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The Wigner function and quantum state tomography Boris Braverman Department of Physics, MIT-Harvard Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: May 16, 2012) We present a pedagogical introduction to the subjects of Wigner functions and optical homodyne tomography. The basic formalism of Wigner functions is given and its connection to tomographic reconstruction of the quantum state of light discussed. The experimental signatures of nontrivial quantum states such as Schrödinger cats are presented, along with a survey of notable experimental measurements of Wigner functions. A complete simulated quantum tomography experiment is given, highlighting both the robustness and limitations of the procedure. INTRODUCTION WIGNER FUNCTION Quantum mechanics is the most complete theory of Nature currently known, but it often conflicts with our classical intuition. A notable deviation between quantum and classical theories is the manner in which the states of systems are specified. Quantum systems are described as vectors (wavefunctions) |ψi within a Hilbert space, while classical systems are properly characterized as a point in phase space. A powerful use of the classical phase space formalism is the Gibbs ensemble commonly used in statistical mechanics, in which the states of a fictitious ensemble of systems are described as a probability distribution in phase space [1]. Time evolution in this picture is given by Hamilton’s equations, and the expectation values of any physical observables can be obtained by averaging over the phase space. Such an ensemble picture is often used to introduce the inherently uncertain and statistical nature of quantum mechanics and quantum measurement. Despite all its strengths, a Gibbs ensemble cannot replicate all the predictions of quantum mechanics, in particular ones where interference between different components of the ensemble can occur. In addition, while a classical system can be perfectly localized, the state of a quantum mechanical system must satisfy Heisenberg’s uncertainty principle. Nonetheless, representations of quantum states within a phase space can be made, of which the Wigner function [2] is the most notable. The Wigner function provides an accurate description for certain measurements performed on quantum states, but only at the cost of being potentially negative. Therefore, the Wigner function cannot be interpreted as a regular probability distribution for a random variable, and is usually referred to as a “quasiprobability distribution”. While the possibility of negative probabilities having physical meaning seems bizzare, this possibility has been studied by notable physicists, such as Dirac [3] and Feynman [4]. If one comes to terms with the possibility of having a negative-valued quasiprobability distribution for the states of a system, then the entirety of quantum mechanics can be reformulated in terms of the dynamics of the Wigner function within phase space [5, 6]. Consider a system that exists in a continuous 1dimensional Hilbert space. Any state |ψi in the space can R be represented as a complex wavefunction: |ψi = |qi ψ(q)dq. Similarly, a Rgeneral mixed state is expressed as a density matrix ρ̂ = ρ(q1 , q2 ) |q1 i hq2 |. The Wigner function W (q, p) provides an equivalent representation of any quantum state in the quadrature phase space (q, p). In the general case of a mixed state, the Wigner function equals [2, 7]: Z x x ipx/h̄ 1 ρ q − ,q + e dx, (1) W (q, p) ≡ 2πh̄ 2 2 while for the special case of a pure quantum state, the expression is evidently Z 1 x x ipx/h̄ ψ∗ q − W (q, p) ≡ ψ q+ e dx. 2πh̄ 2 2 Since this relationship is nothing but an unusual Fourier transform, one can straighforwardly invert it to obtain the density matrix from the Wigner function: Z x1 + x2 ρ(x1 , x2 ) = h̄ W , p e−ip(x2 −x1 )/h̄ dp. (2) 2 Note that while the Wigner function is most commonly used in discussions of the quantum states of a 1-dimensional harmonic oscillator, such as a single mode of the electromagnetic field, this representation is quite general and can be used to represent any system. The only thing that changes then is the time evolution of the Wigner function, governed by the Moyal equation [5]. In subsequent discussion, we assume that the Hamiltonian of the system is that of a harmonic oscillator, Ĥ = P̂ 2 mω 2 Q̂2 + , 2m 2 with energy eigenstates 1 Ĥ |ni = h̄ω n + |ni , 2 2 n = 0, 1, 2, . . ., and using the coherent states, defined as |αi ≡ ∞ X αn √ |ni . n! n=0 The most important feature of the Wigner function is that it provides a faithful prediction for measurements of quadratures of the quantum mechanical state. That is, if we measure the quadrature operator M̂ (θ) ≡ Q̂ cos θ + P̂ sin θ, then the statistical distribution of the measurement outcomes m will equal the Radon transform of the Wigner function, given by Z∞ W (m cos θ − p sin θ, m sin θ + p cos θ)dp. p(m, θ) = −∞ (3) where S(Q̂, P̂ ) is summed over all possible orderings of Q̂ and P̂ . For example, if m = 2, n = 2, then S(Q̂2 , P̂ 2 ) = (Q̂2 P̂ 2 + P̂ 2 Q̂2 + P̂ Q̂P̂ Q̂ + Q̂P̂ Q̂P̂ + P̂ Q̂2 P̂ + Q̂P̂ 2 Q̂)/6. Because of these limitations in the intuitive predictive power of the Wigner function, it is most of all useful as an intermediate tomographic stage en route to the reconstruction of the quantum state, which can then of course predict the outcome of any measurement. Finally, we note that the Wigner function is by no means a unique way of representing a quantum state in phase space. Two other most commonly used representations of quantum states in phase space are the Q and P functions. The Q function equals the convolution of the Wigner function of a state with the Wigner function of the vacuum, which turns out to be the same as the overlap of the quantum state with different coherent states: Q(Re(α), Im(α)) = 1 hα| ρ̂ |αi . 2π (5) On the other hand, the P function is the deconvolution of the Wigner function of a state with the Wigner function of the vacuum, which turns out to be the coefficients of a decomposition of the state into coherent states: Z ρ̂ = P ((Re(α), Im(α)) |αi hα| dα. (6) FIG. 1: The Radon transform R(m, θ) of the function f (x, y) is found by integrating the function along the line connecting A and A0 . The Radon transform is a familiar operation from tomography, finding uses in fields such as medical imaging [8] and pattern recognition [9]. Its action is very simple, projecting out a 2-dimensional image onto different axes, as illustrated in Figure 1. For an example, compare the function in Figure 4d and its Radon transform shown in Figure 6. Note that this integral is precisely the manner in which one would predict the statistics of measurements performed on an ensemble of classical harmonic oscillators based on a statistical distribution of their locations in phase space. Beyond the potential negativity of the Wigner function, this quantum-classical analogy is limited in that one can predict the outcomes of only a subset of all possible quantum mechanical measuremnts in such a straightforward manner. In particular, one can predict the outcomes of measuring symmetrized operators: h m tr ρ̂S Q̂ , P̂ n i Z∞ Z∞ = −∞ −∞ W (q, p)q m pn dqdp, (4) These representations are sometimes useful, although mostly in a complementary role to the Wigner function. Especially notable is the fact that the P function can be used to define a rigorous classicality criterion: A state is classical if its P function is non-negative and is not more singular than a Dirac δ function anywhere. An equivalent set of nonclassicality criteria exist for the Wigner function. A state is nonclassical if its Wigner function is non-negative anywhere, which immediately implies that it cannot be expressed as a mixture of classical coherent states, or if the variance of any quadrature operator M̂ (θ) is smaller than its value in the vacuum state. These nonclassicality criteria for the Wigner function are useful because a deconvolution operation is necessary to find P , making it extremely difficult to reconstruct accurately in experiments. Wigner function examples The Wigner functions for the Fock states |0i, |1i, |2i, |3i are shown in Figure 2. Several common features are immediately apparent between these functions. The Wigner function of |ni resembles the wavefunction for |ni in having n zero-crossings; the functions are all radially symmetric; the even n states have a peak at the origin, while the odd n states have a dip at the origin. The magnitudes of the peak and dip are both equal to 1/π. 3 (a) Vacuum state |0i (b) Fock state |1i (c) Fock state |2i (d) Fock state |3i FIG. 2: Wigner functions of the first four Fock states. Each Wigner function is shown as a contour plot and as a 3D surface map, with red corresponding to positive values and blue to negative ones. Contours are spaced at 0.02. In fact, this value bounds any possible Wigner function: |W (q, p)| ≤ 1/π for any physical W (q, p). The vacuum state |0i is the only state that is simultaneously a coherent state and a Fock state, and indeed the Wigner function of any coherent state |αi is identical to that of the vacuum state, with a translation in phase space to P = Re(α), Q = Im(α). This embodies the quasi-classical, minimum-uncertainty nature of the coherent states–their quasi-probability distributions are as close to a δ function as quantum mechanics will allow. The Wigner functions of these Fock state have been successfully measured in many experiments. Some notable examples are the first reconstructions of a |1i Wigner function for the motion of an atom [10] and for an optical mode [11], as well as the |2i state in an optical mode [12]. Notice also that the Wigner function is linear in the density matrix of a quantum state. This means that the Wigner function is additive for the incoherent mixture of two states, but is not additive for a coherent superposition, which allows it to encode both phase and coherence of quantum states. For an example, compare Figures 3a, 3b, and 3c. For the coherent superpositions |0i + |1i and |0i + i |1i, the Wigner function gains an evident directionality which encodes the phase between the two states. On the other hand, the incoherent superposition ρ̂ = |0i h0| + |1i h1| has a Wigner function that is radially symmetric, with no phase to encode, and non-negative anywhere, indicating that the state is now classical. This destruction of quantum behaviour due to decoherence and state mixing is a familiar trend in quantum mechanics, and can be readily visualized and understood in the language of Wigner functions [13]. Finally, we consider some experimentally important examples of nonclassical states. Figure 4a shows a squeezed vacuum state, which in general is a state where the variance of one of a pair of canonically conjugate variables has been reduced below the vacuum noise level, while the variance of the other variable has correspondingly increased to satisfy the Heisenberg uncertainty principle. For the specific state in Figure 4a, the variance of the momentum hP̂ 2 i has been reduced by a factor of eζ ≈ 5, with a concommitant increase in the variance of the position variable by the same factor. These states can notably be generated using nonlinear optical processes [14], which makes them experimentally acessible, and they function as the primary source of nonclassical light in most current quantum optics experiment. Some of the first reconstructions of Wigner functions were indeed those of squeezed states [15, 16]. The state shown in Figure 4b is a photon subtracted squeezed vacuum state. This is a squeezed vacuum state that has been projected onto the subspace {|1i , |2i , . . .}, and then all energy eigenstates are reduced by one: |ni → |n − 1i. Again, the significance of this state comes from its experimental accessibility; one can generate such a state by taking a squeezed vacuum state and impinging it on a beamsplitter with near-unity transmission. If a photon is detected in the reflection mode from the beamsplitter, then this state will be generated in the transmission mode. These states have also been recently observed [17, 18]. 4 In Figure 4c, we show the Wigner function for a Schrödinger kitten state, which is simply a superposition of the two coherent states |α = 1i and |α = −1i. These two states are usually taken to represent the cat’s macroscopically distinguishable states |alivei and |deadi from Schrödinger’s famous thought experiment [19]. As can be seen, the two states |α = 1i and |α = −1i are not very distinguishable, and there are not two quasi-classical humps corresponding to the two states. However, if we increase the amplitude of the coherent states to 3, as done in Figure 4d, then we see all the features typical of Wigner functions of macroscopic superposition states: there are two humps corresponding to the Wigner function of each state independently, and in between there are rapid oscillations with large negative values that encode the coherence of the superposition of the two states. If the state was mixed rather than coherent, these fringes would not be present. The the generation and quantum tomography of cat states has only been realized recently [13, 20] because of the extreme difficulty of creating such large nonclassical states. (a) Coherent superposition state |0i + |1i (b) Coherent superposition state |0i + i |1i HOMODYNE DETECTION Above, we mentioned the remarkable property of the Wigner function to predict the statistics of the quadrature measurements of quantum light, which can be measured through the technique of balanced homodyne detection. Indeed, the popularity of the Wigner function as a representation of quantum optical states can be directly connected to its usefulness in understanding this type of experiments. The basic setup of a balanced homodyne tomography experiment is shown in Figure 5. The quantum state under study ρ̂ is interfered on a 50% transmission beamsplitter with a strong coherent state |αi = ||α|eiθ i, called the local oscillator (i.e. |α| nm , where hn| ρ̂ |ni is negligible for n > nm ). The phase θ of the local oscillator is varied, and the statistics of the difference in the currents I1 and I2 through the photodiodes PD1 and PD2 are recorded for different θ. If the current is proportional to the number of photons hitting each detector, then we have I2 − I1 ∝ n̂21 = n̂2 − n̂1 = â†2 â2 − â†1 â1 , where a1 and a2 are the annihilation operators for the light modes hitting the two photodetectors. Using the beamsplitter Hamiltonian, we have the operators for the photodiode modes 1 â1 = √ (â − α) 2 1 â2 = √ (â + α) 2 (c) Incoherent mixture |0i h0| + |1i h1| FIG. 3: The Wigner function encodes information about the amplitudes, phases, and coherences of quantum states. where we are treating the coherent state classically, so its operator is replaced by its classical amplitude. √ Recalling that α =√ |α|(cos θ + i sin θ), Q̂ = (â + ↠)/ 2, P̂ = (â − ↠)/ 2, we find √ √ (7) n̂21 = 2|α| Q̂ cos θ + P̂ sin θ = 2|α|M̂ , where M̂ is the quadrature operator defined earlier. Notice that in addition to allowing measurement of the operator M̂ , the method also amplifies the weak photonic signal by a factor of |α|. This is a hugely important experimental consideration, as it is a technical challenge to detect single photons, especially over a background of experimental noise. The addition of this enhancement factor allows the signal to noise ratio in current experiments to greatly exceed the signal to noise ratio possible with single photon detectors. Physically, this enhancement comes about because the amplitudes of the quantum state ρ̂ and the coherent state α that are added on 5 (a) Squeezed vacuum state with squeezing parameter ζ = 1.6 FIG. 5: Configuration of a balanced homodyne tomography experiment. (b) Photon subtracted squeezed vacuum, ζ = 1.6 qubits encoded in atomic systems. Thus, if one encodes a qubit in the spin of a nucleus, it is readily possible to perform arbitrary rotations of the state and perform measurements in any basis on it. On the other hand, for a photonic qubit encoded in the different Fock states, the equivalent operations are extremely challenging. (c) Schrödinger kitten |α = 1i + |α = −1i (a) (b) FIG. 6: (a) Radon transform of the Wigner function of the cat state |ψc i in Fig 4d, which equals the ideal experimental distribution of quadrature measurements. The vertical axis is normalized to 1. (b) The same, after the addition of simulated experimental noise. (d) Schrödinger cat |α = 3i + |α = −3i FIG. 4: Examples of Wigner functions for some typical nonclassical states. the beamsplitter, and when the amplitude is squared to obtain an intensity, the inteference term is proportional to the magnitude of both ρ̂ and α. Another notable reason for the popularity of homodyne tomography for the measurement of the quantum states of light is the fact that there are no robust ways to transform the quantum state of light, unlike for example FIG. 7: Reconstructed Wigner function of the cat state |ψc i. 6 (a) (a) (b) (b) FIG. 8: (a) Real and imaginary parts of the density matrix ρ(x1, x2) of the cat state |ψc i. (b) Same, for the reconstructed state obtained from the Wigner function in Figure 7. RECONSTRUCTION EXAMPLE After the discussion of the basics of Wigner functions and homodyne tomography, we present a simulated quantum tomography experiment to illustrate the whole procedure. In this experiment, we simulate the measurement of the Schrodinger cat state |ψc i = |α = 3i + |α = −3i. The Wigner function of this state is shown in Figure 4d, while the Radon transform of the Wigner function, which is equal to the statistical distribution of quadrature measurements, is shown in Figure 6a. Note the narrow interference features near θ = π/2, which encode the phase coherence between the two coherent states that make up the Schrödinger cat state. To simulate an experimental reconstruction, we add noise to the quadrature components and attempt to recover the original quantum state. First, we coarse-grain the quadrature measurement to only 80 possible outcomes, such that the width of each Gaussian hump near θ = 0 is approximately 10 quadrature bins, and each sharp interference fringe near θ = π/2 is about 5 bins wide. We add a constant background technical noise giving a signal to noise ratio of 10 for the Gaussian humps near θ = 0, and a photon shot noise corresponding to having 3000 measurements per phase setting. The number of phase settings used for reconstruction is 30, and we simulate flucuations in the phase by averaging over a π/20 FIG. 9: (a) Real and imaginary parts of the density matrix of the cat state |ψc i expressed in the Fock state basis, i.e. ρmn = hm| ρ̂ |ni . Note that only when both m and n are even is the matrix element non-zero, because of the destructive interference between the odd Fock components of the two coherent states making up the Schrödinger cat. (b) The same, for the reconstructed state. FIG. 10: Diagonal elements ρmm of the density matrices from Figure 9, comparing the actual state with the reconstructed one. In particular note that the reconstructed state is unphysical, since it has negative diagonal matrix elements. 7 angular interval around the phase setting, corresponding to the experimental reality that while the experiment is running, phase fluctuations are not recorded, but simply averaged over. The resulting simulated quadrature measurement is shown in Figure 6b. Note that these noise values are quite conservative, with state of the art experiments achieving much higher quality signals. Next, we perform an inverse Radon transform on the quadrature distribution to reconstruct the Wigner function, which is shown in Figure 7. As can be seen, the reconstruction is very noisy, but recovers the key features of the state: the two humps at P = 0, Q = ±4, and the narrow interference fringes near the origin. Despite the large amount of noise added to the simulated measurements, the nonclassicality of the state is still very evident in the reconstruction. This illustrates the robustness of quantum state tomography using the Wigner function – even with a large level of noise, one can recover indicators of interesting quantum states from the experiment. We can then use (2) to recover the density matrix of the quantum state, which is compared to the true density matrix in Figure 8. Again, the reconstructed density matrix maintains the key nonclassical features of the original state (notably the humps at x1 = −4, x2 = 4 and x1 = 4, x2 = −4 which indicate the coherence between the two coherent states), while suffering from added noise from the reconstruction procedure. The conventional way of presenting these density matrices is in the basis of harmonic oscillator eigenstates, which is shown in Figure 9. The RMS error of the elements of the density matrix is 0.012, which is consistent with the level of noise introduced into the homodyne measurements. This straightforward method of recovering the quantum state by performing an inverse Radon transform on the measured quadrature values suffers from two limitations. First, the reconstruction relies on physically arbitrary parameters in the filtering necessary to perform the backprojection for the inverse Radon transform. More importantly though, the reconstruction can lead to nonphysical final states. This can be seen even in our reconstruction above. A physical density matrix must have all non-negative eigenvalues, making it positive semidefinite. In particular, this implies that the elements of the density matrix on its diagonal must be positive in any basis. However, as can be seen in Figure 10, this is not the case, with ρ̂21,21 and ρ̂23,23 being negative. The most popular method of dealing with these limitations is maximum likelihood estimation [21, 22]. The basic idea of the method is to ask: “what is the physically allowed state that most likely would have produced the observed distribution of quadratures”? This approach guarantees that the reconstructed state will be physically meaningful. From the perspective of Bayesian statistics, this approach is equivalent to starting with a uniform prior distribution over the allowed density matrices, and using Bayes’ theorem for each measurement result to up- date the distribution. This correspondence to a Bayesian approach also leads to another advantage of maximum likelihood estimation over the inverse Radon transform – that one can find a reliable estimate of the uncertainties in the reconstructed state. CONCLUSION The field of quantum state tomography has blossomed in the last 20 years, driven by an increased understanding of the quantum nature of light and the creation of highrate nonclassical light sources, primarily through spontaneous parametric downconversion. The Wigner function’s ability to visually display the behaviour of quantum states, as well as its straightforward experimental interpretation have given it a central role in this new and exciting field. Clearly, only a taste of the richness of the field can be given here, but many excellent review papers and books have been written on the subject, such as [7, 23, 24]. [1] L. D. Landau and E.M. Lifshitz. Statistical Physics. Butterworth-Heinemann, third edition, 1980. [2] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40:749–759, Jun 1932. [3] P. A. M. Dirac. Bakerian lecture. the physical interpretation of quantum mechanics. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 180(980):1–40, March 1942. [4] Marlan O. Scully, Herbert Walther, and Wolfgang Schleich. Feynman’s approach to negative probability in quantum mechanics. Phys. Rev. A, 49:1562–1566, Mar 1994. [5] J. E. Moyal. Quantum mechanics as a statistical theory. Mathematical Proceedings of the Cambridge Philosophical Society, 45(01):99–124, 1949. [6] Cosmas K. Zachos, David B. Fairlie, and Thomas L. Curtright. Quantum Mechanics in Phase Space. World Scientific, 2005. [7] Ulf Leonhardt. Measuring the Quantum State of Light. Cambridge Studies in Modern Optics, 1997. [8] Gabor T. Herman. Fundamentals of Computerized Tomography: Image Reconstruction from Projections (Advances in Computer Vision and Pattern Recognition). Springer, second edition, 2009. [9] J. Illingworth and J. Kittler. A survey of the Hough transform. Computer Vision, Graphics, and Image Processing, 44(1):87–116, October 1988. [10] D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland. Experimental determination of the motional quantum state of a trapped atom. Phys. Rev. Lett., 77:4281–4285, Nov 1996. [11] 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., 87:050402, Jul 2001. 8 [12] Alexei Ourjoumtsev, Rosa Tualle-Brouri, and Philippe Grangier. Quantum homodyne tomography of a twophoton fock state. Phys. Rev. Lett., 96:213601, Jun 2006. [13] Samuel Deleglise, Igor Dotsenko, Clement Sayrin, Julien Bernu, Michel Brune, Jean-Michel Raimond, and Serge Haroche. Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature, 455(7212):510–514, September 2008. [14] D. F. Walls. Squeezed states of light. Nature, 306(5939):141–146, November 1983. [15] 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(9):1244–1247, Mar 1993. [16] G. Breitenbach, S. Schiller, and J. Mlynek. Measurement of the quantum states of squeezed light. Nature, 387(6632):471–475, May 1997. [17] Alexei Ourjoumtsev, Rosa Tualle-Brouri, Julien Laurat, and Philippe Grangier. Generating optical schrödinger kittens for quantum information processing. Science, 312(5770):83–86, April 2006. [18] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik. Generation of a superposi- [19] [20] [21] [22] [23] [24] tion of odd photon number states for quantum information networks. Phys. Rev. Lett., 97:083604, Aug 2006. E. Schrodinger. The current situation in quantum mechanics. Naturwissenschaften, 23:807–812, 1935. Z. Leghtas, G. Kirchmair, B. Vlastakis, M. Devoret, R. Schoelkopf, and M. Mirrahimi. Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity. ArXiv, 1205.2401, May 2012. Konrad Banaszek. Maximum-likelihood estimation of photon-number distribution from homodyne statistics. Phys. Rev. A, 57:5013–5015, Jun 1998. A I Lvovsky. Iterative maximum-likelihood reconstruction in quantum homodyne tomography. Journal of Optics B: Quantum and Semiclassical Optics, 6(6):S556–, 2004. A. I. Lvovsky and M. G. Raymer. Continuous-variable optical quantum-state tomography. Rev. Mod. Phys., 81:299–332, Mar 2009. G. Mauro D’Ariano, Matteo G.A. Paris, and Massimiliano F. Sacchi. Quantum tomography. In Peter W. Hawkes, editor, Advances in Imaging and Electron Physics, volume 128, pages 205–308. Elsevier, 2003.