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arXiv:quant-ph/0101051v2 15 Jun 2001

Quantum state reconstruction of the single-photon Fock state

A. I. Lvovsky∗, H. Hansen, T. Aichele, O. Benson, J. Mlynek†and S. Schiller‡

Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konstanz, Germany

(February 1, 2008)

We have reconstructed the quantum state of optical

pulses containing single photons using the method of phase-

randomized pulsed optical homodyne tomography.

single-photon Fock state |1? was prepared using conditional

measurements on photon pairs born in the process of para-

metric down-conversion.A probability distribution of the

phase-averaged electric field amplitudes with a strongly non-

Gaussian shape is obtained with the total detection efficiency

of (55 ± 1)%.

structed from this distribution shows a strong dip reaching

classically impossible negative values around the origin of the

phase space.

The

The angle-averaged Wigner function recon-

a. Introduction

completely described by their Wigner functions (WF),

the analogues of the classical phase-space probability dis-

tributions. Generation of various quantum states and

measurements of their WFs is a central goal of many ex-

periments in quantum optics [1–3]. Of particular inter-

est are quantum states whose Wigner function takes on

negative values in parts of the phase space. This clas-

sically impossible phenomenon is a signature of highly

non-classical character of a quantum state.

Quantum states containing a definite number of energy

quanta (Fock states |n?) are paradigmatic in this respect.

Their WFs exhibit strong negativities and their marginal

distributions are strongly non-Gaussian (Fig.1).

property reflects the fundamentally non-classical nature

of these states as carriers of the particle aspect of light.

States of quantum systems can be

This

FIG. 1. Theoretical phase space quasiprobability density

(Wignerfunction)of

state |1?:

ˆ X = (ˆ a + ˆ a†)/√2 andˆP = (ˆ a − ˆ a†)/√2i are normalized

non-commuting electric field quadrature observables.

gle-quadrature probability densities (marginal distributions)

are also displayed.

thesingle-photon

W(X,P)=

2

π

?4(X2+ P2) − 1?e−2(X2+P2).

Sin-

Generation and complete measurements of the Fock

states’ WFs were performed on vibrational states of a

trapped Be+ion [1].In the electromagnetic domain,

Nogues et al. [2] recently reported the measurement of

the WF of a single-photon state in a superconducting

microwave cavity at a single point (origin) of the phase

space. A full characterization of a Fock state of the elec-

tromagnetic field has not been achieved so far.

In this paper we present a measurement of the com-

plete (phase-averaged) Wigner function of the propagat-

ing single-photon state |1? in the optical domain. We per-

form a direct measurement of the dynamical variables of

the electromagnetic field, the electric field quadratures,

whereby their probability distributions are obtained. The

Wigner function is then reconstructed from the measured

distributions. This method, homodyne tomography, has

been established as a reliable technique of reconstructing

quantum states in the optical domain. Previously, it has

been applied to classical and weakly non-classical states

of the light field, such as vacuum, coherent, thermal and

squeezed states, in the continuous-wave as well as in the

pulsed regime [3].

The main challenge associated with a tomographic

characterization of the single-photon state is the prepara-

tion of this state in a well-defined spatio-temporal mode.

We solve this task by employing conditional state prepa-

ration on a photon pair born in the process of parametric

down-conversion [4,5]. The two generated photons are

separated into two emission channels according to their

propagation direction (Fig. 2). A single-photon counter

is placed into one of the emission channels (labeled trig-

ger) to detect photon pair creation events and to trigger

the readout of a homodyne detector placed in the other

(signal) channel [6].

FIG. 2. Simplified scheme of the experimental setup

b.Theory

The process of pulsed 2-photon down-

conversion produces strongly correlated photon pairs.

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The generated biphoton state can be written as

|Ψ? = N

?

|0,0? +

?

d?ksd?ktΦ(?ks,?kt)

???1?ks,1?kt

??

, (1)

where N is a normalization constant,?ks and?kt denote

the signal and trigger beam wave vectors, respectively,

and the function Φ(?ks,?kt) carries the information about

the amplitude as well as the transverse and longitudinal

structure of the photon pair generated [8].

A single-photon Fock state is prepared from |Ψ? by

projecting this state onto a photon count event in the

trigger beam path:

ˆ ρs= Trt|Ψ??Ψ| ˆ ρt,

(2)

where ˆ ρtdenotes the state ensemble selected by the trig-

ger and the trace is taken over the trigger states.

The trigger state ensemble ˆ ρtis determined by the spa-

tial and spectral filtering in the trigger channel:

ˆ ρt =

?

d?ktT(?kt) |1?kt??1?kt|,

(3)

where T(?kt) is the spatiotemporal transmission function

of the filter. Note that although |Ψ? represents a pure

state, ˆ ρt and hence ˆ ρs are statistical mixtures.

ever, if sufficiently tight filtering is applied to the trigger

channel (so that T(?kt) is much narrower than the spatial

and spectral width of the pump beam), ˆ ρswill approach

a pure single-photon state [7]. In this case, the signal

photons are prepared in a relatively well-defined optical

mode suitable for homodyne detection.

It is important to understand that the “signal beam”

as shown in Fig.2 is not an optical beam in the traditional

sense. The down-converted photons are in fact emitted

randomly over a wide solid angle. The optical mode of

the signal state is created non-locally only when a photon

of a pair hits the trigger detector and is registered. The

coherence properties of this mode are determined by the

optical mode of the pump and the spatial and spectral

filtering in the trigger channel.

Once the approximation ˆ ρs of the Fock state is pre-

pared, it is subjected to balanced homodyne detection.

The signal wave is overlapped on a beamsplitter with a

relatively strong local oscillator (LO) wave in the match-

ing optical mode. The two fields emerging from the

beamsplitter are incident on two high-efficiency photo-

diodes whose output photocurrents are subtracted. The

photocurrent difference is proportional to the value of the

electric field operatorˆE(θ) in the signal mode, θ being

the relative optical phase of the signal and LO.

For each phase θ one measures a large number N of

samples ofˆE(θ) ∝ˆ Xθ≡ˆ X cosθ +ˆP sinθ, so that their

histogram (i.e.the marginal distribution) pr(Xθ) can be

determined. The latter is related to the WF as follows:

How-

pr(Xθ) = ?Xθ|ˆ ρmeas|Xθ?

?∞

−∞

(4)

=

W(X cosθ − P sinθ,X sinθ + P cosθ)dP,

where ˆ ρmeasis the density matrix of the state being mea-

sured. The marginal distribution pr(Xθ) can be envi-

sioned as a density projection of the WF W(X,P) onto

a vertical plane oriented at an angle θ with respect to the

plane P = 0 (Fig.1). From the set of marginal distribu-

tions pr(Xθ) for a large number of phase angles θ the WF

of ˆ ρmeascan be reconstructed via a procedure similar to

the one used in medical computer tomography [9].

In a perfect experiment, ˆ ρmeas = |1??1|, where the

single-photon state is in the optical mode which matches

that of the local oscillator. In reality, various imperfec-

tions (such as optical losses in the signal arm, inefficient

photodiodes, dark counts, non-ideal matching between

the signal and the LO optical modes [10]) cause an ad-

mixture of the vacuum |0? to the measured state, so that

ˆ ρmeas= η|1??1| + (1 − η)|0??0|,

(5)

η being the measurement efficiency.

that all these effects act upon ˆ ρmeas in a similar way,

so that their effect can be expressed in a single number

η which is a product of efficiencies associated with indi-

vidual parts of the setup. The value of η is crucial for

this experiment as it strongly influences the shape of the

measured marginal distributions and the reconstructed

Wigner function (Fig.3) [6].

It is remarkable

pr( )

X

X

X

0

0

0

0

-2

0.4

0.4

0.8

-0.4

2

2

-2

?=0

0.25

0.5

0.75

?=1

(a)

?=0

0.25

0.5

0.75

?=1

(b)

W?(X,?P=0)

FIG. 3. Effect of the non-perfect measurement efficiency

η on the marginal distribution (a) and the reconstructed WF

(b). For the WF, cross-sections by the plane P = 0 are shown.

Negative values require η > 0.5.

In our experiment we used a simplified scheme in

which the phase θ varied randomly, so that we only

measured a single phase-randomized marginal distribu-

tion prav(X) = ?pr(Xθ)?θ.

measurement result for quantum states with rotationally

symmetric Wigner functions such as those described by

Eq.(5). The phase-averaged WF W(R) is obtained from

prav(X) via the Abel transformation [9,11]:

This does not change the

W(R) = −1

π

?∞

R

dprav(X)

dX

(X2− R2)−1/2dX.

(6)

From the phase-randomized marginal distributions one

can also directly infer diagonal elements ρnnof the state

density matrix in the Fock basis,

ρnn= π

?∞

−∞

prav(X)fnn(X)dX,

(7)

where fnn(X) are the amplitude pattern functions [13]

which are independent of the optical state being sampled.

The statistical uncertainty of the reconstructed ρnnis

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?σ2

nn? =

1

N

?∞

−∞

prav(X)(πfnn(X))2dX,

(8)

where N is the total number of field samples acquired.

c. Experimental Setup

Ti:sapphire laser (Spectra Physics Tsunami) in combi-

nation with a pulse picker to obtain transform-limited

pulses at 790 nm with a repetition rate 816 kHz and a

pulse width of 1.6 ps. Most of the radiation was fre-

quency doubled in a single pass through a 3-mm LBO

crystal yielding 100 µW at 395 nm and then passed on

to a 3-mm BBO crystal for down-conversion.

Down-conversion occurred in a type-I frequency-

degenerate, but spatially non-degenerate configuration,

with 790-nm photon pairs emitted at angles ±6.8◦with

respect to the pump beam. The BBO crystal was cut at

θ = 35.7◦, φ = 0◦, so that the direction of the walk-off

of the 395 nm pump beam inside the crystal coincided

with the direction of the signal beam so as to minimize

distortions of the signal spatial mode (“hot spot” down-

conversion, [12]). The short crystal lengths allowed us to

avoid group-velocity mismatch effects which would have

complicated temporal mode matching to the LO pulse.

The trigger photons passed through a spatial filter and

a 0.3-nm interference filter centered at the laser wave-

length. They were then detected by an EG&G SPCM-

AQ-131 single-photon detector (quantum efficiency 60%,

dark count rate < 15 s−1) at a rate of about 0.25 s−1.

Such a low pair production rate made the effect of Fock

states with n > 1 negligible. Precise (within 0.6 ns) gat-

ing of the count events with the laser pulses allowed us

to eliminate most of the dark counts, thereby reducing

their contribution to about 2% of all trigger events.

We used a small fraction of the original optical pulses

from the pulse picker — split off before the frequency-

doubler — as the local oscillator for the homodyne sys-

tem. Achieving a good spatial and temporal mode match-

ing between the LO and the photons in the signal channel

constituted a major challenge in this experiment due to

extremely low intensity of the field in the signal mode.

To this end, a fraction of the laser output power was di-

rected into the BBO crystal from the back along the trig-

ger beam path so that it passed through the spatial filter

in the trigger channel. Inside the crystal these alignment

pulses were temporally and spatially overlapped with the

pump to produce a difference frequency (DFG) emission

into an optical mode which modeled, to a good precision,

that of the conditionally prepared signal photons. This

mode was then matched to that of the local oscillator by

observing an interference pattern between the two beams.

A visibility on the level of v = 83 ± 1% was reached.

The method of conditional state preparation also es-

tablished special requirements for the homodyne detector

electronics. The detector needed to resolve quantum shot

noises of individual laser pulses at a 0.8-MHz repetition

rate. Details on design and performance of the homodyne

system developed will be published elsewhere [14].

We employed a mode-locked

FIG. 4. Experimental results: a) raw quantum noise data

for the vacuum (left) and Fock (right) states along with their

histograms corresponding to the phase-randomized marginal

distributions; b) diagonal elements of the density matrix of

the state measured. c) reconstructed WF which is negative

near the origin point. The measurement efficiency is 55%.

d. Results and discussion

run about 200,000 vacuum state and 12,000 Fock state

samples were acquired (Fig.4a).

then binned up to obtain their statistical distributions.

A Gaussian distribution was fit to the vacuum state noise

spectrum by varying its X-scale and point of origin.

The best fit parameters of the vacuum state were used

to scale the Fock state data.

by the theoretical marginal distribution of the ensemble

(5) to find the measurement efficiency η. The best fit

efficiency value was η = 0.55. Using the Abel transform

(6) the phase-randomized WF of the observed quantum

state was reconstructed (Fig.4c). As expected, it exhibits

negativity around the origin point, with a minimum value

W(0,0) = −0.062.

The diagonal elements of the density matrix have been

evaluated along with their statistical errors by apply-

ing the quantum state sampling method as defined by

Eqs.(7, 8) directly to the rescaled raw data. We found

ρ11 = 0.553 ± 0.013 in agreement with the value of η

obtained by fitting the marginal distribution (Fig.4b).

Substituting this quantity into Eq.(5) and calculating

the corresponding value of the WF we find W(0,0) =

−0.067±0.016, in agreement with the above value deter-

mined from the marginal distribution. The uncertainty

of ±0.016 gives an estimation for the accuracy to which

the negative value of the Wigner function has been de-

termined in this experiment.

The values of ρ00 and ρ11 for the vacuum state were

found to be 0.9975±0.0029 and 0.0021±0.0032, respec-

tively. These quantities being equal to their ideal values

within the statistical errors indicates that no unknown

conditions (e.g. mechanical vibrations) are present that

might bring the apparent measurement efficiency above

its actual value. The systematic experimental errors(e.g.

technical noise in the homodyne detector) were insignif-

In a 14-hour experimental

Both data sets were

The latter was then fit

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icant and would result in decrease of the measured effi-

ciency. It is thus impossible that neglecting these errors

could bring about a too optimistic estimate of η.

What are the main factors reducing the measurement

efficiency? The non-perfect match between the optical

modes of LO and DFG fields diminishes the efficiency

by a factor of v2= 0.69 ± 0.02. The fact that the DFG

wave does not perfectly mimic the conditionally prepared

mode of the single photon causes an additional reduction

by ×0.95. A factor of 0.90 arises from losses in the sig-

nal beam path and non-perfect quantum efficiency of the

photodiodes in the homodyne detector. Finally, a 2%

reduction occurs due to false trigger count events. Com-

bining all the above factors we obtain the upper limit

estimate of the quantum efficiency as (57±2)%, which is

in good agreement with the experimental value of 55.3%.

e. Conclusion and outlook

phase-averaged Wigner function and the density matrix

diagonal elements of an optical single-photon Fock state

|1? with a total measurement efficiency of 55.3 ± 1.3%

using the method of phase-randomized pulsed optical ho-

modyne tomography. The reconstructed WF is of non-

Gaussian shape and exhibits negative values around the

origin of phase space, reflecting the strongly non-classical

character of the state |1? as a particle state of the light

field. Single-photon Fock states were prepared in a well-

defined electromagnetic mode by conditional measure-

ments on photon pairs created in the process of para-

metric fluorescence. The measurement technique and er-

ror analysis were checked by performing a simultaneous

measurement on the vacuum state. Major experimental

inefficiency factors have been identified and quantified.

This experiment represents the first quantum tomogra-

phy measurement of a highly nonclassical state of the

electromagnetic field.

Relatively straightforward modifications of the setup

would allow tomography measurements of displaced Fock

states [15] and 1-photon added coherent states [16]. Of

special interest is the entangled state |0,1?−|1,0? gener-

ated when a single photon is incident on a beamsplitter.

This state can also be used to demonstrate nonlocality of

a single photon [17]. In a more distant future, arbitrary

quantum states of the light field might be generated via

repeated down-conversion [18].

We thank S. Eggert, C. Hettich and P. Lodahl for their

help in building up the experimental setup.

supported by the Alexander von Humboldt Foundation.

This work was funded by the Deutsche Forschungsge-

meinschaft.

We have reconstructed the

A. L. is

∗

email: Alex.Lvovsky@uni-konstanz.de

Present address:

Berlin, D-10099 Berlin, Germany

†

President, Humboldt-Universit¨ at zu

‡

Present

physik, Heinrich-Heine-Universit¨ at D¨ usseldorf, D-40225

D¨ usseldorf, Germany

[1] D. Leibfried et al. Phys. Rev. Lett. 77, 4281 (1996)

[2] G. Nogues et al. Phys. Rev. A 62, 054101 (2000).

[3] D. T. Smithey et al. Phys. Rev. Lett. 70, 1244 (1993);

G. Breitenbach, S. Schiller, and J. Mlynek, Nature 387,

471 (1997); M. Vasilyev et al., Phys. Rev. Lett. 84, 2354

(2000)

[4] C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58

(1986)

[5] P. Grangier, G. Roger and A. Aspect, Europhys. Lett. 1,

173 (1986)

[6] B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987)

[7] Z. Y. Ou, Qu. Semiclass. Opt. 9, 599 (1997)

[8] A. Joobeur et al., Phys. Rev. A 53, 4360 (1996)

[9] for details on quantum tomography and inverse Radon

transformation, see U. Leonhardt, Measuring the quan-

tum state of light, Cambridge University Press, 1997

[10] F. Grosshans and P. Grangier, Eur. Phys. J. D 14, 119

(2001)

[11] U. Leonhardt and I. Jex, Phys. Rev. A 49, 1555 (1994)

[12] K. Koch et al., IEEE J. Quant. El. 31, 769 (1995)

[13] G. M. D’Ariano, U. Leonhardt, and H. Paul, Phys. Rev.

A 52, R1801 (1995); U. Leonhardt and M. G. Raymer,

Phys. Rev. Lett. 76, 1985 (1996)

[14] H. Hansen et al. quant-ph/0104084

[15] K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857

(1969)

[16] G. S. Agarwal and K. Tara, Phys. Rev. A 43, 492 (1990);

C. T. Lee, Phys. Rev. A 52, 3374 (1995)

[17] S. M. Tan, D. F. Walls, and M. J. Collett, Phys. Rev.

Lett. 66, 252 (1991); K. Banaszek and K. Wodkiewicz,

ibid. 82, 2009 (1999); K. Jacobs and P. L. Knight, Phys.

Rev. A 54, 3738 (1996)

[18] J. Clausen et al., quant-ph/0007050

address: Institutf¨ ur Experimental-

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