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On-demand Parity-Time symmetry in a lone oscillator through complex, synthetic gauge fields

What is the fate of an oscillator when its inductance and capacitance are varied while its frequency is kept constant? Inspired by this question, we propose a protocol to implement parity-time (PT) symmetry in a lone oscillator. Different forms of constrained variations lead to static, periodic, or arbitrary balanced gain and loss profiles, that can be interpreted as purely imaginary gauge fields. With a state-of-the-art, dynamically tunable LC oscillator comprising synthetic circuit elements, we demonstrate static and Floquet PT breaking transitions, including those at vanishingly small gain and loss, by tracking the circuit energy. Concurrently, we derive and observe conserved quantities in this open, balanced gain-loss system, both in the static and Floquet cases. Lastly, by measuring the circuit energy, we unveil a giant dynamical asymmetry along exceptional point (EP) contours that emerge symmetrically from the Hermitian degeneracies at Floquet resonances. Distinct from material or parametric gain and loss mechanisms, our protocol enables on-demand parity-time symmetry in a minimal classical system -- a single oscillator -- and may be ported to other realizations including metamaterials and optomechanical systems.

Introduction. Non-Hermitian Hamiltonians that are invariant under combined operations of parity and timereversal (or more generally, an antilinear symmetry) are called PT symmetric. Since their discovery in the late 1990s [1,2] and first experimental realization in coupled optical waveguides a decade later [3,4], the field of PTsymmetric systems has grown increasingly diverse [5][6][7]. This expansive growth is driven by the realization that a PT -symmetric Hamiltonian represents an open system with balanced, but separated, gain and loss. A PTsymmetric Hamiltonian has purely real spectrum when its non-Hermiticity is small; at large non-Hermiticity, it changes into complex-conjugate pairs. This PTsymmetry breaking transition occurs at an exceptionalpoint (EP) degeneracy where the non-orthogonal eigenmodes of the Hamiltonian also coalesce [8]. EPs occur at the ends of a branch cut of Riemann manifolds that represent complex eigenvalues, and are responsible for enhanced sensing and topological effects [9].
These results are applicable to any system governed by a linear, first-order differential equation with a non-Hermitian generator of motion. This observation has led to the Cambrian explosion of PT -symmetric systems, and more generally, systems with antilinear symmetries. The resulting platforms are as diverse as two waveguides [3,4], two mechanical oscillators [10], two coupled electrical oscillators [11,12], two fiber loops [13], two or more coupled micro-resonators [14][15][16], acoustics [17], diffusive systems [18], damped and driven shallow fluids [19], and two coupled, time-delayed semiconductor lasers [20]. In the past two years, these classical realizations with gain and loss have been superseded by realizations in minimal quantum systems [21][22][23][24] governed by post-selected, lossy Hamiltonians (or Lindbladians [25]) with similar EP degeneracies.
Fundamentally, the quantum noise in a linear amplifier [26] introduces a time-reversal asymmetry between gain and loss mechanisms [27], and makes it impossible to create PT -symmetric systems with small number of excitation quanta [28]. In the semiclassical domain, customarily, the dissipation has been introduced through material impurities [4], coupling to a cold reservoir [24], or resistive elements [11,12]; the amplification, through nonlinear gain media [4,[13][14][15], four-wave mixing [29], parametric driving [30], or temporal modulation of two non-reciprocally coupled reservoirs [31]. Balancing the gain and loss requires independent control over physically different mechanisms in spatially separated parts of the system. Therefore, engineering dynamically tunable PT -symmetric systems is extremely challenging.
Here, we present a radically new protocol for PT symmetry based on a time-dependent similarity transformation on the system of interest. The concomitant complex gauge potential, from the non-unitary change of basis, generates balanced gain and loss potentials. An added benefit of this approach is that we can implement PT symmetry in a minimal system with one variable i.e. a single oscillator. In contrast, all gain-loss realizations to date have required two or more coupled, separated degrees of freedom, one with the gain and the other, loss.

Static PT
Dynamical PT The state |φ(t) of the circuit encodes the voltage V (t) across the capacitor and current I(t) in the inductor, and satisfies a linear, first-order equation given by Kirchoff laws. A static change-of-basis to |ψ(t) = A 1/2 |φ(t) leads to a Hermitian Hamiltonian H0 = ω0σy for the new state |ψ(t) . Thus, an LC circuit is mapped into a two-site model. (b) When C(t), L(t) are exponentially varied subject to the constraint L(t)C(t) = L0C0 (hyperbola), the timedependent change of basis generates balanced, constant gain and loss ±iγ. This leads to a purely imaginary, PT -symmetric Hamiltonian HPT = ω0σy + iγσz. (c) When the constrained variation is not exponential, arbitrary, but balanced gain and loss potentials ±iγ(t) can be generated. Here, we focus on the special case when γ(t) is periodic.
Therefore, although the protocol is applicable to arbitrarily large systems, we will focus on the illustrative example of a single LC oscillator.
PT -symmetry in a single oscillator. Figure 1a shows a schematic circuit in which the voltage V (t) across the capacitor and the current I(t) in the inductor satisfy Kirchhoff laws, Equivalently, it is described by a "state vector" |φ(t) ≡ [V (t), I(t)] T which satisfies the equation i∂ t |φ(t) = M |φ(t) where the real, non-symmetric matrix M has eigenvalues ± = ±ω 0 = ±1/ √ LC. Under a static, nonunitary change of basis to |ψ(t) = A 1/2 |φ(t) , the equation of motion becomes i∂ t |φ(t) = H 0 |ψ(t) with a Hermitian Hamiltonian [32,33] Here A = diag(C/2, L/2) is the positive-definite bilinear form that encodes the circuit energy E(t) = φ(t)|A|φ(t) , σ y is the Pauli y-matrix, and the unitary evolution under H 0 indicates the constancy of energy in the LC circuit.
When the basis transformation depends on time, the new state vector satisfies i∂ t |ψ(t) = H eff (t)|ψ(t) . The effective Hamiltonian H eff is the sum of (possibly timedependent) Hamiltonian H 0 (t) and a gauge potential iΓ(t) that arises from the non-constant nature of the change of basis matrix [34], In fundamentally quantum systems, the matrix A 1/2 is unitary and gives rise to a Hermitian gauge term iΓ = (iΓ) † . In contrast, for effective models like ours, a nonunitary A 1/2 (t) can be tailored to create non-Hermitian, gain and loss potentials. Now, let us consider constrained variations of the form C(t) = C 0 exp[+2f (t)] and L(t) = L 0 exp[−2f (t)] to ensure that the frequency of the oscillator remains unchanged. Such variations give rise to a traceless, anti-Hermitian gauge potential iΓ = iγ(t)σ z = i(df /dt)σ z that represents balanced gain and loss in a solitary oscillator. The effective Hamiltonian then becomes H PT (t) is invariant under combined operations of parity P = σ x and time reversal T = * (complex conjugation). In contrast to quantum systems with a complex state vector, the realness of the elements of |ψ(t) = [ C(t)/2V (t), L(t)/2I(t)] T is guaranteed by an H PT with purely imaginary entries. This requirement also constrains the most general form of the non-Hermitian Hamiltonian for this system to With a suitable choice of the dimensionless function f (t), Eq.(4) provides the protocol for arbitrary, balanced gain and loss for the energy dynamics. When f (t) = γt is linear in time, Eq.(4) gives the static PT -symmetric Hamiltonian H PT (γ) = ω 0 σ y +iγσ z . As shown in Fig. 1b, when γ > 0, the capacitor acts as the "gain site" and the inductor acts as the "loss site" for the circuit energy. On the other hand, surfing the hyperbola L(t)C(t) = L 0 C 0 back and forth leads to a time-periodic γ(t) where each "site" acts as a gain for fraction of the period and a loss for rest of the time (Fig. 1c). Experimental results for a static H PT . We exper- imentally demonstrate this protocol with a state-of-theart fully reconfigurable electronic oscillator comprising functional blocks synthesized with operational amplifiers (op-amps) and passive linear components [32,33,35]. We thus electronically reproduce the dynamics described by Eqs. (1) and (2) in the presence of a parasitic resistance R = 10 3 Ω in parallel with the LC circuit. Figure 2a shows for the circuit schematics, while the actual device is shown in Fig. 2b. C 0 = 100 µF is minimum capacitance and L 0 = 0.01 H is the maximum inductance that our electronic platform can efficiently simulate. Their combination gives ω 0 = (2π)×159 Hz as the fundamental frequency of the oscillator. By increasing the capacitance at different speeds, different gain-loss strengths are realized. The eigenvalues of H PT (γ) are ± ω 2 0 − γ 2 , and they change from real to complex-conjugate pair at the exceptional point marked by γ EP = ω 0 .
The time-dependent evolution of the circuit energy E(t) = ψ(t)|ψ(t) in the PT -symmetric phase is shown in Fig. 2 (experimental data: blue dots, theory: red dashed lines). When the gain-loss strength is doubled from γ = 0.375ω 0 , Fig. 2c, to γ = 0.75ω 0 , Fig. 2d, the period of oscillations increases by √ 2, and the amplitude of oscillations also increases. It is worth pointing out that the fast fluctuations in the experimental data are due to oscilloscope's inherent noise; as the circuit energy E(t) increases from tens of micro-Joules (µJ) to a milli-Joule (mJ), the relative effect of the noise is suppressed. When γ = 1.05ω 0 , Fig. 2e, the system goes into the PT -broken phase, as indicated by a monotonically increasing circuit energy. The temporal range of our simulation of a static H PT (γ) is limited the maximum value of capacitance, and not by the gain saturation of op-amps at high circuit energies.
For this set of experiments, using C 0 = 400 µF and L 0 = 1 mH fixes the oscillator frequency at ω 0 /(2π) = 251 Hz. We use an experimentally friendly parameter [32,47] Λ amp (α, ν T ) = lim obtained from the circuit energy to characterize the strength of the PT -broken phase. Since E(t) oscillates in the PT -symmetric phase, Λ amp = 0, whereas its exponential growth in the PT -broken region gives Λ amp > 0. Figure 3b shows the emergent triangular PT -broken regions at ν 2 = 100 Hz (cyan circles), ν 3 = 71 Hz (yellow circles), ν 4 = 55 Hz (pink circles), and ν 5 = 45 Hz (green circles). The gray surface is theory. In the PT -broken regions, at high circuit energies, E(t) does not grow exponentially due to op-amp saturation. This leads to a suppression of the effective amplification rate Λ amp . This suppression is maximum in the deepest PT -symmetry broken region and leads to Λ amp → 0 as the circuit energy saturates at short times. It is clearly seen in Fig. 3c at ν 1 = 167 Hz (orange circles) at large α, but is absent at smaller values of α. The mauve surface is a theory prediction with gain saturation +iγ(V ), whereas the gray mesh is one without gain saturation. This suppression is almost complete at the primary resonance which occurs at ν 0 = 2ω 0 /(2π) = 502 Hz (not shown) limiting the use of Eq. (8) to distinguish between PT -symmetric and broken phases.
In the Floquet case, the conserved quantities are the expectation values of Hermitian, indefinite operatorsη F that satisfy the intertwining relation G † Figure 3d shows η 1F obtained from the experimentally measured state-vector |ψ(t) at times t = t m . Since ν ∼ 70 Hz−40 Hz, there are ∼ 5 stroboscopic data points available. These are representative results in the PTsymmetric (circles) and PT -broken (diamonds) phases in the vicinity of ν 3 and ν 5 domes with color-coded symbols (Fig. 3b). Experimentally measured η 2F (t m ) results are shown in Fig. 3e. The deviation from flat gray lines (theory) at long times t 2τ is due to gain saturation in the PT -broken phase (diamonds). Since G F (T ) = −1 2 at the DP degeneracies ν k = 502/(2k + 1) Hz, we see that η 2F (t m ) = −η 1F (t m ) holds, surprisingly, irrespective of the gain saturation. Similar results are valid, of course, across the entire α−ν T plane. This remarkable ability to map out the entire Floquet PT -phase diagram across five domes showcases the tremendous versatility of the synthetic electronic platform, and the distinct advantage of complex gauge-field induced gain and loss mechanism over traditional approaches [4, 11-15, 24, 29-31, 44]. This unparalleled versatility has also enabled the first demonstration of conserved quantities in the Floquet dynamics of a PT -symmetric system. Walking the EP contours. As a last demonstration of the synthetic oscillator platform, we investigate the temporal dynamics at numerous points along the EP contours. This has been extremely challenging in gainloss systems due to the requisite fine-tuning of multiple mechanisms. In loss-only PT -systems, it is a challenge because decay rate is maximum at the EP.
A complementary way to show the PT -phase diagram in the α−ν T plane is the Dirac inner-product of the right eigenvectors of G F (T ). By expressing the real, Floquet evolution matrix as G F (T ) = G 0 1 2 + G x σ x + G z σ z + iG y σ y where G k ∈ R, it is straightforward to obtain the inner-product as I P (α, ν T ) = min(r, 1/r) where .
(9) Figure 4a shows the heat-map overlaid with EP contours, I P = 1, where blue is for the plus sign and red corresponds to the minus sign in Eq. (7). Figure 4b shows the region near ν 1 = 167 Hz extended to negative values of α. We "park the system" at 9 points with equidistant α EP values along the blue (filled circles) and red (open circles) contours each, and obtain the circuit energy evolution E(t). Since the system has second-order EP contours, the stroboscopic circuit energy E(t m ) = ψ(0)|G † F (mT )G F (mT )|ψ(0) grows quadratically with time t m . Figure 4c shows the experimentally measured circuit energy E(t) at α = 0.2 on the blue contour (blue trace) and the red contour (red trace) over m ∼ 25 periods. It is constant except at integer and half-integer periods when the δ-function gain-loss potential is active. Surprisingly, E(t) also shows an order-of-magnitude asymmetry for the two contours that emerge symmetrically from the DP at ν 1 ; this asymmetry persists at all α EP .
We quantify the growth of stroboscopic, normalized circuit energy E(t m ) with two dimensionless coefficients, that depend only on α EP since it uniquely determines the corresponding ν T via Eq. (7). This approach allows us to investigate their dependence on the proximity to the DP degeneracy at α = 0, and the dynamical asymmetry between temporal evolution along the blue (plus) and red (minus) EP contours. At the second-order EP, ≥ 0 for the coefficients in Eq. (10). Figure 4d shows the stroboscopic, normalized circuit energy E(t m )/E(0) as a function of m along the blue contour for |α| = {0, 0.1, 0.3}. The error bars on E(t m ) are obtained by averaging its value over the constant region. As the DP at α = 0 is approached from either side, the coefficients A + (α) and B + (α) are monotonically suppressed to zero. The slight negative slope of E(t m ) at α = 0 (no gain or loss) is due to the parasitic resistance in the circuit.
We extract the coefficients A − (α) (Fig. 4e)  show approximate α ↔ −α symmetry, vanish at the DP as expected, and show that E(tm) growth along the red contour is ∼ 25-fold larger than along the blue contour. (f) Similar results, including a giant 25-fold asymmetry, are obtained for B±(α).
( Fig. 4f) from the experimental data along the red contour. Similar results, with dramatically smaller values of A + and B + , are obtained for a walk along the blue contour. They clearly demonstrate the order-of-magnitude dynamical asymmetry that arises when the same initial state |ψ(0) , with a fully charged capacitor, is evolved along the two symmetrical EP lines that emerge from ν 1 = 167 Hz. Discussion. Most of the transformative ideas in non-Hermitian physics-Riemann surfaces, bi-orthogonal basis, exceptional points, to name a few-have been wellknown in mathematics. Yet, their reinterpreation in the context of open systems has lent novel insights that underpin the advances such as enhanced sensing [15,16], chiral mode switch [48,49], or topological braiding [50].
We have, instead, contextualized time-dependent nonunitary transformations into a novel protocol to implement balanced gain and loss in a single oscillator. Simple harmonic oscillator based models are all-pervasive in nature, and our protocol provides a recipe for their non-Hermitian generalization. For example, in a metamaterial, the non-unitary change of basis is given by the permittivity and the permeability µ. A constrained variation of the two, with a constant product (and therefore a constant index of refraction), can lead to a new class of PT -symmetric metamaterials [60] without material gain or loss. It is also easy to generalize this protocol to a network of oscillators, where the gain and loss "sites" are localized in different nodes or are distributed throughout the network. For example, with reconfigurable synthetic LC circuits, this method can lead to non-passive, PTsymmetric extensions of topoelectrical circuits [61].