothersystems,likeanatom Lindblad environment. For this particular time, choos-
inside an optical cavity, or a superconducting qubit cou- ing a probe operatorBˆ = e e and probe-systeminitial
| ih |
pled to a coplanar waveguide resonator,can also enjoy a state ρ(0)= g g ρ , we obtain
f
| ih |⊗
similar behavior.
dP (τ)
Fromtheunitarytimeevolutionofthetotaldensityop- e =0, (7)
erator, ρˆ˙ = [HN,ρˆ]/i~, the first and second time deriva- dτ (cid:12)(cid:12)τ=0
(cid:12)
tpirvoebseoofpetrhaetoberxBpˆecctaantiboenexvaplrueesseodf aas time-independent withdimensionlesstimeτ(cid:12)≡ΩtandPe(τ)≡h|eihe|i. For
the second derivative, we obtain
d Bˆ 1 d2 Bˆ 1
hdti = i~h[Bˆ,HN]i, (3) dht2i =(i~)2h [Bˆ,HˆN],HˆN i+Tr(cid:20) κm(AˆmρˆAˆ†m
(cid:2) (cid:3) Xm
d2 Bˆ 1 b
dht2i = (i~)2h(cid:2)[Bˆ,HˆN],HˆN(cid:3)i. (4) −21Aˆ†mAˆmρˆ− 21ρˆAˆ†mAˆm)[Bˆ,HˆN](cid:21). (8)
Westudyhereamoregeneralcase,allowingdecoherence This is a different situation and the trace will not van-
in the (field) systembut disregardingdecoherence in the ish in general, as before. However, considering again
probe. We do that based on the fact that most physical the observable Bˆ = e e and probe-system initial state
| ih |
setups use probes with long lifetimes compared to the ρ(0) = g g ρ , it follows that the second term in
f
| ih | ⊗
ones of the systems to measure. Under this assumption, Eq. (8) vanishes again, and we have
we consider the most general master equation ρˆ˙ = ρˆin
L d2P (τ)
the Lindblad form [18], e
=P . (9)
dτ2 (cid:12) N
(cid:12)τ=0
1 (cid:12)
ρˆ˙ =i~hHˆN,ρˆi Here, PN = Tr[ρf|NihN(cid:12)|] is the probability of finding
Fock state N in the initially unknown harmonic oscil-
1 1
+ κm(AˆmρˆAˆ†m−2Aˆ†mAˆmρˆ−2ρˆAˆ†mAˆm), (5) lator state.| i
Xm Equation (9) describes a remarkable result, it shows
that the curvature of the function P (τ), at vanishing
whereκ expressdecayratesandLindbladoperatorsA e
m m τ =0, contains undisturbed information about P . This
and A† are associated with the (field) system. N
m valuablefieldinformationisencodedcorrectlyinthelevel
We calculate first
statisticsofthetwo-levelprobeeveninpresenceofafield
d Bˆ 1 reservoir of a general kind. Needless to say, all previous
hdti =i~h[Bˆ,HˆN]i results hold when the usual thermal bath is considered
as a reservoir for the harmonic oscillator. At zero tem-
1 1
+Tr κ (Aˆ ρˆAˆ† Aˆ† Aˆ ρˆ ρˆAˆ† Aˆ )Bˆ . perature, for example, Lindblad operators Aˆ and Aˆ†
(cid:20) m m m−2 m m −2 m m (cid:21) m m
Xm would have to be replacedby aˆ and aˆ†, respectively,and
κwouldrepresentthedecayrateofthesinglefieldmode.
Thesecondtermonther.h.s. canberewrittenunderthe
The counter-intuitive results of (6) and (9) are of an in-
trace as
finitesimal nature and, without harming their theoreti-
calimportance,shouldsufferhigh-ordercorrectionswhen
1
Tr2(cid:20)Xm κmρˆ(cid:16)[Aˆ†m,Bˆ]Aˆm+Aˆ†m[Bˆ,Aˆm](cid:17)(cid:21) dweilallibnegewxiptlhaianeddislcarteetre.sampling of interaction times, as
Inorderto measurethe completefieldpopulation, P ,
and, since [Bˆ,Aˆ† ]=[Bˆ,Aˆ ]=0, we have n
m m n, one just needs to tune resonantly the other selected
∀
subspaces and follow a similar procedure. The mea-
d Bˆ 1 Hn
h i = [Bˆ,Hˆ ] . (6) surement of all Pn allows the estimation of the complete
dt i~h N i Wigner function W(α) of ρ, conditioned to the realiza-
tion of previous arbitrary field displacements D( α) in
In this expression, see similarity with Eq. (3), the dy- −
phase space. For that, we haveto recallthat the Wigner
namicsofprobeoperatorexpectationvalue Bˆ doesnot
h i function can be expressed [1], among other possibilities,
seem affected by the system decay. This is certainly not
as
thecase,asthetime-dependentexpectationvalueonthe
∞
r.h.s. of Eq. (6) will in general be susceptible to deco-
W(α) 2 ( 1)nP ( α), (10)
herence: thecalculationinvolvestime-dependentρ(t)fol- ≡ − n −
nX=0
lowing Eq.(5). However,at t=0 the noise terms vanish
identically,i.e.,thetimederivativeofanyprobeexpecta- whereP ( α)standsforthe fieldpopulationaftera dis-
n
−
tionvalueatt=0isindependentofanyfield-decohering placement D( α). Another rather original manner of
−
3
reconstructing the quantum field state in phase space is have∆2(τ)= Bˆ2 Bˆ 2 =P (τ) P (τ)2 P τ2,fol-
q h i−h i e − e ∼ N
viatheinstantaneousmeasurementoftheQ-function[1], lowing the Taylor expansion of P (τ) around zero time.
e
defined as Q= αρ α . It can be shown that Performing M measurements at each of the three times
f
h | | i
ofEq.(12),wecanreducetheuncertaintyinthethreeex-
Q(α)=Tr[D( α)ρfD†( α)0 0]=Tr[ρf( α)0 0] pectation values by a factor 1/√M. Thus, our estimate
− − | ih | − | ih |
=P ( α). (11) for the second derivative comes with an error variance
0
−
6P τ2+4∆2
This means that, following Eq. (9), measuring instanta- ∆2 N t, (13)
∼ Mτ4
neouslytheprobabilityofhavingFockstate 0 afterfield
| i while the signal is P . Then, in order to achieve a
displacements D( α), amountsto a full measurementof N
∼
− signal-to-noise ratio larger than unity, P /∆ > 1, we
the Q-function. Note that this will only require the tun- N
need a number of measurements
ing of a single selective subspace, reducing enormously
the experimental efforts when compared to the case of M >τ−2+∆2τ−4, (14)
t
the Wigner function. Typically, coherent displacements
approximately. Inthisway,aswereduceτ byafactorf,
can be realized at a very high rate, depending mainly
in order to improve the approximationof P¨ (0), we need
on the intensity of the excitation fields, so this is not a e
to increase the number of measurements by f2 (or f4 if
critical issue.
technicalerrorsdominate)tomaintainthedesiredsignal-
From a fundamental point of view, our proposal sug-
to-noise ratio. Consequently, the total probe-system in-
gests that, no matter how short the lifetime of a certain
teraction time Mτ summed over all measurements in-
system is, there would always exist the possibility of en-
creases,while eachmeasurement outcome represents the
codingitsquantuminformationintwo-levelprobestatis-
effectofanarbitrarilyshortinteractiontimeτ. Although
tics at infinitesimal interaction times. In other words,
thisisahighpricetopay,contaminationoftheoscillator
measuring quantum states may not require long-living
by decoherence can be kept arbitrarily small, no matter
systems or strong probe-system coupling, against com-
how fast it is.
mon belief. In this work, we have not proved this con-
Quantitatively, the determination of P¨(0) is associ-
jecture in general. However, we have given a particular ... e
example, the case of a quantum harmonic oscillator or ated with an error Pe(0)τ due to the finiteness of τ.
∼
a single mode field, that is applicable to many physical In this case, we can use Eq. (5) to obtain an expres-
...
setups. sion for Pe(0) and observe two contributions: a unitary
Turning to more practical considerations, we stress component, present even in the absence of noise, and an
that our scheme employs exclusively, as a final readout additional term κ/Ω due to the noise [20]. In order
∝ ...
mechanism, the measurement of the population of the to determine P¨(0) to accuracy q, we need τ < q/P (0).
e ... e
excitedstateofatwo-levelprobeatdifferenttimes. This This,inturn,requiresM >q−2P (0)2 foridealmeasure-
... e
probe population is measured directly by ionization, in ments and M > q−4P (0)4∆2 in the case of technical-
e t
the case of cavity QED, or fluorescence,for trapped ions
dominated errors. Hence, the number M of required
and atoms, with suitable additional fields, different from
measurementsincreasespolynomiallywithκ(inthe case
theonesusedinEq. (1). Thisisastandardmeasurement κ Ω like κ2 resp. κ4). Remarkably, no matter how
inmanyquantum-opticalexperimentalsetupsofinterest ≫
strong κ is, we can always make sure that it does not
and has been realized routinely with high accuracy and
affectthemeasurementresultbyshorteningτ. Thispro-
efficiency [6, 9, 19].
cedure finds its natural limit τ > Ω/ω, where ω is the
Estimatingderivativesfromafinite-time experimental
smallest dominant frequency of the specified dynamics
sampling will produce higher order corrections and an (ω = Ω2/∆ in our example, which implies τ > Ω /Ω ).
2 1 2
overhead in the number of repetition measurements. In
Note that we only need to know an upper bound of the
consequence, the benefit of instantaneous measurements
decoherence rate to perform an accurate measurement,
demands animprovedmeasurementaccuracy,which can
while alternative methods involve specific decoherence
be seen as follows. To determine the discrete second
models.
derivative in Eq. (9), P¨ (0), we measure P at 0,τ,2τ,
e e The proposed method can be applied to any physical
for small τ, and calculate
systemenjoyingJCselectiveinteractions,andmayallow
for the complete measurement of elusive fast-decaying
P (2τ) 2P (τ)+P (0)
e − τe2 e =P¨e(0)+o(τ), (12) systems,asisthecaseofintracavityopticalfields,among
others. Furthermore, since the scheme is based on in-
The measurement results are scattered around each P finitesimaltransferofinformation,thismethodcouldob-
e
with variance ∆2 = ∆2 + ∆2, referring to quantum- serve initial state conditions. For example, it could be
e q t
mechanical and technical noise, the latter arising from used to test the initial purity of a system, as well as its
imperfections in preparation, timing, and measurement. unavoidable entanglement with external degrees of free-
For small values of τ and probe operator Bˆ = e e, we dom at zero interaction time. This may prove useful for
| ih |
4
testing the validity of usual assumptions taken in deco- 78, 2547 (1997).
herence theories, such as system and environment being [8] P. Lougovski, E. Solano, Z. M. Zhang, H. Walther, H.
initially in a separable state or markoviandynamics. Mack, and W. P. Schleich, Phys. Rev. Lett. 91, 010401
(2003).
The authors thank Benasque Quantum Information
[9] D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe,
Workshop 2005 for sporting and also for scientific hos-
W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 77, 4281
pitality. E.S. acknowledgessupport from DFG SFB 631,
(1996).
EU RESQ and EuroSQIP projects. M.F.S. received fi- [10] P. Lougovski, H. Walther, and E. Solano, Eur. Phys. J.
nancial support from Infoquant/CNPq. D 38, 423 (2006).
[11] T. Bastin, J. von Zanthier, and E. Solano, J. Phys. B:
At. Mol. Opt.Phys.39, 685 (2006).
[12] M.Fran¸caSantos,E.Solano,R.L.deMatosFilho,Phys.
Rev. Lett.87, 093601 (2001).
∗ Electronic address: msantos@fisica.ufmg.br [13] E. Solano, Phys. Rev.A 71, 013813 (2005).
† Electronic address: [email protected] [14] M. Fran¸ca Santos, Phys.Rev.Lett. 95, 010504 (2005).
‡ Electronic address: [email protected] [15] E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89
[1] K. E. Cahill and R. Glauber, Phys. Rev. 177, 1882 (1963).
[16] M. G. A. Paris and J. Rˇeh´aˇcek (eds.), Quantum State
(1969).
[2] U. Leonhardt, Measuring the Quantum State of Light Estimation, Springer 2004.
[17] Z. Hradil, D. Moligevstev, and J. Rˇeh´aˇcek, Phys. Rev.
(Cambridge University Press, Cambridge, 1997).
Lett. 96, 230401 (2006).
[3] See, for example, J. M. Raimond, M. Brune, and S.
Haroche, Rev.Mod. Phys.73, 565 (2001). [18] H.-P. Breuer and F. Pretruccione, The theory of open
quantum systems (Oxford University Press, Oxford,
[4] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.
2002).
Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.
Schoelkopf, Nature431, 162 (2004). [19] G. R. Guth¨ohrlein, M. Keller, K. Hayasaka, W. Lange,
and H. Walther, Nature 414, 49 (2001); H. H¨affner,
[5] D.Leibfried,R.Blatt,C.Monroe,andD.Wineland,Rev.
Mod. Phys. 75, 281 (2003). W. H¨ansel, C. F. Roos, J. Benhelm, D. Chek-al-kar,
M. Chwalla, T. K¨orber, U. D. Rapol, M. Riebe, P. O.
[6] P.Bertet,A.Auffeves,P.Maioli,S.Osnaghi,T.Meunier,
Schmidt, C. Becher, O. Gu¨hne, W. Du¨r and R. Blatt,
M. Brune, J. M. Raimond and S. Haroche, Phys. Rev.
Lett.89, 200402 (2002). Nature 438, 643 (2005).
[20] Possible terms ∝(κ/Ω)2,(κ/Ω)3 can be seen to vanish.
[7] L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett.
