Engineering qubit dynamics in open systems with photonic synthetic lattices

The majority of experiments in quantum mechanics require, to some extent, to account for the influence of external effects over which the experimenter has limited knowledge or control. At the same time, the interaction of a system of one or few quantum bits with an environment involves exchanges of energy, work, and information, which are at the basis of quantum thermodynamics [1, 2, 3]. Engineering qubit-environment interactions is thus a pivotal task for devising energy transfer processes, such as quantum heat engines [4, 5, 6, 7, 8], quantum batteries [9, 10, 11, 12], and quantum refrigerators [13, 14, 15, 16].

Experimental setups where such interaction can be partially controlled have recently been demonstrated. For instance, the environment can be cooled down using active feedback control loops on a qubit [17], thus allowing for reducing decoherence effects in quantum devices [18, 19].

In recent years, it has also become common to describe open systems in terms of non-Hermitian processes, wherein the system evolution is dictated by a Hamiltonian operator that admits a complex spectrum, accounting for the possibility of gains and losses in subsets of modes [20, 21, 22]. Non-Hermitian dynamics have demonstrated a wide variety of applications in quantum physics [23, 24, 25, 26, 27] and photonics [28, 29, 30], in particular when exploiting phenomena related to the topological properties of the Hamiltonian [31, 32, 33]. The connection between the description of open systems as non-Hermitian processes and via Master equations has been established [34], proving that it is generally possible to reduce the problem to a set of Kraus operators acting on the principal system [35].

Engineering system-environment interactions can also be of interest for investigating fundamental questions, for instance, in the context of emerging developments in understanding the interplay between geometric phases and quantum back action [36], wherein the variation of the geometrical phase has been demonstrated to be connected with the system-environment coupling strength in a topological fashion [37].

Different schemes have been proposed to engineer open quantum systems, but they still lack full generality [38, 39, 40, 41]. In this work, we demonstrate a step forward in this direction. We realize a platform where arbitrary sets of Kraus operators acting on a single qubit can be implemented, thus allowing for the simulation of any noise operator. Our strategy is based on coupling the qubit with an environment modeled as an infinite, discrete set of modes, visualized as a one-dimensional (1D) lattice space. By leveraging translational invariance, we derive an effective relation between the global (system-environment) unitary and the target set of Kraus operators. Once such a unitary is identified, it can be implemented in a compact photonic circuit consisting of a stack of three patterned waveplates [42]. In so doing, we can implement arbitrary operations on light polarization, encoding the qubit, by coupling it with the transverse spatial degree of freedom. This method is validated in a proof-of-principle experiment where we simulate common noise processes with different strengths. Our results demonstrate that this approach could also be employed to benchmark quantum thermal machines’ functionality and test the fundamental properties of open quantum systems.

The dynamics of a quantum system (here, a single qubit) is most generally described by a quantum channel ${\mathcal{N}}$, describing the noisy evolution of the system, that maps an input state $\rho_{\text{in}}$ into the final state ${\rho_{\text{f}}=\mathcal{N}(\rho_{\text{in}})}$. Mathematically, ${\mathcal{N}}$ corresponds to a completely positive, trace-preserving map, which can be specified by a set of Kraus operators ${\{A_{k}\}}$ such that

under the trace-preserving condition ${\sum_{k}A_{k}A_{k}^{\dagger}=\sigma_{0}}$, where $\sigma_{0}$ is the ${2\times 2}$ identity operator [35]. An equivalent description is to consider the qubit as a system $S$ interacting with an environment $E$. The qubit and environment define a closed system ${\Omega=S+E}$, described by a pure state undergoing a unitary evolution (see Fig. 1a). This corresponds to choosing a purification of the system ${S}$. The choice of purification (or, equivalently, of the environment $E$) is not unique [35]. We will show in the following that it is always possible to model the environment as an infinite-dimensional Hilbert space $\mathcal{H}_{E}$ spanned by a discrete set of states $\{\ket{m}\}$, with ${m\in\mathbb{Z}}$. These states can be interpreted as the eigenstates of the position operator on an infinite 1D lattice. This suggests that the system-environment interaction occurs in the form of a translation-invariant unitary operator $U$. Accordingly, the latter can be block-diagonalized in the reciprocal space:

where ${\ket{q}:=\sum_{m}\exp(iqm)\ket{m}/\sqrt{2\pi}}$ is a quasi-momentum eigenstate and ${\mathcal{U}(q)}$ is an SU(2) operator that acts on the Hilbert space of the qubit $\mathcal{H}_{S}$.

Without loss of generality, we can assume that the environment is initialized in a pure state ${\rho_{E,\text{in}}=\outerproduct{m=0}{m=0}}$, so that the global initial state is ${\rho_{\Omega,\text{in}}=\rho_{S,\text{in}}\otimes\rho_{E,\text{in}}}$. Applying $U$ to the input state, we obtain the following:

where the integrals are taken over the interval ${[0,2\pi]}$. After the interaction, the state of the qubit can be retrieved by taking the partial trace of the evolved density operator with respect to the environment in the quasi-momentum representation:

where the system label $S$ has been omitted. Our goal is to design $\mathcal{U}(q)$ such that the evolved density operator $\rho_{\text{f}}$ of the qubit is equivalent to the outcome of a target noise operation $\mathcal{N}$. Mathematically, this corresponds to solving

under the constraint that $\mathcal{U}(q)$ is an SU(2) matrix at each $q$.

To this aim, we decompose ${\mathcal{U}(q)}$ and each Kraus operator $A_{k}$ into linear combinations of the identity and Pauli matrices ($\sigma_{1},\sigma_{2},\sigma_{3}$), a set of complete basis for the $2\times 2$ matrices:

which yields (see Eq. (7)),

where $*$ denotes complex conjugation. Equation (10) corresponds to a system of 16 equations of the following form:

for ${\{{i,j\}}=0,1,2,3}$, that can be solved numerically for the coefficients $\{u_{i}(q)\}$ under the unitary constraint

for all $q\in[0,2\pi]$ (see Methods for details). Importantly, the solution holds for any qubit initialization; therefore, we engineer an effective simulation of the target noise.

Once the global unitary has been identified, an experimental setting can be created to implement such an operator. In our photonic setup, the qubit is encoded into light polarization, while the environment state $\ket{m}$ corresponds to an optical mode carrying $m$ units of transverse momentum ${\Delta k_{\perp}=2\pi/\Lambda}$, where $\Lambda$ is the spatial period of the beam transverse profile. This corresponds to mapping the quasi-momentum coordinate into the photon transverse position: ${q=2\pi x/\Lambda}$ [43]. With this encoding scheme, arbitrary unitary operators in the form of Eq. (2) can be simulated using a minimal stack of three patterned waveplates (see Fig. 1b). This technique has been introduced in Ref. [42] as a compact solution to simulate ultra-long quantum walks. It is well-known that a sequence of a half-wave plate (HWP) sandwiched between two quarter-wave plates (QWPs) can implement an arbitrary polarization transformation by adjusting the orientation of the optic axes ${\theta_{i}}$ of each waveplate [44]. In the circular polarization basis, where ${\ket{L}:=(1,0)^{T}}$ and ${\ket{R}:=(0,1)^{T}}$ are the left and right circular polarization states, respectively, and $T$ stands for transpose, the general transformation matrix associated with a waveplate can be written as

where the subscript ${i=1,2,3}$ labels individual waveplates, ${\delta_{1}=\delta_{3}=\pi/2}$ are the optical retardations of the QWPs $W_{1}$ and $W_{3}$, ${\delta_{2}=\pi}$ is the retardation of the HWP $W_{2}$, and $\theta_{i}$ are the optic-axis orientation angles. The required patterns $\theta_{i}(x)$ can be determined by solving ${W_{3}W_{2}W_{1}=\mathcal{U}}$ at each $q$ (see Methods). Patterned waveplates are fabricated as liquid-crystal metasurfaces, tunable nematic liquid-crystal samples, with the optic axis locally patterned via a well-established photoalignment technique [45].

We demonstrate our approach by fabricating stacks of plates simulating different instances of quantum noise, specifically, phase flip, bit flip, bit-phase flip, and depolarization channels. These are defined by the following sets of Kraus operators. For the first three channels, ${A_{0}=\sqrt{1-p}}\,\sigma_{0}$ and ${A_{1}=\sqrt{p}\,\sigma_{i}}$, with ${i=1,2,3}$ for bit flip, bit-phase flip, and phase flip, respectively. The depolarization channel is instead characterized by the Kraus operators $\{A_{0},A_{1},A_{2},A_{3}\}$, where ${A_{0}=\sqrt{1-p}\,\sigma_{0}}$ and ${A_{i}=\sqrt{p/3}\,\sigma_{i}}$ [35]. Each channel has been implemented for three values of the system-environment coupling parameter, ${p=1/8,\,1/4,\,1/2}$, corresponding to the weak, intermediate, and strong interaction regimes, respectively.

The experimental setup is outlined in Fig. 2a. A 810 nm laser beam crosses a state-preparation stage consisting of a polarizer (P), a half-wave plate (H), and a quarter-wave plate (Q). For each noise realization, we choose as input states $\ket{L}$, $\ket{H}$, and $\ket{D}$, where ${\ket{H}=(\ket{L}+\ket{R})/\sqrt{2}}$ and ${\ket{D}=(\ket{L}+i\ket{R})/\sqrt{2}}$ are the horizontal and diagonal polarization states, respectively. The noise $\mathcal{N}$ is implemented by three liquid-crystal metasurfaces $W_{1}$–$W_{2}$–$W_{3}$. The output qubit-polarization state is analyzed with a sequence Q–H–P, measuring the transmitted intensity with a power-meter detector (D). This allows us to retrieve the three Stokes parameters providing the representation of the qubit state on the Bloch sphere [46]. The optic-axis patterns of the metasurfaces used to simulate phase flip, bit flip, and bit-phase flip are plotted in Fig. 2b. These channels feature the same Kraus operators decomposition up to a basis rotation that can be implemented with uniform waveplates, and can therefore be simulated with the same sets of plates. The patterns for the depolarization channels are provided in Fig. 2c. It is evident that the complexity of the patterns increases with the complexity of the simulated interaction.

The experimental results are illustrated in Fig. 3, where we plot the measured Stokes parameters (see Fig. 3a) and the corresponding trajectories (see Fig. 3b) of the qubit states on the Bloch sphere as a function of the noise level $p$. Deviations from the theory are systematic errors due to imperfect alignment of the plates and fabrication defects, in agreement with Monte Carlo simulations whose results are reported as error bands around the theoretical lines in Fig. 3a. Overall, the average fidelity $\mathcal{F}=\Tr\left(\sqrt{\sqrt{\rho_{\text{th}}}\rho_{\text{exp}}\sqrt{\rho_{\text{th}}}}\right)^{2}$ between the measured states ${\rho_{\text{exp}}}$ and the expected ones ${\rho_{\text{th}}}$ is always above 98%, proving that our devices reproduce the target channels with high accuracy. The results for phase flip, bit flip, and bit-phase flip show that the action of the noise keeps unaltered the qubit states initialized in the eigenstate of $A_{1}$, while it pushes other inputs toward the origin of the Bloch sphere. The depolarization channel displays a similar behavior for all pure input states [35].

In this work, we demonstrated a general approach for engineering arbitrary noise operations on a qubit system, based on suitably coupling the qubit with a lattice-like environment. We provided a systematic procedure for constructing a global unitary operator corresponding to the desired process on the qubit state. In our photonic demonstration, light polarization played the role of a qubit, and the environment was encoded into a set of transverse modes, with the interaction occurring in the form of a space-dependent polarization transformation, engineered via liquid-crystal metasurfaces. The patterns of the metasurfaces have been inverse-designed to simulate typical noise operations in different interaction regimes.

Our findings pave the way for innovative solutions for studying open quantum systems. Potential applications involve the simulations of quantum machines, such as quantum refrigerators, protocols of quantum thermodynamics, chiral dynamics in open systems, and geometric-phase and topological effects in non-Hermitian evolutions. Furthermore, the controlled simulation of common forms of noise could foster the development of effective strategies for quantum error correction [47]. Another interesting prospect concerns the generalization of our scheme to a larger number of qubits, where the emerging concepts of local passivity [48] and ergotropy [49, 50] could be experimentally observed. Nevertheless, one could use the same setup to implement a specific noise on a single qubit and test its effects on entangled states.