Tomography of a multimode quantum black box

We report a technique for experimental characterization of an $M$-mode quantum optical process, generalizing the single-mode coherent-state quantum-process tomography method [M. Lobino et al., Science 322, 563 (2008); A. Anis and A.I. Lvovsky, New J. Phys. 14, 105021 (2012)]. By measuring effect of the process on multi-mode coherent states via balanced homodyne tomography, we obtain the process tensor in the Fock basis. This rank-$4M$ tensor, which predicts the effect of the process on an arbitrary density matrix, is iteratively reconstructed directly from the experimental data via the maximum-likelihood method. We demonstrate the capabilities of our method using the example of a beam splitter, reconstructing its process tensor within the subspace spanned by the first three Fock states. In spite of using purely classical probe states, we recover quantum properties of this optical element, in particular the Hong-Ou-Mandel effect.

Introduction. Precise understanding of the performance of individual quantum systems is a key requirement for the development of compound devices, e.g. quantum computers or secure communication networks. This requirement gives rise to the problem of experimentally characterizing quantum systems as "black boxes": learning to predict their effect on arbitrary quantum states by measuring their effect on a limited number of "probe" states. The art of solving this problem is referred to as quantum process tomography (QPT).
A straightforward approach to QPT consists of measuring the action of the black box on a set of states whose density operators form a spanning set in the space of all operators over a particular Hilbert space. Because any quantum process is a linear map with respect to density operators, this information is sufficient to fully characterize the process [1]. However, such a direct method requires a large set of difficult-to-prepare probe states, and is consequently restricted to systems of very low dimension. Another possibility is the ancilla-assisted method [2] utilizing an input state that is a part of a fully entangled state in a larger Hilbert space. Although in this case only a single input is necessary thanks to the Jamiolkowski isomorphism [3], preparation of this state is, again, complicated, which dramatically limits the scalability of the method.
In application to optics, the coherent-state QPT (csQPT) [4][5][6] offers a practical solution. This method uses only coherent states for probing the black box, relying on the fact that these states span the space of operators over the optical Hilbert space (the optical equivalence theorem) [7]. Because coherent states are readily obtained from lasers, and their amplitudes and phases are easy to control, csQPT is relatively easy to implement in an experiment. However, its experimental accessibility is complicated by the generalized nature of the Glauber-Sudarshan function, leading to a sophisticated data processing procedure.
The procedures proposed in Refs. [4,6] evaluate each element of the process tensor individually, and can hence lead to unphysical (non-trace preserving or non-positive) process tensors. This shortcoming is absent in a method known as MaxLik csQPT, which exploits the Jamiolkowski isomorphism to reduce the QPT problem to the well-studied problem of the quantum state estimation, and applies the likelihood maximization technique to estimate the process tensor [8]. In this way, one can perform the reconstruction without leaving the physically plausible space. MaxLik csQPT has been proposed in Ref. [9] and successfully realized for nondeterministic singe-mode processes [10].
In this work, we expand csQPT beyond the "single input -single output" case, which covers only a few of practically relevant quantum optical black boxes. The need for the multimode case is dictated by the growing fields of quantum optical communication and logic, which are impossible without multimode processing. Examples include multimode quantum memories [11] and logic gates for processing photonic qubits [12], to name a few.
Multimode MaxLik QPT. Our method generalizes the single-mode MaxLik csQPT approach [9]. We work in the Fock basis and represent a general M -mode quantum process E by a tensor of rank 4M which maps the input density matrix into the output one: where underlined symbols |i = |i 1 , . . . , i M refer to multimode Fock states. In practice, the infinite dimension of the optical Hilbert space is truncated to the N + 1 lowest Fock states, so that i k ∈ 0 . . . N . In the experiment, the black box is fed with a set of coherent probe states |α = |α 1 , . . . , α M , while the output channels are examined by homodyne measurements. The obtained quadrature data {X i , θ i } define a set of projectors {Π θ i (X i )}, where θ i = (θ i1 , . . . , θ iM ) is the set of local oscillator phases associated with the ith measurement. Subsequent maximum likelihood reconstruction consists of finding the physically plausible process E arXiv:1403.0432v2 [quant-ph] 10 Mar 2014 which maximizes the log-likelihood of the harvested data set: Although this is a highly nonlinear optimization problem, it can be effectively solved by an iterative algorithm [8,9].
Tomography of beam splitting. The process of choice for testing the capability of our method is beam splitting. Its paramount importance in quantum optics needs no proof: all linear optical devices (interferometers, waveguide couplers, loss channels, etc.) are equivalent to single beam splitters or sets thereof. Accompanied by single photon sources and photon detectors, beam splitters enable quantum computation [13]. In some form, a beam splitter is present in any imaginable optical setting. Although the operation of the beam splitter (BS) is consistent with classical physics (coherent state inputs lead to coherent state outputs, and vice versa), its response to nonclassical input gives rise to quantum phenomena. A striking example is the Hong-Ou-Mandel effect: when two photons impinge upon a symmetric BS, they appear only in pairs at one of its outputs [14]. Our technique reveals this quantum effect in spite of using only classical states in measurements.
The beam splitter has previously been characterized by QPT [15], albeit only for a heralded single photon input. As such, this method realizes incomplete QPT, since the response of the black box beyond the single-photon subspace of the Hilbert space remains undetermined and the losses are unaccounted for. Our technique is free of these shortcomings. It allows one to predict output of the process for any arbitrary Fock states and their superpositions in the input, up to a certain cut-off photon number. We use the truncation Fock number N = 4 in our iterative reconstruction, which is conditioned only by the computational power of our machine.
Our technique is different from a recently developed method for characterizing linear optical networks [16] in that it makes no assumptions about the content of the "black box". Although we do use a linear optical device for the experiment, our approach can be successfully applied to a multimode process of any nature.
The light source in our experiment is a mode-locked Ti:Sapphire laser (Coherent Mira 900), which emits pulses at 780 nm with a repetition rate of 76 MHz and a pulse width of ∼ 1.8 ps. We realize symmetric beam splitting with respect to the horizontal and vertical polarization modes in the same spatial channel, marked 1 in Fig. 1, using an electrooptical modulator (EOM) with its optical axis oriented at 45 • to horizontal and a λ/4 voltage applied to it. A polarizing beam splitter (PBS) subsequently separates the output modes spatially for detection. Our black box is thus implemented by combination EOM + PBS. In the Heisenberg picture, this process where a in,out

1,2
are photon annihilation operators of the input and output modes. The relative amplitudes and phases of the input coherent states are set using a λ/2 + λ/4 waveplate pair.
The measurement of the output is performed using balanced homodyne detectors (BHDs) [17] in both output channels. To this end, we introduce two local oscillators (LOs) in orthogonal polarizations in spatial mode 2, so the central PBS directs them into the two output spatial channels (Fig. 1). In each output channel of the PBS, we then find the signal and LO in orthogonal polarizations. For homodyne detection, these polarizations are mixed in each channel using a combination of a λ/2 plate oriented at 22.5 • to the horizontal and an additional PBS. The relative phases the LOs can be controlled by two wave plates, while their common phase is slowly scanned using a piezo-mounted mirror in the signal channel.
Evaluating the process tensor. The process reconstruction is simplified by its invariance with respect to the global phase shift. That is, if both input phases are shifted by some phase θ, so will be the output state. This invariance is a consequence of the symmetric nature of time: a global phase shift by θ is equivalent to a shift in time by θ/ω, where ω is the optical frequency. If   the "black box" is not connected to any external clock (such as in our case), it will respond to a signal that is shifted in time by the same amount in the output. The effect of phase invariance on the process tensor can be determined from the fact that a phase shift of both modes will transform density matrix elements according to ρ in n1,n2,m1,m2 → ρ in n1,n2,m1,m2 e iθ(n1+n2−m1−m2) , ρ out j1,j2,k1,k2 → ρ out j1,j2,k1,k2 e iθ(j1+j2−k1−k2) .
Reconciling this with Eq. (1), we find that only elements such that j 1 + j 2 − k 1 − k 2 = n 1 + n 2 − m 1 − m 2 can be nonzero in tensor E n,m j,k . The process reconstruction requires knowledge of the LO phase vector θ i at each moment in time. We acquire this knowledge by periodically setting the EOM voltage to zero, so the black box becomes the identity process and the quadrature measurements correspond to the input states. The quadratures acquired during this period are divided into sets of 4600 measurements, so each set is acquired within 120 µs and the LO phases within each set can be assumed constant. The inverse sine of the mean quadrature value for each set yields the differences θ i −θ in,i of the LO and input state phases for both modes.
The switching between the BS and identity processes is performed with a period of 0.1 s, which is much faster than the characteristic time of phase fluctuations caused by air movements in the two interferometer channels. In this way, the evaluated LO phases can be translated to the process output measurements by taking into account the linear motion of the piezo.
We acquire a total of 48 sets of 10 6 quadrature samples for three different relative phases of LOs: 0.67, 2.64 and 5.29 rad and 16 pairs of input coherent states, obtained by setting each waveplate at 0 • , 15 • , 30 • and 45 • . Each pair of the input states has the same total energy corresponding to 0.9 photons.
We implement a two-step reconstruction process as prescribed by Ref. [9]. In the first step, we choose the cut-off point in the Fock space at N = 4, i.e. sufficiently high to make sure both the input probe states and the output states are well accommodated in the reconstruc-tion Hilbert space. Doing so ensures existence of the process tensor consistent with the experimental data. The phase invariance property of the process kills about 90% of ≈ 4 × 10 5 tensor elements. The resulting dimensionality of the optimization space is close to that in the 8-ion tomography done in work [18] and is computationally intensive. The iterative algorithm runs on an Intel Core i7 processor. Paralleled onto 4 of 8 computing cores, each iteration takes about 2 hours. The maximum-likelihood reconstruction algorithm appears to converge at around 100 iterations.
Once the iterations are completed, we apply the second step of the reconstruction process by truncating the tensor to a lower maximum photon number N . This is necessary because the fraction of high number states in the probe states is low, and the tensor elements associated with these states are determined inaccurately [9].
Results. Fig. 2 shows the result of the process reconstruction with N = 2 in comparison with the theoretical expectation according to Eq. (3) with an additional common phase delay of 0.8 rad. The elements of the process tensor associated with the diagonal elements of the input and output density matrices [ Fig. 2(a)] have transparent physical meaning as probabilities of the corresponding transitions. In particular, the Hong-Ou-Mandel effect is represented by the probability of |1, 1 → |1, 1 transition, which is zero in ideally symmetrical beamsplitting and amounts to ≈ 0.01 in the reconstructed tensor. This error, as well as errors in other matrix elements, is likely due to imperfect EOM setting and waveplate manufacturing, limited size of the dataset, and imperfectness of the reconstruction due to a finite number of iterations.
The data in Fig. 2(a) are only a small fraction of the full tensor shown in Fig. 2(b), which has ∼ 10 3 non-zero, generally complex elements. These elements determine the phase behavior of the black box, and are equally important in the description of the process. The left and right columns of the grid present, respectively, the real and imaginary parts of the tensor, while the top and bottom rows correspond to the reconstruction result and the theoretical expectation. The insets in each panel shows the response of our black box to the Hong-Ou-Mandel query, the |1, 1 input state. The diagonal of the left (real) panel in Fig. 2(b) corresponds to the full panel in Fig. 2(a).
To characterize the quality of the reconstruction, we estimate the fidelity between the ideal and reconstructed processes, in the Jamiolkowski state representation: When the full reconstructed tensor with (N = 4) is truncated to N = 3 and N = 2 Eq. (4) gives F (E, E est ) = 0.88 and 0.95, respectively. Without this truncation, a fidelity of 0.7 would obtain. Summary. We presented experimental csQPT reconstruction of the most common multimode optical process, the beam splitter. Our technique can be readily generalized to other processes. Unique scalability of csQPT to higher number of channels and larger state spaces is warranted by the simplicity of the required optical measurements and probe state preparation.