quantum wave function

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LETTERdoi:10.1038/nature10120

Direct measurement of the quantum wavefunctionJeff S. Lundeen1, Brandon Sutherland1, Aabid Patel1, Corey Stewart1 & Charles Bamber1

The wavefunction is the complex distribution used to completelydescribe a quantum system, and is central to quantum theory. Butdespite its fundamental role, it is typically introduced as an abstractelement of the theory with no explicit definition1,2. Rather, physicistscome to a working understanding of the wavefunction through itsuse to calculate measurement outcome probabilities by way of theBorn rule3. At present, the wavefunction is determined throughtomographic methods48, which estimate thewavefunctionmostcon-sistent with a diverse collection of measurements.The indirectnessofthese methods compounds the problem of defining the wave-function. Here we show that the wavefunction can be measureddirectly by the sequential measurement of two complementary vari-

ables of thesystem. The crux of our method is that the first measure-ment is performed in a gentle way through weak measurement918,so as not to invalidate the second. The result is that the real andimaginary components of the wavefunction appear directly on ourmeasurement apparatus. We give an experimental example bydirectly measuring the transverse spatial wavefunction of a singlephoton, a task not previously realized by any method. We show thatthe conceptis universal, being applicable to other degrees of freedomof the photon, such as polarization or frequency, and to otherquantum systemsfor example, electron spins, SQUIDs (super-conducting quantum interference devices) and trapped ions.Consequently, this method gives the wavefunction a straightforwardand general definition in terms of a specific set of experimentaloperations19. We expect it to expand the range of quantum systems

that can be characterized and to initiate new avenues in fundamentalquantum theory.

The wavefunction Y, also known as the quantum state, is con-siderably more difficult to measure than the state of a classical particle,which is determined simply by measuring its position X andmomentum P. According to the Heisenberg uncertainty principle, inquantum theory a precise measurement of X disturbs the particleswavefunction and forces a subsequent measurement of P to becomerandom. Thus we learn nothing of theparticlesmomentum. Indeed, itis impossible to determine a completely unknown wavefunction ofsingle system20.

Consider instead performing a measurement ofXon an ensemble ofparticles, allwith the sameY. The probability of getting resultX5 xisjY(x)j2. Similarly, the probability of P5p would be jW(p)j2, where

W(p) is the Fourier transform of Y(x). Even these two probabilitydistributions are not enough to determine Y(x) unambiguously (seethe one-dimensional phase retrieval problem21). Instead, Ymust bereconstructed by performing a large set of distinct measurements (forexample, of quadratures Q(h)5Xcos(h)1Psin(h), for mixing angles,h, ranging from 0 to 2p), and then estimating a Ythat is most com-patible with the measurement results. This method is known asquantum state tomography48. In contrast, we introduce a method tomeasure Y of an ensemble directly. By direct we mean that themethod is free from complicated sets of measurements and computa-tions; the average raw signal originating from where the wavefunctionis being probed is simply proportional to its real and imaginary com-ponents at that point. The method rests on the sequential measure-ment of two complementary variables of the system.

At thecentre of thedirect measurement methodis a reductionof thedisturbance induced by the first measurement. Consider the measure-mentof an arbitraryvariableA. In general, measurement canbe seen asthecoupling between an apparatusand a physicalsystem that results inthe translation of a pointer. The pointer position indicates the resultofa measurement. In a technique known as weak measurement, thecoupling strength is reduced and this correspondingly reduces thedisturbance created by the measurement918. This strategy also com-promises measurement precision, but this can be regained by aver-aging. The average of the weak measurement is simply the expectation

value YjAjY, indicated by an average position shift of the pointerproportional to this amount.

A distinguishing feature of weak measurement is that it does notdisturb a subsequent normal (or strong) measurement of anotherobservable Cin the limit where the coupling vanishes. For the particu-lar ensemble subset that gave outcome C5 c, one can derive the aver-age of the weak measurement of A. In the limit of zero interactionstrength, this is called the weak value and is given9 by:

Ah iW~c Aj jYh i

cjYh i1

Selecting a particular subset of an ensemble based on a subsequentmeasurement outcome is known as post-selection, and is a commontool in quantum information processing22,23.

Unlike the standard expectation value A, the weak value AW canbe a complex number. This seemingly strange result can be shown to

have a simple physical manifestation: the pointers position is shiftedby ReAW and receives a momentum kick of ImAW (refs 2426).The complex nature of the weak value suggests that it could be used toindicate both the real and the imaginary parts of the wavefunction.

Returning to our example of a single particle, consider the weakmeasurement of position (A5px; jx xj) followed by a strong mea-surement of momentum givingP5p. In this case, the weak value is:

pxh iW~pjxh i xjYh i

pjYh i2

~

eipx=BY(x)

W p 3

In the case p5 0, this simplifies to

pxh iW~kY x 4

where k5 1/W(0) is a constant (which can be eliminated later bynormalizing the wavefunction). The average result of the weak mea-surement ofpx is proportional to the wavefunction of the particle at x.Scanning the weak measurement through xgives the complete wave-function. At each x, the observed position andmomentum shiftsof themeasurement pointer are proportional to ReY(x) and ImY(x),respectively. In short, by reducing the disturbance induced by mea-suring X and then measuring P normally, we measure the wave-function of the single particle.

As an experimental example,we performed a direct measurement ofthe transverse spatial wavefunction of a photon. Considering a photontravellingalong the zdirection,we directly measure the xwavefunction

1Institute for National Measurement Standards, National Research Council, 1200 Montreal Road, Ottawa, Canada, K1A 0R6.

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of the photon, sometimes called the spatial mode (see SupplementaryDiscussion). The Wigner function of the spatial mode of a classicalbeam has been measured directly but not for a single photon state27,28.

We produce a stream of photons in one of two ways, either byattenuating a laser beam or by generating single photons throughspontaneous parametric down-conversion (SPDC; see Supplemen-tary Methods for details). The photons have a centre wavelength ofl5 783 nm or 800 nm, respectively. The experiment (details and dia-

gram in Fig. 1) canbe divided into four sequential steps:preparation ofthe transverse wavefunction, weak measurement of the transverseposition of the photon, post-selection of those photons with zerotransverse momenta, and readout of the weak measurement.

An ensemble of photons with wavefunction Y(x) is emitted from asingle mode fibre and collimated. We begin by directly measuring thiswavefunction (described in detail in Fig. 1). We then further test ourmethod by inducing known magnitude and phase changes to thephotons here to prepare a series of modified wavefunctions.

We weakly measure the transverse position of the photon by coup-ling it to an internal degree of freedom of the photon, its polarization.This allows us to use the linear polarization angle of the photon as thepointer. At a position x where we wish to measure px5 jx xj, werotate thelinear polarizationof thelightbya. Consider ifa issetto90u.

In this case, one can perfectly discriminate whether a photon hadposition x because it is possible to perfectly discriminate betweenorthogonal polarizations, 0u and 90u. This is a strong measurement.Reducing the strength of the measurement corresponds to reducinga,which makes it impossible to discriminate with certainty whether any

particular photon hadX5 x. The benefit of this reduction in precisionis a commensurate reduction in the disturbance to thewavefunction ofthe single photon.

We then use a Fourier transform lens and a slit to post-select onlythose photonswithp50.ThisconstitutesthestrongmeasurementofP.

In this subset of the photon ensemble, we find the average value ofour weak measurement ofpx. The average rotation of the pointer, thelinearpolarization, is proportional to thereal part of theweak value.Its

complementary pointer variable, the rotation in the circular polariza-tion basis, is proportional to the imaginary part of the weak value25.Formally, if we treat the initial polarization as a spin-1/2 spin-down

vector, then the weak value is given by

pxh iW~1

sin as sxj jh is{i s sy

s

5

where sx and sy are the Pauli xand ymatrices, respectively, and js isthe final polarization state of the pointer25. We measure the sx and syexpectation values by sending the photons through a half-wave plate ora quarter-wave plate, respectively, and then through a polarizingbeamsplitter (PBS). Thus, we read out ReY(x) (half-wave plate) andImY(x) (quarter-wave plate) from the signal imbalance betweendetectors 1 and 2 at the outputs of the PBS (Fig. 1).

With a5

20u, we scan our measurement ofpx in 1-mm steps andfind theweak value pxW at each step. In this way, we directly measure

the photon transverse wavefunction, Y(x)5 jY(x)jexp(iw(x)). Wenormalize the sx and sy measurements by the same factor, so thatY x j j2dx~1, which eliminates the proportionality constant,

sin a/W(0).To confirm our direct measurement method, we test it on a series of

different wavefunctions. Using ourSPDC single photonsource, we startby measuring the initial truncated Gaussian wavefunction (Fig. 2)described in Fig. 1. Switching to the laser source of photons, we thenmodify the magnitude, and then the phase, of the initial wavefunctionwith an apodized filter and glass plate, respectively, to create two newtest Y (Fig. 3). We conduct more quantitative modification of thewavefunction phase by introducing a series of phase gradients and then

phase curvatures (Fig. 4). For all the test wavefunctions, we have found

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Figure 1 | Direct measurement of the photon transverse wavefunction. Tobegin with photons having identical wavefunctions, we transmit them throughan optical fibre (Nufern PM780-HP) that allows only a single mode (SM) topass. This mode is approximately Gaussian, with a nominal 1/e2 diameter of5.361.0mm. The photons emerge from the fibre and pass through a micro-wire polarizer (Pol.; Edmund Optic NT47-602) to be collimated by anachromaticlens (f15 30cm, diameter 5 cm,ThorlabsAC508-300-B), onefocallength (f1) away from the fibre. The lens was masked off with a rectangularaperture of dimension x3y5 43mm3 11 mm. Thus our nominal initialwavefunction was a truncated Gaussian with a 1/e2 diameter of 56.4mm and aflat phase profile. We modify the magnitude and phase of the nominal Y(x) to

create a series of test wavefunctions (see Figs 3, 4). At 45 mm past the lens, arectangular sliver of a half-wave plate (l/2 sliver) (x3y3 zdimensions of1 mm3 25mm3 1 mm) at position xis used to weakly measure px5 | x x|(see SupplementaryMethods for details). The photons then undergo an opticalFourier transform (FT) induced by an achromatic lens (f25 1 m, diameter5 cm,ThorlabsAC508-1000-B), placedone focal length(f2) from thewaveplatesliver. In the Fourier transform plane, one focal length f2 past the lens,we post-select those photons with p5 0 by accepting only those that passthrough a 15-mm-wide slit on axis. We collimate the photons emerging from the slit with anf35 3 cm focal length lens. The photons pass through either a half-wave plate(l/2) or quarter-wave plate (l/4) and then through a polarizing beamsplitter(PBS). At each outputport,the photons arefocused onto a detector (Det. 1 andDet. 2): for the single photons, a photon counter (silicon avalanchephotodiodes, PerkinElmer SPCM-AQHR-14); and for the laser, a siliconphotodiode (Thorlabs, DET10A). The imbalance in counts or signal betweenthe two detectors is proportional to the real (l/2) or imaginary(l/4) part of the

wavefunction.

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Figure 2 | The measured single-photon wavefunction, Y(x), and itsmodulus squared and phase. a, ReY(x) (solid blue squares) and ImY(x)(open red squares) measured for the truncated Gaussian wavefunction.b, Using the data in awe plot the phase w(x)5 arctan (ReY(x)/ImY(x)) (opensquares; right axis) and the modulus squared |Y(x) | 2 (solid blue circles; leftaxis). There is good agreement between the latter and a strong measurement ofthe xprobability distribution Prob(x) (solid line; left axis) conducted byscanning a detector alongxin theplaneof thesliver. The phase is relatively flat,as expected from the fibre mode. The slight variation is consistent with themanufacturer specification of the first lens and the phase curvature measuredwith a shear plate. We also removed the slit completely. In this case, there is nopost-selection and the weak value px becomes equal to the standardexpectation value Y|px|Y5 |Y(x) |

2. We plot themeasured Repx (openredcircles; left axis) after it is normalized so that

Re Y x dx~1 and find it is in

good agreement with Prob(x).We findthatImpx is tentimessmaller,makingpx largely real, as expected. Error bars are 61 s.d. found from statistics in

repeated scans. In b, only every third error bar is shown for clarity.

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good agreement between the expected and measured wavefunction,including its phase and magnitude (see the figure legends for details).

We now describe how the technique of weak measurement can beused to directly measure the quantum state of an arbitrary quantumsystem. We have the freedom to measure the quantum state in anychosen basis {ja} (associated with observable A) of the system. Themethod entails weakly measuring a projector in this basis, pa; ja aj,and post-selecting on a particular value b0 of the complementary

observable B (see Supplementary Discussion for a precise definitionof complementarity). In this case, the weak value is

pah iW~b0jah i ajYh i

b0jYh i~ ajYh i=v 6

where vis a constant, independent ofa. Thus the weak value is pro-portional to the amplitude of state ja in the quantum state. Steppingathrough all the states in the A basis directly gives the quantum staterepresented in that basis:

Yj i~vX

a

pah iW aj i 7

This is the general theoretical result of this Letter. It shows that in anyphysical system one can directly measure the quantum state of thatsystem by scanning a weak measurement through a basis and appro-priately post-selecting in the complementary basis.

Weak measurement necessarily trades efficiency for accuracy orprecision. A comparison of our method to current tomographic recon-struction techniques will require careful consideration of the signal tonoise ratio in a given system. Inorder to increase thisratio in the directmeasurement of the photon spatial wavefunction, future experimentswill investigate the simultaneous post-selection of many transversemomenta.

In our direct measurement method, the wavefunction manifestsitself as shifts of the pointer of the measurement apparatus. In thissense, the method provides a simple and unambiguous operationaldefinition19 of the quantum state: it is the average result of a weakmeasurement of a variable followed by a strong measurement of thecomplementary variable. We anticipate that the simplicity of the

method will make feasible the measurement of quantum systems

(for example, atomic orbitals, molecular wavefunctions29, ultrafastquantum wavepackets30) that previously could notbe fullycharacterized.The method can also be viewed as a transcription of quantum state ofthe system to that of the pointer, a potentially useful protocol forquantum information.

Received 22 March; accepted 14 April 2011.

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Figure 3 | Measurements of modified wavefunctions. We further test ourability to measure Y(x) by changing Prob(x) by placing a reverse bulls-eyespatially apodized attenuator (RB in Fig. 1) (Edmund Optics, NT64-388) afterthe fibre. a, We calculate |Y(x) | 2 from the data (solid blue circles) along with adetector scan of Prob(x) (solid line) and find good agreement between the two.b, With the reverse bulls eye still in place, we modify the phase profile w(x) ofthe wavefunction by creating a phase discontinuity at x5 0 imposed with aglass plate half-way across Y(x). At the bottom, we show ReY(x) (solid bluesquares; left axis) and ImY(x) (open red squares; left axis), which exhibit adiscontinuity at the plate edge. This discontinuity is even clearer in the phasedifference between the wavefunctions measured with and without the glassplate, shown at the top (open black diamonds; right axis). Despite theirdiscontinuities, if we use ReY(x) and ImY(x) to calculate |Y(x) | 2 (open redcircles in a), we find that it is largely unchanged by the glass plate. This is asexpected,as theglass hasa transmissionnearunity. Error bars are61 s.d. foundfrom statistics in repeated scans. In a, every third error bar is shown for clarity.In b, those bars smaller than the symbols are not shown.

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Figure 4 | Phase modification of the wavefunction. a, We displace the slittransversely byDxslit5230,220,210, 10, 20, 30 and 40mm (curves in mainpanel,top to bottom). This effectively redefinesthe zero momentum axisof thesystem. Our photons now travel at an angle to this axis or equivalently thewavefunction has a linear phase gradient, w(x)5mx, where m5Dxslit2p/f2l.We plot thephase difference between the original wavefunctionand those witha phase gradient. For clarity, the curves have been offset to cross at 220 mm.This corresponds to shifting the arbitrary global phase ofY(x). Inset, gradientm as a function ofDxslit (circles) along with theory (line), which show goodagreement. b, We introduce a quadratic phase by displacing the first lens byDz5 200, 400, 600 and 800 mm (curves in main panel, bottom to top) along

with theoretical fits (lines). The phase w(x)5 rx2

, where the phase curvaturer~pDz

f21 l. Inset, the phase curvature rfrom these fits (circles) as a function

of lens displacement, Dxslit, which shows good agreement with theory (line).Statistical error bars (61 s.d.) are smaller than the symbols in all plots.

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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements This work was supported by the Natural Sciences and

Engineering Research Council and the Business Development Bank of Canada.

Author Contributions Theconcept andthe theory weredeveloped by J.S.L. All authorscontributed to the design and building of the experiment and the text of themanuscript. J.S.L, B.S. and C.B.performed the measurements and the data analysis.

Author Information Reprints and permissions information is available atwww.nature.com/reprints . The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to J.S.L. ([email protected]) .

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