Fresnel Representation of the Wigner Function: An Operational ...

Fresnel Representation of the Wigner Function: An Operational Approach P. Lougovski1,2, E. Solano1,3 , Z. M. Zhang1,4 , H. Walther1,2 , H. Mack5 , and W. P. Schleich5

1 Max-Planck-Institut f¨ ur Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany Sektion Physik, Ludwig-Maximilians-Universit¨ at M¨ unchen, Am Coulombwall 1, D-85748 Garching, Germany 3 Secci´ on F´ısica, Departamento de Ciencias, Pontificia Universidad Cat´ olica del Per´ u, Apartado 1761, Lima, Peru 4 Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China 5 Abteilung f¨ ur Quantenphysik, Universit¨ at Ulm, D-89069 Ulm, Germany

arXiv:quant-ph/0206083v3 16 Apr 2003

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We present an operational definition of the Wigner function. Our method relies on the Fresnel transform of measured Rabi oscillations and applies to motional states of trapped atoms as well as to field states in cavities. We illustrate this technique using data from recent experiments in ion traps [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)] and in cavity QED [B. Varcoe et al., Nature 403, 743 (2000)]. The values of the Wigner functions of the underlying states at the origin of phase space are W|0i (0) = +1.75 for the vibrational ground state and W|1i (0) = −1.4 for the one-photon number state. We generalize this method to wave packets in arbitrary potentials.

Currently the Wigner function [1] enjoys a renaissance in many branches of physics, ranging from quantum optics [2], nuclear [3] and solid state physics to quantum chaos [4]. This renewed interest has triggered a search for operational definitions [5] of the Wigner function, that is, definitions which are based on experimental setups [6]. In the present paper we propose such a definition based on the resonant interaction of an atom with a single mode of the electromagnetic field. In contrast to earlier work it is the atomic dynamics that performs the major part of the reconstruction. Our definition is well-suited for the Jaynes-Cummings model [7] but allows a generalization to other quantum systems. Many methods to reconstruct the Wigner function [7] of a cavity field or the motional state of a harmonic oscillator have been proposed [8]. In particular, the Wigner function of a cavity field can be expressed in terms of the measured atomic inversion [9]. This operational scheme [10] lives off the dispersive interaction between the atom and the field. Since this scheme requires long interaction times, an operational definition of the Wigner function based on a resonant interaction is desirable. The method of nonlinear homodyning [11] and quantum state endoscopy [12] fulfill this need, but require rather complicated reconstruction schemes. In contrast, the Fresnel representation proposed in the present paper is rather elementary. It expresses the value of the Wigner function at a phase space point α as a weighted time integral of the measured atomic dynamics caused by the state displaced by α. The weight function is the Fresnel phase factor. Since this method can be applied to a cavity field as well as a trapped ion [13] we use a harmonic oscillator as a model system. However, we show that this approach can easily be generalized to non-harmonic oscillators. Our definition relies on controlled displacements [13, 14] of the quantum state of interest and the observation of Rabi oscillations of a two-level atom interacting resonantly with this field. We record the probability [15] Pg (τ ; α) =

∞ √ 1 1X − Pn (α) cos (2 n + 1τ ) 2 2 n=0

(1)

of finding the atom in the ground state |gi as a function of dimensionless interaction time τ and complex-valued ˆ ˆ † (α)|ni dedisplacement α. Here Pn (α) ≡ hn|D(α)ˆ ρD notes the occupation statistics of the state ρˆ displaced ˆ by the displacement operator D(α). The Wigner function W (α) of the original state ρˆ is determined [16] by the alternating sum W (α) ≡ 2

∞ X

(−1)n Pn (−α)

(2)

n=0

of the probabilities Pn . Hence, we can find the Wigner function from the atomic dynamics when we measure the probabilities Pg for different interaction times and solve Eq. (1) for the occupation probability Pn (α). However, there is no need to evaluate the probabilities Pn explicitly [17]. We can obtain the Wigner function directly from the atomic dynamics Pg without ever calculating Pn , making use of the integral relation Z ∞ √ 2 √ (3) dτ exp (iτ 2 /π) cos(2 nτ ) = (−1)n . π i 0 When √ we multiply Pg − 1/2 from Eq. (1) by 2/(π i) exp [iτ 2 /π] and integrate over τ , Eq. (3) yields 2 √

π i

Z

∞

0

dτ eiτ

2



/π

Pg (τ ; −α)−

 ∞ 1 1X = (−1)n Pn (−α). 2 2 n=0

We recall the connection, Eq. (2), between W (α) and Pn (α) and find the Wigner function W (α) ≡ lim W (α; τm ) in terms of the truncated Fresnel trans-

τm →∞

form [18] W (α; τm ) ≡ 4

Z

0

τm

  1 2 dτ iτ 2 /π √ e Pg (τ ; −α) − 2 π i

(4)

of the Rabi oscillations. Hence, the Wigner function at the phase space point α is determined by a weighted time integral of the atomic dynamics due to the initial state displaced by −α. The weight function is the Fresnel phase factor exp(iτ 2 /π).

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FIG. 1: Fresnel representation of Wigner function illustrated by the first excited energy eigenstate of a harmonic oscillator. The atomic dynamics Pg (τ ; α), Eq. (1), as a function of interaction time τ and real-valued displacement α (top) serves as input to the truncated Fresnel representation, Eq. (4). For a finite measurement time τm the so-obtained function W (α; τm ) is complex-valued. The real part of W (α; τm ) (bottom) approaches the correct Wigner function, W|1i , already for moderate measurement times.

FIG. 2: Asymptotic approach of the truncated Fresnel representation for the first excited state. Real part (black line) and imaginary part (gray line) of W (α = 0; τm ) oscillate around their asymptotic values −2 and 0, respectively, with slowly decreasing amplitudes. The asymptotic values emerge already after a measurement time of the order of 2π.

It is instructive to compare the Fresnel representation, Eq. (4), to the Fourier method used in Ref. [13]. The latter relies on the analysis of the Fourier transform of the

Rabi oscillations Pg and thus needs the full functional dependence on frequency. In contrast, the Fresnel representation gives the Wigner function as a single integral of Pg without the need to analyze an intermediate function. The Fresnel representation seems to suffer from three disadvantages: (i) It contains the complete time evolution of Pg , that is, from τ = 0 to τ = ∞. However, any experiment can only record the dynamics for a finite measurement time τm . (ii) It can take on complex values. However, the Wigner function is always real. (iii) It relies on a continuous time evolution. However, the experiment can only provide a discrete sampling. We address each of these problems separately and first demonstrate that the phase factor exp(iτ 2 /π) makes the integral insensitive to the long time behavior of Pg . For this purpose we show in Fig. 1 the Fresnel reconstruction for the first excited energy eigenstate of a harmonic oscillator. The corresponding Wigner func
tion [7] W|1i (α) = −2 1 − 4|α|2 exp(−2|α|2 ) is rotationally symmetric. Therefore, it is sufficient to depict W|1i along the real axis. The top of Fig. 1 displays the atomic dynamics Pg as a function of interaction time τ and real-valued displacement α. Sinusoidal Rabi oscillations appear along α = 0. Moreover, for appropriately large displacements we find collapses and revivals. In the bottom part we present the real part of W (α; τm ). The time axis corresponds to the upper limit τm of the truncated Fresnel representation, that is the measurement time. For very short times the function has no similarity with the correct Wigner function, W|1i . However, for longer measurement times the curves approach the exact Wigner function. In order to study the asymptotic behavior in more detail we show in Fig. 2 a cut of the truncated Fresnel representation along α = 0. Here we depict both, real and imaginary part of W (α; τm ). We note that after τm ≈ 2π the real part, indicated by the black line, begins to oscillate around the correct value W|1i (α = 0) = −2 with decreasing amplitude. In a similar way, the imaginary part, denoted by the gray line, approaches its asymptotic value zero. Unfortunately, this convergence is slow. This feature is a consequence of the familiar Cornu spiral [7], also used by R. P. Feynman and J. A. Wheeler to add up scattered waves — a precursor of the path integral [19]. Despite the slow asymptotics the knowledge of the Rabi oscillations up to a measurement time of a least τm ∼ = 2π is sufficient to obtain the asymptotic Wigner function. The upper part of Fig. 3 shows measured Rabi oscillations of an ion due to its center-of-mass motion [20, 21]. Due to the high sampling frequency we can directly interpolate the measured data to evaluate the Wigner function. In the bottom part we display the approaches of the real part (black line) and imaginary part (gray line) of the truncated Fresnel integral towards their asymptotic values (dashed lines). For the ion in the motional ground state with Wigner function [7] W|0i (α) = 2 exp(−2|α|2 ) we find the asymptotic value W|0i (0) = +1.75 + i 0.05 whereas the ideal value is W|0i (0) = 2. The presence

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FIG. 4: Dependence of the Wigner function W (N) (α = 0) at the origin of phase space on the cutoff N of the overdetermined system, Eq. (5). The values scatter around W|1i (0) = −1.4. The Rabi oscillations [23] (inset) result from a cavity field prepared in a one-photon number state.

FIG. 3: Wigner function at the origin of phase space obtained from experimental Rabi oscillations (top) of a stored ion [20] using the truncated Fresnel representation, Eq. (4): Approach (bottom) of real (black line) and imaginary parts (gray line) of W (α = 0; τm ) towards their asymptotic values (dashed line) yielding W|0i (0) = 1.75+i0.05. The ion was prepared initially in the atomic and vibrational ground state.

of an imaginary part in the estimated Wigner function does not represent a fundamental drawback, as long as it comes from a complex integral of realistic data that ideally should produce a real value. It is quite remarkable that the method works so well considering the fact that in the particular experiment [20] the atomic dynamics is governed by the nonlinear Jaynes-Cummings model [22]. In this case the relation between the atomic population and the photon statistics is not of the form of Eq. (1). However, for low excitations the deviations from the familiar JaynesCummings model are small as shown in Fig. 1(b) of Ref. [20]. Therefore, superpositions of low excitations can be successfully reconstructed. The inset of Fig. 4 shows Rabi oscillations of an atom due to a cavity field [23]. Since in this case only a few data points are available we have to employ a discrete version of the Fresnel representation. The data [23] consist of a finite set of M interaction times τj and measured probabilities Pg (τj ; −α). We now choose kernel coefficients fj such that M X

√ cos(2 n + 1τj ) fj = (−1)n+1 ,

(5)

j=1

multiply Eq. (1) evaluated at discrete times τj by fj , and sum over j. With the help of Eq. (5) we then arrive at the discrete representation W (α) = 4

M X

  1 fj Pg (τj ; −α) − 2 j=1

(6)

of the Wigner function. Since the photon number n runs from zero to infinity, Eq. (5) describes a system of infinitely many equations for M unknown coefficients fj . When we consider the truncated system of N > M equations we find the so(N ) of this overdetermined system by minimizlutions fj ing the sum of quadratic deviations. Substitution of the (N ) so-calculated coefficients fj into Eq. (6) results in the Wigner function W (N ) (α). Figure 4 displays the dependence of W (N ) (α = 0) on the cutoff N . The values scatter around W|1i (0) = −1.4 indicated by the dashed line. We recall that the ideal value of the Wigner function corresponding to a onephoton number state is W|1i (0) = −2. For N = M = 10 the method is extremely sensitive to uncertainties in the measured data resulting in a large deviation from -1.4. The error bars δPj translate into a standard deviation i1/2 hP 2 2 of the Wigner function. AccordδW = 4 f δP j j j ing to Eq. (5) the coefficients fj are solely determined by the set of τj and can take almost any values. Obviously large values give rise to a large error in the Wigner function, unless the corresponding uncertainties δPj are small and compensate this effect. For the experiment of Ref. [23] performed for a different purpose and therefore not taking advantage of this feature, we find δW ≈ 4.3. However, a future experiment can either choose the interaction times τj such as to minimize the fj or increase the accuracy of the measurement of Pg at times where fj is large. The Fresnel representation, Eq. (4), of the Wigner function as well as the discrete version, Eq. (6), rely on the resonant Jaynes-Cummings model. However, the technique to obtain the Wigner function directly by an appropriate integral transform of the time dependence of a measurable quantity is much more general. Indeed, our method even allows us to obtain P the Wigner function of a wave packet |ψ(0)i = n ψn |φn i consisting of a superposition of energy eigenstates |φn i with en-

4 ergy En ≡ ~ωn of a potential V ≡ V (~r). Here, we do not restrict ourselves to a harmonic oscillator potential. Our only assumption is that V is symmetric with respect to the origin. In this case, the representation Eq. (2) of the Wigner function still holds true. Moreover, we recall that P the autocorrelation function C(t) ≡ hψ(0)|ψ(t)i = n |ψn |2 exp(−iωn t) is measured routinely in wave packet experiments, e.g. [24]. ˆ to the initial When we apply the displacement D state ρ ≡ |ψ(0)ihψ(0)|, the autocorrelation function P C(t; α) = n Pn (α) exp(−iωn t) is similar to Eq. (1) with ˆ ˆ † (α)|φn i. We multiply both sides Pn (α) ≡ hφn |D(α)ˆ ρD of the equation for C by Z ∞ 1 dω eiωt cos[n(ω)π] (7) f (t) ≡ 2π −∞ and integrate over t. Here n(ω) is an arbitrary and continous function such that n(ωn ) = n. When we make use of the symmetry relations f (−t) = f ∗ (t) and C(−t) = C ∗ (t), we find the Wigner representation Z ∞  W (α) = 4 Re dtf (t)C(t; −α) (8) 0

of the initial wave packet in terms of the measured auto-

[1] M. Hillery et al., Phys. Rep. 106, 121 (1984). [2] M. Freyberger et al., Phys. World 10 (11), 41 (1997); D. Leibfried et al., Phys. Today 51(4), 22 (1998); D. G. Welsch et al., in: Progress in Optics XXXIX, ed. by E. Wolf (North-Holland, Amsterdam, 1999). [3] H. Feldmeier et al., Prog. Part. Nucl. Phys. 39, 393 (1997). [4] M. V. Berry, in: Les Houches Lecture Series Session XXXVI, eds. G. Iooss et al., (North Holland, Amsterdam, 1983); S. Habib et al., Phys. Rev. Lett. 80, 4361 (1998). [5] W. E. Lamb, Phys. Today 22(4), 23 (1969). [6] D. T. Smithey et al., Phys. Rev. Lett. 70, 1244 (1993); Ch. Kurtsiefer et al., Nature 386, 150 (1997); G. Breitenbach et al., Nature 387, 471 (1997); K. Banszek et al., Phys. Rev. A 60, 674 (1999); A. I. Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001). [7] See for example W. P. Schleich, Quantum Optics in Phase Space (VCH-Wiley, Weinheim, 2001). [8] K. W´ odkiewicz, Phys. Rev. Lett. 52, 1064 (1984); A. Royer, Phys. Rev. Lett. 55, 2745 (1985); K. Banaszek et al., Phys. Rev. Lett. 76, 4344 (1996); S. Wallentowitz et al., Phys. Rev. A 53, 4528 (1996); U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, Cambridge, 1997). [9] B.-G. Englert et al., Optics Comm. 100, 526 (1993); L. G. Lutterbach et al., Phys. Rev. Lett. 78, 2547 (1997). [10] G. Nogues et al. [Phys. Rev. A 62, 054101 (2000)] measured the Wigner function at the origin of phase space

correlation function C and the integral kernel f . Equation (8) is the generalization of the Fresnel representation, Eq. (4), to a quantum system with arbitrary discrete spectrum. For wave packets in a harmonic oscillator or in a box with energy spectra ωn ≡ 2πn/Tcl or ωn ≡ 2πn2 /Tr , respectively, the integral kernels from Eq. (7) are delta functions at half of the classical period Tcl , or half of the revival time Tr . For these potentials, n the time evolution itself creates the function (−1) , that √ is the parity operator. For a spectrum ωn ≡ n + 1 2Ω, Eq. (7) predicts a quadratic Fresnel-like phase. Complications arise for spectra with degeneracies, such as the hydrogen atom. Nevertheless, we can reconstruct Rydberg wave packets measuring the autocorrelation function at half of the revival time. These and other relevant examples will be discussed elsewhere. They suggest a wide application of integral (Fresnel) transforms in the search for a practical reconstruction of the Wigner function. We thank M. Freyberger for fruitful discussions and D. Wineland for allowing us to use his data. The work of H. M. and W. P. S. is partially supported by the Deutsche Forschungsgemeinschaft. H. W. and W. P. S. acknowledge financial support from the European Commission through the IHP Research Training Network QUEST and the IST project QUBITS.

[11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

[22] [23] [24]

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