Download Generation of nonclassical states from thermal radiation

Invited Paper
Generation of nonclassical states from thermal radiation
Valentina Parigia , Alessandro Zavattab and Marco Bellinia,b
a LENS,
b Istituto
Università di Firenze, Via Nello Carrara 1, Sesto Fiorentino, Italy;
Nazionale di Ottica Applicata - CNR, Largo Enrico Fermi 6, Firenze, Italy
ABSTRACT
We show the experimental observation of quantum states of light exhibiting nonclassical features obtained by
single photon excitation of a thermal state. Such single-photon-added thermal states are the result of the single
action of the creation operator on a mixed state that can be fully described classically. They show different degree
of nonclassicality depending on the mean photon number of the original thermal state. The generated state is
characterized by means of ultra-fast homodyne detection which allows us to reconstruct its density matrix and
Wigner function by quantum tomography. We demonstrate the nonclassical behavior of single-photon added
thermal states by an analysis of the negativity of the Wigner function.
Keywords: Nonclassical quantum states, quantum homodyne tomography, thermal light, single-photon Fock
states, entanglement.
1. INTRODUCTION
In the early works of Raymer and coworkers1 the possibility of reconstructing the quantum state of light was
demonstrated by using optical homodyne tomography. Subsequently, this technique showed its great importance
for the characterization and study of nonclassical field states in the continuous variable domain.2, 3 Nowadays,
the generation and the analysis of nonclassical light is the starting point to generate even more nonclassical
states,4, 5 or the entangled states which are essential to implement quantum information protocols with continuous
variables.6, 7
As a general definition, a quantum state is said to be nonclassical when it cannot be written as a mixture of
coherent states. In terms of the Glauber-Sudarshan P representation,8, 9 the P function of a nonclassical state
is highly singular or not positive, i.e. it cannot be interpreted as a classical probability distribution. In general
however, since the P function can be badly behaved, it cannot be connected to any observable quantity.
A conceptually simple way to generate a quantum light state with a varying degree of nonclassicality consists
in adding a single photon to any completely classical one. This is quite different from photon subtraction which,
on the other hand, produces a nonclassical state only when starting from an already nonclassical one.10, 11
Here we report the generation and the analysis of single-photon-added thermal states (SPATSs), i.e., completely classical states excited by a single photon, first described by Agarwal and Tara in 1992.12 We use the
techniques of conditioned parametric amplification recently demonstrated by our group3, 13 to generate such
states, and we employ ultrafast pulsed homodyne detection and quantum tomography to investigate their character. The peculiar nonclassical behavior of SPATSs had been described in many theoretical papers11, 12, 14–17
and their experimental generation had already been proposed, although with more complex schemes,11, 15 but
never realized. Thanks to their adjustable degree of quantumness, these states are an ideal benchmark to test
the different experimental criteria of nonclassicality recently proposed, and to investigate the possibility of entanglement generation using linear optics. The nonclassicality of SPATSs is here analyzed by reconstructing
their negative-valued Wigner functions.
Further author information: (Send correspondence to M.B.)
V.P.: E-mail: [email protected]fi.it, Telephone: +39 055 457 2215
A.Z.: E-mail: [email protected], Telephone: +39 055 457 2215
M.B.: E-mail: [email protected], Telephone: +39 055 457 2493
Quantum Communications and Quantum Imaging IV, edited by
Ronald E. Meyers, Yanhua Shih, Keith S. Deacon, Proc. of SPIE
Vol. 6305, 63050Z, (2006) · 0277-786X/06/$15 · doi: 10.1117/12.680505
Proc. of SPIE Vol. 6305 63050Z-1
2. THEORY
In order to describe the state generated in our experiment we give a general treatment of photon addition
based on conditioned parametric amplification. By first-order perturbation theory, the output of the parametric
amplifier when a pure state |ϕm is injected along the signal channel is given by
|ψm = [1 + (gâ†s â†i − g ∗ âs âi )] |ϕm s |0i ,
(1)
where g accounts for the coupling and the amplitude of the pump and â, ↠are the usual noncommuting
annihilation and creation operators. For a generic signal input, the output state of the parametric amplifier can
be written as
Pm |ψm ψm |
(2)
ρ̂out =
m
where the input mixed state is ρ̂s = Pm |ϕm ϕm | and Pm is the probability for the state |ϕm . If we condition
the preparation of the signal state to single-photon detection on the idler channel, we obtain the prepared state
ρ̂ = Tri (ρ̂out |1i 1|i ) = |g|2 ↠ρ̂s â.
(3)
When the input state ρ̂s is a thermal state with mean photon number n̄, we obtain that the single-photon-added
thermal state is described by the following density operator expressed in the Fock base:
ρ̂ =
n
∞ n̄
1
n |n n|.
n̄(n̄ + 1) n=0 1 + n̄
(4)
The lack of the vacuum term and the rescaling of higher excited terms is evident in this expression. The P
phase-space representation can be easily calculated and is given by (see also Ref. 12)
P (α) =
2
1
[(1 + n̄)|α|2 − n̄]e−|α| /n̄ ,
πn̄3
(5)
while the corresponding Wigner function reads as
W (α) =
2 |2α|2 (1 + n̄) − (1 + 2n̄) −2|α|2 /(1+2n̄)
e
π
(1 + 2n̄)3
(6)
where α = x + iy. SPATSs have a well-behaved P function which is always negative around α = 0; this feature
is also present in the Wigner function and assures their nonclassicality, however both P (0) and W (0) tend to
zero in the limit of n̄ → ∞.
3. EXPERIMENT
The main source of our apparatus is a mode-locked Ti:Sa laser which emits 1.5-ps pulses with a repetition rate
of 82 MHz. The pulse train is frequency-doubled to 393 nm by second harmonic generation in a 13-mm long
LBO crystal. The spatially-cleaned UV beam then serves as a pump for a 3-mm thick type-I BBO crystal which
generates spontaneous parametric down-conversion (SPDC) at the same wavelength of the laser source. The
SPDC crystal slightly tilted from the collinear configuration in order to obtain an exit cone beam with an angle
of ∼ 3◦ thus the pairs of SPDC photons are emitted in two distinct spatial channels called signal and idler.
In order to remotely select a pure state on the signal channel, idler photons undergo narrow spatial and
frequency filtering before detection; indeed the remotely-prepared signal state will only approach a pure state if
the filter transmission function is much narrower than the momentum and spectral widths of the pump beam
generating the SPDC pair.18–21 Along the idler channel the photons are strongly filtered in the spectral and
spatial domain by means of two etalon filters (total width ≈ 50 GHz) and by a single mode fiber (core/cladding
5/125 µm with collimation lens of 6.2-mm focal length) which is directly connected to the single-photon-counting
module (Perkin-Elmer SPCM AQR-14), further details are given in Refs. 22 and 13.
Proc. of SPIE Vol. 6305 63050Z-2
Loser
HT
SPCM
Filters
LBO N
Pump
Figure 1. Experimental setup. HR (HT) is a high reflectivity (transmittivity) beam splitter; all other symbols are defined
in the text.
When no seed field is injected in the SPDC crystal, conditioned single-photon Fock states are generated from
spontaneous emission in the signal channel.22, 23 We have recently shown that, if the SPDC crystal is injected
with a coherent state along the signal channel, stimulated emission comes into play and single-photon excitation
of such a pure state is obtained.3, 13 However, a coherent state is still at the border between the quantum and
the classical regimes; it is therefore extremely interesting to use a truly classical state, like the thermal one, as
the input, and to observe its degaussification.10 In order to avoid technical problems connected to the handling
of high-temperature thermal sources, we use a rotating ground glass disk (RD) inserted in a portion of the laser
beam to inject the parametric amplifier (see Fig. 1). A superposition of diffracted contributions with random
amplitudes and phases is thus realized which yields the photon distribution typical of a thermal source.24 The
mean number of photons for the thermal state n̄ is varied by a controlled attenuation (not shown in figure,
composed of a polarizer and a half-wave plate) of a small portion of the laser emission impinging the rotating
round glass disk.
Balanced homodyne measurements are performed on the signal output of the SPDC crystal: the signal output
of the SPDC crystal is mixed with a strong local oscillator (LO) by means of a 50% beam splitter (BS). The LO
is an intense coherent state obtained from a portion of the original laser pulses which is spatially mode-matched
to the conditionally-prepared signal state by the insertion of appropriate lens combinations (not shown in the
figure) along its path. In order to finely adjust the alignment and the synchronization between the signal and
LO pulses, we use the stimulated beam produced by injecting a different seed pulse into the idler channel of the
parametric crystal. Under appropriate conditions,19 the amplified seed beam appearing on the signal channel
due to stimulated emission can spatially well simulate the target signal beam and can be used for alignment
purposes.
4. TIME-DOMAIN HOMODYNE MEASUREMENTS
The two output beams of the BS are detected by two photodiodes (Hamamatsu S3883, with active area 1.7 mm2 )
whose difference signal is amplified and sent to a fast digital oscilloscope whose acquisition is triggered by the
detection events in the idler channel. Each acquisition frame spans two consecutive LO pulses where only the
first one is synchronized with the detection of an idler photon and contains the “information” about the desired
state, while the second one can be used as a reference. If the seed thermal state is blocked, the single-photon
Fock state |1s and the LO shot-noise distributions corresponding to the vacuum state |0s are measured (for
further details see Refs. 25 and 22). About 5000 acquisition frames can be stored sequentially in the scope at a
Proc. of SPIE Vol. 6305 63050Z-3
Figure 2. Quadrature measurements: a) Vacuum; b) and c) Thermal state and SPATS with n̄ = 0.093, respectively.
maximum rate of 160,000 frames per second. Each sequence of frames is then transferred to a personal computer
where the areas of the pulses are measured and their statistic distributions are analyzed in real time.
If a narrow temporal gate with the laser pulses is used, the typical rate of state preparation for vacuum
input is about 300 s−1 , with less than 1% contribution from accidental counts. A typical sequence of about 5000
acquisition frames can thus be captured and analyzed in about 20-30 s when no thermal seed is fed in the signal
mode. When the thermal states are injected into the parametric amplifier stimulated emission takes place and
the count rate in the idler channel is proportional to n̂ = Tr(ρ̂out â†i âi ) = |g|2 (1 + n̄). The mean photon number
values n̄ reported in the following are obtained from the ratio between the trigger count rates when the thermal
injection is present (Ct ) and when it is blocked (Co ) as n̄ = Ct /Co − 1 (see Refs. 26 and 27).
In Figs. 2 and 3 are shown some of the quadrature measurements for the vacuum state, thermal state and
SPATS for n̄ = 0.093 and n̄ = 1.1, while the phase between the signal and LO is scanned. The Figures 2a) and
3a) relative to the vacuum are the quadrature measurements obtained from the integration of the homodyne
signal relative to the second acquired pulse (reference) when the thermal seed is blocked. When the thermal seed
is turned on – see Figs. 2b) and 3b) – the quadrature measurements relative to the second acquired pulse contain
the thermal state which is present in the signal channel. Finally, both Figs. 2c) and 3c) show the quadrature
measurements obtained from the synchronized pulse, i.e. when conditional measurements are performed on the
SPATS. The variance increases when the single photon excitation of the thermal state takes place. It is also
important to note that the analyzed states do not show any phase dependence as expected from Eq. (4). The
full acquisitions for each fixed n̄ consist of about 5 × 105 quadrature values obtained with random phases.
5. QUANTUM STATE RECONSTRUCTION
Balanced homodyne detection allows the measurement of the signal electric field quadratures x̂θ = x̂ cos θ+ ŷ sin θ
as a function of the relative phase θ imposed between the LO and the signal, where the two orthogonal field
quadratures x̂ and ŷ are defined as x̂ = 12 (â + ↠) and ŷ = 2i (↠− â) and [x̂, ŷ] = i/2. By performing a series of
homodyne measurements on equally-prepared states it is possible to obtain the probability distributions p(x, θ)
of the quadrature operator x̂θ = 12 (âe−iθ + ↠eiθ ) that are simply seen to correspond to the marginals of the
Wigner quasi-probability distribution W (x, y):28
p(x, θ) =
+∞
−∞
W (x cos θ − y sin θ, x sin θ + y cos θ)dy.
Proc. of SPIE Vol. 6305 63050Z-4
(7)
Figure 3. Quadrature measurements: a) Vacuum; b) and c) Thermal state and SPATS with n̄ = 1.1, respectively.
Given a sufficient number of quadrature distributions at different values of the phase θ ∈ [0, π], one is able to
reconstruct the quantum state of the field under study.29
We have performed the reconstruction of the diagonal density matrix elements using the maximum likelihood
estimation.30 This method gives the density matrix that most likely represents the measured homodyne data.
Firstly, we build the likelihood function contracted for a density matrix truncated to 15 diagonal elements
(with the constraints of Hermiticity, positivity and normalization), then the function with 15 free parameters is
maximized by standard numerical procedures over the experimental data. The Wigner function is then obtained
from the density matrix elements using the straightforward transformation:
ρnn Wnn (x, y),
(8)
W (x, y) =
n
where ρnn is the diagonal element of the density matrix and Wnn (x, y) is the Wigner function of the projector
|n n|.
The reconstructed Wigner functions for two thermal states with n̄ = 0.093 and n̄ = 1.1 detected in the signal
channel using the reference (un-synchronized) pulse are shown in Fig. 4.
When the trigger-synchronized homodyne measurements are used to reconstruct the state we expect to
observe the single-photon excitation of the thermal state and the results are shown in the Fig. 5. From Eq. (6)
the Wigner function of the SPATSs should be always negative. It is evident that for the SPATS of Fig. 5b) with
n̄ = 1.1 the negativity of the Wigner function completely disappeared while it is still evident in the case with a
lower n̄. Here the finite experimental efficiency in the preparation and detection of SPATSs results in a mixture
of the prepared state with vacuum and with a portion of the injected thermal state (the latter arising from
limited purity of the remote preparation and from the SPCM dark counts). The measured homodyne detection
and preparation efficiency for the single-photon Fock state (n̄ = 0) is η = 0.64.
In order to observe how the nonclassical character of the Wigner function disappears, we have performed
different reconstructions of the SPATSs while gradually varying the n̄ of the injected thermal field. It is observed
that the measured negativity of the Wigner function at the origin - see Fig. 6 - tends to vanish as the mean photon
number of the input thermal state is increased. The nonclassical features of the final states are thus expected to
get progressively weaker for large n̄ and be finally hidden by unwanted vacuum and thermal components.
The negativity of the Wigner function is a sufficient but not necessary condition to prove the nonclassicality
of a quantum state. Recently, several nonclassical criteria have been proposed17, 31–33 and the SPATSs can play
the role of benchmark state to put such criteria to an experimental test.
Proc. of SPIE Vol. 6305 63050Z-5
Figure 4. Wigner functions obtained by quantum homodyne tomography for thermal states: a) n̄ = 0.093; b) n̄ = 1.1.
Figure 5. Wigner functions obtained by quantum homodyne tomography for SPATSs: a) n̄ = 0.093; b) n̄ = 1.1.
6. CONCLUSION
We have demonstrated the generation of nonclassical light states by adding a single photon to a completely
classical thermal field. We have used quantum homodyne tomography to reconstruct the Wigner function which
shows negative values around the origin. The SPATSs show a tunable degree of nonclassicality that gives the
possibility to compare several criteria of nonclassicality recently proposed.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Koji Usami for giving the initial stimulus for this work and Milena D’Angelo
for useful discussions. This work has been performed in the frame of the “Spettroscopia laser e ottica quantistica”
project of the Department of Physics of the University of Florence and partially supported by Ente Cassa di
Risparmio di Firenze and MIUR, under the PRIN initiative and FIRB contract RBNE01KZ94.
Proc. of SPIE Vol. 6305 63050Z-6
Figure 6. Sections of the reconstructed Wigner functions for the SPATSs for different values of the mean photon number
of the injected thermal state.
REFERENCES
1. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the wigner distribution and the
density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the
vacuum,” Phys. Rev. Lett. 70, pp. 1244–1247, 1993.
2. G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, pp. 471–475, 1997.
3. A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent
states of light,” Science 306, pp. 660–662, 2004.
4. A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent
states with inefficient photon detection,” Phys. Rev. A 70, p. 020101(R), 2004.
5. H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical
fields with inefficient photon detection,” Phys. Rev. A 72, p. 013801, 2005.
6. M. S. Kim, W. Son, V. Bužek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a
prerequisite for entanglement,” Phys. Rev. A 65, p. 032323, 2002.
7. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77,
pp. 513–577, 2005.
8. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, pp. 2766–2788, 1963.
9. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light
beams,” Phys. Rev. Lett. 10, pp. 277–279, 1963.
10. J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-gaussian statistics from individual pulses of squeezed
light,” Phys. Rev. Lett. 92(15), p. 153601, 2004.
11. M. S. Kim, E. Park, P. L. Knight, and H. Jeong, “Nonclassicality of a photon-subtracted gaussian field,”
Phys. Rev. A 71, p. 043805, 2005.
12. G. S. Agarwal and K. Tara, “Nonclassical character of states exhibiting no squeezing or sub-poissonian
statistics,” Phys. Rev. A 46, pp. 485–488, 1992.
13. A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary
step of stimulated light emission,” Phys. Rev. A 72, p. 023820, 2005.
14. C. T. Lee, “Theorem on nonclassical states,” Phys. Rev. A 52, pp. 3374–3376, 1995.
15. G. N. Jones, J. Haight, and C. T. Lee, “Nonclassical effects in the photon-added thermal state,” Quantum
Semiclass. Opt. 9, pp. 411–418, 1997.
Proc. of SPIE Vol. 6305 63050Z-7
16. L. Diósi, “Comment on nonclassical states: An observable criterion,” Phys. Rev. Lett. 85, p. 2841, 2000.
17. W. Vogel, “Nonclassical states: An observable criterion,” Phys. Rev. Lett. 84, pp. 1849–1852, 2000.
18. Z. Y. Ou, “Parametric down-conversion with coherent pulse pumping and quantum interference between
independent fields,” Quantum Semiclass. Opt. 9, pp. 599–614, 1997.
19. T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by
means of conditional measurements on a biphoton state,” Eur. Phys. J. D 18, pp. 237–245, 2002.
20. M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, “Nonlocal pulse shaping with entangled
photon pairs,” Phys. Rev. Lett. 90, p. 043602, 2003.
21. S. Viciani, A. Zavatta, and M. Bellini, “Nonlocal modulations on the temporal and spectral profiles of an
entangled photon pair,” Phys. Rev. A 69, p. 053801, 2004.
22. A. Zavatta, S. Viciani, and M. Bellini, “Tomographic reconstruction of the single-photon fock state by
high-frequency homodyne detection,” Phys. Rev. A 70, p. 053821, 2004.
23. A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction
of the single-photon fock state,” Phys. Rev. Lett. 87, p. 050402, 2001.
24. F. T. Arecchi, “Measurement of the statistical distribution of gaussian and laser sources,” Phys. Rev.
Lett. 15, pp. 912–916, 1965.
25. A. Zavatta, M. Bellini, P. L. Ramazza, F. Marin, and F. T. Arecchi, “Time-domain analysis of quantum
states of light: noise characterization and homodyne tomography,” J. Opt. Soc. Am. B 19, pp. 1189–1194,
2002.
26. D. N. Klyshko, “Utilization of vacuum fluctuations as an optical brightness standard,” Sov. J. Quantum
Electron. 7, pp. 591–595, 1977.
27. G. K. Kitaeva, A. N. Penin, V. V. Fadeev, and Y. A. Yanait, “Measurement of brightness of light fluxes
using vacuum fluctuations as a reference,” Sov. Phys. Dokl. 24, pp. 564–566, 1979.
28. K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, pp. 2847–2849(R), 1989.
29. U. Leonhardt, Measuring the quantum state of light, Cambridge University Press, Cambridge, England,
1997.
30. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, “Maximum-likelihood estimation of the
density matrix,” Phys. Rev. A 61, p. 010304(R), 1999.
31. J. K. Asboth, J. Calsamiglia, and H. Ritsch, “Computable measure of nonclassicality for light,” Phys. Rev.
Lett. 94, p. 173602, 2005.
32. T. Richter and W. Vogel, “Nonclassicality of quantum states: A hierarchy of observable conditions,” Phys.
Rev. Lett. 89, p. 283601, 2002.
33. T. Richter and W. Vogel, “Observing the nonclassicality of a quantum state,” Quantum Semiclass. Opt. 5,
pp. 371–380, 2003.
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